Promotion of Plastic Design for Steel and Composite Cross Sections

the design value of plastic bending moment for the resistance of cross-sections : MplJld = Mpi/TMo . with partial safety factor ymo according to Eurocode 3 (Ref. 1) ( Ymo = 1,1) or to NAD's values, and, the design value of available inelastic rotation for ductility of formed plastic hinges : 9av.d = 9av / ΤΜφ . with the proposed design model in chapter chapter 4.3.1.2 for
Formulas

Load cases

Ύίνΐφ

of chapter 4.3.1.2

1,52

j My

:

Bending about yy major axis

(4.1) & (4.3)

1,73

j Mz

:

Bending about zz minor axis

(4.2)

Bending about yy major axis combined with axial compressive force

(4.4)

^-y

2,02

My- Ν

:

(10) Steel grades:

All steel grades available in Eurocode 3 Part 1.1

can be applied in the formulas

:

235 MPa < fy < 460 MPa. Indeed the reliability of the model has been demonstrated by tests results with steel grades m that range (see Annex 12).

Eurocode 3 (Ref. 1) allows for plastic global analysis with steel materials characterized according to EN 10025 (Ref. 21) and EN 101 13 (Ref. 22) as given in following table :

Thickness t (mm) *)

Nominal Steel grade

t<40mm

40mm
EN 10025

fy

fu

fu/fy

A5.65

standard

[Mpa]

[Mpa]

[-]

[%]

235 275 355

360 430 510

1.53 1.56 1.44

26 22 22

390

1.42 1.38 1.19 1.15

24 22

S 235 S 275 S 355

£y

fu/fy A5.65

fy

fu

[Mpa]

[Mpa]

[-]

[%]

[%]

0.11 0.13 0.17

215 255 335

340 410 490

1.58 1.61 1.46

24 20 20

0.10 0.12 0.16

0.13 0.17 0.20 0.22

255 335 390

370 470 500 530

1.45 1.40 1.28 1.23

24 22

0.12 0.16 0.19 0.20

ey

EN 10113-3 Standard S 275 M S 355 M S 420 M S 460 M

275 355

420 460

490 500 530

19 17

Note:

*) t is the nominal thickness of the element : - of the flange of rolled sections (t = tf ), in general, or - of the particular elements of welded sections

39

430

19 17

^7^ * 1

J,

t

1

t

^ -» Λ

1

1

"t

Remark 1: about nominal values of yield strength fy

:

Eurocode 3 Part 1.1 (Ref. 1) gives a simplified variation of fy in function of the material thickness (thickness ranges: lower than 40 mm, from 40 mm to 100 mm), with values of fy - for certain thickness ranges - greater than the values specified by EN 10025 (Ref. 21) and EN 10113 (Ref. 22) (thickness ranges: lower than 16 mm, from 16 mm to 40 mm, from 40 mm to 63 mm, from 63 mm to 80 mm, from 80 mm to 100 mm, from 100 mm to 150 mm) (see following figure). As Eurocode 3 always provides yield strength f y equal or greater than values guaranteed by the specifications of delivery conditions (EN 10025, EN 10113), the available inelastic rotation av resulting from fy of EN 10025 or

EN 10113.

Minimum guaranteed yield strength RcH (or Rp0,2) m function of nominal thickness t of material Steel grades' RcH(orRpO^) [Mpa] 4 60

S 460

460

430

440 430

-42ΰ

S 420

420

410

-

400

390

400 390

370

355

S 355

360

355

335

345

340 335

325 315

-ZU

S 275

^

1

275

1

255

265 255

235

S 235

245

1

235

235

2/5

225

225

215

t {mm]

195

0

40

]6

80

63

100

150

Legend: .·

-EN 10113 ( for S 275, S 355. S 420 & S 460 steel grades). Euroc ode 3 specifications (for S 235, S 2?5,S3SS,S420& S 460 steel grades).

Remark 2: about overstrengthening of steel material

:

The real yield stresses of the steels delivered by mills are always greater than the nominal values of the steels indicated in the specifications EN 10025 (Ref. 21) and EN 10113 (Ref. 22). As presented in this chapter 4.3.1.1 clause (9), a statistical procedure has been used to evaluate partial safety factors γ^φ related to available inelastic rotation
40

Decreasing of the rotation requirement in case of actual yield stress which are greater than the

nominal yield stress

1

iHiiimmimnfTTTTrmr -·»-

i^reqWi

φ

Ψγ«Γ

4.3.1.2 Feldmann's formulas for values of
A. type of load: concentrated load with bending about major axis y-y of cross-section (My), B. type of load: concentrated load with bending about minor axis z-z of cross-section (M£), C. type of load: concentrated load with combined axial compressive force, bending about major axis y-y of cross-section (My - N).

In all formulas

:

Δο

=150 MPa,

fy.fi

= yield strength in flange must be introduced in MPa,

fy.w

= yield strength in web must be introduced in MPa,

E

= modulus of elasticity (= 210 000 MPa),

Npi

= plastic axial resistance of cross-section (=Afy),

Mpi

= plastic resistance moment of cross-section (=Wpi fy),

A = sectional area of cross-section. Available inelastic rotations
41

Α..

type of load:

concentrated load with bending about major axis y-y of cross-section

Plastic folding mechanism in compression flange of Η beams submitted to bending about major axis yy (Ref. 11):

5,2

-

f ,

^ M

+ 0,25 (

Y

3

4Et>4

17"

- ^.whtw + -J(^.whtw )

+ 4fy-wbtf twMa

V400

15.

type of load:

concentrated load with bending about minor axis z-z of cross-section

42

[rad]

(4.1)

Plastic folding mechanism in H beams submitted to bending about minor axis zz (Ref. 11)

This formula is only valid when stiffeners are applied at the loading point.

-

Wpi.zfy

(h-2tf)twfy-w

2tf
where

= Φκ,ρΐ +

[rad]

}|ιγ.Α+Δσ)

φκ?ρ1 =

0.2

('-f)

;

(4.2)

i = length of the beam.

In practical cases the ratio 2b/l are rather small. One can therefore assume the safesided value of Φκ,ρΐ = °<2·

v.".

type of load:

concentrated load with bending about major axis y-y of cross-section combined with axial compressive force

43

Plastic folding mechanism in H beams submitted to bending about major axis yy combined to axial compressive force (Ref. 11) :

Limit of application of these formulas is set to N/Npi < 0,5. To apply the appropriate formula, the following criteria should be considered

©if

Ν A-2btf f _ A ^web^ < sN Pl V A total J

then the neutral axis should be in web and


evaluated from (4.3)

;

>-^ <-N

neutral axis

r
=

j

f 4Ebr3

5,2

ry.fl

ρ

CI)

^

)My

i

5h

+ 0,25 (fy-fl +Aa)bh

\

fy.w ht w + (fy.w ht w J

+ 4fy .w bt f l w Μσ

[rad]

(4.3)

400

©if

Ν A-2btf > Ν Pi

_ Aweb V

Λ

Atotal )

,

then the neutral axis lies in flange and φ3ν is calculated

from formula (4.4):

f* <-N v.

JMy

neutral axis >

[rad] where:

(4.4)

Ν is the applied axial load

MN.y = WpLyfy

1-

i-JL a_^, A-2btf

with the limit MNy
(4.5)

2A (see Eurocode 3 [3]

44

or Appendix

1

of present Annex

13)

Nfl=(tf-tM)bfy.fl XM

-' 2b

1-

(4.6)

N

(4.7)

Af,

yj

ρ _ 4EbtS

Pc 51?" Pu

(4.8)

= btf(fy.fl+Äa)

(4.9)

The following value of Niimit defines the border between formulas (4.3) and (4.4) for q>av- Niimit is related to the case where neutral axis hes in the border between the web and the flange of the crosssection.

h-tf

<-N

JMy

neutral axis

Niimit=(A-2btf)fy.w 4.3.2

(4.10)

Kemp's model for steel sections

(1) Kemp and Dekker have developed two models for predicting the rotation capacity of steel beams (Ref. 15). The first proposed model is based on a theoretical consideration of the strains in the section and of the length of plastic region of the flange for which local buckling is assumed to occur. Complicated formulae of the first model are given for information in Annex 11. The second model is a semi-empirical approach, which is based on comparisons with tests results and results produced using the first theoretical model. More details are given about the second simplified model in Annex 1 1 .

(2) ProfilARBED has applied formulae of Kemp's simplified model for I- and H-cross-sections submitted to bending according to the major axis (3 point bending beams), in order to evaluate available rotation capacities (Rav) (see Annex 11).

4.3.3

CRM's model for steel sections

(1) The available inelastic rotation
4.3.4

Comparison between models of
(1) Comparison has been made between 26 experimental results on steel section and calculation models determining available rotation capacity, allowing local buckling: the Kemp & Dekker theoretical and simple models (K & D simp.), the Johnson & Chen best fit (J & Β best) and lower bound (J & Β lower) models, and the Spangemacher & Sedlacek (S & S) model (see Figure 21a), issued from document 3263-1-19: "Design rules for continuous composite beams using Class 1 and 2 steel sections Applicability of EC 4"). 45

Kemp & Dekker simple model appears to be more appropriate, because : the calculated rotation capacity is nearly always less than the tests results (meaning that the model is conservative),

the spread of errors is less than for the other models (better scatter of values).

Kemp model and others give relative precise values of the ultimate load of the tested beams (differences of ± 5 %) but provide less precise values for φ>0ι, the largest rotations which are decisive to evaluate the available rotation capacities.

(2) On the other hand, Feldmann's model which has been recently developed (Ref. 11), intends to determine as accurately as possible the values of
(3) ProfilARBED compared the efficiency of both models on the basis of a databank of tests reults (see chapter 4.2 and Annex 1 1). Therefore rotations capacities are compared with tests results in Annex

11

as

follows

:

R experiments are plotted versus R models (Feldmann's model, Kemp's simplified model and CRM's model) (Figure 21 b)) and Annex 1 1) experimental results with >60, meaning !zc*e occurrence of lateral-torsional buckling (see clause 3.2 (5)), are excluded, assumptions for Feldmann's model

assumptions

for

Kemp's

:

simplified

model

^-<20οτ^-> 100 are excluded (see Annex 1zc·^

11

:

experimental

results

with

or Ref. 15).

ïzc·^

in (Ravaiiable - Lspan(= span length)) graphs (Annex 11), the curves are calculated with nominal dimensions of profiles and measured yield points in using Feldmann's model, Kemp's simplified model and Kemp's theoretical model; experimental results are given and local values of models with measured geometry are sometimes evaluated. (4) Conclusions

:

In order to elaborate practical tables or charts for designers, the working group of this research project has preferred Feldmann's model to Kemp's model and CRM's model for the following reasons, on basis of presented databank of tests results (see chapter 4.2) : more results from Kemp's model are on the unsafe side, even if extreme cases of tests are excluded (see Figure 21b) and Annex 11),

with Feldmann's model the evaluation of ductility of plastic hinges can be carried out by comparisons with either inelastic rotations (pav or rotations capacities Rav; and, on the other hand as explained in chapter 4.4.7.1, the criterion (3.1) comparing inelastic rotations
46

av. experiments

20 -ι

SAFE

15

M

>o

·

y

w · 10

-

5

-

^ /

0

χ

°

K&D simp. J&C best J&C lower

"VM** fl* η

·

S&S

UNSAFE|

fill (

1

)

10

5

15

20

R av. model (Spangemacher and Sedlacek values (S&S) given for fu / fy = 1,4) a) : Comparison

of rotation capacity from tests results and from various models

R av. experiments 24 23 22

X CRM's model

21

Δ Kemp's simplified model

20 19 18 17

Feldmann's model

16 15

14 13 12 11

10 9 8

7 6 5

4 3

2 1

0 0

1

2

3

4

5

6 7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

R av. model b) : Comparison of rotation capacity from tests results and from models of CRM, Feldmann and Kemp

Figure 21 : Comparison of rotation capacity 47

Author Test N° dl6b3m dl9a4m d20b4m d01a4m d02b4m d09a3m

dl0b3m EA22312 EA2233 EA2234 EA22412 EA2243 EA2244 EB28312 EB2833 EB2834 EB28412 EB2843 EB2844 EA28312 EA2833 EA2834 EA28412 EA2843 EA2844

CRM

Rm

Rav

17.0 11.4 6.8

13.6 18.7 15.6 5.1 2.8 5.7 1.6

22.5 20.0 13.2 6.4 7.8 18.9 19.8 32.9 12.0 9.3 15.4 2.8

0.6

1.5

18.9 19.8 16.5

45.4 34.1 20.5

4.3 4.3

15.8 9.5 8.3 50.4 19.0

5.0 6.0

4.6 14.8 8.3

4.6

6.4

5.3

16.5

1.5 1.3

6.4 4.1 11.25 23.4 12.2 28.4 17.7 17.6 23.3 40.4 15.2 11.4 21.5 13.5 9.3 11.6 11.5 12.8 9.5 8.3 14.35 12.9 12.5 8.6 8.7 6.1 6.75 8.5 5.7

2

6.0

3

15.0 7.9 19.1

4 5

6 7 8

9 10 11

12 13 15 17 18

19

20 21

22 23 24 25 26 27 28 29 30 31

32

7.8 6.9 13.3

20.7 9.2 6.9 12.2 9.1 6.4 8.0 8.95 9.2 8.5 5.6 10.49 8.2 8.1

6.0 6.25

4.6 4.6 8.5

4.4 10.5 17

Author

TestN°

Ymax

φ(ρΐ)

φ(ρ1)

A-2

0.122 0.028 0.048 0.051 0.030 0.029 0.023 0.048 0.029 0.021 0.031 0.062 0.036

0.272 0.042 0.120 0.092 0.052 0.048 0.049 0.112 0.078 0.045 0.068 0.136 0.083

B-l B-2 B-3 B-4 B-5

C-l C-2 C-3 C-4 C-5

D-6 E-6

M À

φ max

(PD

_?rot Rav-

R_ m Lm

rmax

Φρί

Φρί

R Kav _ «¿M 9rot

9max

»m

20

I

i5

S

io

fa.

¿«¿o õnSl ~ _*

15.3

«« ¿5 ¿"i ¿o J,o ¿o m »ι ÍS s- ο <"ì S <". «η «ξ. S «ξ. -*-1 _·«^ cí
Rav/Rm

Figure 22 : Ratio of Rav/Rm for experimental data

48

o o

Λ

4.3.5

Couchman's model for composite cross-sections

(1) The purpose of the thesis of Mr. Couchman (Ref. 14) is to investigate the behaviour of continuous composite beams with plastic (Class 1), compact (Class 2) or semi-compact (Class 3) critical sections. (2) The software Compcal developed at EPFL has been used to compute the available rotation capacities of beams with plastic and compact sections by introducing Kemp's model (Ref. 15). A wide variety of parameters have been chosen to study the influential factors to the available rotation capacities. These factors were : slendemess of the cross-section,

reinforcement at the support, represented by the ratio of hogging to sagging plastic resistance moment (the plastic moment ratio), structural steel characteristics,

slip between the steel and concrete, degree

of shear connection between the steel and the concrete,

ductility of shear connectors, span lengths,

number of spans,

ratio of adjacent span lengths, type and arrangement of loading,

propping of the beam during construction. The influence of parameters which are related to the composite section, or length of beam in hogging, on available rotation capacity has been shown, and a single variable can be used to represent all such individual parameters (see Annex 2 (3/9) and (4/9)). All parameters which affect available rotation capacity should therefore be allowed for in a design model, which is not the case for simplified methods of analysis such as those proposed in Eurocode 4 (Ref. 2). (3) For semi-compact composite sections, the available rotation capacity depends on cross-section properties but also on the arrangement of spans and loads (see Annex 2 (5/9)). The Kubo and Galambos model is used (see Annex 2 (6/9)).

4.4.

Numerical simulations of available rotation capacity

4.4.1

Simulations of tests results

(1) In order to check the efficiency of the finite elements modelization LABEIN has carried out numerical simulations of different 3-point bending tests from RWTH (Aachen) (Ref. 5 and Ref. 13) (Tests documentation: Annex 5 (6/46)) : the tests n° 16 to 18 of table 2 (n° EA28412, EA2843, EA2844) concerning HEA 280 profiles, with yield strength of 504 Mpa, which are in class 4 according to Eurocode 3 and nevertheless which furnish available rotation capacities from 4,1 to 16,5 ! * other tests specimens as n° 4 to 6 of table 2 (n° EA22412, EA2243, EA2244) concerning HEA 220 profiles, with yield strength of 420,5 Mpa, which are in class 3 according to Eurocode 3 and neverheless which furnish available rotation capacities from 1,5 to 15,4 ! * here is the list of profiles related to the numbers of tests specimens: . HEB 220 for tests n° 1 to 8 of table 1 (class 1 for S235 and S460 steel grades), . HEA 220 for tests n° 1 to 6 of table 2 (class 1 for S235 and class 3 for S460), HEB 280 for tests n° 7 to 12 of table 2 (class 1 for S235 and class 2 for S460), HEA 280 for tests n° 13 to 18 of table 2 (class 2 for S235 and class 4 for S460). The tables and figures included in Annex 5 present a summary of these simulations :

*

49

a) Assumptions of

finite element modelization are given in Annex 5 (1/46 to 5/46)

b)

The tables of Annex 5 (7/46 and 10/46) shows a synthesis of the results presented in an appropriate way to be compared with the experimental results. Beside this, the available rotation capacity given by Kemp's and Feldmann's models have been also included together with the computed and tests results. It can be observed how the finite element results correlate reasonably well with test data. Feldmann's model always provides conservative results and Kemp's model is often unsafe.

c)

The graphical layouts of Annex 5 (9/46 and 1 1/46) present the correlation between the calculated and measured values of different parameters for every case: maximum moment (Mu), rotation at the maximum moment (Qmax = Φπμιχ), inelastic rotation (Qr0t = 9av)>

f rotation capacity defined with Qmax · Rmax

~ « _9max-9pl

"\

Φρί and rotation capacity as usually defined with Qrot : Rpi

=R

^Φκ^-Φρΐ
ν,

d)

J

The figures of Annex 5 (8/46 and 12/46 to 29/46) give the comparison between calculated and measured moment-rotation curves for the cases where experimental relations were available. The Figures of Annex 5 (30/46 to 46/46) also provide deformations of beams related to the most right hand intersection between respective moment-rotation curve and plastic moment level. Those simulated deformations can be compared to tested specimens.

(2) CTICM has performed 5 types of numerical simulations in order to evaluate the available rotation capacity of a I-profile. The ahn of these simulations was to identify the main parameters which govern the available plastic rotation. More information is given in Annex 6 (document 3263-2-9) : a)

Firstly, comparisons were made between experimental results and numerical simulations from LABEIN (ABAQUS Program) and from CTICM (ANSYS Program). These results were not different in the increasing part of the curves, but after the top, significant deviations have been noted.

b)

4.4.2

The influence of a global imperfection, the influence of an eccentric load (biased load) and the case of a cantilever beam have been investigated (see also chapter 4.4.6.2).

Simulations of nominal cases without tests results

(1) In order to check if Feldmann's model is valid for all I or Η steel cross-sections, simulations were performed for different nominal cases for which no tests results are available. As numerical simulations have been successfully calibrated to existing tests results (see chapter 4.4.1), they are considered as representing the reality to which Feldmann's model has been compared (see Annex 14).

(2) Labein simulated following 3-point bending beams without tests results using nominal values of dimensions and yield strengths :

*

providing Ravailable > Ö, in case of class 3 or 4 cross-sections

:

- IPE A 300, with fy = 460 Mpa (EC3: class 3f) and span = L = 2,80 m ((L/2)/(iz.e) = 58,6 < 60), [Feldmann's model:
50

providing very high values of Ravailable^ hi case of class 2 cross-sections

*

:

- HE A 280, with fy = 235 Mpa (EC3: class 2f) and span = L = 7,9 m ((L/2)/(iz£) = 56,4 < 60), [Feldmann's model:
*

providing very low values of Ravailable-· m case of class 1 cross-sections

:

- HE A 650, with fy = 460 Mpa and span = L = 5,9 m ((L/2)/(iz.e) = 59,2 < 60), [Feldmann's model: (pav = 0,07 rad = 4,01°; Ravailable = 3,3, for L = 5,9 m], - IPE A 500, with fy = 460 Mpa and span = L = 3,7 m ((L/2)/(iz.e) = 59,1 < 60), [Feldmann's model:
*

providing very high values of Ravailable^ hi case of class 1 cross-sections

-

:

HEM 100, with fy = 235 Mpa and span = L = 3,0 m ((L/2)/(iz.e) = 54,7 < 60), [Feldmann's
model:

*

in case of high cross-sections -

*

:

HE AA 1000, with fy = 460 Mpa (EC3: class 2w) and span = L = 4,9m ((L/2)/(iz.e) = 59,1 < 60), [Feldmann's model: (pav = 0,04 rad = 2,29°; Ravailable = 3,33, for L = 4,9 m].

for HE A 200 and Ή EAA 200 profiles, for spans of respectively L = 2 m & 4 m and L = 2 m, for fy = 235 MPa, classifying the sections respectively in class 1 and class 3 and, for fy = 460 MPa, classifying the sections respectively in class 3 and class 4, in order to quantify the influence of the yield strength fy on the rotation capacity and on the classification of cross-sections.

(3) The tables and figures included in Annex 14 present a summary of these simulations a)

:

the tables and graphical layouts of Annex 14 (3/13 and 10/13) provide a synthesis with comparison between Abaqus simulations and Feldmann's model results.

Following symbols are used : Phi pl. =


Phi rot. = %0{, theta rot =


of the

results

and Rpl = R.

Feldmann's model fits very well with simulations results and moreover if partial safety factor Tm

b)

4.4.3

the Figures of Annex 14 (from 4/13 to 9/13 and from 11/13 to 13/13) show moment-rotation curves and deformed shape of beams for several singular cases.

Simulations of beams submitted to My-N loading

(1) In order to check Feldmann's model about My-N interaction loading, simulations have been carried out to determine the available inelastic rotation for different (N/Npi) values especially when the neutral axis moves from web to flange providing big differences of available inelastic rotations
^1

-(max)
if

51

Npi

<^^A

(see formula (4.3)),

- (min) ^2SL ΎΜφ

A°2

Npl

A

(see formula (4.4)),

where Aweb = A - 2btf , A = area of total cross - section and

'N^

-Aweb=0j257-

VNPlAimit

A

of Annex 15 (7/10) shows the plot of
(3) Figure 3

transition between those 2
4.4.4

|

N/T

)

ll/^Aimit

= 0,257 J

Simulations of LTB limits for Feldmann's model

(1) An important topic concerns the rules delimiting the application field of Feldmann's model in regards to lateral-torsional buckling (LTB) of members. In the formulas of Feldmann's model the length of member is not considered. But it is well known that for different levels (L/iz), the collapse of bended members with plastic hinge(s) is influenced : - either, only by LTB of members without occurrence of plastic hinge(s), - or, by an interaction between LTB of members and local buckling in plastic hinge(s), - or, by local buckling in plastic hinge(s) followed by LTB of members. (2) In order to check if Feldmann's model is valid within lateral-torsional buckling limits (of Eurocode 3 (Ref. 1) or CM66 (Ref. 18)) and on the other side in order to determine which limits from Eurocode 3 or CM66 are more realistic, numerical simulations have been realized by Labein. (3) Concerning the results for HEA 200 cross-section (see Annex 16 (9/25)) available inelastic rotation
52

LTB rules

:

χ

-Ξ-

/777Γ

Yy * Lateral restraints \i

S-r-Zr îl

Β

ψ

ψ=0

ψ=0

CM6:

if ψ = 0 and Lj/iz < 60 ε, then lateral-torsional buckling is prevented.

Eurocode 3

if Xlt á 0,40 , then lateral-torsional buckling is prevented. LTB rules: Lj/iz £

Standards

HEA200 CM66

Steeel grades

IPE 200 60 S 235

Eurocode 3

63,3

61,4

CM66

42,9 S 460

44,2

Eurocode 3

42,4

In these cases, the maximum length Lj between lateral restraints is quite similar according to CM66 and Eurocode 3 rules. (5) The differences between prescriptions from both design standards CM 66 and Eurocode 3 are presented for all standard I and H profiles (ΓΡΕ, IPE A, TPE O, HE AA, HE A, HE Β and HE M) and for both steel grades S 235 and S 460 (see Annex 16 (24/25) and (25/25) : Figures with L/^ = ί(ψ)), for simple cases with linear moment gradient and with ratio between extreme bending moment ψ :

-1< ψ ν

M2

<1.

As first global approximation, it can be said that CM 66 seems to be -

safer than Eurocode 3 when

0 < ψ < +1,

-

similar than Eurocode

ψ = 0,

-

less safe than Eurocode 3, when

3

when

-1 <

ψ < 0.

53

:

4.4.5

Numerical simulation of double clamped beam case

(1) A 6 meters long beam clamped at both ends and subjected to distributed load has been modelled by LABEIN in order to evaluate how the behaviour of the plastic hinge in this configuration differs from the simply-supported case. The data of the test case "17-EA2843" included in document 3263-1-8 (Spangemacher & Sedlacek) have been taken for the cross-section (HE 280 A ; b = 281 mm; tf = 12,6 mm; h = 275,6 mm; tw = 9 mm).

In this case, a first hinge is developed at the end locations, while the central section is still subjected to half of the plastic moment From this moment, the end hinges have to rotate up to the point in which the plastic hinge appears at the central section.

The figures included in the working document 3263-3-10 (see Annex 7) show, on one hand, the deformed shape after the formation of the first plastic hinge just before the maximum load (P/Po = 1 .46) (Annex 7 (1/7)) and, on the other hand, the deformed shape after the maximum of the load when all plastic hinges (at supports and at mid-span) have been developed (P/Po. = 1-29) (Annex 7 (2/7)) . Po indicated the load at which the plastic moment is reached.

A slightly buckled lower flange can be appreciated at the end plastic hinge just before the maximum of the load (the figure is affected with a magnification factor of 5) although the maximum end moment Mi is not reached yet, indicating that the local buckling starts to happen before the maximum moment Nevertheless, the figure representing the evolution of both moments, Mi (end section) and M2 (central section) show clearly how the first plastic hinge is capable of rotating without falling into instability until the last plastic hinge starts to form at the central section. The rotation has been taken at the section of zero moment to be consistent with the curves coming from the 3-point bending tests. (2) The same 6 meters long beam described above has been subjected to central load, extracting the same kind of results. In this case, both the end section and the central one reach the plastic moment at the same time. Because of this, both plastic hinges are simultaneously developed as shown in the figure representing the deformed shape (Annex 7 (5/7)) . (3) The comparison between the moment (at end section) - rotation (at zero moment section) curves obtained for the two cases, distributed and concentrated load, can been appraised in the figure (Annex 7 (6/7)) : the resulting available rotation capacity is very close one to each other. It is worth to note that the equivalent span for the concentrated load case (Leq = 3 m) is a bit larger than that for the distributed case (Leq = 2,5 m), due to the different sections where the zero moment appears. In the last figure (Annex 7 (7/7)) , the two curves have been represented together with the moment rotation curve coming from the 3-point test simulation and with the experimental one. It is interesting to note the good correlation with the experimental curve obtained for the clamped concentrated load case simulation.

4.4.6

Moment gradient and LTB restraints influence on available rotation

4.4.6.1 Introduction This section presents some conclusions about the combined influence of the moment gradient and the lateral restraints on the resulting available rotation capacity on the basis of interpreting the results of a number of numerical simulations carried out by LABEIN.

4.4.6.2 Simulation of simple cases (1) The working document 3263-3-9 includes results concerning different simulations carried out by LABEIN to appraise the influence of some factors on the available rotation capacity: type of load, position of the load and lateral restraints. (2) Position of the load:

Four different models have been considered moving the concentrated load from the center towards one of the beam ends but keeping constant the total length. The results are in good agreement with other simulations carried out on the basis of changing the beam span at the same time that the load was moved (document 3263-3-7) : the moment-rotation response, the predicted rotation capacity and the ultimate moment do not change significantly when moving the position of the load from the beam center provided that the load is not placed very close to the beam end. In this case, the deformation 54

mode changes suddenly due to shear effects. In this case when the load is near the support (25% for example), high shear deformation dramatically diminishes the available rotation capacity if the reference level remains Mpi.

But in case of high shear the reference level to evaluate the rotation capacity should not be the plastic moment resistance Mpi but the assumed ultimate resistance defined by the reduced plastic resistance moment allowing for the shear force My. For example that value My could be obtained from Eurocode 3; then the interaction between shear force and bending moment should be considered if the applied shear force is greater than half the plastic shear resistance of the cross-section. In that case of lower reference level (My < Mp]) , if moment-rotation curves are similar, the available inelastic rotation
Elastic rotation

Total rotation

Inelastic rotation

Φρί

9rot

9av =
100 % (CENTER)

1,2046

29,99

28,79

75%

1,2048

33,90

32,69

50%

1,2049

31,81

30,60

25 %

|

1,2049

?heJ£n?lusions of CTICM simulations about biased concentrated loads (see Annex 6) are similar to

LABEIN results.

(3) Type of load and LTB restraints: The following table presents the moment-rotation curves associated to the same simulation case (S10) but considering different load cases and different lateral restraints (see Annex 8 (2/3)):

Calculation rotation capacity, Rn

1

Concentrated Load

Distributed Load

No lateral restraint at hinge

24

19

Lateral restraint at hinge

34

33

The simulation results corresponding to cases with distributed load allow to observe a general trend in their behaviour in contrast with the concentrated load cases. The distributed load increases the sensitivity of the beam to suffer from lateral-torsional buckling (LTB) simultaneously interacting with the local buckling. As a consequence of this, the combined inelastic buckling mode (global and local deformation) makes the instability to happen at lower levels of load and rotation providing that no provision is made to avoid the global LTB deformation mode. The results is a lower rotation capacity caused by a combination of both factors: a very flat moment diagram and an insufficient lateral restraint ot the beam.

This effect can be clearly appreciated in the calculated moment-rotation curves (see Annex 8 (3/3))· if we compare the cases with concentrated and distributed load, the rotation capacity keeps constant in case of restraining the lateral displacement at the plastic hinge location. However, if this lateral restraint is removed, the rotation capacity becomes lower for the distributed load than for the concentrated one. Thus, there seems to be an interaction between the moment gradient and the lateral torsional restraints in such a way that one of the factors strongly affects the influence of the other on the calculated rotation capacity. A critical minimum length (Lb) for providing lateral restraints may be assumed to exist in order to ensure the calculated rotation capacity to be unaffected by the type of loading or the moment gi duldlL.

It is worth to note the presence of the stiffener at central section in the cases of concentrated load provides an additional torsional restraint compared to the distributed load case. This effect has been proved by applying the distributed load to a model with central stiffener. The calculated rotation capacity (Rav = 23) is increased compared to the case without stiffener (RaV = 19) 55

4.4.6.3 Interaction between moment gradient and LTB (1) A general trend could be observed when comparing simulation results corresponding to cases with distributed load in contrast with the concentrated load cases: the distributed load increases the sensi¬ tivity of the beam to suffer from lateral-torsional buckling (LTB) simultaneously with the local buck¬ ling. As a consequence of this, the combined inelastic buckling mode (global and local deformation) makes the instability to happen at lower levels of load and rotation providing that no provision is made to avoid the global LTB deformation mode. The result is a lower rotation capacity caused by a combi¬ nation of both factors: a very flat moment diagram and an insufficient lateral restraint of the beam.

The interaction between both factors can be appraised by defining the following two parameters:

- Lltb- Lb:

I)

Π)

*-!=-<

length between lateral restraints. length of the region of significant yielding determined by the moment gradient (flatter moment diagrams give rise to higher Lb values).

When sufficient lateral restraints are provided to avoid the global lateral deformation to interact with the local buckling (Lltb « Lb)» a single local flange buckling occurs and the available rotation seems to be unaffected neither the restraint length (Lltb), nor the moment gradient (Lb).

IOC« BaMPPOIICWtNRflinrilwOTR)

ΰίψ"

ΙΠ) In the intermediate case with a restraint length Lltb partially avoiding the global lateral buckling, a combined dependence of the available rotation on Lltb and Lb seems to happen.

r

j

Σ

Σ

1

ftF*{U,Xt>)

locai n»NPP»qqm lairaai auno hot ιρ^ππ}

In case of high Lltb values in comparison to Lb (Lltb » Lb), the combined inelastic buckling mode (local flange and global lateral) appears and the available rotation seems not to be depending on the restraint length (Lltb)» being only influenced by the moment gradient (Lb).

1

U>U>

Χ

Σ-Χ

«.-OS

toc« π aure» pappan YPPSTPAiNFn p¡ (wat utfpm rup

¿ 11

χ χ-κ-χ τ Figure 23 : Buckling mode related to unbraced length Lltb

Figure 24 : Inelastic rotation θρ of plastic hinge in function of unbraced length Lltb 56

(2) If we compare the cases with concentrated and distributed load, the available rotation keeps nearly constant in case of providing sufficient lateral restraints at the proximity of the plastic hinge location. However, if the lateral restraint is relaxed, the available rotation becomes lower for the distributed load than for the concentrated one. Concentrated

Load

Distributed

Load

Lb'

m

lili.

Mmax

-i.

μιιιι A

,γττ

£

U.»Lb

Σ

S

*

-Χ

L,.

ϊ -X

U-,

"

+

Γ

τ

i^vi

Ν^

4

4-

i À

Δ L,,.-«Lb

K^

>

-E

K *

Ut,

X-

x-x-

Ή

x

Figure 25 : Unbraced lengths Lltb for different moment gradients The question that arises is the determination of the critical minimum length Lltb for providing lateral restraints in order to ensure the calculated rotation capacity to be unaffected by the type of loading or the moment gradient

(3) Deformation buckling modes

:

The simulation cases modelled with the lowest values of Lltb (providing lateral restraints along all the upper flange or restraining the rotation at the plastic hinge location) gave, unexpectedly, lower available rotations than those obtained for the higher Lltb values. The reason for this behaviour is a change in the buckling deformation mode. The additional lateral restraints have avoided the lateral global buckling to interact with the local one but, at the same time, it has introduced a different mode con¬ sisting in a coupling between the flange local deformation and a local web buckling (B).

Therefore two different deformation buckling modes have been identified:

57

A) Flange local + lateral global Ducklings

Figure 26 : Flange local + lateral global bucklings B) Flange local + web local bucklings

Figure 27 : Flange local + web local bucklings (4) Summary of simulation results

:

The following cases have been modelled combining different lateral restraint conditions (varying the length between lateral restraints) with different moment gradients (varying the length between vertical supports) in order to evaluate the interacting influence of both factors on the calculated available rota¬ tions. The total length of the beam has been adapted, for each case, to the maximum required distance between vertical or lateral supports. The table shows the calculated available rotation (
HE 280 A

Material:

fy = 504 N/mm2 , fu/ fy =

Load:

Concentrated central load

58

1,35

Available Rotation


Type of beam

1

f"

S m

Lateral restraints X

X

«-

4 m

Buckling Deformation Mode

Li /M

0,21

0,10

All

41,8

0,17

0,10

AH

31,4

0,20

0,10

AH

12,6

Χ

0,24

0,30

AI

X XXX X XXX Χ

0,13

0,18

ΒΠ

0,29

0,36

BI

0,30

0,38

AI

4 m

-X-

3 m

-x-

1.2 m

S

1

ι SS- S3

Σ XXX Χ XXX Χ ttíppor fiLis«)

S-

X

(upper

II* Aft·)

Buckling deformation modes:

*AII

"^

*AI

Χ

*ΒΠ

Ι)

r

_37

Local flange + limited global lateral

Γ&

^"""W^ ® r*er -

*BI

Local flange + free global lateral

-s-^

ζ

Local flange + free local web

,

Local flange + limited local web

A further detailed description of the results is found on working document 3263-3-12 (Annex 3). 4.4.7

Equivalent length evaluation : continuous beam simulation

4.4.7.1 Inelastic rotation φ and rotation capacity R of a steel cross-section (1) As presented in chapter 3.1, the validity and the consistency of a plastic analysis of a frame under a given loading can be checked in all cross-sections by evaluating the ductility of plastic hinges either (3.1) with plastic rotation, or (3.2) with rotation capacity :


9av ΥΜφ

59

(3.1)

<

R

or,

av

(3.2)

Ymr

in reference to the 3 points-bending system

:

φ

7^ rot

mnnrtpnt 3α moment Hi diagram

^^*^1^-*"^

with

-^av Kav -

1?

φρι=

and

Φρί

Mpl.L

E.I

Figure 28 : Moment diagram and moment rotation curve for 3 points-bending system (2) The concept of rotation capacity R introduces a supplementary parameter such as the length L of a reference beam. It must be noted that this parameter is a structural and not local parameter. Rreq is to be calculated from frame plastic analysis as explained hereafter. The available rotation capacity Rav can also be given in tables but length because (ppi depends on the length.

it must refer to a particular beam

Because the verification has to be consistent, the required rotation capacity Rreq must be computed by considering the same length of reference which will be used to determine Rav

(3) By analogy with the 3-point bending system, this length of reference (or equivalent length) L related to a plastic hinge in a frame can be defined as the distance between zero-moment points on each side of the hinge : Plastic hinge ν

A "^ujjiipf

.Üb.

<

"Δ

τ

^¡W Figure 29 : Equivalent length L to 3-point bending system (4) For a beam where a plastic hinge occurs under a single concentrated load, the zero-moment points are often easy to identify. For more complex configurations, it is not so easy to determine the reference length, such as the following common examples in Figure 30 :

60

l\

\

A\\

Plastic hinge

\

Plastic hinge ..

m I

Figure 30 : Plastic hinge in structures (5) In chapter 3 of Annex 10, a method is proposed to calculate the required rotation capacity if the shear force is known on each side of the plastic hinge. The case of a plastic hinge at the end of a beam in a portal frame is also investigated (Figure 31).

Method of evaluation for required φ arid R

Bending moment gradients

or Rreq(Ti,T2) < Ravail(Leq) 9p.avail or Ravail(D given by formulae or tables


*req(Tl,T2) = "A ^.f M J, Ti+T2 Leq = Mpl

f J_

«Pjueq

_1_

σι + τ2

or RreqCO ^ Ravail(Leq) 9p.avail or Ravail(L) given by formulae or tables

9p.req- 9p.avail

ρKTeq

- ELT " 9p.req

M Pi Leq=2.Mpl/T

-req -

Çp.avail or RreqCO ^ Ravail(Leq) 9p.avail or Ravail(L) given by formulae or tables

τι

_ 2.E.I.T M2 ^P-req

Rreq ~

Mpi

Leq=2.Mpl/T Figure 31 : Summary table (issued from Annex 10) (6)

At supports, when the moment is linear in the vicinity of the plastic hinge, the problem is the same that for a concentrated load on a beam. TI /fff|Yiv T2

il iibl W

TL

.Hastic hinge

i Figure 32 : Equivalent length L for different moment gradients 61

When the moment diagram is not linear (due to distributed loads), the reference length is given by the tangent on each side of the plastic hinge as shown in Figure 32. Therefore it does not correspond to the distance between zero-moment points, but this seems to be more realistic since the required rotation capacity depends only on the shape of the moment diagram near the plastic hinge. (7) Conclusions : It is difficult to clearly identify the length of reference (or equivalent length) L by analogy of 3 point bending beam in order to calculate φρί. On the other hand, computer softwares usually provide required inelastic rotations (preq at each formed plastic hinge for each load level till structural collapse (mechanism,...). Thus the most simple way to check ductility of plastic hinges clearly consists in comparing directly required and available inelastic rotations φ of plastic hinges instead of converting those values in rotation capacities R.

Therefore it would be advisable to refer to the criterion (3.1) dealing with inelastic rotations (Φ) and not with rotation capacities (R) (criterion (3.2)). This conclusion is highlighted by the results of LABEIN numerical simulations presented in chapter 4.4.7.4.

4.4.7.2 Introduction to numerical simulations

Two cases of 2-span continuous beams with concentrated and distributed load respectively have been modelled and simulated by LABEIN in order to evaluate the equivalent length to be considered for the checking of required rotation capacity in comparison with the available rotation capacity evaluated from the equivalent 3-point bending test in cases of different moment gradients. 4.4.7.3 Model description Geometry & cross-section:

Two span beam. IPE 270, Steel grade S 235

* Concentrated load case: L=10m λ = 0,5 α = 0,8

«L

* Distributed load case: L=10m λ = 0,4 γ=1

. "

A *

L

.1

t

m

XLÎ n

Figure 33 : Concentrated Load

* 3-point bending cases: 1) Li = 1 m L2 = 5 m 2) Li = 3 m 3) Li = 1,6 m 4) Li = 1,9 m

L2 = 3m L2 = 2,17m L2 = 4m

Figure 34 : Distributed Load

Material:

Elasto-plastic model with fy = 235 MPa and fu/fy = 1,58

Meshing:

Type of element: 8-node shell with parabolic shape functions (S8R Abaqus) A more refined mesh has been provided at the areas where local buckling behaviour is expected to happen.

62

Concentrated

Distributed

Number of elements

640

616

Number of nodes

1973

1913

Degrees of freedom

11.838

11.478

Fillet radius modelling: By means of web and flange overthickness calculated for keeping constant the total section area.

Initial imperfection: Pattern:

the first positive elastic buckling eigenmode is used as the initial imperfection shape.

Magnitude:

maximum of

1

%

of the flange thickness.

Load application:

*

Concentrated load:

Incremental-iterative solution procedure has been employed using displacement control. Appropriate constraints to degrees of freedom at upper flange of the loaded section have been applied to ensure the same vertical displacement of mese nodes.

*

Distributed load: Load control has been used in the load-displacement analysis together with an arc-length integration method (Riks) to cope with the post-buckling unstable response. The load has been applied by means of equivalent node forces at upper flange central line.

*

3-point bending cases: As described for concentrated load case.

Boundary conditions

Vertical supports have been introduced at every node of the lower flange located at the corresponding support sections.

Longitudinal support has been provided in the loaded section at the centre of the upper flange.

Lateral restraints have been introduced on the compressed flange of the support and load application sections and additional sections according to Additif 80 - DPU Ρ 22-701 (CM66) (Ref. 18):

*

concentrated load case: a moment ratio of 0 has been considered in all the length (leading to L = 1,81 m between lateral restraints) excepting the part between both plastic hinges, where the moment ratio is - 0,87 (L = 2,87 m).

*

distributed load case: a moment ratio of 0 (L = 1 ,8 1 m) has been taken for the right part of the beam, the first hinge influence area, while a conservative moment ratio of 1 has been considered for the left part (2nd hinge zone) leading to L = 1,05 m between lateral supports.

*

3-point bending cases: a moment ratio of 0 is assumed leading to L = 1,81 m between lateral supports.

63

The bending moment diagrams are as follows

:

Concentrated Load

Distributed Load

1

Γ

15

>^JL+

15

ν

--wáu*

^UJJILP^

Ά £_!_ y/ìW %l17 S

1

Î

N-U

fr-**L

1-L-5

/

2.17 J

1.6-L-2.1

"'

Ê

i

â

JL

3

ì

m

3-L-3

1.9-L-4

Figure 35 : Bending moment diagrams for concentrated and distributed loads The equivalent lengths selected for the 3-point bending tests have been determined according to two different criteria: the length between the two zero moment sections adjacent to the plastic hinge,

the length determined by the tangent to the bending moment diagram on each side hinge.

of the plastic

4.4.7.4 Results Comparison with 3-point bending case results

= Φτοί - Φρί

Φρί (deg)

Φτοί (deg)

1,7

8,3

6,6

3,9

1,65

8,7

7,1

4,3

1,65

8,5

6,9

4,2

Continuous beams with distributed load

14

15

13,9

12,6

3-point bending beam 1.9-L-4 Li = l,9;L2 = 4

1,62

8,7

7,1

4,4

3-point bending beam 1.6-L-2.1 1,6; L2 = 2,17

1,03

9,1

8,1

7,8

CASE Continuous beams with concentrated load

3-point bending beam l-L-5

Li = l;L2 = 5 3-point bending beam 3-L-3

Li = 3;L2 = 3

Li =

64


Rav

(deg)

Concerning the concentrated load case (Figure 38), a very good agreement can be observed between the moment-rotation curve of the continuous beam and that of the corresponding 3-point testing case (Li = 1 m; L2 = 5 m). Also the case of central load (L\ = 3 m; L2 = 3 m) gives the same result as expected.

In the distributed loads case (Figure 37), however, a clear disagreement is observed when comparing the moment-rotation curve of the continuous beam and the corresponding curves of the 3-point bending cases

In view of simulation results it seems better to evaluate rotation capacity R with equivalent lengths determined by tangent to the bending moment diagram on each side of the plastic hinge (case 1.6-L-2.1) The discrepancy seems to be caused by the interaction taking place between the lateral deformations induced by both hinges in the continuous beam and giving rise to a stiffening effect This interaction is caused by the fact that the continuous beam has got positive and negative moments along its length. The lower flange, compressed near the first hinge location (negative moment region), is subjected to tension stresses in the positive moment region. This tension, acting on the lower flange at second hinge location, is responsible for the suffer post-buckling response appreciated in the moment-rotation curve of the first hinge in comparison to the 3-point bending case. It can be seen as a sort of stress stiffening effect that produces a positive influence on the side of increasing the actual rotation capacity of the beam.

This effect is avoided in the concentrated load case since the lateral deformation at both hinges is mainly local and does not interact one to each other. Thus, the rotation capacity of continuous beams subjected to distributed loads (this load case is more likely to induce coupled lateral deformations) may be significantly higher than that given by the 3-point bending test due to the described stiffening effect

(Moment distribution) /N.

<- Compressed

Tensionsd lower flange

lower Hange -

(Lower fange lateral deformation}

S-8-X Figure 36 : Lower flange lateral deformation

A further detailed description of the results is found in Annex 4 (working document 3263-3-17).

65

16

τ

^-,

8

10

12

14

16

13

Rotation (dag)

Figure 37 : Continuous beam with distributed loads

o

4-

ι

10

8

12

14

Rotation Weg)

Figure 38 : Continuous beam with concentrated loads

66

16

18

5.

5.1

Required rotation capacity

Introduction The work performed by CTICM has mainly consisted in developing practical tools in order to facilitate the verifications of the plastic rotations obtained from a plastic global analysis of a steel frame. For several simple cases of continuous beams, formulas and abacuses have been developed. For more complex cases, a general methodology has been established. All these results are explained in the "Guide for plastic analysis". CTICM has written the second chapter of that guide : "Required plastic rotation of plastic hinges in structures" (see chapter 2 of Annex 13). The content of that chapter is : 2.1

2.2 2.3 2.4

Introduction Influence of parameters on required inelastic rotation Continuous beams Frames

We give hereafter the main results of that chapter 2 of the "Guide of the plastic analysis" (see Annex 13).

5.2

Influence of second order effects Through the simple following example, it can be shown the influence of the second order effects on the required inelastic rotation (influence of parameter 10.1 of Figure 4). 1.25

320 kN

320 kN 17 kN/m 23 kJjL· nniiiiiiuiinyniiuiUHiiiTTi.

Λ

1.22J

1.20

2*"" Jlst order

1.15

4

|

1.10

IPE 300

1.05 1.00

1.000

Λ

h =6 m

IPE 240

0.95

Γ!

l/^ 3 TZp-

L=8m

2

Z£k

0.80

/' ^ y\

3

' S S*

A

¡10.9

ΤΛ

= plastic hinges

|2nd order

10

15

]*~

F20ϊβ] ¡22525

32.7' 30

10"3rad

35

Required plastic rotation

Figure 39 : Portal frame

Figure 40 : (λ-φ) curves

Due to the second order effects, the collapse is reached before a full plastic mechanism is obtained. Therefore, only three plastic hinges occur instead of four and the required inelastic rotation is much lower, at the ultimate limit state, from a second order plastic analysis than it is from a first order plastic analysis. But for a given load factor (λ = 1 for instance), the second order effects generally increase the required inelastic rotations in plastic hinges.

5.3

Influence of load level (design loads or collapse)

(1) Influence of the parameter 10.2 of Figure 4 : practical methods give directly the plastic mechanism for frames, and also the plastic rotations in the hinges and the internal forces just before the last hinge occurs. So, an usual way is to check TJLS criteria just before collapse, comprising rotation capacity in hinges. However, design rules only require to check these criteria under the design loads; especially, the designer has to check that a plastic mechanism is not reached. Computer programs can calculate the plastic behaviour of frames under design loads by increasing applied loads proportionally to a common load factor, α for instance. So, it can be shown that the inelastic rotation in the first plastic hinge can be far greater at collapse than it is under the ULS design loads. For most cases, checking the plastic rotation at collapse can be too conservative if so far as the frame design is never accurately optimised for all the limit states (ultimate limit states, serviceability limit states, ...).Therefore it is sufficient and economical to check the rotation capacity criterion under the ULS design loads. 67

Load factor = λ Collapse

Design loading at ULS

Plastic rotation
Figure 41 : Evolution of the plastic rotation in a first plastic hinge (2) A remark arose from the F6 committee about the influence of the parameter 10.2 of Figure 4, which influences the required rotation capacity of plastic hinges: the ratio between design load level (k¿) and complete plastic mechanism load level (Xmax).

In view of that parameter, the required rotation capacity chosen by the designer (Rrequiredl; as an example related to λι = X¿) must be clearly specified in the calculation note. Indeed, afterwards the building could be used or refurbished with other conditions (higher loads, other boundary conditions,...) and, the required rotation capacity in that context (Rrequired2; as an example related to Λ.2 with λ<ι < %2 < %max) could be greater than the previous one (Rrequiredl) and so the available rotation capacity of the members should be checked (Ravailable ^ ^required^ )· On the other hand, at ultimate limit state, the design of a building would not be optimised if the difference between design load level (λ<0 and complete plastic mechanism load level (Amax), was too high. But mat big difference between load levels could also be explained by restrictions at serviceability limit state or other reasons (instability of the frame,...).

5.4

Influence of loading path

(1) On one hand Eurocode 3 (Ref. 1) states, about global analysis, that "It may be assumed to be sufficient, in the case of building structures, to adopt simultaneous proportional increases of all loads" (clause 5.2.1.1 (5) of Eurocode 3). (2) On the other hand, in case of plastic global analysis, for a given loading arrangement the mechanism and the ultimate load factor do not depend on the loading path, but the order of occurrence of the plastic hinges in the structure depends on the loading path. The required inelastic rotations also depend on the loading path under the design loading and at collapse.

(3) The examples of Figure 42 shows a continuous beam with different loading path

:

path A corresponds to a proportional increase of all loads as allowed by Eurocode 3 (Ref. 1) and, path Β defines a specific increase of loads which is not always proportional. The collapse mechanisms are reached for the same load level (Pi ; P2 )whatever the loading path. But the required inelastic rotations of plastic hinges (
(4) It can be concluded that whatever the loading path is, required inelastic rotations in plastic hinges
68

Different loading paths lead to the same φ at collapse if the same mechanism is reached and if no elastic return occurs in any plastic hinge during loading. The following examples highlight differences in (preq at collapse because of elastic returns :

Pl

P2

Qi = 8P1

ΓΡΕ160 S 235

*~~Z L

Path A : Path Β :

2m

=

4m

*H-

All loads increase proportionaly Pi and Qi increase together with P2 = 0, then P2 starts while Pi and Qi are also increasing

mechanism

1746

Path A
(preq.2

=0 = 0,002623

Path Β (preq.i = 0,01142 (preq.2 = 0,008333

[daN] 2196

Pl

[daN]

3265

P2

3534

Çjjxamplëj^

1717

Path A (preq.l = 0 (preq.2 = 0,006926 Path Β (preq.l

= OJ004170

(preq.2

= 0,009012

[daN] 2917

3583

4364

Figure 42 : Influence of the loading path on (preq at collapse 69

P2

Continuous steel beams

5.5

Formulas and abacuses have been developed for simple configurations of two span continuous beams (see chapter 2 of Annex 13):

Two span beam with a concentrated load

5.5.1

Required inelastic rotation

F

i ,

ocL

0
l

Λ

r_ Mpl L -ct2-2a + 2h + l " Δ If α<ν2(η + 1)-1,Φπχι=γ^7·

Δ TlL

Η-

1

ι

:

,

if os-^cñ+5-.

Figure 43 : Continuous beam

Φ^=^^·(<*2+2α"2ιΗ

In the "Guide for plastic analysis" (see Annex 13), other formulas are given to allow the designer to calculate the ultimate load and the ratio of the ultimate load to the load for which the first plastic hinge occurs. Moreover, abacuses supply these results in relation to the relevant parameters.

5.5.2

Two span beam with uniform distributed loads 0

<η<1 ©SO

km»

coq

JELL

Figure 44 : Continuous beam Required inelastic rotation

If ωη2<

1

First plastic hinge on intermediate support _ Mp! L +ωη3) :

Φη^ =

2!±ί_2ίΜ(ι

EI

3

12

/

V

First plastic hinge in the first span :

If ωη2>1 First plastic hinge on intermediate support

_Mpl_L

Φπ*ι- EI

η+1

3 + 2λ/2

1

12

First plastic hinge in the second span

:

+ anf ατη :

.MplLV2 + l η + 1-(ν2+ΐ) 2 l + corf 3

ΦΓβς- EI

4α>η2

70

5.5.3

Two equal span beam with a concentrated load and an uniform distributed load

The relative influence of a concentrated load and an uniform distributed load has been evaluated by using the PEP micro program because for this case, formulas are too complex.

iF |||||||||j|]|||||||]|lillilllllllllllli!lllilllll ^ I


Δ

.

Ll

,

Δ L

,

Figure 45 : Continuous beam Two parameters have been taken into account : the position (a) of the concentrated load in the first span and the ratio between the concentrated load and the uniform distributed load (μ = F / (qL))· By varying these parameters, the following abacus has been plotted in Figure 46. The required inelastic rotation may be expressed as

req ~ *r

:

E.I

-F/CqL) Figure 46 : Design curves

71

5.5.4

Two span beam - General case

A more general method is proposed to calculate the required inelastic rotation when the position of the plastic hinge is known from an elastic analysis of the beam, whatever the load arrangement is. It is assumed that the maximum moment Mmax in the beam exceeds the plastic moment Mpi. So : ΔΜ = Mmax ~ Mpi ι

BL

|ο<β<ι

I 1

Δ

Δ 1

L

ι

HL

ι

ι

Figure 47 : Continuous beam From that, the expression of the required inelastic rotation is given by :

_AML

Φι^" ΕΙ

η+1 ·3β2

The abacus plotted in Figure 48 gives the factor kr versus β for several values of η , with ,

AML

Figure 48 : Design curves 5.5.5

Conclusions

The results show that it is difficult to give a value of the required inelastic rotation to a type of structure (continuous beam, portal frame, ... ). This value could be too conservative for some cases or on the unsafe side for other cases.

5.6

Frames

5.6.1 Plastic analysis using specific programs In order to make a plastic design of a frame and determine the required inelastic rotations in plastic hinges, a global elastic-perfectly-plastic analysis (see §5.2.1.4 of Eurocode 3 (Ref. 1)) of a frame can be performed with the help of a plastic analysis computer program. Those programs are generally based on the step-by-step method and can automatically take into account particular aspects such as elastic returns, second order effects, semi-rigid behaviour of connections and also N-M interaction in bending resistance of sections. All these aspects have direct influence on the required inelastic rotations (see chapter 3.1 for the influence of second order effects). 72

5.6.2

Plastic analysis of simple frames using an elastic analysis program

The aim of this part is to explain how to perform a plastic analysis of a frame with the help of a simple elastic analysis program provided the axial force and the shear force do not reduce the bending resistance of the cross-sections in which plastic hinges may occur. This method is known as the "step-by-step method". The following flowchart is a summary of the method. Geometry - Steel Support conditions

I

Γ

Load

* i=0

Initializations Moment diagram : Shear force : Displacements : Hinge rotations :

"~'

t

Mø = 0

V0 = 0 Dn = 0 Φβ = 0

'

ELASTIC ANALYSIS i (with i hinges) (with applied design loads)

-"^^^

^^~^*«

Is the structure stable ?

no

If i = 0 : Support conditions are not valid

If i > 0 : A plastic mechanism is reached

*

yes

Results of the elastic analysis i (with i hinges)

Moment diagram : Shear force diagram : Displacements : Rotations in the hinges :

Elastic return Suppression of the perfect hinge

An - Ai

Θ

m¡ v¡ 8¡

φ;

o

^

**""^

hor each plastic hmge, ^^^^

^Μφ>0?

^^^

^"lyes Step number : j = i + 1 Research of die cross-section for which

AAj = (Mpi - Mj) /m; is minimum V

Load factor

State

:

λ

= Aj + AL·

of the structure at the end of the

step

j

(for the loading λ: F) Moment diagram : M; = Mj + Akt mj Shear force diagram : V: = Vj + Δλ v¡ D: = Dj +

Displacements :

Δλ; 8¡

*

Rotations in the lunges at the end of the step j Φ} =

Φχ+Ατ\}%

ί i=j

ί A new perfect hinge is introduced in the structure

Figure 49 : Flow-chart for "step-by-step method'

73

Particular phonema such as elastic returns in plastic hinges may happen during a plastic analysis. This is explained in the "Guide for plastic analysis" (see chapter 2 of Annex 13). A simple example is also included to this guide (see chapter 4 of Annex 13).

5.6.3

Indicative values of (preq for pre-design of steel frames Rreq Η

Type of frame

-i'-

Min

Max

Mean

Min

Max

Mean

15

0,06

0,49

0,21

0,46

0,85

0,66

19

0,00

2,12

1,11

1,04

4,02

2,58

14

0,06

2,41

1,10

0,66

3,59

1,75

15

0,57

2,08

1,07

0,83

2,52

1,37

12

0,05

1,34

0,41

0,40

1,36

0,76

0,23

0,55

0,34

0,64

1,42

0,90

ss

-Γ

Uli

xb

^

-j^

LL

^

.V?

^

ι

"ÄV

i {

+

" i

¿

,·Α,-Α,Αι w

At collapse

Under design loads

au

^

¿

η

w

:w

w

Figure 50 : Indicative values of required rotation capacity Rreq of plastic hinges in steel frames (1) A study has been conducted in order to determine the required inelastic rotation (Preq and the required rotation capacity Rreq of formed plastic hinges for different types of frames (Ref. 23). Six types of frames have been identified, and for each type, a parametric study has been made. So, 79 frames have been studied (see Annex 10). A simplified method was adopted in order to obtain a realistic design. An elastic analysis was first performed in order to check Serviceability Limit State requirements. Then a plastic analysis was made in order to check Ultimate Limit State criteria and to determine required inelastic rotations under design loads (λ = 1) and at collapse (Xu) : mechanism or instability. Some of the frames were sway frames according to Eurocode 3 and the second order effects were taken into account. All analyses have been conducted with the PEP micro program of CTICM (Ref. 16). Required rotation capacities Rreq have been calculated from required inelastic rotation (preq directly given for PEP-Micro program and with the relations summarized in Figure 31 (chapter 4.4.7.1) and detailed in Annex 10.

74

(2) Figure 50 issued from Annex 10 gives a summary of the values of required rotation capacity Rreq of plastic hinges under design loads and at collapse for different steel frames (n = number of calculated frames). It is not possible to consider all types of structures, so the results are not necessarily fully representative. These results can show that it is rather difficult to associate a value of the required rotation capacity to a type of frame : the values are strongly scattered. If this was to be done, the maximum value should be considered but this leads to too conservative methodology. Type of frame

Under design loads:

At collapse

max φ^.<1 [rad]

max %eq.u [rad]

0,009

0,020

0,015

0,032

0,026

0,035

0,034

0,040

0,013

0,014

0,006

0,008

:

ill* ^ !_jfc_3r_i_3 '

*

ä :

^

au ss>

Si

to

ft

±

sx-

,-Μ-Α,Λλ ä

?*?

TO

sic

Figure 51 : Indicative values of required inelastic rotation

q>req

of plastic hinges in steel frames

(3) As explained in chapter 4.4.7.1, it is more simple to evaluate the ductility of plastic hinges with inelastic rotation (Preq instead of rotation capacity Rreq. Figure 51 issued from Annex 13 provides a summary of the maximum values of inelastic rotation (Preq (in radian) of plastic hinges under design loads and at collapse for the different steel frames identical to the ones given in Figure 50.

75

5.7

Evaluation of required rotation capacity for composite beams

(1) In the document 3263-1-20 following comments may be given about the 2 diagrams ("Ototal / öelastic" in function of "moment redistribution") concerning continuous composite beams: the required rotation capacities of continuous composite beams is much greater than for continuous steel beams :

for 3 spans composite beams: ^required (to reach the full span plastic moment resistance) = ± 10 (Figure 52),

for 2 spans composite beams: ^required (to reach the full span plastic moment resistance) = ± 20 (Figure 53),

for 3 spans steel beams: ^required = ± 3 (see document 3263-1-2: Figure 3.6). Those differences between composite and steel required rotation capacities may be explained by the non-linearities of the concrete which induces large non-linear response in "moment-rotation curves" in span behaviour of continuous beams (the shape factor, α = ( Wpiastic / Weiastic)> is greater for a composite cross-section than for a steel I section), while no differences appear in support behaviour (nearly ideal elastic-plastic behaviour for both composite and steel crosssections). the rotation capacities required to form a mechanism in continuous composite beams are dependent of the degree of shear connections ("Fc / Fpf" or "N / Nf") which defines the limit of the plastic bending moment resistance of the cross-sections and so influences the moment redistribution.

design curves with rotation capacities given in function of moment redistribution could be established; consequently for a given rotation capacity the required moment redistribution related to the given "M'p / Mp", could be deduced from those abacuses. (2) In the scope of the design method proposed by Mr. Couchman G. (Ref. 14; see chapter 3.1), the required inelastic rotation 8req (=req) has been graphically represented as a function of moment redistribution (Δ). The use of such curves allows for the parameters which affect the rotation capacity required by a beam to achieve a given moment redistribution. These parameters are:

Elastic moment ratio (μ6ι) and span type (external or internal). These two parameters affect the basis form of Oreq vs. Δ curves. Plastic moment ratio (μρι) , which affects values of moment redistribution but not the form of 9req vs. Δ curves.

Degree of shear connection and construction method (propped or unpropped). These two §arameters may necessitate modification to the value of moment redistribution which is given by a req vs.

Δ curves.

( See Annex 2, (7/9) to (9/9)).

76

7

ί?Ν

-

O"

-

o o o

_


ir

s

S

J4S

^

o o IO tn

o o

o

©

_

o q in ^#

-

o o

-

o

; -

ο ο in*

co

&

\

o o o co

: \

*.

\

·.

\

o o

3

ΐ\Φ

o

Vf»

o

<$

»7-

S

n

τ

Γ

lil

τ

o-O LO

·*

τ

n

r η

r

r r r

O o

τ

r

ι

τ

o CVi

o

o

O«

ID

o

o

l8

Figure 52 : Diagram of Ototal / ©elastic in function of moment redistribution for 3 spans composite beams

77

àH

J S

ξ

o

in

<£ τ

Γ

τ ο ο

j

rr

Τ ο

ο

ο

o

o

tv»

Oí

ο

o y/i

V) Γ"

j

r r τ r τ

ι

ι

Γ

τ o

o o

00

ι

ι

Γ

o o o c\2

o

rõ

Figure 53 : Diagram of Ototal / öelastic in function of moment redistribution for 2 spans composite beams

78

6.

Review of Eurocode 3 rules

(1) The clause 5.3.3 (4) (which is a principle) of Eurocode 3 (Ref. 1) may be unsafe in certain cases

:

" 5.3.3 Cross-section requirements for plastic global analysis (...) (4) For building structures in which the required rotations are not calculated, all members containing plastic hinges shall have class 1 cross-sections at the plastic hinge location."

Indeed it may happen that class 1 cross-sections would have not enough available rotation capacity Rav compared with rotation requirements depending on the percentage of redistribution of moments and the structural arrangement.

The two following examples (2 span beam with a concentrated load) illustrate this problem in using Feldmann's model (see Annex 13) and formulae of required rotation capacity for continuous beams (see chapter 5.5 and Annex 13) : Example

1 :

IPE 400, S 460 : class

α = 0,37
cross-section

1

η = 1 L = 5,8m

= 0,067/1,5 = 0,045 rad

Rav = 2,708/1,5 = 1,805

Example 2

:

IPE A 600, S 355 : class

α = 0,33
η=

1

1

YMR=l,5

= 0,069 rad

<


<

Rreq= 2,792

Vsd/Vpl.Rd = 0,59 (not fulfilled !) (not fulfilled !)

cross-section

L= 11,0m

= 0,077/1,5 = 0,051 rad

Rav = 3,293/1,5 = 2,195

ÏM

YM
Vsd/VpLRd = 0,47

<

(Preq = 0,0794 rad

(not fulfilled !)

<

Rreq = 3,389

(not fulfilled !)

(2) The clause 5.2.1.3 (3) (which is a principle) of Eurocode 3 (Ref. 1) may be unsafe in certain cases

:

" 5 .2. 1 .3 Elastic global analysis (...) (3) Following a first order elastic analysis, the calculated bending moments may be modified by redistributing up to 15% of the peak calculated moment in any member, provided that : a)

the internal forces and moments in the frame remain in equilibrium with the applied loads, and

b)

all the members in which the moments are reduced have class 5.3)."

1

or class 2 cross-sections (see

In Annex 9, the problems of that rule are precised for 2 common examples : a)

2 spans continuous beam with a uniform distributed load;

b)

2 spans continuous beam with a point load.

In example a), the required rotation capacity is shown to be quite limited (Rrequired < 0,6) and so class 2 cross-sections may be used in that case. In example b), for certain cases (concentrated loads,...) the required rotation capacity may be very large (Rrequired > 4) and the use of class 2 or even class 1 cross-sections with a 15% redistribution of peak moment may be unsafe regarding the rotation capacity.

79

Conclusion

7.

(1) Generalities

:

The plastic global analysis of structures may be more competitive than elastic global analysis but, despite "easy" evaluation of structural resistance, some difficulties limit or prevent its common use : practical tools for checking the ductility of plastic hinges are missing. Previous researches on the rotation capacity of plastic hinges (Ref. 3, Ref. 4, Ref. 5) have allowed to understand the behaviour of plastic hinge formation for conventional steel. The results of these researches have shown that the requirements of Eurocodes 3 and 4 (rules for classification of crosssections : b/t - ratio and its yield stress dependence; material properties : fu/fy ^ 1,2, eu > 20 ey, A5.65 ^ 15 %) are very safesided and could be substantially reviewed especially for high strength steels (S460)).

(2) New rules for Eurocode 3 (Ref. 1)

:

In the scope of this research project the ductility of plastic hinges is evaluated by inelastic rotation (
:

"5.3.3 Cross-section requirements for plastic global analysis (...) (2) At plastic hinge locations, the cross-section of the member which contains the plastic hinge shall have a rotation capacity of not less than the required rotation at that plastic hinge location. (3) To satisfy the above requirement, the required rotations should be determined from a rotation analysis."

In order to respect these Eurocode 3 requirements, a new limit states criterion is presented in detail in this research project : a comparison between the required inelastic rotation
Φκχι4 ^ Φ3ν.ά

.

with


=

Φ av

ΎΜφ

This new criterion could be directly proposed as an Application Rule to Eurocode 3 as explained in the Background Document 5.09 for chapter 5 of Eurocode 3 (see Annex 12) :
*

"5.3.3 Cross-section requirements for plastic global analysis (...) (4) For building structures in which the required rotations are not calculated, all members containing plastic hinges shall have class 1 cross-sections at the plastic hinge location."

*

"5.2. 1.3 Elastic global analysis (...) (3) Following a first order elastic analysis, the calculated bending moments may be modified by redistributing up to 15 % of the peak calculated moment in any member, provided that :

80

a) the internal forces and moments in the frame remain in equilibrium with the applied loads, and

b) all the members in which the moments are reduced have class

1

or class 2 cross-sections

(see 5.3)."

On the other hand this new criterion applies to all cross-sections able to develop their plastic bending moment resistance, then it is explicitly applicable for class 1 and class 2 cross-sections. According to tests results and numerical simulations, present rules of Eurocode 3 for cross-section classification are shown to be too many conservative. Those rules could be improved and thus could extend the application field of plastic global analysis (Ref. 17).

In order to help designers involved in plastic global analysis of structures, a guide has been developed (see separate Annex 13) presenting a simple tool for the evaluation of ductility of steel plastic hinges. Beside generalities (concept, concerned parameters,...) explanations and recommendations are provided in detail for the new aforementionned limit state criterion : about required inelastic rotation (preq of plastic hinge

:

the influence of parameters is shown,

formulas and charts are given for simple cases of continuous beams,

indicative values are furnished for several frames,

in general, reference is made to design softwares that always detiver hinge, about available inelastic rotation

at each plastic

:

assumptions and limitations of Feldmann's model are given : the model is applied to class 1 & 2 cross-sections (according to present rules of Eurocode 3) and to conventional steel grades (S 235, S 275, S 355, S 420 and S 460 : with f^/fy > 1,15, ε > 20 ey, Α5.65 ^ 17 %); different load cases are considered (bending about major and minor axes (My;M^), combined bending about major axis and axial compression (My - N)), tables and graphs are provided with
different design examples are included, an appendix comprises tables with values sections and for conventional steel grades.

of plastic resistance for standard I and Η hot-rolled

The guide clearly specifies that provisions for sufficient lateral restraints are obligatory in order to avoid occurrence of lateral-torsional buckling of members with plastic hinges. Lateral-torsional buckling mat influences a lot the values of inelastic rotation
A lot of numerical simulations that were calibrated to tests results, have been carried out to check the validity of Feldmann's model, to understand the influence of parameters and to investigate cases without tests results (see chapter 4.4).

(3) New rules for Eurocode 4 (Ref. 2)

:

A specific design method has been proposed by Couchman (Ref.

14)

for continuous composite beams

(see Annex 2).

That design method based on the idea of rotation capacity : includes the influences of all relevant parameters, is applicable to beams with plastic (Class 1), compact (Class 2) or semi-compact (Class 3) sections, 81

gives a uniform margin of safety for all cases, is suitable for everyday use by practising engineers.

That design method allows considerable increases in beam load capacity for beams with compact or semi-compact critical sections.

Available rotation capacities of composite beams have been computed in relation with Kemp's model and with Kubo and Galambos model. A wide variety of parameters have been selected to study the influential factors to available rotation capacities (see chapter 4.3.5).

For several continuous composite beams, required inelastic rotation (Preq of plastic hinge have been computed and graphically represented as a function of moment redistribution. Differences of required inelastic rotation between steel and composite continuous beams have been highlighted (see chapter 5.7).

But, at present state, more developments (tests results, numerical simulations, statistical evaluations,...) are necessary to exploit that method for continuous composite beams in order to elaborate design aids or charts which will help designers in their daily works. (4) Proposals for numerical simulations

:

Finite element modelling for the simulation of unstable inelastic behaviour of beams

:

Study of suitable modelling strategies concerning the shape and magnitude of initial imperfections to be introduced in the original geometry with the aim of attaining the right deformation pattern when the instability occurs during the load-displacement analysis.

The first elastic buckling eigenmode has been used until now for this purpose but, in some cases, important differences have been detected between this elastic deformation pattern and the right one corresponding to the non-linear behaviour. Also the magnitude of imperfection changes the postbuckling behaviour of the beam.

FE modelling of composite beams by correlation with available experiments.

Numerical simulation of continuous beam and frame cases considering the formation of successive plastic hinges for different theoretical situations : Çieq > tyav and ^eq< φβν - Appraisal of the effect of strain stiffening and yielding spread on the required rotation. Analysis of the collapse by instability of the plastic hinge taking into account those effects. Parametrical analysis on composite beams in order to obtain the typical deformation modes and to evaluate the effect of different parameters on the available rotation capacity.

Additional simulation series to evaluate, in more detail, the influence of lateral-torsional buckling restraints and moment gradient on the deformation mode appearing and, therefore, on the available rotation capacity and local buckling sensitivity.

82

Annex

1

"Complete set of distributed documents (10.95)' (7 pages)

Complete set of distributed documents (Date: 14.12.95)

Concerning: ECSC research project:

** Ρ 3263: "Promotion of plastic design for steel and composite cross-sections: new required conditions in Eurocodes 3 and 4, practical tools for designers (rotation capacities of profiles,...)" ** 3263-1. From ProfilARBED-Recherches : 3263-1-1 the draft final report of the ECSC research nr. 7210-SA/508, realized by ARBED-Recherches, Esch/Alzette, August 1992Arbed: "Elasto-plastic Behaviour of Steel Frame Works",

3263-1-2 the summary - written by a working group - of 3 complementary ECSC researches dealing with plastic hinges and their rotation capacity: " Résumé of the coordinated Project: Elastic-Plastic Behaviour of Steel Structures Requirements and Material Properties", Mars 1993, common résumé of the 3 ECSC researches: - nr. 7210-SA/l 13: RWTH (ffiHK, LfS), Aachen, - nr. 7210-SA/204: ULG (CRM, MSM), Liège, - nr. 7210-SA/508: ARBED-Recherches, Esch/Alzette. 3263-1-3 the data (extracted from the report 3263-1-1) for the numerical simulation n° 1 to be carried out by CTICM and LABEIN (simulation n° S 10 of the report)

3263-1-4 paper issued from "Acier-Stahl-Steel" nr. 3 / 1978: "Résultats de recherches CECA: Effet de la forme du diagramme contrainte-dilatation sur la performance des ossatures métalliques ", A. Bernard and M. Da Rin. 3263-1-5 excerpt from a paper issued from the journal "Construction métallique" nr. 1-1983: "Flexion gauche des poutres-colonnes métalliques", G.H.P. Opperman and P.-A. Matthey. 3263-1-6 paper issued from "Structural Engineering International" n° 3 / 93: "Structural analysis: elastic or plastic ?", J. Blaauwendraad and J. Schneider. 3263-1-7 paper of Prof. Sedlacek G. (RWTH, Aachen University) presented in TU Delft in 1993: "Stability aspects of high strength steels". 3263-1-8 paper issued from "Stahlbau" n°61 (1992): "Zum Nachweis ausreichender Rotationsfähigkeit von Fliessgelenken bei der Anwendung des Fliessgelenkverfahrens", R. Spangemacher and G. Sedlacek.

3263-1-9 paper issued from "Bauingenieur" n°66 (1991): "Abschätzung maximaler Randdehnungen bei Anwendung der Traglastverfahrens im Stahlbau", U. Vogel. 3263-1-10 paper from the Swiss Federal Institute of Technology, Icom - Steel Structures, Lausanne, Switzerland: "Continuous composite beams using class 1 and 2 steel sections - Applicability of EC4, and a proposition for a new design method", G. Couchman and J.-P. Lebet

84

Complete set of distributed documents (Date: 14.12.95)

3263-1-1 1 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 4 (1984): "Slendemess Limits Normal to the Plane of Bending for Beam-columns in Plastic Design", A.R. Kemp.

3263-1-12 list of "references for tests on continuous composite beams, and studies of required rotation", by G. Couchman (Icom) on 9.12.1993. 3263-1-13 work paper with figures about the effect of axial load on the rotation capacity obtained from 3 points bending tests on high strength steel section, by M. Feldmann (RWTH) on 9.12.1993.

3263-1-14 excerpts from Mr. Spangemacher' s thesis about the formula of available rotation capacity, by M. Feldmann (RWTH) on 9.12.1993

.

3263-1-15 work paper with figures showing the influence of fu/fy ratio on different theoritical models compared to tests results, by G. Couchman (Icom) on 9.12.1993.

3263-1-16 data and results about numerical simulations to evaluate available rotation capacities of hypothetic steels (fu/fy = 1,20, 1,10 and 1,00): excerpts from the report 3263-1-1 and figures (Bild 4.6 up to Bild 4.10) issued from the draft final report of RWTH - Aachen european research n°7210-SA/118, September 1993. 3263-1-17 paper issued from technical report of University of Cambridge, 1971, England : "Moment-rotation curves for locally buckling I-beams and composite beams", by JJ. Climenhaga and R.P. Johnson.

3263-1-18 paper "Composite Beams", by A. M. Price and D. Anderson, University of Warwick, Coventry; UK. 3263-1-19 paper from the Swiss Federal Institute of Technology, ICOM-Steel Structures, Lausanne, Switzerland : "Design rules for continuous composite beams using Class 1 and 2 steel section- Applicability of EC4", by G. Couchman and LP. Lebet 3263-1-20 work papers from ICOM about rotation capacities in composite beams, 1994, by Couchman G. 3263-1-21 work paper from RWTH about models of available rotation capacity for different types of loading, 1994, by Feldmann M.

3263-1-22 work paper from ProfilARBED applying the model of available rotation capacity (developed by Feldmann M.) to H and I hot-rolled shapes, by Chantrain Ph. 3263-1-23 paper issued from "The Structural Engineer", volume 69, N°. 5/5, March 1991, "Available rotation capacity in steel and composite beams", by Kemp A.R. & Dekker N.W.

3263-1-24 paper issued from the journal "Construction métallique" nr. 3-1994: "Analyse des poutres et dalles mixtes au moyen d'un modèle numérique et comparaison avec des résultats d'essais", by P. Ren, G. Couchman et M. Crisinel. 3263-1-25 paper issued from "Stahlbau" n° 63 (1994): "Experimentelle Untersuchungen zur Rotationkapacität von Verbundanschlüssen ", by R. Kindmann & KKathage.

85

Complete set of distributed documents (Date: 14.12.95)

3263-1-26 letter from Icom about the relation between the length of beam and the available rotation capacity, 09.1994, by Couchman G. 3263-1-27 work paper (14.12.1994) from ProfilARBED applying Kemp's and Feldmann 's models of available rotation capacity to H and I hot-rolled shapes: "Available rotation capacity of plastic hinges R available - Tests results and models", by Chantrain Ph. & Klosak M. 3263-1-28 paper issued from "The Structural Engineer"/volume 64B/N°2/June 1986, "Factor affecting the rotation capacity of plastically designed members", by Kemp A.R.

3263-1-29 EPFL thesis n° 1308 (1994) of Mr. Couchman G., Lausanne, EPFL, 1995: "Design of continuous composite beams allowing for rotation capacity". 3263-1-30 RWTH thesis of Mr. Feldmann M.: "Zur Rotationskapazität von I-Profilen statisch und dynamisch belasteter Träger" (Aachen; Heft 30; 1994; ISSN 07221037).

3263-1-3 1 floppy-disk from Mr. Feldmann M. with a databank of worldwide tests results in the field of rotation capacity (for Excel application on IBM or compatible PC).

3263-1-32 publication: "Plastic Design in Steel - A Guide and Commentary", published by ASCE, second edition, 197 1. 3263-1-33 workpapers (3 pages given on 16.12.1994) from ICOM about parameters influencing rotation capacities in beams, by Couchman G.

3263-1-34 work paper (1 page given on 16.12.1994) from RWTH about evaluation of available rotation capacities from CTICM tests, by Feldmann M. 3263-1-35 working document from ProfilARBED about the problem of overstrengthening of steel profiles in plastic global analysis.

3263-1-36 new version of the document previously given in Bilbao, on 23 March 1995 (first draft): work document from ProfilARBED : "Ductility of plastic hinges in steel structures - Guide for Plastic Analysis", by Chantrain Ph. & Klosak M, second draft (June 1995). 3263-1-37 see document 3198-1-15. 3263-1-38 paper issued from "Stahlbau"n° 56, Heft 12 (Dezember 1987): "Experimentelle Ermittlung der Rotationskapacität biegebeanspruchter I-Profile", by Κ. Roik and U. Kuhlmann.

3263-1-39 paper issued from JCSR Journal ("Journal of Constructional Steel Research") n° 32 (1995) published in 1994: "Prediction of Local Buckling and Rotation Capacity at Maximum Moment", by MX. Daali & R.M. Korol. 3263-1-40 document for COST Cl/ECCS TCI 1 draft group of Annex J of EC4, 25 March 1995, Luxembourg: "Available and Required Rotation Capacities for Composite Beams and Frames", by Dr R.Y. Xiao. 3263-1-41

paper issued from JCSR Journal ("Journal of Constructional Steel Research") n° 32 (1995) : "Determination of Rotation Capacity Requirements for Steel and Composite Beams", by T.Q. Li, B.S. Choo & D.A. Nethercot

86

Complete set of distributed documents (Date: 14.12.95)

3263-1-42 thesis n°1991:15L from Lulea University (Sweden), Division of Steel Structures, by Agneta Wargsjö, May 1991:"Plastik rotationskapacitet hos svetsade stalbalkar" (Investigation on rotation capacity and postbuckling behaviour of girders with slender webs). 3263-1-43 paper issued from "Stahlbau" n° 62 Heft 1 1 (1993): "Ein dehnungsorientiertes Verfahren zur Ermittlung der Duktilität bzw. Rotationskapazität von Trägem aus I-Profilen", by Vayas I. & Psycharis I.

3263-1-44 report from S AES about Projekt 124 :"Traglast von Verbund-Durchlaufträgern für den Hoch- und Industriebau unter besonderer Berücksichtigung einer nachgiebigen Verdübelung", by Prof. Bode H., 1987. 3263-1-45 report N° 95Fe05 from RWTH (Lehrstuhl für Stahlbau): "Safety Evaluation for Rotation Capacities of I- and -shaped Sections and for Correlated Strength", by Feldmann M., Aachen, June 1995.

3263-1-46 paper presented in the Third International Workshop on Connections in Steel Strucutres (Trento, May 1995): "Safety considerations of Annex J of Eurocode 3", by Feldmann M., Sedlacek G. and Weynand K. 3263-1-47 working document presenting document 3263-1-36 and open questions, by Klosak M., June 1995.

3263-1-48 paper issued from JCSR Journal ("Journal of Constructional Steel Research") n°9 (1988) : "Strength and Deformability of -Shaped Steel Beams and Lateral Bracing Requirements", by Takeshi Nakamura.

3263-1-49

paper issued from JCSR Journal ("Journal of Constructional Steel Research") n°34 (1995) : "Differences in Inelastic Properties of Steel and Composite Beams", by A.R. Kemp, N.W. Dekker & P. Trinchero.

3263-1-50

paper issued from JCSR Journal ("Journal of Constructional Steel Research") n°34 (1995) : "Factors Influencing the Strength of Continuous Composite Beams in Negative Bending", by A.R. Kemp, N.W. Dekker & P. Trinchero.

3263-1-51

paper issued from JCSR Journal ("Journal of Constructional Steel Research") n°35 (1995) : "Required Rotations and Moment Redistribution for Composite Frames and Continuous Beams", by D.A. Nethercot T.Q. Li & B.S. Choo.

3263-2. From CTICM

3263-2-1 paper issued from the JCSR Journal ("Journal of Constructional Steel Research") nr. 13 (1989) published in 1989: "Rotation capacity of H -section members as determined by local buckling", by Ben Kato.

3263-2-2 paper issued from the journal "Construction métallique" nr. 1-1990: "Sécurité des assemblages dans les structures calculées en plasticité Condition sur la résistance des assemblages pour éviter leur plastification' by J. Berthellemy.

87

Complete set of distributed documents (Date: 14.12.95)

3263-2-3 work paper from CTICM (given on 09.12.1993) : "Proposal for the required rotation capacity R req"+ 1 figure about "load-deflection curve" from ANS YS numerical simulation.

3263-2-4 work paper from CTJCM (given on 18.03.1994) : "About Rotation Capacity verifications"; "Computation step" (evaluation of R required in portal frame); 1 figure about "moment-rotation curve" from ANS YS numerical simulation of "S10". 3263-2-5 Report CTICM nr 8.009-3, April 1993, "Evaluation of partial safety factors for the check of resistance of steel cross-section from UNTMETAL profiles", "Final Report n° 2", "Analysis of tests series, by A. Bureau. 3263-2-6 new version of the document previously given in Bilbao, on 27.06.94 and in Paris, on 16.12.94; this new version was given in Bilbao, on 23.03.95 and includes the document n° 3263-2-4: work paper from CTICM: "Evaluation of required rotation capacity for various types of portal frames'V'Working document for ECSC Project Ρ 3263". 3263-2-7 work paper from CTICM (given on 27.06.1994) : "Projet CECA SA/520woriring document: Influence of second order effects and the load factor of reference on the required rotation capacity of frames", YG 8/6/94.

3263-2-8 work paper from CTICM (given on 27.06.1994) : "Numerical simulations with

ANSYS". 3263-2-9 working document from CTICM (given on 16.09.1994) : "Numerical simulations performed by CTICM". 3263-2-10 new version of the document previously given in Esch-sur-Alzette, on 16.09.1994; this new version was given in Paris, on 16 .12.94: working document "Moment-rotation curves", issued from the results detailed in document n° 32632-5, (Report CTICM nr 8.009-3, April 1993). 3263-2-11 paper issued from "Engineering Journal'XAISQ, 1994 "Local Buckling Rules for Rotation Capacity", by M.L. Daali and R.M. Korol.

3263-2-12 work paper from CTICM (given on 16.12.1994): "ECSC Projects SA/319-SA 321 working document: Required rotation capacity for a 15 % redistribution of elastic peak moment", YG 14/12/94. 3263-2-13 new version of the document previously given in Paris, on 16.12.1994: work paper from CTICM about "Required rotation capacity for continuous beams", 04.01.1995.

3263-2-14 work paper (1 page) from CTICM (given on 16.12.1994): "Plastic Analysis of a Portal Frame with variable V/H ratio". 3263-2-15 work paper from CTICM, Bilbao, 23.03.1995: "ECSC Projects P3263: Required rotation capacity for continuous beams - Working document based upon document 3263-2-13". 3263-2-16 working document from CTICM, Esch, 21.06.1995: "Guide for plastic analysis Chapter 2: Required rotation capacity of plastic hinges in structures".

Complete set of distributed documents (Date: 14.12.95)

3263-2-17 work paper (1 page) from CTICM, Esch, 21.06.1995: "Influence of the loading path on (Prequired at collapse".

3263-3. From LABEIN :

3263-3-1 work paper from LABEIN (given on 09.12.1993) : results from numerical simulations with ABAQUS (5 figures).

3263-3-2 paper issued from "Computers & Structures" Vol. 48, nr 3, 1993 "On the correlation of analyses and tests of the inelastic flexural behavior of wide-flange steel beams", by G. Greschik, D. W. White, W. Mc Guire and J.F. Abel. 3263-3-3 work paper from LABEIN (given on 18.03.1994) : results from numerical simulations with ABAQUS (S10 simulation with concentrated local and distributed load).

3263-3-4 paper about "Experiments on Wide-Flange Beams Under Moment Gradient", by A.F. Lukey, R.J. Smith, M.U. Hosain and P.F. Adams. 3263-3-5 paper about "Local buckling of I-sections in plastic regions of highmoment gradient", by A.R. Kemp. 3263-3-6 paper issued from "Journal of Structural Engineering" Vol. 1 14, No 1, January 1988:"Moment-rotation tests of steel bridge girders", by Charles G. Schilling. 3263-3-7 work paper from LABEIN (given on 27.06.1994): "Inelastic buckling of beams: numerical approach", results from numerical simulations with ABAQUS (S10 and S 4 simulations with concentrated and distributed loads; biased concentrated load). 3263-3-8 work paper from LABEIN (given on 27.06.1994) : summary of the results from numerical simulations with ABAQUS (S10 simulations with concentrated load).

3263-3-9 work paper from LABEIN (given on 16.09.1994) : summary of the results from numerical simulations with ABAQUS (simulations with centered concentrated and distributed loads; biased concentrated load). 3263-3-10 work paper from LABEIN (given on 16.12.1994): "Numerical results: Spangemacher tests, clamped-clamped supported beam, moment gradient & LTB restraints influence". 3263-3-1 1 work paper from LABEIN (given on 16.12.1994): "Plastic hinge stability considerations - Proposal of practical rules based on rotation capacities".

3263-3-12 work paper from LABEIN (given on 23.03.1995): "Conclusions from simulation results: deformation buckling modes, moment gradient & LTB restraints influence".

89

Complete set of distributed documents (Date: 14.12.95)

3263-3-13 work paper from LABEIN (given on 23.03.1995): "Numerical simulation of RWTH tests & nominal cases results".

3263-3-14 work paper from LABEIN (given on 23.03.1995): "Numerical simulation of CTICM tests & norninal cases results". 3263-3-15 work paper from LABEIN (given on 23.03.1995): "Numerical simulation of singular cases results".

3263-3-16 work paper from LABEIN (given during May 1995): "Numerical simulation of CTICM tests: updated results". 3263-3-17 work paper from LABEIN (given during May 1995): "Continuous beam simulation". 3263-3-18 new version (August 1995) of work paper from LABEIN (firstly given on 21.06.1995): "Numerical simulations of 3 point bending beams in case of M-N interaction".

3263-3-19 workpaper from LABEIN (given during December 1995): "Promotion of plastic design for steel and composite cross-sections: new required conditions in Eurocodes 3 and 4, practical tools fordesigners (rotation capacities of profiles,...); Technical report n° 4 : Period from 01.01.95 to 30.06.95 (ENSIDESA-LABEIN contribution) ".

3263-3-20 work paper from LABEIN (given during December 1995): "Continuous beam simulation (Additional results)". 3263-3-21 work paper from LABEIN (given during December 1995): "Feldmann's model checking within LTB limits".

90

Annex 2

Excerpts from Mr. Couchman's thesis (Ref. 14)

"Design of continuous beams allowing for rotation capacity" (9 pages)

b) Semi-compact sections (according to EC4):

- Determine 0av. This is

a function equation 7.1 or taken as 2.0.

of the

AVAILABLE ROTATION CAPACITY ea

sections, spans and loads, and is either given by

REQUIRED ROTATION CAPACITY

.W^rj.Î'-Vj.'rtÎjj.-j

jar

JUL

M*

M

PLASTIC

SEMI COMPACT

or

COMPACT

M Mel

=

M*

M

Ht7.2

ea

HJ.7.S

¿

EQ7.1 or

eav

2.0

fig. 73

ea ^«.Τΐ-τ

a»«

Kp 7.6

e 'av

©

COMPARISON OF Θ

VERIFICATION OF SECTION

Resistance moments > applied redistributed moments

Figure 7.1- Overview of design method.

Tnèse EPFL 1308

92

Web or flange slender?

Design method not applicable

yes

no

Calculate Mp] and Mp]

Preliminary calculations

Calculate Mel and Me]

J no

Calculate

Web or flange N. semi-compact? y^

yes

Calculate

θ.

Available rotation capacity

Déterminée,

Determine Δ \ >

Required rotation capacity

Identify ©q vs. ί(Δ) curve

Determine Δ from ί(Δ) curve

©req vs.

>

Comparison of rotation capacity

Allow for η and propping

Calculate applied moments

Verification of the section Revise section

no

Resistance moments >

Applied moments.

yes

Section

OK

Figure 7.7- Flowchart of the detailed procedure used to verify that a given beam can support a given load.

Thèse EPFL 1308

93

Table 7.1 - Identification of curves in Figure 7.2 to be usedfor different combinations ofLjjrand a. 1.0

1.25

1.5

1.75

2.0

2.5

3.0

3.5

4.0

5.0

6.0

8.0

10.0

a=0.50

1

2

3

4

5

7

8

9

10

11

12

14

14

o=0.75

4

6

8

9

10

11

13

15

16

17

-

-

-

a=1.00

8

10

11

13

15

17

LLr= (m)

values are not given for cases where ©a is less than 2.0, since plastic or compact sections will normally exceed this value. Available rotation capacity of steel section alone (© )

Note

:

2

1

20.0

3

/

18.0

/

16.0

14.0

12.0

10.0 ^,

8.0 9 10

6.0

11

12

4.0

-

-^.

13

14

-

15

2.0

17

0.0

200

250

300

350

400

f =235 N/mm2

450

500

550

600

Steel section (IPE)

Figure 7.2 - Relationship between Θα and steel section. Example for IPE sections with Fe E 235 steel. Curves showing ©a as a function of the steel section are not only useful in simplifying the new design method, they also enable the influence on ©a of changing the steel section or the proportion of the web depth in compression to be appreciated. For curves 1 1 to 17 a small change in section size does not Thèse EPFL 1308

94

produce much change in ©a, whereas this is not the case for curves 1 to 7. For a given curve there is generally an increase in ©a as steel section size increases, but because section size is related to span length the same curve would not normally be used for a wide range of sections. Curves are given in Figure 7.2 for values of α equal to 0.50, 0.75 and 1.00. Definitive design curves would consider smaller intervals of a, and because it is possible to group curves this would not lead to a large increase in the number of curves needed.

©_/©. d, = 175 mm

0.95 -ι

d4= 100 mm d. = 50 mm

0.90-

For Fe E 355 increase Θ/Θ4 by:

0.85 -

0.6

0.7

0.8

1.0 0.05

0.45 -

0.00

ί

200

-] 250

i

1

i

1

1

1

\

300

350

400

450

500

550

600

Steel section (IPE)

Figure 7.3- Relationship between Θ(ρ/Θα and steel section. Example for IPE sections. Figure 7.3 gives values of the adjustment factor ©aV©a ^ a function of ^e sieá section for IPE sections. This adjustment factor takes into account the influence of compòsite action on the steel section. Values of ©av/©a depend on the slab reinforcement lever arm (dg), the proportion of the web depth in compression (a) and the yield strength of the steel (fy). Curves are given for Fe E 235 steel. To allow for the use of Fe E 355 steel, values of ©av/©a can be increased by an amount which is independent of the section and given for each curve on the figure. For example when α = 0.6 values of © y/ can be increased by 0.05, 0.04 or 0.03 appending on the reinforcement lever aim, as noted on the respective curves. In deriving the curves in Figure 7.3 it was assumed that S500 reinforcing steel is used. Separate curves are needed for different families of sections.

Thèse EPFL 1308

95

beam load capacity, so an exact knowledge required.

7.3.2

of the

length which influences inelastic rotation is not

Semi-compact sections

It is shown in chapters 4 and 6 that to enable calculation of load capacity for beams with semi-compact sections it is necessary to calculate either the peak resistance moment (Μ^χ1) at which buckling occurs, or the post-buckling resistance moment which corresponds to an available rotation capacity of the composite section (©av) of 2.0. ©av and the resistance moment of the section are dependent not only on the section properties but also on the arrangement of spans and loads.

As explained in chapter 6, two distinct cases can be identified. When the arrangements of spans and loads give an elastic moment ratio of 1.3 the shape of the curve showing required rotation capacity as a function of moment redistribution is such that, as required rotation capacity increases from 0.0 to 2.0, the gain in moment redistribution leads to an increase in beam load capacity which is balanced by the loss in load capacity due to the decrease in section resistance moment For an elastic moment ratio greater man 1.3 there is a greater increase in moment redistribution, and for an elastic moment ratio less than 1.3 there is less increase in redistribution. The consequences of this are that· - "When span and load arrangements give an elastic moment ratio less than 1.3 ultimate load may be assumed to be reached when the section buckles, so the resistance moment of the section is taken as titie peak resistance moment QA^^ which is given by equation 4.11, 4.13, 4.15 (LRFD), or 4.2 (EC4). The value of ©av which corresponds to the rotation at which buckling occurs is derived from equation 4.21 and given by:

Θ =2 'av

M,-max

Mpl'

:

M Mpl'

(7.1)

1

available rotation capacity of the composite section peak resistance moment plastic resistance moment

- When span and load arrangements give an elastic moment ratio equal to or exceeding 1.3 ultimate load may be ?sCT""^ to be reached when the rotation capacity equals 2.0. The resistance moment which corresponds to this value of ©av is the post-buckling value M^', given by equation 4.17. A summary of the values of resistance moment and available rotation capacity to be used for different cases with semi-compact sections is given in Table 7.3. Table 7.3 - Values of resistance moment and Θαν to be usedfor different cases with semi-compact sections.

Pd*

Elastic moment ratio

Pel < 1-3

Resistance moment

Mmax

M2'

®av

3Mmax',Mpl')

2.0

1-3

[Equation 7.1]

In conclusion, using the procedure described above ©av and resistance moment can be determined for a beam with a semi-compact support section. Because these are the same two parameters which define Thèse EPFL 1308

96

λςΐ Ljjj i2

: largest value of λ for which buckling is inelastic : xmrestrained length of beam in hogging : radius of gyration about minor-axis

Flange local buckling,

μ·=μ·-(μ/-μ·)

(4.13)

λ =-

(4.14)

The elastic resistance moment again allows for a residual stress of 69 N/mm2 in the flanges. Web local buckling,

M_'=M/-(M/-Md')|^^L

(4.15)

λ =-

(4.16)

d depth of web. No allowance is made for residual stresses in the steel section when calculating the elastic resistance moment for this failure mode. The resistance moment of the section is taken as the lowest of the three values for these failure modes, i.e. it is determined by the most critical of lateral torsional buckling, flange local buckling or web local buckling.

Kubo and Galambos

Kubo and Galambos [33] showed that resistance moments calculated using the LRFD model agree well with peak moments measured in tests. They also considered results from three-point bending tests to show that a linear falling branch on the moment vs. rotation curve may be used to represent postbuckling behaviour. The form of this linear falling branch is given by.

M'=Mpl' M' Mpl' M'max θ*

«d'

Μ^-Ο.Ι( θ' λί

Λ

(4.17)

support moment

plastic resistance moment, calculated using a stress-block model peak resistance moment, calculated using LRFD model rotation at the support rotation at the support which corresponds to the attainment of Mp]', assuming elastic rigidity.

A typical moment vs. rotation curve predicted by this model is shown in Figure 4.3.

Thèse EPFL 1308

97

behaviour of a beam with a plastic or compact support section, the same design method can be used for beams with any of these three types of section.

7.4

REQUIRED ROTATION CAPACITY

The graphical representation of required rotation capacity (©req) as a function of moment redistribution (Δ) is considered in detail in chapter 6. The use of ©req vs. Δ curves is an effective way of allowing for all the parameters which affect the rotation capacity required by a beam to achieve a given moment redistribution. These parameters are: - Elastic moment ratio (jy) and span type (externai or internal). These two parameters affect the basic form of ©,^, vs. Δ curves. - Plastic moment ratio (Up]), which affects values of moment redistribution but not the form of ©req vs. Δ curves. - Degree of shear connection and construction method (propped or unpropped). These two parameters may necessitate modifications to the value of moment redistribution which is given by a ©rea vs. Δ curve. The way in which they are taken into account is discussed in section 7.5.

To identify the appropriate ®ieq vs. Δ curve for a given example, the distribution of "uncracked elastic" moments must firstly be determined. This distribution shows whether a mechanism would fonn first in an external or internal span, and gives the elastic moment ratio. Knowledge of the span type and the elastic moment ratio allows the moment redistribution which corresponds to a required rotation capacity of 1.0 to be found from Figure 7.5. This figure shows moment redistribution as a function of elastic moment ratio, and is basically the same as Figure 6.4 except that, specific Compcal results are not presented. Two curves are shown, one for external spans and the other for internal spans, for beams with a plastic moment ratio of 0.57. Curves are given for this value of plastic moment ratio so that they agree with the choice of axis used for the curves shown in Figure 7.6. Any value of plastic moment ratio could have been used to establish these curves provided the two figures are in agreement. Curves shown in Figure 7.5 can be used for a beam with any plastic moment ratio. As stated above, both curves in Figure 7.5 relate to a required rotation capacity of 1.0, but any value of rotation capacity could have been chosen since the purpose of Figure 7.5 is merely to fix both co-ordinates of a point on Figure 7.6. The required rotation capacity value of this point is not important, provided that it corresponds to the correct value of moment redistribution. Knowing the value of moment redistribution which corresponds to a required rotation capacity of 1.0 for a plastic moment ratio of 0.57 enables the appropriate curve to be chosen from Figure 7.6 a) for external spans, or 7.6 b) for internal spans. These figures are derived from Figures 63 and 6.5 respectively, and the form of the curves is fully described in section 6.3.1. So that tiae curves shown in Figure 7.6 are applicable to beams with any value of plastic moment ratio, required rotation capacity is not simply given as a function of moment redistribution, rather required rotation capacity is shown as a function of

Δ-(θ.57-μρ1)Δ* μρΐ Δ*

(7.2)

moment redistribution plastic moment ratio constant given in Table 7.4 as a function below.

of Uçj. The derivation of this constant is explained

The choice of abscissa comes from the fact that curves shown in Figure 7.6, although being ©^q vs. Δ curves for a plastic moment ratio of 0.57, may be used for beams with any value of plastic moment ratio. This is possible because changes in plastic moment ratio merely lead to a series of paralld curves Thèse EPFL 1308

98

on a ©req vs. Δ diagram, as discussed in chapter 6. These parallel curves can all be represented by the same curve if the abscissa is revised to allow for the differences in moment redistribution between them.

Moment redistribution (Δ) at ©=1.0 I%] 50 -ι

40 -

30-

20 -

External span

Internal span 10

-

0-h7/-/n 0.0

0.6

ι

ι

ι

i

!

1

0.8

1.0

1.2

1.4

1.6

1.8

Elastic moment ratio (μ^)

Figure 7.5 -Moment redistribution vs. elastic moment ratio for required rotation capacity of 1.0. Firstly it is necessary to calculate the rate of change in moment redistribution with plastic moment ratio, given by the parameter Δ*. Knowing this rate of change, differences in moment redistribution can be ra1m1at<>ri for given differences in plastic moment ratio. To establish values of Δ* for various arrangements of spans and loads, the difference in moment redistribution between curves for plastic moment ratios of 0 5 and 0.6 was calculated using equation 6.5. Plastic moment ratios of 0.5 and 0.6 were chosen since they are representative of values likely to be found in practice. Values of Δ* were then calculated by dividing the difference in moment redistribution by the change in plastic moment ratio, i.e. 0.6 - 0.5. It was found that Δ* varies as a function of the elastic moment ratio (u^j) of the beam. Results are presented in Table 7.4, which gives Δ* for various values of elastic moment ratio. T^rer interpolation is possible to calculate Δ* for other arrangements of spans and loads. It should be noted that values of Δ* given in Table 7.4 are only valid when the elastic moment ratio exceeds the plastic moment ratio, so that redistribution is away from the support.

Having calculated Δ* for different arrangements of spans and loads, the value of plastic moment ratio can be calculated for a given beam from a consideration of span and support resistance moments. Knowing Δ* and the plastic moment ratio, allowable moment redistribution can be calculated from the value of Δ-(0.57-μρΐ)Δ* derived from Figure 7.6.

Table 7.4 - Values of A* as afunction of elastic moment ratio.

W

1.8

1.6

1.4

1.2

1.0

0.8

0.6

Δ*

65

69

74

82

92

108

133

Thèse EPFL 1308

99

Required rotation capadty (©,) 14.0 -ι

40

50

Δ-(0.57-μρ1)Δ* [%]

a) Required rotation capacity (θ^,)

40

50

60

70

Δ-(0·57-μρ1)Δ* [%]

b) Figure 7.6 - Required rotation capacity as afunction σ/"Δ-(0.57-μρΐ )A*, a) external span, b) internal span.

Curves shown in Figure 7.6 represent the results of specific calculations using Compcal. These differ from definitive design curves which would be based on a large number of simulations, adopting small load steps, using the procedure described in section 6.3.3 to group the curves. Definitive curves would not contain the irregularities evident in the curves shown for specific cases. However, definitive curves

Thèse EPFL 1308 100

Annex 3

Document 3263-3-12

"Conclusions from simulation results : Deformation buckling modes moment gradient & LTB restraints influence" (20 pages)

IDENTIFIED DEFORMATION BUCKLING MODES

Θ

FLANGE LOCAL & LATERAL GLOBAL COUPLED BUCKLING

i'

A

¿ST

X-

g^_---'X

4r

4r

Æ

X'""~~-Z---,--S

FLANGE LOCAL & WEB LOCAL COUPLED BUCKLING

Λ

ΔΤ

S-

4rX^K

X

102

yr

yr

"A

MOMENT GRADIENT & LTB RESTRAINTS

Concentrated Load

(I)

Distributed Load

Mmax

i

μΐϋΐπ

Æ

~z\

A

L,TB^Lb "LTB

X

-----X

-----X

^ A

ΖΓ"

A"

¿±

Lb

x-x-x «

X~^ ΖΓ k

M

Ut

w -LTS

103

Χ ¿i

MOMENT GRADIENT & LTB RESTRAINTS

7\

*-LÈL·,

a

Θρ

Plastic hinged

b

j

rotation

(II)

Ν

Ν

\

-LTB

_\

V

(Lb^
1

Lb \ N.

(u/>Lb) Lb

-Lb

-LTB

FLANGE LOCAL BUCKLING

I.

Lb

A

zsr

-LT8

&¿(UyB,]_b)

COUPLED FLANGE LOCAL & LATERAL GLOBAL BUCKLING

LLTB

^ Lb

Θρ

X

=

Φ (Lb)

--Z~--')

FLANGE LOCAL & PARTIALLY RESTRAINED LATERAL GLOBAL BUCKLING

Là. < L

^

Lb

A

a

-Χ-Χ

104

X

=

*(Lb.LtTt)

COUPLED LOCAL FLANGE + GLOBAL LATERAL BUCKLING MODE (ISO-ROTATION LINES)

FREE LTB -LTB

t/ROT '=> t/HOT

-> t/HOT

Lb

LUTB

Lb

:

:

length between lateral restraints

length of the region of significant yielding (moment gradienti

105

COUPLED LOCAL FLANGE + GLOBAL LATERAL BUCKLING MODE (SIMULATION CASES)

Lb

CASE (θ,οτ)

4 m

3 m

1.2 m

A

Æ

3 m

J

->

m

k

H

C34

(0.21)

C44

(0.10)

C33

(0.17)

C43

(0.10)

C312

(0.20)

C412

(0.10)

C3R

(0.24)

C4R

(0.30)

106

AVAILABLE ROTATION

3 m

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3

ϊ

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1

1

A

1.2 m

χχχχχχχχχ

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BUCKLING DEFORMATION MODE

0.21

0.10

Al

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0.10

Al

0.20

0.10

Al

0.24

0.30

AH

0.13

0.18

Bl

0.29

0.36

Bil

0.30

0.38

All

(upper flange)

XffiSS ¿±

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£%φ££ A (upper flange)

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*)

© l^@f

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$

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"ξ ξ

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Annex 4

Document 3263-3-17 'Continuous beam simulation' (25 pages)

L- DESCRIPTION OF THE FINITE ELEMENT MODEL - Geometry & cross-section:

* Concentrated load

Two span beam. IPE-27Ü. Sieel S235

case:

F

0< α <

α L

L= 10 m landa=0.5

1

Ψ 2Γ

alpha=0.8

& XL

* Distributed load

case:

0< λ <1

L= l()m

γ>ο

rq

landa=0.4

jjjiiílji!»^

gainma=l

Δ" XL

L

* 3-point bending

-

cases:

1)

Ll=lm

3)

Ll = l,6m L2=2,17m

L2=5m

2) Ll=3m

L2=3m

4) LI = 1,9m L2=4m

Material:

Elasto-plastic model with fy=235 MPa and fu/fy=1.58

-

Meshing: Type of element: 8-node shell with parabolic shape funcuoas (SKR Ahaqus)

A more refined mesh has been provided

at the areas where local buckling behaviour is

expected to happen.

Concentrated

Distributed

Number of elements

640

616

Number of nodes

1973

1913

11.838

11.478

Degrees

of freedom

122

CONTINUOUS BEAM SIMULATION (Equivalent length) Concentrated Load

Distributed Load

8

r

I

A

~2K

Ii

10

^wc

5

r

>

is

<

1-

à

zs 5 -

«

>i

3-POINT BENDING TEST

n1

1

t

5

M n

ΖΓ

LBO^

ri

1-L-5

Δ 3

>4

2·17

<

1.6-L-2.1

Δ <

2îi

3

4Γ

nI^P^

J -Ρ\

3-L-3

1

123

.9-L-4

»ι

yr

yr

- Fillet radius modelling:

By means of web and flange overthickness calculated for keeping constant the total section area

- Initial imperfection: Pattern: the first positive elastic buckling eigenmode is used as the initial imperfection shape

Magnitude: maximum of 1% of the flange thickness

- Load application:

* Concentrated load: Incremental-iterative solution procedure has been employed using displacement control. Appropriate constraints to degrees

of freedom at upper flange of the loaded section have been applied to ensure

the same vertical displacement of these nodes.

* Distributed load: Load control has been used in the load-displacement analysis together with an arc-length integration method (Riks) to cope with the postbuckling instable response. The load has been applied by means

of equivalent node forces at upper flange central line. * 3-point bending cases: As described for concentrated load case.

-

Boundary conditions:

Vertical supports have been introduced at every node of. the lower flange located at the corresponding support sections.

Longitudinal support has been provided in the loaded section at the center of the upper flange.

Lateral restraints have been introduced on the compressed flange of the support and load application sections and additional sections according to Additif 80 - DPU Ρ 22-701 (CM66):

* concentrated load case:

a moment ratio

of 0 has been considered in all the length (leading

to L=l,81 m between lateral restraints) excepting the part between both plastic hinges, where

124

the moment ratio is -0.87 (L = 2,87 m). The following figure shows the precise location of the restraints.

* distributed load

case: a moment ratio

of 0 (L=l,81 m) has been taken for the right part

of the beam, the first hinge influence area, while a conservative moment ratio of 1 considered for the left part (2nd hinge zone) leading to

has been

L=l,05 m between lateral supports.

The following figure shows the precise location of the restraints.

* 3-point bending

cases: a moment ratio

of 0 is assumed leading to L=l,81 m between

lateral supports

125

BOUNDARY CONDITIONS

Concentrated Load

A

7Σ

«

«

ZX

1.7

1.8

1.8 <

25

p\ <

1.7 «

>\

Distributed Load

XXI

I

126

2.- RESULTS

2.1.- Concentrated load case The following results are included in the following figures: 1) Deformed shape

of

1st elastic eigenmode

2) Deformed shape after the beginning of the plastic hinge instability (A) 3) Deformed shape with updated initial imperfection magnitude (B)

4) Deformed shape after the beginning of the plastic hinge instability: 2nd hinge 5) 1st hinge moment-rotation curves for both deformation cases A and B. 6) Evolution

of Ml (1st hinge moment)

and M2 (2nd hinge moment) in function

of the

rotation at both hinges

The deformation (A) initially obtained shows a 2nd order lateral deformation mode suggesting that a bifurcation point is

likely to have

been overpassed during the load process. The analysis

of

an

updated model with a higher initial imperfection magnitude has lead to the valid result (B). In this case the

initial imperfection has been calculated on the basis of providing

a magnitude

of

1%

of the

flange thickness at the compressed lower flange, where the instability is to be developed, instead of evaluating the magnitude to be the 1% at the maximum point.

The moment values at the plastic hinge locations have been evaluated from the corresponding values

of the reactions and load.

127

ttivnewl Jfff fff

CONTINUOUS BEAM Monwn-ratation rant

CONCENTRATED LOAD

8

10

12

14

16

IS

Rotation 1st hing· (dog)

CONCENTRATED LOAD

14

12 +

2

.^S^k^^E^aao^

-

Mill«

C

M2 «2nd hing·)

-·

Mpl

hing·)

x

10

8

Rontwn 2nd hing« (dag)

128

12

14

16

18

ΟΠ

OJ

rK]

129

Q.

coi

m

130

131

CQ

en

ûa_

cvj

ϊ 32

α

o co

LO
O CNJ

rj\ ω Ό
r-î CD

LO

c C O -H 4-3

*



O

O

T-i

C¿

LO

O

Η

^>

+

lO

LO t-i

O 2 Φ C -¡->

'·*-»

C

+

ca ω o o rf^.

LO

CO co CO η

o

CM

cr> ^T

<"Π

C\J

2-

τ-.

ta*

....

w.

H


«!:£

ε s

133

O *-l o o -τ + tu ca

>2*

ε ε

2.2.- Distributed load case The following results are represented in the following figures: 1) Deformed shape

of 1st elastic eigenmode

2) Deformed shape after the beginning of the plastic hinge instability (A) 3) Deformed shape after the beginning

of the plastic hinge instability: 2nd hinge

4) Evolution of Ml (1st hinge moment) and M2 (2nd hinge moment) in function of the rotation at both hinges

134

baaesaiLaum

CONTINUOUS BEAM Mcmtrw otation ram

DISTRIBUTED LOAD

a

10

14

12

18

16

Rotation lit hing· (d*g)

DISTRIBUTED LOAD

8

10

12

Rotation 2nd hing· (d«g)

135

14

16

18

20

iH

%

m

136

ca

i

_C-t.

137

138

2.3.- COMPARISON WITH 3-POINT BENDING CASE RESULTS

CASE

pi

rot

rot -

pl

(deg)

(deg)

(deg)

Concentrated

1,7

8,3

6,6

Distributed

1,1

15

13,9

R

4 i.

8,6

Ll=l

;

L2=5

1,65

8,7

7,1

4,3

Ll=3

;

L2=3

1,65

8,5

6,9

42

Ll=l,9

;

L2=4

1,62

8,7

7,1

4,4

Ll=l,6

;

L2=2,17

1,03

9,1

8,1

7,8

:

ij.

Concerning the concentrated load case, a very good agreement can be observed between the

moment-rotation curve of the continuous beam and that of the corresponding 3-point testing case

(Ll=lm

;

L2=5m). Also the case of central load (Ll=3m

;

L2=3m) gives the same result

as

expected.

In the distributed load case, however, a clear disagreement is observed when comparing the momentrotation curve of the continuous beam and the corresponding curves of the 3-point bending cases.

The discrepancy seems to be caused by the interaction taking place between the lateral deformations induced by both hinges in the continuous beam and giving rise to a stiffening effect. This interaction

is caused by the fact that the continuous beam has got positive and negative moments along its length. The lower flange, compressed near the first hinge location (negative moment region), is subjected to tension stresses in the positive moment region. This tension, acting on the lower flange

at .second hinge location, is responsible for the stiffer postbuckling response appreciated in the moment-rotation curve of the first hinge in comparison to the 3-point bending case. It can be seen

139

as a sort

of stress stiffening effect

thai produce a positive influence on the side

of increasing

the

actual rotation capacity of the beam.

This effect is avoided in the concentrated load case since the lateral deformation at both hinges is

mainly local and does not interact one to each other.

Thus, the rotation capacity of continuous beams subjected to distributed loads (this load case is more

likely to induce coupled lateral deformations) may be significantly higher than that given by the 3point bending test due to the described stiffenins effect.

<

Tensioned lower flange

«-Compressed lower flange-

(Lower flange lateral deformation)

140

DISTRIBUTED LOAD

8

10

12

14

16

18

Retallen (deg)

CONCENTRATED LOAD

16

141

IS

OJ

o-j

LO i

<0

Û4

LO CO

ín

X\

'0 /


o CN

Ol M

Λ

142

co I

1-H I

CO

Cu

s LO co

o Γ*CN

Cu M

J

143

1-q I

Cu

S LO

co CN

>. O

Π

c-LO

J

cu

144

ii?

1

4 I

VÛ

<0

CL,

ε LO CO CN

>. O r--


w

145

Annex 5

Documents 3263-3-10 & 3263-3-13 (LABEIN) (excerpts)

"Numerical simulations of Spangemacher and Seldacek tests" 'Numerical simulations of RWTH tests & nominal cases results" (46 pages)

FINITE ELEMENT MODEL Meshing The following figure shows the mesh containing 616 elements and 1913 nodes.

Element type S8R-Abaqus (parabolic 8-node shell element)

Load application

* Control displacement

* Constraints: Vertical

displacement of the central section upper flange nodes linked

together

Boundary conditions

LOAD

Vertical supports: both ends

Lateral restrains: both ends and central section RESTRAINS

Initiai imperfection First linear-elastic buckling eigenmode (antimetrical deformation) has been used as an initial imperfection shape.

148

-^

ta

G o ca

α

P E-f

W W E-i

m

φ

H

O

ca

CU

>1

w ta

id

Eh

I

I

rji

C -H i

1

ω

ta f¿4

03

rü Q 1

M

O OS

Q

149

e

o

φ H

m o

m o

-ö

E-i

-H

C φ

M

Φ

ÊH ,-t

Φ

Φ

M

ε

u>

FINITE ELEMENT MODEL

**

Modelization of root fillet

:

In order to simulate the roof fillets, an overthickness has been used for the elements near the web to flange connection. The overthickness is evaluated from equivalence of area of the web to flanges junction for the real hot-rolled beam and for the beam discretised into finite elements. We have supposed that thickness of elements (web or flanges) are constant along the first 1/5 of the fillet. For the evaluation of statical characteristics (A, Wpi, I,...) no recovering of shell finite elements is taken into account (Ref. 3). Area of the real connection

:

tf

1FI K-ä lw

Ar = tw.tf + 2.R.tf + R.tw + (4 - %).RZ ¡2 Shell finite element modelization

tf + 2a .

μ=U,

a

Λ'

"l·-

:

»

/

/

ν

Ν

'

/

ι

,

Ι

.

L2 *

*

α

tw

ι

+ 2a

h

1

a

..

Π"

e

fl

tw Characteristics

R = nominal radius of root fillet of hot-rolled profile, lengths of overthickness (a) (Ref. 3)

L!=4/5R

+ tw/2,

L2=4/5R

+ tf/2,

:

overthickness (Ref. 3) obtained from equivalence of Ar and area of finite element modelization of root fillet :

-(tf +tw -2L2 -4-Li)- J(tf + tw -2L2 -4Li)2 -8.

-^.tf + l,5.tf.tw-twJL2 + 2.tf .R + tw.R+R2j -^J

a=

150

sectional area of the cross-section

A = 2btf

:

--a)

+(h-2tf )tw +4L1a+2a(2L1 -tw)+4a(L2

radius of gyration about minor axis z-z

:

moment of inertia according to the minor axis z-z

16L3a + tfb3+(tw+2a)3

:

[l2-*+ lw_(h_tf_2L2),

Iz=^ moment of inertia according to the major axis y-y

ly = -jJO» - tf - 2L2)3 +

:

i(v + 2a)ÍL2 - 1· -

-Ìl_YÌ_2, 2 4 '

+ 2(tw+2a)[L2-Ä-a](|-|.f

»_JaY

2

l)

+ -L!(tf + 2a)3+L1(tf+2a)(h-tt)2

elastic section modulus according to the major axis y-y

:

2Iy

WeLy=^, plastic section modulus according to the major axis y-y

Wply =2aL1(h + a) + tfb(h-tf) + 2aL1(h-2tf

+ (2L2

:

-a)

-tf -2a)(tw +2a)-(2h-3tf -2a-2L2)

+ ^tw(h-tf-2L2)2

151

**

Mechanical characteristics : The following stress-strain curves have been used for non-linear numerical simulations

ty= 235 MPa

0,05

0,15

0,1

02

0.25

0,3

02

025

0.3

Strain

fy= 460 MPa 600 500 + 400 Stress (MPa) 300

-|

200-· 100

-

0*0

+-

0.05

0,15

0,1

Strain

152

Documentation of RWTH tests (Ref. 10 and 11)

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Annex 6

Document 3263-2-9

"Numerical simulations performed by CTICM" (12 pages)

INTRODUCTION CTICM has performed 5 numerical simulations of a 3-point bending test in order to obtain additionnai information on the rotation capacity of a i-section at plastic hinge location.

Calculations are executed with the ANSYS program on power station HP 9000 (serie 700). The profile is modelled by shell elements. The global analysis takes into account the material and geometrical non linearities for reproducing, as far as possible, the real behaviour of the beam. The first simulation (1.1) has been that of the test S10 (HEB 200 section 3m span - load at mid-span - global sinusoidal imperfection) performed in a previous research project1 This simulation has been made in order to calibrate the ANSYS program and determine the best values of the different parameters of the global analysis (ANSYS uses the Newton-Raphson method), and to compare the numerical results with the test results. For the 4 other simulations, the influence of some structural parameters on the rotation capacity have been investigated : - In the second simulation (1.2), the global imperfection has been removed in order to evaluate its influence on the rotation capacity, and to be taken as reference for the further simulations. - in the third simulation (2.1), the distance of the left support of the beam to the load point has been half reduced (150 cm to 75 cm) in order to verify the assumption of the dependance of the rotation capacity only on the greater distance of the load point to the supports. - in the fourth simulation (2.2), the load point has been eccentred and set to 1/5 of the span. - in the fifth simulation (3.1), the mid-section has been fully restrained except against the vertical displacement in order to study the rotation capacity in a

plastic hinge occuring near a beam-column joint.

ARBED Recherches : ELASTO-PLASTIC BEHAVIOUR OF STEEL FRAMEWORKS - Draft of the final report. August 1992 1

196

!

PRESENTATION OF THE MODEL

..1

Geometrical model

The model is automatically generated from a paramétrica! process. The model contains only one type of element. It is a 4-nodes shell element with 6 degrees of freedom by node. This element allows the non-linear material behaviour and large deformations. Shear deformations are taken into account. The cross-section has the following dimensions which were measured on the test specimen : = = = = =

h

b

tf tw re

200.4 mm 200.4 mm 14.6 mm 9.6 mm 18 mm

In order to simulate the fillets, an overthickness has been used for the elements near the web to flange connection. Thus the cross-section is modelled as shown in figure 2.

364

12 58

17 59

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200 4

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f tw

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Figure 2

1

a = 0,25 (4,8 r + tw -

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λ/δ) )2 - 4 (4 -

π)

r2

197

Material behaviour

2.2

The stress-strain relationship can be input point by point. It is different for the web, the flanges and the stiffeners.

Figure 3

E

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(MPa)

fy

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(MPa)

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Stress-Strain curves : point coordinates Ame 1

2

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6 7 8 9

Semelle

Raidisseur

ε

σ

ε

σ

0 0,0014195 0.013 0,03025 0,0475 0,0837 0,11983 0,156 0,228

0 291 291 364 385,9 415,1 429,7 437 437

0 0,0012731 0,0144 0,031725 0,04905 0,0837 0,11835 0,153 0,292

0 261 261 335,5 357,85 387,65 402,55 410 410

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0 350 350 433 458,55 491,95 508,65 517 517

2.3

Imperfections No local imperfection has been introduced in the model.

For the simulation 1.1, a global imperfection has been introduced with a sinusoidal shape in the plane of the web (a) and in the plane of the flanges (β) as shown hereafter.

X

V

ctsin(pi x/L)

ß

\

ßsin(pi x/L)

F'iQure 4

2.4

Stiffeners

Transverse stiffeners are located at each support and under the point of load application. The thickness is taken equal to 14.6 mm.

2.5

Support conditions and load application

The application of the load is reproduced by imposing a displacement to the nodes on the upper flange in the loaded cross-section.

ίιΓΡΏΓΓΙ· -

0 uz »

U

Displacement uz

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imposed

uy -

U

uz * 0

Figure 5

Restraints to lateral displacements are located on the nodes of the lower flange at both ends. For the simulation 3.1. all the nodes in the section under the load are fully restrained except against the vertical displacement.

199

RESULTS

The following table gives the values of the parameters and the main results φ Sum of the rotation at both ends Rav Available rotation capacity _ Φυ - Φρί

Rav-

Φρί

^ ψ

Figure 6

Ll

L2

Imp. α

Imp. β

ΦρΙ

(cm)

(cm)

(cm)

(cm)

(deg)

(deg)

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150

150

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0.07

1,24

36

28,0

1.2

150

150

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0

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34

26,5

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30

32,3

2.2

60

240

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25

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150

150

0

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Simulation 2.2

Rotation (degrees) I

0

J2

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10

20

15

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Rotation (degrees) 0

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10.00

20.00

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Annex 7

Document 3263-3-10 (LABEIN) (excerpts)

'Numerical results: Spangemacher tests, clamped-clamped supported beam, moment gradient and LTB restraints influence" (7 pages)

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Document 3263-3-9 (LABEIN) (excerpts)

Numerical simulations with centered concentrated loads, distributed loads and biased concrentrated loads" (3 pages)

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»

100% I

"Ξ _S

in

tf*

-Pi^"

cs cs

US

CM

OS

(de

Λ¿f

JO

<3

iP &J ñ

¿r JTx S JT X

*»

o

e CC

CM OS

o

O^,

«

in

e

1^

«ρ u

<\

s»

*

Ξ Ο

*

O

ΙΟ

^

^.

w

* m

C*

CS

]

ι

:

1 >

J

<>

in

t 1

'

.

in

o

CM

CM

^

",

** °*-¡

1

1

1

in

o

m

(DI IQ) W

216

'

"

Ci es

cs

J3

εα

E

o

(UIIQ)W

217

INFLUENCE OF MOMENT GRADIENT AND LTB RESTRAINTS ON AVAILABLE ROTATION CAPACITY

® Concentrated

* If"

L L-

* If

L

load

Distributed toad

-* e ROT

LTB

LTB

< l;

-Ρ

θ ROT

'ROT

^-

e

vROT

Ma

II,

Θ ®

L

llc

-

Concentrated load

->

Distributed load

-ρ

No lateral restraint at plastic hinge location

(

L

LTB

Ila

θ ROT

-

·

Ban «ROT

Lateral restraint at plastic hinge location (L^^L/a) VROT

VROT

218

^

L

)

Annex 9 Document 3263-2-12

Required rotation capacity for a 15% redistribution of elastic peak moment" (8 pages)

CTICM

YG 14/12/94

ECSC PROJECTS SA 31 9-SA 321

Working Document

Required rotation capacity for a 15% redistribution of elastic peak moment Following an elastic analysis, the paragraph 5.2.1.3 of Eurocode 3 allows to redistribute up to 5% of the elastic peak moment, provided that 1

- the internal forces and moments remain in equilibrium with the applied loads - ail the members in which the moments are reduced have Class

1

or Class 2 cross-

sections

procedure is equivalent to a plastic analysis, in which the redistribution of moments would be limited and compatible with the reduced rotation capacity of Class 2 cross-sections.

This

The question which comes up is : is "this allowance of 15% redistribution always safe for Class 2 cross-sections, that is to say, is the required rotation capacity always small? The required rotation capacities for two common cases are studied hereafter.

A - Two span continuous beam with an uniform distributed load

q I

lllllit mimimi muli

ω

m

min

©

A

nun λ<1

XL

L 1

I

1

Figure I The first plastic hinge always occurs at the centrai support for 8 MP|

ql

=

ί2(λ2-λ+1)

and the mechanism (2nd plastic hinge in span

1

2 Mp|

qu=-[r-(3

+ 2V2)

220

)

for :

:

Required plastic rotation

Between these two levels, at q = η ep.req = 3

+ η)

(1

q\ . the required plastic rotation in the first hinge is

Mp[L

0 + λ)

El

θρ req is plotted versus λ and η.

η \

/^-H

\

ί

!

i

El

θ

H

l

1

_^

f \_^Jr~"~"Ί 0.176S

1

I t

ι

«chanisrT

!

V\

1

k^fc^ii

preq MplL

t^i^lrri^-

0.2

'

¿-

.

^-1

i

1

r^i

L--_ ---Ί 1

i 1

j 0.05

J

1

i

i

!

,

1

0.1

0.3

0.2

a<

es

o.»

o.?

c.«

0.9

λ Figure 2

Of course, this expression is valid if

q
is

to say if

η
(3 + 2>/2J (λ2 -

The limiting value of η 1

η

= 0,85

1

is

λ +1 ) -

1

(dotted line in figure

2)

plotted versus λ on the figure below. For example, with

=0,1765

0.12 0,1

1

0

0.15

(max value allowed by EC3)

the valid range of λ is

λ < 0,2604 or λ > 0.7396

221

l

0.5 i

1

0.45

TI 1 1

'

- . -" o.*

\l

0.35

κ ι ι

0.3

Ν\

1

I

1

I

1

1

1

1

1

1

1

'

'

'

'

1

I

1

1

1

1

1

1

i

VI

I

I

r-r-J I

I

0

3

/.

X

1

1

1

1

1

I

I

I

1

1

1

M )

/

1

--H

^-a

1

f

/

1/ A

-Æ ¡1

Ι

I

1:

0.2

1

1

1

Μ

l:

1

1

~}--r-W~

I

II

I

0.5

0.4

0.3

X

0.2604

i

O.i

0.7

ι

o.a

0.9

1

0,7396

Figure 3

Required rotation capacity The plastic rotation of reference φρ|

is

given by

:

Mp| d (Ppl= 2 El

where

d = d] +02 is

assumed to be the distance between zero moment points at the current load level q =

η)Ρΐ

A ^^UilJJliiiUU^

"^UUiiiiiiiiJ^1 XL

Figure 4 L

λ2 -

λ+

1

di=4-TTir-L and

L

d = ^(l+Ti)

The rotation

1

+ λ3

capacity at q =

(1

+

-^

and

d d2- λ

with

d
n).qi

is

given by

θ p.req vreq

Φρί

222

(1

+

that

Rreq

is

is

to say

:

λ<Λ/Ξ-ι

^βα

λ>Λ/2-1

Rreq =

=

§η0+η)4λ(1+λ)++λ*.λ + 1 3η(1 + ηΐ

λ

(1 + 1

λ)

+ λ3

plotted on the figure below. 0.6

R

req

Notes

:

- The parts of curves above the dotted line are invalid because in this zone, the

mechanism occurs before the load - For

λ < λ/2 -

1

,

d2 is limited at

(1

+ -qi

is

reached.

XL.

Conclusion For this sketch, the required rotation capacity is quite limited Class 2 cross-sections may be used in that case.

223

(

Rreq < 0,6). Therefore,

Β

- Two span continuous

L

aL

ϊ

'

beam with a point load

Ζ L

'

ι

XL

ι

Figure 6 Two ranges for a are to be considered

a < yJ2{X+

1)

-

a > ^/2(λ + lj -

a < \)2(X

B.l

+~Y) -

1

1

:

first plastic hinge

at loading point

first plastic hinge

at central support

first plastic hinge at loading point

1

The first plastic hinge occurs at

Wpi

F^-f-ffca) For

F

= (1 +

tiI.Ft

:

The plastic rotation in this hinge is given by

MplL Öp.req = "εΓ *(α.λ).η(αλ,η) The distance d between zero moment points each side of the plastic hinge

d = Lg(aA,Ti)

and the required rotation capacity is given by 1

τ(α,λ).Γΐ(α,λ,η)

Rreq = 3

where

g(a.X^)

:

ί(α.λ)=α(α_1)[α(α

9(α.λ.η) -

2(λ+1) + 1).2(λ+ιη

α(1+η) a(a2-i) η + α"2(λ+1)

η(α.λ.η)=η (1-α)

[2λ + 3(1 - α) +~(1 -

224

α)(λ+ -α)] 1

is

All these expressions are valid if the mechanism

is

not reached, that is to say

Fu

and this condition becomes a(a + l)

^(a-MHl-jTTTT)-]-1 a>-v/2fX +

B.2

11

-

first plastic hinge at support

1

The first plastic hinge occurs at

MP|

F^-f-flca) For

F

= (1 + ti).Ft :

The plastic rotation in this hinge

is

given by

Mp,L

εΓ

ep.req =

{(α.λ).η(α,η)

The distance d between zero moment points each side of the plastic hinge is

d = Lg(a,X.Ti) and the required rotation capacity

given by

Κα,λ).η(α,λ,η)

Ί

9(α,λ,η)

Rreq ~ 3

where

is

:

2(λ+1)

f(a^=a(l-a2) g(a,M)

1

^7^

=

1+2(λ+1)(1+η) Η(αλη1=ηα(1

-a2)

All these expressions are valid if the mechanism is not reached, that is to say

F^1+Tl and this condition becomes (1

^

12 +O.V

- 2(λ+1)"1

225

For

the max value of η allowed by EC3

for the λ values : 0,1 12

11

1U

¡MUNII

R req

8

I

MM MM

M

I

I

1

ι imi in III URI II

1

!

Μ

Ι

1

1

1

1

1

Kl

III

1

Ν

|

>

ι

Il

I

ι

|

1

1

L·

ι

1

I

I

I

η = 0,1765 (15°/oMpeak)

ι

i

IL

II

I

I

i

1

b

II

Hl -__._!

1

I

ι

TT

._

it

1

II /

II Ml || MIM Mill

m

M

I

1

ι

b

ΙΙ"Γ\|_ΤΓ m

I

\\ IN

1

plotted on the figure below

"

I

Mill |v mn HIM 1 M.\! 1 πιιιινΛΐ r \. mn IMI i\r\i INJIIII' l'i *m mix MII κ IKK illl IU, Jill Μ,Ι l\l V-iUllf«. rm 1 MIHI Tv mi 1

is

1

Μ

ί

1

Mimmi m m im m.

= 0,1765), Rreq

II

II

1

ι

1

MM II

II Ml Ml II MIM

I

1

/

1

III Ml

I

II UN IN

MM

1

- 1 ,5 - 2. The range of a values is limited to [0,2 - 0,8].

I

jilillllll in umi >l

ϋ

1

[q~õc -

I

Ι

ι

ι

I

4

1

1

1

3

m

IMI

1

1

II

1

ft**ÍXL/HJ

Ví^2A >..

'-Å.

rm rtkìK -kl L- - - .[ Jill TrM-.L 1.. um li μι ι ^ôiiiiim4XH43^L ri- 1. 'Κ |lîî__iï,-rTr7 III ITU-mil T^^ötótK INR M Ml fm 0Li ii unir L jrtr 1

1

I

2

ι

1

I

Ι

mim

n

0,2

0.25

ι

Hi

ι

nimmm.

..iT-:rr-H-w-u-LLMiiimTJi

I

1

1

1

1

0.3

0.35

0,4

0.45

0,5

0,55

0.6

1

I

1

t

0,65

0,7

0,75

0,8

α

Figure 7

Conclusion For values of X £ land if the point load acts in the left half-span of the first span, the required rotation capacity may be very large (> 4), and the use of Class 2 crosssections with a 15% redistribution of peak moment may be unsafe regarding the

rotation capacity.

226

Example :

Section Mp| =

- L=10m - λ =1,5 -a = 0,27

IPE 270

11 3,74

kN.m -

I

= 5790 cm4 -

E

= 21000 kN/cm?

The first plastic hinge occurs at the loading point for

:

fy= 235 N/mm2

Ft =

The second plastic hinge (mechanism) occurs at support for The

él, 95 kN Fy = 73,3 kN

:

peak moment is at the loading point.

The maximum allowable moment redistribution (15% of peak moment) F

is

given for

:

= 61,95/0,85 = 72,88 kN

At this load level

F :

plastic rotation

:

[ep = 0,189 rad

length of equivalent isostatic beam Mpl d

d

:

= 640

cm

ΛΛΟ

φΡΐ= 2ΕΓ = 0·03 Rreq = 0,189/0,03 = 6,3

At SLS (F,j/1 ,5), the maximum deflection is in the first span

:

fmax = 5,05 cm ( « L/200) a) It should be noted that the available plastic rotation model for IPE 270 is φΓΟϊ = 0,11

which

is

φΓΟ{

calculated with the Feldmann's

rad

much lower that θρ req

Therefore, the 15% redistribution

= 0,189 is

rad

not valid in that case, although

IPE

270

is

a Class

1

cross-

section. b) It would be possible to find Class 2 (near Class 3) cross-sections with the same characteristics Mpl and I as IPE 270 for which the rotation capacity or available plastic rotation could be still lower. For example

:

h = 26 cm - b = 7,5 - tw = 0,6 - tf = 0,85 - r = ,5 > Class 2 flanges and Class 2 web 1

1

227

Annex 10

Document 3263-2-6

Evaluation of the required rotation capacity for various types of portal frames" (38 pages)

NOTATIONS 1

Frame n°

Units

h

Frame height

cm

L

Length of span

cm

Column/R

Section for edge column

Column/C

Section for intermediate column

Beam

Section for beam

H

Horizontal load

daN

V

Vertical load

daN

δ

Horizontal displacement (elastic analysis)

cm

«CT

Critical load factor

Load

α1

Load factor at the first plastic hinge

factors

°ηι

Load factor at collapse

Nsd ULS

α =1.0

Axial force in the column to be checked

MSdl

daN.cm

Msd2

Bending moment at the top of the column

daN.cm

Tf

Formula EC3 §5.5.4

δ

Horizontal displacement at load factor 0.7 (SLS)

SLS

Γδ

α = 0.7

deflection

^deflect

M] >v

Hinge n° hinge

ep.d

Td

Maximum deflection at SLS Plastic moment of the beam Second moment of area of the beam

daN.cm cm4

Number of the plastic hinge (with maximum rotation) Plastic rotation in the hinge under load factor 1.00 Shear force at plastic hinge location

Rdreq

middle

Su

Plastic rotation just before collapse

of a beam

Tu

Shear force at plastic hinge location at collapse

*^u.req

Required rotation capacity just before collapse

radian daN

Required rotation capacity under load factor 1.00

radian daN

Number of the plastic hinge (with maximum rotation)

plastic

ep.d

hinge

Td

at the end

Rdreq

of a beam

epu

Plastic rotation just before collapse

Tu

Shear force at plastic hinge location at collapse

^u.req

Required rotation capacity just before collapse

Mechanism

cm

Deflection criterion

in the

Hinge n°

cm

Horizontal displacement criterion

1

plastic

daN

Bending moment at the basis of the column

Plastic rotation in the hinge under load factor 1.00 Shear force at plastic hinge location

radian daN

Required rotation capacity under load factor 1.00

Reference to collapse mechanism

230

radian daN

SCOPE The aim of this study is to determine the required plastic rotation 8req and the required rotation capacity Rreq for a given type of portal frame. First a simplified method has adopted in order to obtain a realistic design. The assumptions and the method are explained in the paragraph 2. Then six types of portal frames have been identified, and for each type, a parametric study has been made.

It must be pointed out that the selected criteria for the design are somewhat arbitrary. Moreover the chosen types of frame do not cover all the real structures. However, the results give a good idea of the plastic behaviour of frames and they lead to some 79 frames have been studied.

interesting conclusions.

DESIGN METHOD In order to facilitate the design of the frames, some simple design criteria have been selected. An elastic analysis is executed in order to check Serviceability Limit State requirements and to calculate the critical amplification factor. Then a plastic analysis is made in order to check Ultimate Limit State criteria and to determine required plastic rotations. The two analyses are made for one reference loading corresponding to ULS loading (weighted loads) with the program PEP micro.

2.1

Serviceability Limit State (SLS> The global elastic analysis allows us to calculate the deselections and the horizontal displacement at the top of the frame. The following SLS requirements are checked for 0.7 times the reference loading.

f
- Deflection condition - Horizontal displacement

The value of 0,7 is arbitrary. One can consider that it is close to the value corresponding to the non weighted loads.

2.2

Ultimate Limit State (ULS) The ULS requirements are checked for the reference loading with the results obtained after the plastic analysis.

First the critical amplification factor acr is calculated from the results of the elastic analysis, for the reference loading, as follows :

(H +
vcr acr~Vsd

δν

231

where

:

H V h

Φ δ

Horizontal load Total vertical load Frame height Global imperfection Horizontal displacement

The value of acr allows us to classify the frame and to deduce the type of analysis

txcr<10

acr>

10

Non sway fíame

First order plastic analysis

Sway fíame

Second order plastic analysis

The buckling resistance in the plane of the frame can be checked with the non-sway buckling length for each column in conformity with Eurocode 3. By simplification the buckling length is taken as the system length (this is conservative) : £

= system length

The columns are checked according to the criterion of EC3

Nsd

i

XyAfy/ It necessarily assumed that it is Class

§

5.5.4

:

kyMy.sd

wpl.yVïMl" 1

cross-sections (plastic analysis).

It is also assumed that no buckling phenomena can occur out of the plane of the frame.

3

REQUIRED ROTATION CAPACITY

3.1

Definition of the available rotation capacity The available rotation capacity can be determined from a three point bending test. It can be an experimental test or a numerical simulation. It gives the available plastic rotation 6p-avaii :

Figure Öp.avail

- "{"avail - Φρί

1

(D

232

where

φρ]

Sum

:

of the rotations at the supports when the theoritical plastic moment

Mpi is reached (increasing part)

_Mpi Φρ1

= 2EI .L

"tavail

Sum

:

(2)

of the rotations at the supports when the theoritical plastic moment

Mpi is reached (decreasing part) Mi , τη

χ"*" ""**"1/

pi

I p.avail

T = "ivail

V^

y

w

ΦίνβΗ

Φρί

Φ

Figure 2 The available rotation capacity can be defined as follows

:

_ ^p.avail _ 2 EI Ravail

M^jT ep.avail

ψ^

(3)

From (2) and (3), and when 9D_avaii is reached, the shear force is equal to

'

_2Mpi ~~ Τ avail ~ L

(4)

and the available rotation capacity can be also expressed by

:

EI Tavaji 9p_avaji Ravail ~

(5)

M pl

By making a sufficient number of tests, it is possible to propose an empiric or semi-empiric formulation of the required rotation capacity, in particular for a given cross-section, in function of L or Τ (moment gradient). This can be supported by a mechanical model.

This formulation of 6p>ava¡¡ and RaVaiI is assumed hereafter to be known; therefore, one can determine for a given cross-section (L is known) :

MpiL Qp.avail

2 EI

(6)

availvw

233

32

Required rotation capacity in a statically unrleterminate system - Symmetrical local hendinff configuration There is a "symmetrical local bending configuration" when the cross-section, the material and the moment gradient are identical for each side of the plastic hinge. These are the conditions of the three point bending test.

We consider the equivalent three point bending system that gives the similar conditions at the plastic hinge location.

-*

ep.req (Computed by

Statically undeterminate

PEP micro)

system

~* "p.avail (Computed by

Equivalent 3-point bending system

an empiric

2.MEÍ

-.ς

formula)

Figure 3 The available plastic rotation 6p.avan can be computed from (6) with Ravail determined for L = Leq calculated from the shear force Τ from (4). GpTeq is directly computed by the analysis program (here PEP micro). The configuration in the vicinity of the plastic hinge is similar in the undeterminate system and in the 3-point bending test, so the following requirement must be checked :

öp.req

- ^p.avail

Each side

t7)

of the inequality can be multiplied by

ΕΙ Τ and considering (5), we obtain

:

Mpl ΕΙ Τ

(8)

ep.req < R-avail(Leq) M:pl

The notion

of the required rotation capacity Rreq can be introduced as follows

ΕΙ Τ 6p-req ^ 9p-req

^req "* H

2

M2

pl

where

Φρί

So, the condition (7) can be expressed as

RreqCD

* Ravail(Leq)

Mpl Leq Φρ1= J EI

(9)

:

^en

234

Leq

2 Mpi -γ

(10)

By definition the required rotation capacity Rreq associated to the frame for a loading level is the largest one for all the plastic hinges existing at this loading level. This required rotation capacity could be defined either when at collapse, or under design loads for Ultimate Limit States checks.

33

Required rotation capacity in bending configuration

a

statically undeterminate system - Non symmetrical local

We only consider hereafter the case of a different moment gradient for each side of the plastic hinge.

Direct approach We also consider hereafter the equivalent three point bending system that gives the similar conditions at the plastic hinge location. The length of this equivalent system is given by :

Leq^MplOfï + f^)

(Π)

ii

ι

ττ

ι

ii

, -

2

l-e,-Mpl(J-J-) '1

«2

Figure 3 bis

Noticing that the sum φρι of elastic rotations at supports is independent on the load position in the equivalent system, we also decide to adopt hereafter the definition of Rreq adopted for symmetrical three-point bending system, that is

:

6p.req

Rre
where

φρι =

ΕΙ

ΕΙ

Τ,m

(12)

Τ! Τ2

It yields

Rreq "~

ΕΙ Μ

τ1τ2 Tm

θ. 'Ρ·"1

where

pi

235

Tm = ai+T2)/2

(13)

Rreq is to be compared to Ravail calculated for a length Leq

Rreq(Tl>T2) ^ RavailfLeq)

(13 bis)

where Leq given by (1 1) and Rreq by (13)

Indirect approach This approach only uses the definition of R

adopted for a symmetrical three-point bending

system (expre >sion (9)). Because

of the strain hardening, the moment exceeds Mpj at the loading point.

\/ ε.χ *2 approximation

Figure 4 In the part of strain hardening, there is an increase of the moment

:

ΔΜ = Mm - Mpi The plastic elongations are assumed to be developped on the following distances

(approximativeiy)

,

.ΔΜ

£1=2TJ-

:

, and

£2 =

ΔΜ

2jj

(14)

The plastic rotation is expressed as a function of the plastic elongation Sp in the extreme fibres, and by considering 8pj from the part 1 and θρ2 from the part 2 of the plastic zone.

236

2£r

ep.req = £ J £p(xi).dxi +

öpl.req

τ1} -J £p(x2).dx2

θρ1 + θρ2 = θρ

(15)

6p2.req

The approximation about the distribution of the plastic elongation ερ leads to

epl.req = K^l where

:

and

ep2.req =

(16)

κ·^2

Κ = proportion factor

From the relations (14) and (16),

epl.req-Tl = ^.req·^

(17)

and from the relations (15) and (17)

.req

<

ep.req iι \+ T2

(18)

"6p2.req ~ ^p.req " ^pl.req

Similarly to the case of a constant shear force for each side of the plastic hinge (in absolute values), we look for an equivalent 3-point bending system. This equivalence is searched for each side of the plastic hinge with the corresponding shear force.

eq

T,

Figure 5 From (9), and condidering each symmetrical system, it yields

EiTj epl-req

Ri

req

EI T2 ep2.req

and

M

R2. req

Pl

and taking (17) into account

Rl.req = R-2.req

237

M:pl

Let us note Rreq the common value of Rl.req ^

R req =

2

ΕΙ

M

2

Pl

τ1τ2

R2.req- Taking (18) into account,

Λ

(19)

Τι + Τι θτP-re
it yields

2

Keeping relation (12) in mind, Rreq can also be expressed by

:

_ 6p.req

re(l"

Φρί

where φρί is the sum of elastic rotations at supports for a equivalent beam of length

Leq = (M.eq + L2.eq)/2 = Mpi -γρζ-

(20)

From the following basic requirement to be checked öp.req

:

- öp.avail

the rotation capacity criterion to be satisfied is

:

Rreq(Tl,T2) ^ Ravail(Leq)

(21)

where Leq given by (20 or 1 1) and Rreq by (19)

3.4

Summary tabk ep.req

* ep.avail

or

KreqCTi.Tj) ¿ Ravail^eq)

3p.avail or ^availO-) S'vcn °>' formulae or tsbles

EI T1T2 n , j. + j, 9p.req Mpl

2

Rreq(Tl»T2) ~

Leq-Mp^-U + i-)

Figure 6 a

®p.req

- ep.avail

or

RreqCr) ^ Ravail(Leq)

ep.avail or RavailiL) given by formulae or tables

El Τ θ P-rcq

Rreq~'

M Pl 2M,pl

Figure 6 b -eq

θ preq

ι

«p.req^ep.avail

<>r

RreqCT) *

^vaild-eq)

ep.avail or Ravail^) given by formulae or tables Mpi

2ΕΙΤΘ P-rcq ^req

M 2M

Figure 6 c

238

pi

pi

Application to our study on rotation capacity of frames PEP micro program gives

:

9pd

:

Required plastic rotation for the reference loading (load factor α = 1 )

0pU

:

Required plastic rotation at failure (ay)

The required rotation capacity is computed as follows

Plastic hinge at mid span

»

^ T(j]

Tjj2

:

:

Plastic hinge at beam end

TdlTd2

_ "

:

EITdl+Td2ep
Rd-req

Mpl2

shear force on the left of the plastic hinge T¿ : at load factor 1

shear force on the right hinge at load factor 1

Shear force in plastic hinge at load

factor

1

(U.L.S)

of the plastic

TulTu2

^ E 1 Tu 6pU

^.req Ru-re
2

^EiTdepd x. -> ΜρΓ

2

Mpl2

Tu]

:

shear force on the left of the plastic hinge Tu at failure

Ty2

:

shear force on the right hinge at failure

of the plastic

239

:

2

x. 2 Mpl

Shear force in plastic hinge at failure

ANALYSIS OF FRAMES Six types of portal frame have been studied. They are submitted to concentrated loads indicated in Figure 1.

ι

ί

1

J

1

1

ι

1

1

ι

1

i

1

Jί

ι

1

1

ι

1

1

1

i

illllii

li li lili 1

1

·«

Type Numbe r

as

1

1

2

3

4

5

6

15

19

14

15

12

4

1

of cases

Figure 7

For each type of portal frame, many cases are considered by varying the parameters such the height, the span, the profiles ... The following results are given

as

:

- Table containing all the data and all the results

of the design criteria

- Tabic containing the mean values and the maximum values of the plastic rotations and of the plastic rotation capacities, under the design loads and at collapse. -

Graph of the plastic rotations

- Graph of the plastic rotation capacities

4.1

Portal frames of type

1

r

ι-

1-

Figure 8

240

1-

1-

PORTAL FRAMES : TYPE

1 <=

*

s

*

¿BO Frame n°

|

h

L

1

|Coiumn/R j Column/C Beam

H V δ aer al

j

1

1.4

1.3

1.6

1.5

300 300 IPE 240

300 400 IPE 240

300 500 IPE 240

300 600 IPE 240

IPE 140 IPE 220

IPE 140 IPE 220

IPE 140 IPE 220

IPE 140 IPE 220

850 34000

750 30000

300 700 IPE 240 IPE 140 IPE 220 650 26000

1,86 4,76

0,968 1,014

1.7

1.81

300 800 IPE 240 IPE 140 IPE 220

300 300 IPE 240

300 400 IPE 240

IPE 140 IPE 220

IPE 140 IPE 220J

575

1600

1333«

23000

48000

1,81

1,78

2,41

4,90

4,99

4,72

0,980 1,022

0,987 1,024

0,983 1,050

0,962|

6343 568020

5614 563850

12283 616030 0,93

10018 627840

1325

1025

41000

2,04 4,35

1,92

1,85

4,62

4,79

0,939 1,018

0,982 1,046

0,993

13248 639870 0,98

10101

558780 0,84

8326 540900 0,79

7377 602870

δ

1,69

L54

1,48

Γδ

0,85 0,62 0,41

0,77 0,89 0,44

0,74 0,50

1,49 0,75 1,75 0,58

0,72 2,26 0,65

1,14 0,57 2,83 0,70

670680 2772

670680 2772

670680 2772

670680 2772

670680 2772

670680 2772

670680 2772

6706801

Node 3 0,00645 4539

Node 3 0,00240 3825 6414 0.15

Node 3 0,00120 3336

Node 3 0,00880 3036 4463

Node 3 0,00350 2428 3322

Node 3 0,00174

Node 3 0,00510 3549

0.01511 3567

8931

0,01257 4042 6542

0,67

0.81

5567 0.85

0,01700 3170 4570 0,82

Node 3 0,00470 2685 3815 0,19 0,01840 2844 3944 0,79

1

2

2

2

2

au Nsd facti Msd facti

I Deflect. 1

TDeflect.

I

Mpl

iy Hinge 9pd

Tdl Td2 Rd.req Opu

Tul Tu2 J Ru.req |Mechanis S

1.2

53000

Tf

|

Ll

8710 0,49

0,00864 4563

-

1,047

1,25

5163 0,06

0,41

m

Table

1.1

241

1,44

40000 2,39J 4,761

1,028

0,911

1,93

1,92

0,97

0,96| 0,96 0,47

0,61

0,40

27721

0,13

4021 7978 0,12

0,01975 2572 3572 0,76

0,00950 3673 8949 0,64

0,00920

2

1

2

6451

0,30 3601

6705 0,56

PORTAL FRAMES : Type 1 (continued)

Frame n°

1.9

1.10

1.11

1.12

1.13

1.14

1.15

300 500 IPE 240 IPE 140 IPE 220

300 700 IPE 240

25000

300 800 IPE 240 IPE 140 IPE 220 750 22500

300 400 IPE 270 IPE 160 IPE 220 1050 42000

300 500 IPE 270 IPE 160 IPE 220

34000

300 600 IPE 240 IPE 140 IPE 220 967 29000

35000

350 500 IPE 270 IPE 160 IPE 220 850 34000

aer

2,37 4,80

2,31 4,93

2,23 5,10

2,23 5,10

1,74 5,09

1,72 5,16

2,36 4,39

al

0,957

au

1,012

0,970 1,018

0,991 1,035

0,983 1,022

0,981 1,036

0,984 1,030

0,978 1,027

8408 625060 0,89

7147 604120

6154 573510

5532 588820

10506 615630 0,71

8696 607680 0,67

1,90

1,84

1,77

1,38

0,95 1,35 0,54

0,92

0,69 0,86 0,43

1,36 0,68

0,60

0,88 2,30 0,66

1,77 0.88 2,93 0,73

670680 2772 Node 3

670680 2772 Node 3

670680 2772 Node 3

670680 2772 Node 3

670680 2772 Node 3

670680 2772 Node 3

670680 2772 Node 31

0,00729 3172 5327 0,38

0,00601 2834 4416 0,27

0,00211 2539 3710 0,08

0,00445 2329 3295 0,16

0,00247 4112 6388 0,16

0,00256 3564 5186 0,14

0,01209 3259 5356 0,63

0,01410 2942 4455 0,65

0,01570 2668 3812 0,64

0,01710 2432 3328 0,62

0,00713 4189 6706 0,48

0,00876 3673 5358 0,49

0,00362 3329 5170 0,19 0,00864 3385 5362 0,46

2

2

2

2

1

2

2

h

L Column/R Column/C Beam H

V δ

Nsd facti Msd facti

Tf δ Γδ Deflect. TDeflect.

Mpl

iy Hinge 6pd

Tdl Td2 Rd.req 8pu

Tul Tu2 Ru.req Mechanis

1133

1,81

IPE 140 IPE 220 833

m

Table 12

242

875

1,23

0,49

8487 620560J 0,69 1,91

0,82 1,28 0,51

For type

1,

one mechanism has been obtained, but plastic hinges can appear in two different ways.

Mechanism n°1

5Π

-α

Δ.

Rd mean

Rdmax

Rumean

Rumax

0,21

0,34

0,54

0,64

9pd mean

Qpdmax

Gpu mean

Bpumax

0,00298

0,00446

0,00842

0,00950

Rdmean

Rdmax

Rujnsaii

Rumax

0,19

0,41

0,59

0,69

9pd mean

Opdmax

9pu mean

e_£u_max.

0,00398

0,00880

0,01296

0,01970

Δ.

Mechanism n°2

Table 13

These graphs show the required rotation capacities and the plastic rotations for the frames at load factor 1 .00 and at collapse.

2

ι I

1

ι

1

'V

IB

IiiIIIIjii Graph

1.1 :

hiII

-1- -

J--4JÌ 1 η g IS

PM*

B

¡M1

Graph 1.2 : Required rotation capacity

Required plastic rotation

243

4.2

Portal frames of type 2

r

ι-

1- 1-

Figure 9

244

r

PORTAL FRAMES : Type 2

Frame n°

|

h

1

1

300 400 LPE240

300 700 IPE 240

ΓΡΕ180

300 500 IPE 240 IPE 180

IPE 220

ΓΡΕ220

2·61 300 800 IPE 240 IPE 180 IPE 220 600 24000J

1425

1125

45000

950 38000

δ

1,26

1,26

1,16

1,06

1,00

acr al au

7,04

7,06

7,63

8,37

8,90

0,918

0,918 1,042

0,889

0,898

0,896

1,065

1,018

1,025

1,015

0,928 1,044

30142 193210 0,93

23435 153200 0,93

19345 159311 0,85

16316 133730 0,70

14166 128970 0,66

12357 86140 0,50

14400

11372 569890

9734 633680 0,93

8119 614490 0,88

7124 635610 0,90

5965 563920 0,79

0,79 0.40

0,68 0,34 2,5 0,63

670690 2772

Tf 1

Nsd Edge Msd Edge

1

rf

Ι

δ

jΙ

Deflect.

Γδ

| rDeflect, I

2.5

300 300 IPE 240 IPE 200 IPE 220 5700Ö

Nsd Central Msd Central

8

2.4

v

Beam

1

2.3

H

L

|

2.2

300 600 IPE 240 IPE 180 IPE 220 800 32000

Column/R C0lumn/C

|

2.1

Mpl

ly

527290 0,85

0,86

IPE 180 IPE 220 700 28000

0,97 0,49 0,47 0,31

0,97 0,48 0,77 0,39

0,88 0.44 1.13

1,53

0,45

0,51

0,75 0,38 2,03 0,58

670690 2772

670690 2772

670690 2772

670690 2772

670690 2772

0,91 9,75

Hinge

Node 3

Node 3

Node 3

Node 3

Node 3

Node 3

I

6pd

0,00574

1

Tdl

0,00159 5297 8953 0,14

0,01317 4147 5352 0,80

0,01147 3544 4456 0,59

0,01458 3190 3810 0,66

0,00376 2659 3340 0,14

0,0219 6284 8928 2,09

0,0271 5061

0,0306 4357 5328

0,02876 3782 4432

0,02769 3330 3791

0,02716 . 2968

2,11

1,65

1,36

Node 5 0,00230

Node 5

Node 5

Node 5

Node 5

H

I }

Td2 Rd.req 6pu

Tul Tu2 Ru.req 1

4543 6707 0,40

6680 2,34

3311 1,16

Rd.req

8953 0,53

0,00452 6707 0.78

0,00908 5352 126

0,00856 4456 0,99

0,01054 3810 1,04

Node 5 0,00487 3340 0,42

6pu Tu Ru.req

0,01460 8908 3,37

0.01753 6664 3,02

0,02030

0.0189C 3784

0,01862 3303

2,80

0,01947 4422 2,23

1,85

1,59

1

1

2

2

2

2

Hingen0 ePd

Td

Mechanism

5327

Table 2.1

245

PORTAL FRAMES : Type 2 (continued)

Frame n°

h L Column/R Column/C Beam

2.10

2.11

2.12

2.131

350 300 IPE 240 IPE 220 IPE 220

400 300 IPE 240 EPE 220 IPE 220

450 300 IPE 240 IPE 240 EPE 220

500 300 EPE 240 PE 240 EPE 220

350 400 EPE 240 EPE 200 EPE 220

400 400 EPE 240 PE 200 IPE 220

450 400 IPE 240 IPE 200 EPE 220

1125

1100

1050

45000

44000

42000

1,65

3,01 4,41

1425

1400

1375

57000

56000

55000

1325 53000

δ

1,69 6,13

2^8

2,85 4,67

3,66 4,04

6,27

2,31 5,13

0,866 1,044

0,854

0,820 1,019

0,823 1,017

0,875 1,026

0,872 1,014

0,891 1,026

29682 279530 0,90

28956 336720

27524 440070 0,94

23158 221650 0,88

22651 253890 0,96

22014 250940

1,00

28377 409650 0,90

14864 577660 0,92

14964 625140 0,99

14939 620760 0,99

14544 621800 0,99

11646 610730 0,92

631840 0,95

1,32

1,91

231

2,471

0,72 0,50 0,33

0.77 0,48 0,32

1,29 0,55

1,84

0,56 0,48 0,32

3,05 0.92

0,69

040 033

0,77 0,39

0,81 0,41

0,82Ì 0,82

670690 2772 Node 3

670690 2772 Node 3

670690 2772 Node 3

670690 2772 Node 3

670690 2772 Node 3

670690 2772 Node 3

670690 2772 Node 3J

0,00620 5300 8950

0,00968 5052 8950

0,00984 4801

0,00834 4299

0,53

0.81

8948 0,80

8951 0,63

0,01075 4546 6704 0,75

0,01284 4296 6074 0,84

0,00783 3790 6710 0,49

0,01429 5317

0,01479 5067 8943

0,01334 4531 8946

0,017 4096 6702

1,23

U4

1,04

0,03094 4889 6679 2,26

0,0175 4473 6699

Ru.req

0,02444 5979 8930 2,27

Hinge

Node 5

Node5

Node 5

Node 5

0,00573

Td

.8950

0,00780 8950

8951

6703

0,00936 6704

0,00663 6709

1,33

1.81

0,00915 8948 2.12

Node 5 0,00816

Node 5

God

Node 5 0,00854

9pu

0.0174C

0.0104C

Tu

893C

a.u

Nsd/C Msd/C

Tf Nsd/R Msd/R

Tf δ Γδ Deflect.

rDeflect. Mpl

iy Hinge 8pd

Tdl Td2 1

Rd.req

1

Öpu

Tul Tu2 1

2.9

V

a.l

1

2.8

H

a.cr

I

2.7

Rd.req

Ru.req

Mechanism

4,97 1,017

1

4,02 ι

8943

894] 2.41 3

0.01202 8943 2,71ί

109251

11531

591850 0,90

0,41

Ui

1.12

1,91

1,42

1,62

1,15

0,01 144 8946,

0,02132

0,01165

667Sι

669S)

2,6ί

3.6S

2,ο:

>

0,01165 6702 2,02

3

; 2

;5

3

3

Table 22

246

0,95J

>

>

α

PORTAL FRAMES : Type 2 (continued)

Frame n°

|

h

L Column/R Cólumn/C Beam

Η

v

1

1

δ aer

1

ai

Ì

au

!

|

Nsd Msd

I

rf Nsd

| I

Msd

rf

1

δ

1

Γδ

Ι

I

Deflect.

j rDeflect. !

Mpl

I|

ly Hinge

1

6pd

1

Tdl

1

j

Td2

Rd.req

1

6pu

1

Tul

i |

Tu2 Ru.req

2.14

2.15

2.16

2.17

2.18

500 400 IPE 240 IPE 220 IPE 220 1000 40000

300 400 IPE 270 IPE 200 IPE 240

300 500 EPE 300 IPE 220 IPE 270

300 600 IPE 300 EPE 200 PE 270

300 700 IPE 330 IPE 200 IPE 300

1425

1625

1375

1500

1350

57000

65000

55000

60000

54000

337

1,09

4,39

8,14

0,94 9,44

0,97 9,14

0,89 9,97

0,85 10,44

0,889

0,938 1,063

0,891 1,018

0,904 1,011

0,946 1,036

0,905 1,047

21411

29 980

27 852 217 280

27 656

171 490

33 005 266 900 0,92

30 924

260350 0,77

174 020

178 970

0,94

0,90

0,85

1,065

14 223

508630 0,79

681 820

2,77 0,83 0,77 0,38

0,83 0,42

670690 2772

861 5101

\ Mechanism

5789,8

137 400

5789,8

1,55

0,44 1

13 451 273 000 0,84

0,64 0,32 2,03

0,66 0,33

0,73 0,37 1,32 0,44 1

1

476 740 8356,1

0,51 1

476 740 8356,1

Node 3

Node 3

Node 3

Node 3

0.11

0,01320 6208 7442 0,84

0,00555 6602 8398 0,33

0,01062 6171 7329 0,57

0.0176 3967 6704

0.02340 6592 8554

0,02289 7512 9036

0,02840 6393 7507

0,02410 7189 8346

0,02598 6474 7271

5661

8589

1,14

1,92

1,76

1,84

1,50

1,43

Node 5

Node 5

Node 5

Node 5

0.00205 8589

0,00765 9064 130

0,00827 7442

Node 5 0,00384

0,00317 6662 0.55

I

3891,6

137 400

14 953 283 100 0,85

0,01143 7186 9064 0,86

6pd

Tu Ru.req

1

1

0,94

0,70 0,35 0,94 0,38

0,69 0,35

108 600

Node 3

Hinge

1

1

0,00150

|

8pu

0,80

086 900 0,95

Node 3

|

Rd.req

14 047

16 665! 1

300 800 EPE 330 IPE 200 IPE 300

0,00000 3338 6662 0,00

Node 5

Td

0,88

10241

2.19|

039

0,01349 6704 2.34

0.0 1520

3

1

8554 2.86

8398 0,52

0,00669 7379 0,79

0,01510 9036 2,56

0,01822 7507 2,57

0,01466

0,01890

8346

7271 2,21

2

2

1

1

Table 23

247

1,16

1,9:

2

Rd mean

Rd may

Ru mean

Ru max

0,24

0,40

1,96

2,34

9pd mean

9pdmax

θρ\ι mean

Opumax

0,00360 Rd mean

0,00570

0,00241

0,0271

RiLmax.

Rumean

Rmnax

0,57

0,86

1,58

2,26

3pd mean

9pdmax

9pu mean

9pu max

0,0095

0,0140

0,0247

0,0310

Rdmean

Rdmax

Rumean

Rumax

0,59

0,83

1,16

1,23

9pdmean

gpdmax

9pumean

Qpumax

0,00810

0,0128

0,0158

0,0170

·*2

Δ.

Δ.

Mechanism π°1

-û.

ù.

Mechanism n°2

·<--

¿L

¿L

Mechanism n-°3

Table 2.4 : plastic hinge at mid span These graphs show the required rotation capacities and the plastic rotations for the frames at load factor 1 .00 and at collapse. a

li.i.. . iirmirrprriiirni I

r rip Graph 2.1 : Plastic rotation for hinge at mid

1

1

1

1

1

1

1

!

1 1

1ρ

ir

ι

Graph 22 : Required rotation capacity for hinge at mid span

span

248

--"

Rdmean

Rdmax

Rumean

Rumas

0,55

0,78

2,81

3,36

9pd mean

9pd max

9pu mean

9pu max

0,00320

0,00450

0,0155

0,01750

Rd mean

Rd max

Kiunsan

Klimax

0,97

1,41

2,35

4,02

9pd mean

9pdmax

9pu mean

9j2u_max.

0,0070

0,0105

0,0168

0,0213

Rdmean

RsLmax.

Rumean

Rumax

1,54

2,12

2,37

2,78

Qpdmean

fìpdjnax.

9pu mean

fipujnax.

0,0074

0,0094

0,0118

0,0135

1

Mechanism π°1

Mechanism n°2

·<-

A Mechanism n°3

.0.

Table 2.5 : Plastic hinge at beam end These graphs show the required rotation capacities and the plastic rotations for the frames at load factor 1 .00 and at collapse.

D

fis jJjLiIii,

flllN ΓΙ I III ; V

Graph 2.3 : Plastic rotation at beam end

\'

Ι<

\' Γ

II ΠΙΜΙΠΡΙΙ lililí ir

ι πι V Ι

li \' ϋ

V

\'

Ι

Γ

Ι Χ

Ι' Ι

Graph 2.4 : Required rotation capacity at beam end

249

4.3

Portal frames of type 3

Figure 10

250

PORTAL FRAMES : Type 3

Frame n°

|

h

L Column /R Column /C Beam H

| | |

v

1

dx5f=0.7 i 1

ft

EPE 220

EPE 220

2267 68000

2267 68000

132 8,61

1,047

PE 200 PE 270

PE 220

1933

1933

1850

18001

58000

58000

2467 74000

74000

72000|

3,09 6,13

0,89 12,78

2,85 6,64

2,64 7,18

0,72 12,34

0,827 1,048

0,939

0,884

0,896

0,960

1,046

1,035

1,053

1,064

14 378 -1 066 500

18 251

17 983

17 585J

-1 293 500

-845 380 1 184 700 0,80

-1 272 9001

0,17 0,09

0,52 0,15

14 194

0,84

0,82

0,50

130

034

1,00

0,90

035

0,12

030

037

0,58 0,39

0,36 0,67 0,45

670 690 2 772

670 690 2 772

N° Hinge à mi travée

Node 3

Öpd

0,01924 8078

rf dx5f=0.7

S

Tdl

S

Td2 Rd.req

657 720 0,98

13.09J

0,929| 1,050|

1

577 7001 0,8

lj

1,13

1,16

1,14

1,12

130

038

0,39

0,38

0,37

030!

476 740 8 356

890 1051 11767J

Node 3

Node 3.

Node 3.

Node 3 j

0,02550 6972 7528

0,01950 8716 9783

0,013241

2,11

2,41

0,00912 6917 7582 0,62

0,00505

8921

0,02193 8072 8928

1.74

1,45

O37

8553 9447 0,82

0,03111 8908 8913 3,59

0,01565 7582 7582

0,03178 9741 9745 2,49

0,01310 9837 9838

0,02160 9446 9447

1,12

0,03480 7502 7504 2,45

5

Node 5

Node 5

Node 5

Node 5

Node

5

Node 5

0,00821 8550 0,91

0,00861 7529 0.84

0,00475 7352

0,00119 9480

0,00902 9735

0,00268

033

0,11

0,71

03c

0,00861 8502 0,51

0,01336

0.00938 8913

0,01515 7504

1,54

1.0Í

MS

0,0083îí 974Í 0,6i

0.0127Sl

8898

0,01672 7582

1

1

1 J

Ru.req

N° Hinge sur

Node

Mechanism

1,13

Node 3

Tul

6pu Tu Ru.req

IPE 3301

Node 3

1

Td

341 700 0,87

PE200J

476 740 8 356

6pu

Rd.req

1

8001

IPE 360

5 790

1

appuis öpd

076 800 0,89

3.71

50θ|

5 790

0,02870 8894 8898 3,30

Tu2

1

3.6 300 600 PE 330 PE 200 PE 300

PE 300

-770 160 969 990

ly

1

PE 200 PE 270

16 983

Mpl

]

3.5 500 600 PE 330

-977 070 672 100

Tdeflect

1

3.4 500 600 PE 300

16 931

j

I

3.3 300 600 EPE 300

-664 890

dy3f=0.7

f

EPE 180

0,860

j

1

EPE 180

α.1 a.u Nsdf=l.

rdx

1

33 500 300 EPE 300

αχΓ

Msdl f=l. Msd2 f=l. ft

3.1

300 300 EPE 240

1

137 400

1

137 400

>

1

1,01 1

Table 3.1

251

1

1

8661

9838

1,04

9389

>

0,01368 9446 0,89

:I

2

>

983S

>

1,01

]

1,41

PORTAL FRAMES : Type 3 (continued)

Frame n° h

L Column /R Column IQ. Beam

H V

Ι 1

PE 180 PE 240

ΡΕ

180

PE 240

PE 270

PE 180 PE 300

1250 50000

55000

2100 84000

750 30000

68000

66000

54000

1,10 8,06

0,94 9,44

0,91

a.u

0,878 1,048

0,908 1,036

Nsd

16 790

Msdl

-582235

16228 -664 180

Msd2

615 100 0,94

a.cr

rf δ(5) Γδ

Deflect

ly

1

\

Hinge

I

Õpd

1

Tdl

J

Td2

|

Rd.req

1

6pu

1

TuI

1

Tu2 Ru.req

Hinge ôpd Td Rd.req

9pu Tu Ru.req Mechanism

3.12 300 800 PE 330

3.11 300 700 PE 300 PE 180

1350

\ rDeflect. t Mpl

|

3.10 300 500 PE 270

1650

a.l

Ι

PE240 PE180 PE220

3.9 300 400 IP E 270

1700

dx5 |

3.8

300 300

3.13 300 700

ΡΕ 360 ΡΕ 200 PE 330

1375

3.14 300 700 PE 240

ΡΕ 180 ΡΕ 220

0,70 12,67

0,68 13,04

0,92 9,64

0,906

0,79 1133 0,942

0,980

1,011

1,040

1,073

0,936 1,028

0,914 1,010

12226 -680 510 963 010 0,80

13 328 -722 440 1 161000 0,76

20216

760 340 0,86

370 -649 580 807 040 0,88

-1 103 300 1 578 000

6 922 -440 950 629 760 0,87

0,39

038

035

0,14

0,12

0,17 0,09

0,14 0,07

0,15 0,07

032

030 0,55 0,37

0,77 0,38

1,11

1,17

1,19

1,14

0,44

033

0,30

2,07 0.59

670 690 2 772

861 510

861 510

3 892

3 892

5 790

476 740 8 356

Node 3 0,01680 8070 8930

Node 3 0,01330 7936 8564

Node 3 0,01543 6666 6833

1,21

1,15

Node 3. 0,00109 6371 7379 0,06

Node 3. 0,00645

1.84

Node 3 0,00788 6001 6499 0,46

0,47

Node 3 0,01600 3705 3794 0,64

0,02498 8906 8910 2,88

0,01860 8544 8547

0,01711 6825 6827

1,75

139

0,01377 6498 6498 0,84

0,01233 7378 7379 0,73

0,01171 10796 10796 0,87

0,01823 3788 3788 0,73

Node 5 0,00735 8929 0.85

Node 5 0,00622

Node 5 0,00859 6857 0,65

Node 5 0,00458 6498

Node 5 0,00346

038

Node 5 0,00106 7268 0,06

0,01147 8910 2,65

0,00932 8582

0,00999 6853

0,00948 6498

0,00880 7378

0,00784 10796

Node 5 0,01076 3810 0,43 0,01260 3806

1,76

1,51

1.«

1,02

1

3

1.16 2

1,05

1

2

2

3

9,75

8501

0,58

13

1

137 400

Table 32

252

1

0,83

033 1

890 105

11767

10203 10797

10797

036

0,11

670 6901 2 272

Rd mean

Rd max

Rumean

Ru max

1,79

2,40

2,74

3,58

9pu mean

Spdjnsx

9pu mean

0,0194

0,0255

0,0283

0,0348

Rd msan

Rd max

Rumean

Ru max

0,46

0,82

1,01

1,41

6pd mean

âgd-max

9pu mean

âpiunax

0,0071

0,0132

0,0147

0,0216

Rdmean

Rd max

Rumean

Ru max

0,89

1,14

1,01

1,28

Qpdmean

9pd max

9pu mean

âpiLmax

0,0157

0,0160

0,0176

0,0182

ΙΠ Π I II III

I

firn-m^

Mechanism π°1

?ΡΓ.

Mechanism n°2

Ψ^~-

Mechanism n°3

Table 33 : Plastic hinge at mid span

l PI

II Graph 3.1 : Plastic rotation at mid span

Ι Ι Ι

1

1

Graph 32 : Requ. rotation capacity at mid span

253

Rdmean

Rdmax

Rumean

Rumax

1,33

1,81

2,18

3,07

9pd mgan

9pdmax

9pu mean

9pumax

0,0067

0,0090

0,0111

0,0151

Minean

Rdmax

RujnejQ

Rumax

0,54

1,01

1,59

2,38

9pdmean

fìpdmax

flpu mean

9pumax

0,0041

0,0110

0,0086

0,0167

RjLmsan

Rdmax

Rumean

Ru max

1,08

1,29

1,26

1,50

9pd mean

9pdmax

9pu mean

Opumax

0,0096

0,0107

0,0112

0,0126

?Γ,

Mechanism n°l

F*!

Mechanism n°2

^r-

Mechanism n°3

Table 3.4 : Plastic hinge at beam end

χI

fc

ill I Pilli 1

Ι

I Ι

1

il

uni

Γ1Ι li I

FI II IF ΓΙ

ΡI I I I Γ |: V V I Ivl· 1

Graph 33 : Plastic rotation at beam end

pr I

Ulf

I

Ι

PI

I

Graph 3.4 : Requ. rotat capac at beam end

254

4.4

Portal frames of type 4

Figure

255

11

PORTAL FRAMES : Type 4

Frame n°

h

L Column Beam H

V

δ aer

al au

4.6

4.7

4.8

350 600 IPE 270

400 600

500 600

600 600

300 600

500

PE270 IPE 240

400 600 IPE 270 IPE 240

1125

1125

11251

22500

22500

22500J

O33

7237

0,49 44,72

0,90 30,73

0,89 1,019

0,861 1,019

0,836

PE240

PE270 PE240

PE240

PE270 PE240

562,5 22500

562,5 22500

562,5 22500

562,5 22500

0,12 72,58

0,12 84,68

037

0,48

44,78

3135

0,79 22,78

0,89 1,019

0,875 1,019

0,861 1,019

0,836

0,814

1,019

1,019

PE270

600

PE270| IPE 240J

1,019

11340

11342

-494 990 860 100 0,93

-513 200 860 090 0,92

11345 -548 290 860 090 0,92

-548 290 860 090 0,92

11335 -559 730 860 080 0,92

11341 -625 460 860 100 0,92

11344 -688 700

0,09 0,04

0,13 0,06

0,19 0,07

0,34 0,10

0,55 0,14

0,16 0,08

O34

0,62 0,19

Deflect.

1.58

1,84

1.95

1,58

1,71

1,84

0,53

1,65 0,55

1,71

rDeflect. MdI

0,57

0,61

0,65

0,53

0.57

0,61

861510

861 510

861 510

861 510

861510

861 510

861 510

861 510

3891,6 Node 3

3891,6 Node 3

3891,6 Node 3

3891,6 Node 3.

3891,6 Node 3.

3891,6 Node 3

3891,6 Node 3

3891,6 Node 3

0,00925 5516 5734 0,57

0,01154 5516 5734

0,01399 5516 5734 0,87

0,01932 5516 5734

0,02522 5516 5734

0,01769 5516 5734

0,02511 5516 5734

130

1,56

0,01134 5516 5734 0,70

1.10

1,55

0,01330 5734 5734 0,84

0,01590 5734 5734

0,01870 5734 5734

0,02480 5734 5734

0,03140 5734 5734

0,02255 5734 5734

0,03070 5734 5734

1.00

1.18

1.57

1,98

0,01540 5734 5734 0,97

1,42

1,94

40

40

40

40

40

20

20

20

1

1

1

1

1

1

1

ly Hinge epd

Tdl Td2 Rd.req 6pu

Tul Tud

1

4.5

300 600 IPE 270 IPE 240 562,5 22500

11337

δ

S

4.4

-475 750 860 100 0,93

Γδ

1

4.3

Nsd.

Tf

1

4.2

Msdl. Msd2

1

4.1

Ru.req

V/H

Mechanism

1

0,71

Table 4.1

256

11347

0,13

860 100 0,91

PORTAL FRAMES : Type 4 (continued)

Frame n°

|

h

L |

|

1

1

Column Beam H

v

I 1

|

I

1

4.13

4.14

4.15J

300 300

300 500

300 600

300 700

3001

PE270 PE240

PE300 IPE 240

300 400 IPE 270

PE270

PE270

PE300

800 IPE 300

PE240

PE240

IPE 270

1125

2250 45000

22500

PE240

IPE 270

1700

1350

1125

1275

1125

34000

27000

22500

25500

22500

0,30 54,64

036

033

037

0351

63,95

72,37

60,66

66,001

0,89 1,019

0,901 1,02

0,9051

11335 -559 730 860 080 0,92

12 844 -700 890 1 137 400 0,94

11281 -702 400 1 137 400

0,13 0,07 1,85 0,53

0,12 0,06 2,16 0,54

035

αχΓ α.1

0,814 1,019

0,856 1,019

0,841

0,871

1,012

1,019

Nsd.

11346

Msdl. .Msd2

-751 280 860 100 0,90

22 691 -746 670 860 080

17 076 -675 290 860 100 0,97

13 670 -597 850

0,19 0,10 0,70 0,47

031

0,16 0,08

0,41

0,18 0,09 1,16 0,46

861.510 3891,6 Node 3.

861 510

861 510

861 510

861 510

3891,6 Node 3

3891,6 Node 4

3891,6 Node 5

3891,6 Node 3

5789,8 Node 3

5789,8 Node 3.

0,03355 5516 5734 2,08

0,01139

0,01558 8399

1.41

1,46

0,01256 6619 6880 0,93

0,01134 5516 5734 0,70

0,00958 6250 6500 0,57

0,01070 5563 5686 0,57

0,03994 5734 5734 2,52 20

0,01390

0,01770

11468

8601

11468

8601

0,01636 6880 6880

1,76

1,68

134

0,01362 6500 6500 0,83

20

20

20

0,01540 5734 5734 0,97 20

20

0,01319 5686 5686 0,70 20

1

1

1

1

1

1

1

Γδ

Deflect.

Mpl

Iy

Hinge 6pd

Tdl Td2

Rd.req 8PU

Tul Tud

|

4.12)

600 600

66,80

| rOeflect

|

4.11

1,46

δ

3

4.10

22,60

Tf

I

4.9

depl

a.u

1 1

j

Ru.req

V/H Mechanism

1,02

036 1.95

0,65

0,80

11032 11468

0,11

0,82

8601

860 100 0,94

Table 4.2

257

1,58 0,53 1

137 400

1,0111

0,93

1

137 400

Only one mechanism has been obtained.

Mechanism

Rd mean

Rdmax

Rumean

Rumax

1,06

2,08

1,37

2,52

9pd mean

6pd max

9pu mean

Bpu max

0,0159

0,0330

0,0202

0,0399

Table 43

Graph 42 : Required rotation capacity

Graph 4.1 : Plastic rotation

258

4.5

Portal frames of type 5

f

r

I

r ι

Ν a...

¿

Figure 12 I

Frame n°

j

L Column/R

j

Beam

h

H

v

1

5.1

52

5.3

5.4

5.5

5.6

5.7

300 600 IPE 270 IPE 240 500 20000

350 600 IPE 270 IPE 240 500 20000

400 600 IPE 270 IPE 240 500 20000

450 600 IPE 270

500 600

550 600

PE270

PE270

600 600 IPE 270

500 20000

IPE 240 500 20000

IPE 240 500 20000

1,11 10,81

1,50 9,02

1,95

7,69

2,48 6,65

0,915 1,040

0,900

10492 793230 0,88

ι

δ

0,55

0,83

9

acr

16,36

12,65

al

0,969

0,934

au

1,084

0,951 1,074

|

Nsd Msd

I

rf

Ι

δ Γδ

j

Deflect. f TDeflect. 1

1

Mpl

ly |

Hinge

i

6pd

1

Tdl

S

|

i

Td2

Rd.req Qpu

j

Tul

1

Tu2

|

Ru.req

1

Mechanism

1,064

PE240

PE240 500 20000

3,80l 4,74 0,874

1,028

0,887 1,016

10559 813020 0,91

10630 834180 0,93

10705

10288 728500 0,81

10340

10391

743500 0.82

758490 0,84

0,39 0,19

0.56 034

0,78

1,12

1,48

1,91

039

0,37

0,44

0,52

1,003

856630 0,96 2,43 0,61

1.51

1.59

1.66

1,73

1,80

1,86

1,92

0.50

0,53

0.55

0,58

0,60

0,62

0,64

860100 3891.6 Node 3

860100 3891.6 Node 3

860100 3891.6 Node 3

860100 3891,6 Node 3

860100 3891,6 Node 3

860100 3891,6 Node 4

860 100 3891,6 Node 3

0.00237 4700 5300 0.13

0.00421 4650 5350

0.00585 4600 5400

032

0,00977 4433 5566 0,53

0,01158 4363 5636 0,63

0,01340 4288

033

0,00798 4500 5500 0,44

0.00905 5093 5743 0,54

0.01030

0.01 140

4941

4892 5743 0.67

0,01170 4677 5723 0,67

0,01250 4555 5725 0,70

0,01318 4429 5726 0,73

0,01370 4300 5728 0,74

1

1

1

1

1

1

.

5743 0.61

1

Table 5.1

259

5711

0,73

PORTAL FRAMES : Type 5 (continued)

Frame n°

5.8

5.9

5.10

5.11

5.12

h

400 600

500 600

500 600

600 800

600 900

PE270 PE240

PE300 PE240

PE330 PE240

PE330 PE270

PE300

1600 16000

1475 14750

5,70 11,05

L Column/R Beam

H V

434

1232

12,38

0,961 1,064

0,875

0,993

0,993

1,001

1,081

1,071

rf

9964 861500 0,94

9391 861500 0,72

9979 861500 0,58

9250 1137400 0,73

8398 1137400 0,73

δ

239

330

3,99

0,90

0,96

2,97 0,89

4,03

Γδ Deflect rDeflect Mpl

1,01

1,00

1,46

138

134

0,49

0,43

0,41

2,00 0,50

2,52 0,56

860 100 3891,6 Node 4

860 100

3891,6 Node 4

860 100 3891,6 Node 4

0,00104 5600 0,13

0,00322 5400

0,01060 5737

038

134

0,00316 5743 0,40

0,00849 5743

0,01070

1,08

2

2

Nsd Msd

ly Hinge 6pd

Td Rd.req

1

3,41

0,988 1,026

au

I

1700

17000

5,77 10,92

acr al

j

1600

16000

4,57 11,49

5 1

1750

17500

IPE 270

6pu

Tu

Ru.req

| Mechanism

260

137 400

1

137400

5789,8 Node 4

5789,8 Node 4

0,00050 5260 0,05

0,00069 4720 0,06

136

0,00831 5686 0,89

0,00803 5055 0,76

2

2

2

5743

Table 5.2

1

ó

Rd mean

RgLmax.

RiLTJKâû

Rmnax.

9pd mean

Opdmax

9pu mean

9pu max

ô Mechanism

Table 53

Id lì li I 1^ li lã 1^ η li η u îî η κ iä ;

3ffi

i ΓI

Ι<β|4ΜΜ?βΗΗΡΡ

lUil^niMíPUHPP

Graph 5.1 : Plastic rotation

I

Graph 5.2 : Required rotation capacity

261

4.6

Portal frames of type 6

il

»

Ijü

J

Sfc:

^

3&

1

Figure 13

Three cases have been treated with full restrained supports and one case has been treated with pinned supports.

262

Frame n°

6.1

62

63

6.4

h

300

300

300

350

LI

500

500

500

700

L2

500

500

500

1000

PE 270 PE 270

IPE 270

IPE 300

Β 3-4

PE270 PE270

ΡΕ 270

Ρ 1-2-7-8

IPE 300

IPE 300

IPE 300

PE330 IPE 300j

Ρ 3-4-5-6

PE300

PE 300

PE 300

IPE 3301

75000

90000

84000

7200o|

4500

2400

Β 1-2-5-6

Η

V Full restrained

7500

X

6000

X

Χ

supports

Pinned supports

Χ

δ

1,97

134

1,13

2,07

31

25,9

8,54

29

!

acr

I

ai

0,923

0,922

0,921

0,857

1

au

1,189

1,078

1,113

1,008

rf p3 Tf P7

0,99

0,86

0,89

0,87

1

0,8

0,7

0,75

0,94

Tf

0,75

0,78

0,73

0,78

0,97

0,6

0,99

0,87

0,6

0,33

0,36

0,5

Tdeflect (node 10)

038

034

03

0,36

031 034

0.51

rdeflect (node 9)

P8

Γδ between nodes 1-3

Γδ between nodes 2-3

| j

Hinge

Mpl

Iy 1 1

j 1 1

|

6pd

Td Rd.req 6PU

Tu Ru.req

noeud 7 1

137 400

noeud 7 1

137 400

0,62

noeud 12

noeud 7 1

137 400

1

889 400

5789

5789

5789

11770|

0,0015

0,0015

0,002

0,0053

7978

8598

8415

7557

0,232

035

0,31

0,55

0,00831

0,0084

0,0068

0,003"'

9095

9099ι

1,1"

0,64[

Table 6.1

263

9ΐι:

!

7557

1,4:»

0,87

CONCLUSION

4

Type of frame

j

1

1

ι

1

1

0,06

0,49

0,21

0,46

0,85

0,66

0,00

2,12

1,11

1,04

4,02

238

0,06

2,41

1,10

0,66

3,59

1,75

0,57

2,08

1,07

0,83

2,52

137

0,05

1,34

0,41

0,40

1,36

0,76

0,23

0,55

034

0,64

1,42

0,90

.

ι

At collapse Min Max Mean

j

1 1

Under design loads Min Max Mean

j

1

2 1

1 1

1

I

1

1

j

3 1

.

I

j

1

4

ι

1

5 L

i S

1

i

6 y

Γr

1

i

I

Ij t ç

¡

^>

Table 7 : Summary of the results for the required rotation capacity Rreq The table 7 shows that the required rotation capacity at collapse can vary from 0,4 to 4. The values are rather scattered. It is difficult to associate a value of Rreq to a type of frame because of many reasons.

First in order to be on the safe side, we could keep the maximum value for each type of structure. This can be too conservative in most of the cases and this does not agree with our purpose which is to promote the plastic design of steel structures. This study is not exhaustive and one can easily find cases with greater required rotation capacities than those found here. However the values of Rreq are not generally very high. This is to say that some slender profiles could be used in a plastic design. It could also be sufficient to check the required rotation capacity under design loads instead of a checking just before collapse.

264

Annex

11

Document 3263-1-27 (ProfilARBED) (revised in December 1995)

'Available rotation capacity of plastic hinges Ravailable - Tests results and models." (49 pages)

flange

ΙΟ IO O en OJ o o O O o IO IO IO IO IO o o o Tf Tt CO CM CM «o IO IO IO IO CM CM CM IO IO Tf Tf Tf IO io

t

τ

IO

IO IO IO O CO CO co Tf

IO

CO CM

cn o IO CO Tf co o o

>>

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CO CO

CM o IO O O N- CO Tf Ν Ν- N- CO N- N- CD ID Ν- ΙΟ

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oc oc LU UI UJ UJ UJ UJ UJ UJ UJ UJ UJ UJ UJ

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co Tf

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co Tt «η m m

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m to o (0 (O tø Ν

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DC

< < < < CO TT CT TT < ffi ffl

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cc cc

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>

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ε E £ 3

φ u c

φ φ Φ

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>> >

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κ

κ

3 _l

3 _l

3 _l

> > > > φ φ φ φ κ κ. κ κ 3 3 3 3 _Ι _1 _Ι -Ι

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CM

CO

Tf

Ο

Ο O Ο ο

φ

φ

> φ

2 2 2 2 2 2 2 2 2 2 2 2 2 Ü

Cj

ÇJ

ξ0s

O O ce oc

OC

(0

£ 3

> > Φ Φ φ Φ φ Κ Κ κ Κ κ 00 oØ 3 3 3 3 3 _1 _1 -I _l _Ι Κ

*

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ce oc cc

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co Tt CM -3T

CN

1 .

Ravailable for beams submitted to My.Sd according to Feldmann's model ζ I

tf

té

,-y - X

y"

282

*

M y.Sd

Summarv of formulas from Feldman's model to evaluate plastic rotations φΡ(= Φ ) of I or Η steel profiles for different load cases.

Ρ

I å --ξ·1

-I

Ξ

4k,

CpO-

a

r-~

f4Ebw}

\

5

Π-ξΗ

---1-i

2t

(pi)

Him

Μμ -

(h-2t)w"f

1

(fy*Ao)

b3

Π-ξΗ-

I

JP

JP

(p0=

Γ - -tUeff

J^.,

H

α

--à-H

/Γ

t

(f7HTAo)bh\ 5h"

1/2f-

Y

"^

^

^

J

\ϊ_ώτ L

Τ.Μ·2

1

_

^Mp^

h

_

-(Nh-Pç)!

p_ha

P h

;

1/2-

Φι

"=(Ä^+^W)2+4f^hA°-f^

Jk^, Acr-^-irJ/^i, iv

-

1,4

K. - W -

Pc =

LOI.'t4L

^40*. i/4,0

/

^

\0J5

0,38(£)

4Ebw' Sh2 283

y

f.

*f*

yLá ^VX

^/^

1.1) Presentation of Feldmann's model: 1.1.1) Kav.

Φρί

Mi /

Mp/

>---

φ >-

Φρί

τ available

Rotations can be evaluated from the following formulas : where

»SP-

4.E.O. .3

5,6

i

5.h·

-^Sl+0,25 (fvJ1+Äa).h.b 400

)

2

J

τ-Λ/ίίγ^.Ιι.^] +4.fyW.h.b.tf.tw.Ao ίγ^.ίίΛψ

yM

with Δσ = 150 N/mm2 ΦR

Φτ

a.

= ί<Μρ£Μχ φΡ1 2- Ε.Ι

9l+9r

1.1.2) Limitations :

Li

Li

Class 4 sections (which mean elastic local buckling) should not be evaluated with Feldmann's model but the present tables do.

1.2) Legend of tables and graphs :

1.2.1) "Class"

= classification of cross-section submitted to pure bending, according to Eurocode 3 ("Γ = class 1; "3w" = class 3 web; "2f ' = class 2 flange)"

1.2.2) "S 235, S 355, S 460"

= Steel grades (fy = 235, 355 or 460 Mpa)

1.2.3)

1.2.4)

"b/tf.£" "a.d/tw.E"

= flange slendemess = web slendemess with α = 0,5 (pure bending) ; ε =^235/ f y (fy in [Mpa])

"Ψ.Ι? trad]"

= available - Φρί (in radians) (φ^Ρ written as "Phirot.pi" in tables)

"RavaLxLM"

= available rotation capacity multiplied by the span (that value is given in meters).

=> to obtain the available rotation capacity for a given span, the number furnished in the table shall be divided by that span [given in meters].

284

1.2.5) Particular case 1:

1.2.5.1)

"(ULS, SLS) Lmax Μ

maximum span [in meters] allowable according to ULS and SLS criteria of Eurocode 3, for 3 point-bending beams, (with partial safety factors γρ =1,0 for SLS and yp =1,5 for ULS):

ι

SLS criterion: arbitrary limitation elastic deflection)

,

,

P.LJ

L

48E.I

250

of the maximal deflection

:

(here,

L^Æ

250.P

ULS criterion: limitation of the maximal bending moment : Class land 2:

M max

1,5-P.L

_ Wpf .f y
4.Wpf.fy max

YmO τ

_i

c 12-E-LYmO

250.Wpf.fy

'

l,5.L.Ym0

(where jmQ is taken equal to 1,0 in the

tables) Class 3 and 4:

M max

lvlrtf V

.

-'fm« max TfrnO

τ

Hnax

1.2.5.2) 'Ravail. min

-

l>5-L.Ym0

_i

c 12.E.LYm0 1»-λ~,-Λ ,.

250.Wef.fy

minimum available rotation capacity related to the maximum span

(=

"Lrnax J'

1.2.6) Particular case 2:

1.2.6.1) "LmaxKemp"

equals to the minimum of:

- Lmax =

'-^

(see Part 2, items 2.2.2 and 2.3.6).

- Lmax ULS, SLS according to ULS and SLS criteria of EC3 (see above item 1.23.1) was introduced here to easily compare Feldmann's model with Kemp's model (see Part 2, item 2.3.6). According to ECCS recommendations concerning graphs on pages 18-25, we introduce Li / (iz . ε) = 60, which is maximum distance of distance between two adjacent lateral bracing in order to avoid LTB of 3-point bending beams.

1.2.6.2) "RavaiL (Lmax Kemp)" 1.2.7)

285

2

.

R available for beams submitted to My.Sd according to Kemp's model

tf X

*

286

M y.Sd

2.1. Presentation of Kemp's theoretical model (extract from the paper of Kemp and Dekker "Available rotation capacity in steel and composite beams", The Structural Engineer/Volume 69/N" 5/5 March 1991)

X : lolcrot

Ar

Lb

It,

177!

tí

li

.lp

ι

restraint

-Λ

lg. pHp

(cl tetero! restraint)

HpV./Cli-lp) Strain el etnt/« of

,

compression nanç«

t

rty.
Curvature - strain r" a* dt

Actual «cuîvelcM strut Unetfc « Cj l¡

Λΐ'.υαΐ tottrat ceil. lettre! ¿ri-ai cquïv. strut

T~V

ßlocx ι strainhorcwnina mocutvs. E j

Cn&ct of

stratn-hcrdenínq

Stoetxmoouka

fig. 8

el elasticity. E <*Ei£,

Strain

Appendix A. Theoretical model for predicting available rotation capacity due io local and lateral buckling of steel «elioni Local bueillnt The theoretical models proposed in this section refer to the ease or linear moment tradicnt shown in Fij 7, representînx approximately the rejion adjacent to an internai support ora continuous beam or to an caves joint in portal frames. Foltowint the proposals of Lay & Galambos", the local buckle is assumed to develop when the length of plastic rcjion of the ftanje

287

Piper: Kemp/Dekker (1 in fix 7(o)) extends sufficiently fir to accommodate the futi wavelength of the cucile. Xerap* is» rearranged the expressions and modified the assumption! of Lay it Gabmooi, Southwsrd . xnd Stowtll". resulting ín the following formulae which may be solved jccraiivcly to give the plxstie length L^ at which fiante buekling would occur theoretically in the absence of membrane restraint:

'r

"V

on the strain condition in the compression flange are evaluated separately and thea coeibincd. The maximum vertical out-of-planc deflection *, of the extreme fibre of the flange due to local buckling may be assessed as:

*.<-

where

l-/t¿ Is the flange widih-to-ihiekaess ratio C¡ 0-3 for no web restraint or » 1-0 for optimum web restraint «, is the longitudinal strain at buckling in the comprcssîoo fiante at the «aire of the buckled length and may be determined from ?ϊχ 7 M as:

.,,(,+0-3« i-)

....(A J)

3 and «_<

e are the strain-hardening characteristics in Πχ is the yield strain of the flange

Ζ

"Web slendemess his been observed in tests as not causing significant strain-

weakening in isolation and having only a secondary effect ea strain* weakening behaviour due to local buckling of the liante aad distortion of the cross-section. From an assessment of test results it is proposed that he intención between web aad fiante budini be considered by deiinint C, in eqn. (A.I) as:

C,- l-0-{«eVJ0rO< out > 0 where «β*, is the clear depth 2. and t. is web thickness.

i-Urehtersienai

fc*,
(A-3)

of web in compression, as defined ia Table

Je.i -.-i-S-h.

_ χ,^ imm--Jo-6*9-*e L,'- CL,

....(Α.«)

Plastic

- L,- (m-1) ^~ ' J dp-Sta+0-4) L, - C/m

,...(AJ)

....(A.S)

This reflects the interaction between local and lateral bucking and should be added to th« ntaximum amplitude of sinusoidal deflection of the flange due to lateral buckling t^ which is assessed as:

AucJtÆnr

LatcraJ-tonional buckling is considered' in terms of the lateral buckling of a strut comprising the compression flange and portion of the web in « compression over the length'Q /*)£. between lateral restraints ia Tig 7. The results are adjusted nominally to allow for the cahaaccacnt of resistance due to torsional rigidity. This length is subdivided Into elastic and plastic portions as shown in Hg 7(d) which are adjusted to ν« effceive lengths X-, and Lm allowing for moment gradient In the normal way and for the reduction from elastic modulus JTto strain-hardening modulus Σ, sn the plastic region as follows:

-slist«:

....(A.1)

where the initial imperfceåon is assumed to be 0-Olè. ¿^ is the full ws»efcigth of the local buckle given by eqn. (Al), and L> is the length bcr-c:a seesior-s of maximum and aero moment at which lateral buckSng is predicted by eejn. (At) for a denned length of the plastic region £,. This reflects the common observation, confirmed in "Dekker'j tests" and described above, that local buckSng Itnot itself an unstable strain-weakening process ín the absence of lateral buckling which, ¡nierai!. , release the membrane restraints to local buckling. . The shortening In length of a strut with a deflected shape represented by * sin(«aVZO Is given approximately by A7*£. in which L. a the half wavelength. The buckling of a flange with maximum venieaí amplitude *, in eqn.
....{Λ.«

U-5S-C/.["/íji

0-014

Xi-tL/LJHm/LJi

. _ .-caiLfi-e-i

(A.10)

*. U-(L/Lfl in -which the initial imperfection is assumed to be -CtJlLß -fi)

and

L%

is again given by eqn. (A4).

.The lateral curvature* a,* across the width of the flange due to these défierions at distane:: from the yielded end of the equivalent strut Is given

.by:

«í-Aí.+ ajrintrce/^AÇ^1 ..'..(A.!!. At the «entre of the local buckle in the equivalent strut a: - 0-3Ο,£, from

oqa. (A3) and the criterion of failure Is the lowest value 0f ¿ ct

iii · irAfr from "fig 7 at which the minimum strain on one extreme fibre of the flange «A falls below the yield strain «,. Le.

where at is the ratio of maximum moment to resistance moment » M_/Mr L/Um, L¡) for Ener moment gradient In fig 7 and t » E/E,

. *.C-«»-«-3Mf-.«¿. " ....(Α.Π) in which «, Is the average strain in the flange at this section given by cqa.

¡uriti. These two lengths are combined to reflect an elastic dement subjected to a uniform axial force equal to that required to fully yield this strut. The critical buckling length i, between sections of maximum and zero moment is obtained by determining the value of L{ at which this equivalent strut reaches Its elastic buckling load: .

(A2). Alternatively, In

eases of relatively compact flanges and large lateral slendemess ratios, the criterion or eqn. (A12) is applied at the centre οΓ the equivalent strut where 0-3<^i, from eqn. (Aí) to give a lower

value of i^andm.

.(A.Í) where lm â the radius of gyration of the flange and portion of web in compression and C, a the enhancement due to distortion-! restraint by the web and based en the analyses of Svensson", and the second author is given by:

C, - 1-0 + 0-07 11-33

-if.

hot > 1-0

.(A.7)

la which i Is the overall depth of the section. Interaction btttttn /oca/ «nef ¡aleni bueiUnt To define the onset of strain-weakening. It Is necessary to Identify those regions la the beam where local flange and lateral buckling displacements cause a significant change m tb« strains resisting major-axis bendine. The requirement that the minimum strain at the critical section should not fall below the yield strain has been adopted as a conservative criterion of strain weakening. The effects of deformations due to local and lateral buckling

χ-

Rotation capacity Assuming the idealised stress-strain curve for steel shown In Fig t and simplifying the elasto-plastic transition, it is convenient to adopt trilinear moment-strain and moment-curvature relationships Illustrated In Figs 7(b) and (c) along each length i, between sections of maximum and aero moment. These distributions are based oa Lay's discontinuous yield theory"' and reflect the conditions at maximum load prior to the onset of strain-weakening behaviour. The following relationship between available rotation capacity and length of the plastic region Lf may then be derived from the area under the simplified curvature distribution in Fig 7(c). assumine the rotation capacity at which the moment falls below M, in "Fig 1 Is twice that at maximum moment:

'V-^é;1 where «' is the proportion of the depth centres of th« two flanges

(A-»)

of section in compression between the

The Structural Engineer/Volume 60/No.S/S March

SG

288

1301

2.2Ì Presentation of Kemp's simplified model 2.2.1)

vav

= 2a.
2.2.1.1)

* Equivalence of svmbols (Kemp - Feldmanni: (pa (Kemp)
=


(Feldmann)

(Kemp) = (ppfÇFeldmann)

ι
φR

1

Π-

Mpf.Rd-L 2.E.I

2.2.1.2) Stress-strain curve 1

Yield Onset of strain-hardening ,

j

Slope = strainhardening modulus, E s

. Slope = modulus of elasticity, E

e=E/Es s=Ss/6y

εs

Strain

Kemp assumes the following values of e and s:

e

= 50

,s

= 10.

2.2.1.3) Kemp proposes to evaluate Rav as follows:

where

λβ (Lambda e) is the effective lateral slendemess :

Xe-Kf.Kw. Ct,

f

L· ^

where

ε = ^235/fy (f y in [Mpa])

Kw are empirical factors :

Kf = ^f-£>

for class 1 and 2 flanges,

20

289

txd

KW2=-

for class 2 web,

OJalnf.tr

Kwl = 460-

f L,

|WKw2

{izc.z)}

for class 1 web.

400

a' is the propotion of the depth flanges ; here a' = 0,5 .

of section in compression between the centres of two

2.2.2) Limitations of the simplified model: a) applicable to class

1

and 2 only, according to the old EC3 (1988).

^<:< 100 , L¡ = 2

b) 20 < --

ι,Ρ.ε 2c c

for 3 - points bending beams

*

2.3) T^?end of tables and graphs : 2.3.1) "Class EC3"

"Class Kemp"

.

= classification of cross-section submitted to pure bending, according to Eurocode 3 ("1" = class 1; "3w" = class 3 web; "2f" = class 2 flange) = class of sections according to the ancient Eurocode 3 (1988) used in Kemp's model ("l/2f ' = class 1, 2 flange; "3/4f ' = class 3, 4 flange; "lw" = class 1 web; "2w" = class 2 web; "3w" = class 3 web).

Type of section

Web in pure bending

Maximum width/thickness ratio

Code

Eurocode 3 (1988) - Kemp present Eurocode 3 (1992)

Flange in pure bending

Eurocode 3 (1988) - Kemp

present Eurocode 3 (1992)

Classi 66.ε 72.ε

20.ε 20.ε

Class 2

Class 3

76.ε 83.ε 22.ε 22.ε

Elastic form

124.ε 30.ε 30.ε

·

Ι

2.3.2)"S 235, S 355, S 460"

= Steel grades (fy = 235, 355 or 460 Mpa)

2.3.3) "b/tf.

= flange slendemess web slendemess with α = 0,5 (pure bending) ; ε = J235/fy [Mpa]

ε" "α.ά/ΐν,.ε"

2.3.4)Particularcase 1:

"(ULS,SLS)Lmax[m]"

:

maximum span [in meters] allowable according to ULS and SLS criteria of Eurocode 3, for 3 point-bending beams, (with partial safety factors γρ =1,0 for SLS and yp =1,5 for ULS):

. SLS

criterion: arbitrary limitation of the maximal deflection : (here,

elastic deflection) P.L3

48 E.I

^ L

=>L2<48-EJ

250

250.P

290

.

ULS criterion: limitation of the maximal bending moment

- Class

1

:

and 2 :

Mm« max =

WDf.fv

1.5.P.L

4-Wpr.fy max

l,5.L.7mo

YmO

_

12.E.LYm0

250.Wpf.fy

(where 7m0 is taken equal to 1,0 in the

tables) Class 3 and 4:

Mmax =

1.5.P.L Wci-.fv i27^^Mpf=-^ 4

=>P max

TmO

_ 4-Wef.f y '

l,5.L.YmO

250.Wcf.fy 2.3.5)

n

-Μ

ι

ίΖε·ε.

= lateral slendemess of beam where

izc is the radius

of gyration about minor axis z-z

ε = ^235/ f y (f y in [Mpa]) 2.Lj = L in case of 3-points bending beams.

< 60 (ECCS recommendations to avoid LTB) .

2.3.6) Lmax Kemp equals to the minimum of:

- Ljmax

*

J-c'

(see limitations

of Kemp's model in present item

2.2.2)

- Lmax ULS» SLS according to ULS and SLS criteria of EC3 (see present item 4) 'R-avail. min

= minimum available rotation capacity related to the maximum span (= "Lmax Kemp")

2.3.7) "Lp"

= the length of plastified region of the beam

¿TMp

Mrn=m.MD

*

291

=MpJLi/(Li-Lp)

* Ravailable for beams submitted to My sd according to CRM model (Ref. 4).

in [radians]

Ravailable

=

(ppl

-1.

_ Mpl.L

where φρ1 = F

^

2.E.I

292

4.3

Reliability of models (R experiments vs. R models)

Rotation capacities according to Kemp's simplified model which are presented in these graphs respect its limitations (see chapter 2.2.2).

293

l

.

e φ

E \_

+m+

φ Ol

tn

Φ

>

E

,

Χ

|

φ

φ "O o

φ

Q. .

%

m

C

.

<0

>

E

Χ φ

»--m

Φ

o

vi

Τ3 Ο

_φ

>

ε

"φ

«

to

to

'o. E

ì> te

Ε

]θ

φ mt

o χ

φ LL

<

ΙΟ

mod

<ο c%

oc

pli

expe sim

ο (Ο ment nn's

ΙΟ ΙΟ ο ΙΟ ΙΟ «a-

3

ο τΐ-

ιο co o 00 IO CJ

o

t

CM

IO

C

φ

IO

Ε φ α. χ φ

i

IO

o



IO IO

O to

IO *

O «ί·

IO co

© co

294

IO

o

IO

Ο

evi

CM

τ-

τ-

IO

.

mode

.

c

φ

Ε U

+m*

C

m

φ

O

tn

tn >

φ

Τ3

E α.

χ φ <0

>

F

Τ3 Φ «^

15

<ο

k.

φ Q. Χ Φ

φ

w

a.

.

>

»---

φ Τ3 Ο L·

CS

F

Ζ>

E

σ

CE

φ rnt

"φ

ϋ Χ

u.

< φ

*

mod

pli

expe sim

CM

Ε. >

CO CM

CC

CM CM ment nn's

CM

O CM

en CO

hCD

IO

* CO CM

en

co

hco io

co φ

CM

ε m.

Φ α. χ

φ

i

295

Ο

<°

c

φ

ε ι

rnprn^

C

Φ

ε

φ

Q.

Χ φ

vi >

β

tn

>

φ XJ o

F

TJ Φ

ì>

tn CL

ε

ϋ

φ -C

Χ

<

OC

φ α. χ φ .

tn

> M

φ TJ o

ε

"Φ

tn

k.

CO

ε

TJ φ

"

φ

Τ5 Ο

κ <χ

mod

pli

expe sim

CO ment nn's

CM

en

CO

co

--

io

co

CM

£ t-

CO

CM

τ-

Ο

CO

CO

296

IO

CO

CM

τ-

O

4.4. Influence capacity

of beam length

and lateral slendemess on available rotation

In

some graphs, beside curves corresponding to nominal cross sections, there are "·" (black points) which equal to Rav calculated for measured sections of experimental beams.

297

CRM Liège: Tests N° 2,6,27,28

Feldmann model nom

Kemp simplified model

Kemp theory model

Feldmann - measured profile

40

80

60

100

120

Li/(Ì2*epsylon) 1

1

t

I I

|

1

4

6

8

10

L span

==

1

2*Li (m)

Kemp application range

HE 200B, fyf = 445 MPa, fyw = 462 MPa, class

298

1

(EC3)

CRM Liège: Tests N° 3, 9, 31

Feldmann model nom

Kemp simplified model

Kemp theory model CRM tests

100

120

Li/(iz*epsylon) 1

I

I

1

1

I

1

t I

I

1

0

2

4

6

8

10

12

I

1

1

Lspan = 2*Li {ml

Kemp application range

HE200B, fyfi = 261.0 MPa, fyw = 291.0 MPa, class

299

1

(EC3)

CRM Liège: TestN°7,

15

40

35 Feldmann model nom

Kemp simplified model

30

Kemp theory

*

CRM tests

25 Feldmann - measured profile ja

5 20 > a

15

10

O

-r 60

40

20

80

100

120

Li/(iz"epsylon) T"

0

ι

2

4

10

6

Lspan = 2· Li {m)

Kemp application range

HE200B, fyf=404MPa,fyw = 426MPa, class

300

1

(EC3)

CRM Liège:

Tests N° 4, 8, 17, 25, 26

40

35

Feldmann model nom

30 Kemp simplified model

Kemp theory model

25

CRM tests

Feldmann - measured profile

S3

5 20

15

10

0 -r 0

40

20

80

60

100

120

Li/(iz*epsyIon) I

1

1

1

1

1

ι

I

1

I

0

2

4

6

8

10

L span = 2* Li {m}

Kemp application range

HE200B, fyf=375MPa, fyw = 421 MPa, class

301

1

(EC3)

CRM Liège: Tests N° 10, 18, 19,29,32

Feldmann model

Kemp simplified "model nom Kemp theory model CRM tests

Kemp s. - measured profile

o -

20

Τ

ι

ι

40

60

80

100

120

Li/Uz'epsylon) 1

I

4

8

-1_ I

10

Lspan = 2· Li (m)

Kemp application range

HE200B, fyfl = 303MPa, fyw = 342MPa, class

302

1

(EC3)

12

RWTH Aachen: Tests EA2243, EA2244

4

6

Lspan = 2*Li

{m)

Kemp application range

HE 220A, fyf = 420.5 MPa, fyw = 437.5 MPa, class 3f (EC3)

303

10

RWTH Aachen: Tests EA 2833, EA2834

40

20

60

100

80

120

Li/{iz*epsylon] 1 1

2

1

»

4

!

'

1

1

ι

I

I

1

I

6

8

10

12

14

1

Lspan = 2* Li (m)

Kemp application range

HE 280A, fyf = 276.5 MPa, fyw = 31 1.5 MPa, class 3f (EC3)

304

t 1

16

RWTH Aachen: Tests EB 2843, EB2844

40

Feldmann model

·

Kemp simplified model Kemp theory model

20

0

40

60

120

100

80

L5/(îz*epsylon) I

ι

-i

τ

0

2

I

1

I

I

4

6

I 1

8

I I

10

Lspan = 2* Li Im)

Kemp application

range

HE280B, fyf=489MPa, fyw = 539MPa, class 3f(EC3)

305

12

RWTH Aachen: Tests EB2833, EB2 834

60

Li/{iz*epsylon] i

i

1

6

8

10

Lspan = 2* Li

12

14

{m)

Kemp application range

HE280B, fyf= 248.5 MPa, fyw = 252.5 MPa, class

306

1

(EC3)

16

RWTH Aachen:

Tests EA2233, EA2234

HE220A, fyf= 282.5 MPa, fyw = 308.0 MPa, class 3f (EC3)

307

RWTH Aachen: Testd09a3m

HE220B, fyf= 278.5 MPa, fyw = 286.1 MPa, class

308

1

(EC3)

RWTH Aachen: Test d01a4m

Li/liz^epsylon) 1

1

1

i

0

2

ι

i

4

Lspan = 2*Li

ι

1

1

1

ι I

6

8

10

(m)

Kemp application range

HE220B, fyf= 486.2 MPa, fyw = 531.7 MPa, class

309

1

(EC3)

RWTH Aachen:

Tests EA 2843, EA 2844

Feldmann model

Kemp simplified model Kemp theory model

ι

40

60

80

Lî/(iz*epsylon) î

>

10

468 L span = 2e Li

lm)

Kemp application range

HE 280A, fyf = 504 MPa, fyw = 535 MPa, class 4f (EC3)

310

12

Lukey - Adams: Test N° Al

-i

20

1

40

60

100

80

Li/Uz'epsylon) τ

4

6

L span

8

= 2* Li (m)

Kemp application range

fyf=285MPa, fyw = 309.3 MPa, class 2f(EC3) 311

10

Annex 12

Document 3263-1-48 (ProfilARBED-RWTH)

'Background document to Eurocode 3 about ductility evaluation of plastic hinges" (66 pages)

1.

Analysis models and limiting b/t-ratios of Eurocode 3 - 1.1

(1 )

Eurocode 3 - Part

(ENV 1 993-1-1) presents a classification system for cross-sections according to the slendemess b/t of cross sectional parts in compression, that allows to take profit of plastic reserves of the cross-section or the plastic reserves of systems as for as moment redistribution may have an effect, see fig. 1. 1.1

M

Mpl M el

¿

ι

æ æ

b/t

ι

Definition of the classification of cross-section

21

^ujjÍjj^

Global analysis Class

Behaviour model

Design resistance

Available rotation capacity of plastic

of structures'

hinge

PLASTIC

M

across full section

Mp/j--/

PLASTIC

iM Mpi,

elastic or.

plastic

f

kxal·^"^ buckling

limited

elastic

none

elastic

none

elastic

across full section

local budding


f ELASTIC

M

across full section

Mp< /c

Λ" f

important

local

bc buckling?

r ELASTIC

M

across effective

section

Uit] local

buckling?

'

Fig. 1: Plastic reserves of beams ace. to EC 3

314

(2)

So far the sections allowing for moment redistribution due to the formation of plastic

hinges are class 1 -section. The strength level is the full plastic moment M^ of the cross-

section.

(3)

Class 2 sections need to consider the maximum bending moments from an elastic analysis without moment redistribution but allow to exploit the plastic resistances of the

cross-section as for class 1-sections.

(4)

For class 3-sections yielding in the extreme fibre of the full cross-section is limiting the resistance, whereas for class 4-sections yielding in the extreme fibre may only be allowed when local buckling to teken into account by reducing the cross-sections to the effective zone.

(5)

As the controlling parameter for the classification of a cross section the b/t-ratio of its compressed parts is considered. The numerical values for the b/t-ratios for the different cross sectional classes as presented in Eurocode 3 are given in fig. 2.

(6)

The values have been determined for 235 N/mm2 only. An adjustment to other steel

Classification of cross-section: limiting widuVto-thickness ratios for class I ¿.class 2 J cross-sections Types

Stresses

of

distribution for

classi

i "comiiprcssion

Mv

M,

+

I

3Ξ

&.

Class

classi

Flange

Web

Flange

d/twS

c/tfS

d/twS

C/tf¿

fy

Ν

38ε W

U-^

ε (jf 40 mm < t á 100 mm)

lie

9ε

W

W

W

W

9ε 10e

α>0,5:

9ε

456ε 13α-1

10ε

11ε W

10e

275

355

420

460

0,92

0,81

0.75

0,71

0,96

0.84

0,78

0,74

Fig. 2: b/t-ratios for different cross-sectional classes ace. to Eurocode 3

315

10ε 11ε

10ε

396e

10e

83ε W

13a-l

+ : Stiesses in compression R = rolled sections W = welded sections - : Stresses in tension 235 fy (H/mm2) e=^235/f, ε fif t¿ 40 mm)

W

10ε

cofAS:

J

9ε

72ε

iL_ljL~J" C

11ε

10ε 33ε

-Ξΐ-7

rtçtjnm, " My

Class 2

1

Web

grades may be performed by factoring with

235

e =

Ν

*

values which were derived from the Euler-buckling formula.

°"ki

(7)

=fy

π2 =

/t\2

·Ε

12 (1

-

lb

μ2)

The transition from plastic cross sectional resistances Mp, to elastic resistances M^, is stepwise. Only for class 4-sections, where local elastic buckling reduces the crosssectional resistance, the influence of such local buckling an "effective" cross-sections by the reduction factor is continuous:

MR=a4ff M^cc^

WeI

i;

where

aeff= (8)

The classification in fig. beams ace, to fig. 3.

1

£

<_

1.0

is justified by moment rotation curves

Fig. 3: Justification of cross-sectional classification by tests

316

of three point loaded

(9)

The moment-rotation curves as shown in fig. 3 are the result of bending tests. These tests are able to give information about which level of resistance can be achieved and how great is the rotation. As aforementioned the certain sorts of moment-rotationcharacteristics gave rise for the classification system. In particular the b/t-limits for class 1 are such that they guarantee a certain plastic rotation φΜ on ,,,-level so that plastic zones, which can be modeled as plastic hinges, are able to rotate.

(10)

One can introduce a value which is able to describe the ability of a plastic hinge which

may be the inelastic rotation of a plastic hinge,


, which is that rotation on ,,,-level

up to the point of intersection of the moment-rotation curve after that it drops below the ,-level. We also can introduce the rotation capacity R, which compares the magnitude of the inelastic rotation φ^ with the elastic rotation
p ^available

_
(1 1)

-



_


1

x

Φρί

Introducing a general value for the rotation requirement, coming out from the static analysis, and which might be in the range of R^, = 3.0, one can identify those section allowing for moment redistribution along the structure, i.e. those sections with

R» * 3,0 (12)

The sectional rotation capacity depends on the geometry, slendemess and the steel grade

of the cross-section and also depends on the loading arrangement Slendemess limits in terms of b/t should comprise all these demands to allow for plastic analyis. (13)

However the design models used in Eurocode 3 - part 1 .1 are in some parts extremely safe sided and not realistic. The diagrams in fig. 4 show the rotation capacities of a great amount of so called "Three-Point-Bending"-tests (both rolled and welded sections) versus the b/t-slendemess as well as the present b/t-regulations of Eurocode 3 (for hot rolled sections) , see also annex (3) in which the data are given in detail.

(14)

In fig. 4a to fig.4e the test results are presented. The conclusions for the present EC 3 rules are as follows:

The presentation of the available rotation capacities terms of angles ("
317

The present classification rules for the web slendemess show little more

correlation than the present classification rules for the flange slendemess, however not satisfying at all, see fig. 4ax and fig. 4b.d. The present consideration ace. to EC 3 of the higher yield stresses than fy =235 MPa by introducing e = (235/fy) os shows only tendencies but not more and is also not satisfying, see fig. 4ax and fig. 4b.d. Having the most tests in the steelgrade range from 235 N/mm2 up to more than 500 N/mm2 and also some up to more than 950 N/mm2 it is visible that even in the higher steelgrade ranges sufficient rotation capacities in most cases can be provided, see fig. 4e.

318

25.0

20.0

Λ!

ί

3-μ

15.0

ι

Rex

^4

10.0

m

5.0

ι

\ψ

0.0

10.00

S

15.00

20.00

25.00

30.00

35.00

80.00

100.00

b/t

25.0

%

20.0

Ι

J

15.0

IΙ

·:

Rex 10.0

Η

t

Ικ'-" ë

*

ι

5.0

0.0

0.00

?: 20.00

-

f

Η

40.00

60.00

(h-2t)/w

Fig. 4a: Rotation Capacities R of three point bending tests versus simple siendemesses

319

25.0

J_L

-i+

20.0 .w

i 15.0

τ*"

Rex

\-Λ

1

10.0

_

:*v

5.0

m~. m

R

4-

0.0 10.00

j 25.00

20.00

15.00

30.00

35.00

b/t * (fy/235)*0.5

25.0 I

1 ta 1

I m

20.0 1

! ,7_ %

15.0

B

Rex 10.0

_

m.

B fl

5.0

a

f

.

l

ι

1

I

1

0.0

0.00

20.00

40.00

60.00

(h-2t)/w*(fy/235)'0.5

'.

S

80.00

100.00

Note:

= class 1 2 = class 2 1

3

= class

3

Fig. 4b: Rotation Capacities R of three point bending tests versus siendemesses ace. EC3

320

0.6 ι

0.5

I a

j'

Y

ι

W

-,

V fl

^S-

,

f

I

c

d

ι

\ *·

Β

I

BB

0.2

1

0.1

'i

a

*

0.3

b

h

0.4

phi av

ι

γ

ι

β

li Γ

s-.'í-

0

10.00

20.00

15.00

25.00

30.00

35.00

80.00

100.00

b/t 0.6 fl

0.5 .

0.4

phi av

,m

i

0.3 m

0.2

ki li

0.1

!"

1 0.00

20.00

40.00

β

60.00

(h-2t)/w

Fig. 4c: Rotation Capacities φ of three point bending tests versus simple siendemesses 321

0.6

11 0.5 fl

0.4

'T

Vfl

f

phi av

0.3

't-

¡r^A

-e

A

0.2

I

Ai«

0.1

' u

I

U

1, 10.00

3, 25.00

20.00

15.00

30.00

35.00

b/t * (fy/235)"0.5 0.6 !

1

i

0.5

1

I*

0.4

phi av

0.3

0.2

tø.

%

·;,af

0.1

"1 Β

0.00

Β

20.00

Β

Β

IJ*

VΚ 60.00

40.00

(h-2t)/w* (fy/235) '0.5

Β

2,:

>

80.00

100.00

Note:

= class 1 2 = class 2 3 = class 3 1

Fig. 4d: Rotation Capacities φ of three point bending tests versus siendemesses ace. EC3

322

25.0

20.0

li -4c

ί 1S.0

Rex

'Κ--

W4

Χ^ηΛ

10.0

Τ'.

m

5.0

m

-

Ί 0.0

400

200

600

800

1000

fy [N/mm2] (mean of flange and web)

0.6 fl

0.5 ι

-

0.4

phi av

1I

κ

0.3

k't

Γ' 02.

m

"f.. ! l·

0.1

I

Ί

*

ι mt 200

400

600

800

1000

fy [N/mm2] (mean of flange and web)

Fig. 4e: Rotation Capacities of three point bending tests versus yield stress

323

(14)

Not only b/t-regulations are unrealistic, but also the maximum resistance is limited to MpI. Many tests have shown, that sections often exhibit more resistance than Mp, even

α · Mp,, depending on moment shape, slendemess, ect., see fig.

5

1.80 ! ί

1.60

1

ι

1

1.40

ffl

1.20

1

1.00

Mu/Mp!

!

;

;

-ll- /

Λ

I

b

1

Jc

i

0.80

Γ

t

0.60

t

! i

'

0.40

A

3

;

15.00

4r

4t-

.1

t>

t>

10.00

d

0

1 Ζ 0.00

-

*

ι

0.20

/

25.00

20.00

30.00

I 1

35.00

40.00

45.00

b/t*(fy/23S)A0.5

1

60

1

40

-

1

20

.

1

00

Γ Β

I

JU-

__

ί·- tï'

fll

*

1

Mu/MpI

.

1 1

I

0 80

i ι

0 60

-

040

·

No te:

= 2 = 1

cias;sl

i

cla<:s 2

1

1

0 20

:s3

·

0.000.00

!

10.00

20.00

30.00

40.00

50.00

60.00

1,.

2 t>

70.00

80.00

h/w* (fy/235)A0.5

Fig. 5: Ultimate resistance of bending tests versus slendemess (15)

The causes of the safesidenesses are:

324

90.00

(15)

The causes of the conservativisms are: 1.

2. 3.

4. 5.

The b/t- and h/w- values are developed from elastic theory basis with extension

to plasticity which originally are intended to have a criterion for resistance but with poor consideration of the rotation capacity. The important interaction between flange and web has not been considered in the present rules. Resistances are limited to Mp,, though the moment rotation curves show that higher resistances are possible. The magnitude of resistances is defined stepwise neglecting the actual continuous resistance distribution. The minimum rotation requirements for class 1 -section has been assumed to R = 3.0 for usual buildings without taking into account the actual rotation requirements which are often smaller.

2.

Experimental and numerical investigations on cross-sectional resistance and rotation capacities of beams in bending

(1)

In order to overcome the shortcomings of the present b/t- and h/w- rules of Eurocode 3, Part 1.1, bending tests, which were performed in the past have been re-evaluated and also new test series on bending tests have been established. Single span beams with several kinds of loading were tested. Three-Point-Bending tests with bending about the strong axis. Three-Point-Bending tests with bending about the weak axis. Bending tests with normal forces, with bending about the strong axis. Four-Point-Bending tests with bending about the strong axis. The test-setup, the test-procedure and the test-evaluating of typical three-point-bending tests are described in annex (1) of this background document.

(2)

Additionally a lot of computer-simulations were carried out to obtain more and "synthetic" results of the plastic behaviour of beams in bending, in particular for high strength steels the computer-simulations are described in annex (2). The simulations are welcome for those investigations for which experiments would produce much effort, e.g. for the question of the influence of the load position along the span of the three point bending beam. The simulated rotation capacities for this question are also shown in annex (2).

325

3.

Improved design models for accounting of plastic crosssectional resistances

(1)

The most realistic and homogeneous procedure to predict the strength capacity of a structure is to introduce the true moment-rotation curve of the section, depending on moment shape, type of loading, geometry, slendemess and steel grade, see fig. 6a. No further concern about slendemess of section or rotation capacity must be made. However this may increase evaluation time due to strong nonlinearities not only due to nonlinearities in the hinge but also in cases of large and complicated structures.

(2)

Alternatively a uniform definition of the cross-sectional for all b/t- and h/w- ratios can be achieved by applying the format of class 4-sections:

MR=a Μ,, = α Wd Î, where M¿ is the elastic moment resistance α is a modification factor that may attain

(3)

values cc>1.0e.g.

aP>

or values a<1.0e.g.

«eff

=

see fig. 6b.

=

In general the behaviour of plastic hinges in statically nondetenninated structures is deformation controlled as a function of the rotation capacity, except for the last plastic hinge that leads to a completion of the plastic mechanism, the behaviour of which is force controlled reaching the ultimate load of the frame. Due to this fact not only the stable plastic part of the --curve, but also the instable part ofthat curve may be used in the design, see fig. 3. By the choice of resistances cc¡ < αρ, the rotation capacities can be significantly increased thus allowing for complete moment redistribution on a reduced level cc¡. This concept is applicable to all cross-sectional classes, e.g. also for class 4-sections.

(4)

In the present Eurocode 3, Part 1 the plastic hinge method is only allowed and defined for α = αρ, for class 1 -sections. For this case one can reduce the spectrum of -cases, described in clause 3(2), to one case α = αρ with an a priori nonlimited deformation of plastic hinges on Mp, -level, but with an a posteriori check of the ductility by slendemes limits (class 1 sections only, see EC 3, 5.3.3 (4)).

326

Fig. 6a,b,c:

Different design models for accouting realistic behaviour of plastic zones

(5)

To obtain the true --curve of a beam in bending either experiments are necessary or sophisticated Finite Element simulations have to be performed. Furthermore the use of the whole --curve in the global plastic analysis of a structure costs time, as indicated.

(6)

Thus the use of modelized --curves as they are in fig. 6b and in fig. 6c will be necessary. In the following a new rotation check of plastic hinges is presented using constant Mpl-level of the plastic hinges,see fig. 6c.

4.

The procedure of the rotation capacity check

(1)

The verification of the rotation capacity comprises the comparison of the rotation requirement with the available rotation capacity in the plastic hinges or plastic zones.

Both the rotation requirements and the rotation capacities are to be obtained using the defined ,-resistance level. (2)

The rotation requirement results from the global analysis of a structure.

(3)

The rotation capacity results from experiments or, if not available, from computer simulations or, if also not available, from engineering models.

(4)

Fig. 7 illustrates thè procedure of the rotation assessment Having a two-spancontinuous beam with distributed load it is visible that the first plastic hinge occurs

327

above the support, whilst the load-deformation curve is still increasing to ultimate load

at that point where the second plastic hinge forms in the span.

Beyond the last hinge the system becomes kinematic. From the beginning of plastification of the first hinge up to the last hinge the first hinge has to rotate. This rotation φ,^ can be mathematically determined either by hand for simple cases or by computers.

The required rotation in terms of dimensionless R«, can be calculated by deriving φ rcq by the sum of the plastic rotations at the ends of the single equivalent beam. The length of this equivalent beam can be taken as the distance either between the points of inflexion (points of zero moment) of each side of the relevant plastic hinge (1st option) or the tangents to each side of the plastic hinge's moment peak in the moment diagram (see fig. 7. 2nd option) in the ultimate limit state:

(5)

^

Φρι

rq

,

M

τ

Έ

2ndPUiticHinge

-*3

1 it

-φ

Plaide Hinge

"«υ*

2nd Plastic Hinge

/

.Λ

Ά χι

opfionl

lrtPbutlc Hinge

Equivalent single beami

φ

Fig. 7: Obtaining the required rotation

(6)

The shape of the moment distribution can vary from "peakshaped" to "parabolicsmooth". However the determination of the rotation capacity is based on having constant moment gradient byside the plastic hinges, that means also with moment peak.

If no lateral buckling phenomenae occur the peakmoment-hinges are to be assumed to produce the smallest available rotation capacities for I- and H-sections. 328

If no lateral buckling phenomenae occur the peakmoment-hinges are to be assumed to produce the smallest available rotation capacities for I- and H-sections.

(7)

The rotation check can be performed by the comparison

φΙβς<φ«/γΜφ or

Rfeq
(the partial safety factors γΜφ are described in chapteró "Safety evaluation"). Byside the verification of full ,-level this rotation check should be performed for all plastic hinges which have to fulfill certain rotations up to ultimate load except the last one having structures with more than one plastic hinge.

5.

Determination of the available rotation capacity of plastic hinges

(1)

The determination of the rotation requirement is the individual result of the global analysis, whereas the rotation capacity depends on the cross-section to be used.

(2)

As aforementioned the rotation capacity can be determined by experiments, computer simulations and by engineering models.

(3)

The demand for useable and practical tools gave rise to suggest the development of a simple model which does not make much effort to use. Such an engineering model might be the following formulae for I- and -shaped sections for different kind of loadings. The basis of these formulae is the modelling of the nonlinear buckling phenomena in the yielding zones by a plastic folding mechanism that allows to determine the rotation capacity in a reliable way. The formulae have been developed for the rotation capacity of I- and -profiles in bending about the strong and the weak axis, for bending about the strong axis with axial forces. The reliability of this formulae is proved by the comparison with test results, see fig. 8 and annex (3).

(4)

However in fig. 8 the comparison is given in terms of R whereas the following formulae are expressing the available inelastic rotation in terms of angles φ.^. That is no problem considering the relationship Rav =
329


So no further uncertaincies are

(5)

Thus the following formulae can be used for the determination of the rotation capacity R of a beam and also for the determination of the available inelastic rotation
(6)

As the rigid plastic model describes the whole instable part of the --curve, it also can be used for further plastic approaches with non constant Mpf-level, ace to fig. 6a and fig. 6b.

(7)

For pure bending about the strong axis the available inelastic rotation of plastic hinges (given in radians) can be determined as: 4kfy Tav

with Δ σ =

w3 Í4Ebw3

(fy.Fi+Aa)bh ^

5h:

ff \2 + V WwebhwJ + 4£y.webbtwhA
" ^.webhw

150 N/mm2 (for all cases) and with 1,3

kfv=r^Fj/ 400 + 0,25 (yield stress in N/mm2) (8)

For three point bending with about the weak axis with a load introduction stiffener the available inelastic rotation of plastic hinges (given in radians) can be determined with Mp, and with Las the whole span of the simple beam as: f 0,2_ 2b

,

4

2t

L

(9)

For bending with normal for as about the strong axis the available inelastic rotation of plastic hinges (given in radians) can be determined as given in sentence (5) if the amount of axial load fulfills: Npl

A

with Aweb = A - 2bt. If the axial load is greater (ν / Ν , > Aweb / a) then the following formula applies:

330


t(Nn-Pc)

tM.pl,N

=2

bt^

h

+ AaJh2

i

bt(fyF1+Aa)h

with Mp, N acc. e.g. EC 3, 5.4.8.1 (4) and with Nn as the part of the axial load which is in "the flange: Nn = t

N

1-

2b

N.

pi

\^

Wy.n

JJ

and with the web restraint influence Pc:

ρ _ 4Ebw3 C

(10)

In fig.

9 sample values

5h2

of the theoretical available inelastic rotation


(given in

radiants) are presented varying slendemess and yield stress. (1 1)

In annex (3) of this Background Document all necessary information on all available bending tests being properly documented for rotation capacity evaluation are given. These are three point bending test with bending about both the strong and the weak axis and bending tests with bending about the strong axis with axial forces.

(12)

All

applied in the proposed engineering model: 235 N/mm2 ¿ fy ¿ 460 N/mm2. Indeed the reliability of the model has been demonstrated by test results with steel grades in that range. Eurocode 3 allows steel grades available

in Eurocode

3 Part 1.1 can be

for plastic global analysis with steel materials characterized according to EN 10025 and EN 10113 as given in table 1. (13)

As given in annex (2) Finite Element simulations have shown that the rotation capacity does not strongly depend on centric or excentric location of the load of the equivalent three point bending beam.Consequently rotation capacity values obtained from the symmetrically loaded three-point bending beams are always safesided even for high shear forces (according to EC 3-rules: 0.5 ν^Κ(1 Ypl Rd) the cross-section is rejected by insufficient resistance capacity and then there is no need of rotation capacity values which might be not safe enough.

(14)

Lateral Torsional Buckling phenemonae have to be avoided not only when Mp, - level is reached even during the whole rotational process. The proposed engineering model is applicable if lateral torsional buckling is prevented e.g. according to Eurocode 3 rules (ÃLT s 0,4 )(or CM 66 rules: CLTQ <; 60 with ψ = 0, for instance).

331

(15)

Local instabilities e.g. due to introduction of concentrated loads also have to be avoided during the whole rotational process in the plastic hinges, by introducing at location of plastic hinges: sufficient stiffening of webs for My- and My-N - load cases, sufficient stiffening of flanges for M. - load cases. Sufficient stiffening can be provided with usual welded steel plates but also by other means or by considering of real service conditions in order to ensure economical design: support of secondary beams, support of slabs, masonry, concrete walls, concrete infillments, steel sheet decks (floor, wall, root) with sufficient stiffness. In case of distributed load stiffening of location of plastic hinges should not be necessary.

Thickness t (mm) *)

Nomimal Steel grade

t<40mm EN 10025 standard S 235 S 275 S 355 EN 10113-3 Standard S 275 M S 355 M S 420 M S 460 M

fu

fu/fy

40 mm
[Mpa]

[Mpa]

235 275 355

360 430 510

1.53

1.56 1.44

26 22 22

0.11 0.13

275 355 420 460

390 490 500 530

1.42 1.38

24 22

0.13 0.17

1.19 1.15

19

0.20 0.22

17

0.17

A*5.65

*y [Mpa]

fu

fu/fy

[Mpa]

[Mpa]

215 255 335

340 410 490

1.58 1.61 1.46

24 20 20

0.10 0.12 0.16

255 335 390 430

370 470 500 530

1.45 1.40 1.28 1.23

24 22

0.12 0.16 0.19 0.20

19

17

Notes:

JL

*) / is the nominal thickness of the element - of the flange of rolled sections (t = tf)

Tt

- of the particular elements of welded sections

Table 1:

Steel grades for plastic design ace. EC

332

3

24

Β /

Β Ji

η

20

Β Hup, 16

Γ 4-1-1

-1

R,ex 12 α-5κ

BP ^ ÄSE"/"

[-

8

til

4 ΓΊΙΊ

.ΐ 8

12

20

16

24

R,th β.οο

i ι

7.00

4_

i

«.00

1/

1

j

I

coo

_

1

i

i

H

Α -Ι

Ι' τη j

3Ξ-& n w

I

S.O0

!

i

4.00

j-

>

/

,

ι

;

-î

3.00

«

j

1.00 0.00

i

...j

2.00

7^

0.00

1.00

1

ί

3.00

2.00

4.00

5.00

Ϊ.Ο0

7.00

«OO

S.00

Rth

*

;

,'

120

/

1

:-

:"7

._.

.

1

*

100

._.

/

y,

Γ

80

-H-m

4-^-4 --j-

R

...

ex

j

60 «

[

;

40

/

.' j

ί

«

/ 0

I

/

i

20

1

'.

V\

/*

i

!

1

; ;

ι

20

40

80

60

100

120

140

Rth

Fig. 8: Comparison of the experimental rotation capacities with the theretical rot. capacities

333

Variation of b/t, t=const

Variation of b/t, b=const 0500 τ

τ

ΟίΟΟ

04SO

O4S0 ·

OjKO

ΟΛΟ ·

0353

O3S0

φ av

Φ av 0300 -·

··

OJCO

0253 ·

0.2S0

0200 ··

02»

-i

CIS) &0

&.0

1

10Ό

1-

IM

1

13.3

1

IojO 200

H-

·

-t

aiso

26J

1

1

1

1

l·-

1

- 1-1Äh-19.v*aoitv.24

- &·20.Ν·19.νΜ1ΐ5Λ«24

1-2mH-30.w.l.l0.fr-24

b-3CU>C0.v<*l.iaf»*24

- U3.G0.h-42.w.l ΛΧίν-24

- t>-tah-4aw>i/«ifr-34

Variation of fy, HE 300 Β

Variation of h/w τ

O350

O4S0 ·

OJOO

0.400

O2S0

(X3S0

1

b/I

b/I

CLSOO

1

7.3 IOjO 12.7 15.3 16J0 20.7 23.3 24.0 28.7 31.3

·

0.200

·

(pav

Φ av osco

0.150

0.253 -

0.103

0200

aoso

0.150

omo

·

23SO

33SJ0

-4-

-+-

-+-

-+-

435.0

S3S.0

63SJ0

73SJ0

- 1>3αΝΟΰ.1-1.90Λ«24

t>mnrmAmUÍ-Q.tr-3*

-

b-aah-sai-i.çgw-i.io

- t~zxtmXíu\.ta.tr-2a

Fig. 9: Functions of theoretical rotation capacities, dependend on slendemess and yield stress

334

Safety evaluation

6. (1 )

The verification procedures of Eurocode 3 - Part 1 . 1 take into account semi probabilistic safety approaches in the Ultimate Limit State. So far the reliability is considered by introducing partial safety factors.

(2)

The Annex Ζ of Eurocode 3 - Part 1.1 presents a procedure which allows to evaluate partial safety factors γΜ from experimental results. Thus, from the comparison between the design model values and the experimental values, it is possible to determine a design value of the resistance and a partial safety factor.

(3)

Annex (3) of this background gives an overview of the statistical analysis, performed for all available rotation tests with different sort of loading.

(4)

In general the real yield stresses of the steels delivered by the mills are greater than the nominal values of the steels indicated in the catalogues. The following statistical procedure takes that difference between nominal and expected value of steel grade into account, even considering decreasing of rotation capacity by increasing the yield stress of the steel. That is safesided as another fact is that for many structures the rotation requirement is decreasing by increasing the yield stress whilst the ultimate load target (i.e. the design load level) remains using the nominal yield values [40], see fig 10. Thus with respect to safety the unexpected overstrenghtening is well considered.

.

1

A freq »-

tmnHinuimrrrrTTTTnr -^-

i^reqjiverj Fig. 10 :

(5)

φ

Decreasing of the rotation requirement in case of actual yield stresses which are greater than the nominal yield stresses.

Eurocode 3 Part 1.1 gives a simplified variation of fy in function of the material thickness (thickness ranges: lower than 40 mm, from 40 mm to 100 mm), with values of fy - for certain thickness ranges - greater than the values specified by EN 1 0025 and EN 10113 (thickness ranges : lower than 16 mm, from 16 mm to 40 mm, from 40 mm to 63 mm, from 63 mm to 80 mm, from 80 mm to 100 mm, from 100 mm to 1 50 mm).

335

As Eurocode 3 always provides yield stresses fy equal or greater than values guaranteed by the specifications of delivery conditions (EN 10025, EN 10113, see fig. 11). the available rotation capacities resulting from fy of Eurocode 3 are always safesided in comparison to the rotation capacities resulting from fy of EN 10025 or EN 10113.

yWditiingdifyJtccoollneioTiMtl.l

ffmiufd

(EC3|

yWd ttangOi (R^) according to EN 1002S

/«to

315

f «430

"Ss~

f· MO

ftlfylKltS

250

INckrMu (mm)

yWd ttrangdl fy according to Tabla 3.1

« OA« of EC3 guanriwd yWd annetti (jL^J according lo EN 10 1 !3

(N/mm1)

FiE460 fi

E 420

Ft£355

Fl£27S Sic«! grada

0

Fig. 11:

(6)

If

4«

43

M

100

tfilcVnaat (mm]

Nominal yield stresses ace. to EC EN 10113

3 and yield stresses ace.

to EN 10025 and

However, for the statistical safety evaluation of the rotation capacity in the frame of resistance safety there are several options:

336

1.

The first option is considering the assessment of the rotation capacity as a single-standing Ultimate Limit State check as follows: Φιος <


^ Ύμ<ρ

This check has to be fulfilled seperately in addition to the check of the resistance of the cross-section.

For that the wide spread horizontal distribution, see fig. 12. for the determina¬ tion of the partial safety factor will be relevant That wide spread distribution is due to the in general very small negative slope of the descending branch of moment rotation curve and for that any good prediction formula will produce a more or less large scatter. The comparison is graphically given in annex (3), also the statistical distribution

for three point bending in linear presentation and as well on gaussian paper is shown in annex (3). For this evaluation to which all the data of all kinds of loading and bending are available, the safety elements are shown in table 2.

of the rotation capacity simply as a means of where the resistance has to be determined. It is just that rotation on the Mp, - level at which the section is fully rotationally exploited

The second option would be regarding the prediction

(fig. 6c) in the last expected rotation degree. That means detenriining the rotation capacity with the formulae given in chapter 5, and determining with that rotation the resistance My = f

This resistance function describes the descending branch of the moment rotation curve (which is in reality at the point of
That resistance design model Mp,=Wp,-fy now can be compared with the resistances of the moment-rotation curves at the vertical rotation-capacities" levels. That would be presented by the vertical distribution in fig. 12. The safety elements for that vertical distribution has been determined for pure three point bending and is also given in table 2 for information. Thus within this option a safety factor related to rotations φ will always be γΜφ= 1,0.

As both option 1 and option 2 are safety considerations only at the cross-section they do not take into account the global structural behaviour except for statically determinant structures. However for statically non detenninant structures the 337

interrelationship between option 1 and option 2 is linked with the slope of the descending branch of the 1. to the (n-1). plastic hinge. The magnitude of this slope is indeed an indicator whether the structure fails rapidly or not if the rotation capacity has been fully exploited. Fig. 13 gives an example for that: even if one uses a class 2 -section for the first hinge rotations of the descending branch can be exploited such that the global load is increasing, however having benefit from the strain hardening of the second hinge. Certainly this depends also on the remaining elastic stiffness of the structure before ultimate load has been reached.

This approach has not been yet investigated within this report Thus the model approach ace. to fig. 6a and fig. 6b has to be attended with respect to the global structural response.

338

Op-

tion 1

Statistical Parameters

load cases My

Ym,2

k '

Dmcan

s*

vR

kc

Y Mo

γ mo

1,3

1,229

0,080

0,159

1,239

1,52

1,10*

1,3

1,041

0,070

0,110

1,004

1,00

1,17

-

1,090

0,050

0,210

1,500

2,02

1,10*

-

1,397

0,329

0,360

1,120

1,73

1,10*


My Ym
1

My + Nx

Ύμ,Γ 1

M, Ύμ»-

Table 2:

Fig 12:



Safety elements for rotation capacities (* ace. ENV 1 993-1-1 , or NAD values)

Horizontal scattering of rotation capacities and vertical scattering

of resistances at the point of the rotation capacity

339

Example: 2-Span-Continuous-Beam

γ =1,0 τ

^r

ι

XL

I

»t Plastic Hinge

2nd Plastic Hinge

Equivalent single beam

Constant MpI-LeveI

1 1

it Plastic Hinge

M

...

ς"(21 ι*)

2nd Plastic Hinge

Mr

2-3

-φ

Nonconstant Mp,-Level

2

1st Plastic Hinge

M

23

4

M

bh 2 nd Plastic Hinge

-φ

q/qr

-φ

ΛΟ

"{ι} Ί

ftì

φ ^

Fig 13:

(1

* Χ)<Χ2-

λ

3

Χ

"β, feqF+

ί

- 11)^ 8vf

-2y2~

Example of a two span continuous beam using constant Mp, - level and nonconstant Mp, - level

340

7.

List of symbols

b

flange width

h

cross-section depth

d

= h-2t-2r, depth of hot rolled sections

t, tF w, t^

flange thickness web thickness radius of rolling beam length sectional area

r LL A Α I

area of web

second moment of inertia

Wd

elastic moment of inertia

Wpl

ξ

plastic moment of inertia plastic benefit factor excentricity of load position along the beam span

E

elastic modulus

fy

yield stress of material tensile coupon tensile strength of material tensile coupon POISSON's ratio stress amplification =150 Mpa to consider material hardening

α

fu μ Δσ φ phi φ p,

rotation

-IIelastic limit rotation of the beams' ends reaching plastic moment Mpl (e.g.= Mpl { / (2 E


I) for three point bending beams) elastic limit rotation of the beams' ends reaching Mp, Q elastic limit rotation of the beams' ends reaching Mpl v elastic limit rotation of the beams' ends reaching Mp, N inelastic rotation of a plastic hinge or plastic zone inelastic available rotation of a plastic hinge on Mp, - level in terms of radiant

φm

inelastic rotation requirement in terms of radiant

cprot

elastic and inelastic available rotation of a beam on Mp, - level in terms of radiant

φαρ

-ii-

R^

rotation capacity in terms of ratio of plastic rotation and elastic rotation rotation requirement .- // rotation capacity theoretically derived from formulae rotation capacity experimentally detemuned


φ p, y
φ (pl)

Rjjq R,h

R^

341

M

NH

moment plastic moment resistance ace. EC 3, exeptions are explained plastic moment resistance, interacted ace. EC 3 with axial load ace. EC 3 plastic moment resistance, interacted ace. EC 3 using Qpl plastic moment resistance, interacted ace. EC 3 using VpI axial load full plastic axial load resistance part of axial load which is in the flange, having plastic stress distribution

Q,V

shear force

Qp,

= (h-t) w fy , as approximation for plastic shear force resistance

Vp,

full plastic shear force resistance ace. to EC 3

Pc

part of the resistance in the flange which is due to web restraint while plastic buckling

Mp, Mpi¿,

Mp, q Mp, v

Ν Np,

mechanism forms «

η

nom

".a,"

-li¬

M

C<

-π «

«<

web

indicator for nominal values instead of measured as indicator for values of the flange as

as

indicator for values of the web

342

8.

References

[I]

ENV 1993 -1-1 (Eurocode 3), Design of Steel Structures, Part 1.1: General Rules and Rules for Buildings, 1993 prENV 1994 Part 1-1 (Eurocode 4), Design of composite steel and concrete structures, Part 1.1: General Rules and Rules for Buildings, 1992 DIN 18800, Tl: Stahlbauten, Bemessung und Konstruktion, 1990 DASt 008: Richtlinien zur Anwendung des Traglastverfahrens im Stahlbau, 1973 SIA 161: Verständigung; Grundsätze der Projektierung und Ausführung; Berechnung,

[2] [3] [4] [5]

Bemessung und Nachweise; Bemessung von Bauteilen; Werkstoffe; Herstellung und

Montage; Aufgaben der beteiligten Fachleute, Entwurf 1990

[6]

BS 5950: Part 1: British Standard, Structural use of steelwork in building, Code of practice design in simple and continuous construction: hot rolled sections, 1985

[7]

Sedlacek, G.; Feldmann, M.; Dahl, W.; Kalinowski, B.: Anforderungen bei der

plastischen Bemessung von Stahlkonstruktionen aus hochfesten Stählen, EGKS F6,

Projekt Ρ 2954, 1995

[8]

Sedlacek, G.; Spangemacher, R.; Dahl, W.; Langenberg, P.: Elastisch-Plastisches

Verhalten von Stahlkonstruktionen, Anforderungen und We±stoffkennwerte, EGKS

F6, Projekt Ρ 7210-SA/113, 1992

[9]

Sedlacek, G.; Feldmann, M.; Dahl, W.; Langenberg, P.: Elastisch-Plastisches

Verhalten von Stahlkonstruktionen, Anforderungen und Werkstoffkennwerte, EGKS

F6, Projekt Ρ 7210-SA/118, 1993

[10]

Sedlacek, G.; Spangemacher, R.; Dahl, W.; Langenberg, P.: Untersuchung der

Auswirkungen unterschiedlicher Streckgrenzenverhältnisse auf das Rotationsverhalten

von I-Trägern, Abschlußbericht Projekt Ρ 169, Studiengesellschaft Stahlanwendung, Düsseldorf 1992

[II]

D'Haeyer, R.; Delooz, M.; Defourny, J.: Elastoplastic behaviour of metallic

frameworks - Interaction between strength and ductility; ECSC agreement 7210SA/204; Draft of final Report, 1992

[12]

Sedlacek, G.; Feldmann, M.; Spangemacher, R.; Dahl, W.; Langenberg, P.; Schleich,

J.B.; Chantrain, Ph.; Gerardy, J.-C; Maquoi, R.; Lognard, B.; Defourny, J.; D'Haeyer,

R.: Resümee of the coordinated Project "Elastic Plastic Behaviour of Steel Structures, Requirements and Properties", EGKS-F6 Projects SA 113, SA 204, SA 508, Aachen,

Liege, Luxemburg, 1993

[13]

Lukey, A.F.: Rotation Capacity of Wide-Flange-Beams under moment gradient, Structural Engineering Report No. 10, Department of Civil Engineering, University of Alberta, Edmonton, Canada, 1967

[14]

Lukey, A.F.; Adams; P.F.: Rotation Capacity of Wide-Flange-Beams under moment gradient ASCE Journal of the Structural Division, Vol. 95, No. ST 6, pp. 1173-1188, Paper 6599, 1969

343

[15] [16] [17] [18]

[19] [20]

[21]

[22]

Lay, M.G.: Yielding of uniformly loaded Steel Members, ASCE Journal of the Structural Division, Vol. 91, No. ST 6 pp. 49-66, Paper 4500, 1965 Lay, M.G.; Galambos, T.V.: Inelastic Steel Beams Under Uniform Moment; ASCE Journal of the Structural Division, Vol. 91, No. ST 6, pp. 67-94, Paper 4566, 1965 Lay, M.G.: Flange local Buckling in Wide-Flange Shapes, ASCE Journal of the Structural Division, Vol. 9, No. ST 6 pp. 95-116, Paper 4554, 1965 Kuhlmann, U.: Rotationskapazität von I-Profilen unter Berücksichtigung des plastischen Beulens, Dissertation Bochum 1986 Roik, Kl·, Kuhlmann, U.: Rechnerische Ermittlung der Rotationskapazität biegebean¬ spruchter I-Profile Teil 1, Der Stahlbau (56), Heft 11, 1987 Roik, KL; Kuhlmann, U.: Rechnerische Ermittlung der Rotationskapazität biegebean¬ spruchter I-Profile Teü 2, Der Stahlbau (56), Heft 11, 1987 Roik, K.; Kuhlmann, U.: Rotation Capacity of I-Profiles considering the Effects of Plastic Plate Buckling, Contribution to the International Colloquium "Stability of plate and shell structures", Ghent 1987 Petersen, Ch.: Bericht über Versuche zur Rotationskapazität von Walzprofilen in Fließgelenken für die Fließzonen- und Fließgelenktheorie, 7. StaWbau-Seminar 1988, Fachhochschule Münster, 1988

[23]

Petersen, Ch.: Rotationskapazität

in Fließgelenken als Grenzzustandsgröße, DAST -

Deutscher Ausschuß für Stahlbau, Bericht aus Forschung, Entwicklung und Normung 17

[24] [25]

/ 1990

Bureau, Α.: CTICM Report Essai 92-S-155, 1992 Sedlacek, G.; Spangemacher, R.: Zur Bestimmung der Querschnittsausnutzung und

Rotationsfähigkeit bei I-Profilen mit Biegung um die schwache Achse, Abschlußbe¬

richt ATF-7751, 1991 [26]

Sedlacek, G.; Feldmann, M.; Golembiewski, D.; Berger, K.: Rotationskapazität von

Drei-Punktbiegeträgem unter Normalkraft aus hochfestem Stahl, Abschlußbericht

[27]

[28]

Projekt Ρ 235, Studiengesellschaft Stahlanwendung, Düsseldorf, 1994 Kulak, G.L.; Perlynn, MX: Web Slendemess Limits for Compact Beam-Columns, Structural Engineering Report No. 50, Department of Civil Engineering, University of Alberta, Edmonton Canada, 1974 Spangemacher, R: Zum Rotationsnachweis von Stahlkonstruktionen, die nach dem Traglastverfahren berechnet werden, Dissertation Lehrstuhl für Stahlbau, RWTH Aachen, 1992

[29]

Feldmann, M.: Zur Rotationskapazität von I-Profilen statisch und dynamisch belasteter Träger, Dissertation Lehrstuhl für Stahlbau, RWTH Aachen, 1994

[30] [31]

Roik, K; Lindner, J: Einführung in die Berechnung nach dem Traglastverfahren, Stahlbau-Verlags-GmbH, Köln, 1972 Murray, N.W.: Introduction to the theory of thin-walled structures, Oxford Engineering Science Series, 1984

344

[32] [33]

[34]

[35] [36]

[37] [38] [39]

[40]

Murray, N.W.: Das aufnehmbare Moment in einem zur Richtung der Normalkraft schräg liegendem plastischem Gelenk, Die Bautechnik, 50, 1973 Zhao, X.L.; Hancock, G.J.: Theoretical Analysis of Plastic Moment Capacity of an Inclined Yield Line Under Axial Force, Research Report, No. R648, School of Civil and Mining Engineering, University of Sydney, 1992 Zhao, X.L.; Hancock, GJ.: Experimental Verification of the Theory of Plastic Moment Capacity of an Inclined Yield Line Under Axial Force, Research Report, No. R649, School of Civil and Mining Engineering, University of Sydney, 1992 Bakker, M.C.: Yield Line Analysis of Post-Collapse Behaviour of Thin-Walled Steel Members, Heron, Vol. 35, No. 3, 1990 Dahl, W.; Hesse, W: Auswirkung der Beurteilung von Stählen auf die Anwendung im Hochbau und im Anlagenbau, Stahl und Eisen 106, Heft 12, 1986 Dahl, W.; Hesse, W; Krabiell, A: Zur Verfestigung von Stahl und dessen Einfluß auf die Kennwerte des Zugversuchs, Stahl und Eisen 103, Heft 2, 1983 Haaijer, G.; Thürlimann, B.: On Inelastic Buckling in Steel, Journal of the Engineering Mechanics Division, Paper 1581, 1958 Adams, P.F.; Lay, M.G.; Galambos, T.V.: Experiments on High Strength Steel Members, Welding Research Council, Bulletin No. 1 10, 1965 Feldmann, M.; Sedlacek, G.; Weynand, K.: Safety considerations of Annex J of Eurocode 3, Third International Workshop on Connections in Steel Structures, AISC, ECCS, Trento, 1995

[41]

Maier-Leibnitz, H.: Beitrag zur Frage der tatsächlichenTragfähigkeit einfacher und durchlaufender Balkenträger aus Baustahl St 37 und Holz, Die Bautechnik 6, 1928

345

Annex (1): Experiments on Rotation Capacity (1)

To obtain the rotation capacity of sections with momentgradient the three point bending test may be performed; to obtain the rotation capacity of sections with momentgradient and axial forces the three point bending test with axial load may be performed; to obtain

the rotation capacity of sections with moments but without gradient the four-point bending test may be performed. (2)

The following will focus on the typical three-point-bending tests.

Documentation of bending tests (1)

Bending experiments are time and cost intensive. For that it is necessary to take care about full and precise documentation.

(2)

For three-point bending tests the following values should be recorded and documented. 1.

Cross-sectional parameters

1.1

Beam length and position of load Measured thickness of both flange and web, possibly at several locations along

1

.2

1 .3 1 .4

the beam length Nominal thickness of both flange and web Measured depth and width of the sections, possibly at several locations along the beam length

1 .5

Nominal values of depth and width of the section

1 .6

1.7

Possible excentricities of the section Whole Measured sfress-stram curve of both flange and web

1.8

Norninal steel grade

2. 2.1

Experimental parameter Load-Deflection curve (Ρ-δ, Ρ = applied load

2.2

deflection) Moment-Rotation curve (Μ-φ, M = Peak moment in the beam, φ = total

2.3

2.4 2.5 (3)

of the jack, δ = corresponding

rotation, obtained from rotation of each beam end) Eventually strain measurements of deflection measurements of buckles in the flange and in the web Eventually strain measurements of the tension flange Further observations

In following pages these parameters and measurements of interest are illustrated by Data Sheets for the 3-Point-Bending Test

346

DS1

Datasheets on 3-Point-Bending Tests Test-Setup *"ΉΙ<

Θ

Α - Λ

I

HØ

Γ·*

jI

©

©Θ

"«

Θ

U;

Θ Θ

zr>

_B&z2BL·

ϊ^

k

-l/4-t

-

t*7z.-r-r -Vac*·

-Wr

Z£: ττ - tyntf-yfitw

Cross-Sectional Shape, measured and nominal Beam-No. 2S

Test-No.

Φ

r-|-yvi-h3 b-j,

[mm]

1

2

© tl-rÌHb1,b2

,

t, «2

t, t<

br

h,

πη

h, ha

b,

G= 3 b2 *"-

b2 s

L

347

2Σ

Θ

l^-tøn b-j,b2^

b2,s

4

3

I

J-

DS2

Datasheets on 3-Point-Bending Tests Material Data No. of Stress-Strain-Test

Test-No. Location across the section Geometry of coupon Strain rate of the test

1.

2. 3. 4. 5. 6.

Test temperature

Documentation of the measured and the true stress strain curve (whole curve) Values for the lower yield stress and tensile strength both for flange and web as well as values for fracture strain A5 and strangle Ζ two 700

soo co

o.

soo

«

300

£ Ο C) C

Φ* E

(SO

tn tn

f Ionsch©

100

|Φ 88

»40 Flansch

Γ©

o ®1 Lower Flange

Web -.,4

m.

SI

Φ © Φ

Upper Flange

\

200

Ι

BS«2S Steg


ω

st ε

460

Λ

(00

tn o

220

Zugproben:

--

/

HEB

2

ta i. O)

Ό

Ίη

tNWl

too

TOO

"

«S3

«oo

r

0

20

10

(0

30

Sí -

Strain ε [%]

Location of Coupons HEB uuu

700

(S

o.

600

S

SOO

/

1

/

8S«2S Steg

lUpper Ff lange

Ci i.

φ*

Θ ©I

Π© ι·/-Μ

Web

Lower Flange

f/.l

tn

(00

<

300

ω o 3

φ

ÎS

D tn tn

ιφ @~Ί

220

St E 460 Zugproben: B 8 »40 Flansch

2 200

Sii Flore

g

rt®

30

·

« Ν

(60

1

20

70

(0

Ç}

a

σι

S

i:

κ

Jit True Strain ενί [%]

348

sv

LU.

»?

Datasheets on 3-Point-Bending Tests

DS3

Records of Moment-Rotation

Beam-No.

Test-No. No. of -e curve

E <

α

ε ε ε ε c

O

Q c f

WMMfO) 11 II II

εo i.

It

α.

Ξ

= ο οΓ α>" «Ί 2 ΙΟ τ- τ- CD mm

φ»

ce

α.

|εεε| 0 ι- o ο

Τ"

<Ν _

to'

Ν.

**>

8«! ττ τ¬

II

c2«

Ξ

η ιι

«Ο

~

m ο to τII ιι

δ! si

s

m a c

2

o CN CS

° *=Ι-χ- ι> 3 > co

m ui

χ

Χ

CD

5

*^

0)

«*-

s-

«-»**

φ*

ε«.2**

ζ !.?·£ϊ

X12C "1

·Λ «^1

« «t.

** c-;.

Ο ·η_

«η

<Μ

Sili

ο

ο

/

? «**

ο

ο

Ι

8

t^V S

349

*" (Α

II

II

D Χ

Ι

,.»

y

lm* 73

Η-

DS4

Datasheets on 3-Point-Bending Tests

'

a CM

(S

co CM

CM

o

*->

CD

t

cc

3

E E

CM

*

CO CM

Φ Q. O &. Q.

CM CM

cc

φ

cn

> CO

.5 sz

L.

*

θ-

co

CM

75 mm

v.

f

Φ Η-·

σ

3

O CM

CO

ε

ο

Ό

CD

C

co

CS

"cõ M

υ

E E

#-»

Φ

σ

co

*E Z

N-

«5

ε

o φ Uî

z"

c "c

E E

»»-

o ,φ

E E

Sì cc

z

co

co

Η

*

E E

S5 CM

c

ζ

4-1

V. CC

t-

0)

350

1?

ta

Datasheets on 3-Point-Bending Tests

Observations (Failure modes, buckling shape, etc)

351

DS 5

Annex (2): Finite Element Simulations of Bending Tests (1)

Finite Element Simulations may be a tool to produce artificial results on the rotation behaviour of beams in bending without and with axial forces. Using nonlinear constitutive laws and nonlinear calculation procedures the whole moment-rotation curve including the elastic, the stable plastic and the nonstable plastic part beyond ultimate moment should be produced.

(2)

The Finite Element results should be compared and calibrated with test results.

(3)

Having calibrated and improved the finite element model, parameter studies can be carried out such that combinations of parameters which have not been tested in experiments so far now arteficially can be derived. Thus the population of results can be enlarged and all the phenomenae can be investigated more clearly. A further advantage is that the resulting data files provides stresses, strains, forces and deformations of each point of the specimen.

(4)

As there are different kinds of finite element computer programs in the following recommendations, not to regard as rules, are established for the modelling of the Finite Element mesh and for the calculation procedure.

(5)

Basis for a computer-simulation is the whole measured data of geometry and material.

If a program calculates stresses on the actual deformed section, then the true stress strain curve has to be introduced.

(6)

Not only for that nonlinear elements should be used. These elements should perform large deformations and also large strains with nonlinear geometry consideration and nonlinear material consideration.

(7)

The elements might be shell elements with three degrees of freedom for translations and three degree of freedom for rotations per node or volume elements with three degrees of freedom for translations per node. The shape function may be linear or quadratic, however in case of linear deformation the mesh has be refined very well and very condensed. By the way this also controls the number of nodes per element The number of integration points (gaussian points) should be sufficient such that accurate extrapolation to the boundaries of the element can be provided. In case of shell elements sufficient procedures to calculate the normal and shear stress distribution across the thickness should be provided.

352

(8)

The mesh should be modelled such that in areas of high strain and stress gradient (buckling process-zones) a proper and high refinement is provided with length to width ratio of about 1.0 to 2.0.

(9)

In order to simulate buckling the imperfections have to be considered either as geometrical "out of straightnesses" or producing them by little imperfection loads, perpendicular to the flange. The local equivalent imperfections might have 1/500 1/2000 (as indicative values) of the flange thickness. Also the global out of straightness imperfections (in y-direction) should be considererd with e.g. t/1000 (as indicative value). However all the imperfections should be calibrated to known tests, thus the magnitude of the initial imperfections can differ from the above given values. In cases where small rotations are expected, the residual stresses might be respected.

(10)

The buckling shape in the process zone might be symmetric or antimetrie.

(11)

The calculation procedure i.g. is deformation controlled such that calculating the

dropping part of the curve is possible. (12)

Steering parameter of the solver, the error terms, amount of steps and increments etc.

should also be precisely considered.

(13)

In the following figures an example of finite element modelling of a beam in bending with axial forces.

353

ε [%]·

Fig. Α(2) - 1: True and conventional stress-strain curve aw

= σ (

ew

= In (

1 1

+ +

e

¤

)

)

RWTH Aachen

354

S8R

Ν

Fig. A(2) - 2: 8-node shell element, FE-model, deformed buckling zone

RWTH Aachen

355

ΙίΕβΖδο

Fig. Α(2) - 3: Imperfection-loads, antimetrie (example)

RWTH Aachen

356

o in cv

LEGENDE

X P23501S FEM α versuch Test o

o cu

Ζ

lm

O

tn

\s

4J

C CU

o

10.10 6.00 19.80 31.67 3.00

t

^

s

t)/t h/w

o

Laenge

·

nun nun

m

Hatería] : P23501 Fliesspannung : Flansch· 480.00 N/mm2 >

Steg .05

.00

RWTH Lehrstuhl

Aachen fuer

Stahlbau

.10

.15

Rotation

rad

-

502.00 N/mm2

.20

Comparison: Experimental Test - FEM Simulation HEA 200, N/Na = 0,15, S 460

LEGENDE

X P23502S FEM versuch Test

α o o

CM

i¿

o

t '

C (U

ε

o

s

b/t h/w

o <=

Laenge

.05

.00

R W

T.H

nm ewn

m

Material : P23502 Fliesspannung : Flansch« 480.00 N/nwi2 565.00 N/mm2 Steg

/ Lehrstuhl

«

15.40 7.70 11.69 23.38 3.00

Aachen

fuer

Fig. A(2) - 4:

Stahlbau

.10

.15

Rotation

rad

.20

Comparison: Experimentai Test - FEM Simulation HEB 180, N/N,, = 0,15, S 460

Comparison of M-phi-curves: FEM - Test Example for bending with axial load

RWTH Aachen

357

12,0

HE

lá

9,0

^:,&^:

t R

220

6,0

B

A

-ol=í.,50;St37

· 1=ί«,50,·$Μ} α

3,0

-b135Q;SU7

0,0 2,0

3,0

WI

í=í,,50;St37

U^STEISQ Bl=3t50;St37

U3¿0,STE1¿0

Fig. A(2)-5:

FEM-investigation on rotation capacities dependent on load position of the equivalent single beam span

358

Statistical evaluation on bending tests Abstract This report presents safety considerations which allow for determining partial safety factors for resistance functions in structural engineering. They are based on Annex Ζ of Eurocode 3 [1]. This Annex Ζ of Eurocode 3 represents a standardized evaluation procedure for test results in comparison with strength functions. It simultanously gives criteria for the reliability of the resistance function in (and out of) the range of the experimental parameter-field. This report gives an example for this procedure on the problem of the rotation capacity of I- or H- shaped sections using a prediction formula for the rotation capacity

However this report recognizes the problem whether rotation capacity is an alone-standing Ultimate Limit State or not, i.e. the safety considerations on rotation capacity then should be regarded with respect to the safety considerations on resistance.

The Procedure An Ultimate Limit State Design procedure should in general based upon a statistical distribution of both the action effects S and the resistancy effects R. This statistical distribution is represented by the distribution-function, the mean value m and the standard deviation σ. The safe condition is

R4-Sd>0 fig.

In Eurocode 3

a safety index is defined to guarantee that there is sufficient distance between the loading effect S and the resistance effect R:

see

1.

Λ

ß =

mR - m« s

fi[ + 0s

>

3,8

In this definition of β the resistances and the action effects are combined in In a safe sided approach the resistance can be seperated from the action side: (mR

with

ccR

and

as

+

QfcftrR)

- (ra. + «^

a nonlinear way.

)>0

= -0,8 = 0,7.

Due to this separation the resistancies can be investigated independently if experimental results are available. The resistance functionian be calibrated with experimental results . By comparing the resistance function with the the experimental results the mean value correction^ and the error term Ss can statistically determined, see fig. 2.

359

With this distribution the resistance design can be formulated:

bmRe(*-c^

Rd =

with

aR β

= -0,8 · 3,8 = -3,04

The characteristic value RK can be considered as the 5%-fractile of the log-normal distribution: RK =

bmRe<"1·64^"0^

The variation coefficient cR considers the errorterm S4 and the individual coefficients of variation of the different geometries and strength parameters which have not yet been considered in the test population:

'R-/E

tflCLi

+

SS

The values for the individual coefficients of variation are of = 0,07 for stress and strength, σ, = 0,05 for thicknesses t and ob =0,005 for width and depth. If then Rk and RD are known, the pariai safety factor applies:

Normally the nominal value of the yield stress f}. is used for the design. However that is not the 50% - tractile but it is the 5% or less - fractile of the distribution of the material property. This fact is considered by a correction-factor Ak:

Ak= -5 be<-1*«** " °*%>

The final partial safety factor reads:

ΪΜ

=

ÄkYM

360

S*

Hgure

1

S«,

S.R

Rrf R*

Statistical distribution of the action effect and the resistance

ιi

'λ-

-

yA k

/

mean value correction

S ij-

standard deviation oí error term

Design values (0,8 -G)

rx

Hg.

2

Population of test results 361

Three Point Bending Tests (Strong Axis)

362

Remarks on the three point bending test evaluation (strong axis)

(i) (2)

(3)

Test data obtained from [7][8][9][10][11][14][18][22][23][24]. Some tests (No. 1,2,3) have been excluded. However they provided sufficient great rotation capacity but away from the test population (may be due to high shear effects).

of Aachen and Munich series) have been corrected such as they now are evaluated on real M^ - level of the --curve. That means for cases where Some tests (some

Mp], calculated from the yield stress is obviously lower and thus not in accordance

with that resistance which the moment rotation curve shows, as for example: ifi (real)

That effect (4)

can be due to dynamic effects

of the testing jack speed.

The Mp[ - value has been determined with Mpl = btfy>F(h-t) +

(h-t)2/4wfyiW

the error to the EC3-value is very small. Both the welded and the rolled specimen were calculated with that formula.

(5)

Test beams with lateral torsional buckling (LTB) during the plastic rotation do sometimes produce low rotation capacity results. These tests have been deleted from

the test population, and they are not in the chart anymore.

363

Distribution for "horizontal" evaluation (see chapter 5):

33.33-

99.90-

99.B099.70 -

99.50 -j 99.00

-j

98.00 -j 97.00

'

/'

!

00.00

|

3-.00

,

.

!

90.00 -J

!

Ι

ί

1

80.00-

1

70.00 -

Ί

/

I

1

/

1

/

1

30.00-

Ι¬

1

ιι

20.00-

'

16.00 -

ι

II

4.00 3.00

-

2.00

-

1.00 0.50

-

0.30 0.20

-

0.10

-

0.05

- Η

iΡ

i

ί

y /

/

/

ι

t

(_/

I

/

f s

h 0.6

0,7

03 0,9

1JD

tl

13

12

fai

364

^4

15

1j6

17

1,8

Λβ

2,0

Distribution for "vertical" evaluation (see chapter 5)

99.9S

---

1

!

99.90

/

1

99.80 99.70 99.60 99.50

.

t

1

1

1

-4-

1

99.00

98.00

ι

!

1

:

i

I

Ì

1

97.00 96.00 9S.00

/

j

1

90.00

.

I ¡

84.00

i

80.00

I

j I

ι

1

70.00

/

:

|

60.00 -t

/ / /

I

ι

50.00

40.00

ι

30.00

i

20.00

16.00 10.00

I

1

I

/j f

1/ i /

1

5.00 4.00 3.00

/

/

1 1

,

1

EZ

1

2.00 1.00

0.50 0.40 0.30 0.20

-·~

0.10 0.05

I

0.7

0,8

0° 10

V

12

365

13

No

Project

facta

ΕΛ28312 EA2833

I 1200 1200 1200 3500 3S00 3500 3500 1200 3000 000 1200 3000 4000 3000 4000 1200 3000 4000 1200 3000

EA2834

4000

EA28412 EA2843 ΕΛ2844 EB2843F3 EA2234b EA2244b

1200 3000 4000 3000 4000

ETB3S33 ETB3S43 ETO3563 ETB3SS3 ETB3543

3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000

Designation

P169


2

facta P169

416b3m

1

3

facta Pir»

d19*4m

4

facta PI 69

S

Anchen P169

6

tonta PIM facta PI 69

d01(4m 402b4at Ò09*3m d10b3rn

7 a

facta »11 3 facta «113 facta »113 facta »113 facta »113

s to ιι 12 13 14 15

facta»! 13 facta «113

EA22312

EA2233 EA2234 EA22412

EA2243

facta SM 13

EA2244 EB2833 EB2834

If

facta

EB28412

17 IS 1$

facta »113 facta »113 facta SMU facta »1U facta »113 facta »113 facta »113 facta »113 facta »1)3 facta »113 facta »113 facta »113 facta »113

20 21 22 23 24

25 2S 27 28

29 30

.

SM υ

facta »113

31

fach./üege»118

32

fcctv./lige»11ä

33 34 35

fach/liege SU 18 fachjtitge »118 fachjljege »116

EB2S43 EB2844

EIB2883 ETB2863 EIB2S43 ETB2S33 04<%M Ν·2

36

fadi/ljegeSma

37

fach/Ueqe»1U

3a

liege CRH »204

39

liege

CSI»2W

Η·3

40 41 42

liegt CRܻ204

Ν·4 Ν·5 Η·6 Ν·7

43 44

4S 46

47 48

CBI»204 CMI»204 liege CBI»2M liege liege

liege CRU»204 liege 091» 204

»204 liege CRU »204 Uge Oui »204 liege CSU

49

urge Obi »204

SO

liege OSI »204

SI S2

liege CRII »204

S3

54 55 SB 57

»204 lige CSU »204 liege CRU »204 lige C»i »204 liege

CRU

liege CRU»204 liege CSU »204

53

liege CRU»204

59 60 61 62

UgeCKU»204

Ν·11 Ν·12 Ν·13 NMS ΝΜ7 Η·18 ΝΜ9

Ν·20 Ν·21 Ν·22 Ν·23 «24 «2S

67

wichen

68

lunchen Petersen

69 70

lunchen rel etsen

ΗΕ400Α

lunchen Petersen

71

lunchen Petasen

HE 300 Α HE 340 Α

72 73 74 75 75 77 78 79

lunchen Petenen Bochum Km/fatàmoMi

ΗΕ400Α

Bochum KM/Kuhfcnonn

2

Bochum Roit/Kuhlmonn

4

Bochum RoiyKuhfcnonn

S

Bcchum nM/Kuhknoftn Bcchum RA/Kunknonn

6 7

63

Petersen

2000 2000 2000 2000 2000 2800 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3800 3800 3800 3800 3800 3800 3800

Ν·9 Ν·10

64 65 66

CSU

3000

Ν·8

»204 liege CRU »204 liege CSU » 204 liege CRU »204 liege CSU »204 liege CSU »204 liege OBI » 204 lege

4000

Ν·26 Ν·27

Ν·28 Ν·29 Ν·30 Ν·31 Ν·32

2000 5000

HE 300 Α

ΗΕ340Α

5000 5000 sooo 5000 5000 3404 3704

1

oochufn ηΜφΚυπηιοηη

S

SO

Bochum Kov/Kuninonn

9

81 82 83

oochufn ncik/iCunhTsnn

10

Bcchum RrA/KiMmcm

11

Bcchum Roft/Xuhlmonn

84

Bcchum Ro^KunÉnonn

as

Bochum Koc/Kuhfenonn

12 13 1«

86

Bochum RrA/KuMmm

IS

87 88 89

Bochum

SO

Bochum

2540 2636 2716 1796 2196 2598 2802 3002 3400 »

3000 3200

RoVfaNrrm

16

Bochurn HotyruMtncra

17

3S08 2304 2204

Bochum RotyKuNrran

IS 19

2100 2000

/Kiraiuii

t>

220.50

220.60 219.20 218.80 218.60 219.00 218.40 220.S0 220.00 221.00 22S.S0 225.50 222.00 279.30 279.00

281.30 283.30 284.00 280.00 280.00 280.00 280.50 281.00 281.00 283.00 225.50 225.50 300.50

30030 300.50 300.00 300.00 280.00 280.00 280.00 280.00 202.00 201.50 200.40 200.20 200.50 201.50 199.90 200.40 200.60 200.40 200.30 199.80 200.10 199.90 200.20 200.30 200.30 200.30 200.00 200.30 200.40 200.20 200.40 200.20 201.20 201.10 200.80 200.20 199.90 200.30 300.00 300.00 300.00 299.00 299.00 301.00 141.00 150.00 160.00 160.00 160.00 160.00 160.00 160.00 170.00 182.00 190.00 141.00 150.00 160.00 160.00 160.00 161.00 160.00

bnom 220.00 220.00 220.00 220.00 220.00 220.00 220.00 220.00 220.00 220.00 220.00 220.00 220.00 280.00 280.00 280.00 280.00 280.00 280.00 280.00 280.00 280.00 280.00 280.00 280X0 220.00 220.00 300.00 300.00 300.00 300.00 300.00 280.00 280.00 280.00 280.00 200.00 200X0 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 300.00 300.00 300.00 300.00 300.00 300.00 140.00 150.00 160.00 160.00 160.00 160.00 160.00 160.00 170.00 180.00 190.00 140.00 150.00 160.00 160.00 160.00 160.00 160.00

t

tnorn

1S.50 15.70 16.00 16.30 16.20

16.00 16.00 16.00 16.00 16.00 16.00 16.00 10.50 10.50 10.50 10.50 10.50 10.50 18.00 18.00 18.00 18.00 18.00 13.00 13.00 13.00 13.00 13.00 13.00 18.00 1030 10.50 15.00 15.00 15.00 15.00 15.00 18.00 18.00

16 JO

16.10 10.50 10.50 11.00 11.00 11.00 10.70 17.80 17.70 17.40 17.40 17.40 12.70 12.60 12.80 12.60 12.60 12.70 17.30 10.30 11.10 15.00

15.00 15.00 15.00 15.00 18.00 18.00 18.00 '18.00 18.00 18.00 14.00 14.00 15.10 15.00 14.60 15.00 14.70 15.00 14.60 15.00 15.10 15.00 14.90 15.00 14.70 15.00 14.60 15.00 14.50 15.00 14.70 15.00 14.50 15.00 15.30 15.00 14.80 15.00 14.80 15.00 14.60 15.00 14.60 15.00 14.80 15.00 14.80 15.00 14.50 15.00 14.80 15.00 14.70 15.00 14.80 15.00 14.80 15.00 15.00 15.00 15.10 15.00 14.50 15.00 15.10 15.00 14.30 15.00 14.50 15.00 13.30 13.50 15.80 16.00 18.30 18 JO 13.70 13.50 15.60 16.00 19.40 18.50 8.00 8.00 3.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 3.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 10.20 10.00 10.00 10.00 10.40 10.00 10.20 10.00 10.00 10.00 10.00 10.00 10.00 10.00

366

h 220.00 219.50 220.40 219.10 217.30 218.70 217.40 208.30 209.00 211.00 215.50 210.30 214.00 276.80 282.00 283.00 281.20 284.50 255.20 266.00 269.00 276.10 275.60 275.00 281.50 206.00 208.30 350.00 350.00 350.00 350.00 350.00 280.00 280.00 280.00 280.00 200.00 199.70 200.40 198.00 197.60 199.70 204.30 198.40 200.50 19S.20

198.50 203.10 19S.80 204.50 198.60 197.50 197.50 198.90 204.50 197.60 198.60 197.80 198.20 198.20 138.80 199.10 197.90 203.60 201.80 197.40 292.00 330.00 390.00 290.00 328.00 389.00 294.00 294.00 277.00 274.00 275.00 296.00 296.00 295.00 295.00 294.00 294.00 260.00 259.00 258.00 169.00 220.00 289.00 298.00

hnewn

w

220.00

9.80 10.00 9.40 9.80 9.40 9.60 9.40 7.50 7.50 7.40 7.50 7.50

220.00

220.00 220.00 220.00 220.00 220.00

210.00

210.00 210.00 210.00 210.00 210.00

280.00 280.00 280.00 280.00 280.00 270.00 270.00 270.00 270.00 270.00 270.00

7.50 10.90 10.80 11.50 11.40

1130 7.80 8.00

7.50 8.80 9.00

wneffl

9.50

hrt

h/w

Lfb

14.23

19.29 18.81

2.72 2.72 2.74 8.00 8.01 739

9.50

14.05

9.50

13.70 13.42 13.49 1144 13.57 21.00 20.95

9.50 9.50 9.50 9.50 7.50

7.50 7.S0

7.50 730 7.S0 10.50 10.50 10.50 10.50 10.50

8.50 8.50 8.50

8.50 8.50 8.S0 10.50

20X9 2030 2030 20.7S 15.69 15.76 16.17 16.28

16.32 22X5 2222 21.88 2226 22.30 22.13 16.36 21.89 2032 20.03 20.03 20.03 20.00 20.00 15.56 1536 1S36 15.56 14.43

210.00 210.00

9JO 11.35 7.25 7.65

350.00

10.00

10.00

350.00

10.00 10.00 10.00 10.00 10.00 10X0 10.00 10.00 9.00 9.50 9.60 9.S0 8.80 9.45 9.40 9.S0 9.50 9.00 9.10 9.60 10.00 9.40 9.20 8.80 8.80 9.15 9.40 8.90 9.20 9.10 9.05 9.00 9.30 9.20 9.00 9.40 9.10 8.70 9.40 10.30 11.90 9.20 10.40 12.10 5.00 5.00 6.00

10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00

s.oo

5.00

4.00 5.00

4.00

20.00 20.00 20.00

5.00

20.00

S.OO

5.00

S.OO

5.00 5.00

20.00 20.00 21.25 22.75 23.75

280.00

3S0.O0

350.00 350.00

280.00 280.00 280.00 280.00 200.00 200.00

200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00

200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00

200.00 200.00 200.00 200.00 200.00

200.00 200.00 290.00

330.00 390.00 290.00

330.00 390.00 295.00 295.00

275.00 275.00 275.00 295.00

295.00 295.00 295.00 295.00 295.00 260.00 260.00 260.00 270.00 220.00

5.00 530 s.so s.so s.so S30

730 730

13.34

13.73 13.62

13.73 13.34

13.42 13.63 13.74

13.82 13.63 13.78 13.08 13.51

1333 13.72 13.72 1353 1331 13.81 13.S4

13.62 13.S4

1353 13.41

10.00 1130

13.32 13.85 13.26 13.98 13.81 22.56 18.99 16.39

9.00

21.82

10.00 11.50 5.00

19.17 1S32 17.63 18.75

9.00

5.00 6.00

S.SO

530

5.50 5.50 S.SO

s.so

S.50

530

290.00

5.Ï0 5.50

300.00

6.00

6.00

S.SO

13.82 15.00 1S.38 1S.69

16.00 16.10 16.00

20.04 19.03 19.67 19.39 19.70 24.97 25.07 2534 25.80 25.11 25.6S 22.13

22.83

2138 21.61 21.71

29.46 30.10 32.45 2831 27.82 26.84 21 .7S. 2537 2433 32.00 32.00 32.00 32.00 32.00 24.40 24.40 24.40 24.40

8.01

2.72 6.82 9.05 2.66 6.65 9.01 S.37 7.17 2.13 5.29 7.04 2.14 5.36 7.14 2.14

534

7.12 530 8.87 8.87

4.99 439 4.99 5.00

5.00 5.36 SJ6 S.36 5.36

19.11 17.84 17.83 17.75 19.14 17.94 1856 17.79 18.03 18.80 18.58 18.14 16.52 18.61 18.37 19.13 19.13 18.50 18.61 18.94 18.37 18.51 18.63 18.73 18.1S 18.36 18.77 18.4S 19.03 19.36 28.23

7.43 7.44

2837

8.33 833 8.36 836 831

29.70 2834

2834 28.94 55.60

55.60 4330 51.60 64.7S

56.00 56.00 55.80 55.80 50.55 50.55 43.56 43.45 43.13 27.02 3636 48.91 46.33

7.49 7.49 4.99 4.96 S.0O

4.99 4.99 639 7.49 7.51

730 730 7.49

7.49 7.49 7.49 730 7.49 7.49 7.49 9.48 9.49

9.44 9.45

9.46 9.49

950 4.99 8.33

12.07

12.35 734

8.24 8.49 5.61

6.86 8.12 8.24 8-2S

35 10.64

10.67 1036 7-20 6.89 632 6.25

No

A

1

88.40

2 3

89.65 89.36

4

SUO

S 6 7 8

89.73

9 10 11 12

13 14 15 16 17 18 19

20 21

22 23 24 ~2S

26 27 28 29 30 31

32

33 34

35 36 37

90.82

89.25 61.14 61.09 63.42 64.95 6436 62.76 127.66 127.31 128.44 128.66 12935 90.04 90.83 90.90 93.87 94.48 95.77 12730 60.64 65.1 S

123.65 123.65 123.65 122.00 122.00 125JO 125.20 12S30 125 JO 73.30

38

74.39

39 40

76.35 76.27 74.6S 78.30 77.37

41

42 43 44

45 46 47

48 49 SO

51 52 S3 54 SS 56 57 58 59 SO

61 62

63 64 65 66

67

68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89 90

76.37 76.24 74.6S 75.61 76.05

79.28 77.00 76.17 74.58 7438 76.13 77.03 74.38 76.23 7532 75.92 75.77 77.45 77.66 74.74 78.18 74.23 74.00 106.00 127.16 154.03 107.35

I 7858.68 7912.18 801034 8030.95 7813.32 799030 7778.36 5017.12 5044.04 5360.24 5726.27 5426.10 S438.56 18279.18 18935.54 190*4.41

18920.91 19478.38 1139234

12421.00 12823.23 13620.73 13618.72 13684.76 18855.78 4904.63 5360.93 2844236 28442.56 2844236 27998.17 27998.17

18536.07 18536.07 18536.07 18536.07 5383.73 5693.82 5573.76 5442.14 5361.44 5691.19 588534 5471.91 5579.73 5378.02

fy.n»

fy.flanom

27430 27430

235.00 235.00

525.20

455.00 455.00 4SS.00 235.00

486.20

486.20 27850

278.50 28230 28230 282.50 420.50 42030 42030 24830 24830 489.00 489.00 489.00 27630 276.50

27630 604.00 504.00 504.00 489.00 275.00 43030 248.00 486.00

817.00 S60.50

490.00 98130 864.50 468.00 277.50 25730

445.00 261.00 375.00 303.00

445.00 409.00 375.00 261.00

303.00

5454.94

375.00

5699.32 S489.28 5868.83 5491.66 5350.45

261.00

S350.45 S51036 S871.49 5333.96 S496.67 S409.30 5464.07 5456.51 5590.30

375.00 409.00 375.00 303.00 303.00 375.00 409.00 303.00 375.00

375.00 375.00

fr.wcb 34830 348.50 541.30 S31.70 531.70 286.10 286.10 308.00 308.00 308.00 43730 43730

fy.w*bnom

43730

41 S.OO

135

235.00 235.00 455.00 455.00

1.55 1.55

455.00 455.00 235.00 235.00 235.00 455.00 455.00

25230 252.50 539.00 539.00 539.00 31130 31130 31130 S35.00 535.00

455.00

53S.O0

455.00 235.00 4SS.00 235.00 455X0 685.00 88S.00 455.00

539.00 302.00 44830 323.00

235.00 235.00 235.00 235.00 415.00 415.00 415.00

235.00 235.00 455.00

685.00 455.00 235.00 235.00 455.00 235.00 355.00

421X0

295.00 455.00

342.00 462.00 426.00

415.00

355.00 235.00 295.00 355.00 235.00 355.00 415.00

355.00 295.00 295.00

355.00 415.00 29S.00

355.00 355.00 355.00 355.00

536932 5906.89 5534.45

375.00 445.00 445.00 303.00 409.00 261.00

5311.64

303.00

17203.40

26079.16 43048.45 17265.88

245.00 217.00 255.00 336X0

12S.78

25422X5

418.00

16131 36.46

4501131 5509.70 S8043S 5521.45 524S31 514239 6224.45 6224.45 6177.88

375.00 236.00

355.00

236.00 287.00

235.00

3730 41.26

38.50 35.96 39.60 39.60 3935 41 .15 44.41

45.69 4134 43.15 46.33 40.81 43.00 47.00 48.68

S629.S1

6507.44

6941.03 7202.84 5120.12 5278.29 571S.32 221037 3897.33 7161.03 7712.43

287.00

287.00 287.00 287.00 287.00 236.00

236.00 236.00

455.00

455.00

295.00 415.00 235.00 295.00 235.00 235.00 235.00 355.00 355.00 235.00

235.00 235.00 235.00 235.00

235.00 235.00 23S.00 235.00

333.00

235.00 235.00

333.00

235.00

333.00

235.00

333X0 333.00

235.00 235.00

333.00 333.00

235.00

235.00

455.00

455.00 235.00 235.00 235.00 235.00 235.00 415.00 415.00

455.00 235.00 235.00 235.00 455.00 455.00 455.00 455.00 235X0 455.00

990.00

813.00 969.00 S3630 984.00 813.50 535.60 323.00 268.00 462X0 291.00

885.00

235.00 235.00 455.00

291.00 342.00

40533 396.78

1.50

151.97

150 1.50 1.25 135

15239 160.14 247.61 240.46 236.97 366.13 37136 745.02 742.79

1.40 1.40 1.40 131 1.51 1.51 1.35

135 135 1.40 131 133

Mpl.Q/Mpl

CW

O/Qpl

0.88 0.88

0.94 0.93 1.19 0.38 039

321.0 312.6 263.8 264.7 263.2

0.41 0.41 0.96

75539 274.15 287.19 292.23 551.09

202.60 327.69 402.12 394.37 228.40 223.49 132.83 151.06 156.65 203.77 239.72 231.95 364.53 371.43 460.70 742.42 751.15 172.57 287.09 291.31 351.00

403.2 410.1 600.5 610.1 S80.3

552.58

SS2.S7

55736

551.56 738.94

22935

22437

73938 14536 245.85 465.10 1011.62 1461.78 1719.82

142.44

24137

885.00

1.06

455.00 885.00 685.00

139 1.06

455.00 235.00

131 133

235.00

158

455.00 355.00 295.00

139 137 1.40 139

455.00

139

415.00 355.00

455.00 295.00 415.00 235.00 295.00

1.16 1.40 1.57 1.39 1.40 157 1.40 1.16 1.40 1.39 139 1.40 1.16 139 1.40 1.40 1.40 1.40 139 139 139 1.16 137 139

23S.O0

1.47

317.16

185.01 312.72

235.00 235.00 355.00 355.00

158 1.54 1.50

378.34

374.80

61933

355.00

1.42 1.55 1.55 1.65 1.65 1.65 1.65 1.65 1.65 1.55

625.17 439.28 715.16 964.30 98.32 103.18 127.04 120.01 116.62 13133 13133 131.38 114.46 122.68 127.00 180.47 184.82 196.96 110.88 154.88 225.47 196.87

421.00 291.00 421.00 426.00 421.00 342.00

415.00 355.00 29S.O0

342X0 421.00 426.00 342.00 421.00

295.00 355.00 415.00 295.00 355.00 355.00 3SS.00 355.00

421.00 421.00 421.00 462.00 462.00 342.00 426.00 291.00 342.00 245.00 217.00 255.00 336.00 418.00 375.00 217.00 217.00 260.00 252.00 260.00 252.00

139 139 130 130

Mpl.Q 199.84

1.49

235.00 295.00 355.00 235.00 355.00

'

Mpl 22736 229.94 429.65

236.00 885.00 685.00

235.00

421.00

fu/fy.fla 1.45 1.45 1.30

455.00

235.00 235.00 235.00 235.00 235.00 235.00

252X0

235.00

252.00 217.00 217.00 217.00 709.00 709.00 709.00 709.00 709.00 709.00 349.00

235.00 23S.00

235.00 235.00 235.00 235.00 235.00 235.00

235.00 685.00

235.00

367

130 1.05

1.04

135

155 1.55 1.41 1.41 1.41 1.41 1.41 1.41 1.41

465.10

98736 1448.69 170634

88930 1464.92 1281.16

886.14 1448.45 1260.12

70930 421.86 156.31 28734 166.00 235.89 18731 287.14 266.64 236.68 165.99 187.71 23S30 167.45

70730 420.60

24132 26537 236.93 18733

18733 237.30 265.68 186.64 237.14 234.18 236.02 235.64

283.13 284.61 187.68 268.64 163.17 185.84

1SS.51

286.11

1643S 233.67 186.70 285.10 264.94 236.22 16S.65 186.88 233.61 165.69 239.09 26433 235.18 186.08 186.08 235.61 264.35 185.33 235.40 232.50 23139 231.00 278.78 280.54

183.67 264.25 1S9.43

434.41 707.41

957.90 9S.S2 100X1 126.18 119.93

11633 127.70 131.36 1313S 11437 12137 12S.S7 160.55 163.93 175.65 105.65

14731 217.87 196.12

0.76 0.99 0.99 1.00 1.00 0.87 0.99 038 0.82 1.00 0.98 1.00 1.00 0.62 1.00 0.99 0.63 1.00 1.00 0.64 1.00 039 1.00 038 0.98 1.00 0.98 0.99 039 1.00 0.99 038 1.00 1.00 0.99 1.00 039 0.99 1.00 039 0.99 1.00 1.00 1.00 0.99 0.99 0.99 0.99 0.99 0.99

0.99 039 039 039 0.99 039 038

387.4

377.6 385.1 411.6 416.1

038 0.30 1.07 0.42 0.31

059 0.45

9505

1.31

9353

0.53 0.40 1.34

955.9 340.2 364.6 345.6 716.2 731.1

7S33 933.2 247.4 390.6 624.7 1914.8 1572.4 18743 1037.7 14883

033 0.42 138 0.50 037 053

0.30 031 0.50

035

0.62 0.61 037 0.66

12305 8103 488.6 259.0

0.69

467.8 299.7

0.41

423.3 318.0 465.3 437.9 424.2

296.7 326.5 4063 304.2 438.7 438.6 411.0 317.8 317.8

058 0.58 0.40 0.37 0.37 0.59 0.62 0.61 0.56 0.56 0.41

0.39 0.37 037 0.40

038 0.39 0.39

409.4

039

438.6 321.8 411.0 405.0 403.4

0.40 039 038 0.39 031

038

401.2

031

038 0.99 0.98 0.98 0.98 1.00

4SS.9

0.33

4513

0.33

325.9 435.8 286.7

0.30 032 0.30 0.59

0.99

370.6 40S.S

0.99 0.99 0.99 0.99 0.99 0.97 0.97 0.99 1.00 1.00 0.97 1.00 1.00 1.00 0.99 0.99 0.89 0.89 0-89 0.95 0.9S 0.97 1.00

314.2

0.34

1793

0.37 0.38 0.36 0.36 0.40 0.32 0.31

242.3

0.41

193.5

0.47

160.3

034

2093 209.5

0.70

208.8 179.8

0.48 0.45

197.1

0.41

197.1

0.38 0.21 031 030 0.27

651.2 493.1 784.1

9683 179.2

562.4 560.6 SS7.4

357.5 472.8 628.1 348.2

057

0.30 0.34 0.57

ut** 0.0083 0.0083

0.01S3 0.0420

8 9 10 11 12

13 14

IS 16 17 18 13

20

0.0423 0.0239 0.0240 0.0087 0.0216 0.0285 0.0124 0.0317 0.0415 0.0143 0.0187

0.0112 0.0280

0.0370 0.0069 0.0165

21 22

0.0217

23

0.0290 0.0388 0.0280 0.0283

24 25 26 27

0.0116

0.0437

28 29 30 31 32 33 34

0.0117

35

0.0274

36 37 3β 39

0.0163

40 41

42 43

0.0254

0.0367 0.0439

0.0227 0.0565 0.0494

0.0207 0.0360 0.0213 0.0310 0.0167 0.0240

51

0.0308

52

0.0250

S3 54

0.0250

62 63 64 6S 66

0.0308 0.0323 0.0250 0.0308

0.0309 0.0391 0.0391 0.04S8 0.0457 0.0316 0.0411

0.0267 0.0167

67

0.0219

4»

0.0173 0.0173 0.0303 0.0335 0.0255 0.0145 0.01S7 0.0139 0.0144 0.0147 0.0091 0.0111 0.0132

69

70 77 72 73 74

75 76 77 78 79 80 81 82 83 84 8S 86 87

90

0X406

3.1

0.0142 0.0187 0X069 0.0280 0.0367 0.0043 0.0165

23.4 17.0

0.0117

0.0126 0.0143 0.0252 0.0267

0.0288 0.0275 0.0209 0.0157 0.0122

203 S.O

3.8 43.1

103 7.7

0.0216 0.0074

11.4

0.0290 0.0384

3.0 2.4

0.0280

43

0X277 0.0429 0.0117 0.0248 0X364 0.043S 0.0226 0.0SS8

8.1 3.1

16.7 3.4 0.9

15.5 30.3 10.5 9.3 12.3 23 23 31.5 20.5 14.5 8.6 7.9 43.5 1S.0 8.9 10.3 5.7 4.0 10.1 93 2.9 14.1

0.7

3.4 1.0

13

4.7 16.0 19.9

5.0 20.0

6.2

9.8 23.4

213 9.4 20.4

193

lil

0.1762 0.1676 03651 0.3553 0.2266 03253

03219 0.1273 0.1325 0.1273 0.3333 03174

0.1337 0.1394 0.1382 0.1866 0.1800 0.1664 O.OS36

0.0861 0.0903 0.1379 03251 0.1344 0.1947 0.0833 0.0344

0.0263

0.0769 0.0398 0.0487 0.1289 0.2598 0.4104 0.2216 0.4609 0.2876

28.4 17.7 17.7

0.3381

13.9 32.0 1S.0 8.7 21.6 105

23.3 40.4 153

03864

73

9.5 11.8 11.8

93

103

8.9 13.6 13.6 8.8 7.3 13.8 8.9 8.7 6.9 6.9 4.8 4.7 11.3 S3 15.8 20.0 10.1 14.7 12.1

4.9 3.4

S3 7.7 7.0

73 6.2

S3 9.4

73 63 8.7 7.7

0.0224

5.1

0.0237 0.0257 0.0262 0.0199 0.0152 0.0121

4.8 4.5

73 63 6.5 73

368

6.4

7.7

1S3

23 0.9 0.6 3.3 1.5

0.6

0.0486 0.0273 0.0162 0.0206

0.0311 0.0322 O.0306 0.0248 0.0248 0.0305 0.0322 0.0248 0.0306 0.0307 0.0383 0.0383 0.0451 0.0451 0.0309 0.0405 0.0261 0.0166 0.0216 0.0171 0.0171 0.0300 0X331 0.0253 0.0141 0.01S2 0.0138 0.0143 0.0147 0.0088 0.0110 0.0132 0.0117 0.0126 0.0141

0.0323

60 61

0.0316

0X208

0.0233

SO

56 57 58 59

0.0214 0X278 0X102

0.0314

0.0142

4.0 153 14.8 30.0 10.5 8.0 12.5 43

0.0076

0.0210

0.0206

45 46 47 48 49

4.2

0.0421 0.0238 0.0239

0.0308

44

55

0.0073 0.0073 0.0117 0.0417

0.0359 0.0211 0.0307 0.0166 0.0239 0.0214 0.0206 0.0141 0.0232 0.0306

0.0216

pN.av.th

pujío

11.4

22.0 13.4

13.1

9.7 8.9 14.6 133 12.5 8.8 8.7 63 6.9

8.7 5.8 1S.6

285 8.0 13.9 12.4 3.6 4.9 6.1 8.0

7.0 12.7 8.7 4.6 13.7 113 7.8

8.9 7.6 S.l 3.8 3.6 10.5 95 6.6 12.1

0.2197 0.2345 0.4513

0.3479 0.2663 0.4478 0.3265 0.2335 0.2716 0.3385 0.3385 0.2684

0.233S 0.3436 03715 03681 0.2654

0.2630

phi. av. *x

0.2667

03253 03692

03711 03290 03246 03588 0.1251 0.0694 0.0894 0.4487

03829 0.1000 03410 03901 αϊ 883 03476 0.1926 0.0758 0.1652 0.1 S3S

03838 03628 0.1243 0.1647 0.0570

0.0326 0.0261 0.0746 0.0837 0.0728 0.1363 0.3242 0.3961 0.3524

0.4918 03715 0.4713 0.4229 0.3790 0.4785 0.5707 0.3522 03487 0.4562 0.4163 03061 0.3615

0.2922 0.3250 0.2971 03862 03634

0.4032 03853 0.3360 0.3342

0.2153 0.211S 0.3489 0.2372 0.4113

0.2818

03317

0.4732

0.2186 03517

0.1731 0.2378 0.2124

03073

0.3107 0.2685 0.2359

0.4058

0.1475 0.1120 0.1337

0.1078

0.1079

0.1124

0.1066 0.1009 0.0896 0.0799 0.0825 0.0825 0.0828

0.1064 0.1755

0.1089 0.1080 0.1150 0.1130 0.1159

0.1118

0.1974

0.2752 0.1891

0.1366 0.0989 0.0909

0.1623 0.1S45

0.1244 0.0674

0.1198

0.1269 0.1026

0.1073

0.1142 0.0899 0.0924

0.1004

0.1463

No 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109

Project

Bochum Rot/Kuhhrom Bochum RrA/KuNmom

Bcchum Rct/fcrtnom Bochum KcVKuNrnoMi

Designation 20 21 22 23

I 2402 2804

2406 2500

24 Al A2

2700 3480 2946

U» luke//A
SI B2 83

U»liier/Adoms

84

U»Ufar/AiJcm= U»Ute>/Ar!oms U»Uitar/Adoms

SS

CI

1554 1036 1254 1396 1448 1372

U»U*er/faorre U»b*ef/Moms U»tuker/Momi

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ClCU-Uetz

155

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158 160 161 162

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110 111 112 113 114 115 116 117 118 }19 120

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121 122 123 124

facta »122 facta »122 facta »122

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166 170 171 172 176 177 180 EVHA3A EVHA4A EVH83A EVH84A

960 1168 1296 1240 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

b 160.00 160.00 170.00 183.00 190.00 203.45 176.02 103.00 73.91 86.11 93.98 96.77 101.85 73.66 85.8S 93.47 8932 20135 200.45 200.55 200.10 201.65 200.40 200.60 199.80 20030 201.65 201.05 200.60 200.85 224.00 224.00 280.00 279.00

bftoin

t

160.00 160.00 170.00

10.00 10.00 10.00 1030 10.20 10.80 10.80 538 538 5.28 5.28 S.28 536 536 536 536 536 9.70 9.55 933 9.68 9.70

180.00

190.00 203.45 176.02 103.00 73.91 86.11 93.98 96.77 101.85 73.66 85.85 93.47 89.92 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 200.00 220.00 220.00 280.00 280.00

9.SS

9.60 9.55 958 938 9.68 9.75

9.70 12.00 12.00

1930 18.00

369

h 10.00 299.00 10.00 299.00 10.00 299.00 10.00 299.00 10.00 299.00 10.80 250.44 10.80 250.44 538 200.0S 538 200.05 538 200.05 538 200.05 538 200.05 536 250.44 536 250.44 5.26 250.44 536 250.44 5.26 250.44 10.00 190.00 10.00 ' 189.20 10.00 189.20 10.00 189.80 10.00 19030 10.00 18930 10.00 189.40 189.00 10.00 189.90 10.00 10.00 189.60 10.00 18930 10.00 189.30 189.90 10.00 12.00 212.00 12.00 213.00 19.50 284.00 18.00 280.00 tnorn

300.00

6.00

6.00

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300.00

6.00

6.00

16.00

300.00

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6.00 6.00 6.00 7.65 7.65

17.00 17.77 18.63 18.84 16.30

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19.51 14.00 1631 17.80 1833 1936 14.00 16.32 17.77 17.09 20.78

hnem

300.00

300.00 250.44

250.44 200.05 200.0S 200.05

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250.44 250.44 190.00 190.00 190.00

190X0 190.00 190.00 190.00 190.00 190.00

190.00 190.00 190.00 190.00 210.00 210.00 280.00 280.00

vr

730 7.60 7.40

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730 730 8.00 8.00 10.50 10.00

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20.67 20.79 20.98 20.90 20.92 2038 21.06 20.78 2057 20.71 18.67 18.67 1436 15.50

h/w 4650 46.50 46.50 46.40

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46.43 29.91

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29.91

42.58 42.58 4238 42.58 4238 52.16 52.16 52.16 52.16 52.16 23.37 23.63 23.63 22.43 23.12 2430 23.64 2333 24.05 24.01 23.31

2338 23.36 23.50 23.63 23.33 24.40

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333.00 333.00 333.00 333.00 333.00

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311.90

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Three Point Bending Tests with Axial Load

372

Remarks on three point bending with axial load test evaluation (1)

Test data obtained from [26][27]

(2)

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in test series [27] are neglectible, whereas in [26] the --effects

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Annex 13

Document 3263-1-36 (ProfilARBED)

Ductility of plastic hinges in steel structures Guide for plastic analysis" PART Π of this Final Report (Available Separately) (203 pages)

Annex 14

Document 3263-3-15 & 3263-3-14 (LABEIN)(excerpts)

"Numerical simulations of singular cases" 'Numerical simulations of CTICM tests & nominal cases" (13 pages)

FINITE ELEMENT MODEL

Meshing The following figure shows the mesh containing 616 elements and 1913 nodes.

Element type S8R-Abaqus (parabolic 8-node shell element)

Load application

* Control displacement

* Constraints: Vertical displacement of the central section upper flange nodes linked together

Boundary conditions

LOAD

Vertical supports: both ends

Lateral restrains: both ends and central section RESTRAINS

Initial imperfection First linear-elastic buckling eigenmode (antimetrical deformation) has been used as an initial imperfection shape.

382

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Annex 15

Document 3263-3-18 (LABEIN) (excerpts)

"Μ- Ν interaction" (simulations) (10 pages)

Finite element modelling Cross-section:

HEA-200

Span:

2m

Material:

S235

Meshing:

616 elements, 1913 nodes.

Element type:

S8R-Abaqus (parabolic 8-node shell element)

Loads:

Two différents loads have been applied. At first, a compression load for several values of N/Npl and next a bending load controlled by displacement (no proportional case load)

Bending load application: LOAD

* Control displacement * Constraints: Vertical displacement of the central section upper flange nodes

linked together RESTRAINS

Boundary conditions

* Vertical supports: both ends * Lateral restraints: both

ends and

central section

Initial imperfection Erst linear-elastic buckling eigenmode (antimetrical deformation) has been used as an initial imperfection shape.

397

Evaluation of MNjIld The level of the reduced design plastic resistance moment taken for each simulation has been calculated according to the following precised formulas (where Aw = A - 2-b-tf ):

* If

N/NpLBd

< Aw/A

MNyXd

*If

1

,

f =

,

/ 1

MNy£d

1

- Mpfy£d

> N/NpLRd > Aw/A

meaning neutral axis in the web:

-

-

Ν

W*

Ν

J

/

meaning neutral axis in a flange:

Ν ΚplJRd

1

-

Ν Κpl&d

2b

Mpfy£d 2WP..y

According to this formulas the value which corresponds to the transition of neutral axis from web to lower flange would be 0.256 (Aw/A for HEA 200).

398

Position of the neutral axis corresponding to (pav

M

At the moment when the curve M-phi above crosses the line Mpl for the second time, axial stress value has been taken at several points on the same beam section (closest possible to the

central section). The position of the neutral axis has been calculated interpolating the relative distance between the nearest two points with reverse sign stress (tension and compression

axial stress).

N/Npl

0.1

0.2

0.3

0.4

0.5

Ζ (mm)

17.5

13.2

7.9

7.5

6.5

Concerning the results, it is also important to mention scattering that has been obtained for each simulation, since we have used local stress values at several points

regardless

of the beam section

of the influence of some factors over these stresses such as overthicknesses and the

local deformation mode.

399

Figures

* Figures

1

and 2 show for each simulation, that is for N/Npl values from 0.0 to 5.0, the plot

of Moment (Tn-m) versus Rotation (degrees). The reduced design plastic resistance moment Mny.Rd is also plotted in the same colour as its corresponding M-R curve.

* Figure 3 shows the plot of Phi available (radians) versus N/Npl as well as the maximun and minimun Phi-av values corresponding to the Feldmann model for HEA 200 and S235 with this type of charge. Figure 4 shows the plot of Phi-max versus N/Npl value.

* Figure 5 shows the plot of axial stress at the intersection between the web

and the lower

flange versus N/Npl value. By interpolating, the transition of the neutral axis between web to

lower flange would take place at about N/Npl = 0.21.

* Finally, figure 6 displays the local deformation mode corresponding to N/Npl = 0.15. The mode obtained for the other cases considered is similar to this one.

400

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406

Annex 16

Document 3263-3-21 (LABEIN) (Excerpts) "Feldmann's model checking within LTB limits" (25 pages)

FINITE ELEMENT MODEL

INTRODUCTION

The following series of simulations have been carried out in order to check Feldmann's model

within LTB limits for HEA200 and IPE200 cross sections. Two steel grades have been used S235 and S460.

MODELLING Meshing The mesh used in the simulations is made up of 616 elements and 1913 nodes.

Element type S8R-Abaqus (parabolic 8-node shell element)

Load application

* Control displacement * Constraints: Vertical displacement of the central section upper flange nodes linked together Boundary conditions * Vertical supports: both ends

* Lateral restraints: both ends and central section

Initial imperfection First linear-elastic buckling eigenmode (antimetrical deformation) has been used imperfection shape.

409

as an

initial

INITIAL IMPERFECTION

The tables bellow show for different beam lengths the relative deformation at three points of interest corresponding to the first positive antisimetric linear buckling mode.

-----

\s

'j: '"f

f = Local flange deformation

Δ

Δ \

g = Global lateral deformation

w = Local lateral web deformation

Kzl

w

HEA 200

'L(m)

g

w

f

1.5

0.010

0.254

1

2

0.014

0.240

1

2.5

0.022

0.232

1

3

0.031

0.233

1

4

0.065

0.237

1

5

0.230

0.258

1

6

1

1

0.5

7

1

1

0.46

8

1

1

0.44

Note that the local lateral web deformation is coupled with the global lateral deformation for beam lengths longer than 5 metres.

410

IPE 200

L(m)

g

1

0.253

0.759

1.5

0.936

0.417

f

w

2

0.283

2.5

0.248

3

0.233

4

0.217

5

0.204

6

0.193

8

0.172

1

For this beam type local lateral web deformation is coupled with the global lateral deformation

for all beam lengths tested except for the shorter ones.

The initial geometry has been modified in the shape

of the first positive antisimetric buckling mode

by introducing different levels of imperfection in each of the axes.

1%

of the flange thickness

has been taken as the maximun

nY

imperfection amplitude in the Y direction whereas the criterion

followed in the Ζ direction is shown in the next table.

Ί-

Í

Ζ direction Type of web deformation

Imperfection Amplitude

coupled

1

ν» of the beam length (L) for HEA200

0.1 °/M

uncoupled

of the beam length (L) for ΓΡΕ200

1%

411

of the web thickness (tw)

|

A sensitivity analysis for the HEA200 beam has been done modifying the magnitude of the global lateral imperfection between 0.1

0/ L and 1,5 ·/ L. A maximum difference of 16% and

phi-available has been obtained for S235 and S460 respectively.

412

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in

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413

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414

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415

RESULTS

Concerning the results for HEA200 cross section, phi-available seems to show a uniform

behaviour for li/iz values between 50 and 80. More detailed analysis is required to obtain a description of the phi-ay curve for li/iz values higher than 80.

On the other hand a sudden fall in the phi-av is obtained for S235 and li/iz values lower than 30 (beam length = 3 m), due mainly to a change in the deformation mode. For this range, local web

deformation is uncoupled from the global lateral one whereas for the rest of values (higher than

30) this local deformation does not appear within the deformation mode. This change in the deformation mode occurs now for lower li/iz values in comparison with the linear buckling case

for which this modification in the deformation pattern appeared for beam lengths higher than 5 metres.

A mesh refinement at the web (doubling the number of elements) has been introduced in

one

of the cases ( L = 2m) for which a local web deformation mode appeared. The results confirm

the

fall of phi-av, giving a 10% lower value of phi-av in comparison with the coarse mesh.

The obtained results for both profiles, HEA200 and ΓΡΕ200, show that the Feldman's model is

in the safe side within the LTB limits (CM66) for most of the analyzed cases. If the partial safety

factor γΜφ = 1.5 is used Feldman's model is safe in comparison with all the simulation cases.

416

SIMULATION RESULTS

HEA200-S235 -CM66Umit -A- -Feldmann's Model Feldmann's model / 1 ,5

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30

40

50

Ι

60

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0,25

417

1

1

70

80

90

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0

50

100

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150

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fy= 460 MPa Abaqus

20

40

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phLFeldman

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Promotion of plastic design for steel and composite cross sections: new required conditions in Eurocodes 3 and 4 practical tools for designers

Partii Guide for plastic analysis

Table of Contents

437

List of References

438

1.

Generalities

441

l.L

Introduction

441

Concept of plastic analysis

444

1.3.

Concept of inelastic rotation of plastic hinges

446

1.4.

Influence of lateral restraint

448

1.5.

Parameters influencing the inelastic rotation

450

1

2.

3.

4.

.2.

.

1

.6.

Design rules for ductility of plastic hinges

453

1

.7.

Concept of plastic analysis based on inelastic rotation φ

461

Required inelastic rotation of plastic hinges in structures

467

2.1.

Introduction

467

2.2.

Influence of parameters on required inelastic rotation φ req

467

2.3.

Continuous beams

47 Γ

2.4.

Frames

487

Available inelastic rotation of plastic hinges in cross-sections

492

3.1.

Introduction - presentation on Feldman's model

492

3.2.

Limitations and assumptions of Feldman' s model

492

3.3.

Influence of parameters on inelastic available rotation cpav

496

3 .4.

Feldman's formulas for values of cpav

498

3.5.

Extension to other load cases

501

3.6.

Tables with values of cpav for I and H cross -sections

504

3.7.

Graphs with values

of cpav

538

Design examples

553

4.1.

Introduction

4.2.

Design example

4.3.

Design example 2 : continuous beam with uniform distributed load

556

4.4.

Design example

558

4.5.

Design example 4 : Step by step method with the help of an elastic

553

Continuous beam with concentrated load

1:

3

:

Simple portal frame

analysis program

553

562

Appendix I : Plastic resistance of I and H cross-sections

437

568

LIST OF REFERENCES I]

"Elastic plastic behaviour of steel structures. Requirements and material properties", Résumé of the three coordinated project ESCS-F6-7210, March 1993

SA 113 (12/88 -7/91); SA 204 (7/88 -12/91): SA 508 (7/88 -12/91):

IEHK/LfS RWTH Aachen C RM Liège ARBED Recherches Luxembourg

2]

"Plastic design in steel. A guide and commentary", by a joint committee of the Welding Research Council and the American Society of Civil Engineers, Second Edition, ASCE, New York 1971

3]

ENV 1993-1-1 Eurocode 3: "Design of steel structures; Part 1.1: General rules and rules for buildings", CEN, February 1992

4]

ENV 1993-1-1: 1992/A1, Annex D of Eurocode 3 (ENV 1993-1-1 [3])

:

"The use of steel

grades S460 and S420", CEN, December 1994

5]

"Règles de calcul des constructions en acier", Additif 80 - DPU P 22 - 701 Construction Métallique n° 1-1981 France

6]

Merchant W., "The failure load of rigid jointed frameworks as influenced by stability", The Structural Engineer, volume 32, 1954, 185-190

7]

"Zur Rotationskapazität von I-Profilen statisch und dynamisch belasteter Träger", Ph.D thesis, Markus Feldmann, Stahlbau, RWTH-Aachen, 1994, heft 30, ISSN 0722-1037

8]

Background Document 5.09 for chapter of Eurocode 3, Part 1.1, "The b/t ratios controlling the applicability of analysis models in Eurocode 3, Part 1.1", G. Sedlacek, M. Feldmann, Aachen, December 25th, 1995.

9]

"Available rotation capacity in steel and composite structures", A.R.Kemp, N.W.Dekker, The Structural Engineer/Volume 69/n° 5/5 March 1991

10]

"Lateral stability of steel beams and columns - common cases of restraint", D.A.Nethercot, RM.Lawson, The Steel Construction Institute 1992

II]

EN 10025 + Al: "Hot-rolled products of non-alloy structural steels - Technical delivery conditions (includes amendment Al: 1993)", CEN, March 1990 (EN 10025), August 1993

(CM66),

(Al) 12]

EN 10113: "Hot-rolled products in weldable fine grain structural steels", Part 1,2,3, CEN, March 1993

13]

"Capacité de rotation d'une section plastifiée : Extension du calcul plastique", Frédéric Mazuy, Mémoire de fin d'études. CUST - CTICM 1994

14]

"Application de l'Eurocode 3 : classement des sections transversales en I", by Yvan Galea & Alain Bureau, Construction Métallique n° 1-1991, (EC3-STA 1-91)

15]

PEP-Micro Version 2.01, Plastic Analysis Computer Program, User's manual by Yvan Galea, Alain Bureau, CTICM, Saint Rémy-lès Chevreuse - FRANCE

16]

"Promotion of plastic design for steel and composite cross-sections: new required conditions in Eurocodes 3 and 4, practical tools for designers (rotation capacities of profiles, ...)", C.E.C. agreements 72 10-SA/520/32 1/935, Final Report Part L ProfilARBED-Recherches, Luxembourg, February 1996.

17]

"Safety considerations of Annex J of Eurocode 3 ", M. Feldmann, G. Sedlacek, Third International Workshop on Connections in Steel Structures, AISC, ECCS, Trento, 1995.

18]

"Essentials of Eurocode 3 - Design Manual for Steel Structures in Building", ECCS Advisory Committee 5, Application of Eurocode 3, First edition, 1991, ECCS n° 65.

439

1.1

Introduction [1][2][3]

An engineering structure is satisfactorily designed if it can be built with the required economy and if, throughout its useful life, it carries its intended loads and otherwise performs its intended function. In the process of selecting the members for a steel frame structures it is necessary, first, to make a general analysis of structural strength and, second, to examine certain details (usually covered by codes or specifications) to ensure that premature local failure does not occur. The design of a steel frame can be based on a number of criteria, any of which may actually constitute a "limit of structural usefulness". These criteria are: 1/

attainment of a specified minimum yield-point stress or of maximum plastic strength,

2/

excessive deflections and/or vibrations,

3/

global and local instability,

4/

fatigue,

5/

fracture.

It becomes more and more common to utilise design models taking into account plastic behaviour of steel, thus replacing the usual design concept based on elastic range of steel strength. It has been long known that an indeterminate steel frame has a greater load-carrying capacity than that indicated by the allowable-stress concept. Such frames are able to carry increased loads about the yield value because structural steel has the capacity to yield. Although the allowable-stress concept is satisfactory for simple structures, its extension to indeterminate steel structures has overemphasized the importance of stress rather than strength as the basis of engineering design. Furthermore, introduced a complexity that is unnecessary for many structures.

it has

Indeed, mere is no basis for an assumption that at no time should the stress in a steel structure go beyond the elastic range. As the matter of fact, it is necessary to consider plasticity in all structural design. An actual structure is a complex body with an extremely complicated state of stress. It is an assembly of many individual members joined together to form a working unit. The individual structural elements, such as beams and columns, come from the mills with residual stresses which are often over one-third the yield stress. In connecting the parts local stresses are produced by welding, by other fabrication and erection methods, and by misfits. There are over-all assembly stresses. The structure is sometimes pierced by many holes, reinforcement of all kinds are present, and many secondary stresses arise owing to continuity of the structure. Because of the deformations caused by the loading, bending and torsion may occur in what are assumed to be simple tension members, and axial force and torsion may occur in beams. As a consequence of these factors (the combination of unknown initial stress, stress concentration, and redistribution due to discontinuities of the structure), it is inevitable that local plastic flow will take place in any kind of design.

Numerous examples have been given elsewhere in which benefits of plasticity are used consciously or unconsciously in allowable-stress design. In continuous beams, for instance, a reduction of the negative moment is permitted at points of interior support. This is to recognize the moment redistribution which results from ductility.

From the foregoing it is evident that local yielding undoubtedly will occur in most steel frames by the time full service loads are reached. Furthermore, parts of these structures will enter appreciably into the plastic range before reaching their assumed limit of carrying capacity.

This situation will also exist in a structure designed on the basis of plasticity. Nethertheless it is important to note that at working load, the plastically designed structure is normally in the so-called

441

"elastic" range. A plastic analysis is based on a load which is equal to the working load multiplied by a load factor. Local inelastic deformations at working load may develop at first loading, just as such deformations may develop in a structure designed by the allowable-stress method. With either method of design, when the working load is removed from the structure a permanent set remains, and with it a corresponding system of residual stress. The change in stress and strain from the residual state produced by subsequent reapplications and removals of the working load would then be purely elastic.

It

is emphasized that plastic design is not a technique that is intended to replace all other design procedures. Factors such as buckling, fatigue, and deflections may become the design criteria. As an example of a limitation that comes because of column buckling, most trusses would be excluded from plastic design (with the exception of Vierendeel type); the method discussed herein requires that "hinges" form, and chord members would not exhibit the necessary deformation capacity in compression. In ordinary building construction, limitation such as fatigue and buckling are usually the exception and not the rule. Therefore plastic theory is finding considerable application in continuous beams and low building frames where the members are stressed primarily in bending.

A

simple example of a beam fixed at both edges highlights the benefit of plastic analysis in comparison with elastic analysis either assuming plastic resistance (Mpi = Wpi fy) or elastic resistance (Mei = Wei fy) of the cross-section (see figure 1.1 and table 1.1).

Figure 1.1

442

Table 1.1 Type of design

Type of global analysis

Stress

Redistribution

distribution

of bending

over cross-

moments

Benefit of strength

Limit load

(*)

section

ASD

elastic

ULS

elastic

ULS

plastic

>P *iP tyg

l
no

P'el=12Mel/L

100

no

Pei=12Mpi/L

109 to 124

yes

Ppl = 16Mpi/L

145

to 165

Notes to table 1.1:

ASD = allowable stress design ULS = ultimate limit state design Mpi = Wpify Mel = Welfy (*) : Wpi/Wd = from 1.09 to 1.24 for available H and I profiles

Modern rules of design of steel structures, for example Eurocode 3 [3], are based on limit state design. Limit states are states beyond which the structure no longer satisfies the design performance req uirements. In general, three categories of limit states are recognized: serviceability limit states (deformations, vibrations), ultimate limit states (resistance of cross-sections and members) and fatigue limit states. The present Guide deals with ultimate limit states design.

Ultimate limit states (ULS) are those associated with collapse, or with other forms of structural failure which may endanger die safety of people. For ULS assessments one can use models which are able to describe the actual failure mechanism like buckling, plastification or cracking. Depending on the chosen concept of resistance, different types of global analysis of a structure may be used to determine the effects of actions (internal forces and moments - N,V,M, deformations - δ, rotations - φ ...) - see table 1.2.

Table 1.2 Concept

of resistance

Types of global analysis of a structure 1st order theory

2nd order theory

EGA

PGA

EGA

PGA

Allowable stress design

X

-

X

-

Ultimate limit state design

X

cgl

X

®

Notes to table 1.2: 1st order theory uses the initial geometry of the structure 2nd order theory takes into account the influence of the deformation of the structure (Ρ-δ effects) EGA= elastic global analysis PGA= rigid-plastic or elastic-plastic global analysis "x" = applicable "-" = not applicable "cgi" = applicable, scope of the present Guide

443

1.2

Concept of plastic analysis [1] [2]

1.2.1

Assumptions

The important concepts and assumptions with regard to die plastic behaviour of structures according to "simple plastic theory" are as follows:

of

a/

The structure and die loads are all in die same plane, and each member has an axis symmetry lying in die plane.

b/

The material is ductile. fracture.

c/

Each member cross-section has a maximum resisting moment (die plastic moment, Mpi), a moment drat is developed tiirough plastic yielding of die entire cross-section.

d/

Because of me ductility of steel, rotation at relatively constant moment considerable angle; in other words, a plastic hinge will form.

e/

Connections proportioned for full continuity will transmit die calculated plastic moment. The condition is idealized as a plastic hinge at a point.

il

Plastic hinges will first form at sections where die moments under elastic condition reach Mpi. With diese sections rotating at constant moment, additional loading will be accompanied by a redistribution of moments in die structure, so that plastic hinges will appear at some other locations where die moments under elastic conditions were less tiian Mpi.

g/

The plastic limit load is reached when enough plastic hinges have formed to create a

It

has die capacity

of undergoing

large plastic deformation without

will occur through a

mechanism.

hi

The deformations are small, and dierefore die equilibrium equations can be formulated for the undeformed structure (as in ordinary elastic analysis). Similarly, virtual-work expressions for mechanism displacement are based on small deflections (first order theory meaning no allowance for die influence of die deformation of die structure or meaning no Ρ-δ effects).

i/

No instability will occur before die attainment of die plastic limit load.

j/

The loading is proportional, tiiat is, die ratios between different loads remain constant during loading.

k/

The influence of normal force and shearing force on die plastic moment is not considered.

That "simple plastic tiieory" can be refined by die use which concern the following items:

of more complex plastic

analysis methods

h/ with improved hypothesis Ρ-δ effects could be taken into account (second order theory),

k/ influence of Ν (axial force), V

(shear force) on plastic moment resistance could be

considered.

This can be done thanks to application of sophisticated computer softwares.

1.2.2

Plastic hinge theory

A usual method for a realistic elastic-plastic modelling of steel structures is die plastic hinge metiiod. In tiiis method die material behaviour of steel is modelled by an idealized elastic-plastic contitutive material law, 'which neglects die strain hardening behaviour of steel. From tìiat simplified stressstrain curve die plastic section resistance Mpi at die location of plastic hinges may be defined (see figure 1.2).

444

Material model

Ultimate cross-section resistance

ι

f 1

"y

1

fy

'

1

I,

IVlpl-

ε

c fy VYp

[fy

Figure 1.2 The principle of the formation of plastic hinge mechanisms can be best shown with a single span beam (see figure 1.3). At a certain deformation 6e\ the full plastic moment Mpi is reached. The fibres of die section on the load level Ppi are fully plastified. The other parts of die beam remain still fully elastic.

System

J^/7/

+Fomiing of the plastic hinge

4Mpl Ppi =

=t8-

f.

"plastic hinge"

P,M4

y^_

Kinematic

-ζ-

pi

mechanism

X

'δ = δ,.ι+δJpl

4X ψ

Figure 1.3 445

elastic i{ Mpl 12EI

plastic ρ'

4

A kinematic mechanism forms when further energy is introduced. Further deformation is possible without any increase of die resistance. During this process die elastic parts do not change their deformation line. The moving system may be interpreted as a rigid body mechanism or as civil engineers say. a plastic hinge mechanism (see figure 1.4).

Figure 1.4 1.3

Concept of inelastic rotation of plastic hinges

The behaviour of a real beam is quite different from the assumption made in the plastic hinge theory. The moment-rotation diagram presents not only a curve shape between the elastic and die plastic moment, but it also goes over the value of the plastic moment because of die strain-hardening of steel. The curve reaches a maximum which corresponds to the maximum bearing capacity of the beam, and then the moment must decrease because of local instability phenomena (see figure 1.5). The fact of reaching the plastic moment corresponds to die full plastification of a cross-section, i.e die apparition of the plastic hinge. When the moment continues to increase, die plastic hinge rotates inelastically. This available inelastic rotation denoted as tpav is a feature characterising the ductility of plastic hinge or, in other words, die usefulness of the given cross-section for plastic analysis. The available inelastic rotation cpav of plastic hinge is clearly defined in figure 1.5 which presents a typical Μ-φ curve for 3-point bending beam and can be expressed by die following formula:

Φ*ν rav where

tprot


(1.1)

= φ«*-ΦPP1

is the reached rotation when the bending moment gets back to die plastic moment value (Mpi) in the descending part of die moment-rotation curve, expressed in radians, is die elastic rotation corresponding to die theoretical plastic bending moment (Mpi) of the beam, expressed in radians,

MpiL

for 3-point bending beam.

In die plastic analysis of structures, one deals with so-called "required inelastic rotation" and "available inelastic rotation"

of plastic hinges. They could be defined as follows:

(preq

the required inelastic rotation which corresponds to die greatest rotation which is necessary in plastic hinge to ensure die formation of a plastic mechanism; it is a feature of the structure.

cpav

the available inelastic rotation is the rotation furnished by a given cross-section and feature of this cross-section (its dimensions and steel grade).

446

it

is a

Mi

Actual behaviour of tested beams

ψ

M

Usual bilinear concept

M pi

φ=



M pl

Figure 1.5 Inelastic rotation φ could also be defined as the parameter necessary to allow redistribution bending moments M.

of

Inelastic rotation φ of plastic hinge is mainly influenced by local buckling in the concerned crosssections and often by lateral-torsional buckling (LTB). In this Guide, no interaction between local buckling and LTB is considered (see chapters 1.4 and 1.6.3).

A common check consisting on the comparison of both values 1.6.2 of this Guide.


and φ3ν, is presented in chapter

In international references [1] [2][9], one often deals with the term "rotation capacity R" which is defined as (based on tests results

of 3-point bending beams):

R^
(1.2)

Φρί

447

In general, dus nondimensional parameter is not recommended, because it contains die term "Leci¬ tile equivalent length L of a reference beam which is difficult to evaluate. For more information about rotation capacity R, refer to [1], [7], [9], [13].

On die other hand, die physical parameter (pav (expressed in radians or degrees) is easily evaluated without reference to die length L of a reference beam as presented in chapter 3.

1.4

Influence of lateral restraint [2]

In order to realize die necessary inelastic rotations at plastic hinge locations a member must have sufficient lateral support to assure that the plastic moment at diese hinges locations is not reduced by lateral-torsional buckling (LTB) before a mechanism has formed. Local buckling is controlled by limitting die width/thickness ratios of the flanges and die web, and lateral-torsional buckling is controlled by limiting die unbraced length of die member. The schematic relationship between die unbraced length and die moment capacity is illustrated in die figure 1.6. When the unbraced lengtii is very large, die member will fail by elastic lateral-torsional buckling. When die lengtii is relatively short, die full plastic moment will be attained or exceeded. Between diese two extremes tiiere is a transition range in which part of die member has yielded, but buckling occurs before Mpi is reached. The lower part of figure 1.6 shows die variation of available inelastic rotation witii die unbraced lengtii. In die region of elastic buckling, die available inelastic rotation is small. When die unbraced length is small, the available inelastic rotation is much larger.

Moment Capacity

Available Inelastic Rotation

Figure 1.6

448

The available inelastic rotation depends largely upon its unbraced length on either side of a plastic hinge. Figure 1.7 shows, qualitatively, for different Lj/iz ratios, die moment-rotation relationship of a beam under uniform moment. As L¡/iz decreases, die available inelastic rotation increases. In order to maintain die plastic moment and provide adequate available inelastic rotation, die 1φζ ratio must be controlled.

In general, lateral-torsional buckling shall be avoided in plastic global analysis and this restriction is applicable to this Guide.

M

J!

Mpi

1.0

-

/>) r

/ f\

u

"-·«.

A \

/L\

Ori

Oä

Λ \

(θ.

W4>W3>W2>\Wi

.

"φ

Ι

Figure 1.7

449

Parameters influencing the inelastic rotation

1.5

This chapter gives a general view of parameters which influence die required and available inelastic rotation (φ^, φ&ν). More detailed explanations are provided in chapters 2 and 3, dealing with tibe required rotations and die available rotations, respectively. The parameters are presented in die table 1.3.

Table 1.3 Parameters influencing die inelastic rotation φ of plastic hinges (*) Parameters

1.

Φ required (structure)

Φ available (cross-section)

Geometry of die structure (storeys, span, boundary conditions)

YES

NO

of die cross-sections witii

YES

YES

YES

YES

YES YES

YES YES

YES

YES

YES YES YES

YES YES YES

YES

NO

YES

NO

2. Shape and dimensions or without stiffener)

3. Material characteristics 3.1 Yield strength, fy and ultimate strength fu (constant or not in die structure) 3.2 Maximum elongation, A5 £5 3.3 Modulus of elasticity, E

4. Load configuration (distributed, concentrated) 5. Interaction between internal forces and bending moments 5.1 Interaction M-V

5.2 Interaction M-N 5.3 Interaction M-N-V 6. Ratios to load level of die complete mechanism 6.1 Instability load level / complete mechanism load level (sway frames 6.2 Design load level / complete mechanism load level

plastic

plastic ...) plastic

Note to table 1.3:

(*)

Parameters influencing die available inelastic rotation are considered with die assumption tiiat no LTB (lateral-torsional buckling) occurs in members, tiiat is, if sufficient lateral restraint exists in die members (see chapter 1.6.3).

450

Comments on table 1.3:

Parameter

1: see

chapter 2 for details.

- Parameter 2:

T^-fe-T

«f-^

I ¿£

*'/ />

stiffeners

I

t(-


^

2/3 h

m^

Λ

m

^

3ΘΕ

-¿ ^_

or

stiffeners

Figure 1.8 Parameter 3: table 3.1

of chapter 3.2 provides all

steel grades allowed by Eurocode 3 [3]

plastic global analysis in accordance with EN 10025 and EN101 13 specifications.

i

^^^^

fy-

H\E

\

ί

ey

eu

Figure 1.9

451

^N ;

a565

ε

for

Parameter 4: see chapter 2 for details concerning cpreq or chapter 3 for details about cpav .Chapter 2.2.1 presents some remarks about die influence of loading patii that is different from usual assumption of simultaneous proportional increase of all loads.

- Parameter 5: interaction formulas between M-V, M-N or M-N-V are provided in Eurocode or in other standards.

Ν.

V, i

3

[3]

ι

Vpl

Npi -

pai rameter

5.1

M pi

Mpi

M

M

parameter 5.2

Figure 1.10

Parameter 6: see chapter 2 for details. For parameter 6.1, see explanations in chapter 2.2.2 (influence of second order effects on required inelastic rotation of sway portal frames). For parameter 6.2, see explanations in chapter 2.2.3 (influence of die load factor of reference).

452

1.6

Design rules for ductility of plastic hinge

1.6.1

Classification of cross-sections

According to chapter 5.3 of Eurocode 3 [3], steel profiles are categorized in section classes (see table 1.4) which depend on: slendemess

of cross-section members (width/thickness ratios of flange and web),

yield strengdi whh die factor ε = J235 / fy , die internal forces and bending moments applied to die cross-section (NcompressicHb My, Mz, My-N, ... : all inducing normal stresses). The cross-section of class 1 allows die full plastic strengdi and a moment redistribution due to die formation of plastic hinges which have sufficient available inelastic rotation. For class 2 crosssections, die available inelastic rotation becomes so limited that die plastic cross-section capacity Mpi may be taken into account, but die redistribution of moments has to be neglected. Hence die system is assumed to behave elastically though the plastic capacity Mpi of die cross-section is exploited locally. Classes 3 and 4 concern cross-sections with a distribution of elastic stresses, respectivelly on full or effective cross-sections. Tables 1.5a and 1.5b summarize die existing rules in Eurocode 3 [3] [14].

The aim of this Guide is to introduce new tools permitting to calculate a real value of plastic hinge rotation, replacing die existing terms "sufficient" and "limited" in regard to the available inelastic rotation. Eurocode 3 allows plastic global analysis of structures if plastic hinges occur in class 1 crosssections, but not in class 2 cross-sections. However, test results [1] have delivered available inelastic rotation of class 2 cross-sections which were not negligible. Therefore plastic global analysis might be used when die full plastic section capacity and die required inelastic rotation can be reached by die used steel profiles provided that cross-section is class 1 or 2. Subsequent improvements are required in cross-section classification, because recent researches have revealed tiieir conservatism (which exists mainly due to simple assumptions made for setting diese limits): even class 3 and class 4 cross-sections delivered non negligible inelastic rotations according to 3-point bending tests. More developments are necessary, but die scope of tiiis Guide is limited to class 1 and 2 cross-sections defined by present rules of Eurocode 3 Γ31.

Figure 1.11

453

Table 1.4

Definition of die classification of die cross-section

«Ppi

Class

Behaviour model

q>

Design resistance

Global

=
analysis

Available rotation capacity of plastic

of structures

hinge

PLASTIC across full section

*M Mp,

local budding

Ψ

I

plastic fy

PLASTIC 4M

-r^ / L

across local buckling

1

Mp,

limited

elastic

none

elastic

none

elastic

full section

£

fy

ELASC across full section

M

elastic or,

7sr-

Ζ Mpi

important

M¿¡

local budding

fy

ELASTIC M

across effective section

Mp| M,

Md r<

L

local buckling

1

ι

fy

454

Table 1.5a Classification of cross-section : limiting width-to-thickness ratios for class Types

Stresses

of

distribution for

Web

& class 2

"to*

loading

class

+

I

1

N,compression

+

fy

ιοε

72ε

^>My

My

r-kkr

Mz

c/tfá

W

I

N,comp.

My

fy

I

S

d/tw¿

C/jfS πε

ιοε

W

83ε

ιΐε

w

9ε

w

ιοε

R

ιοε

R

ΐΐε

W

9ε

W

10ε

R

Πε

W

ιοε

456ε

ιοε

13α-1 W

9ε

where : 0,5 < α

R = rolled sections W = welded sections

fy (N/mm2)

t- fes ft,

Flange

9ε

13α -ι

5

ced:

+ : stresses in compression - : stresses in tension

Web

38ε

10ε

396ε ~*

Class 2

Flange

33ε

I

+

I-cross-sections

N

4-

I

& class 2

Classi

fy

I

1

Ν

2dtfL ν w y

+

1

2

< ι

)

Values of d, c, tw and tf are defined in figure 1.11 275

355

420

460

ε (if tá 40 mm)

0,92

0,81

0,78

0,71

ε (if 40 mm
0,96

0,84

0,78

0,74

235

455

Table 1.5b Classification of cross-section : limiting width-to-thickness ratios for class 3 I-cross-secüons Class 3

Types

Stresses

of

distribution for

loading

class

I

+

1

& class 2

Web

Flange

d/tw*

c/tf<

fy

I

15ε

42ε

N,compression

Ν +

I

Ι

+

Ι

fy

15ε

124ε

^My

My

+Í-7

Μζ

+

w

14ε

r

23zJi\S7

W

2teV0,57

15ε

I

Τ

Ν,comp. .-My

14ε

w

I

fJ

(lvfy|<|fyl)

5

"*-\My

0 >

ψ >

-1

:

W

14ε

42ε 0,67+0,33ψ

+ : stresses in compression - : stresses in tension

fv(N/mm2)

^fes/fy

Values of d, c, t^, and tf are defined in figure 1.11

R = rolled sections W = welded sections 235

8(ift^40mm)

ε (if 40 mm á t < 100 mm)

456

275

355

420

460

0,92

0,81

0,78

0,71

0,96

0,84

0,78

0,74

Application rules

1.6.2

The text quoted hereafter is issued from die present Eurocode 3 [3]. concerning rotation requirements of plastic hinges.

It provides

general rules

"5.3.3 Cross-section requirements for plastic global analysis

(...) (2)

At plastic hinge locations, ue cross-section of die member which contains the plastic hinge shall have a rotation capacity of not less tiian die required rotation at that plastic hinge location.

(3)

To satisfy the above requirement, the required rotations should be determined from a rotation analysis.

(4)

For budding structures in which die required rotations are not calculated, all members containing plastic hinges shall have class 1 cross-sections at die plastic hinge location."

hi order to respect diese Eurocode 3 requirements, a comparison of die required inelastic rotation 9req of each plastic hinge appeared in a structure submitted to design loads with die available inelastic rotation qw for each relevant cross-section should be proceeded. It can be expressed by die following ultimate limit state rotation check :

Φτβς,α

^

Φαν.ά

or

rn

<

av

(1.3)

YMq>

In die formulas, die symbols denote:

-


-


-


-

YMcp

design value of required inelastic rotation in die considered plastic hinge (calculated from plastic global analysis of die structure submitted to design loads) (see chapter 2), design value of available inelastic rotation of plastic hinge appeared in cross-section,

characteristic value of available inelastic rotation section (see a model proposed in chapter 3),

of plastic

hinge appeared in

cross-

partial safety factor; in die scope of die model proposed in this Guide (see chapter 3), die values of γΜφ have been determined by statistical evaluation of available tests results [8]. The values of partial safety factor γΜφ applied to
of cross-section resistance and

* die design value of plastic bending moment for die resistance of cross-sections : Mpl.Rd = Mpi/yMo, witii partial safety factor γ\ω according to Eurocode 3 [3](γΜ0 = 1 5 1)

or to NAD's values, and,

* die design value of available inelastic rotation for ductility of formed plastic hinges : 9av.d = , with die proposed design model in chapter 3 for
457

Guide for plastic analysis

1. Generalities

Table 1.6 Load cases

?Μφ 1,52

1,73

1

2,02

Ι

φ

Formulas

j My

:

Bending about yy major axis

(3.1) & (3.3)

J Mz

:

Bending about zz minor axis

(3.2)

Λ My-N

:

Bending about yy major axis combined witii axial compressive force

(3.4)

Such a rotation check so far could not be carried out, if die available inelastic rotation could only be determined from tests or sophisticated numerical simulations. But die present Guide provides a possibility of a quick check of dus condition (1.3) which makes it much easier to satisfy die quoted requirements of Eurocode 3 [3]. References shall be made to chapters 2 and 3 to find die needed tools permitting evaluation of required inelastic rotation (ft-eq and available inelastic rotation
Chapter 1.7 presents the new concept of plastic analysis based on inelastic rotation φ. To facilitate understanding of this concept, first an overview is presented, tiien more details are given in a flow¬ chart and related comments.

The example in figure 1.12 highlights the need of checking die ductility of each formed plastic hinge. The figure shows the evolution of die lateral displacement δ of a portal frame and of die inelastic rotation φ} in die first plastic hinge versus loads. Soon after die occurrence of die titird plastic hinge, die required inelastic rotation in die first plastic binge exceeds die available inelastic rotation
Figure 1.12

458

Conditions of lateral restraint

1.6.3

Some general remarks concerning lateral restraint can be found in the Eurocode 3 [3]:

"5.4.2 (3) When plastic global anlysis is used, lateral restraint shall be provided at all plastic hinge locations at which plastic hinge rotation may occur under any load case.

5.4.2 (4)

The restraint should be provided within a distance along the member from the theoretical plastic hinge location not exceeding half the depth of the member."

Presently lateral-torsional buckling is not explicitiy forbidden by Eurocode 3, but strength and limited as follows (see 5.5.2 (7) of [3]):

it

is related to

if die relative slendemess Xuj > 0.4, then the ultimate bending moment resistance of die beam (=%LT-Mpi) is lower tiian die plastic moment resistance of class 1 and 2 cross-sections (=Mpi), because die reduction factor

χιτ

is lower than 1.0. The factor XLT includes die

unbraced length of the member (¿lt)·

In order to respect the process of plastic global analysis which excludes lateral-torsional buckling as regards strengdi

of

of plastic

frames and inelastic rotation

XLT < 0.4 shall be fulfilled in all cases (see figure

1.

hinges, the Eurocode 3 condition

13).

M Mpi, I

1,0

0,8 0,6

-

0,4

-

0,2

Effective slendemess λ^ 0 0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

Figure 1.13

In alternative to Eurocode

3 rule, otiier existing rules defining maximum distances between lateral restraints should be provided to designers. As an example, the ones of CM66 - Additif 80 [5] are quoted below, which seem realistic for present concept of design:

"In the case of the member in bending which contains at one of its ends the plastic hinge allowing for redistribution of sollicitations, the conditions of lateral restraints given by §5.21 of Additif 80 - DPU Ρ 22-701 (CM66) should be satisfied in the neighbourhood of the plastified section to avoid die lateral-torsional buckling (LTB).

459

If die moment varies linearly along a member with the length L¡ which is free to buckle laterally, the conditions are as follows:

L:

< 35ε

L: τ1- < (60-40ψ)ε where

ίί0.625<ψ<1

(1.4)

ϊί-1<ψ<0.625

(1.5)

Li

is die lengtii

of die member or of die portion of that member where the linear distribution of bending moment is applied,

ψ

is die ratio of bending moments at both ends tiiat member (-1 < ψ <1),

iz

is the radius

with

of die member or of die portion of

of gyration about minor axis (= ^Jlzl A) Iz - moment of inertia about zz minor axis of the cross-section,

A - total area of die cross-section." Figure 1.14 is a graphical presentation of CM66 rules. An example on die evaluation of die ratio of bending moments ψ is shown in figure 1.15.

Ill Ei7

-Ψ -1 -0,8 -0,6 -0,4 -0,2

0

0,2 0,4 0,6 0,8

0-> ¿tf

35Γ Λ-Α

Figure 1.14

460

1

Ü

0.5 L

0.5 L

-3XÌ

-m+e-

(LR = lateral restraint) LR

LR

Χ"

-zi"

Ln = 0.5L

ψ, =-5/6

LR L¡2

χ

= 0.5 L

Ψ2 = 0

0.5 L ^93,3

0.5 L

i, ε

ι, ε

i

>. "-S z

L

60 ε

^60

for constant cross-section over the whole span

Figure 1.15

This Guide does not consider interaction between local buckling in plastic hinge and lateral-torsional buckling of elements where plastic hinges occur. Therefore designer should avoid die appearance of lateral torsional buckling (LTB) by applying sufficient lateral bracing to fulfil the requirements of Eurocode 3 [3] (or CM 66 - Addditif 80 [5], for instance). Lateral bracing should possess adequate strengdi and stiffness. General requirements and practical solutions concerning strengdi and stiffness of lateral bracing can be found in [2],[5] and [10].

1.7

Concept of plastic analysis based on inelastic rotation φ

This chapter gives a general view of die proposed concept of plastic analysis based on inelastic rotation of plastic hinge, φ. An overview is first presented, then a flow-chart furnishes more details. Hints and references are given in comments on the flow-chart.

461

Flow-chart n°

1

for ULS plastic global analysis (general)

based on inelastic rotation of plastic hinges φ

Input data

for the analysis

Global analysis of the structure

Check of resistance of the structure

Check of ductility of plastic hinges

462

Flow-chart n°

1

for ULS plastic global analysis (details)

based on inelastic rotation of plastic hinges

φ row:

. .

Type of frame or beam (geometry, boundary conditions) Load cases for ULS

1

Choice of : cross-sections, yield OTr^ÄT^Äth and joints (pinned oi

[

Assumption of lateral restraints )

ULS plastic global analysis of the structure Results: . maximum load multiplicator of design loads : Xmax . internal forces and bending moments : NSd, V^j , MSd . required inelastic rotations of plastic hinges : φ rreq

yes

Check of bearing capacity of the structure (see row 4a)

*rnax-

yes

no

1

Check of resistance of . cross-sections . members .

not fulfilled

joints

Check of local effects

Evaluation of available inelastic rotation of plastic hinges φ from cross-sections yiels strength and joints (see row 2) and from values of N^, VSd , MSd (see row 4b)

yes

Check of rotation requirements

:

^req ~Tav' ·Μφ

-»»(Adopt chosen cross-sections, steel grade and joints}

463

4a 4b 4c

no

OVERVIEW OF PLASTIC ANALYSIS METHOD (based on inelastic rotation

of plastic hinges, φ) AVAILABLE INELASTIC ROTATION

REQUIRED INELASTIC ROTATION

φ available

φ required l.Data

Cross-section and joints

Structure

λΡ '

λΗ

,.

i

1

type of frame prediction from LTB

C

±

f choice of cross-sections and joints j

)

2. Plastic global analysis *»»

3. Results of analysis and checks

*{

maximum load factor

λ,^ J

»-f internal forces and moments (N, V, M)

V

Check of bearing capacity (λ^, > 1) Check of resistance of members and joints

required rotation Φ

Check of rotation requirements for each plastic hinge

Notes

:

Ψ^

- required inelastic rotation

Ρ

vertical load

Φ,ν

- available inelastic rotation

Η

- horizontal load

LTB

- lateral-torsional buckling

Ν

- normal force

Ύ*Μφ

- partial safety factor

V

- shear force

λ,

- load factor

M

Xma

- maximal load factor

464

bending moment

Comments on the flow-chart n°l

row

1:

ULS means Ultimate Limit State ULS load cases are defined in Eurocode 3 (chapter 2.3.3) [3].

row 2: This flow-chart concerns structures using pinned and/or rigid joints. In the case of semi-rigid joints whose behaviour is between pinned and rigid joints, the designer shall take into account the moment-rotation characteristics of die joints (moment resistance, rotational stiffness and rotation capacity) at each step of the design. The semi-rigid joints should be designed according to chapter 6.9 and the Annex J of Eurocode 3 [3]. Indicative values of (preq for plastic hinge in portal frames are given in chapter 2 table 2.3,

of
row3:

witii local buckling in plastic hinges which influences the value of available inelastic rotation (see chapter 1.4 and 1.6.3). For rules defining maximal

members and its interaction

unbraced length reference should be made to [2],[3],[5] or [10].

row 4: In order to determine in a structure: die maximum load multiplicator of design loads Xmax, XP

λΗ»

-7777

'-max

die internal forces and bending moments (Nsd, Vsd, Msd), die required inelastic rotations of each plastic hinge

q>req,

Eurocode 3 ([3] : chapter 5.2) allows to use different types of plastic global analysis:

either, first order analysis using the initial geometry of die structure,

or, second order tiieory taking into account the influence structure.

row 5: If Xmax >

of die deformation of die

dien die structure has sufficient bearing capacity. To optimize the solution (^max « 1), new proposals of cross-section, steel grade and/or joints (row 2) should be checked by a new global analysis. 1,

465

bearing capacity of the structure

^max

λ=1

ULS design loads level

δ If

Xjnax < 1, dien there is a need for strengthening section, stronger joints and/or higher steel grade.

λ=1

of the structure with stronger cross-

ULS design loads level bearing capacity of the structure

»max

If λιηπχ = l,then the structure is optimized as far as resistance is concerned. row 6: The classification of cross-sections (chapter 5.3 of Eurocode 3 [3]) have to be determined before all ULS checks of members, cross-sections, webs and joints. The sequence of ULS checks is not imposed and it is up to the designer to choose die order of die ULS checks which are anyhow all necessary to be fulfilled. row 7: For each formed plastic hinge, the available inelastic rotation models or tests in function of:


shall be determined by

the cross-section, the steel grade,

die joint (if die plastic hinge occurs in die joint), the internal forces and bending moments applied in plastic hinge (Nsd, Vsd, Msd)·

row 8: The ductility of plastic hinge shall be checked by the requirement

cpreq

< qWyMcp (see

chapter 1.6.2) where: (preq has been cpav has

calculated from plastic global analysis (see chapter 2),

been evaluated from models (see chapter 3) or tests ,

?Μφ is the partial safety factor affected to the model (see chapter 1.6.2 and [8]). 466

2.1

Introduction

This chapter deals with die calculation of required inelastic rotations that plastic hinges develop when performing an elastic-perfectly-plastic analysis (see §5.2.1.4 of Eurocode 3). Formulas and abacuses are provided for two-span continuous beams under basic loadings and procedures or flow charts are also given for simple frames. For two-span continuous beams under basic loadings, information are also provided on collaspe loads and on first occurrence of plastic hinges. General procedures are explained for more complex loading cases or frames, based on die use of classic analysis programs (linear elastic global analysis) or specific ones (plastic global analysis). Particular aspects concerning frames are also discussed.

2.2

Influence of parameters on required inelastic rotation


Some general remarks on influence of parameters have been presented in table 1.3 of chapter 1. Among all parameters which are to be taken into account for the determination of required inelastic rotations, die questions about the influence of die loading patii, the second order effects and die load level at which these required rotations are calculated are detailed hereafter.

2.2.1

Influence of the loading path

During tiieir life, structures are submitted to permanent and variable actions. Eurocode 3 defines ULS combinations which allow to determine the ULS loading acting on die structures and to check tiieir resistance. To reach a given ULS loading, different loading patiis may be followed, with expected consequences on die calculated values of φ«,.

Firstly, it can be said tiiat whatever die loading patii is, φ,,, at collapse will be the same if the same mechanism is reached and if no elastic return (see 2.4.2.2 of this chapter) occurs in any plastic hinge during loading [16]. Secondly, about global analysis, Eurocode 3 states tiiat "it may be assumed to be sufficient, in the case of building structures, to adopt simultaneous proportional increases of all loads" (§5.2.1.1 (5)). Where necessary, this assumption will be adopted in tins chapter, and λ will be die common load factor for all loads.

2.2.2

Influence of second order effects on required inelastic rotation for sway portal frames

The example given hereafter illustrates parameter 6.1 frames (see §5.2.5.2 of Eurocode 3),

of table

1.3 and shows, that

for sway portal

a) die collapse load level determined widi a second order plastic analysis may be much lower tiian die one obtained from a first order plastic analysis. The decreasing of die collapse load level depends on the critical load factor

Κ -^where

V Ve-

(2.1) is the total vertical load applied to die frame at the considered loading level is die value of the total vertical load for elastic instability in a sway mode

To give an idea, the Rankine-Merchant formula [6]

467

λ
(2.2)

withXCT>4,

)

where λ^} and λ^ are collapse load factors for first order and second order plastic global analysis, gives a good approximation of the collapse load level taking into account die second order effects (this formula is valid for portal frames with one or two levels).

b) consequently die required inelastic rotation obtained witii a second order analysis may be much lower than the one obtained from a first order analysis. In die latter case, a complete plastic mechanism is reached at collapse, while in die first case, an "elasto-plastic instability" often occurs with a reduced number of plastic hinges. The number of plastic hinges at collapse in a second order analysis may be evaluated by considering

die different

λ^

values determined for die frame wfth the η first plastic hinges (n = 1, 2, ... ) here

treated as perfect hinges. When λ^. falls down below die current load factor λ, all supplementary plastic hinges to die η first ones tiiat occur in die frame in die first order analysis are not valid because, from stiffness considerations, die -hinged frame becomes unstable under the reached loading. The required inelastic rotation may then be taken as tiiat determined in die governing plastic hinge when the nth hinge occurs, and may sometimes be much lower than die one at die complete plastic mechanism.

The following example (see figure 2.1) is an industrial portal frame supporting a total vertical load

V=776kN.

λ^ = 5,47. Therefore, the frame is classified as a sway frame according to §5.2.5.2 of Eurocode 3 (λ^ < 10). Under die SLS loading Get us say related to load factor λ = 0,7), the horizontal deflection δ is equal to h/260 and the vertical deflection of die beam is

The critical load ratio λ« is equal to

L/396. 17kN/m

320 kN

320 kN

23_kN

IPE 300 (S235)

h=6m

IPE 240 (S235)

^Z w

L = 8m

Figure 2.1 - Data Figure 2.2 shows die horizontal deflection versus die load factor λ plotted according to a first order and a second order plastic analysis. The collapse load factor is λρ = 1,222 for die first order analysis and λ = 1,000 for die second order analysis.The fall down of die collapse load factor is 18,2% in that case due to second order effects. The number of plastic hinges at collapse is 4 for the first order analysis and 3 for the second order analysis. The locations and die numbering of die plastic hinges are shown on the Figure 2.3. Figure 2.4 shows die evolution of the critical load factor λ,,, for the different configurations of die frame : 0, 1, 2, 3 and 4 plastic hinges. Of course, dus critical load factor decreases and falls below

468

die first order behaviour (λ-δ) curve. Therefore, die 4tii plastic hinge is not valid and die collapse load factor taking into account die second order effects will be equal or less than die load factor at which die 3rd hinge occurs. Then, the required inelastic rotation may be determined at die 3rd hinge occurrence if first order plastic analysis is performed.

4

6

12

10

S

14

δ

[cm]

Horizontal displacement

Figure 2.2

Horizontal displacement versus load factor 1

Ml

'

ΙΙΙΙΙΗΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΓ

4

<

<

L3

1

L2

Figure 2.3 Locations and numbering of plastic hinges

λ °=5.47

1

2

1.39

= 1.222 4 1

st

λ

= 0.39

=0

λ

horizontal displacement

0

n-0

n-1

n=2

il ir

n-3

4.3

λ <λ

λ>λ

δ

n-4

¿2

¿3

¿2

plastìchinge4 is not valid

Figure 2.4 Evolution of the critical load factor witii plastic hinges occurences

469

Considering die required inelastic rotation, let us assume tiiat die governing hinge is the hinge number 1 . Figure 2.5 shows die evolution of its inelastic rotation versus the load actor in a first and in a second order analysis. From this figure, it can be said that :

If the required inelastic rotation is determined at collapse ((preq.u), die value fells down from 32,7 IO"3 rad to 18,8

IO3 rad when considering the second order effects (-42,5%). The value obtained at die occurrence of the 3rd plastic hinge in the first order analysis is 22,5 10~3 rad and is a better approximation of the true required rotation (18,8 10~3 rad).

125

4

1222

1.20

*&"* 1.15

list order

r

I

j

\

1.10 1.05

3

1.000

1.00

1Í

^

1

0.95

;

i OSO

l/^X

0.85

0.80

/~2

\

|2ndordcr

10

j

hi Λ

jlOS

T\

h ϊ

!

15

25

φ

32.7Ì

¡22.5

20

30

[io"3i»d]

35

Required inelastic rotation

Figure 2.5 Required inelastic rotation versus load factor 2.2.3

Influence of the load factor of reference

This paragraph explains parameter 6.2 presented in table 1.3 of chapter 1. Eurocode 3 imposes to check the ductility in plastic hinges only under design loads


^

:

(2.3)


ΥΜφ

where φ,^ is the required inelastic rotation under design loads and γΜφ is the partial safety factor (see chapter 1.6.2).

But, for simplicity, the designer may want to check this ductility at collapse (mechanism of structure, instabüity,...) :


^

(2.4)


ΥΜφ

where <»,,. is the required inelastic rotation at collapse.

This latter verification is more severe because 9roq.u

^ Φι«).«'

An advantage of checking at collapse is tiiat, under die conditions mentionned in 2.2.1, tp^.u is independent of die loading patii.

Assuming that the first order analysis in die example presented in chapter 2.2.2 (see figure 2.5) is valid, then die difference between required inelastic rotations at collapse (tpreq.u) (here 32,7 10"3 rad, for λ = 1.22) and under design loads (
470

Continuous beams

2.3

This paragraph gives, for two-span continuous beams, formulas for the required inelastic rotation φ,«,.,, at collapse (plastic hinge mechanism) when performing an elastic-perfectiy plastic analysis (see §5.2.1.4 of Eurocode 3). The formulas are exact. Abacuses are provided to help die designer. Information are also given for ultimate load and first plastic hinge occurence.

If elastic return occurs in plastic hinge, then the formulas for (preq given hereafter are not applicable and particular detailed analysis shall be made (see chapter 2. 2. 1 and 2. 4. 2. 2). Table 2. 1 is a summary table which presents the different load cases and die references to formulas and charts given further in this chapter. Summary table of load cases, formulas ("Eq.n) and charts ("Fig.") for two-span continuous beams

Table 2.1

Fi,qi

Loading case aL

Fu>qu

Eq.2.5

0<
ΤΓ

JT

0<η<1 ω>0

I UT

Eq.2.9to

Eq. 2. 12 to

2.11

2.14

Ί^

Fig. 2.10

Fig. 2.11

Eq.2.19,

Gùq

τττ?

Eq. 2.18, 2.20, 2.24 or 2.26

Eq. 2.16 or 2.22

2.21, 2.25 or 2.27

Eq. 2.28 to 2.31

Fig. 2.15

Fig. 2.16

Eq. 2.34

JT

1ST f

Æ Γ

"Zr-

Eq. 2.32

2.36

Eq. 2.33

Fig. 2.21

am

I

-Η

Fi, qi : loads related to first plastic hinge occurrence :

Fu, qu : ultimate loads

required inelastic rotation at collapse

any loading

Procedure based on elastic analysis in §2.3.4 J7J7'

ιI

777?

7779"

L

I 1

or

Fig. 2.19

I


| Eq. 2.35

Fig. 2.18

φ

k gL

Φ««*«

Fig. 2.8

nL

7//τ~

Fu/Fh q^

*

(preq.d under design loads : Eq. 2.38 and Fig. 2.24

I

1

471

For information, Table 2.2 allows to cany out quick plastic global analysis. The table provides for end spans and intermediate spans of continuous beams whh constant cross-section.and steel grade :

- die plastic bending moment resistance Mpl required to form die plastic mechanism at collapse, - die collapse loads Fu or qu and, - die position of plastic hinge in span at collapse. Table 2.2 Plastic global analysis of continuous beams with constant cross-section Loading case

Intermediate span

End Span

M*

Hi

Mpi *%>

M*

Mp!

required Mpi

Collapse load F|i»Qu

required Mpi

Collapse load

ï"u»flu 16 M pl

ll,66Mpl 16

11,66 a

a = 0,5l

= 0,414 j

fi

F£

6M pl

6

8M,pl

8 a = 0,5l

c

F£ 4

c-

=

6

a=

C

:·

C

FFF 1

:'

I-

C

C

!

i

¿

3

3

3M,Pi

>'


21 3

El

4M

Fab li

2¿M,Pi

4

3

É.

,

a

(b+l)Mpl

Fab b+e 2.3.1

6 M pl

Hi

4M pl

ab

Two span beam with a concentrated load

Figure 2.6 472

= 0,5

ί

ab

2.3.1.1 Ultimate load

Only one plastic mechanism may be reached. The ultimate load Fu can be determined by using the kinematic theorem : the external work We done by the load as it moves through a virtual displacement Δ is equal to die internal work W¡ absorbed at die plastic hinges as they rotate through corresponding angles (see figure 2.7). We = W¡

with:

We = FaA

and:

W¡ = Mp, (
Figure 2.7 From above expressions and geometrical considerations, we can obtain M pl

1+a

F.=" L a(l-a)

:

M_

=k

(2.5)

L

"

Fu does not depend on r\

Note :

Figure 2.8 gives the coefficient k as a function of the position parameter a. m¿

91

-21 __

20 1Q

ty

17

17

18

16

ku

15 14 14

-

14 13 12

^

\\ \ \

11

lu >

^Y

9 S

Ν

7

7

6

r 0. ι

o.:2

0. ì

0.'y

o.:5

0.
o: 7

0.Si

5

0.<

a Figure 2.8 Let us assume that the first plastic hinge occurs for the load Fj. The expression of this load Έχ can be derived by considering that the maximum moment in the beam calculated from an elastic analysis, reaches the plastic moment Mpl. Two cases must be investigated depending on where the maximum moment occurs :

473

Case A

First hinge at the loading point if

:

M pi

F, = -

then

α<^2(η+1)-1

:

2(η+1)

(2.6)

L α(α-1)[α(α+1)-2(η+1)]

Case Β : First hinge at the intermediate support if:

then

α>^2(η+1)-1

M Pi 2(η+1) =L α(1-α )

F,

For the particular case ij =

(2.7)

1 :

M pi

F,=-

L α(α-1)(α

(2.8)

+α-4)

(α<^2(η + 1)-1 = 1) 1

0,8 -

η

(Λ)

0,6 -

>I^

2

£-

0,4 -

>

^.

Γ]

oa . 0

0 0, ι

oa

0,3

0,4

0,5

0,6

1

-£. 0,7

ί 0,8

0, 9

α Figure 2.9 The ratio FJF1 is an indicator for die interest of die plastic analysis. This ratio is easily derived from die previous expressions. Case Α

α<^2(η+1)-1 f

-^

= (α + 1) 1-

r νΛ α(α+1)

V

Case Β

:

α>τ]2(τ\+ΐ)-1

(2.9)

2(η+1) :

_ (q + ir

Fu

(2.10)

2(η+1)

F,

= 0,25(α + 1) (4 - α - α2 )

ΡθΓη = 1 Fi

With the restriction : 0

<α<1

474

(2.11)

17

1.6

1

-

-

]L

5 1.5

m.

1.4 -

Ά

Fa/Fi 1.3 -

Jm>^_^IS1.2 -

0?

l

2

4

1.0

-CÃ \£2 L/

f / /

- 1.4

/

0.3

1

£-

- 1.5

/

0.1

1.6

i

(p^ ^sT

-

~·

- 1.3

/

0.4 - 1.2

0.5

[

^fo .5 ^

1.1 -

- 1.1

0.1 N ι

τ

.

0. 1

0.2

-

0.4

0.3

0.5

0.6

0.7

0.8

1

0. 9

a Figure 2.10 Note :

Figure 2.10 shows that the loading capacity is increased up to about 22% for equal spans owing to a plastic analysis, when the concentrated load is near the middle of the span.

2.3.1.2 Required inelastic rotation

φ^α

The required inelastic rotation may be obtained by integrating the bending moment under the load increment (Fu - Fj) along the beam in which a perfect hinge has been introduced at die plastic hinge location. Therefore the required inelastic rotation is die rotation in the perfect hinge under (Fu - Fj).

If α<Λ/2(η+1)-1

(case A

:

1st plastic hinge at loading

point)

MpiL -α2-2α+2η+1 Φ req.u

6EI

α

475

(2.12)

If α>>/2(η+1)-1

Β : 1st plastic hinge at support)

(case

MnlL , Φ»,Λ=-7=-·(« 2+2α-2η-1) 6EI

(2.13)

If η=1 Mp,L (q + 3)(l-a) Treq.u

6EI

(2.14)

α

The required rotation φ^^ may be expressed by:

MP,L Φ«η.υ=1εΓ·

Figure 2. 1 1 shows curves giving k, versus α for various values

Figure 2.11

476

(2.15)

m

of die ratio η.

2.3.2

Two span beam with uniform distributed loads

Figure 2.12 Note : Uniform loads are assumed not to vary independently in each span (proportional loading). 2.3.2.1 Ultimate load

Three possible mechanisms may occur (see figure 2.13), but we assumed that ω £ 0. Therefore the mechanism C is not considered. Depending on die values of ω and η, mechanism A or mechanism Β will be formed witii the first hinge occuring eitiier at support or in span (see figure 2. 14).

Figure 2.13 The methodology is similar to the one described in

If ω η2 < 1

Case

§

2.3. 1 for a concentrated load.

A : The mechanism occurs in the first span M. qu= (6+4^2) -f

(2.16)

a) First plastic hinge on the intermediate support

The first plastic hinge occurs on the intermediate support condition is fulfilled :

4 (3

or

- 2V2 ) < ^^-

0,6863 <

Ι+η

1+ωη 1+η

< 4 (3 + 2 V2 )

if the following

(2.17)

< 23,3 137

Let us assume that qt is the loading for which the first plastic hinge occurs, then :

477

q'

"

qu

8Mpl

ι+η

L2

1+ωη3

(2.18)

3+2-JÏ 1+ωη3

so

10-

(2.19)

1+η

4

<1ι

\

\

9

\

\

8-

ν \ \

\

7-

\

2

\

£

Ν

\

6-

^

1

\ \ Ν

5

ω

-

V

4

1

ν^Λί

\ 2^-58 Γ"*--.

^"--.^

32

"""""--·

=

c-i 1

Sí-

-

'-"

0()

0.1

0.2

0.4

0.3

0.5

*J -*<· 0.6

Ja

Μ-

0.7

0.8

-mW-

0.9

1

η Figure 2.14 b) First plastic hinge in the first span

If the first plastic hinge occurs in the first span, then

qi=-

128M pi

1+η

(2.20)

3+4η-ωη

q^ 3+2V2 so

64

478

3+4η-ωη

1+η

(2.21)

If ωη2>1

Case Β : The mechanism occurs in die second span

Μ

l

._

,

u=-|-(6+4V2)

Qu"

L2

(2.22)

-.

ωη

a) First plastic hinge at the intermediate support

The first plastic hinge occurs at the intermediate support if the following condition is satisfied :

4(3-2^2) <

or

£^

ωη (1+η)

1+ωη

0,6863 <-

ωη (1+η)

<4(3+2λ/2)

(2.23)

< 23,3 137

Let us assume that qi is the loading for which die first plastic hinge occurs, then

:

i+

8Mp,

qi~ so

(2.24)

1+ωη3

L2

iL·

3 + 2V2

1+ωη3


4

ωη (1+η)

(2.25)

b) First plastic hinge in the second span

If die first plastic hinge occurs in the second span, then :

128 4_.

ωη

1+ωη 3 Y

(2.26)

ωη2(1+η).

so

qu_3+2j2 q,

If ω η2 = 1

64

4 κ.

1+ωη

ωη'(1+η);

Plastic resistance is reached in die two spans at the same time.

479

(2.27)

Figure 2.15 shows the evolution of the ratio q^ related to η for various values of ω (boxed values). The curve peaks indicate die limit between mechanism A and mechanism B. 1.5

1.457

1.4

1.3

%^l 1.2

-£

1.1

0

0.1

»

0.2

0.4

0.3

0.5

0.7

0.6

0.8

0.9

1

η

1.5

1.457

Figure 2.15 2.3.2.2 Required inelastic rotation φ«,,., Case Α:

ωη2<1

If the first plastic hinge occurs at the intermediate support, then MplL Φ,req.u

EI

+l

-!

3+2-JÏ

3

12

480

,

(1+ωη3)

:

(2.28)

If the first plastic hinge occurs in die first span, then MpiLn+i Φreq.u

Case Β Γω τι2 >

EI

3

:

. , 1+ωη (V2 + I) 1-(V2 + 1)2 4(1+η).

(2.29)

11

If the first plastic hinge occurs on the intermediate support, then : MplL

η+1 3+2^2 1+ωη3

EI

12

Φ,req.u

If the first plastic hinge occurs in the second span, dien MpiL λ/2+1 Φ,req.u

(2.30)

ωη :

, 11+ωη

,

η+1- (λ/2+1)2

EI

4ωη

3

L

(2.31)

Figure 2. 16 shows curves giving a factor k,. versus the ratio η for various values and so die required inelastic rotation φ«, u can be obtained by :

of the ratio ω

Mp,L

Φ.ϋ = Κ

0.3

,Ί

J.

-

m

lio

Ί

Γ

ι

"Λ

ν-""'

ν-\

.-

ΚΙ

kr 0.2

\\

-

·"""

0.1

-

α>

0.1

0.2

ν

ín

3

ι

7~Ρ

1

ΠΠ

ΓΗ

4)

\

[.

[Γ

Π

Γ

Γ?

r

-».Ι ΪΠ

4

Κ\ c

TJ

0.3

ι

Ι

\

|3

1/

0.4

0.5

η

481

0.6

0.7

0.8

0.9

ι

1

0.4

m

ii-

0.3

kr IÕ51

0.2

0.1

-

ϋ] IOTI

0 0

0.1

0.2

0.3

0.4

0.8

0.7

0.6

0.5

0.9

1

η Figure 2.16 Note

In the case of increasing the loading from zero to reach the plastic mechanism, the first plastic hinge may occur in one of both spans. This first plastic hinge is accurately located at the maximum moment location which is determined from an elastic analysis. While the loading increases to reach the ultimate loading, the yielding spreads in the zone of this first plastic hinge because of redistribution of bending moment. At collapse the final position d of the concentrated plastic hinge in the span will be differentfrom its initial position:

d = fsj2-l)L1'span The designer should be aware of this phenomenon when applying stiffeners. Stiffeners must be applied at the position of plastic hinges according to the state of collapse.

d = 0,414 L

X

^Q_ Mechanism A 0,414η L

X-

-X

&

Mechanism Β

X-

r,L

-X-

-X

Figure 2.17 2.3.2.3 Same uniform distributed load for both spans (ω = 1)

For this case, the mechanism A always occurs since :©η2ί1 The first plastic hinge always occurs at the intermediate support 8M pi


L2

η2

482

-η+1

:

(2.32)

The collapse is obtained for die load q,

:

= ^L(3 + 2VI) qu=-¡I

(2.33)

The second plastic hinge is located at the distance (λ/2 - 1) L = 0,414 L from the origin

of die

first span.

The following ratio is an indicator for die interest of die plastic analysis :

1.5

,

1,4

.

1,3

.

q u/q,1 1,2

qu

=

3+2-JÏ t ,

ν

(η2-η+ΐ)

(2.34)

1,1

1

.

()

0,1

0,2

0,4

0,3

0,5

0,6

0,7

0,8

0,9

η Figure 2.18

Required inelastic rotation φ_

u

:

Mp]L 3+2λ/2

%*.»

=-=ΕΙ

12

2 lα+η) (η -η-11+*Λ/2)

(2.35)

M..L

Φ»,.» = 0,4857 -f ΕΙ

or:

(1+η) (η2 -η+0,3137)

The required rotation φ«,^ may be obtained from figure 2. 19 with:

(2.36)

φ^, u

= kr

035

03 0,25

Κ

.

0,15 0,1

0,05

0

.

C>

0,1

θα

0β

0,4

0,5

η Figure 2.19 483

0,6

0,7

0,8

0,9

1

M,L

-

EI

1

2.3.3

Two equal span beam with a concentrated load and an uniform distributed load

In order to evaluate the relative influence of a concentrated load and an uniform distributed load, a numerical study has been carried out using die PEP micro program of CTICM [15] instead of a complex analytical study.

Figure 2.20 As for the previous cases, the required inelastic rotation ^.may

be expressed as:

MplL

Φ=ία req.u

ι

EI

In figure 2.21, the factor k, is plotted as a function of the ratio

:

(2.37)

μ=

0.5 0.45

0.4 \

0.3 r

0.35

0.3 0.2S

0.2

α

\?s ^r \s

^>

0.8

X >*

I0.7L

a. V ^^ yjo-sl

0.15 0.1

y

?

-"

Γ02 J

y£"U _u

\~á

_

Jõst

'ΎοΣ

0.05

i^^v i-St/^

0 0.1

0.2

0.3

0.4

0.5

Figure 2.21

484

^0.6

0.7

0.8

0.9

1

In die left part of the curves, the distributed load is predominant and the first plastic hinge occurs at the intermediate support. In die right part, the concentrated load becomes predominant and the plastic hinge occurs at the load location.

23.4

Two span beam - General case of loading

It

is rather difficult to give formulas and abacuses dealing witii complex loading cases for which parameters could be :

- Span length ratio - Relative position of the concentrated load - Ratio of the distributed loads for the first span and the second one - Ratio between the concentrated load and the distributed load

A more general method based on elastic analyses is given hereafter in order to calculate the required inelastic rotation ©, d under design loads, that is for a load level between the first plastic hinge and the second one for which the collapse occurs.

After an elastic analysis, we assume that, under design loads, the maximum moment Mm« exceeding die plastic moment Mp, , is located at a distance ßL from the origin of the beam.

,

Figure 2.22 So, ΔΜ = M,.,* - Mp! is to be redistributed. For tiiis, the following diagram has to be added to the one resulting from the elastic analysis.

Figure 2.23

485

This diagram is obtained by considering the beam witii a perfect binge at the maximum moment location, and moments -ΔΜ applied each side of this hinge. The required inelastic rotation is the rotation in this hinge subjected to -AM. Therefore the expression of the required inelastic rotation φ«,^ is given by :

AML η+1 req.d

Note :

ΕΙ

(2.38)

3β2

This expression does not depend on the loading, but only on ¿M.

It

must be checked that the plastic moment is not exceeded somewhere else in the beam mechanism must not be reached.

If λ is the load factor of all the applied loads, then :

λ"-ι

ΔΜ = Μ pi

λ! λ»

Loading factor for the first plastic hinge Loading factor at collapse

The required inelastic rotation φ^.«! may be expressed as

AML

%co.d=K

m Figure 2.24 gives the factor k, versus β and the lengtii parameter η.

M -

\\\W\\ \\ W& \\ W \\\ \ T \\ Λ\ w i ιυ \V w \\ w λ\ ν\ \\ T \\ \\ Ä γ Λ y\ W λ\ \ π V λ> \\ λ 1

V

\

\ Λ- Λ \\ \V NN VN \> y\ ys ss s \\ Sn Xs \\ \s T S/ Λ ^ ν ^. st λ ^ ss Λ s> & Si ^ Ν, ΝΛ ν Xs ^ ^ r-N

10 -

6

-

s

\Ν ΝΙ\

^U\

η

0

Zi

^ ^ s|v TS^sN

\jT_

ÌJ~

S Ν Χ

s

^

3> j*4 s^. \

>x *N

..

-

0.1

0.2

03

0.4

0.5

β

Figure 2.24 486

0.6

0.7

0.8

0.9

:

the

2.4

Frames

2.4.1

Plastic analysis using a specific program

In order to make a plastic design of a frame and determine die required inelastic rotations in plastic hinges, a global elastic-perfectly-plastic analysis (see §5.2.1.4 of Eurocode 3) of a frame can be performed with die help of a plastic analysis computer program. Those programs are generally based on die step-by-step method and can automatically take into account particular aspects such as elastic returns (see 2.4.2.2 of this chapter), second order effects, semi-rigid behaviour of connections and also M-N interaction in bending resistance of sections. All these aspects have direct influence on required inelastic rotations (see paragraph 2.2 for second order effects influence). The M-N interaction in bending resistance of sections is often necessary for columns where the influence of axial force may be not negligible. Then, an interaction curve has to be considered. During die step-by-step method, moment and axial force may vary in such a way that they remain on the interaction curve. Moreover, axial plastic elongations and inelastic rotations have to be evaluated witii respect to a "normality law".

For instance, in case of I sections with bending about strong axis, the interaction curve may be approximated by a bi-linear curve as shown in figure 2.25. This leads to introduce two types of plastic hinge : without (type M) and witii N-interaction (type M-N). Consequently, the type of a plastic hinge may change as the axial force increases during loading and a specific program can take tiiis phenomenon into account. N/Npl

i teraction curve

MMpl

Figure 2.25 2.4.2

Plastic analysis of simple frames using an elastic analysis program

2.4.2.1 Scope

A global elastic-perfectly-plastic analysis can however also be performed with the help of a simple elastic analysis program if the following assumptions are satisfied. Assumptions :

The axial force and the shear force must not reduce the plastic bending moment resistance. The frame is "rigid" according to Eurocode 3 (§5.2.5.2) so that second order effects need not be taken into account in global analysis.

487

These assumptions are needed because a plastic hinge is considered hereafter like a perfect hinge acting under a constant moment Mp,. If it is expected that die bending moment resistance in a section will be reduced by die axial or shear force effect, the user can take that into account by reducing Mp, before starting the method.

2.4.2.2 Methodology

This method is known as die "step-bv-step method" and consists in a succession of elastic linear analyses in which plastic hinges are replaced by perfect hinges to ensure no variation of bending moments in the yielded sections (see figure 2.26). Each analysis, that is each step, is performed on the frame witii a perfect hinge more tiian in die previous step, at die location where a new plastic hinge occured in die previous analysis. Each step is limited at a variation Δλ of the load factor λ by the occurence of a new plastic hinge where the plastic moment is reached in a new cross-section somewhere in the structure. Inside each step, variations of displacements, internal forces and moments, and rotations in perfect hinges are calculated and can be summed through all the previous steps in order to determine tiieir total values.

mechanism

4.

Ä9j=o i

0

δ, D

rotation in the first hinge

Δφ,

Δφ.

Δφ,

A3


φ

Figure 2.26 Figure 2.27 shows a flowchart of die methodology, considering tiiat die loading increases in proportion to the load factor λ. This metiiod allows die designer to calculate the required inelastic rotation in die plastic hinges just before collapse and even for each step of die incrementation.

This metiiod is radier easy to apply for a simple structure (simple frame, continuous beam, ... ). However, in case of more complex structures where the state of the structure under design loads or at collapse may involve a great number of plastic hinges, this methodology may become tedious and a specific plastic analysis program is necessary. The user must be aware of a particular phenomenon which may occur and has to be taken into account : an elastic return of a plastic hinge (see figure 2.28). During the step-by-step procedure, die sign of die rotation in a perfect hinge may change. This means that, because of die bending moment 488

redistribution, the corresponding plastic hinge takes an elastic behaviour. Then die structure must be modified by suppressing the perfect hinge and die elastic analysis is to be started again, hi the plastic hinge where elastic return occurs, die remaining inelastic rotation Remaining (see figure 2.28) constitutes a local discontinuity in the structure and must be kept for following step of calculations. Geometry - Steel Support conditions Loac

I

i =0

Initializations

Moment diagram : M0- 0 Shear force diagram :Vo=0 Displacements :Όο= 0 Hinge rotations : <ζ,-0

ELASTIC ANALYSIS i (with i hinges) (with applied design loads)

If i = 0 : Support conditions are not valid If i > 0 : A plastic mechanism is reached Results

of the elastic analysis (with i hinges)

Moment diagram : m¡ Shear force diagram : v¡

Displacements : 5¡ Rotations in the hinges : ψί

Elastic return Suppression of the perfect hinge

Step number : j - i +

1

* Research of the cross-section for which

Akj

= (Mpl - M y m ¡ is minimum

Χ Load factor:

ákj

Xj = X¡ +

Χ

State of the structure at the end of the step

j

Xj F)

(for the loading

Momentdiagram : M¡= Mj + AXj mj Shear force diagram :Vj = Vj+A^ Vj Displacements : Dj = Dj +

Δλ- g.

I Rotations in the hinges at the end of the step

<&j=

Φ, +

AXjÇ,

Τ i-j

A new perfect hinge is introduced in the structure

Figure 2.27 489

j

M

Elastic return


Mpi

plastic hinge behaviour

_^_

ΔΜ<0 "^^^^

<

'^mm-f'^^T."

"^^Î> Mpl

i

0

inelastic rotation

'remaining

Figure 2.28 2.4.3

Indicative values of φ,*, for pre-design of simple portal frames

A study has been conducted in order to determine the required inelastic rotation φ,«, of formed plastic hinges for a given type of portal frame. Six types of portal frames have been identified, and for each type, a parametric study has been made [13]. So, 79 frames have been studied. A simplified method was adopted in order to obtain a realistic design. An elastic analysis was first performed in order to check Serviceability Limit State requirements. Then a plastic analysis was made in order to check Ultimate Limit State criteria and to determine required inelastic rotations under design loads (λ = 1) and at collapse (λ) : mechanism or instability. Some of die frames were sway frames according to Eurocode 3 and the second order effects were taken into account. All analyses have been conducted witii the PEP micro program of CTICM [15].

Table 2.3 summarises the calculated indicative maximum values of φ^ which have been determined for each type of frame. This table gives to die designer an idea of die magnitude of φ^ for such frames and must be considered as a help in order to choose a starting section in a pre-design procedure.

490

Table 2.3 - Indicative maximum values of φ^ [rad] for pre-design of simple portal fiâmes

Type of frame

[ 1

1

,

[

1

;

3

1

1

l

0,009

0,020

0,015

0,032

1

0,026

0,035

0,034

0,040

0,013

0,014

0,006

0,008

1

1

'

I

.

\

4

9req.ii

i

1

ι

1

fTBq.d

ι

i

2

At collapse

|

l

1

ι

Under design loads

i

l

\

1

j

5 ι

ι

1

i

1

f

i

6

;

f

]

]

τ?

s

j

^s

l\ li

τ s

491

Introduction - presentation of Feldmann's model

3.1

This chapter deals with available inelastic rotation of plastic hinges. It presents a simple procedure for calculating the inelastic rotation
3.2 1/

Limitations and assumptions of Feldmann's model concerned load cases: the available inelastic rotation q>av of plastic hinge can be determined for the following load cases: A. concentrated load, bending about major axis y-y of cross-section : Eurocode 3 [3] provides a formula for the design plastic resistance bending moment about major axis Mpi y r¿ :

Β.

concentrated load, bending about minor axis z-z of cross-section : Eurocode 3 [3] provides a formula for the design plastic resistance bending moment about minor axis Mpi z r¿ :

492

concentrated load combined with axial compressive forces, bending about major axis y-y of cross-section : Eurocode 3 [3] proposes formulas for the reduced design plastic resistance bending moment allowing for the axial force, MN y Rd :

The extension of Feldmann's model to other load cases is presented in chapter 3.5. Tables of Appendix 1 provide design plastic resistance of I and H cross-sections (Mpl y Rd , Mpl 2 Rd , MN.y.Rd > )

2/

classification of cross-sections: this model applies for all Eurocode 3 class 1 and class 2 crosssections (plastic distribution of stresses over the cross-section); Eurocode 3 class 3 (elastic distribution of stresses over die cross-section) and Eurocode 3 class 4 (local elastic buckling) cross-sections shall be excluded (see [3] and chapter 1.6.1 for definition of classes);

3/

steel grades: all steel grades available in Eurocode 3 Part

can be applied in the formulas: 235 MPa < fy < 460 MPa. Indeed the reliability of the model has been demonstrated by tests results witii steel grades in that range [8]. Eurocode 3 [3] [4] allows for plastic global analysis witii steel materials characterized according to EN 10025 [1 1] and EN 101 13 [12] as given in table 3. 1; 1. 1

Remark 1 about nominal values of yield strength fy : Eurocode 3 Part 1.1 [3] gives a simplified variation of fy in function of the material thickness (thickness ranges: lower than 40 mm, from 40 mm to 100 mm), with values of fy - for certain thickness ranges - greater than the values specified by EN 10025 [1 1] and EN 101 13 [12] (thickness ranges: lower than 16 mm, from 16 mm to 40 mm, from 40 mm to 63 mm, from 63 mm to 80 mm, from 80 mm to 100 mm, from 100 mm to 150 mm) (see figure 3.1). As Eurocode 3 always provides yield strength fy equal or greater than values guaranteed by die specifications of delivery conditions (EN 10025, EN 10113), the available inelastic rotation (pav resulting from fy of Eurocode 3 are always safesided (equal or lower) in comparison of the inelastic rotation q>av resulting from fv of EN 10025 or EN

*

10113.

Remark 2 about overstrengthening of steel material : The real yield stresses of the steels delivered by mills are always greater tiian the nominal values of the steels indicated in the specifications EN 10025 [11] and EN 10113 [12]. As presented in chapter 1.6.2, a statistical procedure has been used to evaluate partial safety factors Ytøa, related to available inelastic rotation
493

ultimate load target (i.e. die design load level) remains using die nominal yield values [17] ( see figure 3.2 : (plover <
Nomimal Steelgrade fu [Mpa]

fu/fy

[Mpa] 235 275 355

360 430 510

1.53 1.56 1.44

275 355 420 460

390 490 500 530

1.42 1.38 1.19 1.15

EN 10025 standard S 235 S 275 S 355 EN 10113-3 Standard S 275 M S 355 M S 420 M S 460 M

40 mm
t<40mm A5.65

Sy

[%]

[Mpa]

[Mpa]

Η

26 22 22

0.11 0.13 0.17

215 255 335

340 410 490

1.58 1.61 1.46

24 20 20

24 22

0.13 0.17 0.20 0.22

255 335 390 430

370 470 500 530

1.45 1.40 1.28 1.23

24 22

Η

19 17

19 17

Sy

[%] 0.10 0.12 0.16

0.12 0.16 0.19 0.20

Notes:

*) r is the nominal thickness of the element : - of the flange of rolled sections (t = tf), in general, or - of the particular elements of welded sections for definition of parameters see figure 1.9 (chapter 1.5)

Steel grades

a

Φ

t

Minimum guaranteed yield strength ReH (or Rp0,2) in function of nominal thickness t of material

ReH(orRpO,2)[Mpa] 460

S 460

460 430

440

430

420

S 420

420

410 390

400

400

390 355 S 355

370

360

355

335

345

340

335 325 315

295

275

S 275

275

255

265 255

235

S 235

245

235

235

215

225

225

215 195

0 Legend:

16

40

63 80 100 - EN 10025 (for S 235, S 275 & S 355 steel grades), - EN 10113 (forS 275, S 355, S 420 & S 460 steel grades). ; Eurocode 3 specifications (for S 235, S 275, S 355, S 420 ά S 460 steel grades). :

Euronorms specifications

:

Figure 3.1 494

150

t [mm]

Figure 3.2 : Decreasing of the rotation requirement in case of actual yield stress which are greater tiian the nominal yield stress

4/

lateral restraint: the model does not take into account die lateral-torsional buckling, therefore, the ordinary rules of plastic analysis concerning lateral restraint shall be respected (see chapters 1.4 and 1.6.3 or [3], [5], [10]).

5/

effect of stiffener: the given formulas are only valid when stiffeners are applied at the location of plastic hinges. On the otiier hand, if several stiffeners ( more than necessary) are applied in the zone of plastic hinges, the available inelastic rotation can be multiplied by a factor 2 or 3.

In practice, local instabilities due to concentraded loads have to be avoided during the whole rotational process in plastic hinges, by introducing at location of plastic hinges

:

sufficient stiffening of web, for My and My - Ν load cases and, sufficient stiffening of flanges, for Mz load cases. Sufficient stiffening can be guaranteed witii usual welded steel plates (see figure 1.8) but also by other means or by consideration of service conditions to ensure economical design : support of secondary beam, support of slab, steel sheet deck (floor, wall, roof) with sufficient stiffening, masonry, concrete walls, concrete infilments, .... In case of distributed loads the need of stiffening is to be appreciated by the designer. 6/

effect of shear forces: the model provides conservative results in case of high shear (meaning Vsd > 0.5 VpiRd and reduced plastic design resistance bending moment MyRd (^ Mpi j^j) as defined in chapter 5.4.7 of Eurocode 3). The model shall respect Eurocode 3 rules about shear resistance of cross-sections, Vsd ^ Vpi.Rd (see Appendix 1 for tables witii plastic resistance of I and Η cross-sections (Vpi.y.Rd, Vpi^Rd, ...)).

7/

brittle fracture: no brittle fracture is considered.

8/

fatigue effects: no fatigue effects are taken into account.

9/

type of loading: only static loading is permitted (no seismic loading, no dynamic loading).

495

Influence of parameters on inelastic available rotation

3.3


The values of inelastic rotations of plastic hinges are mainly influenced by die dimensions of die cross-sections as well as by the steel grades of web and flange. They do not depend on die span length (L span) if lateral-torsional buckling is avoided. Some general remarks on influence of parameters have been made in table 1 .3 of chapter 1 . The graphs which follow give an idea of influence of cross-section dimensions and of steel grade on die inelastic rotation (paV. Note tiiat (pav decreases widi increase of slendemess of cross-section members as well as witii increase of yield strengdi. «Pav

[rad]

0.2 τ 0.18

0.16 f 0.14 0.12 0.1

-

fy= 320 MPa h= 0,27 m b= 0,135 m tw = 0,0066 m

0.08 "

0.06

-

0.04

0.02

t

b/tf

0

I

8

10

12

14

18

16

20

22

24

Figure 3.3 Influence of flange slendemess b/tf on (pav (b = const)

0.14

fy = 320 MPa h = 0,27 m b = 0,102 m tw = 0,0066 m

0.13

0.12

-

0.11

-

Figure 3.4 Influence of flange slendemess b/tf on
496

26

0.18

Φ»ν [rad]

0.16 0.14

0.12 0.1

0.08

0.06

fy = 320 MPa h = 0.27 m b= 0,135 m tw = 0,0102 m

0.04

0.02

d/tw

0

1

20

15

25

30

35

40

45

Figure 3.5 Influence of web slendemess d/tw on
0.2

φ .ν I»d] -

0.18

-

0.16

-

IPE 270

0.14 0.12 0.1 -

0.08 0.06 0.04 -

f y [MPa]

0.02 -

235

275

315

355

395

Figure 3.6 Influence of steel grade fy on
497

435

475

Feldmann's formulas for values of φ«ν

3.4

Three types of load case for 3-point bending system are possible:

A. Β. C.

type of load: concentrated load witii bending about major axis y-y of cross-section (My), type of load: concentrated load witii bending about minor axis z-z of cross-section (M~), type of load: concentrated load whh combined axial compressive forces, bending about major axis y-y of cross-section (My - N). The following symbols are used to describe cross-section dimensions:

Figure 3.7

In all formulas:

Δσ =150 MPa, fy A = yield strengdi in flange must be introduced in MPa, fy.w = yield strengdi in web must be introduced in MPa,

E

= modulus of elasticity (=210 000 MPa),

Npi = plastic axial resistance of cross-section (=A fy),

Mpi = plastic resistance moment of cross-section (=Wpi fy),

A

= sectional area of cross-section.

Available inelastic rotations (pav of plastic binge are given in radians[rad]. The following formulas provide characteristic values of available inelastic rotations
498

Ά..

type of load:

concentrated load

witii bending about major axis y-y of cross-section

[rad](3.1)

D,

type of load:

concentrated load

witii bending about minor axis z-z of cross-section

&r

^)MZ

-^>2-

aL

,. (1-C0L

X

v

^

/ tF

tF-

0<α<1

This formula is only vahd when stiffeners are applied at die loading point.

\

2C

f

w

Pl-z y

2t,
where

Φκ,ρΐ=7

(h'2tf)^fy 4

w

[rad]

b3(fyfl+ACT)

0.2

~^\

'>

1= lengtii of the beam.

499

(3.2)

In practical cases die ratio 2b/l are rather small. One can therefore assume die safesided value

of Φκ,ρΐ = °·2·

C/.

type of load:

concentrated load with bendinp; «hont major axis v-v

of cross-section

combined witii axial compressive force

Limit of application of these formulas is set to N/Npi £ 0.5. To apply the appropriate formula, the following criteria should be considered: 0ifJL
A

Np,

^

_ Aweb Afflai

5

men me neutral axis should be in web and


is evaluated

from (3.3): ty

""^

N^My

neutral axis

____J.

fyB

ι

[rad](3.3)

®

if

;>

Npi

A 2btf =-^gsLÌ A ^ AtotaiJ

,

then the neutral axis hes m flange and 9av is calculated from

formula (3.4):

fy

^N^My -t-

^Jl'Ura

imPfÊm^mVmÍ^.

ft,

.

[rad]

where:

Ν is die applied axial load 500

(3.4)

1-

-

M*N.y =W\ "pl.y f. *y

Af.

~ £^p 1--

with the limit MN y * MpI y(= Wpl y fy )

(3.5)

2A (see Eurocode 3 [3] or Appendix 1)

Nfl=(tf-tM)bfyil

*Μ="2b 1-

I

(3.6)

N

(3.7)

*v

Ρ _4Ebtj

(3.8)

5hz

Pu=btf(fy.fl + Aa)

(3.9)

The following value of Niimit defines the border between formulas (3.3) and (3.4) for
1

.si Γ

fy h-tf

neutral axis 1

k

N^)My

fy^

Nlimit=(A-2btf)fy.w

(3.10)

Extension to other load cases

3.5

This chapter contains practical information about application of Feldmann's model to other load cases inducing gradients of moments different from the case of 3-point bending beams: other load cases (distributed loads, concentrated loads witii combined axial tensile forces), other spans arrangements (continuous beams, frames, ...), other load arrangements (alternative distributed or concentrated load, ...).

They are summarized in table 3.2.

It should

tiiat because cpav does not depend on spans arrangement if sufficient lateral restraint is imposed to avoid lateral-torsional buckling (die lengtii L does not interfere in the Feldmann's formulas), the same model is valid to all possible plastic hinges, no matter if they appear in die span (with sagging bending moments) or at die support (with hogging bending moments). be noted

501

Table 3.2 Load case

Applied model & formulas

& span arrangement

^iH

&

Γ

=>

^w

©

i ^H ©

^ujll^

The available inelastic rotation φ av for plastic hinges (h) or (x), (y) and (z) are all identical and equal to : values given by formula A (3.1) fori values given by formula Β (3.2) forH

I

l.J

I

οτ η

1ÍH

££

=>

ΚMJXX>" Λ @

-^jp-

The available inelastic rotation φ av for plastic hinges (h) or (x) are all identical and equal to : values given by formula A (3.1) for I values given by formula Β (3.2) for η

r

=>

h

ws

xqijp7 © The available inelastic rotation
Ν, =>

.Ν

-w

^qjp^

sfr

®

If Ν compression (->H

H*j : The available inelastic rotation φ av for plastic hinges (h) or (x), (y) and (z) are all identical and equal to values given by formulas C (3.3) and C (3.6) fori

If Ν tensioni-

Conservative values for the available inelastic rotation
502

In case of distributed load, a realistic approach consists in considering the case of concentrated load, because there are no experimental results to confirm the model for this kind of load case. On the other hand, test results on 4-point bending beams have delivered greater available rotation capacities in comparison with 3-point bending beams [7]:

Ρ Ρ


trit

ΎΤΤΓ

)

avi ¿cr Jj >
-^r>

)

According to numerical simulations presented in figure 3.8, if sufficient lateral restraint is ensured. cpav (for distributed load) = cpav (for concentrated load). For lateral restraint rules - see chapter 1.6.3.

M

il

MpLj


(-"*)

-DL/LR1

"-Ό

CL/LR1

"Γ

C CL/LR2 DL/LR2

φrav I,»-"-"-»/ ('01i_i_i'>

m.

Φ

load case

y

M

:

distributed load (DL)

concentrated load (CL)

* restraint (LR1) :

Έ

E

7ff7

7π7

·"

E

x

(LRl)

I

I i

Ϊ

χ

-χ

(LRl}

ζ

lateral restraint (LR2)

:

£-

χ

-Κ

(LR2)

Figure 3.8

Therefore,

503

X

I

(LR2)

Tables with values of q>av f°r I and H cross-sections

3.6

The tables which are included in this chapter are derived from the Feldmann's formulas presented in chapter 3.4 and based on assumptions detailed in chapter 3.2. They have been prepared to give a quick and easy estimation of
IPE, IPE A, IPE O, HE AA HE A HE B, HE M, UB and UC. The following steel grades are considered in the tables (see table 3.1 with fy depending on material thickness according to Eurocode 3): 3.6.1 Tables of tpav for S 235 steel grade (fy = 235 MPa for tf < 40 mm), 3.6.2 Tables of
Inelastic rotations can be provided only for "plastic" cross-sections, therefore only classes 1 and 2 cross-sections according to present Eurocode 3 rules [3] [4] are covered in the tables. For crosssections in classes 3 and 4 , there are no available inelastic rotations ( cpav^O) and class 3 (elastic) cross-sections (=> (pav=0) is in Eurocode 3 [3] [4] defined by the highest (the least favourable) limitation of its compressed elements (web and/or flange). These limitations are presented in table 3.3 (issued from table 1.5a).

Table 3.3 : Border between class 2 and class 3 cross-sections according to Eurocode 3 [3] [4] Load cases

1

+

*

1 1

ίy r

Class of elements of cross-sections

L-Å tw

£¿íι

d I

Flange class

Web class

c/+ <11ε N
d/t

which is equivalent to

<83ε

456ε 13α -1

_d_

t...

Ν

where 0,5 < α

V

Note:

:

= ^235/fy

:

w

+ : stresses in compression fy (N/mm2)


-+0,5

2dVy

j

stresses m tension

235

8 (if t <, 40 mm)

8(if40mm¿t£l00mm)

504

275

355

420

460

0,92

0,81

0,78

0,71

0,96

0,84

0,78

0,74

Three load cases are represented in the tables:

A.

concentrated load with bending about major axis v-v of the cross-section (My): given in the column 2, they are calculated from the formula (3.1).

B.

concentrated load witii bending about minor axis z-z of the cross-section (M21: cpav values are given in the column 3, they result from the formula (3.2) (witii safesided φκ.ρι = 0,2).

C.

concentrated load with bending about major axis v-v of the cross-section combined with axial compressive force (My-N): tables provide four columns for this load case, because the values of tpav are not constant for a given cross-section and depend on the applied normal force N. Figure 3.9 shows the influence of that force on inelastic rotation cpaV. In formulas of Feldmann's model, the normal force Ν is also expressed in terms of N/Npi ratio,with Npi meaning axial plastic force (=A fy).

(pay values are

Φ,ν [rad]

neutral axis in web (formula (3.3))

neutral axis in flange ^(formula (3.4))

0,1

0,4

limit of application of φ model ¡

N/Npi [-] 0,3

0,4

ο·5

Ν force [kN] Nlimit

0,5 Npi

Figure 3.9 Column 4 furnishes values of maximal normal force which can be applied to the cross-section, in other words, the limit of application of the model Ν,-αχ is calculated as a minimum of the two

following values:

limit of application of
Values of Numit (see formula (3.10)), listed in column 6, correspond to the transition of neutral axis from web to flange (see figure 3.9). Thus, if values of Ν are lower than Niûnit, Niimit.Obviously N]^ can not be greater than Nmax·

505

For sections in classes 3 or 4 in case of My loading, tables deliver no available inelastic rotations for N-My loading (
All available inelastic rotation (pav given in radians [rad] in tables are characteristic values and must be divided by appropriate partial safety factor ΎΜφ to obtain design values of available inelastic rotation
:

=
where ΎΜφ

For
related to
1,52

in columns 2 & 5

(3.1) & (3.3)

1,73

in column 3

(3.2)

2,02

in column 7

(3.4)

Notation of symbols used in die tables: (Pav

Mv Mz

Ν Nmax

available inelastic rotation of plastic hinge, [rad]; applied bending moment about major axis y-y; applied bending moment about minor axis z-z; applied axial force; limit of application of die formula My-N, [kN], ="-" in tables, if sections are in class 3 or 4 for My loading, =minimum (0,5 A fy; Νΐχ,^ between class 2 web and class 3 web for My - Ν loading),

where Ν^^

= minimum dtwfy; 456ε -5,5

d W j

V

4θ 6.5

j , (see table 3.3),

(Note : if the class of a cross-section submitted to My loading is lower or equal to 2 and if, the web class for Ν loading is also lower or equal to 2, then Ν border does not exist); Niimit border between two Feldmann's formulas : (3.3) and (3.4) (see formula (3. 10)) (with Numit < Nmax , in tables) fkN]; Npi plastic axial force (=A fy). The following flow-chart n°2 explains the elaboration of the tables and the way to use them.

506

Flow-chart n°2 Internal force and/or bending moment applied to the cross-section

U

Β

My.Sd*0 My.Sd

*

Nsd 0

Mz.5d

"0

*0

(compression)


=0

column 2

of tables


column 7

of tables

φ3ν"0 columns of tables

ϊ ^av.d = ^av ;ΎΜφ ' with appropriate safety factor γΜφ

507

3.6.1

Tables of
508

IPE

Available inelastic rotation (pav

IPE A -IPE O

Noie: ÇaV must be divided by ff^a (Μβ chapter 3·6)

1

2

5

Δ -J

L L-

r^

_>y

ι I

ïc

-ί

Nmax

[rad]

[rad]

[kN]

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 χ 137 IPE 750 χ 147 IPE 750 χ 173 IPE 750 χ 196

0,520 0,390 0,321 0.278 0,248 0,227 0,211 0,201 0,191 0,174 0,163 0,157 0,156 0,149 0,144 0,141 0,139 0,141 0,096 0,103 0,127 0,147

0,356 0,343 0,336 0,331 0,325 0,322 0,317 0,316 0,313 0,304 0,299 0,299 0,303 0,304 0,306 0,311 0,313 0,319 0,290 0,289 0,312 0,332

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

0,398 0,307 0,247 0,202 0,178 0,169 0,158 0,159 0,152 0,139 0,134 0,130 0,130

0,326

IPE Ο 180 IPE Ο 200 IPE O 220 IPE Ο 240 IPE Ο 270 IPE Ο 300 IPE Ο 330 IPE Ο 360 IPE Ο 400 IPE Ο 450 IPE Ο 500 IPE Ο 550 IPE Ο 600

0,273 0,245 0,234 0,224 0,212 0.197 0,188 0,188 0.176 0,179 0,175 0,168 0,188

0,335 0,329 0,326 0,322 0,324 0,316 0,315 0,318 0,318 0,327 0,330

0,332 0,348

r

>K L/2

iA

5 My- Ν


0,119 0,119 0,117 0,120

7

I

1


0,122

L/2

ñu IJMZ

0,318 0,310 0,306 0,299 0,299 0,297 0,297 0,295 0,289 0,285 0,286 0,293 0,292 0,295 0,300 0,303 0,310

6

I

Ί

pl

0 Designation

4

3


trad]

for Ν < Nnmit

1148 614 1075 1458 1887

0,520 0,390 0,321 0,278 0,248 0,227 0,211 0,201 0,191 0,174 0,163 0,157 0,156 0,149 0,144 0,141 0,139 0,141 0,096 0,103 0,127 0,147

75 103 130 157 190 230 276 332 391 259 312 346 326 347 380 452 499 582 318 376 439 514 633 738 853 989 1133 1382 1606 1834

90 121

155 193 236 281

335 392 460 540 632 736 855 653 749 851 999

2312

509

Nllmit

[kN] 67 95 121

149 187 221

270 309 367 433 510 606 694

[rad]


for N £ N,Imit 0,149 0,108 0,087 0,074 0,065 0,059 0,054 0,051 0,048 0,043 0,040 0,038 0,038

-

-

-

-

-

-

-

-

-

-

.

.

-

.

-

-

-

-

0,398 0,307 0,247 0,202 0.178 0,169 0,158 0,159 0,152 0,139 0,134 0,130 0,130 0,122 0,119 0,119 0,117 0,120

59 85 106 123 153 182 223 266 315

0,111 0,083 0,065 0,052 0,045 0,042 0,039 0,039 0,037

-

-

0,273 0,245 0,234 0,224 0,212 0,197 0,188 0,188 0,176 0,179 0,175 0,168 0,188

248 296 342 408 485 569 679 789 939 1177 1409 1655 2097

0,073 0,064 0,061 0,058 0,055 0,050 0,047 0,047 0,043 0,044 0,042 0,040 0,045

-

-

-

-

-

.

-

-

-

-

-

-

-

-

-

-

S 235

HE AA

Available inelastic rotation (pav

HEA

Noie: Cay must be divided by Υ^φ (see chapter 3.6)

2

1

4

3

I

Pi

0

Δ

,

Κ »

J My* j

______

H> Ι

ΙΨ

6

|

7

I

pl

0\
Η

Designation

5

I

L/2 *-'-

. L/2 -ΊΔ

·ίΙ« Pf1-. ,

1

1

r

JMy-N

j

Nlimit

trad]



Nmax

[rad]

[rad]

[kN]

0,451 0,309

0.276 0,263

183 218

0,451 0,309

108 126

0,136 0,091

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0,265 0,269 0,271 0,274 0,278 0,283 0,292 0,297

1608 1179 1214 1249 1282 1355 1413 1459

0,133 0,132 0,124 0,118 0,115 0,106 0,104 0,098

1243

0,033

HE 800 AA HE900AA HE 1000 AA

0,133 0,132 0,124 0,118 0,115 0,106 0,104 0,098

HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

0,628 0.434 0,381 0,343 0,280 0,264 0,255 0.247 0,218 0,212 0,209 0,213 0,214 0,214 0,213 0,200 0,191 0,180 0,172 0,165 0,163 0,147 0,140 0,131

0,310 0,292 0,284 0,278 0,273 0,269 0,269 0,269 0,266 0,264 0,265 0,271 0,276 0,281 0,288 0,297 0,306 0,311 0,315 0,320 0,325 0,329 0,338 0,343

250 298 369 456 532 633 756 903 1020 1143 1322

HE 100 AA

HE120AA HE140AA HE 160 AA HE 180 AA HE 200 AA HE 220 AA HE240AA HE 260 AA HE 280 AA HE 300 AA HE 320 AA HE 340 AA HE 360 AA HE400AA

HE450AA HE 500 AA

HE550AA HE600AA HE650AA HE 700 AA

1461 1568

1677 1868 2092 2321 2488 2661 1571 1788 1699 1771 1640

[rad]


for Ν < Nlimit

[kN]


for Ν > Num»

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0,628 0,434 0,381 0,343 0,280 0,264

123 144 179 234 260 325

0,255 0,247

375 452 513 575 670 737 810 887 1057

1797

0,195 0,133 0,116 0,103 0,084 0,078 0,076 0,073 0,064 0,062 0,061 0,062 0,062 0,062 0,060 0,055 0,052 0,048 0,044

-

-

-

-

-

-

0,218 0,212 0,209 0,213 0,214 0,214 0,213 0,200 0,191 0,180 0,172 0,165 0,163

0,147 0,140 0,131

1223 1399 1592

-

"

I

510

-

Available inelastic rotation (pav Nete: (pav must be divided by ffjjm (see chapter 3.6)

2

1

3

s

I

pl

ϋ

0 L/2

Δ

L

-η

_>y

6

I

Ί.

7

I

r

F

Designation

4

Γ-

HV

'T-

L/2

è -Ί

X«->y-N [rad]



Nmax

[rad]

[rad]

[kN]

HE 100 8 HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β

0,879 0,699 0,589 0,579 0,515 0,467 0,431 0,403 0,349 0,330 0,318 0,314 0,308 0,303 0,292 0,265 0,247 0,230 0,216 0,205 0,200 0,178 0,167 0,154

0,338 0,327 0,319 0,312 0,308 0,304 0,301 0,298 0,293 0,289 0,288 0,295 0,299 0,304 0,311 0,320 0,329 0,334 0,339 0,343 0,348 0,352 0,362 0,366

306 400 505 637 767 917 1070 1245 1392 1544 1752 1896 2008 2122 2324 2561 2804 2985 3172 3364 3600 2703 2810 2672

0,879 0,699 0,589 0,579 0,515 0,467 0,431 0,403 0.349 0,330 0,318 0,314 0,308 0,303 0,292 0,265 0,247 0,230 0,216 0,205 0,200 0,178 0,167 0,154

142 179 220 297 349 425 485 573 645 718 824 901 985 1072 1264 1456 1660 1881 2114 2358 2688

0,278 0,220 0,185 0,180 0,160 0,144 0,133 0,124 0,107 0,100 0,096 0,095 0,092 0,089 0,084 0,075 0,069 0,062 0,057 0,053 0,051

-

-

-

-

-

-

HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

3,216 2,381 1,866 1,660 1,393 1,199 1,053 1,153 0,977 0,889 0,974 0,871

0,461 0,431 0,409 0,392 0,379 0,368 0,360 0,379 0,368 0,359 0,374

626 780 947 1140

3,216 2,381 1,866 1,660 1,393 1,199 1,053 1,153 0,977 0,889 0,974 0,871 0,786

255 317 383 486 563 665 750 960 1068 1177 1440 1524 1613 1702 1884 2111 2338 2575 2812 3049 3286 3804 4278

1,039 0,767 0,600 0,531 0,445 0,382 0,335 0,369 0,312 0,283 0,312 0,277 0,248 0,223 0,183 0,148 0,123 0,104 0,090 0,078 0,070 0,056 0,047

-

-

0,786 0,714 0,599 0.497 0,424 0,368 0,325 0,292 0,265 0,224 0,196 0,174

0,379 0,379 0,380 0,381 0,381 0,381 0,381 0,382 0,382 0,382 0,383 0,383 0,383

1331

1543 1756 2345 2581 2822 3561 3667 3711 3746 3828 3941 4045 4164 4273 4391 4500 4750 4978 3646

511


for Ν < Niimit

0,714 0,599 0,497 0,424 0,368 0,325 0,292 0,265 0,224 0,196 0,174

Nlimit

[kN]

[rad]

«Pav

for Ν

2:

N,imit

UB

Available inelastic rotation


Noie: (paV must be divided by ΎίΛψ (see chapter 3.6)

I

1

"Δ

UB 178 X 102 X 19 UB 203 X 102 X 23 UB 203 X 133 X 25 UB 203 X 133 X 30 UB 254 X 102 X 22 UB 254 χ 102 X 25 UB 254 X 102 X 28 UB 254 X 146 X 31 UB 254 X 146 X 37 UB 254 χ 146 X 43 UB 305 X 165 X 40 UB 305 χ 165 X 46 UB 305 X 165 X 54 UB 356 X 171 X 45 UB 356 X 171 X 51 UB 356 X 171 X 57 UB 356 X 171 X 67 UB 406 X 178 X 54 UB 406 X 178 X 60 UB 406 X 178 X 67 UB 406 X 178 X 74 UB 457 X 1S2 X 52 UB 457 X 152 χ 60 UB 457 X 152 X 67 UB 457 χ 152 X 74 UB 457 χ 152 χ 82 UB 457 X 191 X 67 UB 457 X 191 X 74 UB 457 X 191 X 82 UB 457 X 191 X 89 UB 457 X 191 X 98 UB 533 X 210 X 82 UB 533 χ 210 X 92 UB 533 X 210 χ 101 UB 533 X 210 X 109 UB 533 X 210 χ 122 UB 610 X 229 χ 101 UB 610 X 229 X 113 UB 610 X 229 χ 125 UB 610 X 229 X 140 UB 610 X 305 X 149 UB 610 X 305 X 179 UB 610 X 305 X 238 UB 686 X 254 χ 125 UB 686 X 254 X 140 UB 686 X 254 χ 152 UB 686 X 254 X 170 UB 762 X 267 X 147 UB 762 X 267 X 173 UB 762 X 267 χ 197 UB 838 X 292 X 176 UB 838 X 292 X 194 UB 838 X 292 X 226 UB 914 χ 305 X 201 UB 914 χ 305 χ 224 UB 914 X 305 X 253 UB 914 X 305 X 289 UB 914 X 419 X 343 UB 914 χ 419 X 388

3 Mu

fl

Ν max

[rad]

[rad]

[kN]

0,205 0.210 0.192

285 345 376

0.140 0.160

0.308 0,327 0,281 0.300 0,293 0.314 0,336 0.282 0.303 0.320 0.286 0.299 0,314 0,279 0,293 0.305

580

0,192 0,237 0.135 0.158 0.181 0.156 0.184 0.222 0.136 0.159 0.195 0.122 0.140 0.160

0,194

0,326

1005

0.194

0.119

0,285 0,300 0.311 0.324 0,299 0.321 0,335 0,353 0,369 0,293 0.306 0.316 0,328 0.341 0,288 0.303 0.315

426 463 646 1110 336 425 606

0,119

741

0,162 0,183 0,122 0,137

0,324 0,340

1119

1826

0.170 0.191 0.111 0.127 0.142 0.156 0.178

0,290

674

0.108

0,305

823

0,319

0,123 0.139

0,181

0.156 0,184 0.222 0.136 0.159 0.195

0,122

0.134

0,153 0.172 0.107 0.126 0,144

0.162 0.183 0.122 0.137 0.154

0.170 0,191 0.111 0.127 0.142 0,156 0.178 0.108 0,123 0.139

449

280 376 424

466 554 644

277 387 808 374 444

967 502 606 813 967 1472 610 725 900

0,158

0,333

1039 1403

0,132 0.165 0.238 0,106 0.121

0,290

1028

0.307

1764 3564

0.134

0,315 0.329 0,292 0.312 0.332

0.152 0.103 0.126 0.147 0.101 0.114

0.137 0.100 0.116 0.134 0.156 0,151 0.175

0.340 0,289 0,304

0.289

0,303 0,327 0.292 0,309 0,327 0,344 0.306 0,321

£

^My-N


0.135 0,158

>|c L/2

M,


0.237

β

I ïc U2

r¤-

Designation

ι

826 1018 1258 1693 955 1421 1884 1111 1336

1835 1284

1561 2098 3072

3120 5807

512


[rad]

for Ν < Niimit 0,205 0,210

Nlimit

[kN]


Irad]

for Ν > Niimit

194 246

0,054

263

0,051

294

0,065

351 368 342 358 408

0.038

541

0.051

0,055

0,044 0,040 0,049

0,060

0,050

0,134

0.153 0,172 0.107 0.126

871

0,043

1168

0,048

1530

0,044

2533

0.063

4381

0,044

0,144

0.154

0,158

0,132 0,165 0.238 0.106

0.121 0.134

0.152 0.103 0.126 0,147 0,101 0.114

0,137 0,100 0.116 0.134

0.156 0.151 0.175

Available inelastic rotation Note:
2

1

κ Designation

UC 152x152x23 UC 152x152x30 UC 152 χ 152x37 UC 203 χ 203 χ 46 UC 203 χ 203 χ 52 UC 203 χ 203 χ 60 UC 203 χ 203 χ 71 UC 203 χ 203 χ 86 UC 254 χ 254 χ 73 UC 254 χ 254 χ 89 UC 254 χ 254 χ 107 UC 254 χ 254 χ 132 UC 254 χ 254 χ 167 UC 305 χ 305 χ 97 UC 305 χ 305 χ 118 UC 305 χ 305 χ 137 UC 305 χ 305 χ 158 UC 305 χ 305 χ 198 UC 305 χ 305 χ 240 UC 305 χ 305 χ 283 UC 356 χ 368 χ 129 UC 356 χ 368 χ 153 UC 356 χ 368 χ 177 UC 356 χ 368 χ 202 UC 356 χ 406 χ 235 UC 356 χ 406 χ 287 UC 356 χ 406 χ 340 UC 356 χ 406 χ 393 UC 356 χ 406 χ 467 UC 356 χ 406 χ 551 UC 356 χ 406 χ 634

_>y

t

ί

HI

m,

U

[rad]

[rad]

[kN]

.

0,364 0,393 0,266

0,278 0,289 0,300 0,306 0,327 0,350 0,370 0,398 0,427 0,456

7

r

.

'T-

L/2 Δ -Ί

«-^My-N Nmax

0,285 0,303 0,275 0,285 0,296 0,316 0,336 0,277 0,294 0,310 0,334 0,366 0,270 0,284 0,297 0,312 0,339

L/2

Γ-


0,374 0,548 0,283 0,334 0,450 0,506 0,784 0,270 0,364 0,530 0,734 1,125 0,240 0,330 0,421 0,536 0,759 1,077 1,604 0,211 0,273 0,352 0,444 0,537 0,787 1,196 1,568 2,113 2,891 3,609

6

5

Ί


-

f8*· chapter 3.6)

4

3

pl

0


_

450 554 690 779 897 1063 1288 1094 1331 1602 1976 2501 1451 1765 2049 2366 2966 3593 3875 1931

2289 2650 3022 3518 4297 4655 5381 6395 7546 8681

513

[rad]


for Ν < Niimit

N|¡mit

[rad]

for N s NlimIt |

_

0,374 0,548 0,283 0,334 0,450 0,506 0,784 0,270 0,364 0,530 0,734 1,125 0,240 0,330 0,421 0,536 0,759 1,077 1,604 0,211 0,273 0,352 0,444 0,537 0,787 1,196 1.568 2,113 2,891 3,609


[kN] 224 273 328 357

0,112 0,167 0,085

421 447

0,137 0,158 0,246

562 489 579 711 844 1051 691 828 945

1075 1290 1544 1639 830 973 1132 1290 1433 1749 1876 2152 2510 2945 3324

0,101

0,081 0,111 0,164 0,231 0,357 0,071 0,099 0,128 0,166 0,238 0,341 0,482 0,063 0,083 0,108 0,138 0,168 0,249 0,359 0,473 0,640 0,877 1,096

3.6.2

Tables of
S

275 steel grade

514

IPE IPE A -IPE O 1

Available inelastic rotation (pav Note: 9aV must be divided by YfJia> (see chapter 3.6)

pl Ί* Designation

My

5

I

ïc

L/2

Hk

6

I

Ί

ί

!=

j

4

3

2

f

>K L/2

[rad]

Nlimit



^*max

[rad]

[kN]

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 χ 137 IPE 750 χ 147 IPE 750 χ 173 IPE 750 χ 196

0,427

0,352 0,339 0,332 0,327

0,427

79

0,321 0,264 0,229 0,205

111

0,117 0,115 0.117 0,080 0,085 0,105 0,122

0,321 0,318 0,314 0,312 0,310 0,302 0,296 0,297 0,300 0,301 0,303 0,308 0,310 0,316 0,287 0,287 0,309 0,328

105 142 182 226 276 329 392 459 538 632 740 530 586 657 747 844 990 1133 526 1003 1403 1853

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

0,327 0,252 0,204 0,167 0,148 0,140 0,131 0,132 0,126 0,115 0,111 0,108 0,108 0,102 0,099 0,099 0,097 0,099

0,323 0,315 0,307 0,303 0,297 0,296 0,294 0,294 0,293 0,287 0,283 0,284 0,291 0,290 0,293 0,298 0,300 0,307

88 121 152 184 222 269 195 239 252 258 311 344 318 335 360 426 465 541

IPE Ο 180 IPE Ο 200 IPE Ο 220 IPE Ο 240 IPE Ο 270 IPE Ο 300 IPE O 330 IPE Ο 360 IPE Ο 400 IPE Ο 450 IPE Ο 500 IPE Ο 550 IPE Ο 600

0,226 0,203 0,194 0,185 0,176 0,163

0,332 0,325 0,323 0,319 0,321 0,313 0,312 0,315 0,314 0,323 0,327 0,328 0,344

373 439 514 601 740 864 998 1157 1325 1618 1379 1495 2706

0,156 0,156 0,146 0,148 0,145 0,139 0,155

¿A

< ^My-N

[rad] 0,321 0,264 0,229 0,205 0,188 0,175 0,166 0,158 0,144 0,135 0,130 0,129 0,124 0,119

7

I

515


for Ν < Num»

0,188 0,175 0,166 0,158 0,144 0,135 0,130 0,129 0,124 0,119 0,117 0.115 0,117 0,080 0,085 0,105 0,122

[kN]

142 175 219 258 316 361 429 506 597

[rad]


for Ν

at

Numit

0,136 0,098 0,079 0,068 0,059 0,054 0,049 0,047 0,044 0,039 0,037

-

-

-

-

-

-

-

-

.

-

-

-

.

-

-

-

-

-

-

-

-

-

0,327 0,252 0,204 0,167 0,148 0,140 0,131 0,132 0,126 0,115 0,111 0,108 0,108 0,102 0,099 0,099 0,097 0,099

69 99 124 143 179 213

0,101 0,075 0,059 0,047

0,226 0,203 0,194 0,185 0,176 0,163 0,156 0,156 0,146 0,148 0,145 0,139 0,155

0,041 0,039

-

-

-

-

-

-

-

-

-

-

-

-

.

.

-

-

.

-

-

-

-

-

-

-

290 346 400 477 568 666 794 923 1099 1377

0,066 0,058 0,056 0,052 0,050 0,046 0,043 0,043 0,039 0,040

.

-

-

-

2454

0,041

Available inelastic rotation


Note: (f>av must be divided by Υκ/|φ (see chapter 3.6)

2

1

I

ID HV

HE 280 AA HE 300 AA HE 320 AA HE 340 AA HE360AA HE 400 AA

HE450AA HE500AA HE550AA HE600AA HE 650 AA HE 700 AA

HE800AA HE900AA HE 1000 AA HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

7

I

. L/2 Δ

-Τ-

[rad]

-ί

) My -N Nlimit

[rad]

9av [rad]

Nmax

[rad] 0,372

0,274

214

0,372

126

0,123

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-


HE120AA HE140AA HE160AA HE180AA HE200AA HE220AA HE240AA HE 260 AA

L/2

T-'

My

6

I

pl

°Γ-

Η

L

5

I

Δ

,

Κ

HE 100 AA

4

pl

0 Designation

3

[kN]


for Ν < Nnmit

0,267 0,270 0,272 0,276 0,281 0,290 0,294

1186 1211 1233

1253 1300 1325 1334

0,109 0,103 0,098 0,095 0,088 0,086 0,081

0,517 0,359 0,315 0,284 0,233 0,219 0,212 0,206

0,307 0,290 0,282 0,276 0,271 0,267 0,267 0,267

292 348 432 533 622 740 885 1056

-

-

-

~

0,178 0,178 0,179 0,177 0,167 0,160 0,150 0,143 0.137 0,135 0,122 0,117 0,108

0,270 0,274 0,279 0,285 0,294 0,303 0,308 0,312 0,317 0,321 0,326 0,335 0,339

-

-

-

-

-

-

-

-

-

-

-

-

-

0,517 0,359 0,315 0,284 0,233 0,219 0,212 0,206

144 169 209 274 304 380 438 529

0,178 0,122 0,106 0,094

-

-

-

-

-

-

-

-

-

-

-

-

863 948 1038 1237 1431 1637 1863

0,057 0,056 0,056 0,054 0,050 0,047 0,043

-

-

-

-

-

-

-

-

-

-

-

-

1710 1835 1963 2186 2448 2716 2912 1544 1573 1786 1660 1699 1523

516

for Ν > Num»

|

0,109 0,103 0,098 0,095 0,088 0,086 0,081

-


[kN]

0,178 0,178 0,179 0,177 0,167 0,160 0,150 0,143 0,137 0,135 0,122 0,117 0,108

0,077 0,071 0,069 0,067

Available inelastic rotation (pav Itola: (pav must be divided by Υ^φ (see chapter 3.6)

2

1

l^

HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

Δ

-Ί

| |

L'2 >K L'2

I«:

_) My HV

T-' [rad]



Nmax

[rad]

[rad]

[kN]

0,722 0,575 0,485 0,477 0,425

358 468 591 746 897 1074 1252 1457 1629 1806 2050 2218 2350 2484 2719 2997 3281 3493 3712 3937 4213 2717 2789 2601

0,722 0,575 0,485 0,477

0,128

0,334 0,323 0,316 0,309 0,305 0,301 0,298 0,295 0,291 0,287 0,285 0,292 0,297 0,301 0,308 0,317 0,326 0,330 0,335 0,339 0,344 0,348 0,357 0,362

2,620 1,943 1,524 1,357 1,140 0,982 0,864 0,946 0,803 0,731 0,800 0,717 0,647 0,589 0,494 0,411 0,351 0,304 0,269 0,242 0.220 0,186 0,162 0,144

0,454 0,425 0,403 0,387 0,374 0,364 0,355 0,374 0,364 0,355 0.370 0,375 0,375 0,375 0,376 0,376 0,376 0,376 0,377 0,377 0,378 0,378 0,379 0,379

732 913 1108 1334 1557 1805 2055 2744 3020 3302 4167 4291 4343 4384 4479 4612 4734 4873 5000 5139 5266 5559 4102 3623

2,620 1,943 1,524 1,357 1,140 0,982 0,864 0,946 0,803 0,731 0,800 0.717 0,647 0,589 0,494 0,411 0,351 0,304 0,269 0,242 0,220 0,186 0,162 0,144

0,386 0,357 0,334 0,290 0,274 0,264 0,261 0,256 0,251 0,242 0,221 0,206 0,191 0,179

0,170 0,166 0,147 0,139

6

I

7

I

pl

L

\r-

Designation

5

I

Pi

Ü

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β

4

3

I

517


for Ν < Niimit

0,425 0,386 0,357 0,334 0,290 0,274 0,264 0,261 0,256 0,251 0,242 0,221 0,206 0,191 0.179 0,170 0,166 0,147 0,139 0,128

5

ff

My- Ν Niimit

[kN]


[rad]

for N s N|imlt

671 755 841 965 1054 1152 1255 1479 1704 1943 2202 2474 2759 3145

0,254 0,201 0,169 0,164 0,146 0,132 0,121 0,113 0,098 0,092 0,088 0,086 0,084 0,082 0,077 0,069 0,063 0,057 0,052 0,048 0,046

-

-

-

-

-

-

298 371 449 569 659 778 878 1124 1250 1377 1685 1783 1887 1991 2205 2471 2736 3013 3291 3568 3845 4451

0,947 0,700 0,547 0,485

-

-

-

-

166 209 257 348 408

497 568

0,406 0,348 0,305 0,337 0,285 0,258 0.285 0,254 0,227 0,204 0,168 0,136 0,113 0,095 0,082

0,072 0,064 0,051

Available inelastic rotation


Note:
I

1

cr Designation

UB 178 X 102 X 19 UB 203 X 102 X 23 UB 203 X 133 X 25 UB 203 X 133 X 30 UB 254 X 102 X 22 UB 254 X 102 X 25 UB 254 X 102 X 28 UB 254 χ 146 X 31 UB 254 X 146 X 37 UB 254 χ 146 X 43 UB 305 X 165 X 40 UB 305 X 165 X 46 UB 305 X 165 χ 54 UB 356 χ 171 χ 45 UB 356 X 171 X 51 UB 356 X 171 χ 57 UB 356 X 171 χ 67 UB 406 X 178 X 54 UB 406 X 178 X 60 UB 406 X 178 X 67 UB 406 X 178 X 74 UB 457 X 152 X 52 UB 457 X 152 X 60 UB 457 X 152 X 67 UB 457 X 152 X 74 UB 457 X 152 χ 82 UB 457 χ 191 χ 67 UB 457 X 191 χ 74 UB 457 X 191 X 82 UB 457 χ 191 X 89 UB 457 χ 191 χ 98 UB 533 X 210 X 82 UB 533 X 210 X 92 UB 533 X 210 χ 101 UB 533 X 210 X 109 UB 533 X 210 X 122 UB 610 X 229 X 101 UB610X229X 113 UB 610 X 229 X 125 UB 610 X 229 X 140 UB 610x305x149 UB 610 χ 305 X 179 UB 610 X 305 X 238 UB 686 X 254 X 125 UB 686 X 254 χ 140 UB 686 χ 254 X 152 UB 686 X 254 χ 170 UB 762 X 267 χ 147 UB 762 X 267 X 173 UB 762 χ 267 X 197 UB 838 X 292 X 176 UB 838 X 292 X 194 UB 838 χ 292 X 226 UB 914 X 305 X 201 UB 914 χ 305 X 224 UB 914 X 305 X 253 UB 914 χ 305 X 289 UB 914 X 419 X 343 UB 914 χ 419 X 388

1

"Δ

ID Η) Mv

fe L/2

z^

M,



Ν max

[rad]

[rad]

[kN]

0.170

0.305 0.323 0.279 0.297 0.290 0.311 0,332 0,280 0.301 0,317

334 404

0.170 0.174

440

0.159

525 281 328 378 336 649 753 272

0,196 0.111 0.131 0,150 0,130 0,153

0,174

0.159 0.196 0.111

0,131 0,150 0.130 0.153 0.184 0.113

0.132 0.162 0.101 0.117 0.133 0,161 0.099 0.111 0.127 0.143 0,088

0.105 0.120 0.134

0.151 0.101

0.284

0.296 0,311 0.277 0.291 0.302 0.322 0.283 0.297 0.308 0,320 0.297

0.317 0.332 0.349 0,364

0.128

0,290 0.303 0.313

0,141

0,325

0,158 0,092

0.337 0.285 0,301 0.312 0.320 0,336 0.288 0,302 0.316 0,330 0,287

0.113

0,106

0,118 0.129 0.148 0,089

0,102 0.115 0.131 0.110 0.137 0.197 0.087 0,100

387 946

366 440 583 1175 412 451 643 813 309 402 591 733 970 482 591 809 970 1236 579 699 683 1114' 1468 628 783 1009 1391 1001

0.304

1774

0.336 0.286 0.301

4171 767 967 1218 1674


[rad]

for Ν < Niimit

0,184

0.113 0,132 0,162 0.101 0.117 0.133 0.161 0.099 0.111 0.127 0,143 0,088 0.105 0.120

0.118 0.129 0,148 0.089 0,102 0.115 0,131 0,110 0,137 0.197 0.087 0.100

0.104

0.122

1850

0.122

0,083

0.287

1014

0,094 0.113 0.083 0.096 0.111 0.130 0.126 0,145

0.300 0.323 0.290 0.306 0.323 0,341 0,303 0.318

1248 1769 1170

0,083 0.094

518

for Ν > N,|mit

227 288 308 344

0.049 0,050 0.046 0,059

419 477

0,044

634

0,047

855

0.046

0.055

0.106

879

3102 4186

[rad]

0.158 0.092

1364

3041


0,141

0,289

1459

[kN]

0,128

0.309 0.328

2019

Niimit

0.101 0,113

0,104

0,325

Mu-N

0.151

0.085

0,311

ff

0.134

0.111 0,126 0.085

0.111 0,126

>|c L/2

0.113 0.083 0.096 0.111 0,130 0,126 0.145

2964

0,058

Available inelastic rotation

(f>av

Not«:
1

2

l^

UC 152 χ 152 χ 23 UC 152 χ 152 χ 30 UC 152 χ 152 χ 37 UC 203 χ 203 χ 46 UC 203 χ 203 χ 52 UC 203 χ 203 χ 60 UC 203 χ 203 χ 71 UC 203 χ 203 χ 86 UC 254 χ 254 χ 73 UC 254 χ 254 χ 89 UC 254 χ 254 χ 107 UC 254 χ 254 χ 132 UC 254 χ 254 χ 167 UC 305 χ 305 χ 97 UC 305 χ 305 χ 118 UC 305 χ 305 χ 137 UC 305 χ 305 χ 158 UC 305 χ 305 χ 198 UC 305 χ 305 χ 240 UC 305 χ 305 χ 283 UC 356 χ 368 χ 129 UC 356 χ 368 χ 153 UC 356 χ 368 χ 177 UC 356 χ 368 χ 202 UC 356 χ 406 χ 235 UC 356 χ 406 χ 287 UC 356 χ 406 χ 340 UC 356 χ 406 χ 393 UC 356 χ 406 χ 467 UC 356 χ 406 χ 551 UC 356 χ 406 χ 634

Ί

0

^iΔ -Ί

L

Wi

L/2

Γ-


Nmax

[rad]

[kN]

0,451 0,235 0,276 0,371

0,418 0,644 0,224 0,301 0,437 0,604

0,922 0,200 0,273 0,347

0,442 0,624 0,883 1,298 -

0,227 0,292 0,367 0,443 0,646 0,969 1,269 1,707 2,331 2,907

-

_

526 648 808 911 1050 1243 1508 1280 1558 1875 2312 2927 1697 2065

2398 2769 3471 4205 4595

-

0,276 0,286 0,297 0,303 0,324 0,346 0,366 0,393 0,421 0,449

L/2

Δ

-Ί

5 My -Ν


0,283 0,301 0,273 0,283 0,293 0,313 0,333 0,275 0,291 0,307 0,331 0,362 0,268 0,282 0,295 0,309 0,335 0,360 0,387

,

1

[rad] 0,309

r

->K

.Η H>, -

7

6

5

"1

Ü Designation

4

3

-

2679 3101 3537 4117 5028 5521 6382 7585 8950 10296

519

[rad]


for Ν < Nmit .

0,309 0,451 0,235 0,276 0,371 0,418 0,644 0,224 0,301 0,437 0,604 0,922 0,200 0,273 0,347 0,442 0,624 0,883 1,298 -

0,227 0,292 0,367 0,443 0,646 0,969 1,269 1,707 2,331 2,907

Niimit

[kN] _

[rad]


for Ν £ Niimit _

262 319 383 418 493 523 657 572 677 833 988 1230 809 969 1106 1259 1510 1807 1944 _

1139 1324 1509 1677 2047 2225 2552 2977 3492 3942

0,102 0,153 0,077 0,092 0,125 0,144 0,225 0,074 0,102 0,150 0,211 0,326 0,065 0,091 0,117 0,151 0,217 0,311 0,437 .

0,076 0,099 0,126 0,154 0,227 0,326 0,429 0,580 0,795 0,994

3.6.3

Tables of (pav for

S

355 steel grade

520

IPE

Available inelastic rotation (pav

IPE A -IPE O

m\Smm'.

2

1

<&

be divided by γ^φ (see chapter 3.6)

3

4

Ί

Δ

L

Wi

l^

% U2

>\

_>y

I I



[rad]

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 χ 137 IPE 750 χ 147 IPE 750 χ 173 IPE 750 χ 196

0,303 0,228 0,188 0,164 0,146 0,135 0,125 0,119 0,114 0,104 0,097 0,093 0.093 0,089 0,086 0,084 0,083 0,084

0,346 0,333 0,327 0,322 0,316 0,313 0,309 0,308 0,305 0,297 0,292 0,292 0,296 0,296

0,057 0,075 0,087

0,284 0,283 0,305 0,323

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

0,232 0,180 0,146 0,120 0,106 0,101 0,094 0,095 0,091 0,083 0,080 0,078 0,078 0,073 0,071 0,071 0,070 0,072

IPE Ο 180 IPE Ο 200 IPE Ο 220 IPE Ο 240 IPE Ο 270 IPE Ο 300 IPE Ο 330 IPE Ο 360 IPE Ο 400 IPE Ο 450 IPE O 500 IPE Ο 550 IPE Ο 600

0,161. 0,145 0.139 0,133 0,126 0,117

0,061

0,112 0,112 0,105 0,106 0,104 0,100 0,111

L/2

ff

1

^My-N ''max [kN]


[rad]

for Ν < Niimit

Nlimit

[kN]

136 183 234 292 357 425 506 592 694 447 491 529 579 644 721 803 937 1065 313 815 1240 1720

0,303 0,228 0,188 0,164 0,146 0,135 0,125 0,119 0,114 0,104 0,097 0,093 0,093 0,089 0,086 0,084 0,083 0,084 0,057 0,061 0,075 0,087

102 144 183 225 282 333 408 466 554

0,318 0,310 0,303 0,299 0,293 0,292 0,290 0,290 0,289 0,283 0,279 0,280 0,287 0,286 0,289 0,293 0,296 0,303

113 156 196 238 171 186 195 240 251 250 298

0,326 0,320 0,317 0,314 0,316 0,308 0,307 0,310 0,310 0,318 0,321 0,323 0,338

0,299 0,303 0,305 0,311

7

r

>\
"

\*\ M I JM2

[rad]

6

5

pl

ϋ Designation


[rad]


for Ν > Niimit 0,116 0,084 0,068 0,058 0,050 0,046

0,042 0,040 0,037

.

_

-

.

_

.

_

.

.

_

-

-

-

.

-

.

.

_

.

_

_

.

.

-

-

-

89 128 160 185

0,086 0,064 0,050 0,040

297 306 355 377 433

0,232 0,180 0,146 0,120 0,106 0,101 0,094 0,095 0,091 0,083 0,080 0,078 0,078 0,073 0,071 0,071 0,070 0,072

-

-

481

0,161

567

0,145 0,139

374 447 516 616 733

0,057 0,050 0,047 0,045 0,042

327 291

664 776 956 699 772 890

942 1200 1382 1484 2243

521

0,133 0,126 0,117 0,112

0,112 0,105 0,106 0,104 0,100 0,111

-

.

-

_

-

_

_

.

-

.

-

-

.

.

.

.

.

-

_

-

.

-

.

_

.

-

.

.

_

.

.

.

_

_

_

.

.

_

.

-

-

-

Available inelastic rotation


Note: (pay must be divided by Υ^φ (see chapter 3.6)

2

1

κ: j

^

7

I

U

ι

l^t

"

j

L/2

Γ-

Δ

, L/2

'T-

Ζ ι

ι

-ί

r

JMy-N

1

[rad]

Nümit

[rad]


[rad]

9av [rad]

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0,068 0,063 0,061 0,058

0,273 0,277 0,286 0,290

9av

HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

oΔ -ί

HV

LJMy

6

I

S

I

pl

IL

1^

HE100AA HE 120 AA HE140AA HE 160 AA HE 180 AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE 340 AA HE 360 AA HE 400 AA HE 450 AA HE SOO AA HE550AA HE 600 AA HE650AA HE 700 AA HE800AA HE 900 AA HE 1000 AA

4

"l

σ Designation

3

I

ι

Nmax

[kN)

1152 1137 1090 1016


for

Ν < Niimit

0,068 0,063 0,061

0,058

[kN]

for N S Nümit

-

-

-

-

-

-

-

-

0,368 0,257 0,226 0,204

0,303 0,286 0,278 0,273

377 450 558 688

0,368 0,257 0,226 0,204

186 218

270 354

0,152 0,104 0,090 0,080

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1340 1597 1847

0,048 0,046 0,043

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0,130 0,129 0,121 0,116 0,109 0,104 0,099 0,098 0,088 0,084 0,078

0,275 0,282 0,290 0,299 0,303 0,308 0,312 0,316 0,320 0,329 0,333

2534 2822 3160 1512 1521

1527 1527 1725 1522 1487 1214

522

0,130 0,129 0,121 0,116 0,109 0,104 0,099 0,098 0,088 0,084 0,078

Available inelastic rotation (pav Note: (pay must be divided by ΥΜφ (see chapter 3.6)

3

2

1

5

I

P¿

υ

_>y

ι

l

0

Δ ^I -π

ñ« JMZ

Ί,

I

7

9av [rad]

**max

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β

0,512

0,329 0,318

462 604 762 963 1158 1386 1616 1881

HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

0,209 0,198 0,190 0,188 0,185 0,182

0,287 0,283 0,282 0,288 0,293 0,297

0,175 0,160 0,149 0,138 0,130 0,123 0,120 0,106 0,100 0,092

0,303

1,836 1,365 1,073 0,956 0,804 0,694 0,611 0,670 0,570 0,519 0,568 0,510 0,461 0,420 0,353 0,294 0,252 0,219 0,193 0,174 0,158 0,134 0,117 0,103

0,443 0,415 0,395 0,379 0,367 0,357 0,349 0,367 0,357 0,348 0,363 0,367 0,367 0,368 0,368 0,368 0,369 0,369 0,370 0,370 0,370 0,371 0,371 0,371

0,312 0,321 0,325 0,329 0,334 0,338 0,342 0,351 0,355

'T-

L/2

Δ

-Ί

I«->y-N

I

0,311 0,305 0,301 0,297 0,294 0,291

L/2

i1-

9av [rad] 0,410 0,347 0,340 0,304 0,276 0,256 0,240

6

I

t

L L

r^

Designation

4

[kN]

2102 2332 2646 2864 3033 3206 3511 3869 4236

4510 2605 2621 2884

2662 2656 2361

945 1179 1430 1723 2010 2330 2653 3543 3899 4263 5380 5539

5606 5659 5783 5954 6111 6290 6455 6634 6799 4684 4079 3460

523

9av [rad]

Niimit

for Ν < Nümit

[kN]

0,512 0,410 0,347 0,340 0,304 0,276 0,256 0,240

0,209 0,198 0,190 0,188 0,185 0,182 0,175 0,160 0,149 0,138 0,130 0,123 0,120 0,106 0,100 0,092

1,836 1,365 1,073 0,956 0,804 0,694 0,611 0,670 0.570 0.519 0,568 0,510 0,461 0,420 0,353 0,294 0,252 0,219 0,193 0,174 0,158 0,134 0,117 0,103

9av [rad] for N à Nümit

214 270 332 449 527 642 733 866 974 1085 1245 1361 1487 1620 1909 2200 2508 2842

0,216 0,172 0,144 0.140 0,124 0,112 0,104 0,097 0,083 0,078

-

-

-

-

-

-

-

-

-

-

-

-

385 479 579 735 851 1004 1133 1451 1613 1778 2175 2302 2436 2570 2846 3189 3532 3890 4248 4606 4964

0,806 0.597 0,467 0,413 0,346 0,297 0,261 0,288 0,244 0,221 0,244 0.218 0,195 0,175 0,144 0,117 0.097 0,082 0,070 0,062 0,055

0,075 0,074 0,072 0,070 0,066 0,059 0,054 0,049

-

-

-

-

"

-

Available inelastic rotation (pav tifila: (pav

2

1

_>y 9av [rad]

178 X 102 X 19

203 X 102 X 23 203 X 133 χ 25 203 X 133 X 30 254 X 102 X 22 U8 254 X 102 X 25 UB 254 X 102 X 28 UB 254 X 146 X 31 UB 254 X 146 X 37 UB 254 X 146 X 43 UB 305 X 165 X 40 UB 305 X 165 X 46 UB 305 X 165 X 54 UB 356 X 171 X 45 UB 356 X 171 X 51 UB 356 X 171 χ 57 UB 356 X 171 X 67 UB 406 X 178 X 54 UB 406 X 178 X 60 UB 406 X 178 X 67 UB 406 X 178 χ 74 UB 457 X 152 X 52 UB 457 X 152 X 60 UB 457 X 152 X 67 UB 457 X 152 X 74 UB 457 χ 152 X 82 UB 457 X 191 X 67 UB 457 X 191 X 74 UB 457 X 191 X 82 UB 457 X 191 X 89 UB 457 χ 191 X 98 UB 533 X 210 X 82 UB 533 X 210 X 92 UB 533 X 210 X 101 UB 533 χ 210 X 109 UB 533 X 210 X 122 UB 610 χ 229 X 101 UB610X229X 113 UB 610 X 229 X 125 UB 610 X 229 X 140 UB 610 χ 305 X 149 UB 610 X 305 X 179 UB 610 X 305 X 238 UB 686 X 254 X 125 UB 686 X 254 X 140 UB 686 X 254 χ 152 UB 686 X 254 χ 170 UB 762 X 267 X 147 UB 762 X 267 χ 173 UB 762 X 267 X 197 UB 838 X 292 χ 176 UB 838 X 292 X 194 UB 838 X 292 X 226 UB 914 χ 305 χ 201 UB 914 X 305 X 224 UB 914 X 305 X 253 UB 914 X 305 X 289 UB 914 X 419 X 343 UB 914 χ 419 X 388

Δ

L

Γ-

0.122 0.125 0.114 0,141

0.079 0.093 0.107 0,093 0,110 0,133 0.082 0,096 0.117 0.073 0,084 0,096 0,116

0,071 0,080 0.092 0,103 0.063 0,075 0,086 0.096 0,109 0.073 0.082 0,092 0.101 0.114

0.066 0.076 0.085 0,093 0,106 0,064 0,073

0.083 0.094

Kk Nmax

0.301 0.318 0.276 0.293 0.287 0,307 0.327 0.276 0.296 0.312 0.280 0.292 0,306 0.273 0.287 0.298 0.317 0,279 0,293

431

0,304

616 800

[kN] 316 567

678 273 324 378 335 390 972 251 376 635 338 417 571

0,300 0,331 0.283 0.297

0,080

0.307 0.320

0.285 0.305

0.323 0.283 0,296 0,318 0,286

0.302 0.318 0.335 0.299 0.313

L/2 *-' ¿

-.It-

-Τ-

9av [rad] for Ν < Nümit

5

Δ

L/2

'i

My" Ν Nümit

[kN]

9av [rad] for N S Nümit 0,042

0.122 0.12S

294 -

-

0.114 0.141

397 444

0,040 0,050

0.079 0.093 0.107 0.093 0.110 0,133 0.082 0.096 0.117 0.073

-

-

-

-

-

-

-

-

-

-

616

0.047

-

-

-

-

-

-

-

-

-

826

0.116

-

-

368 409

-

-

-

-

948

0,071 0,080 0.092 0,103 0.063 0.075 0.086 0.096 0.109 0.073 0.082 0.092 0.101

1238

0.114

493 621

0,066 0,076 0,085 0,093 0.106

240 338 539 692 948 423 539 774

1066 1450 507 671 912 1322 907 1741 5384 612 825 1092 1581 682 1197

0.099 0.142 0,063 0.072

Pï

0,096

0.315 0,330

0.324 0,284

7

-

818

0.298 0.311

I

0.084

0.307

0.284

6

I

ζ-

9av [rad]

0,315 0.293 0.313 0,326 0.343 0.358 0.287 0.299 0.308 0.319 0.332 0.282 0,297

5

I

°\cΓ-

"il 't

0,079

0,091 0,061 0.075 0.087 0,059 0.067 0,081 0,059 0.069 0.079 0.093 0.091 0.105

&· divided by ΥΜφ (se* chapter 3.6} 4

,

le-

UB UB UB UB UB

3

I

Pi

0 Designation

mu$t

1717

768 1015 1569 881 1185 1780 2876 2966 4140

524

0.064 0.073

0.083 0.094

0.079 0.099 0.142 0,063 0,072 0.080 0.091 0.061 0,075 0.087 0.059 0.067

0.081 0.059 0.069 0.079

0.093 0.091 0.105

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

3826

0.049

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

Available inelastic rotation (pav *

Moje.:
pl

Ü

_>y 9av [rad]

UC 152x152x23 UC 152 χ 152 χ 30 UC 152 χ 152x37 UC 203 χ 203 χ 46 UC 203 χ 203 χ 52 UC 203 χ 203 χ 60 UC 203 χ 203 χ 71 UC 203 χ 203 χ 86 UC 254 χ 254 χ 73 UC 254 χ 254 χ 89 UC 254 χ 254 χ 107 UC 254 χ 254 χ 132 UC 254 χ 254 χ 167 UC 305 χ 305 χ 97 UC 305 χ 305 χ 118 UC 305 χ 305 χ 137 UC 305 χ 305 χ 158 UC 305 χ 305 χ 198 UC 305 χ 305 χ 240 UC 305 χ 305 χ 283 UC 356 χ 368 χ 129 UC 356 χ 368 χ 153 UC 356 χ 368 χ 177 UC 356 χ 368 χ 202 UC 356 χ 406 χ 235 UC 356 χ 406 χ 287 UC 356 χ 406 χ 340 UC 356 χ 406 χ 393 UC 356 χ 406 χ 467 UC 356 χ 406 χ 551 UC 356 χ 406 χ 634

^l Λ

L

Γ1

9av

Nmax

[kN]

0,221 0,321

0,280 0,296

-

-

0,217 0,312 0,429 0,652

679 836 -

1177 1356 1605 1946

-

0,287 0,303 0,325 0,355

-

-

0,196 0,249 0,315 0,443 0,625 0,901

0,279 0,291 0,304 0,329 0,353

-

-

0,164 0,210 0,263 0,316 0,459 0,675 0,881 1,181 1,608 2,003

.

-

0,279 0,289 0,309 0,327

0,379 0,272 0,283 0,293 0,299 0,319 0,339 0,358 0,384 0,411 0,438

6

Ί L/2 L/íí

7

r

,

-il 'T-

L/2

Δ

-Ί

Π>'^My-N

[rad] -

0,199 0,265 0,299 0,457

\r~

HV

-

5

°Γ-

Δ

L

\4

Designation

4

3

2

1

b· divided by Ym9 (see chapter 3.6)

-

2011 2421 2984

3778 -

2666 3096 3574 4480 5428 6037 -

9av [rad] for Ν < Num» .

0,221 0,321 _

0,199 0,265 0,299 0,457 .

0,217 0,312 0,429 0,652 -

0,196 0,249 0,315 0,443 0,625 0,901 -

3458 4003 4566 5315

0,164 0,210 0,263

6491 7253 8385 9965 11757 13526

0,459 0,675 0,881 1,181 1,608 2,003

525

0,316

Nümit

[kN] .

338 412 .

540 636 675 849 _

874 1075 1275 1588 _

1251 1428 1625 1949 2333

2554 ,

1470 1709 1948 2165 2643 2923 3353 3911 4588 5179

9av [rad] for N > Num» _

0,087 0,130 .

0,079

0,107 0,123 0,192 .

0,087 0,128 0,180 0,279 .

0,077 0,100 0,129 0,186 0,266 0,370 ,

0,065 0,084 0,107 0,132 0,194 0,276 0,363 0,491

0,672 0,839

3.6.4

Tables of
S 420 steel

grade

526

IPE

Available inelastic rotation

IPE A -IPE O

Note: (paV must be divided by Ym9 (see chapter 3.6)

2

1


4

3

I

S

I

I

Designation

ff

!=

j

My

Ye

H>,

L/2

7

>l<

ff

L/2

Φ- ^My-N 9av [rad]

9av

9av

Nmax

[rad]

trad]

[kN]

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 χ 137 IPE 750 χ 147 IPE 750 χ 173 IPE 750 χ 196

0,238 0,179 0,148 0,129 0,116 0,107 0,099 0,094 0,090 0,082 0,077

161 217 277 345 422

0,238 0,179 0,148 0,129 0,116

503 598 395 428 445 484 517 562

0,107 0,099

0,067 0,066 0,066 0,045 0,048 0,059 0,069

0,342 0,330 0,323 0,319 0,313 0,310 0,306 0,305 0,302 0,295 0,290 0,290 0,294 0,294 0,296 0,300 0,303 0,308 0,281 0,281 0,302 0,319

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

0.182 0,142 0,115 0,095 0.084 0,080 0,075 0,075 0,072 0,066 0,064 0,062 0,062 0,058 0,057 0,057 0,056 0,057

IPE 0 180 IPE Ο 200 IPE Ο 220 IPE Ο 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

0,127 0,115 0,110 0,105 0,100 0,093 0,089 0,089 0,083 0,084 0,082 0,079 0,088

0,074 0,074 0,071 0,068

I

Ί

Pi %

6

618 682 749 871 982 115 627 1066

for Ν < Nümit

0,094 0,090 0,082 0,077 0,074 0,074 0,071 0,068

0,067

Niimit

[kN]

for Ν ¿ Nümit

120 170 216 267 334 394 482

0,104 0,075 0,060 0,052 0,045 0,041 0,037

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1565

0,066 0,066 0,045 0,048 0,059 0,069

0,314 0,307 0,300 0,296 0,290 0,290 0,288 0,288 0,286 0,281 0,277 0,278 0,285 0,283 0,286 0,291 0,294 0,300

134 184 232 167 172 184 192 235 244 237 281 305 260 257 250 283 290 327

0,182 0,142 0,115 0,095 0,084 0,080 0,075 0,075 0,072 0,066 0,064 0,062 0,062 0,058 0,057 0,057 0,056 0,057

105 152

0,323 0,317 0,314 0,311 0,313 0,305 0,304 0,307 0,307 0,315 0,318 0,319 0,335

569 671 785 918 654 704 774 889 933 1186 1357 1444 2219

0,127 0,115 0,110 0,105 0,100 0,093 0,089 0,089 0,083 0,084 0,082 0,079 0,088

527

9av [rad]

-

-

-

-

-

-

-

-

-

-

189

0,077 0,057 0,045

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

442 528 611

0.051 0,044 0,042 0,040

729 -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

Available inelastic rotation (pav Note: Cpav musl be divided by ΥΜφ (*·· chapter 3.6)

2

1

3

I

4

Pi

Ο Κ 1

Designation

HE140AA HE160AA HE180AA HE200AA

Ι

Ι'Ψ

"

J

9av [rad]

L/2

Γ-

LM H>2 9av [rad]

HE 100 AA HE 120 AA

0

M ^.

6

I

I

pl

Δ

,

S

I

Τ ι

NfTiax

[kN]

è -ί

. L/2

'T-

~i

9av [rad] for Ν < Num«

7

Ìr

JMy-N Niimit

[kN]

9av [rad] for N > Nümit

-

.

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

HE 220 AA

-

-

-

-

-

-

HE240AA HE260AA HE280AA

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

HE 300 AA

-

-

-

-

-

-

HE320AA

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0,048 0,046

0,284 0,288

854 707

0,048 0,046

0,290 0,203

0,300 0,284

446 532

-

-

-

-

-

-

-

-

-

-

-

-

-

-

HE 340 AA HE 360 AA HE400AA HE450AA

HE500AA HE550AA HE600AA HE650AA HE 700 AA HE 800 AA

HE900AA HE 1000 AA HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

-

-

-

-

0,290 0,203

220 258

0,136 0,093

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1889

0,042

-

-

-

-

-

-

0,102 0,096 0,092 0,087 0,083 0,079 0,078 0,070 0,066 0,062

0,280 0,288 0,296 0,301 0,305 0,309 0,313 0,317 0,326 0,330

3339 1509 1511 1498 1479 1455 1635 1366 1264 911

528

0,102 0,096 0,092 0,087 0,083 0,079 0,078 0,070 0,066 0,062

-

-

-

-

-

-

-

-

-

-

-

-

Available inelastic rotation (pav Usus: Cpav mua be divided by

2

1

4

3

Ί

Δ

L

Wi

l^

_>y

r

ϊ< υ2 >κ L/2

-Ί

H>2

7

6

5

Pi

υ Designation

Ymç (see chapter 3.6)

Ζ9av [rad]

ff

^My-N 9av [rad]

9av [rad]

9av [rad]

Nmax

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 8 HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β

0,403 0,323 0,274 0,269 0,240 0,219 0,203 0,190 0,166 0,157 0,151 0,150 0,147 0,144 0,139 0,127 0,118 0,110 0,103 0,098 0,095 0,084 0,079 0,073

0,325 0,315 0,308 0,302 0,298 0,294 0,292 0,289 0,285 0,281 0,280 0,286 0,290 0,294 0,301 0,309 0,317 0,322 0,326 0,330 0,334 0,338 0,347 0,351

547 714 902 1139 1370 1640 1912 2226 2487 2759 3131 3388 3589 3793 4153 4578 5011 2624 2618 2606 2853 2558 2481 2095

0,403 0,323 0,274 0,269 0,240 0,219 0,203 0,190 0,166 0,157 0,151 0,150 0,147 0,144 0,139 0,127 0,118 0,110 0.103 0,098 0,095 0,084 0,079 0,073

254 319 393 531 624 759 867 1024 1153 1284 1473 1610 1760 1917 2259 2603 2967.

-

-

HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

1,433 1,067 0,840 0,748 0,630 0,545 0,480 0,526 0,448 0,409 0,447 0,402 0,363 0,331 0,279 0,233 0,199 0,173 0,153 0,138 0,125 0,106 0,092 0,082

0.437 0,410 0.390 0,374 0,363 0,353 0,345 0,362 0,353

1118 1395 1692 2038 2378 2757 3138 4191 4613 5043 6365 6553 6632 6695 6841 7044 7230 7442 7637 7848 8043 4689 3973 3242

1,433 1,067 0,840 0,748 0,630 0,545 0,480 0,526 0,448 0,409

455 566 685 869 1007 1188 1341 1716 1909 2104

0,447 0,402 0,363 0,331 0,279 0,233 0,199 0,173 0,153

2574 2724

0,721 0,534 0,418 0,370 0,310 0,267 0,234 0,259 0,219 0,199 0,219 0,196 0,175 0,158 0,130 0,105 0,087 0,074 0,063 0,055 0,049

0,344 0,358 0,363 0,363 0,364 0,364 0,364 0,365 0,365 0,365 0,365 0,366 0,366 0,367 0,367

[kN]

529

for Ν < Nümit

0,138 0,125

0,106 0,092 0,082

Nümit

[kN]

for N

2:

0,112 0,101 0,093 0,087 0,075 0,070 0,067 0,066 0,064 0,063 0,059 0,053 0,048

-

-

.

-

-

-

-

-

-

-

-

-

2882 3041 3367 3773 4179 4602 5026 5449 5872

Nümit

0,194 0,154 0,129 0,126

-

-

-

-

-

-

UB

Available inelastic rotation (pav Note: (pay must be divided by ΥΜφ (see chapter 3.6)

1


ID KJ Mv

cr L/2 r*"

^max

[kN]

0.298

0.315

257 318

0.097 0,099

0,112

0.290

802

0,062

0.284 0.304

260

0.074 0,085

0.323

313 371

for Ν < Nümit

Nümit

[kN]

9av [rad] for N 2 Nümit

-

-

-

-

0.112

525

0,045

0.062 0.074

-

-

-

-

0,085

-

-

0,088

-

-

0,106

-

-

0,290 0,310 0,323

387 549 817 321 364 580 772 172 273 482

0.339

641

0.067 0.076 0.092 0,056 0.063 0,073 0,082 0.050 0.059 0.068 0.076

0,353 0,284

909 361 482 726 909 1213 406 538 742 1001

0.086

1404

0.084

387 555 805 1233 804 1676 3931 459 678 955 1464 490 1021 1560 530 783 1355 601

0.051 0.058 0.066 0.075 0,063 0,079 0.112 0,049 0.057 0.063 0.072 0.048 0,059 0.069 0.047 0,053

0.294

0.067 0.076 0.092 0,056 0,063 0.073 0.082 0.050 0,059 0,068 0.076 0.086 0.057

0,285 0,295 0,314 0.277 0.291 0,301

0,065

0.296

0,073 0.080 0.090 0.052 0,060 0.067 0.073 0.084 0,051 0.058 0,066 0.075 0.063 0.079 0.112 0.049 0.057 0.063 0.072 0.048 0.059

0,306 0.316 0.328 0,280

0.309 0.278 0.290 0,304

0,312

0.294

0.304 0.312 0,327 0.282 0.296 0.308

0.321 0.282 0.297 0.327

0.281 0.294 0.304

0.317 0.283 0.302 0.320 0,281

0,053 0,064

0.294

0.047 0.054

0,284

0,063 0.073 0,072 0.083

»^

384 S82 228 358 631

0.088 0.106 0.065 0.076 0.093

0,069 0,047

>lc

9av [rad]

9av

0.097 0.099

I

X«->y-N

M:

[rad]

9av [rad] UB 178 X 102 X 19 UB 203 X 102 X 23 UB 203 X 133 X 25 UB 203 X 133 X 30 UB 254 χ 102 X 22 UB 254 χ 102 X 25 UB 254 χ 102 X 28 UB 254 X 146 X 31 UB 254 X 146 X 37 UB 254 X 146 X 43 UB 305 X 165 X 40 UB 305 X 165 X 46 UB 305 X 165 X 54 UB 356 X 171 X 45 UB 356 X 171 X 51 UB 356 X 171 X 57 UB 356 X 171 X 67 UB 406 X 178 X 54 UB 406 X 178 X 60 UB 406 X 178 X 67 UB 406 X 178 X 74 UB 457 X 152 X 52 UB 457 X 152 X 60 UB 457 X 152 χ 67 UB 457 χ 152 χ 74 UB 457 X 152 X 82 UB 457 X 191 χ 67 UB 457 X 191 X 74 UB 457 χ 191 χ 82 UB 457 χ 191 X 89 UB 457 X 191 X 98 UB 533 X 210 X 82 UB 533 X 210 X 92 UB 533 X 210 X 101 UB 533 X 210 X 109 UB 533 χ 210 X 122 UB 610 X 229 X 101 UB610X229X 113 UB 610 X 229 X 125 UB 610 X 229 χ 140 UB 610 X 305 X 149 UB 610 χ 305 X 179 UB 610 X 305 X 238 UB 686 X 254 χ 125 UB 686 χ 254 χ 140 UB 686 X 254 X 152 UB 686 X 254 χ 170 UB 762 X 267 χ 147 UB 762 X 267 X 173 UB 762 X 267 X 197 UB 836 X 292 χ 176 UB 838 X 292 X 194 UB 838 X 292 X 226 UB 914 χ 305 χ 201 UB 914 χ 305 X 224 UB 914 X 305 X 253 UB 914 X 305 X 289 UB 914 X 419 χ 343 UB 914 X 419 X 388

"Δ

0.315 0.299 0.315 0,331 0,296

0,310

914

1528 2668 2782 4012

530

0.065 0.076 0.093

0.057 0,065 0.073 0.080 0.090 0,052 0,060 0,067 0.073

0,064

0.047 0.054 0,063 0,073 0.072 0.083

-

-

-

-

-

-

.

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

Available inelastic rotation φ3ν tlflls: 2

1

(see chapter 3.6)

4

3

κ

UC 152 χ 152 χ 23 UC 152x152x30 UC 152 χ 152 χ 37 UC 203 χ 203 χ 46 UC 203 χ 203 χ 52 UC 203 χ 203 χ 60 UC 203 χ 203 χ 71 UC 203 χ 203 χ 86 UC 254 χ 254 χ 73 UC 254 χ 254 χ 89 UC 254 χ 254 χ 107 UC 254 χ 254 χ 132 UC 254 χ 254 χ 167 UC 305 χ 305 χ 97 UC 305 χ 305 χ 118 UC 305 χ 305 χ 137 UC 305 χ 305 χ 158 UC 305 χ 305 χ 198 UC 305 χ 305 χ 240 UC 305 χ 305 χ 283 UC 356 χ 368 χ 129 UC 356 χ 368 χ 153 UC 356 χ 368 χ 177 UC 356 χ 368 χ 202 UC 356 χ 406 χ 235 UC 356 χ 406 χ 287 UC 356 χ 406 χ 340 UC 356 χ 406 χ 393 UC 356 χ 406 χ 467 UC 356 χ 406 χ 551 UC 356 χ 406 χ 634

ffj¡a

Ί ff

!=

Ve

J HV 9av

9av

Nmax

[rad]

[rad]

[kN]

-

-

0,175 0,253 -

0,157 0,209 0,236 0,360 -

0,172 0,246 0,338 0.511

_

0,278 0,294

804 989

-

-

0,277 0,287

1392 1604 1899 2302

0,306 0,324 -

-

0,285 0,300 0,322 0,351

-

-

0,155 0,197 0,249 0,349 0,490 0,725

0,277 0,289 0,301 0,326 0,349 0,375

L/2

2380 2864 3531 4470 -

3154 3663 4229 5301 6422 7028

9av [rad] for Ν < Nümit _

0,175 0,253 _

0,157 0,209 0,236 0,360 .

0,172 0,246 0,338 0,511 .

0,155 0,197 0,249 0,349 0,490 0,725

r

>!<:

z-

My

L/2

Niimit

[kN] _

400 487 _

639 753 799 1004 .

1035 1272 1508 1878 _

1480 1689 1922 2306 2760 2974

-

-

.

_

-

-

-

.

.

0,281 0,291 0,297 0,315 0,336 0,354 0,380 0,406 0,432

4736 5402 6288 7680 8444 9761 11601 13688 15747

531

0,166 0,208 0,250 0,361 0,544 0,709 0,949 1,291 1,606

ff

^My-N

-

0,166 0,208 0,250 0,361 0,544 0,709 0,949 1,291 1,606

7

6

5

pl

0 Designation


2022 2305 2561 3126 3403 3903 4554 5341 6029

9av [rad] for N S Nümit ,.

0,078 0.117 _

0,071 0,096 0,111 0,172 _

0,078 0,115 0,162 0,250 .

0,069 0,090 0,116 0,167 0,238 0,336 .

.

0,076 0,097 0,118 0,174 0,251 0,329 0,445 0,608 0,760

3.6.5

Tables of
532

IPE

Available inelastic rotation

IPE A -IPE O

Note: (pav must be divided by ΥΜφ (see chapter 3.6)

3

2

1


4

5

I

Ί

Pi

^ Designation

L L

_>y

ι I

Δ

* ñu JM2

Κ

L/2

9av [rad]

9av [rad]

9av [rad]

Nmax

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 χ 137 IPE 750 χ 147 IPE 750 χ 173 IPE 750 χ 196

0,208 0,157 0,130 0,113 0,101 0,093 0,087 0,083 0,079 0,072 0,068 0,065 0,065 0,062 0,060 0,058 0,057 0,058

176 237 304 378 462 347 377 395 427 441 476 506 546 598 652 708 822 921

0,042 0,052 0,060

0,340 0,328 0,322 0,317 0,312 0,309 0,305 0,303 0,301 0,293 0,288 0,289 0,292 0,293 0,295 0,299 0,301 0,307 0,280 0,280 0,301 0,318

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

0,159 0,124 0,101 0,083 0,074 0,070 0,066 0,066 0,063 0,058

0,313 0,305 0,298 0,295 0,289 0,288 0,286 0,286 0,285 0,280

0.055 0,051 0,050 0,050 0,049 0,050

0,284 0,282 0,285 0,290 0,292 0,298

238 228 212 235 230 255

0,055 0,051 0,050 0,050 0,049 0,050

IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE 0 600

0,112 0,101 0,096 0,092 0,088 0,082 0,078 0,078 0,073 0,074

0,321 0,315 0,313 0,309 0,311 0,303 0,303 0,306 0,305 0,313 0,316 0,318 0,333

623 735 860 611 656 702 770 883 921 1169 1332 1409 2189

0,112 0,101 0,096 0,092 0,088 0,082 0,078 0,078 0,073 0,074 0,072 0,069 0,077

-

[kN]

.

for Ν < Nümit

5

ff

My- Ν Nümit

9av [rad]

[kN]

for N 2 Nümit 0,097 0,070 0,057 0,049

132 186 237 292 366

-

-

-

-

-

0,042 0,052 0,060

147 202 254 167

0,159 0,124 0,101 0,083 0,074 0,070 0,066 0,066 0,063 0,058

182 189 231 238 228

7

0,208 0,157 0,130 0,113 0,101 0,093 0,087 0,083 0,079 0,072 0,068 0,065 0,065 0,062 0,060 0,058 0,057 0,058

500 945 1453

171

I

r

>κ U2

X-

I

6

I

0,042

-

-

-

-

.

-

.

-

.

-

-

-

-

-

.

-

-

-

-

-

-

-

-

-

.

-

-

-

-

-

116 166 207

0,072 0,053 0,042

-

-

-

-

.

-

-

-

-

-

-

-

-

-

-

-

-

_

_

_

-

-

-

-

.

-

0,072 0,069 0,077

533

.

-

-

.

.

-

-

-

-

-

-

-

485 579 669

0,048 0,042 0,040

-

-

-

.

-

-

-

-

-

.

-

-

-

.

-

-

-

-

-

-

Available inelastic rotation (pav Note: Cpav must be divided by ΥΜφ (see chapter 3.6)

2

1

Γ-

_>y 9av [rad]

100 120 140 160 180

AA AA AA AA AA

HE200AA HE220AA HE 240 AA HE 260 AA HE 280 AA HE 300 AA HE 320 AA HE 340 AA HE 360 AA HE 400 AA HE 450 AA HE 500 AA HE 550 AA HE 600 AA HE650AA HE 700 AA HE 800 AA

HE900AA HE 1000 AA HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

5

I

6

I

I

7

Pï I I

0fs.

.jΔ -Ί

L

1^

HE HE HE HE HE

4

pl

σ Designation

3

I

Δ

- L/2

P|s.

^|

X^->y-N

HJM2 M I

9av [rad]

L/2

Nmax

[kN]

9av [rad] for N < Nümit

Nümit

9av [rad]

[kN]

for N > Nlimit

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.042 0,040

0,282 0,287

694 499

0,042 0,040

-

-

-

-

0,254 0,178

0,299 0,283

488 583

0,254 0,178

241 282

0,128 0,087 -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0,287 0,295 0,299 0,303 0,308 0,312 0,316 0,324 0,328

1511 1500 1473 1439 1398 1565 1255 1109 705

0,085 0,081 0,076 0,073 0,070 0,068 0,061 0,058

-

-

-

-

-

-

0,085 0,081 0.076 0,073 0,070 0,068 0,061 0,058 0,054

534

0,054

-

-

-

-

-

-

-

-

:

-

Available inelastic rotation (pav Note:
2

1

3

I

pl

ϋ

Δ

I

w-

-ί

_>y

I I

ftJMZ M 9av [rad]

Nmax

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β

0,352 0,283 0,240 0,235 0,210 0,192 0,178 0,167 0,146 0,138

0,324 0,314 0,306 0,301 0,297 0,293 0,290 0,288 0,283 0,280

599 782 988 1248 1501 1796 2094 2438 2724 3021

-

-

HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

1,248 0,929 0,732 0,653 0,550 0,476 0,420 0.459 0,392 0,357 0,391 0,352 0,318 0,290 0,245 0,204 0,175 0,152 0,134 0,121 0,110 0,093 0,081 0,072

0,127 0,122 0,111 0,104 0,097 0,091 0,086 0,083 0,074 0,069 0,064

[kN]

-

0,285 0,289 0,293 0,299 0,308 0,316 0,320 0,324 0,328 0,332 0,336 0,345 0,349

3711 3931 4155 4549 5013 5489 2633 2609 2579

0,434 0,407 0,387 0,372 0,360 0,351 0,343 0,360 0,351 0,342

1224 1527 1853 2232 2605 3019 3437 4590 5052 5524 6971 7177 7264 7333 7493 7715 7919 8151 8364 8596 5415 4663 3878 3077

0,356 0,361 0,361 0,361

0,362 0,362 0,362 0,362 0,363 0,363 0.363 0,364 0,364 0,364

2815 2473 2350 1907

535

6

I

I

7

pl L/2 L/ ¿

-jit.

9av [rad] for Ν < Nümit 0,352 0,283 0,240 0,235 0,210

0,192 0,178 0,167 0,146 0,138 -

Δ

L/2

'T-

I-

I

9av [rad]

0,132 0,129

5

I

°\cΓ-

-si

r=-

Designation

4

-ι

^My-N [kN] 278 350 430 582 683 832 949 1122 1262 1406

0,182 0,145 0,122 0,118 0,105

0,095 0,088 0,082 0,070 0,066

2851 3249 -

-

1,248

0,072

for N à Nümit

0,062 0,061 0,059 0,056 0,050 0,045

1764 1927 2099

0,653 0,550 0,476 0,420 0,459 0,392 0,357 0,391 0,352 0,318 0,290 0,245 0,204 0,175 0,152 0,134 0,121 0,110 0,093 0,081

.

-

-

0,132 0,129 0,127 0,122 0,111 0,104 0,097 0,091 0,086 0,083 0,074 0,069 0,064

0,929 0,732

9av [rad]

Nümit

2474

-

-

-

-

-

-

-

-

-

-

-

-

498 620 751 952 1103 1301 1468 1880 2090

0,678 0,502 0,394 0,348 0,292 0,251 0,220 0,244 0,206 0,187 0,207 0,185 0.165 0,149

2304

2819 2983 3157 3331 3688 4133 4577 5041

0,122

5504 5968

0,099 0,082 0,069 0,060 0,052

-

-

-

-

-

-

-

"

Available inelastic rotation (pav Note:
1

"Δ

L/2

Γ«"

Designation

Z^

Mv

9av [rad] UB 178 X 102 X 19 UB203X 102 X 23 UB203X 133 X 25 UB203X 133 X 30

UB254X 102x22 UB 254 X 102 X 25 UB 254 X 102 X 28 UB 254 χ 146x31 UB 254 X 146x37 UB254X 146X43 UB305X 165x40 UB305X 165x46 UB30SX 165 X 54 UB356X 171 X45 UB356X 171 X 51 UB356X 171 χ 57 UB356X 171 X 67 UB406X 178 X 54 UB406X 178 χ 60 UB 406 X 178 χ 67 UB406X 178 X 74 UB4S7X 152 X 52 UB457X 152 X 60 UB457X 152 X 67 UB4S7X 152 X 74 UB457X 152 X 82 UB457X 191 X 67 UB457X 191 X 74 UB457X 191 X 82 UB457X 191 X 89 UB457X 191 X 98 UB533X 210 X 82 UBS33X 210 χ 92 UB 533 X 210 X 101 UB 533 X 210 X 109 UB533X 210 χ 122 UB 610 X 229 X 101 UB 610 χ 229 X 113 UB 610 χ 229x125 UB 610 χ 229 χ 140 UB 610 X 305 X 149 UB 610 X 305 X 179 UB 610 X 305 X 238 UB 686 X 254 X 125 UB 686 X 254 X 140 UB 686 X 254x152 UB 686 X 254 X 170 US 762 X 267 χ 147 UB762X 267 X 173 UB 762 X 267 X 197 UB 838 X 292 X 176 UB838X 292 X 194 UB 838 X 292X226 UB914X 305 X 201 UB 914 X 305 X 224 UB 914 χ 305 X 253 UB 914 χ 305 X 289 UB 914 χ 419 X 343 UB 914 χ 419 χ 388

fl 9av [rad]

Ν max [kN]

9av [rad] for Ν < Nümit

0,289 0,283 0,302 0.322

879 250 363

0.098 0.055 0,065 0.074

0,077 0,093

0.293 0.307

378 562

0.077 0,093

0.067 0,082

0.289 0.302

344 624

0.067 0.082

0.058 0.067

0,283 0.294

366 531

0.081

0.313

806

0,056

0.289 0.299 0.311 0.289 0.308 0.321

331 552 749 127 228 440 603 877 318 440 690 877 1189

0.056

0,098 0.055 0,065 0.074

0.064

0.072 0.044 0,052 0,059 0,067 0.075 0.050

0,314

0.337

0.351 0,283

304

0,058 0.067 0,081

0.064

0.072 0.044

0.052 0,059 0.067 0.075 0.050 0.057

0.295

0.070 0.079

0.315 0,327

0.053 0.059

480

0,053

687 952 1365 305

0.059 0.064

0,044

0.293 0,303 0,311 0,325 0.281

0,051

0,294

476

0,057

0.056

0,306 0.319 0.280 0.296 0.325 0.280 0.293 0,303 0,315 0.282 0.301 0.318 0.280 0.292 0.313

730 1166 731 1622 3942 356 578 859 1378 361 898 1448 370 626 1206

0,041

0,283

414

0.041

0,047

0,298

0.055

0.314

0,047 0,055

0.064

0.329 0,295 0,308

729 1352 2513 2643 3903

0.064 0,074

0.065

0.055 0.069 0.099 0.043 0.050 0.055 0.063 0,042 0,052 0.060 0.041 0,047

0.063 0.073

536

N

Niimit

[kN]

9av [rad] for Ν

at

Nümit

0.087

0.057 0.064

0.304

L/2

>y0,085

0.297

*£-

M,

257 317

0.085 0,087

1

0.064

0.070 0,079

0.074 0,044 0.051 0.057 0,065 0,055 0.069 0.099 0.043 0,050 0,055 0.063 0.042 0,052 0.060 0.041 0.047

0.056

0.064

0.063 0.073

575

0,043

Available inelastic rotation (pav Naia: CPav mus, be divided by ΥΜφ (see chapter 3.6)

2

1

r*

j

My

9av [rad] UC 152 χ 152x23 UC 152x152x30 UC 152 χ 152 χ 37 UC 203 χ 203 χ 46 UC 203 χ 203 χ 52 UC 203 χ 203 χ 60 UC 203 χ 203 χ 71 UC 203 χ 203 χ 86 UC 254 χ 254 χ 73 UC 254 χ 254 χ 89 UC 254 χ 254 χ 107 UC 254 χ 254 χ 132 UC 254 χ 254 χ 167 UC 305 χ 305 χ 97 UC 305 χ 305 χ 118 UC 305 χ 305 χ 137 UC 305 χ 305 χ 158 UC 305 χ 305 χ 198 UC 305 χ 305 χ 240 UC 305 χ 305 χ 283 UC 356 χ 368 χ 129 UC 356 χ 368 χ 153 UC 356 χ 368 χ 177 UC 356 χ 368 χ 202 UC 356 χ 406 χ 235 UC 356 χ 406 χ 287 UC 356 χ 406 χ 340 UC 356 χ 406 χ 393 UC 356 χ 406 χ 467 UC 356 χ 406 χ 551 UC 356 χ 406 χ 634

Ί

Δ

L

I I

Y<

ft M IJM2 9av [rad]

L/2

r

sic L/2

-

^My-N »max

[kN]

9av [rad]

Niimit

9av [rad]

for Ν < Nümit

[kN]

for Ν > Nmit

.

.

.

_

-

.

-

.

.

0,293

1084

0,221

_

ι

534

-

-

-

-

_

-

-

-

_

_

0,183 0,207 0,315

0,286 0,304 0,322

-

-

0,151 0,216 0,296

0,284 0,299 0,321 0,349

0,447

1757 2080 2522 -

2606 3137 3867 4896

0,183 0,207 0,315 -

0,151 0,216 0,296 0,447

825 875 1100 -

.

-

-

_

-

-

-

.

4012

0,172

4631 5806 7033 7749

0,218 0,305 0,428 0,628

1850 2105 2526 3023 3279

-

-

-

-

.

-

-

-

.

_

0,146 0,182 0,219 0,316 0,472 0,614 0,821 1,116 1,388

0,280 0,290 0,295 0,314 0,334 0.352 0.377 0,403 0,428

5187 5916 6887 8411 9310 10762 12791 15092 17362

537

0,146 0.182 0,219 0,316 0.472 0,614 0,821 1,116 1,388

_

0,090 0,105 0,162 -

1133 1393 1652 2057

-

0,287 0,300 0,324 0,347 0,372

0,110 -

-

0,172 0,218 0,305 0,428 0.628

ff

1

-

0,221

7

6

5

pl

ϋ Designation

4

3

2215 2525 2805 3424 3752 4304 5021 5889 6647

0,074 0,108 0,152 0,235 .

.

0,085 0,109 0,157 0,224 0,315 .

.

0,071 0,091 0,111 0,164 0,235 0,309 0,417 0,570 0,711

Graphs with values of
3.7

The present chapter provides diagrams with


values for standard European and British

I and

H hot-rolled steel profiles: ΓΡΕ, IPE A, IPE 0, HE AA, HE A, HE B, HE M, UB and UC. These diagrams constitute a graphical presentation values already presented in tables of chapter 3.6.

of Feldmann's formulas

and concern die same

In regard to me load case, the following 2 series of die diagrams are available: 3 .7. 1 Concentrated load witii bending about major axis y-y

of cross-section (My).

3.7.2 Concentrated load with bending about minor axis z-z of cross-section (My.

Five steel grades (see table 3.1 with fy depending on material thickness according to Eurocode 3 [3] [4]) are used to calculate (pav presented in graphs :

(fy = 235 MPa for tf < 40 mm), (fy = 275 MPa for tf < 40 mm), S 355 (fy = 355 MPa for tf < 40 mm), S 420 (fy = 420 MPa for tf < 40 mm), S 460 (fy = 460 MPa for tf < 40 mm).

S 235

S 275

Each graph evaluates the value of φ2ν in function of the cross-section and steel grade. Considering tiiat present Eurocode 3 rules [3] [4] about classification of cross-section are respected in tiiis Guide, all cross-sections having class 3 or 4 have been excluded from the graphs. This is indicated by the missing of marks on the graph curves.

The values of available inelastic rotation

ΤΜφ

Load cases


formulas

1,52

j My

:

Bending about major axis yy

(3.1) & (3.3)

1,73

j Mz

:

Bending about minor axis zz

(3.2)

538

3.7.1

Concentrated load with bending about major axis y-y of I and H crosssections

Æ"

~3>-J

539

zo

M,

IPE cross-sections [rad]

£Γ

"ä-

ZO Φ«

9av

[deg]

0,30

-F-17.2

0,25

14,3

0,20

11.5

8,6

0,15

5,7

0,10

--S

235 l

-D-S

275

--S355

j

-O-S420

ι

-

S 460

2,9

0,05

0.00

M>

4-0,0

IPE

oooooooooooooooooof*-r>«

ajocy^tDcoocj^r^ococpomotooTO^r^

^^^T-^-wcMCjcvjcocoH-e-'WinintO'-'-·^»-

X

X

X

X

o tn

o in

o to

o to

r-

r·»

r»

r-

Available inelastic rotation of plastic hinge
IPE A cross-sections

£F

ZO

^»-

Μλ

9av[deg] 17,2

14,3

11,5

-»-S235 8,6

-D-S

275

--S355 5,7

-O-S420 -

S 460

2,9

0.0 ^

to

to

CO

Available inelastic rotation of plastic hinge cpav for ΓΡΕ A cross-sections (Note:
540

IPE A

Ια ϋ

VJ

cross-sections

Æ"

"ä-

æo

Mv

-«-S

235

-D-S275 --S

355

-O-S420 -S

ο,ο

Available inelastic rotation of plastic hinge
541

460

IPEO

XTjt¡/

AJ\

cross-sections

"ä-

Γ

[rad](j)av

ZO 9av

0,50

M>

(de9l 28,6 ]

25,8

0,45-

\

0,40 1

0,35

\

22,9 ;

A

\

0,30

1

20,1

172

1

0,25

14,3

0,20

11.5

0,15

8,6

-B-S235 -D-S275

»-S355

-O- S 420

5^a1

0.10

Tít^

0.05

5.7

2.9 0.0

οοοοοσοοοοοοοοοοοοοοοοο

cv^cocoocsj^rcocoocM^rtootnoinotnoo

460

·-*! <(=}(

0.00

--S

HEAA

^rtniototot^coo

Available inelastic rotation of plastic hinge (pav for HE AA cross-sections (Note:
JrlJC/

A. cross-sections

Γ

9av

-&~

l·^ 9av

Μλ

0,30

[deg] 17,2

0,25

14,3

0.20

11,5

0.15

8,6

[rad]

-«-S235

-Q-S275 --S355

5,7

0,10

-O-S420 -

2.9

0,05

0,00

S 460

0.0 oooooooooooooogogooooogo OCM-J-UJCOOCMfttJCOOCgtlOOlOOlOCJtOOOOO ^-^.^^Y-cMCMCMCMCMcoconn^r^-toiococo^Gocno

Available inelastic rotation of plastic hinge (pav for HE A cross-sections (Note: cpav values must be divided by yjvfφ (see chapter 3.7)) 542

HEA

HEB [rad]

cross-sections

Γ"

-Ä-J

^

9av

Mv

9av

[deg]

0.30

172

0.25

14,3

0.20

11,5

-»-S235 0,15

8,6

-O-S

275

--S355 -O-S420 0,10

5,7

0,05

25

0,00

0.0

S 460

HE Β

-o-toio«o

Available inelastic rotation of plastic hinge
HEM l«fl

cross-sections

&-

*a-

ZO

(pav

0.30

Ideg] 17,2

025-

14,3

020-

11,5

av

M>

-«-S235 0,15-

8,6

-D-S275

5,7

-O

*-S355 0.10-

S 420

--S460 0.05-

2,9

0.00-

co

oooooooooooooooooooooooo OCM-'J-ÇDdOCiJ'^-tDCOOCMTfiOOlOOtfïOiOOOOO »-'-^«-«-cMcacj
cor^coeo

Available inelastic rotation of plastic hinge (pav for HE M cross-sections (Note:
HEM

UB cross-sections

Γ

-*

a*.*

tradl (pav 0.20

ι

^

M,


Τ |V

Ide9l 11,5

0,15

0.10

MXXXMKMXXKXX

*Μ**ν-*4*<4ν-*Λ~**>**^**-*''*'*ΦΦΦΦΦΦΦ^ΦΦΦ

Available inelastic rotation of plastic hinge
UB cross-sections

Γ

*&-

Z-0

[rad] φ,

[deg]

020

0,15

0,10

W

Oí

^

SI5Sssssg5ç ÈS3SS8SS22

sassassssxxss

Available inelastic rotation of plastic hinge
M>

UC cross-sections [rad]

-3*-

Æ~

9av

Ζ- 3 9av

M.

0,60

[deg] 34,4

0.50

28,6

0.40

22,9 -»-S235

-Q-S275 17,2

-S

355

O-S420 11.5

--S 5,7

0.0 χ

x

xxxxxxx"-"-·-»«»-"·**·*-«


χχχχχχχχ

χχχχχχχχχ

ssasässsaasasSsssss XXXXXXX*X**«»t«W«WP* «

M

Oí

J 3

i

ΧΧΧΧΧΧΧΧΧΧΧΜΧΧΧΧΧ

333S3ISS3

SS8888SS88SS8888S

Available inelastic rotation of plastic hinge
545

UC

460

3.7.2

Concentrated load with bending about major axis z-z of I and H crosssections

££

Ί&-

HO

546

M,

IPE cross-sections [rad]

0.36

η

0,35·

9av

0,33·

Pavífrteg]

20,6

' I

^ ^^^ kpj^-

20,1

's

1

19,5 :

'

185

*

0,32-

M,

<

i <

0,34·

O

~ä--

Æ~

Αι

k

g Βb

T.

0,31

0,30-

il/i

L

--S235

185 -D-S275 17,8

--S355

-O-S420 172 --S460

V

;

0,29·

16,6 j

028-

16.0 "

0,27·

1

1

15,5
CM

CM

CM

CO

CT

CT

IO

IO

tO

·

y

y-

IPE

i-

Available inelastic rotation of plastic hinge
IPE A cross-sections [rad]

¿F

-^-

«Pav

tO
I

033-1

18,9

I 0,32

0 31

<\

^^

"\V

0,30·

0,29-

Mz

18,3

^^

17,8

kl

Ν

1

1

»

-i

^

3

1

fc!

η

^

r^Àψ\ Y

17,2

-^1 ri''/)

16,6 '

028·

16.0

0.27

15.5

KS5?000oooooooooooo

^'-'-'-'-CMcycyoiejriS^Trwìoìo

Available inelastic rotation of plastic hinge (pav for IPE A cross-sections (Note: φ3ν values must be divided by γ^φ (see chapter 3.7))

547

IPE A

lx

kJ cross-sections

JE/

Æ"

1

~3>-

HO

M,


[rad] «Pav 0,35

0,34

19,5

0.33

18,9

-»-S235 0,32

18,3

0,31

17,8

0,30

17,2

0,29

16,6

-O-S

275

--S

355

O-S420

0,28 4 180

200

220

240

270

300

330

360

400

450

500

550

16,0 600

Available inelastic rotation of plastic hinge (pav for IPE O cross-sections (Note: (pay values must be divided by ΥΜφ (see chapter 3.7))

548

--S

460

IPE O

HE AA cross-sections [rad] 0 30

Γ

3

"Ä-

φ,IV


17,2

η

029

028

ιI t 0 27

\

Ι

ι

ι3

026

025

M2

Μ μη

W'

ψ

16,6

16,0

--S

235

-D-S275 15,5

-S

355

-O-S420 14,9

--S

460

14,3

·

0.24

13 s

oooooooooooooooooooooooo OCM'iJ'ÇDCOOCM^'tOCOOCM^CpOlOOlOOmOOOO '-'-^'-'-CMCMCMCMCMCOCOCOCO^f^-lOlOCOCDr-COOÏO

HEAA

Available inelastic rotation of plastic hinge φ^ for HE AA cross-sections (Note: q>av values must be divided by γ^φ (see chapter 3.7))

-LXC/

Ά. cross-sections

&-~

[rad] (pav

Æ.

-&-

HO 9av

0,34

M2

[deg]

'-in 19,5 18,9

18,3

-»-S235 17,8

172

-D-S275 --S35S

-0-S42Q 16.6

--S460 16,0

15,5

14,9 *r-*rioio
Available inelastic rotation of plastic hinge cpav for HE A cross-sections (Note: q>av values must be divided by γ^φ (see chapter 3.7)) 549

HEA

HEB

cross-sections

~4 IO

Γ

M,

[deg]

«Pav

21,2

-«-S235 -D-S275 --S355

O-S420

--S460

15.5

HEB

Available inelastic rotation of plastic hinge
XJll/ IVI cross-sections

O

-&-~

&--

«Pav

[rad] 0,46

«ί'av [deg] -ί«

r

0,45

<

0,44

t

0,43 0,42

l\\

-25 2

ñ^ 4 !

0,40

-24 6

-24

Ú

:229

\

0,38

223 I-

y^ ^^ ^^y/p&$ψ Λ

1

0.37 ;

ί'{

0,36 ;

0,35 ;

1

-235

i i^ fö

0,39

Λ

258

\Ä

0,41

M,

Xv5^

r^Jw

ι

^τ

1

\

ιt

tt

*

1

>=< ι

ι

U)

(Ο

φ

o co

o

o a>

Available inelastic rotation of plastic hinge
550

1

-19,5 o U)

-«-S355|

^S42oj

>212

^20

oooooooo

-D-S275

¡21 8

f=y ì

f=4

0,34 CMCgojcgCM
J

-»-S235

*-S460|

UB cross-sections

¿F

tO

-τ&-~

[rad] «Pav

M,

Ideg]

«Pav

0.37

212

0.36

20,6

0,35

20,1

0,34

19,5

0,33

18,9

0,32

18,3

0,31

17,8

0,30

172

0,29

16,6

0,28

16,0

--S235 -D-S275

0,27
<*

rt

rt

S

S

S

3

S

I^j

y

r«.

r~

r»

r«

r»

S

8

8 8

15,5

«J

Ct

«

*3-

--S355 -O-S420

Φ S 460

UB

*»

Available inelastic rotation of plastic hinge
UB cross-sections

to

[rad] «Pav 0,34

Mz

«Pav [deg]

19,5

0.33

18,9

0,32

18,3

--S235

-D-S275

0.31

17,8

--S355 -O-S420

0,30

17,2

029

»-S460

16,6

028

5

8

S

S

g

o

£

*^

2

«

·

^

«*


*»

-»

e*

<û

«*

S

3

β

N4MO«CJ4*OJO
3

3

3

Available inelastic rotation of plastic hinge
16.0

UB

UC cross-sections Jraí

&-

O

-τ&-~

[deg]

«Pav

«pav

M,

-264

-252 241 229 ^21 8

-S

195

-Ο- S 420

-

183

-S

:

172

:

16 0

:14,9 <*

©

f-

tp

«Μ

O

«

s s ? s

ΧΧΧΧΧΧΧΧΧΧ

»

Õ*

«

,...,,,.,,-,

<0

XXXXXXXXXXXXXXXXX

oicvc«*·*«·!«««'*'·»'****«* ~

r-iMCKtc4çi
W

CH

«t«i*t«w«*«<ø
c*

χ

χ

Λ

χ

χ

χ

χ

χ

χ

χ

χ

χ

χ

χ

x

x

X

x

x

x

«no

Available inelastic rotation of plastic hinge
552

-D-S275

206

-

0?6-

--S235

UC

355

460

Introduction

4.1

Four design examples are given in dús chapter. Only die checking methods for die inelastic rotation and for die bearing capacity of structure are considered under a loading system for verification at die Ultimate Limit State. Otiier limit states should be investigated (deflection at die Serviceability Limit State...).

Design examples 1 and 2 use formulas (developed in chapter 2) to calculate die required inelastic rotations
In all four design examples, die following values of different partial safety factors have been used: for bending moment resistance of cross-sections (Mpi.Rd)

:

for ductility of plastic hinges (
ymo

=

(CEN official value from Eurocode 3 [3] [4]); 1,1

γΜφ = 1,52 (see table 1.6).

As explained in chapter 2.2.3, it is more conservative to check die ductility of plastic hinges at collapse instead of at die design load level (ductility requirement :
4.2 4.2.1

Design example

: continuous beam

1

with concentrated load

Data

X-

Fd

3m

X-

E200

= 45kN

/

6m

6m

-x

Figure 4.1 Steel grade: S 235

fy = 235MPa E = 210000 MPa

Cross-section : IPE 200

Class

1

(for My and S 235)

Iy=1943xl0-8m4

l

b=100

ν=8,5_Γ~Λ

χ J

'

Wpl y = 220,6 χ IO"6 m3 tw=5,6_

Design plastic moment resistance:

M pl.y.Rd

_ WpLyfy _ 220,6x235

ÏM0

= 47,13 kNm

11 ΓΡΕ200

553

h = 200

4.2.2

Structural resistance

From formulas (2.5) and (2.6) (using Mpi = Mpi.y.Rd) Üw load a* collapse (Fu) and the load at occurrence of first plastic hinge (Fl) can be determined (see figure 4.2). 2nd ptatåe hinge : Fu = 47 kN

1tt ptartc hinge :

F1 = 38.SS kN

=

-47.13 kNm

Mp = 47,13 kNm

Bending moment diagram at collapse V = 15,7kN V = 7.8SkN

31.3 kN

Shear force diagram at collapse

Figure 4.2 Bending resistance

The load at collapse is reached for F = 47 kN, greater tiian die applied load Shear resistance

Av.z=14.10-4m2 Vpi.z.Rd = 0,577 Av.z fy / YMO = 0,577 χ 14. ΙΟ"4

χ 235000 /

1,1

= 172,6 kN

So, Vz.sd < 0,5 Vpi ζ Rd : die shear force has no influence on the moment resistance (EC3 §

5.4.7(2)). Moreover, die slendemess of die web is less than 69 ε (d/tw = 28,4), it is not necessary to check die shear buckling resistance (EC3 § 5.6.1(1)).

LTB resistance Let's assume tiiat die distance between lateral restraints is most critical part is at die origin of the second span.

1

m along die compressed flange. The

.1m

Figure 4.3 Therefore, in tiiis less favourable case, the moment ratio between the two lateral restraints is

:

ψ = 0,833 From tiiis, one can calculate the critical moment from EC3 Annex F, and die reduced slendemess

λυτ = 0,39 554

It is not necessary to check die lateral torsional buckling resistance because die reduced slendemess is less then 0,4 (EC3 § 5.5.2(7)). Note :

lateral restraints according to " Additif 80 - CM66 " is Lmin =35ize since the moment ratio is between 0, 625 and 1, 00. So Lmj = 0, 783 cm. The minimal spacing between F(kN) ι

ι

1

M Itw intermediate support ,

rth« toed

0

20

10

|

30

50 S

40

M (kNm)

Mp

Figure 4.4 : Evolution of die bending moment

Required inelastic rotation

4.2.3

The required inelastic rotation is given by formula (2. 14) and must be calculated for die first plastic hinge :

_ Mpl.y.Rd L (α + 3)(1-α)

with α = 0,5

α

χ ΙΟ3 χ 6 6 χ 210000 χ ΙΟ6 χ 1943 χ IO*8 47,13

(Prequ

(0,5 + 3)(l-0,5) 0,5

^req.u^040^ 4.2.4

Available inelastic rotation

In case of bending about major axis y-y of cross-section, die available inelastic rotation is given by formula (3.1) (or can be issued from column 2 of die table of chapter 3.6.1 or from graphs of chapter 3.7.1): 5,6 4 Eb£ _ ^ ht^ +^(fy htw)2+4 fy btf tw hA(T
'4W+0,25)f* + ACT'bh

5h

with : Δσ = 150 MPa Therefore

cpav

:

= 0,211 rad

,pavd=^=MU = 0,139 1,52 ******

*

?Μφ

555

rad

4.2.5

Conclusion

F 47 a) The load factor at collapse is Xu = -^ = = 1,04 > 1,0, tiierefore the structure has the necessary Fd 45 resistance to bear the applied design load Fd (= 45 kN);

b) The inelastic rotation of the first plastic hinge that is required to reach the mechanism is lower than the design available inelastic rotation; therefore regarding ductility of the plastic hinge the

av.d

"req.u

chosen profile is acceptable

:

0,040 rad< 0,139 rad

4.3

Design example 2 : continuous beam with uniform distributed load

4.3.1

Data

Figure 4.5 Geometrical properties of the cross-section are tiiose from die design example

4.3.2

1

(see chapter 4.2).

Structural resistance

From formulas (2.16) and (2.20) (using Mp[ = Mp¡ y Rd) the load at collapse (qu) and the load at occurrence of first plastic hinge (qi) can be determined (see figure 4.6).

£

2nd plastic hinge :qu = 15,23 kNftn

I

A

1st plastic hinge : q1 = 13,74 kN/m -Mp = -47,13 kNm

Mp = 47,13 kNm

Bending moment diagram

y = 38,5kN '""immillili

V = -52,8kN

Shear force diagram

Figure 4.6

556

V = 7,85kN

iiiniimmiiiiiiimiill

I

Bending moment resistance

The collapse is reached for q = 15,23 kN/m, greater than the applied load. Shear resistance

Av.z=14.10-4m2 Vpl.z.Rd = 0,577 Av.z fy / YMO = 0,577 χ 14. ΙΟ"4 χ 235 000 / 1,1 = 172,6 kN So, Vz.sd < 0,5 Vpi z Rd : the shear force has no influence on die moment resistance (EC3 §

5.4.7(2)). Moreover, die slendemess of die web is less than 69 ε (d/tw = 28,4), check die shear buckling resistance (EC3 § 5.6.1(1)).

it is not necessary to

LTB resistance The configuration is similar to the previous example, except tiiat we have to consider tiiat the less favourable portion of the beam is located in die middle of die first span.

In die vicinity of the plastic hinge, the moment ratio is about 1,00 : this is a conservative value. Then die reduced slendemess is about 0,41. In order to obtain a value less or equal to 0,4, die distance between the lateral restrains may be reduced in die vicinity of die plastic hinge, and taken equal to 95 cm for example.

4.3.3

Required inelastic rotation

The required inelastic rotation is given by formula (2.29):

^.u'-^f^UM _
Mpl.y.Rd

i-tÆ+i)2

x l^x 6 " 210000 χ ΙΟ6 x 1943 x

L 3^2 +1 _

El

12

Ι+ωτρ 4(1 + η)

47,13

,

with ω = 0 and η = 1,0

χ 0,4369 HT8

9req.u = 0,030rad

4.3.4

Available inelastic rotation

The design value of available inelastic rotation has been calculated in example 1 (see chapter 4.2.2) for a concentrated load. According to chapter 3.5, the same value of available inelastic rotation can be used in case of a uniform distributed load.

yav.d=0'139rad

557

4.3.5

Conclusion

a) The load factor at collapse is

λ = -^- = 1^1 = i,02 > 1,0 15

therefore the structure has the

,

%

necessary resistance to bear the applied design load q¿ (=15 kN/m);

b) The inelastic rotation of the first plastic hinge tiiat is required to reach die mechanism is lower than die available inelastic rotation; dierefore regarding ductility of die plastic hinge the chosen profile is acceptable : req.u

TSv.d

0,030 rad< 0,139 rad

4.4

Design example 3 : simple portal frame

In this example, it is assumed tiiat the columns and the beam are laterally and continuously restrained in such a way tiiat out-of-plane buckling is not a potential mode of failure. 4.4.1

Data

100

kN

45,45 kN

-X

ΓΡΕ270

6m

IPE 300

ΓΡΕ300

8m

X-

Figure 4.7 Steel grade : S 235

fy = 235 MPa

Cross-section : IPE 270

Class

1

(for My and

S

235)

Wpl.y = 484 x1o-6 m3 Design plastic moment resistance

:

Mpl.y.Rd=

484x235

u

= 103,4 kNm

Classi (for My and S 235)

Cross-section : ΓΡΕ 300

Wpi.y = 628,4 x 10"6 m3 Design plastic moment resistance

:

Mpl.y.Rd =

628,4x235 1,1

558

= 134,25 kNm

Structural resistance

4.4.2

The plastic analysis has been carried out witii PEP micro program [15] and witii die assumption that the loads are increased in a proportional way Goad factor λ). The plastic mechanism is plotted in figure 4.8. Two plastic hinges occur in the beam (IPE 270), then two plastic hinges occur at die bases of ti» columns (TPE 300). 2nd plastic hinge

:

λ 2 = 0,832 λ1 =0,790

1st plastic hinge:

/s

Ν = 38,6 kN

"""

N = 49,7kN

'""""

"""'

INIIMIIHI

g

N = S1.7kN

λ3 = 0,900

3rd plastic hinge:

V = -48.7kN

Axial force diagram M = -103,4 kNm

M = -95.3 kNm

M = 103,4 kNm

V = 51,7kN

V=

V = -e.48kN

M = -134kNm

kN

Shear force diagram

M=

Bending moment diagram

Figure 4.8 Bending resistance

This is a non-sway frame (EC3

§

5.2.5.2(3))

:

ÔV

0,0403x100

hH

6x45,45

= 0,0148 < 0,10

So the second order effects may be neglected.

Influence of axial force

In the columns In the beam

:

:

:

Npi = A fy / ymo = 0,00538 χ 235000 /

1, 10

=

Npi = A fy / ymo = 0,00459 χ 235000 /

1, 1 0

= 980,6 kN

1

149

kN

The applied axial force is very low in comparison witii the axial resistance. So diere is no influence on the moment resistance (lower tiian Npi/4, see EC3 § 5.4.8(3)). The plastic collapse is reached for a load factor equal to 1,014 greater than 1,00.

559

Shear resistance

In the columns :

Vpi.z.Rd = 0,577 Av.z fy / YMO = 0,577 χ 0,002568

In die beam :

Vvl.zXd = 0,577 Av.z fy / YMO = 0,577 χ 0,002214 χ 235000/1,1 = 273 kN

χ 235000/1,1 = 316 kN

The shear force is lower than half the shear plastic resistance, diere is no influence on die moment resistance. Moreover, the slendemess of die web is lower tiian 69 ε, it is not necessary to check die shear buckling resistance (d/tw = 33,3 for ΓΡΕ270 profile, and d/tw = 40,4 for IPE300 profile, see EC3 § 5.6.1(1))

Buckling resistance The buckling lengtii for die columns may be calculated for a non-sway mode. Let's assume that die buckling lengtii is equal to the system lengtii :l- 6 m for die columns. The critical axial force is NCT = 48 1 1 kN. The reduced slendemess is

λ = 0,5126 (> 0,2)

By using die buckling curve a (EC3 table 5.5.3), the buckling resistance is Nb.Rd = Xy A fy / YM1 = 0,92 χ 0,00538 χ 235 000 / 1,1 = 1058 kN Since die plastic moment is reached in die column under die design loads, die buckling criterion which takes into account die interaction between axial force and bending moment is not satisfied (see EC3 § 5.5.4).

The french NAD adds die following condition : if Nsd / Nb.Rd does not exceed 0,10, necessary to check die buckling resistance. This condition is here fulfilled :

it

is not

Nsd /Nb.Rd = 51,7/1 058 = 0,0489

λ 1.1 I

λ,ι»1.014Τ.Ο-

1

0.9

l·

0.8

r-B-i

ΓΤ

0.7

0.6

I

I -τ

120

130

0.5

0.4 0.3

I

I

809

0

0.2 0.1

0 ()

10

2030405060

70

1O0J

110

j

1'η

|M| (kN)

Mp« 103,4 kNm rMp* 134,16 WHm\ IPE 270

IPE 300

Figure 4.9 : Evolution of the bending moment in different cross-sections

560

J

4.4.3

Required inelastic rotation

Just before collapse, die inelastic rotations in the tiiree first plastic hinges are given in table 4.1. It should be mentioned tiiat it is not die first formed plastic hinge which requires the highest inelastic rotation.

Table 4.1 Plastic hinge

Load factor λ

Cross-section

Required inelastic rotation φ at collapse : (λ = 1,014)

u

radi 0,790 0,832 0,900

1st

2nd 3rd

4.4.4

ΓΡΕ 270 ΓΡΕ 270 ΓΡΕ 300

0,06038 0,08383 0,01239

Available inelastic rotation

For the steel grade S 235 and the two used profiles, available inelastic rotations are given by formulas (3.1) and (3.3) or are given in die table of chapter 3.6. 1 . :

for ΓΡΕ 270 : die value of
for ΓΡΕ 300

:


ΓΡΕ 270 and S 235

:


ΓΡΕ 300 and S 235

:

cpaV

rad

= 0,163 rad

design 6" value

=>

:

design value:

φ ^av.d. = 152 = 0,114 rad


0,163 =^^ = 0,107 rad

(witii maximum compressive load in die right column of die frame,N = 52,43 kN < Niimit = 510 kN)

4.4.5

Conclusion

a) The load factor at collapse is λ = 1,014 > 1,0, tiierefore die structure has the necessary resistance to bear the applied design loads;

b) The highest inelastic rotations of different plastic hinges that are required to reach die mechanism are lower tiian die design available inelastic rotations for both profiles; tiierefore regarding ductility of plastic hinges the chosen profiles are acceptable :

for ΓΡΕ 270 (2nd hinge):

<
for ΓΡΕ 300 (3rd hinge):

0,012 rad < 0,107 rad


561

4.5

Design example 4 :

4.5.1

Data

step by step method with the help of an elastic analysis program

A plastic analysis of a three span beam with distributed loads, as shown in figure 4. 10, will hereafter be performed. This example is meant to highlights the calculation procedure given in figure 2.27 of chapter 2. For tiiis example, an elastic analysis computer program can be used, but die elastic calculations could also be made witii die help of a manual metiiod.

30 kN/m

a

X-

6m

25 kN/m

"2Γ

2Γ

b

C

8m

-X-

d 6m

-X-

-X

Figure 4.10

Steel grade S 235

fy = 235MPa

:

Classi (for My and S 235)

Cross-section : ΓΡΕ 270

Wpl.y = 484 x1o-6 m3 Design plastic moment resistance

:

Mpi.y.Rd =

484x235 1,1

562

= 103,4 kNm

4.5.2

Structural resistance 30 kN/m

26 kN/m

2\

2S

6m

8m

Mpl - -103,4

6m

Mpl

-103,4

Mpl = -103,4

93,81

71,169

Final moment diagram

76,82 kN

103,4 kN

rrrC

34,78 kN

kN

-103¿ kN

-110,3 kN

Shear force diagram

Figure 4.11 Bending resistance

The collapse is reached for a load factor equal to 1,03 15, greater than 1,00.

Shear resistance

The shear plastic resistance is Vpi.z.Rd = 273 kN. So die maximal shear force (1 10,3 kN) does not exceed half die shear plastic resistance. Therefore it has no influence on the bending moment resistance.

The web slendemess is lower tiian 69 ε, die shear buckling is not a potential mode of failure.

LTB Resistance The most critical portion is in the middle of the second span : The moment is nearly constant and equal to the plastic moment. If we consider tiiis assumption (constant moment), the condition from Additif 80 (CM66) for the spacing of lateral restraints may be used : L = 35 ε iz = 35 χ 1,00 χ 3,02 = 105,8 cm. It would be recommended to have a distance of 1 meter between each lateral restraints.

If

we consider the rule from Eurocode 3 restraint is about : L = 1,30 m

:

Xlt ^ 0,4 ,

563

the minimal distance between each lateral

4.5.3

Analysis of the beam

4.53.1

First step

Elastic calculation

Figure 4.12 First plastic hinge The maximum moment is at support b, so

^

:

= MpLy.Rd -M0 = :10MzP_= o,7554 mo

^

Μ=λο + Δλι

-136,89

λ! = 0,7554 The state of the structure can be determined at die end shear force, the deflections ..., by the load factor λ}.

of tiiis

The moment diagram at die end of the first step is Ml =

λι mo :

step by multiplying die moment, the

Figure 4.13 There is no inelastic rotation to calculate at the end of die first step.

4.5.3.2

Second step

A perfect hinge is introduced at die place where die first plastic hinge was formed, and a new elastic analysis is performed on die following structure.

564

30 kN/m

25 kN/m

¿-

A

>£-

a

b

c

^

Θ

-162,5

kNm]

0

135

45,75

127

Figure 4.14 We must check that die rotation (obtained from die elastic analysis) in the first hinge at support b has the same sign as the plastic bending moment which was reached in previous step (Ml):

9b = -0,04825 rad

=>

M cpb = (-103,4) χ (-0,04825) > 0

This means tiiat there is no elastic return. The lowest value of Δλ2 =

will be found among the following four cross-sections: ml

Span ab Span

bc :

Support C Span

Δλ = (103,4 - 56,538) / 135 = 0,3471 Δλ = (103,4 - 52,773) / 127 = 0,3986 Δλ = (-103,4 + 93,203) / -162,5 = 0,0628 Δλ = (103,4 - 44,3) / 45,75 = 1,2918

:

:

cd :

The lowest increment is Δλ2 = 0,0628. Therefore, the second plastic hinge is located at support C and it occurs at die load factor :

λ2 = λι + Δλ2 = 0,7554 + 0,0628

=>

λ2 = 0,8182

At the end of step 2, the moment diagram is M2 = Ml + Δλ2 mi

Mpl = -103,4

:

Mpi = -103,4

[kNm]

65,016

60,749

Figure 4.15 565

47,173

The inelastic rotation at die end analysis

(Preq =

4.5.3.3

of step

2 is die rotation in die hinge at support

b given by die

:

Δλ2 φ = 0,0628 χ 0,04825 = 3,0301

χ IO"3 rad

Third step

A new perfect hinge is introduced at the place where the second plastic hinge was formed and a new elastic analysis is performed on die following structure. 30 kN/m

25 kN/m

b

a

d

C

Θ

[kNm] 0

0

135

112,5

200

Figure 4.16 The rotations obtained from die elastic analysis are 1st plastic hinge (at support

b)

2nd plastic hinge (at support C)

:

:

:


= -£,06607 rad


= -0,06237 rad

M cpb = (-103,4 χ -0,06607 rad) > 0 M epe = (-103,4 χ -0,06237 rad) > 0

The rotations in the hinges have the same sign as the plastic bending moments which were reached in previous step (M2). This means that there is no elastic return.

_ Mpl.y.Rd ~M2 will be found among the following threre cross-sections:

The lowest value of Δλ3 = Spaa ab

Span

be

Spaa,

cd

m2

Δλ = (103,4 - 65,016) / 135 = 0,2843 Δλ = (103,4 - 60,749) / 200 = 0,2133 Δλ = (103,4 - 47,173) / 1 12,5 = 0,4998

The lowest increment is Δλ3 = 0,2133. Therefore the last plastic hinge occurs in the middle

be at the load factor : λ3 = λ2 + Δλ3 = 0,8182 + 0,2133 λ3 = 1,0315 The final moment diagram is M3 = M2 + Δλ3 m2

566

of span

Mpi--

Mpi =-103,4

,4

[kNm]

93,81

Mpi =

71,169

103,4

Figure 4.17 The inelastic rotation in the hinge at support b is (Preq

= 3,0301

χ

IO"3 + 0,2133

(Preq

χ 0,06607 rad

= 0,01712 rad

The inelastic rotation in the hinge at support C is (Preq

:

:

= 0,2133 χ 0,06237 rad (preq = 0,01330 rad

4.5.4

Required inelastic rotation

The plastic analysis is now completed because a plastic mechanism is obtained in span be. To reach die mechanism, the highest required inelastic rotation corresponds to the plastic hinge at

support b and is equal to

:

u = 0,01712 rad

(Preq

4.5.5

Available inelastic rotation

For the profile ΓΡΕ 270 and S 235 steel, the design value determined in design example 3 (see chapter 4.4.2) : (Pav d

4.5.6

of available

inelastic rotation has been

= 0,1 14 rad

Conclusion

a) The load factor at collapse is λ = 1,0315 > 1,0, tiierefore the structure has the necessary resistance to bear the applied design loads;

b) The highest inelastic rotation of plastic hinge at support b that is required to reach the mechanism is lower than the design available inelastic rotation; therefore regarding ductility of plastic hinges the chosen profile is acceptable:
0,01712 rad < 0,114 rad

567

Plastic resistance of I and H cross-sections

Appendix 1.

(1) Following tables concern standard European and British I and H hot-rolled steel profiles : ΓΡΕ, ΓΡΕ Α, ΓΡΕ O, HE AA, HE A, HE B, HE M, UB and UC

(2) The following steel grades are considered (see table 3.1 witii fy depending on material thickness according to Eurocode 3 [3] [4]) :

(fy = 235 MPa for tf < 40 mm) (fy = 275 MPa for tf < 40 mm) (fy = 355 MPa for tf < 40 mm) (fy = 420 MPa for tf < 40 mm) (fy = 460 MPa for tf < 40 mm)

S 235 S 275 S 355 S 420 S 460

(3) Classification of cross-sections : only class 1 and class 2 cross-sections can develop plastic resistance; the rules for classification of cross-sections are issued from Eurocode 3 [3] [4]. In the tables, the following legend is assumed : 44 ,

, »»

- "**"

means class 4 cross-section without plastic bearing capacity of Npijr
or class 4 cross-section without plastic bearing capacity of Mpi,y.Rd Mpi.z Rá or MNy .Rd (see table 3.3).

means class 3

(4) Shear buckling

:

if

the web slendemess

(see figure 1.11) and

ε=

ms

(see Eurocode 3 [3] chapter 5.6).

-

"--"

of a cross-section is %

> 69 ε (where d = h - 2 (tf + r)

), then the web has to be checked for resistance to shear buckling

In die tables the following legend is defined :

means cross-section where

% > 69 ε , without plastic bearing capacity of Vpi jr
/lw

(5) Formulas for plastic resistance of cross-sections

:

* Conventions and symbols : for dimensions of profiles,

see figure 1.11; convention forces, shear forces and bending moments applied to the cross-section:

é

,

&

for axial

z.Sd

Nx.sd

N.x.Sd

& * The characteristic values of plastic resistance in following tables (Npi, Vpi.z, Vpi.y, Mpi.y> Mpi.z, MN.y) are calculated assuming die partial safety factor γ\ιο equal to 1,0 in Eurocode 3 [3] rules which define following design values of plastic resistance (Npi.Rd» Vpi.z.Rd> Vpi.y.Rd, Mpi.y.Rd, Mpi.z.Rd, MN.yüd) :

568

column 2

:

design plastic resistance of gross cross-section submitted to tension or compression

N.pl.Rd

:

_Npl

_Afy

Ymo column 3

:

([3] 5.4.4 (2)),

Ymo

where A = 2btf+(h-2tf)tw + (4-3t)r2, design plastic shear resistance related to bending about yy major axis My.sd

:

_ "pl.z _ Ayz fy "pl.z.Rd

where

-

column 4

-

Ymo

([3] 5.4.6(1)),

Ymo^/3

Avz = A-2btf + (tw + 2tr)tf

([3] 5.4.6 (2))

design plastic shear resistance related to bending about zz minor axis Mz.sd

v _ vpi-y _ Avy fy "y-Rd"ïMo"7^?3 where

column 5

:

where

- WpLy fy

Pl.y=-l-+(b-tw)(h-tf)tf+^r2(h-2tf) + ^r3,

W, -

""

"*

2

M pl.z _ "pl.z *y Ymo

:

*

design plastic resistance bending moment about zz minor axis

M pl.z.Rd

:

,

([3] 5.4.5.2(1)),

Ymo

4

column 7

([18]Table 5.14)

design plastic resistance bending moment about yy major axis

Ymo

:

:

([3] 5.4.6(1)),

Ayy= 2btf+(tw + ytw

M, , - MpLy Μρ1.γ .Rd

column 6

,

([3] 5.4.5.2(1)),

Ymo

limit axial force Niimit over which allowance shall be made for the effect of the axial force on the plastic resistance bending moment about yy major axis (=> MN.y.Rd ^ Mpi.y.Rd)>

([3] 5.4.8.1 (3)), where

column

8

:

formula for A is provided for column 2,

ratio between Niimit (column 7) and NpijRd (column 2) over which allowance shall be made for die effect of the axial force on the plastic resistance bending moment about yy major axis (=> MN.y.Rd ^ MpLy.Rd),

Ν limit Ν pLRd

(A-2btf) 4' 2A 1

= minimum

for N-M

569

column 7^

column 2

column 10 : design value of applied compressive axial force Nsd related to the design plastic NSd resistance of die gross cross-section Npi Rd, for the given ratio - = 0, 15 ,

Npi Rå

At,

NSd=0,15

Ymo where formula for A is provided for column 2, column 11: reduced plastic design resistance bending moment about yy major axis allowing for axial force NSd,

ί MN.y.Rd =

where

1

MN.y _ "pLy^y Ymo

NSd

A.fy

Ì

Ymo

([3] 5.4.8.1 (4)),

A-2btf 2A

Ymo

MN.yJld ^ Mpi.y.Rd (defined for column 5),

Nsd is issued from column 10,

Δζ^1<0,25) 2A

formula for Wpi_y is given for column 5, formula for A is provided for column 2, Ncrf

.y

is equal to 0,15 in tiiis case.

A.Iy

Ymo

-

columns 12&13, 14&15, 16&17, and 18&19 are similar to columns 10&11 but related to other NSd ratios, respectively 0,20, 0,25, 0,30 and 0,35 instead of 0,15.

Ν pl.Rd

case of high shear (if Vsd > 0,5.Vpi.Rd) , the design resistance bending moment Mpi.Rd of the cross-section should be reduced to M\r.Rd· The following formulas are not used in tables and are provided for information according to Eurocode 3 ([3] 5.4.7 (3)):

In

-

if Vz.sd > 0,5.Vpi.z Rå , then : 2V.z.Sd

M V.y.Rd -

Where

My.y YMO

wpi.y-

-1

kvz

Vpl.z.Rd

4t w

Ymo

Mv.y.Rd ^ Mpl.y.Rd (defined for column 5), formula for Wpi.z is defined for column 5, formulas for Vpi.zjRd and Ayz are given for column 3, 570

if vy.Sd > 0>5.Vpi.y.Rd , then :

where

Mv\z
(6) Partial safety factor Ymo : As several values of plastic resistance given in tables are characteristic, they must be adapted if partial safety factor Ymo is different from 1,0, in order to obtain design values of plastic resistance needed for checks at Ultimate Limit States (see Eurocode 3 formulas presented in clause (5)). In present version of Eurocode 3 [3] [4], ymo isfixed to 1,1 in general ([3] 5.1.1 (2)) but other values of YMO are provided by several NADs to Eurocode 3.

In practice, all the values given in the following columns of tables must be divided by the partial safety factor ymo"

- columns 2 to 7 : Npi, Vpi.z, Vpiy, Mpi.y, Mpi.zand Niimit for N-My respectively, - columns 10 to 19 : Nsd and MN.y for Nsd / Npi.Rd = 0,15 and respectively 0,2, 0,25, 0,3 and 0,35.

* Remark : In

case of web classification for cross-section submitted to combined N-My, the Eurocode 3 rule that defines die border between class 2 and class 3 webs, is used (see table 3.3) :

456.ε

d ^ Lw

where

(

Ν +1 2 vdtwfy

13

Ν

1

is the compressive axial load applied to the cross-section,

( 0,5 < α

\ Ν + 0,5 <1. 2dtwfy J

In tables, Ν = Nsd from different columns (10, 12, 14, 16 or 18 respectively). As explained above if YM0 differs from 1,0 those values of Nsd must be divided by Ymo · Therefore the web classification calculated in tables is safesided.

(7) Flow-chart explaining the elaboration of tables and the way to use them.

571

4η

*5§·§

¿vi 'S C

jo

o -g

f 3

ιι

α

α sa
GO Ζ ΖB. ©~ ϋ

3« o

3

2

(M

O


o

o

©

5 ο

Ι

o

εο ε

es

à-

¿5

572

Plastic resistance of cross-sections

IPE IPE A 1

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 X 137 IPE 750 χ 147 IPE 750x1 73 IPE 750x196 IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600 IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

-IPE O

Note: all values of columns 2 to 7 must be divided by γΜο

2

3

4

5

6

7

8

Npi

Vp,.Z

vpi.y

Mpl.y

Mpl.z

Nfimit

(Nihnh/Npi.Rd)

for Ν -My IkN]

[kN]

[kN]

[N.m]

[N.m]

[kN]

for Ν -My Η

180 243 310 386 472 563 669 784 919

49 69 86 104

69

5456

91

9 261 14 270

1367 2149 3191

20 761

4 523

29107 39108 51850 67070

6133 8131

34 48 60 75 93 110 135

116 144 174 208 244 289

131

153 190 215 260

1080 1265 1471 1709 1985 2322

300 348 418 477 579 690 812 982

2 715 ++

137

337 393 457 525 614 694 792 912 1033 193

++

1

++

1260 1430 1580 1727

1258 1277 1626 1916

150 206 259

42 60

315 380 460 552 664 783 920

84 106 125 156

56 75 94 116 138 168 200

++ ++ ++

73

184 221

254 302 366 404 485 573 684 818 952

1093 1286 ++

++ ++ ++ ++

++

637

172 210 240 290

751

879

1027 1265 1476 1706 1977 2 265

342 394 473 546

2 765 3 213

806 953

651

3668

1

4 624

1416

122

1

241

285 334 392 456 553 613 705 820 935

1090 237 278 327 378 473 549 624 720 806 965

1095 1225 1538

86162 113739 147664 189018 239 499 307180 399921 515 618

654946 825414 1

143 313

1200 825

10 484 13 656

44 908 53 815 64 949 78 932

94126 114128 144308

461 295

148 241 190 329

1685 932

225318

1

4460 7750 11720 16 826 23 287

31802 42 689 56 449 73 219

1

103

1771 2580

3648 4863 6 571 8586 11394 14 664 19 351

96936 127320 164953 213090 268817 351158 457312 581545 738177

70 881 84 953 103 886

44 450 58 614

12195

75470 96 414

135042 174 797 221555 278 730

353010 480872 614051 766894 1050 694

573

154 183 216 255

17372 22783 29426 36114

25 222

31317 40 386

47486 57750

9 379 15 724 19 833

27661 35 858 43 473 53 326 63 238 80 133 96 005 112921 150 420

303 347 421

509 605 731 851

1001 1

101

1245 1347 30 42 53

0,187 0,196 0,195 0,193 0,198 0,196 0,202 0,197 0,199 0,200 0,202 0,206 0,203 0,212 0,219 0,223 0,231

0,232 0,244 0,250 0,239 0,229

61

0,197 0,206 0,204 0,195

76

0,201

91 111

0,198 0.202 0,200

133 157 184

222 267 292 351

420 506 603 705 124 148 171

204 243 285 339 394 470 588 704 828

1049

0,201

0,200 0,203 0,208 0,194 0,204 0,209 0,213 0,219 0,219

0,194 0,197 0,194 0,199 0,192 0,193 0,199 0,199 0.207 0,213 0,219 0,226 0,227

Plastic resistance of cross-sections

IPE IPE A -IPE O 9

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 χ 137 IPE 750x147 IPE 750x173 IPE 750x196 IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

10

|

Note: all values of Ns
11

Nsd/Npi.Rd-0,15

I

13

Nsd/Np,.Rd=0,2

14

15

|

Nsd/Np,.Rd=0,25

16

|

17

Nsd/NPi.Rd=0,3

18

19

Nsd/Npi.Rd=0,35

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

27 36 47 58

5456 9 261

5369

45 61

5034 8 642

78

13 292

29107

84 100 118 138 162 190

39108 51850 67070 86162 113739 147664 189 018 239499 307180 399921 515618 654946 825 414

4698 8066 12 406 18 016 25 404 34049 45 459 58 448 75 331 99 558 129 487 166661 210 389 272980

63 85 109

71

14178 20 590 29 034 38 913 51850

54 73 93 116 142 169

4 362

9 218

14 270 20 761

36 49 62 77 94 113 134 157 184 216 253 294 342

221

256 298 348 407 474 550 615

543 632 733

113739 147664 189018 239499 307180 399921 515 618 654946 825414

**

*·

881

1200 825 1461295 1685 932

397 464

·*

66 798 86 092

780 884

1200 825 1461295 1685 932

1040

22

4460

30

31

7 750 11720 16 826 23 287

41

11720

73 219 96 936 127320 164953

52 63 76 92 110 133 157 184 219 257

225 258 302 356 413 483

213090

301

268 817

344 402 475

96

44 450 58 614 75 470 96 414 135042 174 797

661

39 47 57 69 83 100 117 138 164 193

113 132 154 190 221

256 297 340 415 482 550 694

31802 42689 56449

351 158

457312 581545 738177

1

179

19303 27 219 36 481 48 706 62 623 80 712 106 670

141

167 196 230 270 316 368

581

138736 178565 225417 292478 384190

679 790 916

497697 638998 806094

427 496

1

201

235 276 324 379 441

513 595 697 814 948 1

100

'· 1.

k*

'· t.

..

*

101

«.

«*

**

..

»ft

·*

**

k*

1529 888

2063

·»

3 887 6 829 10 307 14 626

52 72

3 610 6 342 9 571 13 582 18 944 25 770 34 761 45 879 59 566

37

7 750

52 65 79 95 115 138 166 196 230 273 322 376 429

4165 7317 11043

45 62 78 94 114 138 165 199 235 276 328 386

127320 164953 211599 268 817

1283

120 238 154 757 195361

**

4443

56 449 73 219 96 935

106

1

42 212 54 273 69 950 92 447

**

1560 1768

23 287

596399 752354

31617

**

1440991 1639166

31717 42689

464 518

234 274 322 378 443 515 598 695 813 950

16 729 23 590

*·

1300 1474

16 716

358577

135 165 197

15 671

21858 29 735

40109 52 937 68 730 90 877 119871

156144 »*

««

*·

**

91

110 133

20 401 27 752

161

37435

193

49 408 64 148

232 274

*

**

**

*

*·

**

*

**

*»

*

**

**

*·

*»

.*

**

..

**

*·

**

**

»*

*·

**

»·

**

551

**

**

**

*·

*

*·

*·

644

*·

**

**

*·

*

**

**

38 624 51084 65 582 84 212

223 263 308 360 443 517 597 692 793 968

127 150 176 205 253 295

221555

341

278 730

480 872 614051

395 453 553 643

766894 1050 694

734 925

353010

97 118

7490 11520

44142 58 382 74 951 96 242 133672 173 227 221231 278 546

353010

257 316 369 427 494 566

480 872 614051

766894 1050 694

41383

191

54 733 70 267 90 227 125318 162400 207404 261 137

334005 458144 589867 742 746

225 264 308 380 443 512 593 680 829 964 1 100

019 141

1387

159 188 220

691

803 917 1

156

574

1

»

116963 151573 193577 243 728

311738 427601 550543 693230 951 198

1

124

1284 1618

35 865 47 435 60 898

78197 108609 140 747 179 750

226319 289471 397058 511218 643 714 883256

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

6

7

8

Np,

Vp,.z

Vp..y

Mpl.y

Mpl.z

Nlimit

(Nfimit /NpiRd)

for Ν -My [kN]

[kN]

[kN]

[N.m]

[N.m]

[kN]

for Ν -My Η

HE100AA HE120AA

367 436

158 188

13 714 19 768

HE 140 AA

6684 9546

54 63

0,147 0,144

541

83 94 107

HE160AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA

713 858

1

Designation

HE 1000 AA HE HE HE HE HE

100 A 120 A 140 A 160 A 180 A

HE200A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A

HE400A HE 450 A HE500A HE550A HE600A HE650A HE 700 A HE 800 A HE 900 A HE 1000 A

2505

292 336 373 439 480 525 572

237 316 380 452 527 610 697 789 890 933 977 1021

2 766

651

1

2986 3 217

742 840 986

141

165 210 239

1037 1209 1419 1621

1834 2 089

2223 2 362

3 592 3 855

1

103

++

1226

++

1361

++

1680 1998 2336

++ ++

499 595 738

103 115 137 179 196 245 280 342 390 431

911

1053 1265 1512 1806 2040 2 286

2644 2 923

3137 3355 3 736

4184 4 642

4976 5322 5 678 6121 ++

506 558 610 664 778 893 1014 1

136

1265 1400 1587 1884

++

2 216

++

2504

1 1

105 149 193

1281 1325 1369

1454 1549 1720 1809

*· «· **

·· **

*» *·

·* »·

*· **

··

605414 734986 851419

2 297 531

152585 164180 170250 176404 187932 201289 225 050 238 701

19 508

9668

28 080 40 771

13 830 19 939

977 566 137431

1

1462 828 1879 720

229 272 336 408

57610

27644

481

76 340

36 776

564 680

100929 133587 174986 216146 261372 325069 382601 434862

47897

811

914

1022 1

*·

181

1306 1390 1475 1603 1770 1936 2021 2106 2191 2280 2 371 2 542 2 628

490 791

602023 755 729 927981

1086127 1257340 1442 028 1652 478 2044 380 2540 350 3 012 700

575

621

738 835 937

1045 1284 1482 1658 62 72 89 117 130 163 187 226 256

63 590 82 648

101090

287 335 369 405 444 528

121 761

150674 166 789

177648 188535 205123 226900 248 750

260122 271579 283125 295 334 308 381

332402 345382

0,123 0,121 0,121

0,129 0,122 0,128 0,124 0,125 0,126 0,126 0,127 0,126 0,129 0,132 0,141

611

0,146

700 796 898

0,151

1006 1

0,193 0,206 0,217 0,227 0,233 0,250 0,250 0,250

157

1384 1651 1890

0,160 0,169 0,177 0,189 0,206 0,219 0,232

Plastic resistance of cross-sections Note: all values of Njj and Mj>j y must be divided by Ym0 9

Designation

10

12

11

Nsd/Npi.Rd=0,15

13

i

Nsd/NpI.Rd=0,2

14

Nsd/Np,.Rd=0,25

Nsd IkN]

Nsd [kN]

MN.y [N.m]

Nsd

[N.m]

HE100AA HE120AA HE140AA HE160AA

55 65

13 671 19 635

73 87

12 867

92 109

Mfg.y

[KN]

18480

16

15

|

|

17

Nsd/Npi.Rd=0,3

18

I

19

Nsd/Np,.Rd=0,35

Nky

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd [kN]

[N.m]

12 063

110

17325

131

11259 16170

128 153

10 455 15 015

tttt

·»

tttt

'·

··

**

tt*

tt

**

**

·*

tt

·*

ka

**

*.

tt

*

·*

.*

HE180AA HE200AA

**

**

·*

'*

*·

*.

··

'*

**

*·

**

**

**

k.

**

**

tt

*

HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA

**

··

··

k*

»

*·

··

**

**

tt*

'*

**

··

··

··

·*

·*

k.

**

·*

tt

·*

tt*

**

k»

··

**

tt«

**

**

**

k»

tt

*·

tt

**

*tt

·*

k.

··

tt

tt«

HE 1000 AA HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A

HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

··

t«

·*

**

·

·*

··

·*

··

··

>*

tt·

*«

**

tttt

»·

··

"

tt*

**

··

k.

tt·

··

**

·*

**

k.

*·

··

tt

k«

·*

*·

**

··

**

'*

*·

··

tt

tt*

··

··

**

tt*

**

'*

*·

··

tt

*·

·

··

482 539 578 620 673 770 889 995

137431 1 462 828 1879 720 2 297531

75 89

27157

605 414 734 986 851 419

643 718

600 279

771

851 419

804 898 964

977566

826 897 1027

977566

1033

1

39 436

137 160 190

56194

306 343 397 438 470 503 560 628 696 746 798 852 918

1008 130

1223

1

185

1326

18 914

111

271

1

*·

tt

tt

227

HE300A HE 320 A HE 340 A HE 360 A HE 400 A HE450A HE500A

··

734986

137 431 1462 828 1879 720 2 297 531 1

98 435 129605 170021 210129 254126 316417 372 141 424444

408 457 529 585 627

52 889 69 567 92 645 121 981

451

399477 452469 560973 708044 874114 1034 393

784 839 934

1257340 1442 028 1652 478 2044 380

1064

1

136

1

2540 350 3 012 700

1224 1343 1506 1630

228 266 316 378

160019 197768 239177 297804 350 250

671

1

125 149 185

37116

480 749 596034 752 297 927981 1086 127

747 837 928 995

1284 1482 1658

17801 25 559

100 119 148 182 213 253 302 361

73 915

122

1

210 163 402 074

1630 131 2044 380 2540 350 3 012 700

510 571 661 731

1046 1

161

1244 1330 1420 1530 1679 1883 2 038

576

562762

965

525 244

693 881 815081

1078

647622

157

760 742

948352 112072

1239 1346 1540

885129

1

1

462 828

1

1

126

1257 1349 1446

487727 tt «·

**

··

tt

*·

tttt

··

·*

**

tttt

tt

**

*·

**

**

**

·«

·*

16 688 23 962 34 796 49 583 65 219 86 855

114357 150018 185408 224229 279 191

328360 374510 424190 525912

150 179 221

273 319 380 454 542 612 686 793 877 941

1006 1

121

663 791 819481 969 743 1 134 528 1314 444 1 528 247

1255 1393 1493 1597 1704 1836

1931368

2 015

15 576

175

22364 32 477

208 258 319 372 443 529 632 714 800 926 1023

46 278 60 871

81064

14 463 20 767

30157 42 972 56 523 75 274

1863

99110 130016 160687 194 332 241966 284 578 324575 367631 455 790 575 286 710 217 840444 983 257

·*

·*

·*

»·

*·

«·

..

··

».

106 733

140017 173047 209 280 260 579

306469 349542 395911 490 851 619 539 764 849

905094 1058 893

1098 1

174

1308 1464 1625 1

742

·*

tt

*

··

*·

·*

tt

*

**

..

Plastic resistance of cross-sections Note; aU values of columns 2 to 7 must be divided by γΜο 1

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M

2

3

4

5

6

7

8

Np,

Vplz

Vpl,

Mpl.y

MpU

Nlimit

{Nfimit /NpLRd)

for Ν - My [kN]

[kN]

[kN]

[N.m]

[N.m]

for Ν -My Η

612 799

123 149 177 239 275

286 375 474 589

24 490 38 825

12084 19028 28149

1009 1275 1533 1835 2139

337 379

2 491

451

2 783 3 087

510 558 643 702

3503 3 792 4 016 4 245

761

822 949 1081

4648 5 122

5608

711

847 991 1 149

1281

1417 1603 1729 1814 1899 2 028

2194

57675 83182 113140 150999 194356 247489 301484 360592 439138 505071 565905 630502 759458 935857

1057

338 732

1

351335 364986 403323

1344 1600 1895 2163

2971

++

2883

3 057

3492100

1251 1561 1893

245 287 332 418 470 557 615 815 908 977

614 760 916 1091

55 416 82 394 116049

2 281

2 661 3 085 3 512 4 690

5162 5644 7122 7333 7422 7492 7 656

7883 8 091 8 328

|_

1228 1287 1338 1389 1495 1626 1757 1894

HE600M

8546

2 031

HE 650 M HE 700 M HE 800 M HE 900 Ml HE 1000 M

8 783

2167 2304 2636 2909 3188

9 001

9500 9 955 10 439

2799

1269 1465 1665 2 249

2466 2686 3 417 3 491

3 491

3480 3 469

3469 3458 3458 3 447 3 447

3436 3434 3 423 3 423

451

315168 326899

2 561

2 617 2 706

168629 204483 220688 231644 242635 259448 281449 303 537

8 725

6 729

2 532

71866 92 562 117128 141528

1

7200 7853

6344

2446

39 941 54 288

1

131 423 313 792 1509 907 1 720 172 1 956 876 2 404 050 2 956 300

2 361

71

89 110 149 175 212 243 287 322 359 412

492 536 632 728 830

1219 1358 1503 1656 1860 2195

5 970

[kN]

158523 207610 266760 333570 497482 593049 696 924 958 253 1 042 231 1 108 629 1 172 489 1309 095 1487 789

1667154 1864180 2061440 2 269 385 2476 900 2 935150 3393 400 3 893 950

577

389 710

941

27334

127 158 192 243 282

40 333 56 520 76 483 99 919

127656 159460

332 375 480 534 588 720 762 806

236 394

280229 328 219

449597 458420 458887 456453 454521

179

851

455 713

942 1056

454024

1

169

453 286

1288 1406 1524 1643 1902 2139

455825

2 381

455 267

453639 454883 453 264

453642

0,116 0,112 0,109 0,117 0,114 0,116 0,113 0,115 0,116 0,116 0,118 0,119 0.123 0,126 0,136 0,142 0,148 0.158 0,167 0,175 0.187 0,204 0,217 0,230 0,102 0,102 0.101 0,107 0,106 0,108 0,107 0.102 0.103 0.104 0,101 0,104 0,109 0,114 0,123 0,134 0.144 0,155 0,165 0.174 0,183 0,200 0,215 0.228

Plastic resistance of cross-sections Note: all values of Ns<¡ and Mjj.y must be divided by Ym0 9

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β

10

Nsd/Npuw-0,15 Nsd [kN]

[N.m]

92 120

37156

321

697 768 841

896 952

HE HE HE HE HE HE

100 M 120 M 140 M 160 M 180 M 200 M

HE220M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

1009 1080 178

1

1309 1410 188 234 284 342 399 463 527 704 774 847

1068 1

1 1 1 1

100 113 124 148 182

1214 1249 1282 1317 1350 1425 1493 1566

Nsd IkN]

MN.y [N.m]

122 160 202 255

55 015 80 037 108519 145156 186325

230 275

237713 289841 346854 423038 487202 548 221

613409 747114 927309 128 770 1 313 792 1509 907 1 720 172 1 956 876 1

2404050 2956 300 3 492100 52441 77950 109 757 150822 197355 254117 317438 471088 562 257 661344 906117 988621

1057196 1

124 291

1268 873 1460 151 1656 424 1864180 2 061440 2269 385

13

Nsd/Np,.Rd=0,2

23 546

151

HE600B HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β

M[M.y

191

374 418 463 525 569 602 637

12

11

MN.y [N.m]

22161

153

34 971

200 252 319 383 459 535 623 696 772 876 948

20 776 32 785 48 543 70 621 95 752 128078 164 404

184 240 303 382 460 550 642 747 835 926 1051 1 137

19391 30 599 45 307 65 913 89 368 119 540

214 280 353 446 537 642 749 872 974

18 006 28 414 42 070

51779 75329 102135

458 543 515 972

136 617

175365

577326 703166 872762 1062 372 1247604 1449 420 1668 467 1

924 783

745

2404050 2956 300

1880

3 492100

250 312 379 456 532 617 702 938 1032

49 356 73 365 103 301

1

1

129

1424 1467 1484 1498 1531

1577 1618 1666 1709 1757

141950 185 746 239169 298765 443377 529183 622442 852816 930467 995008

1058156 194 234 1374 260 1

800

1558 987 1764 082 1973 897 2196 803 2423 941

2935150 3393 400

1900

2935150

1991

3 893 950

2088

3 393 400 3 893 950

2 476 900

1

Nsd/Np,.Rd=0I35 Nsd [kN]

758 803 849 930

1269 1346 1440 1571

19

MN.y [N.m]

398153

1

Nsd/Nlpl.Rd=0t3

18

Nsd [kN]

701

122 194

Nsd/N ?l.Rd=0,25

17

16

MN.y [N.m]

223 730 272 791 326 451

1

15

Nsd [kN]

307 367 428 498 557 617

1024

14

1004

429884

195 763 238 692 285 645

348384 401225 451476 505160

1205 1273 1394 1537 1682

1358 831 1564188 1804 484 2264 407

1903 2 019

615 270 763 667 929 575 1091 654 1268 242 1459 909

2160 2356

1684185 2113 447

2 832 397 3 401532

2 617 2 820

2 643 570

878 173

375 468 568 684 798 926 1054

43187 64194

771

46 271 68 779 96 844 133078 174 137 224 221

415 666

162

1281

1402 1493 1586 1682 1800 1963 2181 2350 313 390 473 570 665

1

306 048 373 268

153444

483 724 541243 659 218 818 214 995 973 1 169 629

1061 1

209747 255742

1290 1411 1781

1833 1855 1873 1914 1971 2 023

2082 2136 2196 2 250 2 375

2489 2 610

578

280092

496109 583539 799 515

872313 932 820 992021 1 119 594 1288 369

1461550 1653 827 1850 528 2059 503 2 272 444 2 752 381 3 241 452 3 783 247

1791

1407 1548 1693 2137 2200 2 227 2 248 2 297

2365 2 427

2498 2564 2635

**

1080 1226 1327 1406 1486 1627 1793 1963 2090 2 220 2 355 2 520 2 749 3 054

61205 82 985 111001 142484 181780 221643 265 241

323499 372566 419 228

469077 571322 709119 863177 013 678

1

177654 1355 630 1563 886 1

tttt ··

··

··

438 546 663 798

40 102 59 609 83 932 115 335

162528

931

209 273

1080 1229 1642 1807 1975

150919 194325 242746 360244 429961 505 734 692913 756 004 808444 859 752 970315

90 388 124 206

261419 387955 463035 544636 746 214

814159 870632

2 493 2 567

2598

925 886 1044 955 1202 477

2 622

1364114 1543 571

2832

2680 2 759

1

116 586

1266 677 1433316 1603 791 1784902 1969 452

2 987

727 160 1922 203 2 120 948 2568 889 3 025 355

2 915 2 991 3 074 3 150 3 325 3484

2 385 397 2 809 258

3132

3 531030

3654

**

2 700

2850

1

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by Ym0 2

3

4

5

6

7

8

Νρ,

Vp,z

Vp..y

Mpl.y

Mpi.z

Nlimit

{Nfimit /Np|.Rd)

for Ν - My [kN]

[kN]

[kN]

[N.m]

[N.m]

for N-My Η

570

134 168 174 198

225 266 292

40 245 55 006 60 567 73 876 60 868

1

Designation

UB 178x102x19 UB 203x102x23 UB 203x133x25 UB 203x133x30 UB 254 x 102x22 UB 254 Χ 102x25 UB 254x102x28 UB 254 x 146 χ 31 UB 254 X 146 X 37 UB 254 χ 146 χ 43 UB 305 X 165 χ 40 UB 305x165x46 UB 305x165x54 UB 356x171 x45 UB 356x171 χ 51 UB 356x171 x57 UB 356x171x67 UB 406x178x54 UB 406x178x60 UB 406x178x67 UB 406x178x74 UB 457 χ 152 χ 52 UB 457x152x60 UB 457 Χ 152 χ 67 UB 457 χ 152 χ 74 UB 457 Χ 152 Χ 82 UB 457 Χ 191 Χ 67 UB 457 Χ 191 Χ 74 UB 457 Χ 191 Χ 82 UB 457 χ 191 χ 89 UB 457x191 χ 98 UB 533x210x82 UB 533 Χ 210 χ 92 UB 533 Χ 210 χ 101 UB 533 Χ 210 χ 109 UB 533 χ 210 χ 122 UB 610 χ 229 χ 101 UB 610 χ 229 χ 113

UB610x229x125 UB 610 χ 229 Χ 140 UB 610 Χ 305 Χ 149 UB 610 Χ 305 Χ 179 UB 610 Χ 305 Χ 238 UB 686 χ 254 χ 125 UB 686 Χ 254 Χ 140 UB 686 Χ 254 χ 152 UB 686 χ 254 χ 170 UB 762 Χ 267 Χ 147 UB 762 Χ 267 χ 173 UB 762 Χ 267 χ 197 UB 838 Χ 292 Χ 176 UB 838 χ 292 χ 194 UB 838 X 292 Χ 226 UB 914 Χ 305 Χ 201 UB 914 Χ 305 Χ 224 UB 914 Χ 305 χ 253 UB 914 Χ 305 Χ 289 UB 914 Χ 419 χ 343 UB 914 χ 419 χ 388

691 751

898 658 753 848 932 1

108

1

287

212 227 222 239 274 273 306 360 363 389 427 485 452 469 523 568 495 534 595 639

++

++ ++

1705 2009 ++

++

2 010 2 221 ++ ++ ++

++

2 457

701

++

++

556 593 653 696 759 735 782 840 903 994 907 967

++

1043

++

2 455 2 673

2944 ++ ++ ++

3263 3 652 ++

++

5360 7128

1

153

1069 1279 1681 144

++

1

++

1223 1307 1441 1389 1565 1721 1678 1775 1967 1956 2 079 2 277 2 575 2 593 2 875

++ ++ ++

++ ++ ++ ++

++ ++ ++ ++ ++

10 277

11614

243 289 352 445 522 469 545 638 467 553 628 762 544 637 716 805 469 572 649 738 826 676 773 858

241

1381 1 616

4 188

361 198

71799 82 910 92 370 113559 133087 146423 169 211

2 025 2 740 1 155

198828 182036 210561 237383 284552 247840 281875 316316 352689 257570 302509 341472 382248 425701 345 691 388391 430352 473206 524 616 483775 554620 613823 664680 750964 677065 770990 863831 973388 1079 488 1303 627 1759 258 938 623

1354

1

951

1059 777 917 1026 1

114

1269 947 1

107

1258 1426 1675

1501 1704

1310 1623 1915 1549 1787 2 211

1733 2048 2398

1

071 154 175 107

1323 230 1 211548 1456 454 1684147 1599 811 1

795 427

2 151 381 1962 541 2 240 611 2 570900

2 774

2953950

3 748 4308

3637800 4152 450

579

[kN]

9 774

97

11692 16 672

123

20 733 8 760 10 812 12 890

147 165 175 184

22119

171

28 053 33 153 33 307 38 900 45 973 34 443 40 937 46 707

57098 41898 49126 55 589 62 741

31326 38 317 43 870 50 085 56 490 55 767 63 944 71416 79 518 89 042 70 586 83 562 93 870 102409

117432 94 058 110 277 125825 143690 220 261

268805 369876 127463 149970 166894 190686 152093 189 758 225 269

197844 228844 284 727 230 765

273409 322078 376 296 679048 785110

131

179

204 208 231 271

284 299 326 365 355 364 404 436 392 418 463 493 539 438 463 508 539 584 582 612 653 700 765 723 764 817 898 822 977

1266 911

964 1024 1 1

123 100

1236 1345 1316 1409 1540 1504 1648 1790 2013 1991 2190

0.170 0,178 0.175 0,164 0,250 0,233 0.217 0,183 0,162 0,158 0,172 0,167 0,167 0,210 0,196 0,191

0,182 0.219 0.202 0.201

0,196 0,250 0,233 0,230 0,222 0,219 0,218 0,208 0,207 0,201 0,198 0.237 0,222 0,216 0,215 0.210 0.239 0,226 0,218 0,215 0,184 0,182 0,178 0,243 0.230 0,225 0,220 0,250 0.239 0,228 0.250 0,243 0,227 0,250 0,246 0,236 0,233 0.194 0.189

Plastic resistance of cross-sections Note: all values of Njj and Mjq.y must be divided by Ym0 9

Designation

10

Nsd/Np,.Rd=0.15

UB 178x102x19 UB 203x102x23 UB 203x133x25 UB 203x133x30

UB254X 102x22 UB 254 Χ 102 Χ 25 UB 254 χ 102 Χ 28 UB 254 χ 146 χ 31 UB 254x146x37 UB 254 χ 146 Χ 43 UB 305x165x40 UB 305x165x46 UB 305x165x54 UB 356x171x45 UB 356 Χ 171 χ 51 UB 356 Χ 171 Χ 57 UB 356 Χ 171 Χ 67 UB 406x178x54 UB 406x178x60 UB 406 Χ 178x67 UB 406 χ 178 χ 74 UB 457 χ 152 Χ 52 UB 457x152x60 UB 457 χ 152x67 UB 457 Χ 152x74 UB 457 Χ 152 Χ 82 UB 457 χ 191 χ 67

15

|

Nsd/Np,.Rd=0,25

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

86 104 113 135 99 113 127 140 166 193

40 245 55 006 60 567 73 876 60 868

114 138 150 180 132

38 813 53 532 58 731 70 660 60 868

36 387 50186 55 061 66 244 60 868

71799 82910

151

71799

170 186

82 910 90 485

143 173 188 224 165 188 212 233

222 257

108367 126518

241

141 481

162 540

210 561

276 323 269 305

237383

341

301

284 552

243 270 302 333 235 269 302 333 368

247840 281 875 316316 352689 257570 302509 341472 382248 425 701 345 691 388391 430 352 473 206 524 616 483 775 554620 613 823 664 680 750964

402 324 360 402 444 313 358 402 444

92 370 113559 133087 146 423 169 211 198 828

182036

301

334 368 401

442 369 414 454 490 548 454 507 562 628 670 804

1069 562 629 684 764 660 777 884 790 870

677065 770990 863831 973388 1079 488 1303 627 759 258 938 623

1

1 1

071 154 175 107

1323 230 1211548 1456 454 1684147 1599 811 1

795 427

1017

2 151 381

902 1 007

1962541 2240 611

138

2 570 900 2 953 950 3 637 800 4 152 450

1298 1541 742

1

Nsd/Np,.Rd=0,2

14

MN.y [N.m]

207 242 202 229 256

1

13

|

Nsd [kN]

181

UB457x191x74 UB457x191x82 UB457x191x89 UB 457x191x98 UB 533 Χ 210 Χ 82 UB 533 χ 210 Χ 92 UB 533 Χ 210 χ 101 UB 533 Χ 210 Χ 109 UB 533 χ 210 χ 122 UB 610 Χ 229 χ 101 UB 610 χ 229 χ 113 UB 610 χ 229 χ 125 UB 610 χ 229 χ 140 UB 610 Χ 305 Χ 149 UB 610 Χ 305 Χ 179 UB 610 Χ 305 χ 238 UB 686 Χ 254 Χ 125 UB 686x254x140 UB 686 χ 254 χ 152 UB 686 χ 254 χ 170 UB 762 χ 267 χ 147 UB 762 Χ 267 Χ 173 UB 762 χ 267 χ 197 UB 838 χ 292 Χ 176 UB 838 χ 292 χ 194 UB 838 Χ 292 χ 226 UB 914 Χ 305 χ 201 UB 914 Χ 305 χ 224 UB 914 χ 305 Χ 253 UB 914 χ 305 χ 289 UB 914 χ 419 Χ 343 UB 914 χ 419 χ 388

12

11

Ι

278 264

247840 281 875

316316 350984 257570 302509 341472

402 445

382 248 425 701 345691 388391

491

430352

535 589 492 552 605 653 730 606 677 749 838 893 1072 1426 750 839 912

473 206

491

1019 880 1036 1

178

1053 1

160

1356 1203 1343 1517 1731 2 055 2 323

523519 483 775

554620 613823 664680 750964 677065 770990 863831 973388 1 1

058 375 275 311

1711433 938623 1 1

071 154 175 107

1323 230 1211548 1456454 1684147 1599 811 1795 427 2 151 381

1962 541 2 240 611 2 570 900

2953950 3 609 629 4 094 082

1

1

140

1274 1

100

1295 1473 1316 1450 1695 1504 1678 1897 2164

171

33 961 46 841 51390 61828 56 811 65 513 74 097

200 242 263 314 230 264 297 326 388

31535

207 225 269 198 226 254 280 333 386

**

**

**

*·

·*

566

167 182

'

** a.

155241

**

a.

·*

**

*·

205 519 243481

597 703

226090

**

**

»*

**

**

··

277156 307111

704 777

~

285 175

**

**

»*

·*

aa

».

**

**

**

··

367911 407041 444429 490 799

667 737 802 883

**

·*

**

**

828 907 979 1096

**

704 777 860

381658 379 904 414 800 458 079

**

'·

**

354397 **

859 936 1030

385172 425359

a.

**

*·

a.

**

**

»·

·*

592417 665 032

1 1

142 278

*·

*·

617530

tt*

··

*·

··

**

«tt

··

··

*·

»*

123

*·

»·

»·

195 604 604 468

1

1256

867441

1466

*

**

**

"

*·

1608 2138

115 897 1497 504

1876 2 495

1390 539

1

-*

*·

··

**

*·

**

"*

··

».

"

a»

*·

**

··

1783

**

136 648

1368

1272 997

1529

··

1

··

188 130 *·

.a

**

**

**

2 062

**

**

1434 623 1637028

1554 1767

** **

**

*

..

··

aa

*·

"

**

2 087 734

2034

**

**

**

**

1

527 893

**

··

».

**

*·

**

"* **

2 276

..

».

2 569

2 523 715 2 887079 3 384 027

2904

3838 202

580

43 495 47 719 57 412 52 753 60 834 68 805 73 519 88 048 102796

··

310 568 344 003

tttt

1

**

*·

603 666 737

828 814 929401 1

451

·*

332 751 368 574 408 919

712 534

1

79174 94 821 110 703

·*

260873 238032 265056 296953 329048

634732

116

1340 1782 937 1048

MN.y [N.m]

414 485 404 458 512 603 486 539 603 666

534 544 587216

19

Nsd [kN]

152381 179 124 172926 196452 220199

555 392 448 503 555 614 502 556 614 668 736 615 690 756 816 913 757 846 936 1047

|

MN.y [N.m]

**

426 502 405 450 503

18

Nsd/Np,.nd=0,35

Nsd [kN]

70193

381

17

|

Nsd/Np,.Rd=0,3

79 390 84 829 101594 118610

277 322 302 345 404 337

191065 182036 209549 234 879

16

2596 3083 3484

2 694 607 3 158 425 3 582 322

3 029 3 597 4 065

"

*

2 502135 ··

3 326 442

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

6

7

8

Np,

Vplz

Vy

Mpl.y

Mpl.z

Nfimit

{Nfimft/NpLRd)

for Ν -My IkN]

IkN]

IkN]

{N.m]

IN.m]

for Ν -My Η

687 899

135 157 194 230 254

291

1

Designation

UC 152x152x23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203x203x52 UC 203x203x60 UC 203x203x71 UC 203 χ 203 χ 86 UC 254x254x73 UC 254x254x89 UC 254x254x107 UC 254x254x132 UC 254x254x167 UC 305x305x97 UC 305x305x1 18 UC 305x305 χ 137 UC 305x305x158 UC 305x305 χ 198 UC 305 x 305 x 240 UC 305x305 χ 283 UC 356x368x129 UC 356x368x153 UC 356x368x177 UC 356x368x202 UC 356 χ 406x235 UC 356x406x287 UC 356x406x340 UC 356x406x393 UC 356x406x467 UC 356x406x551 UC 356x406x634

1

107

1380 1558 1795 2125

301

329 416 348 418 517 627 799 483 586 676 778 956

2 576

2188 2663 3 205 3 951 5 002 2 901

3530 4099 4 732 5 932

7186 7 749 3 862 4 578

5299 6045 7037 8594 9 310 10 762 12 791

15 092

17362

1

165

1377 576 682 798 916

1027 1272 1516 1765 2105 2308 2664

·*

402 499 625 712 818 996

1203 1

006

1235 1484 1852 2364 1310 1604 1875 2178 2 769

3 376 3 667 1786

2127 2 464 2 816 3 319

4068 4430 5145 6 162 7 312 8 476

··

··

58 201

26 224

72559 116898 133338

32801

154178 187708 229532 233136 287608 348841 439312 569562 374136 460019 539690 629904 808 395 998062

71754 87814 107215

211

109367 135197

244 289 356 422 525 346 414 473 538 645 772 820 415 487 566 645 716 875 938

1097617 582629 696 711 811981 933 358 1

IkN]

101 328

1365 932 1504801 1767825 2150 000 2 597 200 3 061600

581

112 136 164 179

54 253 62 099

223 281

163 805

206418 267258 170 645

210411 247348 289074 371 472 458388 503564 281812 337170 392 640 451090 560058 693078 761894 893050 1082 375 1302 461 1528 300

1076 1

255

1472 1662

··

0,124 0,123 0,119 0,115 0,117 0,105 0,109 0,112 0,109 0,111

0,107 0,105 0,119 0,117 0,115 0,114 0.109 0.107 0,106 0.107 0,106 0.107 0.107 0,102 0,102 0,101 0,100 0,098 0,098 0,096

Plastic resistance of cross-sections Note: all values of Njj and Mj>j must be divided by γ^0 9

Designation

10

Nsd/Npl.Rd=0,15 Nsd IkN]

UC 152x152x23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203x203x52 UC 203x203x60 UC 203 X 203 χ 71 UC 203 χ 203 χ 86 UC 254 X 254 X 73 UC 254 χ 254 χ 89 UC 254x254x107 UC 254 X 254 X 132 UC 254x254x167 UC 305x305x97 UC 305x305x1 18 UC 305 X 305 X 137 UC 305x305x158 UC 305x305x198 UC 305x305x240 UC 305 Χ 305x283 UC 356x368x129 UC 356x368x153 UC 356x368x177 UC 356 χ 368 x 202 UC 356 χ 406 χ 235 UC 356 χ 406 χ 287 UC 356 χ 406 χ 340 UC 356 χ 406 χ 393 UC 356 χ 406 χ 467 UC 356 χ 406 χ 551 UC 356 Χ 406 Χ 634

··

135 166

207 234 269 319 386 328 399 481

MN.y IN.m] 56 498 70 334 112 743

128024 148476 178 296

218975 223079 274 278 333 534

418069

1078 162

579 687 795 907

1056 1289 1397 1614 1919 2 264

2604

540 952

361030 442974 518 526

604062 770996

13

|

Nsd/Npi.Rd«0.2 Nsd IkN]

··

593 750 435 529 615 710 890 1

12

11

MN.y [N.m]

··

53175 66196

276 312 359 425 515 438 533

106111 120493 139 742 167808

790 1 000 580 706 820 946

206094 209956 258144 313914 393476 509131

Nsd [kN] tt

225 277 345 389 449 531

644 547 666

15

MN.y [N.m] 49 851

62059 99 479 112962 131008 157320 193 213 196 834

242010 294 295

416916

988 1251 725 882

488 025

1025

568529

1

368884 477310 318 556 390859 457523 532996

339 793

1.83

725 643

950 486 1043 330

1437 1550

894575 981958

1483 1797 1937

554872 662647 772674 888100 1042 245 1292 594 1422 398 1669 560

772 916

522 233 623 668

1

727222 835859 980936 1216 559 1338 727

2 026 347

1060 1209 1407 1719 1862 2152 2558

2446 270 2877809

3 018

3 472

2 302 371 2 708 526

1

571 351

1907150

965 144

1325 1511

1759 2149 2 328 2691 3198

3773 4 341

582

16

680 290

838664 920 585 489593 584 688 681771 783 618 919 628 1 140 524 1 255 057 1 473 141

1787953 2 158 473 2 539 243

17

Nsd/Np,.Rd=0,3 Nsd IkN]

··

801

186

1

i

Nsd/Npi.Hd=0,25

tt

180 221

641

14

**

270 332 414 467 538 638 773 656 799 961 185

1

1501 870 1059 1230 1420 1780 2156 2 325 1

159

1373 1590 1813 2111 2578 2 793 3229 3 837 4 527 5209

MN.y [N.m]

18

Nsd [kN] *tt

46 528

315

57922 92 847 105432 122274 146832 180332

387

225876 274 675 344 292 445490 297319 364802 427022 497463 634938 782753 859 213 456 954 545 709 636319 731377 858319 1064 489 1 171386 1 374 932 1668 756 2 014 575 2369 960

19

Nsd/Npi.Rd=0.35

··

183 712

|

483 545 628 744 902 766 932 1

122

1383 1751 1015 1235 1435 1656 2 076 2 515 2 712 1352 1602 1855 2116 2463 3008 3 259

3 767 4 477 5 282 6 077

MN.y [N.m] tt·

43 204 53 785 86 215 97901 113540 136344

167452 170 590 209 742 255 055 319 700

413669 276082 338 745 396 520 461930 589585 726 842 797841

424314 506 730

590868 679135 797011 988 454

1087716 1276 723 1549 560 1 870 677 2 200678

Plastic resistance of cross-sections

IPE IPE A 1

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750x137 IPE 750x147 IPE 750x173 IPE 750X196 IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600 IPE O 180 IPE O 200 IPE 0 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

-IPE O

Note; all values of columns 2 to 7 must be divided by γΜο

2

3

4

5

6

7

8

Npi

Vpu

Vpl,

Mpl.y

Mp|.z

Nfimit

{Nlimit /Np|.Rd)

for Ν - My IkN]

IkN]

IkN]

[N.m]

[N.m]

IkN]

for Ν -My Η

39 56

210 284 363 452 553 659 783 918

57 81

.

100 121

153 179 222

252 304

1076 1263 1480 1722 2000

351

408 489 558 678 807

2 323 ++

81

6385

1600

107 136 169 204 243 286 338 394 460 535 615 719 812

10 837 16 699

2 515

926

24 295 34 061 45 764 60 676

78487 100827 133099 172798 221 191

280265 359465 467993

3735

71

5 293

87

7177 9 515

109 129 158

12 268 15 980

++

1330 1475 1674

μ

1

849

1067 1209 1396 1472 1494 1903

η·

2 021

2 242

++

49 70 86 99 124 146 182 215 259 298 353 428 473 568 671

++

800 957

314573 410929 535152

55 569

++

66 88 110 136 162 197 234 282 333 391 459 533 647 717 825 960

1094 1275

680531 863824

99 413 121568

52016

10 976 14 270 18 400

++

++ ++ ++

951 1 149

175 241

303 368 445 538 645

ΤΠ 916 ++ ++ ++

++

++ ++

1

745 879

114

202 245 280 339

1028 1202 1480 1728 1997

277 326 382 443 554 642 730 843 943

401 461

554 638 762 943

2 314 2 651

3236 3 759

1

4 292 5 411

1313 1657

115

1

129

1282 1434 1800

603 382

766426 965 910

1337919 1405 221 710 026 1972 899 1

5 219 9 070 13 715 19 690 27 251

37 215 49 955 66 057 85 682 113436 148992 193030 249361

181

20 329 26 661 34 435 42 262 52 552 62 975 76 005 92 367

110148 133554 168871 173473 222726 263 669

1290 2 073 3 019

4269 5 691

7689 10 047 13 333

17160 22645 29 515 36 648

47260

67580 82946

214 253 299 355 406 493 596 708 855 995 1

171

1289 1457 1577 35 50 62 72 89 107 130 156 184 215 260 313 342 411 492 593 706 825

204 550 259 266 326 173

41962

413097 562722

74 002 93 772

718571 897429 1229 535

132 142

145 173 200 239 284 333 397 462 550 689 824 968

176023

1227

68 590 88 316 112825

158028

583

23 209 32 369

50 873

62403

112347

0,187 0,196 0,195 0,193 0,198 0,196 0,202 0,197 0,199 0,200 0.202 0,206 0,203 0,212 0,219 0,223 0,231

0,232 0,244 0.250 0,239 0,229 0,197 0,206 0,204 0,195 0.201

0,198 0,202 0,200 0,201

0,200 0,203 0.208 0,194 0,204 0,209 0,213 0,219 0,219 0.194 0.197 0.194 0,199 0,192 0,193 0,199 0,199 0,207 0,213 0,219 0,226 0,227

Plastic resistance of cross-sections

IPE IPE A -IPE O 9

Designation

10

11

Nsd/Np,.Rd=0,15 Nsd [kN]

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750x137 IPE 750 χ 147 IPE 750x173 IPE 750x196 IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600 IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

32 43 54 68

300 348 408 477 554 643 720 773 913

Nsd/Np,.Rd=0,3

18

NSd

6385

42 57 73 90

6 283

53

10 787 16 591

71

5890 10113

24094 33976

113 138 165 196 229 269

63 85 109 136 166 198 235 275 323 379 444 517 600 697 815 953

5498 9439 14 517 21083

74 99 127 158 193 230 274

111

45 536 60 676 78 168 100 746 133 099 172798

15 554 22 589

91

31852 42690

965910

359465 467993 603382 766426 965910

**

**

**

aa

**

1405 221

1031

a.

··

710 026

1217 1380

1710 026 1972899

1522 1724

35 48

5199

44 60 76 92

133099 172 798 221 191

280 265

359465 467993 603382 766 426

314573 410929 535152

112 132 154 180 222 259 300

52 016 68 590 88 316

61

74 89 108 129 155 183 215 256 301

221 191

280 265

9 070 13 715

316 370 430 500 581

679 794 924

1072

19 561 27 251

111

37115 49955

161

66 057 85 682

113435 148992 193030

135

194 229 269

320 376

19

IkN]

MN.y IN.m]

5105 8 765 13 480 19 577

27605 36 998 49 397 63 511

53197 68 397

321

88 153 116 504 151 528 195 029

376 442 518 603

81857 108183

**

**

**

».

**

aa

*·

»*

**

a*

»»

»»

»»

»»

*"

"

*·

**

·*

»»

**

**

»»

··

»»

»»

**

**

**

"

**

**

2069

»»

·*

«*

4 224 7421 11200

56 997 73 282 94 450 124 826 162351

208960 263 786 342262 449 584 582412 747764 943301

1

29 729 39 844

|

Nsd/Np,.Rd=0,35

MN.y [N.m]

302 353 417 484 565

347 398 485 564 644 812

17

Nsd IkN]

5 219 9 070 13 715 19 690 27 251 37 215 49 955 66 057 85 682 113436 148 992 193030 249 361

192 226 264

|

MN.y [N.m]

26 36 45 55 67

161

Nsd/Np,.Rd=0,25

132 157 184 215 253 296 344 400 465 544 635 739 858

1

16

15

|

Nsd IkN]

1972 899

81

14

MN.y [N.m]

1035

97 117 137

13

Nsd [kN]

45 764 60 676 78 487 100827

161

190 222 258

12

Nsd/Np,.Rd=0,2

MN.y [N.m] 10 837 16 699 24 295 34 061

83 99 117 138

Note: all values of Ns
918 173

109

1

1287

4549 7992

61

91

12 062

110 133 162 194 233 275

17116

106 129 156 188

22168 30156

226 272

*

8563 12923 25 579 34 796 46 936

61948

**

53 72

4 874

18 339

140 704

23 873 32 476

43 807

57818

85

15 893

*

··

··

>·

··

*·

··

*·

·

tt

·*

·*

**

"

·*

*·

··

**

·'

80 429

352 402

··

**

*·

··

··

**

a

**

··

··

*·

··

~

·

471

··

*·

tt*

*·

**

»»

"

556

··

*·

tt·

*·

»»

**

k«

··

··

··

**

tt*

**

»»

··

"

·*

··

tttt

**

··

··

»*

··

ka

112825 158 028 204 550 259 266

326173 413 097 562 722 718 571

897429 1229 535

149 176 206 240 296 346

399 463 530 647 752 858

1082

51655 68 319 87 709 112 624

156425 202712 258 887

325958 413097 562722 718571 897429 1229 535

186 220 257 301

370 432 499 578 663 809 940

1073 1353

584

48 427 64 050 82 227 105 585 146 649

190042 242 706

305586 390 857 536126 690 270 869171 1 192 612

224 264 308 361

444 518 599 694 795 971 128

1

1288 1623

45198

261

41970

59 780 76 745 98 546

308 360

55 510

136872 177373 226 526

518 605 699 810 928

421

285 214 364 800

500384 644 252 811 227 1 113 104

1

132

1316 1502 1894

71263 91507 127095 164 703

210346 264841 338 743 464643 598 234 »·

1033 597

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 1

Designation

HE100AA HE 120 AA HE 140 AA HE160AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA HE 1000 AA

2

3

Np,

4

5

6

7

8

Vp,.z

vPi.y

Mpl.y

Mp|.z

Nfimit

{NfimH /Np,.Rd)

[kN]

for Ν -My [kN]

for Ν -My

[kN]

[N.m]

[N.m]

[kN]

M

429 510 633 835

98 110 126 165 193 245 280

185

16 048

7822

220 278 370 445 529 617 714 816 924 1041

23133

-

63 74

0.147 0.144

1005 1214 1415 1660 1897

342 393 437 514 562 614 670

++ ++

2 601 2 764 2 932 3 237 3 494 3 764 4 203

761

1092 1

1

··

»·

tt*

*"

··

··

**

«*

·*

··

··

'*

·*

tt·

·*

"

··

··

**

'·

·*

·· ··

*·

»tt

·*

··

*

**

tt*

**

«

··

**

··

143 195

**

a

»tt

tt

~

«

tt*

-

··

a

*·

*

··

··

154

1293 1345 1396 1499

++

1291

1551

996341

199 229

++

1603 1702 1813

++

1435 1593 1966 2338

727 864 977

2 012

++

2 733

2117

143 960 331 037 1 711 820 2 199 673 2688 600

206430 219921 235 552 263 356 279331

1097 1223 1502 1734 1940

584 697 864

120 134

268 318 393 477 563 660 796 948

22 829 32 860 47711

11314 16185

156 326 204 771

1070

252937

··

197

305 861 380 399

··

72 84 105 137 152 190 219 264 300 336 392

447724

195178 207886 220626 240038

868 983 1

++ ++

HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A

3 671 3 926 4 372 4 896 5 432 5 823

HE600A

6228

HE 650 A HE 700 A HE 800 A

++

HE900A HE 1000 A

161

L_

1066 1244 1480 1769 2113 2388

210 230 287 328 400

457

2 675

504 592 653 714 777 910

3095 3 420

1

1382 1528 1627 1

726

1044

1876 2071

186

2 265

2 365

++

1329 1480 1638 1857 2204 2593

++

2 930

++

++

1

2464 2564 2668 2 775 2 975 3 075

··

ft

708 463 860 090

192126

1

1

67415 89334 118108

508 881 574 330 704 494

884363 1085 935 1270 999 1471356 1687480 1933 750 2392 359 2 972 750 3 525 500

585

··

tt

23 333 32 349 43 036 56 050 74 414 96 715

··

265 521 291091 304 399

317806 331317 345 604 360 871 388 981 404171

431

474 519 618 715 819 932 1051 1

178

1354 1620 1932 2 212

tt

0.193 0,206 0,217 0,227 0,233 0,250 0.250 0,250 0,123 0,121 0.121

0,129 0.122 0.128 0,124 0,125 0,126 0,126 0.127 0,126 0,129 0.132 0.141

0.146 0,151

0,160 0,169 0,177 0,189 0,206 0.219 0,232

Plastic resistance of cross-sections

HE AA

HEA 9

Designation

HE100AA HE120AA HE140AA HE160AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA HE 1000 AA HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

Note: all values of Ngj and M^.y must be divided by Ym0

10

12

11

Nsd/NpI.Rd=0,15

13

|

Nsd/Np,.Rd=0>2

14

15

|

Nsd/NpI.Rd=0,25

16

|

17

Nsd/Np,.Rd=0>3

18

19

Nsd/Npl.Rd=0,35

Nsd IkN]

MN.y IN.m]

NSd

MN.y IN.m]

Nsd

[kN]

IKN]

MN.y IN.m]

Nsd IkN]

MN.y IN.m]

Nsd IkN]

MN.y IN.m]

64 77

15 999

86

15 057

107

14116

129

13175

150

12 234

·*

*·

**

»tt

·*

tt

··

**

*·

*·

tttt

·»

tt

·*

**

··

·»

··

··

*·

**

»tt

**

**

**

aa

**

·*

a

tt

··

Itt

**

tt·

··

**

·*

**

·*

··

tt

»«

**

*·

tt

**

tt

**

tt·

.*

··

tt

**

**

*»

»»

··

*·

*· ··

**

tt*

**

**

··

·*

·*

«·

tt

**

**

**

··

**

··

**

·*

**

·*

tttt

*·

'*

·*

··

··

··

·*

*·

··

tttt

·*

»*

**

**

*·

**

**

**

tt*

'*

**

**

*·

·*

**

**

·*

··

**

tt

··

**

**

~

**

**

**

·*

»·

**

··

·*

aa

tt«

**

**

··

··

*·

**

**

aa

·*

··

**

·*

*·

»·

*·

··

*»

»»

tt*

·*

**

*·

'*

··

··

**

»»

*·

1261

*a

"*

··

1353 1450

a*

··

*·

565 630 677 725 788

860090 996341 1

143 960

1331037

1040

711 820 2 199 673

164

2688 600

901 1

88 105 130 160 187 222 265

317 358

860090

902 967

996 341 1 143 960 1 331 037

1050 1202 1387 1552

22133 31779 46148

1711820

117 139 173 213

65 759 86 496

249 296 354 423

115190 151665 198960

1051 1

811988 953818

128

1208 1313 1502

1

**

»»

··

»»

»»

»»

»»

"

»»

"

»»

»»

tt·

··

109 774

*tt

aa

··

**

»»

»»

**

**

**

·*

*·

"

20 831 29 910 43 434

146 174 216

19 529 28 040 40 719

61891 81408

267

58023

311

76 320

370 442 528

101639

175 209 259 320 373 444

133 822

531

94 863 124901

175553

634

163850

108 414 142 744

187257

tt

18 227 26171 38 004

54 155

71232

204 244 302 373 436 518 619 740

115979 152146

16925 24 302 35 290 50 287

66144 88 087

··

tttt

*·

**

»»

aa

»»

··

*·

401

tt*

**

··

**

»»

»»

»»

**

··

464 513

*·

**

**

a.

*·

·*

**

a*

684 734 785 874 979

409867 467473 529485

384 251 438 256 496 393

1026

358 634

1

101

409039

1

178

463 300

615429

1086

1022899 1210 460

1312 1469 1630 1747

574400 724992 895036 1059152

551

589 656 734 815 874 934 997

1074 1

1

841

*

tt

179

1322 1431

435484 496690 562578 697486 880347 1085 935 1270 999 1471356 1687480 1933 750 2 392 359 2 972 750 3 525 500

1

165

1246 1329 1433 1572 1763 1908

656 458 828 562

1

416 148

1640 725 1907600 2392 359

855 918 981

1093 1224 1358 1456 1557 1661 1791

1965

776 777

958968 1

134 806

*·

197

333017

1285 1374 1530 1714 1901 2038

379 822 430 207

*·

1

533372 673207 831105 983 498

**

**

··

··

aa

»»

··

·*

tttt

·*

**

··

·*

*·

a*

a.

**

aa

··

tt*

··

**

»»

··

a.

**

··

··

**

»»

**

aa

tt

586

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

Np,

Vp|.z

Vy

Mpl.y

Mpl-z

IkN]

IkN]

IkN]

[N.m]

[N.m]

716 935

143 174

28 659 45 433

181

208 279

335 438 555 690 832

67492 97340 132398

991

176 700

962

1159 1345 1499 1658 1876 2023 2122 2222

227438 289615 352801 421969 513885 591041 662229 737822

111

2 373

2568

++

1265 1426 1589 1759 1938 2177 2568 2997

888 728 1 095 151 1324 006

++

3 374

1464 1826

286 336 388 489 550

1

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

1

1492 1794 2147 2504

321

394 443 528 597 652 753 822

2 915 3 257 3 613

4100 4 437 4 700 4 967

5439 5994 6563 6 987

7424 7 874 8 425

9190

891 1

2 215

2669 3114 3 610 4 110

5489 6040 6 605 8 335 8 581

8685 8 767 8 959 9 225 9 468 9 745 10 001 10 278 10 533

651

719 954

1062 1

144

1437 1506 1566 1626 1749 1903 2056 2 216 2 376

11650

2536 2696 3084 3404

++

3 731

11 117

2 763

2863 2 962

3 062 3 167 3 276 3 477 3 577

1537417 1766 912

4 021 4 019

4006 4006

Nfimit

for Ν -My

for Ν -My Η

[kN]

14 141

83

22 266 32 941 46 740 63 529 84 098

105 129 174

204 249 284 335 377 420 482 527 576 627 739 852

108317 137065 165618 197332 239289 258 252 271 073

283935 303610 329356 355 203 368 814 382541

971 1

101

427112 456 044 471 974

2 218 2 531

64849 96418 135802

31986 47198 66141

185 505

89 501

149 185 224 285

242948 312165 390348 582160 693993

116927 149385 186 602

2 631

4034 4034

8

{Nfimft/NpLRd)

2 012 967 2 289 961 2 813 250 3 459 500 4086 500

1072 1277 1485 1714 1948

4 085 4 072 4 059 4 059 4 047 4 047

7

1237 1380 1573 1872

719 889

2886 3143 3999 4085

6

815 549 121360 1 219 632 1

396388 411 137

276 632

327928 384086 526 125 536 449 536 995

1297332 1372062 1531920

534147 531886

741 029

533 281

1

1950925 2181488 2 412323 2655 664 2898 500 3434 750 3 971000 4 556 750

587

330 389 439 562 625 689 843 892 944 996

531304 532 760 530 854

532310 530 416 530 858 530441 533 412

1

102

1235 1368 1507 1645 1784 1922 2226 2503 2786

0.116 0,112 0,109 0,117 0,114 0,116 0.113 0.115 0,116 0,116 0,118 0.119 0.123 0,126 0,136 0,142 0,148 0,158 0.167 0,175 0,187 0,204 0,217 0,230 0,102 0,102 0,101

0,107 0,106 0,108 0.107 0.102 0,103 0,104 0,101

0,104 0,109 0,114 0.123 0.134 0,144 0,155 0,165 0.174 0.183 0,200 0,215 0.228

Plastic resistance of cross-sections Note: all values of Nsa and Mjj.y must be divided by Ym0 9

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 8 HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β

HE800B HE 900 Β HE 1000 Β

10

Nsd/Npl.Rd=0,15

Nsd/Npl.Rd=0.2

14

15

|

Nsd/NpI.Rd=0,25

16

17

|

Nsd/Np,.Rd=0,3

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

107 140 177 224 269 322 376

27554

143 187

25 933 40 923 60 592 88151 119520 159871 205 214 261811 319 224

179 234 295 373 449 537 626 729 814 903

24 312 38 366

215

22 691 35 808

56805 82641 112050

354 448 538 644

53018 77132 104580

751

1084 1230

179562 229085 279321 334 265 407683

1331

469 519

528323

165 501 1368 715

1410 1490 1632 1798 1969 2096

1590 121 1830 433 2111630 2649 838

2 227 2 362 2 528 2 757

2553

3 314 507

2 750 366 457 554 667 779 903

437 489 542 615 666 705 745 816 899 984

1048

43 481 64 379 93 660 126990 169863 218040 278 174

339175 405893 495044 570130 641535 717819 874282 1085149 1320 901 1

537 417

114

1766 912

1

181

1

264

2 012 967 2 289 961 2 813 250 3459 500 4086 500

1

1378 1532 650

1

HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M

220 274 332 400 467 542 616 823 906

HE300M

1250 1287 1303 1315 1344 1384 1420 1462 1500 1542 1580 1668 1747

HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

13

12

11

991

1

832

61367 91218 128 439

176494 230 948

297371 371469 551273 657960 773914 1060 350 1 156 897 1237145 1315 659 1484 852 1708 687 1938 368 2181488 2 412 323 2655 664 2898 500 3434 750 3 971000 4 556 750

236 298 359 429 501

583 651

723 820

887 940 993

1088 1

199

1313 1397 1485 1575 1685 1838 2 042

2200 293 365 443 534 623 722 822 1098 1208 1321 1667

1716 1737 1753 1792 1845 1894 1949 2000 2 056

2107 2223 2330 2443

382017 465924 536593 603798 675 594

822854 1

021 317

1243 201 1459 962 1696130 1952 462 2 252 406 2 813 250 3459 500 4086 500

57 757 85 852 120 884 166112

217363 279879 349618 518845 619 257

728389 997976 1088 844 1 164372 1238 268 1397 508 1608176 1824 347 2 064 351

2309 880 2 570 727 2836 526 3 434 750 3 971000 4 556 750

1025 1

1

109 175

1242 1360 1499 1641

1747 1856 1969 2106 2 297

1027 1372 1510 1651 2084

149 879

192388 245448 299272 358141 436 803 503056 566060 633370 771425 957485 1

874 977

719 997

893652 1087801 1277467 1484113 1708 404 1

970 855

Nsd [kN]

MN.y [N.m]

251

21071 33 250 49 231

327 413 522 628 752 876 1020 1

140

1264 1435 1553 1645 1739 1904 2098

71622 97110 129895 166 736

212722 259369 310389 378 563

435982 490 586 548 920

668568 829 820 010 101

2 297

1

2445

1

2 598 2 756

2949

186 220 378 105

1

1586375 1830080

·*

»*

3063

·*

·*

**

**

"

·*

a*

54147

439 548 665

262386 327767 486 418 580 553 682865 935603 1020 791 1091598

2 165 512 2 410 056 2 659 244 3 220 872

2 912 3 054

3 793188 4 427 204

801

934 1083 1233

1647 1812 1981

2500 2 574 2606

160 876 310 163

2630 2688

1507665 1710 325 1935 329

2 767

1

591 145

Nsd/Np,.Rd=0,35

aa

80 487 113329 155 730 203 777

1

139 887

19

l

»»

2145 2171 2192 2240 2306 2367 2436 2500 2569 2633 2779

588

281

18

2840 2 924 3000 3083 3160

3335 3 495 3665

50 537

75121 105 773

145348 190192 244894 305 916 453990 541849 637341 873 229 952 739 1 018 825 1083 484 1 222819

512 639 775 934

1090 1264 1438 1921 2114 2 312 2 917

227402 284 065 421 562

503146 591816 810 856

3003 3040

884686

3 069

1006 092

1407154 1596 303 1806 307

3136 3229 3 314 3 411

2 021 145 2 249 386 2 481961 3006147 3540309

3500 3 597 3 687 3 891 4 077

**

46 928 69 755 98 218 134 966 176 607

·*

946 052 1

135 475

1306 643 1

482 282

1677285 1876 777 2088 716 2304 678 2 791422 3 287430 tt

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

6

7

8

Np,

Vp,.z

Vpi,

Mpl.y

MpI.Z

Nfimit

{Nfimit /Np,.Rd)

for Ν -My [kN]

[kN]

[kN]

[N.m]

[N.m]

[kN]

for Ν -My Η

667 808 879

156 197 204

47095

11437

64 369

051

231

263 312 342 422

++

1297

248 265 282 260 279

13 682 19 510 24 262 10 251 12 652 15 084

1506

321

++

319 358 422 425 455 500 567 528 549 613 665 579 625 696 748 820 650 693 764 815 888 860 915 983 1057

114 144 154 172 193 205 215 200 210 239 243 270

1

Designation

UB 178 χ 102 χ 19 UB 203x102x23 UB 203x133x25 UB 203 χ 133 χ 30 UB 254 X 102 X 22 UB 254x102x25 UB 254 χ 102 χ 28 UB 254x146x31 UB 254x146x37 UB 254x146x43 UB 305x165x40 UB 305 X 165 X 46 UB 305x165x54 UB 356 X.I 71x45 UB 356x171 χ 51 UB 356x171 X57 UB 356x171 x67 UB 406x178x54 UB 406x178x60 UB 406x178x67 UB 406 Χ 178x74 UB 457x152x52 UB 457x152x60 UB 457x152x67 UB 457x152x74 UB 457x152x82 UB 457x191 χ 67 UB 457x191 χ 74 UB 457x191x82 UB 457x191x89 UB 457x191x98 UB 533x210x82 UB 533 χ 210x92 UB 533x210x101 UB 533x210x109 UB 533x210x122 UB 610x229x101 UB 610x229x1 13 UB 610 χ 229 χ 125 UB 610x229x140 UB 610x305x149 UB 610x305x179 UB 610x305x238 UB 686x254 x 125 UB 686 X 254 X 140 UB 686x254x152 UB 686x254x1 70 UB 762 χ 267 χ 147 UB 762x267x173 UB 762x267x197 UB 838x292x176 UB 838x292x194 UB 838 χ 292 χ 226 UB 914x305 χ 201 UB 914x305x224 UB 914 X 305 X 253 UB 914x305x289 UB 914x419x343 UB 914x419x388

1

881

992 1091

++

1891 ++ ++

1995 2 351 ++

++ ++

2 599 ++

++ ++ ++

2 875 ++ ++ ++

3129 3445 ++ ++ ++ ++

4 273

1

163

++

1062

++

1

++

++

1221 1349 1251 1496 1967 1339 1431 1530 1687 1626 1832 2 014 1963 2 077

++

2 301

++

2289 2433 2665

++ ++

6 272 8 342 ++ ++ ++ ++ ++ ++ ++ ++

++ ++ ++

++

13 591

132

3 014 3034 3364

70876 86 450

231

71229

285 338 412

84 020

97022 108093 132888

521 611

155 741 171 346

549 637 747 546 647 734

198013 232671 213021 246401

277789 332987 290025 329853

891

637 746 838 942 549 669 760 864 967

370 157 412 722 301411

353999 399 595 447311 498161 404 532 454500 503603 553 752 613913 566120 649024 718304

791

904 1004 1

113

1239 909

1073 1201 1303

1484 1

108

1296 1472 1669 1960 2370 3 207 1351 1585

1757 1993 1533 1899 2 241

1812 2091 2 587 2 027 2396 2 807 3 246 4 387 5 042

//7817 878 788 792311

902222 1010 866 1

139 071

25 884 32 828 38 796 38 977 45 521 53 798 40 306 47 905 54 657 66 817 49 029 57 488 65 051 73 420 36 658 44 839

51337 58 610 66 106 65 260 74 828 83 572 93 053

104198 82 600 97 785 109 848 119841 137421

110068 129047 147242 168147

317 332 350 382 428 415 426 473 510 458 489 542 577 630 513 542 595 630 683 682 716 764 820 895 846 894 957 1051 962

1263 231

257 752

525 521 2 058 707 1098 389 1 253 478 1 375 126 1548 460 1417 769 1704 362 1 970 810

314559

1

432 833 149159

1482 1066

1

1872119 2 101 031 2 517574 2 296 591

2621991 3008 500 3 456 750 4 257000 4859 250

589

175497 195302 223143 177981 222057 263613 231519 267796 333191 270044 319 947 376900 440346 794 631 918 746

1 1

143

128 199

1314 1287 1446 1574 1540 1649 1802 1

759

1929 2095 2356 2330 2563

0.170 0,178 0,175 0,164 0,250 0,233 0,217 0,183 0,162 0,158 0,172 0,167 0.167 0.210 0,196 0,191 0,182 0,219 0,202 0,201

0,196 0,250 0.233 0,230 0,222 0,219 0,218 0,208 0,207 0,201

0,198 0,237 0.222 0,216 0,215 0,210 0.239 0.226 0,218 0.215 0,184 0,182 0,178 0,243 0.230 0,225 0.220 0,250 0,239 0,228 0.250 0,243 0,227 0,250 0,246 0,236 0,233 0,194 0.189

Plastic resistance of cross-sections tote: all values of Njj and Mjfy must be divided by Yj^jo

9

Designation

10

Nsd/Np,.Rd=0f15 Nsd IkN]

UB 178x102x19 UB 203x102x23 UB 203x133x25 UB 203x133x30 UB 254x102x22 UB 254x102x25 UB 254x102x28 UB 254x146x31 UB 254x146x37 UB 254x146x43 UB 305x165x40 UB 305x165x46 UB 305X165X54 UB 356x171x45 UB 356x171x51 UB 356x171x57 UB 356x171x67 UB 406x178x54 UB 406x178x60 UB 406x178x67 UB 406x178x74 UB 457x152x52 UB 457x152x60 UB 457x152x67 UB 457x152x74 UB 457x152x82 UB 457x191x67 UB 457x191x74 UB 457x191x82 UB 457x191x89 UB 457x191x98 UB 533x210x82 UB 533x210x92 UB 533x210x101 UB 533 X 210 X 109 UB 533x210x122 UB 610x229x101 UB 610x229x113 UB 610x229x125 UB 610x229x140 UB 610x305x149 UB 610x305x179 UB 610 Χ 305 Χ 238 UB 686 Χ 254 Χ 125 UB 686 Χ 254 Χ 140 UB 686 Χ 254 Χ 152 UB 686 χ 254 Χ 170 UB 762 x 267 χ 147 UB 762 Χ 267 Χ 173 UB 762 Χ 267 χ 197 UB 838 Χ 292 χ 176 UB 838 χ 292 Χ 194 UB 838 χ 292 Χ 226 UB 914 Χ 305 Χ 201 UB 914 Χ 305 Χ 224 UB 914 Χ 305 Χ 253 UB 914 χ 305 Χ 289 UB 914 Χ 419 χ 343 UB 914 χ 419 Χ 388

NSd/Np,.Rd=0,2

14

15

|

Nsd/Np,.Rd=0,25

16

17

|

NSd/Np,.Rd=0,3

18

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd

100

47095 70876

200 243 264 315

39 742 54 813

132 158 116 132 149 164 195 226 212 242 284 236 268 299

71229 84020 97022 105886 126812 148053

167 202 220 263 193 220 248 273 324 377

42580

64 369

133 162 176 210 154 176 198 218 259 301 282 323

45 419

121

264 298 327 389 452

72 352 66 480 76 664 86 709 92 650 110961 129 546

234 283 308 368 270 308 347 382 454 527

**

**

·*

**

**

**

190 206

404 473 394 446 499 588 474 526 588 650

**

»»

**

»»

353 390 431

469 517 432 484 531

573 641

532 594 657 735 784 941

1251 658 736 801

86 450

71229 84 020

97022 108093 132888 155 741

171346 198013 232671 213021 246401 277789 332987 290025 329 853 370 157 412722 301411 353 999 399595 447311 498161 404 532 454500 503603 553 752 613913 566120 649024 718 304 Til 81 7 878 788 792311 902222 1010 866 1

139 071

1263 231 525 521 2 058 707 1

520 575 470 520 575 626 689 576 646 708 764 855 709 792 876 980 1045 1254

1668

329853 370157 410 726

209613

567

195 639

662

36903 50898 55842 67184 61732 71 188

80 516 ·*

103035 120 293 ·· *·

181 665

tt·

tt·

**

·

«tt

tt»

**

tt

»·

a«

**

··

257680 305 277

599 705

284925

823

a«

264 573

«·

**

**

··

·*

**

**

··

··

347498 385056

706 780

** *"

359386

**

910

··

··

*·

*«

«tt

··

*·

·«

·

··

··

··

·

tt

tt·

··

**

**

**

**

*·

399595 447311 498161

389389 431310 478523

706 779 862

404 532

588 650 719 588

**

*·

**

>·

«a

454500

651

**

**

tt«

·*

·*

503 603 553 752

718 782

476324 520 076

862 939

612629 566120 649024 718304 777817 878788

861

574340

720 807 885 955 1068

742772 833816

*·

**

··

··

1010 866 1

139 071

1095 1225

**

**

1492 385

1568 2085

2 002 741

446621

1006

*

·»

*·

*·

··

095

**

1033

485405 536050

1206

497 761

tt

**

**

**

**

·*

**

·*

··

··

··

**

··

**

··

146

«s

*·

**

1

1

1282

778 229

**

**

**

·*

*·

*·

»»

*

··

··

»»

**

**

··

*·

1087597

1470

*·

··

··

·*

··

..

·«

**

tt>

tt

1399111 1877569

tttt

1882 2502

1

752 398

1496

2 920

··

1627 227

··

··

··

**

*»

··

981

**

**

*·

··

**

**

··

375 126

1067

375 126 1548 460

1334 1491

**

**

·*

a.

**

1489 677

1789

a*

**

tt·

1

2 101 031 2 517 574 2 296 591

1332 1519 1804 2039

471

213021 245 217 274859 325629 290025

92 903 99 268 118887 138 799

231

MN.y [N.m]

·

1018

178

470 520 367 419

223587

77520 71229 82140

60137

JkN]

877

1548460 1417769 1704 362 1970810 1872119

190

421

68 728 82 688

58 729 64 433

19

1098389 1253 478

894 772 909 1034 924

1056

378 315 357 399 470 379

62644

|

Nsd/Np,.Rd=0,35

NSd IkN]

431

1

13

|

MN.y [N.m]

353 284 316 353 390 275 314 353 390

1

12

11

2 621 991 3008 500 3456 750 4 257 000 4859 250

1

193

1030 1212 1379 1232 1358 1587 1408 1571 1776 2025 2405 2 718

1

** 1 1

704 362 970 810

*·

1515 1723

1

**

··

**

aa

*·

**

··

»»

**

··

2068

**

**

··

** **

915 671

·*

··

*·

*·

aa

·*

*·

*"

**

*

·*

**

1984

aa

»»

*·

**

··

··

**

aa

*

**

**

*

··

·*

*·

*·

** **

**

** ** **

3 378 497

3 038

3153 264

3545

3960 031

3608

*·

**

*·

4 491 513

4 077

4192079

4 757

**

2 517 574

3008 500 3 456 750 4 224 033 4 790 947

2 219 2 532 3006 3398

590

·· ·· ·*

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by γΜο 1

Designation

UC 152x152x23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203x203x52 UC 203x203x60 UC 203 X 203 χ 71 UC 203x203x86 UC 254 χ 254 χ 73 UC 254x254x89 UC 254x254x107 UC 254x254x132 UC 254x254x167 UC 305 X 305 χ 97 UC 305x305x118 UC 305x305x137 UC 305x305x158 UC 305 Χ 305 x 198 UC 305 Χ 305x240 UC 305x305x283 UC 356x368x1 29 UC 356x368x153 UC 356x368x1 77 UC 356x368 χ 202 UC 356x406x235 UC 356x406x287 UC 356x406x340 UC 356x406x393 UC 356x406x467 UC 356x406x551 UC 356x406x634

2

3

4

5

6

7

8

Np,

VpU

vPi.y

Mpl.y

Mpl.z

Nfimit

{Nfimit /Npi.Rd)

for Ν - My [kN]

[kN]

IkN]

[N.m]

[N.m]

for Ν -My Η

804

158 184

341 471

227 270 298 352 385 487 407 489 605 734 934 566 685

584

1052 1296 1615 1823 2100 2 487

3015 2560 3116 3 751 4 624 5 854 3 395

4131 4796 5 538 6 941 8409 9191 4 519 5 357 6 201 7074 8 234 10 057 11042

791

910 1

119

1363 1611 674 798 934

1072 1

202

156034 180421 219659 268601 272819

166

1407 1

177

1446 1737 2167

336 562 408 218

2 767

666 508

1532 1877 2194 2548 3 240 3 951

437819 538320 631552 737121 945995 1 167945

4350

1

2 091

2489 2884 3296 3884 4 760 5 254

15170 17 899 20 592

2 737 3 160

8 673 10 053

6102 7308

102762 125465 127983 158 210 191 687 241553 312749 199690 246 225 289450 338 278 434 701 536411 597251

514088

301 825

681800 815301 950 191

1092 228 1288 788 1598 431 1784 765 2096 723 2550 000 3080 400 3 631200

591

**

30 687 38 384 63 488 72 668 83 968

84 909 136 796

834 957 1

·*

68108

731

1489 1774 2066 2463

12 765

*·

IkN]

·*

394561 459 472 527871

655387 811049 903 642 059 199 1 283 747 1544 779 1812 634 1

131

159 192 209

246 261

329 286 339 416 494 615 404 484 553 629 755 904 972 486 569 662 755 838 1024 1 1

113 276

1489 1746 1971

·*

0,124 0,123 0.119 0,115 0,117 0,105 0,109 0.112 0,109 0,111 0,107 0,105 0,119 0,117 0,115 0,114 0,109 0,107 0,106 0,107 0,106 0,107 0,107 0,102 0,102 0,101 0,100 0,098 0,098 0.096

Plastic resistance of cross-sections Note: all values of Ngj and Mjty must be divided by Y¡^0 9

Designation

10

Nsd/Np|.Rd=0,15 Nsd IkN]

UC 152x152 χ 23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203x203x52 UC 203x203x60 UC 203x203x71 UC 203x203x86 UC 254x254x73 UC 254x254x89 UC 254x254x107 UC 254x254x132 UC 254x254 χ 167 UC 305x305x97 UC 305x305x1 18 UC 305x305x137 UC 305x305x158 UC 305x305x198 UC 305x305x240 UC 305x305x283 UC 356x368x129 UC 356 χ 368 Χ 153 UC 356x368x177 UC 356x368x202 UC 356x406x235 UC 356x406x287 UC 356x406x340 UC 356x406x393 UC 356x406x467 UC 356 X 406 X 551 UC 356x406x634

MN.y IN.m]

66115

158 194

831

82 305 131933 149 815

173748 208645 256 247 261050 320963 390306 489 229

633029 422482 518373 606786 706881 902229

1041 1261

1

1379

1237438

678 804 930 1061 1 235

1509 1656 1915 2 276

2685 3 089

112 271 **

775438 904193 1039 266 1219 648 1512 610 1687030 1980176 2 403 342 2 901389 3 413 215

|

13

Nsd/Np,.Rd=0,2 Nsd IkN]

·*

tt

242 273 315 373 452 384 467 563 694 878 509 620 719

12

11

|

··

210 259 323 365 420 497 603 512 623 750 925 1

171

679 826 959 1

108

1388 1682 1838 **

1071

1240 1415 1647 2011 2208 2553 3034 3580 4118

MN.y IN.m]

14

196371 241 174

245694 302083 367347 460 451 595 792

397630 487881 571 093

665300 849157 1046 843 1 164 647 «·

729824 851005 978133 1 147904 1423 633 1587 793 1863 695 2261969 2 730 719 3 212 438

16

Nsd/Np,.Rd=0,25 Nsd [kN]

*»

62226 77464 124172 141003 163528

15

|

263 324 404 456 525 622 754 640 779 938

58336

283 203

344387

1463

558555 372778 457388 535399

849

1033 1

199

1384 1735 2102 2298

2 059 2 514

1

'

623 718 796 084

981415 1091857

··

1339 1550 1768

316 389 485 547 630 746 905 768 935

72 622 116412 132190 153307 184098 226101 230338

431 673

·*

684 210

797817 917000 1

076 160

2 761 3191 3 793

1334 656 1488 556 1747 214 2120 596

4 475 5148

3 011661

592

MN.y [N.m]

·*

**

156

1

Nsd/Np,.Rd=0,3 Nsd [kN]

MN.y [N.m]

··

17

|

2 560 050

125

1387 1756 1018 1239 1439 1661

123377 143087 171 825

211027 214982 264323 321428 402895 521 318

2 082 2 523 2 757 **

**

1607 1860 2122 2470 3 017

638 596

744629 855866 1004 416 245 678 1 389 319 1630 733 1979 223 2 389 380 2 810 883 1

19

|

Nsd/Np,.Rd=0,35 NSd IkN]

«·

54 447 67 781 108651

347927 426896 499706 582137 743012 915988 1019 066

3 313 3 829 4 551 5 370 6 178

18

MN.y [N.m]

··

368 453 565 638 735 870 1055 896 1091

1313 1618 2049 1

188

1446 1679 1938 2429 2943 3 217

**

50 558

62939 100890 114565 132866 159 552

195954 199626 245443 298469 374117 484081 323075 396403 464013 540556 689940 850 560 946 276

··

1875 2171 2 476 2 882 3 520 3 865 4 468 5 310

6265 7 207

··

592982 691 441 794 733

932672 1

156 701

1290 081 1514 252 1837850 2 218 710 2 610106

Plastic resistance of cross-sections

IPE IPE A -IPE O

3

Np,

5

6

7

8

Vp,.z

vpt.y

Mpl.y

ΜρΙ,ζ

Nfimit

{Nfimit /Np|.Rd)

for Ν -My IkN]

IkN]

IkN]

[N.m]

[N.m]

IkN]

for Ν -My Η

271

73 104 129 157 198

105 138 176 218 263

8 242 13 989

366 469 583 713 850 1011 1

++

++

631

++

720 875

1631

++ ++

++ ++ ++ ++

-

2161 2 387

++

2609

226 312 392 475 574 695

63

++ ++ ++ ++ ++

++ ++ ++

++ ++ ++

++

962 1

1042 1227 1483 1717

++

++

135

1327 1552 1911

2230 2 578 2 987 3 422 ++ ++ ++ ++

'

91 111

127 160 188 235 278 334

384 456 553 610 733 866

1033 1236 1437 260 317 362 438 517 595 715 824 983 1217 1439 1695 2139

21557 31362 43 970 59 077 78 327 101 319

314 369 436 509 594 690 793 928

231

287 326 392 454 526

185

1389

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600 IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

2

4

1

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750x137 IPE 750x147 IPE 750x173 IPE 750x196

Note; all values of columns 2 to 7 must be divided by γΜο

130 159 171 819

223066 285 537

361797 464037 604136 778912 989387

1048 1

196

1377 1560 1802 1900 1928

1

2 457 2 895

2207488 2546 833

85 114 142 176 209 254

6 737 11708

246 901 1 727 132 1814 012

302 365 430 505 592 689 835 926

1065 1239 1412 1646

1851 2 324

113

12 283 15 837

167 204 233 277

119 237 142190 172405 217997 223 938 287519 340373

1666

9 926 12 970

533 271

726423 927609 158 499 587 218

593

141

327 385 458 524 637 769 915

67840 81295 98115

7347

334 689

91

20 629 26 243 34 417 44 453 54 556

48 041 64 487 85 274 110 608

421060

1

6833 9265

35178

145647 204000 264056

1

72

17 704 25 417

67 148 88 544 114008

1088 1217 1457 1654

51

2 676 3 897 5 510

146435 192335 249184 321902 406085 530472 690832 878504 1 115119

358 420 493 572 715 829 943

2065 3 247 4 821

17212 22152

104

0.231

1285 1512 1664 1881 2035

0.232 0.244 0,250 0,239 0,229

45 64 80 93 115

0,197 0,206 0,204 0,195

138 168 201

0.198 0,202 0,200 0.201 0,200 0,203 0.208 0.194 0,204 0,209 0,213 0,219 0.219

1

38101 47 309

238 278 336 404

61008 71735

441 531

87 240 107076

635 765

128 333 156 934

1065

29 232

14169 18 422 23 753 29 961

41786 54169 65 672 80 557 95 529 121 052

145029 170583 227230

0.187 0,196 0,195 0,193 0,198 0,196 0,202 0,197 0,199 0,200 0,202 0.206 0.203 0,212 0,219 0.223

911

187 223 258 308 367 430 513 596 709 889

1064 1 250 1584

0,201

0.194 0,197 0,194 0,199 0,192 0,193 0,199 0,199 0.207 0,213 0.219 0,226 0,227

Plastic resistance of cross-sections

IPE IPE A -IPE O 9

Designation

10

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600 IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

13

Nsd/Npi.Rd-0,2

14

15

Nsd/N pl.Rd=0|25

16

17

Nsd/Np,.Rd=0,3

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd [kN]

41

8 242

55 70

13 989

54 73 94 117

8111 13925 21418

7604 13055 20 079

141

18 740

29160

143 170 202

43 859 58 783 78 327

175 214

27216 38377 51435

237 278 326 382 445 516 600 702 820 954

100908 130055 171 819 223066 285537 361797 464037 604136

68 92 117 146 178 213 253 296 347 408 478 556 645 750 877

21557 31362

87 107 128 152 178 208 245

287 333 387 450 526 615 716 831

930 998 1

12

11

Nsd/Np,.Rd=0t15 Nsd IkN]

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750x137 IPE 750x147 IPE 750x173 IPE 750 Χ 196

Note: all values of Ngj and Mjj.y must be divided by Ym0

179

1336 34 47 59 71

86 104 125 150 177 208 248 291 341

389 456 538 625 730 144 170 199

233 287 335 387 448 513 627 728 831

1048

43 970 59 077 78 327 101 319

130159 171 819 223066 285 537 361797 464037 604136 778 912 989 387 1

31 104

41 118

55109 73 577 94 601 121926 161 139

209580

18

19

NSd/N 3i.Rd=n>35

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

81

7097

110

12185

95 128 164

6 590 11314 17402

204 250 298 354 415 486

25 272 35 636 47 761 63 767

255 303 355 417 489 573

68 672 88 294 113 798

81987 105669

**

»·

*·

**

t«

··

··

·*

*·

··

**

**

'*

··

·*

**

**

'*

··

··

»»

**

ka

··

**

··

».

··

aa

aa

k.

*·

"·

»»

**

»»

aa

'·

tt

107

··

*·

«·

**

**

ka

·*

··

*·

**

aa

··

aa

*·

ka

**

·*

**

**

a*

"·

»»

**

'·

*·

*·

»»

··

*

**

ka

»»

tt·

**

··

**

**

'·

5 873

79 109 137 166

246 901

1

2207488 2546 833

1571 1781

6 737

45 62 78 95 115 139 167

11708 17 704 25 417

35178 48 041 64 487 85 274

201

110608 146435 192335 249184

237 278 330 389

**

6 711

57

11708 17704

78 98 119 144 174 208

25 252

35178

47913

6 292 11054 16 683 23 673 33 020 44 918

68 94 117 143 172 208

10 317 15 570

22 095

5453 9580 14 458 20 517

*

*

*·

**

*

·*

··

**

aa

'*

··

251

·*

»»

aa

ka

··

296

··

a.

aa

"

··

*·

**

*·

»»

aa

ka

·*

··

**

··

»»

»»

'*

*·

*·

**

*·

**

"*

"

**

*·

··

·*

··

**

**

"

··

**

«a

·*

a.

··

»»

aa

·

tt

··

·*

**

».

tt

»»

»»

'*

"

··

**

**

**

**

»»

»»

·*

**

··

aa

*·

**

··

»»

»»

"

»»

**

*·

··

»»

**

»»

**

**

"

58 347

176 689

337 397 465 543 669

54179 71658 91995 118127 164069

228972

781

'·

64 487 85 274 110608

67148

192

66 682

88 544

227 265 310 382 446 516 597 684 835

88194

114008 145647 204000 264056 334 689 421060 533 271 726 423 927609 1 158 499 1 587 218

113 224

145387 201931 261683 334199 420 783 533 271

726423 927609

971 1 108

1

1397

1587 218

158 499

240 284 332 388 478 558 644 747 855

1044 1213 1385 1746

594

62 515 82 682 106148 136301 189310 245 328

313312 394484 504561

692090 891076 1

122 021

1539 554

289 340 398 466 573 669 773 896

1027 1253 1456 1662 2096

77170 99 071

127214

··

··

ka

*·

tt

ka

tt

··

'·

··

··

ka

··

··

y»

··

**

'»

2445

'*

1436 917

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 1

Designation

HE 100 AA

HE120AA HE140AA HE160AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA

2

3

Np,

Vp,.z

IkN] 554 659 817

5

6

Vpl,

Mpl.y

MpLz

IkN]

[kN]

[N.m]

IN .m]

126

239 284 359 477 574 682 796 922

20 717 29 863

»*

141

1078 1297

162 213 249

++

317

++

361 441

++

507 564 663 726 793 864 983

++ ++ ++ ++

++ ++

4 178

121

++

1

++

++

1269 1489 1666 1853

++

2 056

++ ++

4

++

2538

++

*·

>·

**

··

··

k.

··

··

·*

'*

»tt

'*

*·

tttt

*

2 002

1286186

*

1261

2069 2197 2340 2598

1

283 898

304076

1416 1579 1939 2239

2 733

476 748 718 248 2209 803 2839 578 3470 738

*

1

*

·· »

tttt

tt

*

·*

*·

tt

*

··

**

··

*

**

*·

**

«

··

··

··

*

··

*

115

207

1376 1606

271

508 616 727 852

20 893 30 121

87027 115323 152467 201802 264341 326519

41760

HE800A HE 900 A HE 1000 A

3 995 4 415

764 843

4 738

921

5068 5644

1003 1

6 320

1348

7013

1531

++

1716 1910 2115

++ ++

175

++

2 397 2 845

++

3348

++

3 783

++

1381

1545 1784 1973 2100 2228 2422 2673 2924 3 053

3181 3309 3444 3 582 3840 3969

394 839 491061 577971 656 919

741408 909438 1 141633 1401844 1640 745 1899 387 2178 383 2496 296 3 088 318

3837550 4 551 100

595

tt*

*·

**

42 419 61591

1028 1224

tt«

*»

*·

411

651

tttt

914 562 1 110 298

··

14 605

3453

*·

1053 1192 1344 1410 1476 1542 1670 1736 1802 1935

29 470

424 516 589

0,147 0,144 -

345

HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A

82 95 **

155 173

371

for Ν -My Η

Itt

754 899

1911 2 284 2 728 3 082

[kN]

»·

100 A 120 A 140 A 160 A 180 A

HE200A

for Ν -My

··

++

297

Nfimit

··

HE 1000 AA

1

8 {Nfimit /Np,.Rd)

»·

3 018 3 529

HE HE HE HE HE

7

339 969 360 591

·· ·· ** ·* ** ** ··

«*

**

284809 309867 342764 375 772 392951 410 258 427700 446143 465852 502140 521748

939 1

115

2 505

93 109 135 177 196 246 283 341

387 434 506 557 612 670 798 923

1057 1203 1357 1520 1748 2091 2494 2 855

tt

0,193 0,206 0,217 0,227 0,233 0,250 0,250 0.250

0,123 0.121 0.121 0,129 0,122 0,128 0.124 0,125 0.126 0,126 0,127 0,126 0.129 0,132 0,141 0,146 0.151 0,160 0,169 0,177 0,189 0,206 0,219 0,232

Plastic resistance of cross-sections

HE AA

HEA 9

Designation

Note: all values ofNgj and Mjf-y must be divided by Ym0

10

Nsd/Npl.Rd=0,15 Nsd IkN]

HE100AA HE120AA HE140AA HE160AA HE180AA HE200AA. HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA HE 1000 AA

MN.y IN.m]

13

Nsd/Npi.Hd-0,2 Nsd IkN]

MN.y [N.m]

14

15

Nsd/N pl.Rd=0.25 Nsd [kN]

MN.y [N.m]

16

17

Nsd/Np,.Rd=0,3 Nsd [kN]

MN.y [N.m]

18

Nsd [kN]

··

··

a.

·*

*·

aa

··

**

»»

*·

·*

»»

*·

··

··

a*

**

·*

**

**

··

··

**

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·*

"*

··

**

**

**

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··

**

··

··

*·

**

k*

··

**

·*

··

··

**

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··

aa

**

**

··

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*·

*

*·

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tt*

**

k*

**

**

**

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**

**

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··

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k*

**

**

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··

··

"

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aa

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ktt

··

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*"

·*

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tt*

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a.

·*

**

**

**

»·

·*

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1017

tt*

*·

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»*

··

»»

·*

··

**

É*

··

"

**

·*

»»

ktt

·*

"

**

*·

**

1ktt

··

**

1356

··

»»

ktt

**

»»

163

*·

*·

»»

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··

**

1343 1503

aa

·

"

»·

·*

**

**

··

"

'*

·*

**

1

113 135 167 206 241

287 343 409 462 518 599 662

HE220A HE 240 A HE 260 A HE 280 A

HE300A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A

711

1

718 248

28 572

151

41024

180 223 275

59 573 84 889

226 270 335 413

23 530 33 785 49 060 69 909

264 315 390 482

21849 31371 45 556

64915

tt*

··

a.

tt*

**

**

»tt

··

*·

*·

··

«*

**

··

*·

**

··

**

··

··

»»

··

·*

**

··

tt*

**

··

**

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**

**

tt*

*·

**

**

··

tt

a.

·*

··

·*

··

··

·

··

**

«*

·*

*·

*·

tt

**

**

··

··

**

tt

**

··

**

··

**

**

··

·*

tt

**

··

tt

··

*·

1014

1052

1401844 1640 745 1899 387 2178 383 2496 296

1264 1403 1503 1608 1716 1849

3 088 318

2 029

128

25 210 36198 52 565 74 902

·*

726 237 900 391 1 136 449

1206 1287 1387 1522 1707 1847

188

225 279 344

MN.y [N.m]

tt

760 847 948 1

26 891 38 611 56 069 79 896

19

N&J/N. ¡>1.Rd=0,35

83 99

729 814 874 936

HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A

HE500A HE 550 A HE 600 A HE 650 A HE 700 A HE800A HE 900 A HE 1000 A

12

11

1

129

**

683518 847427 1069 599 1320 469 1562 593

tt*

**

··

**

*»

598078 741499

1774 1975

935 899

2 212

tt

1267

640 798

1411

794463 1002 749

1520 1693 1896

**

*

··

*·

*·

ktt

··

aa

*·

1580 1753 1879

555358 688 535 869 049

··

·*

k·

··

**

**

·*

**

k*

··

**

**

*·

**

»tt

·*

**

*·

·*

aa

t»

··

**

**

*·

*·

·*

**

ktt

··

·*

*·

**

aa

*·

*·

·

··

**

··

596

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by γΜο 1

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β

HE300B HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M

HE300M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M

HE800M HE 900 M HE 1000 M

2

3

4

5

Np,

Vp,.z

Vp..y

Mpl.y

IkN]

IkN]

[kN]

[N.m]

[N.m]

924

185 225 268 361

432 566

36 996 58 650

18 255

28 744

87126

42524

125658 170914 228104 293602 373867 455 433 544 724 663379 762980 854877 952461 1 147267 1 413 741

60 337 82 010

1207 1525 1926 2 316

415 509 572

2772 3 232 3 762 4 205

681 771

4663 5 292 5 728 6 067 6 412 7021

7738

++ ++ ++

1890 2 357

2860 3445

12 223 12 580 12 910 13 268 13 597 14 351 ++ ++

1242 1434 1633

2868 3063 3 315 3 567 3 695 3 824 3 953

4089 4 229

709 171 1984 665 2280 923 2 598 557 2 956 131 1

3869 4355

4488 4 618

5 275 300

370 434

928

841

,_

2 422

3631650 4465 900

501

7085

11078 11212 11318 11565 11908

2 740

632 710

4 020 4660 5 305 7 797 8 526 10 759

2612

2 051 2 271 2 501 2 810 3 315

++

1074 1280 1496 1736 1935 2140

150

1841

9584 10165

716 890

842 972 1061 1

8 472 9 019

.

929 1231 1371 1476 1855 1944 2 021

2099 2 258

2456 2654 2 861 3 067 3 274 3 481 3 982 4 395 4 817

1

147

1383 1648 1918 2 213 2 515

3 397 3 725 4 057 5 5 5 5

163

273 273 257 5 240 5 240 5 224 5 224

83 714

124467 175308 239 471

313624 402977 503904 751 516 895 882 1052 800

1447574 1574 434 1674 738 1771207 1977569

5 171

2 247 511 2 518 466 2 816 102 3114 090 3 428 221 3 741 700 4 433 950 5 126 200

5 171

5882350

5208 5208 5 191 5 188

597

.

6

7

8

Mp].z

Nfimit

{Nfimit /Np,.Rd)

for Ν -My

for Ν -My

IkN]

Η

107 135 166 225 264 321 366

0,116 0.112 0,109 0,117 0,114 0,116 0,113 0,115 0,116 0.116 0.118 0,119 0,123 0,126 0,136 0,142 0,148 0,158 0,167 0,175 0,187 0,204 0.217 0,230

108563 139828 176939

433 487 543 623

213 798

254738 308900 333379 349931 366534 391933 425168

1

458 535

1254

476106 493825 511701 530 741

551363 588 711 609 275

681

744 810 955

1421

1597 1781 2030 2417 2863 3 267

41291 60 929 85 382 115 538 150942 192842 240886 357106 423325

192 239 290 367 425 502 567 725 807 889

495 821

679179 692 507 693 212

689535 686617 688417 685866 687744 685 285

687163 684 718 685 289 684 751 688 587

100

0,102 0,102 0.101

0,107 0,106 0,108 0.107 0,102 0,103 0,104

1088

0,101

151

0,104 0,109 0,114 0.123 0,134 0,144 0,155 0,165 0.174 0.183 0,200 0,215 0,228

1

1218 1285 1423 1595 1766

1945 2124 2303 2 482 2 873 3 231

3596

Plastic resistance of cross-sections Note: all values of ti^a -^ ^N.y must ** divided by f^0 10

9

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β

HE500B HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M

HE450M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

12

11

Nsd/Np,.Rd=0,15

14

13

Nsd/Npi.Rd=0,2

|

15

Nsd/Np,.Rd=0,25

16

17

|

Nsd/Np,.Rd=0,3

18

Nad/Npi.Rd-0,35

Nsd [kN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

NSd

139

35 569

33 477 52 828 78 219 113 795 154289

31385

56130 83108

185 241

231

181

302

49 526 73 331 106682 144646 193480

277 362 457 578 695 832 970

29 292 46 225 68 442 99 570

323 423 534 674

135003 180582 231798

811

229 289 347 416 485 564 631

700 794 859 910 962

1053 161

1

120 907

163933 219 278

281469 359 098

437844 523971 639057 735986 828163 926639 1

128 618

1400 829

1271 1353 1438

1984 665 2280 923

1525

2 598 557

1631

2956131 3 631650 4465 900

1779 1977 2130 283 354 429 517 603 699 796

1063 1

170

1279 1614 1662 1682 1698 1735 1786 1833 1887 1936 1990 2040 2153 2 256

2365

1

705 163

305 385 463 554 646 752 841

933

1058 1

146

1213 1282 1404 1548 1694 1804 1917 2033 2175 2 373

5 275 300

2636 2840

79 219

378

117754 165 803 227838 298132 383 879 479533 711644 849367 999052 1368 815 1493 449 1597041 1698397 1916 809 2 205 760 2 502 257 2 816 102 3114090 3 428 221 3 741700 4 433 950 5 126 200 5 882 350

471

206379 264912 337975 412089 493149 601465 692692 779448 872 131 1062 229 1 318 428 1604 859 1884 679 2 189 549

2520 451 2 907651

3 631650 4465 900 74 559 110 828

2 516 2 582

2654

3 318 575

2 719

3 661698 4 433 950

1417 1559 1705 2152 2 216 2 242

2264 2313 2382 2445

2870 3008 3154

579 693 808 941

248355 316851

1

1051

386 333

1261

166

462327

1323 1432 1517 1603 1755 1935 2118 2255 2396

563 874

1399 1588 1718 1820 1924 2106

1

2 541 2 719

5 126 200 5 882 350

649399 730 732

817623 995840 1236 026 1504 556 1766 886 2 052 702 2362 923

129

295 728 360 578

431506 526 282

606106 682017 763114 929450

MN.y [N.m]

[kN]

27 200 42 923 63 553

1317 1472 1632 1852 2005 2123

92458 125360 167683 215 241 274604 334 822 400684 488690 562813 633 301

2 244 2 457

708 606 863061

970 131

1

2 321

1

153 624

2 708

1071222

2 541 2 706 2 875 3 049

1404 252 1649 094

2965

1303 948 1531302

3 157

··

··

**

**

*·

**

**

·*

aa

2 725 923

3263

2966

**

*

··

**

aa

3 295

aa

.»

**

··

aa

**

**

·*

**

69 899 103901 146 297 201033 263 058 338 717

567 707 858 1034

··

156050 214435 280 595 361298 451325 669 782 799404 940 284 1288 296 1405 599 1503 098 1598 491 1804 055 2 076 010 2355 066 2664 889 2 981844

572 689 804 932 1061

381 481

19

·*

472 589 715 861

1005 1

165

1326 1771 1949 2131 2690 2 769 2803 2829 2 891 2 977

3056 3145 3 227 3 317 3399 3588 3 760 3 942

598

423118 627921 749441

881516 1207 778 1317 749 1409154 1498 585 1691302 1946 259 2 207874 2498 334 2795 479 3 111 164 3 432 842

4157853 4896 661 ··

1206 1398 1592 2126 2339 2558 3 228 3 323 3364 3 395 3 470 3 572 3 667

3 774 3 873 3980 4 079 4305 4 512 tt*

65 239 96 974 136 544 187631 245521

316136 394910 586 059

699478 822 749 1 127 259 1229 899

1315 211 1398 680 1578 548 1816 508 2060 682 2331778 2609114

**

60 579 90 047 126 791

661

825 1

001

1206 1407 1631

1857 2480

2729 2984 3 766 3 877 3 924

3 961

4048 4168 4 278 4 403 4 518

174229 227984 293555 366 702 544198 649516 763981 1046 741 1

142 049

1221267 1298 774 1465 795 1686 758 1913 491 2165 223 2422 749 2696 342 2 975129

2 903 753 3 203 986 3880 663

4644

**

·*

··

aa

**

tt»

4 759 5 023

*·

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

6

7

8

Νρ,

Vp|.z

Vpl.y

Mpl.y

Mpl.z

Nfimit

{Nnmn/Npi.Rd)

for N-My IkN]

IkN]

IkN]

[N.m]

[N.m]

IkN]

for N-My Η

861

340 402

60 796

14 764

83095 91494 111600 91950 108462

17662 25185 31320

++

202 254 263 299 320 342 365 336

147 186 199 222 249 265 278 258

++

361

944

1

Designation

UB 178x102x19 UB 203 X 102 X 23 UB 203 χ 133 χ 25 UB 203 X 133 χ 30 UB 254 X 102 X 22 UB 254 X 102 X 25 UB 254 X 102 X 28 UB 254 X 146 X 31 UB 254 χ 146 χ 37 UB 254 X 146 X 43 UB 305 X 165 X 40 UB 305x165x46 UB 305 x 165 X 54 UB 356x171x45 UB 356x171x51 UB 356x171 X57 UB 356 X 171 χ 67 UB 406 X 178 χ 54 UB 406x178x60 UB 406x178x67 UB 406x178x74 UB 457x152x52 UB 457x152x60 UB 457x152x67 UB 457 χ 152 x 74 UB 457 X 152 χ 82 UB 457x191x67 UB 457 χ 191 χ 74 UB 457x191x82 UB 457 X 191 χ 89 UB 457x191x98 UB 533x210x82 UB 533x210x92 UB 533x210x101 UB 533x210x109 UB 533x210x122 UB 610x229x101 UB 610x229x1 13 UB 610 χ 229 χ 125 UB 610 χ 229 χ 140 UB 610 Χ 305 X 149 UB 610 Χ 305 χ 179 UB 610 Χ 305 Χ 238 UB 686 Χ 254 χ 125 UB 686 Χ 254 X 140 UB 686 Χ 254 χ 152 UB 686 χ 254 χ 170 UB 762 Χ 267 χ 147 UB 762 χ 267 χ 173 UB 762 Χ 267 Χ 197 UB 838 χ 292 χ 176 UB 838 Χ 292 X 194 UB 838 χ 292 Χ 226 UB 914 Χ 305 Χ 201 UB 914 χ 305 χ 224 UB 914 χ 305 χ 253 UB 914 χ 305 χ 289 UB 914 χ 419 χ 343 UB 914 Χ 419 χ 388

1044 1

135

1356 ++

++ ++

1

++

414 412 462 544 549 587 645 732 682 709

++

791

++

858 748 807 898 965 1059 839 895 986 1051

++ ++

2 441 ++

++ ++

++ ++

++ ++ ++ ++ ++

++ ++ ++ ++

++

1

146

++

1

++

1

111 181

++

++

1268 1364 1501 1370 1461

++

1

++ ++ ++

++

576 1741

++

1615 1932 2540 1728

++

1

++

1975 2177

++ ++

10 768

++

848

441

545 299 368 437 532 672 789 708 823 964 705 835 948

477840 532 786

709 864

389095

47323

456 981 515 841

57 883 66 272 75 660 85 337 84 244 96 596

1021 1

1295 1437 1600 1

174

177398 142087 166588 190076 217063 332735 406067 558748 192551 226551 252117 288057 229 757 286656 340300 298870

1022801

1672 1900 2155 2530 3 059

1

164 687

1304 937 1 470 437 1630 716 1969 308

4140 1745 2046 2268

2657603 1417920

2 573

1998 921 1830 211

2 971

3339

2 955 3 140

2 617

4343

134 436

1

++

++

107884 120123 134510 106629 126 232 141803

1431

++

3 917

650106 714844 792505 730809 837831 927265 1004 092

1386 1550 1682 1916

2 699

++

522 213 586 718

167

2 682

++

577438 643080

1248

++

3440 3890

425 811

981 1 115

++

++

255616 300357 274991 318081 358600 429856 374396

151

2 893

++

221 192

822 963

2600 2535

++

201047

33 414 42 378 50 082 50 315 58 764 69 449 52 032

1082 1216

1

1979 2452

++

139538 171 547

61841 70 558 86 254 63 292 74 212 83 975 94 778

2 098 2 365

++

125 246

13 233 16 333 19 472

2339

3 093 3 623 4 190

5663 6508

1 1

618 127 775 162

2 200 176 2 544136

2 416 736 2 712 240 3 249 959 2964 690 3 384 752 3883 700 4462 350 5 495 400 6 272 850

599

154 703

345 700

308 313 349 409 428 452 493 552 536 550

0,170 0,178 0,175 0,164 0.250 0,233 0,217 0,183 0,162 0.158 0.172 0,167 0,167 0,210 0,196 0,191 0,182 0,219 0,202

611

0.201

658

0.196 0,250 0,233 0.230 0,??? 0,219 0.218 0,208 0,207 0,201 0,198 0,237 0,222 0,216 0,215 0,210 0,239 0,226 0.218 0.215 0,184 0,182 0.178 0,243 0,230 0,225 0,220 0.250 0,239 0,228 0,250 0,243 0,227 0,250 0,246 0,236 0,233 0,194 0,189

271

591 631

700 745 814 662 700 768 814 882 880 924

987 1058 1

156

1093 1

154

1235 1357 1242 1476 1913 1376 1456 1548 1697 1661 1866 2 032

1988 2129

430120 348602 413022

2 327 2 271

486 543 568 447 1025 796 1 186 017

2 704

2490 3 041 3008 3309

Plastic resistance of cross-sections Note: all values of Ns
Designation

UB 178x102x19 UB 203x102x23 UB 203x133x25 UB 203x133x30 UB 254x102x22 UB 254x102x25 UB 254x102x28 UB 254X146X31 UB 254x146x37 UB 254x146x43 UB 305x165x40 UB 305x165x46 UB 305x165x54 UB 356x171x45 UB 356 χ 171 Χ 51

UB356X171X57 UB356x171x67 UB 406x178x54 UB 406x178x60 UB 406x178 χ 67 UB 406x178x74 UB 457x152x52 UB 457x152x60 UB 457x152x67 UB 457x152x74 UB 457x152x82 UB 457x191x67 UB457x191x74 UB457x191x82 UB 457 Χ 191x89 UB 457x191x98 UB 533x210x82 UB 533X210X92 UB 533x210x101 UB 533x210x109 UB 533x210x122 UB 610x229x101 UB 610x229x113 UB 610x229x125 UB 610x229x140 UB 610x305x149 UB 610x305x179 UB 610x305x238 UB 686x254x125 UB 686x254x140 UB 686x254x152 UB 686x254x170 UB 762x267x147 UB 762x267 χ 173 UB 762x267x197 UB 838x292x176 UB 838x292x194 UB 838 χ 292 X 226 UB 914x305x201 UB 914x305x224 UB 914x305x253 UB 914x305x289 UB 914x419x343 UB 914x419x388

10

12

11

Nsd/Np,.Rd=0,15 Nsd [kN]

MN.y [N.m]

129 157 170 203 149

60 796 83 095

91494 111600 91950 108462 125246

Nsd/Npl.Rd=0,2 Nsd IkN] 172

58632

215

47638

261

258 313 340 407 298

301

80 868 88 722 106 742 91950 108 462

54 967 75 813

51303

209 227

70 759

93 399

365 397 475

72 087 86 728

**

*·

**

341

**

·*

*·

**

··

271

199

255616 300357 274991 318081 358600

417 488 407

849 950 1033 1

155

997 1

173

1335 1

193

Nsd/Npl.Rd=0.35 MN.y IN.m]

**

1012 1215 1615

19

Nsd [kN]

**

125246 136689 163703

83177

284 339 249 284 320 352 419 486

100071 91950 106036 119929

77632

·*

384

**

**

**

*·

*·

*·

a.

aa

*·

»»

**

179177

583

167 232

··

··

**

**

**

··

·*

**

**

··

«·

·*

**

**

288 631

610

tttt

*·

··

191 122

461

270 591

732

681

155 287

»*

·*

«*

**

**

**

**

··

tt·

·*

**

**

**

**

··

**

**

**

*·

tt

«·

**

354 817

477840

515 607 490 543 607

532 786

671

425 811

Nsd/Npl.Rd=0,3

18

MN.y [N.m]

201047

374396

17

Nsd [kN]

292 273 313 366 305 346 386 455 367 407 456 503 355 406 456 503 557 455 504 556 606 667 557 625 685 739 827 686 767 848 949

429 856

Nsd/Np,.Rd=0,25

16

MN.y [N.m]

139 538 171 547

15

|

Nsd IkN]

211 251

192

14

MN.y [N.m]

227 256 282 335 389

171

13

|

420357

644 759

394 085

910

»*

**

··

**

**

**

tt·

tt

*

tt·

**

tt

·

·*

*«

··

·*

·*

**

530 211

759 839

*«

**

tt«

*·

*·

477840

··

·»

*·

**

**

*·

**

··

**

*"

a

».

a.

·

**

··

**

**

515 841

607

**

aa

**

**

*·

**

tt*

577438 643080

671

577438

**

··

··

**

*·

742

643 080

838 928

113

»·

**

tt·

**

**

**

**

·*

·*

**

**

tt*

586 718

672 742 808 889

**

**

tt

**

tt*

··

tt*

650106

927 1010

tt*

*·

«tt

·*

«*

**

*·

·*

·*

*·

334

*·

··

tt·

650106 714 844 792 505

714 844 790 848

1

617729

741 420

112

1

1

··

··

*

**

**

**

«·

»·

**

**

**

**

··

**

*·

**

**

tt

927265 1004 092

914 986

··

·«

tt

··

"*

**

·*

**

·

·«

**

**

"*

**

tt·

1

134 436

1

134 436

1

076 381

1655

tt

·*

··

"*

**

··

*

**

**

tt*

**

tt

**

**

··

**

**

131

**

tt*

**

*w

··

**

**

1581

**

*·

**

tt*

tt·

**

**

··

**

*·

tt

1926 534

2024

·*

··

·*

**

**

2 585 356

2 692

2 423 771

3 230

2 262 187

3 769

2100 602

470 437

1

1265

**

1969 308 2 657603

1

103

1232 1379

*"

1304 937 1

1

1004 092

··

1619 2154

1

470 437 **

·

··

**

**

·"

**

"*

**

tt·

**

··

tt

tttt

"

*·

*"

··

**

**

««

**

**

"*

tt

**

**

**

**

tttt

*·

775 162

1998 921

1378 1539

··

··

1998 921

1924

·*

tt

**

**

**

··

**

2 200 176 2 544 136

1565

"*

**

"*

**

**

**

**

780

**

**

**

**

··

tttt

**

**

**

tt

tttt

·*

**

··

**

**

··

**

*·

tt*

**

*

**

**

«*

**

**

tt·

**

**

**

**

1

1314 1537

3 249 959

1363

**

*tt

**

*·

**

**

**

**

»*

1521

**

**

**

**

*tt

·*

··

**

··

1719

3 883 700 4 462 350 5 495 400 6 272 850

2 292 2 615 3105 3509

**

··

tt*

**

**

**

tttt

4 462 350

3268

·*

··

··

*a

**

**

tttt

··

**

**

··

4386

**

**

··

*·

**

1961 2 329 2 632

2 049

a·

6 184 678

600

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 1

Designation

UC 152x152x23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203 χ 203 x 52 UC 203x203x60 UC 203 χ 203 x 71 UC 203 X 203 X 86 UC 254 χ 254x73 UC 254 χ 254 χ 89 UC 254 χ 254 χ 107 UC 254 χ 254 χ 132 UC 254 χ 254 χ 167 UC 305 X 305 χ 97 UC 305x305x118 UC 305x305x137 UC 305x305x158 UC 305 X 305 χ 198 UC 305x305 X 240 UC 305 χ 305 χ 283 UC 356x368x129 UC 356x368x153 UC 356x368x1 77 UC 356x368x202 UC 356 χ 406 X 235 UC 356x406x287 UC 356 X 406 χ 340 UC 356 x 406 x 393 UC 356x406x467 UC 356x406x551 UC 356x406x634

2

3

4

5

6

7

8

Np,

Vplz

Vp.,

MpJ.y

Mp|.z

Nfimit

{Nfimit /Np,.Rd)

for Ν -My IkN]

IkN]

IkN]

IN.m]

IN.m]

for Ν - My Η

1038 1358 1672 2085 2353

204 237 292 348 384 454

440 608 753 944 1076

2 711 3 210 3 892 3 305 4 023 4 842 5 969

497 629 525 631 781

**

1236 1505 1817 1520 1866 2 242

**

87921 109610 176591 201425 232907

93 808 108 395

283 560 346 740

132656 161964

352184

*·

434 471 526 973 663 641 860 402

204 234

247450 311822

1978

565185

~

2 423 2 832

7148

1

694 923 815 276 951 557

317855

6 192

730 885 1021

4 382 5 332

8 961 10 856 12 074

221 193

561 159

1507 711

2080

5 714

1

692458 784624

1

5834

870

2699

1030 1205 1384 1552 1922 2290 2666

3 213 3 722

9 131 10 630 12 983 14 507 16 769 19 929

23 515

27053

373 653 436 686

760

1444

6 916

8006

403 731

3290 4182 5101

175

3 180 3 595 4 152

1

710 241

6145 6903

880141 052 479 1226 610 1409 967 1663 708 2 063 429 2 344 691

8 017 9 601 11394 13 207

2 754 518 3350 000 4046 800 4 770 400

4 254 5 014

*·

509 342

1

601

169

206 247 270 318 337 424 369 437 537 638 794 522 625 714 812 975

·*

3 572

7556

··

39 614 49 551

947 1206

2798

IkN]

593137 681434 846 045

1046 991 1 1

187 138 391 497

1686 492 2 029 416 2 381304

1

166

1277 627 735 855 974 1082 1321 1462 1676 1956 2294 2 589

0.124 0,123 0,119 0,115 0.117 0,105 0,109 0,112 0,109 0,111

0,107 0,105 0,119 0,117 0.115 0.114 0.109 0,107 0,106 0.107 0,106 0,107 0,107 0.102 0,102 0,101

0.100 0.098 0.098 0,096

Plastic resistance of cross-sections bte: all values of Ns
Designation

10

Nsd/Np,.Rd=0,15 Nsd IkN]

UC 152 X 152 X 23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203x203x52 UC 203x203x60 UC 203x203x71 UC 203 χ 203 χ 86 UC 254x254x73 UC 254x254x89 UC 254x254x107 UC 254x254x132 UC 254x254x167 UC 305x305x97 UC 305x305x118 UC 305x305x137 UC 305 X 305 χ 158 UC 305 X 305 X 198 UC 305x305x240 UC 305 χ 305 X 283 UC 356x368x129 UC 356x368x153 UC 356 χ 368 χ 177 UC 356x368x202 UC 356 Χ 406 Χ 235 UC 356 χ 406 χ 287 UC 356 χ 406 χ 340 UC 356 χ 406 χ 393 UC 356 χ 406 χ 467 UC 356 Χ 406 Χ 551 UC 356 Χ 406 Χ 634

MN.y [N.m]

·-

85 348 106 249

251

1

133

657 800 929

1072 1

344

1628 1811 875 1037 1201 1370 1594 1947 2176 2 515 2 989 3 527 4 058

Nsd IkN] ··

272 334

·*

**

193398

471

224 293 269341 330 792

542 642 778

**

**

414335 503849 631551 817182

805 968

**

669173 783306 912 519 1 164 695 1435 840 1625 654 »·

1001020 1

167 230

1341598 1574 455 952 641 2 216 294 2 601408 1

13

Nsd/Np,.Rd=0,2

··

204

313 353 407 482 584 496 603 726 895

12

11

j

MN.y [N.m] **

**

182022 211100 253498 311334

588 678 803 973

389962

1511

769113

1066 1238 1430 1792 2171

tt·

2 415

629810 737229 858841 1096184 1351379 1530 027

**

**

1383 1601 1826 2126 2 597

942136 1098 570 1262 681 1481840 1837 780

2 901

2 085 924 2448 384

3157332 3 811629

3354 3986 4 703

4 484 028

5 411

*·

**

474 211 594401

··

Nsd IkN] 340 418

*·

2 971606

3 587416 4 220 262

|

15

Nsd/Npl.Rd=0,25

80 327 99 999

194

1

14

··

1006 1210 1492 1889 tttt

1333 1548 1787 2 240 2 714 3 019 ··

1729 2 001

2283 2657 3 246 3 627 4192 4 982 5 879 6 763

602

MN.y [N.m] ··

75 307 93 749 ·*

170645 197906 237654 291875 ··

16

17

|

Nsd/Np,.Rd=0>3

Nsd IkN]

MN.y [N.m]

*·

18

Nsd/Np,.Rd=0,35 Nsd IkN]

70 287

87499

475 585

··

**

**

159 269 184 712

824 949

221810 272417

··

··

1

124

1362

1791 2 267

**

*·

590447 805164 1027672 1266 918 1434 400

1600 1858 2145 2688 3 257 3 622

551084 645075 751486 959 161 1 182 457 1338 774

**

**

**

·*

883 253 1029 909 1 183 763 1389 225 1722 919 1955 553 2295 360 2 785 881 3 363 202 3 956 495

2075 2402 2739 3189

824369 961249

2 420 2 802

691 152

3 895 4 352 5 031

5 979

7054 8116

341217 414935 520100 672974 w

1408 1695 2089 2645 1866

2167 2 502

3136 3799 4226

104 846

3196 3 720

825 183 2142 336 2 600155 3138 989 3 692 729 1

81249 ·*

147893 171 518

205967 252959 **

316844 385 296

482950 624904

tttt

1296 610 1608 058

1

"

65 266

··

365589 444573 557251 721043

1207 1452

MN.y [N.m]

··

tt*

408 502 706 813 963 1 168

19

|

4544 5 077 5869 6 975 8 230 9469

**

511721 598998

697809 890649 1097995 1

243 147 *·

765486 892588 1025 928 1203 995 1493196 1694813 1989312 2414430 2 914 775 3 428 963

Plastic resistance of cross-sections

IPE IPE A

-IPE O 2

3

4

5

6

7

8

Np,

Vpi.z

Vpl,

Mp|.y

Mp|.z

Nfimit

{Nfimit /Np,.Rd)

for Ν - My IkN]

IkN]

IkN]

[N.m]

[N.m]

IkN]

for Ν - My Η

321

87 123 153 185 234 273

124 163 208 258 311 371

9 751 16 551

2443

434 555 690 844 1006

60 85

196

339 385 464 537 623 747 852

1

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750x137 IPE 750x147 IPE 750 χ 173 IPE 750 χ 196 IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

Note: all values of columns 2 to 7 must be divided by γΜο

1

1402 1643 ++ ++ ++

++ ++

++ ++ ++ ++

1035 1233 1452 754 2 032 1

++ ++ ++

++

2 248

-

2 281 2 907 3 425

268 369 463 562

74 108 131 151

100 135 168 208

++

189

247

++

223 278 329 396 455 539 654 722 868

++ ++ ++ ++ ++

++ ++ ++

++

1

1097 1240 1415 1629 1846 2132

-

2 824 3 086

++

.

436 516 602 702 816 939

1025 1222

++

-

++

-

308 375 428 518 612 704 846 975

138

1342 1570 1836 2 261

2639 ++

52 021 69 894 92 668

10 962 14 532 18 737

119870

24 406

153 991

31048

203 279

40 719 52 592 64 545 80 262

263909 337819 428042 549002 714753 921529 170 542 1 475 207 2043 368

141069 168 225

2146155

203973 257912 264941 340163 402695

7971 13852 20 946 30 071

431

100888 130860 173 247 227551 294810 380842 480438 627601 817323 1039 357 1319 295

20 364 26 208 34 585 45 078 55 971

981

115

1

++ ++

1440 1702

++

2 005

++

2 531

2 749

1082 1306 1520 1788 1969 2 226 2408

72179 84 869 103 213

281

0.201

0.200 0,203 0,208 0,194 0,204 0.209 0.213 0,219 0.219

751

1

905 260

16 763

221

21795 28102 35 447

264 305 365 434 509 606 705 839

77697

1

370 619

201816

1877836

268 835

0.232 0.244 0.250 0,239 0.229

329 397 477 522 628

185668

95 306 113021 143 216 171584

0,231

0.197 0,206 0,204 0,195

1078

49 437 64 087

0,187 0,196 0.195 0.193 0,198 0,196 0,202 0,197 0,199 0,200 0.202 0,206 0,203 0,212 0,219 0.223

53 76 95 110 137 163 199 238

126 681 151 831

134883 172315 241 352 312404 395 970 498155 630912 859431 1097453

603

753 910

6 519 8 692 15 345

79 443 104 756

621

1971

76 295

423 497 584 676 846

276 328 387 456 542

3166 4611

357

1095 1260 1466 1671 1947

241

116 080

1

2 611676 3 013 155

108 133 167 197

96180

11743

++

163

8084

301

509 597 700 815 987

1

37105

41619 56 837

1287 1440 1724 1957 2190

++

25 505

3 841 5 704

1052 1259 1479 1874

0,201

0,198 0,202 0,200

0,194 0.197 0,194 0,199 0.192 0.193 0,199 0,199 0,207 0,213 0,219 0,226 0.227

Plastic resistance of cross-sections

IPE IPE A -IPE O 10

9

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750x137 IPE 750x147 IPE 750x173 IPE 750x196 IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600

1PEO180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

Note: all values of Njj and Mj^.y must be divided by Y^j0 12

11

I

Nsd/NpuM-0,15

13

Nsd/Np,.Rd=0,2

14

j

15

Nsd/Np,.Rd=0,25

Nsd [kN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd [kN]

48 65 83 103 127

9 751 16 551

64 87

9596 16 475

25 505

111

37105

138 169 201

51890

80 108 139 172 211

69 546

251

23 756 34 499 48 647 65 200

92668 119384 153867

299 350

87049 111922

203 279

482 565

151

179 210 246 289 339 394

1

239 280 329 386 452 526

263909 337819 428042 549002 714753 921529

263909

411

17

Nsd IkN]

MN.y IN.m]

Nsd IkN] 112 152 194

8996

96

8396

15 445

130 166

14 416

144 251

207 253 302 359 420 493

18

22172 32199 45 404 60 853

81246

19

Nsd/Np,.Rd=0,35

241

295 352 419

MN.y [N.m]

7797 13 386 20 588 29 899

42161 56 507 75 443

**

··

**

»»

*·

»*

aa

**

»»

··

**

aa

**

.»

*·

**

»tt

··

**

··

»»

··

»*

611

»·

··

**

··

"

*·

**

709 830 970

**

··

«a

··

»»

··

·*

'*

**

**

tt*

**

**

·*

»·

*·

»»

**

**

··

*·

129

»*

tt

»»

·*

»»

··

»»

»»

*·

'*

**

»»

tt*

»»

**

··

100

»»

··

*

**

**

··

**

**

»»

181

»»

**

tt

**

**

··

»»

·«

"

»»

··

'*

tt·

»»

··

»»

·*

«*

**

**

ktt

tt*

**

··

aa

··

**

7971

54 74 93 112 136 164 197 237 280

458 532 623 728 847 983 1

52 021 69 894 92 668 119 870 153 991 203 279

25 339 36 799

MN.y [N.m]

16

Nsd/Np,.Rd=0,3

170 542

1

1394 1580 40 55 69 84 102 123

1

13 852

20 946 30 071 41619 56 837 76 295 100 888

7940 13 852

67 92

20 946 29 875

116

41619 56 685

170 206

141

7444

80

6948

94

13 077 19 737

111

12 206 18 421

129 162

28 008 39 066

139 169 204

**

··

»»

**

6 452 11334 17106 ··

a.

*

·· tt«

ktt

**

a.

**

**

ktt

**

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a.

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ktt

aa

**

aa

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**

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**

**

**

**

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**

ktt

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**

**

k·

tttt

*'

··

"

**

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**

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·

··

·<

··

'*

**

**

**

..

'*

**

**

ktt

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**

aa

*·

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**

1

*·

ktt

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**

**

**

**

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**

·*

ktt

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·*

··

·*

k·

··

**

**

k*

**

*·

··

»·

**

1

**

"

aa

»»

**

»·

··

228 268 314 367 452 528 610 707 810 988

150506

398 470 550 643

**

**

»·

··

**

··

··

k·

··

tt

"

**

tt*

*·

ktt

tt

a.

**

**

ktt

«·

1097453

148 1311

1012 1235 1435

*·

861

78 892 104342 133955 172007 238904 309 596 395391 497827 630912 859431 1097453

69 030 91300 117211

741

79 443 104756 134883 172315 241 352 312404 395 970 498155 630912 859431

»»

tt*

··

ktt

··

370 619

1639

"

*·

*·

*·

··

1877836

2066

1821444

··

ktt

M

148 178

210 247 293 345 403

130860

461

1

1

539 637 739 863 171 201

236 275 339 396 457 530 607

983 1240

370 619 1877 836 1

1

1653

1

285 336 393 459 565 660 762 883

604

tt ·*

73 961

341

97821 125583 161 257 223972 290 247

403

370 679 466 713

471 551

678 792 915 1060

2 479

**

* »·

·*

** **

64100 84 778 108839 139 756

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by Ym0 2

3

Np,

4

5

6

7

8

Vp,.z

Vp..y

Mpl.y

Mpi.z

Nfimit

{Nfimit /Np|.Rd)

for Ν - My IkN]

IkN]

[kN]

[N.m]

[N.m]

for Ν -My Η

HE100AA HE120AA HE140AA HE160AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA

655 779

149 167 192 252 295

283 337 424 564 679 807 942 1091 1246 1411 1590 1668

24 510

·*

97

35330

··

112

HE650AA HE700AA

++

HE800AA HE900AA

++

++

HE 1000 AA

++

1

Designation

HE100A HE 120 A HE 140 A

HE160A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A HE 500 A

++ ++ ++

++

375 428 522 600 667 785 858 938

++

1023

++

1

++ ++ ++

++ ++ ++

IkN]

0,147 0,144

**

·«

»»

*·

k*

*·

**

·*

k*

··

**

·*

**

·*

**

··

>·

**

»*

**

»*

·»

a.

**

ktt

·*

**

··

k*

··

**

··

*

··

**

··

**

*·

k*

··

»»

tt

k·

**

»»

**

*

··

163

1746 1824 1975

**

··

k*

··

++

1326

2 054

»»

··

k·

++

1501

++

1762

++

1971

2192 2433

2132 2290 2369 2448 2599

3 002 3 570

2 768 3 074

++

++

-

892 1064 1319 1628 1901

183 205 245 320

2 261 2 702 3 227

438

3646 4 085 4 726 5 223 5606 5996 6 677 7477 ++

351 501

610 697 770 904 997

1090 1

187

1390 1595 1812 2030

HE550A

++

HE 600 A

++

HE650A

++

2 260 2 502

HE 700 A HE 800 A

++

2836

++

HE900A

++

3366 3960

HE 1000 A

++

-

3234

082 016 1 313 592 1 521 685 1 747 138 2 032 856 2614 415 1

3 612 3 763 3 915 4 074 4 237

4543 4 696

1

··

·* ** tt **

3359 500 4106 225

402 217 426 615

34 865

17 279 24 718

409 486 600 729 860

1009 1216 1449 1633 1827 2110 2334 2 485 2636 2866 3163 3460

··

50186 72868 102962 136438

'*

250

0.151

1423 1606

0.160 0.169 0.177 0,189 0,206 0,219 0,232

'*

'· " »

777199 877159

"

075 955 1350 664 1658 519 1941 163 2 247 162 2 577 242 2 953 364 3 653 785 4540 200 5384 400

366603

'·

405 523 444 575 464 900 485 376 506011 527831 551 149 594 081

617279

0,123 0,121 0.121

1093

232 290 335 404 458 514 599 659 724 793 945

'*

683 797

·*

0.193 0,206 0,217 0.227 0.233 0.250 0,250 0,250

0,129 0,122 0.128 0,124 0,125 0.126 0.126 0,127 0,126 0,129 0,132 0,141 0.146

ka

386304 467134 580974

605

110 129 160 209

..

ka

1

1320 1492 1675 1868 2294 2648 2963

·

180 384 238 752 312 742

110

1

1

798

2068 2 474 2 951 3 378

Plastic resistance of cross-sections Note: all values of Nsd and Mj>j y must be divided by Υμ0 9

Designation

10

Nsd IkN] HE 100 AA

HE120AA HE140AA HE160AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA HE 1000 AA HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A HE 300 A HE 320 A HE 340 A HE 360 A HE 400 A HE 450 A

HE500A HE550A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

11

Nsd/NpLRd-0,15 MN.y [N.m]

12

|

13

Nsd/Npi.Hd-0,2 Nsd [kN]

MN.y [N.m]

14

15

Nsd/NpUM-0,25 Nsd IkN]

«

··

··

·«

··

··

·*

98 117

|

MN.y [N.m]

16

I

17

Nsd/Npi.Rd=0,3 Nsd IkN]

MN.y [N.m]

|

19

Nsd/Npuw-0,35 Nsd IkN]

MN.y [N.m]

··

·*

··

··

··

··

·*

··

·*

·*

**

·*

·*

**

*

··

**

**

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**

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··

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··

*·

**

**

*·

··

**

··

·*

··

·*

»·

»»

··

·*

*·

·*

»·

»»

*·

··

*·

k*

»»

*·

**

··

'·

**

**

1034

··

··

k*

**

**

107

·*

··

k»

aa

a.

**

**

»·

**

·*

**

aa

k·

*

**

**

*·

»*

aa

aa

··

**

»*

**

*·

312 372

25 849

862 963 1

1203 1376 1589 1778 134 160 198

33 803 48 535 ··

·*

244 285 339 405 484 547 613 709 784

**

178 213

31815 45 680

·*

18

*

*

*

223 266

29 826 42 825

268 319

27838

··

ka

**

**

**

·*

·*

'"

**

·«

**

·*

··

ka

**

**

**

·*

ka

**

**

**

**

*·

·*

39 970

37115

**

·*

'*

**

·*

··

k*

**

tt

··

··

k*

··

*·

"*

·»

··

k*

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**

**

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··

··

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**

**

**

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··

"

··

**

tt

*·

tt*

*

*·

··

k*

tt

·*

**

·*

··

"

**

··

k·

*·

··

*·

··

··

"

«·

**

k·

··

·*

841

*·

""

·*

·*

·*

**

**

··

··

899

tt*

«·

**

'"

··

»»

k*

tt

1002 590 1265 441

1669 1869

939928

2003

877 266

··

·*

**

·»

*·

··

**

**

*·

*·

**

**

··

»»

·*

··

*

**

··

«a

»*

·*

*·

*

*·

··

1002 1

122

1244

1334 1427 1522 1641 1801 2019

2185

1065 252 1344531 1658 519 1941 163 2 247 162

1335 1495 1659 1779 1902

2 337

**

814604

··

*·

ν*

**

··

**

tt*

··

*·

»·

**

*tt

aa

··

*·

"

··

**

**

**

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**

**

**

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**

tt

··

**

**

"*

**

»*

>«·

··

*·

··

**

·*

»»

**

··

*tt

**

··

606

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by γΜο 1

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 8 HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M

HE300M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

2

3

4

5

6

7

8

Np,

Vp,.z

Vp..y

Mp|.y

Mp|.z

Nfimit

{Nfimit /Npi.Rd)

for Ν - My IkN]

[kN]

IkN]

[N.m]

[N.m]

IkN]

for Ν - My Η

1094 1428 1804

219 266 317 427

511

21597

1053

127 160 196 266

491

1271

602 677 806 912 996

1514 1770

43 769 69 389 103079 148 665 202 208 269 870

2 279

2 741 3 279 3 824 4 451 4 975 5 517 6 261 6 776

7178 7587 8 307

9155 10 023 10 670

11338 ++

1

150

1255 1360 1469 1697 1932 2178 2427 2687 2959

669 847

347360 442321 538823 644462

2 054 2 289

2532 2866 3090 3 242 3 393 3 624 3 922 4 220 4 372 4 525 4 677

784 843 902 680

1011404 1

126855

1357330 1672 595 2022118 2 348 055 2698 557 3 074 349

4837 5003

++

3 324 3 922 4 577

5 310

3497395 4296 600 5283 600

++

5153

5463

6 241200

2236

437 513

1098 1358 1637 1950

99 041 147257 207407 283 317 371048

++ ++

2 789

3383 4 076 4 757 5 514 6 277

8383 9 225 10 087 12 729

13106 13 265 13 390

13683 14 088 14 461 14 884 15 274 15 697 16 087 ++ ++

++

593 747 840 995

1099 1457 1622 1747 2195 2300

2 269

2618 2976 4 019 4 407

4800 6108 6 239

2 392

6239

2483 2672 2906 3140 3385 3 629 3 873

6 219

4 118 4711

5200 5699

6200 6200 6181 6181 6161 6161 6142 6138 6118

6118

476 762

596168 889117 059 917 1245 566 1 712 623 1

1862 711 1981380 2095 513 2339 660 2 659 027 2 979 594 3 331 727 3 684 276 4 055 923 4 426 800 5 245 800 6064 800 6959 400

607

34 007 50 310

71385 97026

312 380 433 512 576 642 737 805 880 958

128441

165430 209336 252944 301380 365459 394421 414003 433646 463695 503016 542492 563 280 584 244

605393 627919 652316

1

129

1302 1483 1681

1889 2107 2 402

2860

696 503 720 833

3 387 3865

48 851 72 085 101 015

228 283 343 435 503 594 670 858 954

136693 178 579 228151 284992

422492 500 836

586605 803 536 819 304

820138 815 788 812 336

814466 811447 813669 810 759

812982 810 089 810 765 810 128

814666

1052 1287 1362 1441 1521 1684

1887 2 089

2 301 2 513 2 724

2936 3399 3 823 4 255

0,116 0,112 0,109 0,117 0,114 0,116 0,113 0,115 0,116 0,116 0,118 0,119 0,123 0.126 0.136 0,142 0,148 0.158 0,167 0.175 0,187 0,204 0.217 0,230 0.102 0,102 0.101

0,107 0.106 0,108 0,107 0.102 0.103 0,104 0,101

0,104 0,109 0.114 0.123 0.134 0,144 0.155 0,165 0,174 0.183 0,200 0,215 0,228

Plastic resistance of cross-sections Note: all values of Ngd and Mrø.y must be divided by Jm0 9

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β

HE300B HE 320 HE 340 HE 360 HE 400 HE 450

Β

Β Β Β Β

HE500B HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β

HE900B

10

14

15

Nsd/Np,.Rd=0.25

16

17

Nsd/Np,.Hd=0,3

18

19

Nsd/Np,.Rd=0,35

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

164 214

42082

219 286

273 357

328 428

32180

98325

361

631

342

143 045 193 948

456 548 656 765 890 995

34 656 54 688 80 974 117801 159 722 213 646 274 240 349 875 426 599 510 514

383 500

271

39 607 62 501 92 541 134630

37131

66 407

75190 109387 148313 198385 254652 324884 396128 474048 578169 665863 749258 838351 1021086

411

492 574 668 746 828 939 1016

1077 1

138

1246 1373 1503 1601 1701 1804

1930 2105 2339 2 520

HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M

335 418 508

HE 900 M HE 1000 M

13

Nsd/Np,.Rd=0,2

Nsd IkN]

HE 1000 Β

HE800M

12

11

Nsd/Np,.Rd=0,15

611

713

827 941

1257 1384 1513 1909 1966 1990 2008 2 052

2113 2169 2233 2 291

2355 2 413 2 547 2 669

2798

259427 333 006 424 848

518013 619 909 756 067 870 744 979 799 1096 305 1335 267 1 657 319 2 017 376 2 348 055 2 698 557 3 074 349 3 497395 4296 600 5 283 600

103

1

1252 1355 1436 1517 1661 1831 2 005 2134

2268 2 405 2 574 2 807

454167 567335 841944 1004884 181 977 1619 443 1766 897 1889 458 2 009 371 2 267 774 2609 632 2960 417 3 331727 3 684 276 4 055 923 4 426 800 1

5 245 800 6064 800 6 959 400

819 523

922164 1

2 229 761 2 590 452 2 981942 3 440 038

··

447 558 677 815 1

951 103

1255 1677 1845 2 017 2 546

031 817

1256 722 1559 830 1898 707

3119

··

93 724 139 315 196 161 269 554 352 720

182540 244167 313418 399857 487542 583444 711593

451

570 685 820 956 1

113

1244 1379 1565 1694 1794 1897

2077 2289 2506 2668 2835 3007 3217

457070 546979 667118 768303 864 528

967328 1 178177 1462340 1780 037

1335 1492 1655 1878 2033 2153 2276 2 492

2747 3 007

622644 717083 806893 902840 1099 631 1364 851 1661368

797 959 148

1

1338 1558 1741 1931 2191 2372 2 512

2655 2 907

3204 3508

1267362 1542 699 **

aa

··

*·

**

**

··

·*

**

**

**

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**

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**

**

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**

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**

**

·*

**

·

"*

·*

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**

·*

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**

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«·

88 211

559

131120 184623 253698 331972 427451 533963 792418 945 774 1 112449

697 846 1019 1

189

1378 1569 2096 2306 2 522

2 737

2818

2456124

3522

2 892 2 977 3 055

2 786 275 3 152 827

3 615

3139 3 217 3396 3 558 3 731

293 829 374 866

684 822 984 1 147

50 782

*·

3182 3 277

2653 2678

171 131

228906

541

aa

524 182 1662 962 1 778 313 1 891 172 2134 375

2 621

58 595 86 757 126 216

1

3 527 816 3 926 201 4 332149

5 245 800 6064 800 ··

3316 3 347 3 421

3 721 3 818 3 924 4 022 4 245

4448 *·

608

82 698 122925 173084 237842 311224 400 736

500 590

742892 886663 1042921 1

1

428 920 559 027

1667168 1772 974 2000 977 2302616 2 612133 2 955 775

3 307327 3 680 813 4 061390 4 919150

671

77184

837 1015 1223 1427 1654 1883

114 730

2 515 2 768 3 026 3 819 3 932 3 979 4 017 4 105 4 227

4338 4465 4 582 4 709 4 826 5094

161545 221986 290475 374020 467217 693 366

827552 973 393 1333 659 1455 092 1556 024 1654 776 1867578

2149108 2437 990 2 758 724 3086 839 3 435 426 3 790 631

1427 1665 1930 2197 2934 3229 3530 4455

71671 106535 150006 206130 269 727 347304 433 845 643840 768441 903865 1 238398

4 587

1

4643 4686 4789 4 931

1444 879 1536 578 1734 180 1995 601 2263 848 2561672 2866 350 3190 038

783 976 1

184

5 061

5209 5 346

5494 5630

351 157

3 519 871

··

·*

**

·*

*·

··

**

a*

M

··

··

*·

»»

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

6

7

8

Νρ,

Vp,.z

vpi.y

Mpl.y

Mpi.z

Nfimit

{Nfimit /Npi.Rd)

for Ν - My IkN]

IkN]

IkN]

[N.m]

[N.m]

IkN]

for N-My Η

1019

239 300

402 476 522 645 353 435 517 629 795 933 838 974

71928

17468

98 309 108 247 132033 108 786 128 321 148179 165087

20 896

174 220 235 263 294 313 328 306 320 365

1

Designation

UB 178x102x19 UB 203 χ 102 χ 23 UB 203 χ 133x25 UB 203 X 133 X 30 UB 254 X 102 χ 22 UB 254 X 102 χ 25 UB 254 X 102 X 28 UB 254 χ 146 χ 31 UB 254x146x37 UB 254x146x43 UB 305 χ 165 χ 40 UB 305 Χ 165 χ 46 UB 305x165 Χ 54 UB 356 χ 171 χ 45 UB 356 χ 171 χ 51 UB 356 Χ 171 Χ 57 UB 356 χ 171 χ 67 UB 406 χ 178x54 UB 406 Χ 178x60 UB 406 χ 178x67 UB 406 Χ 178x74 UB 457 χ 152x52 UB 457 Χ 152 Χ 60 UB 457 Χ 152x67 UB 457 χ 152x74 UB 457 χ 152 Χ 82 UB 457 χ 191 χ 67 UB 457 X 191 χ 74 UB 457 χ 191 χ 82 UB 457x191x89 UB 457 Χ 191 χ 98 UB 533 Χ 210 χ 82 UB 533 Χ 210 χ 92 UB 533 χ 210 χ 101 UB 533 χ 21 Οχ 109 UB 533 Χ 210 χ 122 UB 61 Οχ 229 χ 101 UB 610 Χ 229 χ 113 UB 610 Χ 229 Χ 125 UB 610 Χ 229 Χ 140 UB 610 Χ 305 Χ 149 UB 610 Χ 305 χ 179 UB 610 Χ 305 χ 238 UB 686 χ 254 χ 125 UB 686 Χ 254 Χ 140 UB 686 χ 254 χ 152 UB 686 Χ 254 χ 170 UB 762 χ 267 Χ 147 UB 762 χ 267 χ 173 UB 762 Χ 267 χ 197 UB 838 χ 292 χ 176 UB 838 χ 292 Χ 194 UB 838 Χ 292 Χ 226 UB 91 4 χ 305 χ 201 UB 914 Χ 305 Χ 224 UB 914 Χ 305 Χ 253 UB 914 Χ 305 Χ 289 UB 914 Χ 419 Χ 343 UB 914 χ 419 χ 388

1

235

1343 1605

311

353 379 405

++ ++ ++

431

++

397 427 490 487 546 644 650 695

++

2300 ++

++ ++ ++ ++

++

764 867 807 839 935

++

1015

++ ++

++

++

++

1

141

1

122

1361 973 1

139

1281 1438 838 1022

954

++

1063

++

1

142

1319

++

1253

1476

993

++

1059

++

1

++

++

1244 1356 1314 1397

++

1501

++

1614 1776

++ ++

++

167

++ ++

++ ++

++ ++

12 740

1729 1864 2060 1910 2 285 3 005

++ ++

2186 2336

++

2 576

++

++ ++

++

2798 3076

++ ++ ++

++

3 515 -

++ ++ ++ ++ ++

4 070 4 602 4634 5 138

261 692

302419 355352

834 988

++

++

202957 237858

1

1

208

1892 1389 1639 1834 1990 2 267 1693 1979 2 248

2549 2 993 3 620

4898 2064 2 421

2683 3045 2 342 2 901 3 423 2 768 3 193

3 951

3096 3660 4 286 4 958

**

424 260 508 562

73164 83477 102047

442948

74 881

503 776 565 331

87800 99351 112132

683166 760827 617830 694145 769139 845 731

937612 864620 991237 1097046

127637 142118 159139 126153 149 345

1377939 543 869

224 878

1739 671 1929 298 2329 886

256807 393658 480418 661054 227807 268032

187 939

1342149 1

1

210 074

3 144 207

1677539 1914 403 2 100 192 2 364 921 2 165 321 2 603 025 3009 964 2 859 237 3 208 847 3 845 022

3 507521 4 004 495 4594 800

5279 400

6699

6 501 600

7 700

7421400

609

651

722 778 700 747 828 882 962 783 828 909 962

55 987 68 482 78 406 89 514 100962 99 669 114 282

167767 183029 209879 168103 197090

1

371

412 484 507 535 583 653 635

82165

376322

460 337 540 654 610 291

1381 1533 1700

**

50 137 59 252 59 528 69 524

325 342

630338

161

**

37054 15 655 19 323 23 037

298 279 340 800

271826 339142 402609 353 593 408997 508 874 412431 488 646 575 628 672529 1 213 618 1 403 175

1043 1041 1094 1

167

1252 1368 1293 1365 1461 1605 1469 1746 2263 1628 1723 1831 2 007 1965 2208 2 405 2 352 2 518 2 753 2 687 2946 3200

3598 3559 3 915

0.170 0,178 0,175 0,164 0,250 0,233 0,217 0,183 0,162 0,158 0,172 0,167 0,167 0,210 0,196 0.191

0.182 0.219 0,202 0,201

0,196 0,250 0,233 0,230 0,222 0,219 0,218 0,208 0,207 0.201

0,198 0,237 0,222 0,216 0,215 0,210 0,239 0,226 0,218 0.215 0.184 0,182 0,178 0.243 0,230 0.225 0.220 0.250 0,239 0,228 0,250 0,243 0.227 0.250 0,246 0.236 0,233 0,194 0.189

Plastic resistance of cross-sections Note: all values of Njj and Mjq y must be divided by Ym0 10

9

Designation

UB 178x102x19 UB 203x102x23 UB 203x133x25 UB 203x133x30 UB 254x102x22 UB 254 Χ 102x25 UB 254x102x28 UB 254x146x31 UB 254x146x37 UB 254x146x43 UB 305x165x40 UB 305x165x46 UB 305x165x54 UB 356x171x45 UB 356x171x51 UB 356x171x57

UB356X171X67 UB 406x178x54 UB 406x178x60 UB 406x178x67 UB 406x178x74 UB 457x152x52 UB 457x152x60 UB 457x152x67 UB 457x152x74 UB 457x152x82 UB 457x191x67 UB 457 Χ 191x74 UB 457x191x82 UB 457x191x89 UB 457x191x98 UB 533x210x82 UB 533x210x92 UB 533x210x101 UB 533x210x109 UB 533x210x122 UB 610x229x101 UB 610x229x1 13 UB 610x229x125 UB 610 χ 229x140 UB 610x305x149 UB 610x305x179 UB 610x305x238 UB 686 Χ 254x125 UB 686x254x140 UB 686 X 254 X 152 UB 686x254x170 UB 762x267x147 UB 762x267x173 UB 762x267x197 UB 838x292x176 UB 838x292x194 UB 838 χ 292 X 226 UB 914x305x201 UB 914x305x224 UB 914x305x253 UB 914x305x289 UB 914x419x343 UB 914x419x388

12

11

Nsd/Npl.fld-0,15

13

Nsd/Np,.Rd=0,2

14

|

15

Nsd/Np,.Rd=0,25

16

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

NSd

153 185

71928 98 309

204 247

69 367 95 674

255 309

65 032 89 695

306 370

201 241

**

**

**

132033 108 786 128321

321

126 287 108 786 128 321

177 202 227 250 297 345 323 370 433

148 179

235 269 303

**

202957 237858

148179

tt

396 460

401

294 336 379

**

118 394

17

Nsd/Npl.Rd=0,3

Nsd [kN]

··

|

IkN]

·*

481

18

!

19

Nsd/Npl.Rd=0,35

MN.y [N.m]

Nsd [kN]

··

«

·*

··

··

··

··

tt*

110 501

562

MN.y [N.m]

*·

102608

tt·

«·

**

··

··

**

··

·*

**

··

**

··

··

**

**

**

tt»

**

**

**

**

»·

**

*·

**

··

··

*·

*·

226117

575

211984

**

**

*·

690

**

tttt

**

**

*·

··

**

**

··

**

**

**

**

··

**

**

**

··

578

341478

722

tt*

*·

**

··

**

355 352

361

·*

tt*

··

tt»

**

*·

*·

·*

·«

409 457 539 434 482 539 595 420 480 539 595 659 539 596 658 717 789 660 739

·*

··

·

**

**

**

··

**

**

tt

»tt

tt»

··

·*

··

··

898

tt*

**

*·

**

«w

424 260 508 562

1

197

1437 1911

1005

497324

**

··

**

··

**

·

··

**

κ·

··

··

**

*"

tt*

**

·*

**

*·

tt*

··

*·

**

··

·*

**

**

**

**

**

··

**

tt»

565 331 630 338

719 794

**

**

·*

**

«·

··

··

**

**

··

"

**

**

**

**

**

**

**

**

**

**

··

**

**

tt«

**

*

**

**

··

**

«w

"*

**

*"

**

**

·*

683166 760 827

tt

794 878

760 827

**

**

**

«·

**

**

**

**

**

··

tt·

··

tt>

**

**

**

*

··

tt·

**

**

**

**

·*

**

··

**

**

**

**

··

··

*·

**

»*

«w

1098

769 139 845 731

878 956

··

937612

1052

935 651

1315

**

·*

**

**

*·

**

**

·*

»·

**

««

·*

*

**

**

*·

··

**

··

**

tt*

**

··

tttt

**

**

166

**

*·

«·

**

**

··

**

1632

··

··

**

**

a«

*·

811

875 979 812 907 1004 1 123

609 718

1

187 939

1

1

342149

1305

1

1

342 149

··

**

**

**

··

*·

*tt

··

·*

**

··

*tt

«ν

**

**

··

**

**

**

··

·»

*·

**

*"

**

··

**

*·

μ*

**

*"

*"

··

**

739 671 ··

2 329 886 3 144 207

1497 ··

1916 2548

·*

tt

*"

*

*·

**

**

*·

**

**

**

**

**

·»

··

·*

3 058 731

3185

2867560

3 822

**

**

··

*·

··

**

**

2 676 390

4 459

**

124

**

··

**

"*

**

**

**

··

**

1223 1366

**

··

**

tt·

**

·*

**

**

**

*·

**

··

**

*·

**

··

1

1

179

1388 1579 1411 1555 1818 1612 1800 2034 2 320 2 755

3 114

2 364 921

1821

··

**

**

**

··

**

·*

·*

*·

**

··

tt·

**

**

**

**

··

**

*«

**

*tt

·*

""

**

·*

**

**

··

··

«·

·*

··

*·

*·

**

**

**

*"

**

*»

**

··

*·

*·

··

tt·

··

··

··

**

**

**

·*

··

·«

·*

**

tt

**

··

··

*·

**

·*

··

**

··

*·

·*

*·

**

**

**

**

**

**

**

··

**

··

*

5 279 400

3 093 3 673 4 151

tt

*·

**

**

·*

·*

**

tt«

**

*«

·*

··

**

*«

**

**

**

*·

**

**

μ«

6 501600 7421400

610

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 1

Designation

UC 152 Χ 152x23 UC 152x152x30 UC 152 X 152 X 37 UC 203x203x46 UC 203x203x52 UC 203x203x60 UC 203 χ 203 χ 71 UC 203x203x86 UC 254x254x73 UC 254 χ 254 X 89 UC 254x254x107 UC 254x254x132 UC 254 χ 254 χ 167 UC 305x305x97 UC 305x305x1 18 UC 305x305 χ 137 UC 305x305x158 UC 305x305x198 UC 305 χ 305 χ 240 UC 305x305x283 UC 356 χ 368 χ 129 UC 356x368x153 UC 356x368x177 UC 356x368x202 UC 356x406x235 UC 356x406x287 UC 356x406x340 UC 356x406x393 UC 356x406x467 UC 356 χ 406 χ 551 UC 356x406x634

2

3

4

5

6

7

8

Np,

VpLz

Vp..y

Mpl.y

Mpi.z

Nfimit

{Nlimit /Npi.Rd)

for Ν -My IkN]

[kN]

IkN]

[N.m]

[N.m]

for Ν -My Η

1228 1607 1979

242 280 346 412 455 537 589 744

521

2 467

2784 3208 3798 4605 3 910 4 759 5 728

7062

621

747 924 1121

tttt

719

«

104019 129679 208924 238306 275 552 335 479 410 227 416669 514 022 623461 785153

891 1 117

1273 1462 1781

2149 1798 2208 2652

··

128 241 156 945 191 619 **

241629 292 758

1427 864

2340

6309 7325 8 457

1047 1208 1390 1708 2082

2 867

3 351 3 892 4 948 6 035

2 461

6 652

1

1030 1219 1426 1638 1836

3193

1041294

··

3 801

1

245 186

**

6 902 9 471 10 803 12 576 15 360 16 888 19 522 23 201

27375 31494

2 274 2 709

3155 3 762

4186 4833

4404 5033 5 932 7 270

8036 9333 177 13 265 15 376 11

368917 477653

1017940 668 669 822 162 964 551 1 125 785 1444 792

·*

376053 442069 516 643 663 907 819 246 913 442

1783 771 991 027

1451201 1668130 1968 330

2441240 2729 640 3 206 752

3900 000

701739 806 203 1000 955 1238 693

1382 041 1

619 952

4 711200

1963 378 2362 603

5553 600

2 772 264

611

200 244 293 319 376 399 502 437 517 636 754 939 618 740 845

110985

8 940 5 185

8182

*·

46 868 58 623

3 310 4 226

10 601 12 843 14 057

IkN]

1

961 153

1380 1487 742 870 1011 1

153

1280 1563 1702 1952 2 277 2 671 3 014

0,124 0,123 0,119 0,115 0,117 0,105 0,109 0.112 0.109 0,111 0,107 0,105 0,119 0.117 0.115 0.114 0,109 0.107 0.106 0.107 0,106 0.107 0,107 0,102 0,102 0,101 0.100 0,098 0,098 0,096

Plastic resistance of cross-sections Note: all values of Ns
Designation

10

Nsd [kN] UC 152x152x23 UC 152x152 Χ 30 UC 152x152x37 UC 203x203x46 UC 203 χ 203 χ 52 UC 203x203x60 UC 203x203x71 UC 203x203x86 UC 254x254x73 UC 254x254x89 UC 254 Χ 254x107 UC 254x254x132 UC 254x254x167 UC 305x305x97 UC 305x305x118 UC 305x305 Χ 137 UC 305x305x158 UC 305x305x198 UC 305x305x240 UC 305x305x283 UC 356x368x129 UC 356x368x153 UC 356x368x177 UC 356x368x202 UC 356x406 Χ 235 UC 356x406x287 UC 356x406x340 UC 356x406x393 UC 356x406x467 UC 356x406x551 UC 356x406 Χ 634

·· 241

297 370 418

MN.y [N.m]

NSd/ISlpI.Rd=0,2

Nsd [kN]

··

100975 125 703

95035

396

118 308

··

·*

587

**

714 859 1059 1341 778 946 1099

490199 596103

1

747 187 966 807

1412 1788

·*

**

791698

1262 1465 1691

1269 1590 1926 2108 1035 1227

926 728 1079 600 1377 950 1 698 740 1 892 552

215 350 249 752 299 913

921

368339

146

2120 2569 2811

Nsd [kN] ··

402 495

461363 561038 703 234

909936 ··

MN.y [N.m]

Nsd/N pl.Rd=0>3 Nsd IkN]

89096

18

Nsd IkN]

83156 103520 **

835 962 1

139

188431 218533 262424

151

345 317

1381

322 296

·*

**

1428 1718 2118

403 693 490 909 615 330

1666 2005 2472

2 682

796194

3 129

1

190

1432 1765 2235

432528 525973 659 282 853 065 «·

**

745 127 872 214

1577

698 557 817701

1893 2198

1016094 1296 894 1598 815

2114 2650

2 537

781 225

3 514

952588 1215 838 1498 889 1669 899

1831

3 211

3180 3853 4 217

tt«

651986 763188 889 082 134 782 1398 963 1 558 572 1

77216 96126

tt

201890 234142 281168 *·

**

562 693

696 802 949 ··

MN.y [N.m]

tt

**

1

19

Nsd/N,3l.Rd=0,35

·*

482 594

**

1

MN.y [N.m]

*·

«·

110914

17

16

··

··

**

tt*

15

NSd/N 3l.Rd=0,25

. **

557 642 760

952

14

··

321

691

570

MN.y IN.m]

tt*

228 809 265361 318 657 391360

481

13

12

11

Nsd/Npl.Rd=0,15

974 1

123

1329 1612

··

174 972

202923 243679 299 275

«·

··

374 858

455844 571378 739323

**

2208 2564 2960 3711

··

605416 708674

4 495 4 920

825 576 1053 726 1299 037 1447 246 *·

·*

·*

**

**

*·

**

**

**

tt

·*

»»

aa

«*

»»

**

**

**

1218 484 1400 508 1643 590

2 841

1

137 252

3 315

1056 020

3 241 3773 4608 5 067 5 857 6960

307 141 534 018 1 902 491 2124 840 2494 063

3 781 4 402 5 376 5911

3 027047 3654 345 4298 998

8 121 9 581

1421 1620 1886

1380 949 1587 243 1862 736

2304 2533 2928 3480

2 310 167 2 580 163 3 028 505 3 675 700

4 106 4 724

4 437419 5 220 212

1894 2161

1299 716 493 876

2 701

2 515 3 072 3 378

1753163

3144 3840 4 222 4 881 5800 6844

3904 4640 5 475 6299

1

2 174 275

2 428 389 2850 357 3459 482

4176395 4913141

2368

7874

612

2 038 383 2 276 614 2 672 210 3243 264 3 915 370 4 606 069

8 213

9448

1 1

6833

11023

1

213 774

1424 445 1766 599 1973 066 2 315 915 2 810829 3 393321 3 991927

IPE IPE A -IPE O

Note: all values of columns 2 to 7 must be divided by Ym0

2

3

4

5

6

7

8

Np,

Vp,.z

Vp..y

Mpl.y

MpI.Z

Nfimit

{Nfimit /NpLRd)

for Ν - My IkN]

IkN]

IkN]

[N.m]

[N.m]

IkN]

for Ν - My Η

95

136 179

10 680

2 676 4 207 6 247 8 853

66 93

1

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750x137 IPE 750x147 IPE 750x173 IPE 750x196

Plastic resistance of cross-sections

352 475 608 756 924 1

135 167 203 256 299

102

1310

372 422 508 588 682 818 933

++ ++ ++ ++ ++ ++

134

++

1

++

1350 1590

++ ++ ++ ++

++

1921 2 225 -

227 282

2499

2350 551

3300122

282475 290173 372560 441047

8 730

2159

15 171 22 941

3 467 5 050

IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450

293 404 507 616

82 118 144 165

110 147 184 228

++

271

++

207 244 305 360 433 498

++

591

++

717 790 950

1PEA500 IPE A 550 IPE A 600

++

-

++

-

++

-

1PEO180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

++ ++

1246 1470 1720 2011 2 476

337 410 469 567 670

++

771

++

926

++

++

1068 1274 1578 1865 2196

++

2 772

++

++ ++

1

2 860 407

32 935 45 583 62 250

329 392 472 557 654 767 892

83 561

110496 143 323 189 747 249 223

322887 417113 526194 687372 895163

1082 1

199

1380 1606 1830 2133

138 343 1444 943 1

464 545 639

223 399

·* *·

18 360

188 726

264338 342157 433682

58 83 104 120 150 178 218 260 308 360 435 523 572 688 823

37879

79 053 92 952 113 043 138 746 166 291 203351

185

1430 1665 1959 2156 2438 2638

7140

23 870 30 778 38 822

147729

1

9 520 12 862 16 807 22 303 28 704

87009

926

1074 1221 1409 1577 1888 2144 2398 3011

184 247

114 733

741

302 359 423 499 594 680 825 997

70 692

87906 105340 127135 154504

2 237974

3380

++

26 731

34005 44 597 57601

2 462

++

++

20 522

2335

3 183 3 751

++

168657 222638 289044 369992 468807 601288 782825 1009 294

118 146 183 216 264

12 006 15 916

282 022 1615 703

3 092

++

40 638 56 975 76 551 101 494 131 287

341

407 478 566 660 769 894 1028 1202 1358 1549 1784 2 022

++

++

18127 27934

991 1

181

1380 242 289 334 399 475 557 664 772 919

545 598

54145 70190 85 097 104383

690999 941281

123 784 156 855

1

201 973 1501 154

187925 221037 294 438

1379 1620 2053

1

2 056 677

613

152

0.187 0,196 0,195 0,193 0.198 0,196 0.202 0,197 0.199 0,200 0,202 0,206 0,203 0,212 0,219 0,223 0,231

0.232 0,244 0,250 0,239 0,229

0,197 0,206 0.204 0,195 0,201

0,198 0.202 0.200 0,201

0,200 0.203 0,208 0,194 0,204 0,209 0,213 0,219 0,219

0.194 0.197 0,194 0,199 0,192 0.193 0.199 0,199 0,207 0,213 0,219 0,226 0.227

Plastic resistance of cross-sections

IPE IPE A -IPE O 9

Designation

IPE 80 IPE 100 IPE 120 IPE 140 IPE 160 IPE 180 IPE 200 IPE 220 IPE 240 IPE 270 IPE 300 IPE 330 IPE 360 IPE 400 IPE 450 IPE 500 IPE 550 IPE 600 IPE 750 x 137 IPE 750x147 IPE 750x173 IPE 750 χ 196 IPE A 80 IPE A 100 IPE A 120 IPE A 140 IPE A 160 IPE A 180 IPE A 200 IPE A 220 IPE A 240 IPE A 270 IPE A 300 IPE A 330 IPE A 360 IPE A 400 IPE A 450 IPE A 500 IPE A 550 IPE A 600 IPE O 180 IPE O 200 IPE O 220 IPE O 240 IPE O 270 IPE O 300 IPE O 330 IPE O 360 IPE O 400 IPE O 450 IPE O 500 IPE O 550 IPE O 600

10

|

11

Nsd/Np,.Rd=0,15

Note: all values of Nsd and Mjj y must be divided by Yj^0 12

|

13

Nsd/Np,.Rd=0,2

14

|

15

Nsd/Npi.Bd-0,25

16

|

17

Nsd/NpLRd«0,3

18

t

19

Nsd/Npuw-0,35

Nsd IkN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd [kN]

MN.y [N.m]

53

10 680 18127 27934 40 638

70 95

10 510 18 044 27 752

88 119 152 189

9853 16916

105 142 182

9196 15 789

123 166

14 661

24 283 35 266 49 728 66 649

213 264 323 386

71 91

113 139 165 197 230 270

317 371

432 502 583 682 797 927 1076 1205

56 975 76 551 101494 131 287 168 657 222 638 289 044 369 992 468 807

601288

122 151

185

220 262 307 360 423 495 576 '669 777

40 304 56 832 76170

101494 130 754

231

275 328 384 450 528

26 018

37785 53 280 71409 95 340 122 581

227 277 330 393

8539 22 549 32 747

46176 *·

*»

»» »»

461

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168521

222638

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1294 1527

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1731

··

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44 61

76 92 112 135 162 195 230 270

8 730 15171 22 941

32935 45 583 62 250 83 561 110 496 143323

59

8 697

73

81

15171 22 941 32 720 45 583

101

101

123 149 180 216 260

306

62084

127 154 186 225

8 153 14 323

21617

88

7 610

103

7066

121

13 368 20 176

141

12 413 18 735

152 185

178 **

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30 675

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441

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504 590 698 809 945

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**

**

75 605 99 995 128 374 164 840

436 515 602 704

70 204 92 852

»*

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**

**

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187 221

258 302 371

434 501 581

665 812 943

1077 1358

87 009 114 733

147729 188 726 264 338

342157 433 682 545 598 690 999 941281 1 201 973 1501 154 2 056 677

249 294 344 402 495 578 668 774 887

1082 1258 1436 1810

86 405 114 280 146 713

188389 261657 339 082 433 047 545 240

690999 941281 1201973 *·

2056 677

312 368 430 503 619 723 835 968

81005

374

107 137

441

137543

516 603 743

176 615

245303

119 204 **

aa

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ta

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108

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1353 1572

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aa

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**

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1

·*

2263

614

**

·*

"

Plastic resistance of cross-sections Note; all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

6

7

8

Np,

Vp,.z

Vpl.y

Mpl.y

Mp|.z

Nfimrt

{Nfimit /Np..Rd)

for Ν - My IkN]

IkN]

IkN]

[N.m]

[N.m]

IkN]

HE100AA HE120AA HE140AA HE 160 AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA

for Ν - My Η

717

163 183 210 276

310 369 465 618 744 884

26 845 38 695

··

106 123

0.147 0,144

,a

**

>*

-

'"

··

ta

**

1032

ka

··

195

ka

··

1365 1545 1742 1827 1913 1998 2164

ka

··

*

·*

'*

*·

2 249

*

··

185 066 1438 696 1666 608

··

·*

++

-

3032 3366

1913 533 2 226 462 2863 407

1216 1445 1634 1834 2046

HE 1000 AA

++

-

3 542

0,193 0,206 0.217 0,227 0,233 0,250 0,250 0.250

977

201

1.165

225 269

448 532 658 798 942

1

Designation

HE 100 A HE 120 A HE 140 A HE 160 A HE 180 A HE 200 A HE 220 A HE 240 A HE 260 A HE 280 A

HE300A HE 320 A HE 340 A HE 360 A

HE400A

++ ++ ++

++ ++

++ ++

++

731

++ ++

860 940

++

1028 1

++

1274 1453 1644 1930 2159 2401 2665

++

3 288

++

++ ++ ++

++

1445 783 2 082 2 476

1

2335 2508 2 594 2 681 2 847

1

351

2960 3534 3994

384 480 549 669 764 843 990

2 001 2 311

1092

2556

194

2 721 2 887

++

5176 5 721 6 140 6 567

7313 8189

HE500A

++

HE900A IHE1000A

120

++

++

HE 450 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A

323 410 468 572 657

++

1

1300 1523 1747 1984

++

2 223 2 475 2 740

++

3107

++

3 687

++

4338

++

++

++

1

105

1332 1587 1789

3139 3464 3789 3 955

4122 4 288 4 462 4 641 4 976

5144

«

'*

«

ka

··

** ·»

1

3679453 4497 294

·· ·»

·· tt

2 513

440523 467245

3 245

38186

18 925

120

54 966 79 808 112 768 149 432

27072

141

*«

197563 261490 342 527 423094 511 623 636 305 748921 851218 960697 1 178 427 1479 299 1816 474

·*

*·

175 229 254 318 367 442 502 563 656

**

721

**

793 868

2126 035 2461 177 2 822 693 3 234 637 4 001764 4 972 600 5897 200

615

2901

«·

*· ·* ·· **

** ·*

444144 486916 509176 531602 554 203 578 101

603639 650660 676068

1034 1

197

1369 1558 1759 1970 2265 2 710

3 232

3699

-

0,123 0,121 0,121 0,129 0,122 0,128 0,124 0,125 0,126 0,126 0,127 0,126 0,129 0,132 0.141 0,146 0.151 0,160 0,169 0,177 0,189 0,206 0,219 0,232

Plastic resistance of cross-sections Note: all values of Ngj and Mfj.y must be divided by Ym0 9

Designation

10

11

|

Nsd/Npl.Rd=0,15 Nsd [kN]

HE100AA HE120AA HE140AA HE160AA HE180AA HE200AA HE220AA HE240AA HE260AA HE280AA HE300AA HE320AA HE340AA HE360AA HE400AA HE450AA HE500AA HE550AA HE600AA HE650AA HE700AA HE800AA HE900AA HE 1000 AA HE HE HE HE HE

100 A 120 A 140 A 160 A 180 A

HE200A HE 220 A HE 240 A HE 260 A HE 280 A

HE300A HE 320 A HE 340 A HE 360 A HE400A HE 450 A HE 500 A HE 550 A HE 600 A HE 650 A HE 700 A HE 800 A HE 900 A HE 1000 A

MN.y [N.m]

108 128

12

|

13

Nsd/Np,.Rd=0,2 Nsd [kN3

MN.y [N.m]

14

|

15

Nsd/Np,.Rd=0,25 Nsd IkN]

MN.y [N.m]

16

|

17

Nsd/Np,.Rd=0,3 Nsd [kN]

MN.y [N.m]

18

|

19

Nsd/Npl.Rd=0,35 Nsd [kN]

·*

*

aa

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MN.y [N.m]

*·

··

·*

**

··

**

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**

··

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··

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944

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1055

··

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ktt

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342 408

40 650

1

1213 1317 1508 1740 1947

tt

147 175

37022 53158

217 268 312

** **

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371

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444 530 599

**

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671

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776 858

aa

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985 1097

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aa

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k·

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1228 1363

195 233

34 845 50 031

244 291

32 667 46 904

293 350

30 489 43 777

28 311

1472 581

1638

'*

··

··

**

**

816 474 2126 035

1817 1948

"

**

·*

·*

**

*·

·*

·*

"

**

1563 1667 1797 1972

**

**

·*

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aa

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2 212 2 393

**

».

··

**

**

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**

**

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*

**

**

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**

1461

1

616

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by γΜο 2

3

4

5

6

7

8

Np,

Vp|.z

Vp..y

Mpl.y

Mp|.z

Nfimit

{Nfimit /Np,.Rd)

for Ν - My IkN]

IkN]

[kN]

[N.m]

[N.m]

[kN]

for Ν -My Η

198

240

23654

291

560 733 928

47938

1564 1976 2496

75 998

37 245

112896 162824 221466 295572 380442 484447 590139 705839

55 101 78 183 106 266

139 175 215 291

277034 330083

859 590

**

988650

431984 453432 474945 507857 550922 594158

1

Designatton

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 8 HE 700 Β HE 800 Β HE 900 Β HE 1000 Β HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M

HE400M HE 450 M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

1

3 002 3 592 4 188

4 875

5448 6043 6858 7422 7 861

8309 9098 10 027 10 977 11687

347 467 538 659 742 882 998 1091 1260 1375 1490 1609 1858

1

154

1392 1658 1939 2 249 2 507 2 773

3139 3384 3550 3 716 3 970

789483

108474

53 504 78 950 110 635 149 711

249 310 375 476

195587

551

249 880 312 135 462 729 548 534

650 734 940 1045

4296 4 622 4 788 4 955

2 214 700 2 571679 2 955 562

3 241 3 641

5123 5298

3 367145

++

4 705 800 5 786 800

++

4296

++

5 013

++

5643

5 479 5 815 5 984

2449

479 562 650 818 920

1203 1487 1792 2136 2485

1090 1203

2 867 3 259 4 402 4 827 5 257

3 055 3 706 4464 5 210

6039 6 874

9181 10 104

11048 13 942 14 354

14 528 14 665 14 986 15 430

1

595

1777 1913 2404

973 795 160 861

1

1364

191

642472 880063 897333

6 812

2 295 086

6790 6790

2562 484

889 701

2 912 267

892034

3263 365 3649 034

891 162

++

6 769 6 748 6 748 6 727 6 722 6 701

++

6 241

6 701

++

522168 652946

2 926

6 769

17192 17619

227160 310 300 406 386

762 836

2720

3 707 3 975 4 242 4 510 5159 5 695

15 838

161 282

663050 687720 714442

2040112 2170 082

3183 3439

16 301 16 728

3830 480

616 925 639 886

6689 6833 6833

2 519 2 619

561 631

6835 600

107 728

1

2116 2385 2658 2943

++

181 185 229 273

703 807 882 964 1050 1237 1425 1625 1841 2069 2308 2 631 3 132 3 709 4233

1234175 1486 600 1831890

++

342 416 475

140674

1

875 730

893483

888 727

4 035 159

887975 890 409 887 241

5 745 400 6 642 400 7 622 200

887981 887283 892 253

152

1409 1491 1578 1665 1844 2066

898 246

4 442 201 4848 400

617

1

2 288 2 520 2 752

2984

!

3 216 3 723 4 187 4660

0,116 0.112 0,109 0,117 0,114 0,116 0,113 0,115 0,116 0,116 0,118 0,119 0,123 0,126 0,136 0,142 0,148 0,158 0.167 0,175 0,187 0,204 0.217 0,230 0,102 0,102 0,101

0,107 0,106 0,108 0.107 0.102 0,103 0,104 0.101

0.104 0.109 0,114 0.123 0,134 0,144 0,155 0,165 0,174 0.183 0,200 0,215 0.228

Plastic resistance of cross-sections Note: all values of Njj and i% y must be divided by Yj^0 9

Designation

HE 100 Β HE 120 Β HE 140 Β HE 160 Β HE 180 Β HE 200 Β HE 220 Β HE 240 Β HE 260 Β HE 280 Β HE 300 Β HE 320 Β HE 340 Β HE 360 Β HE 400 Β HE 450 Β HE 500 Β HE 550 Β HE 600 Β HE 650 Β HE 700 Β HE 800 Β HE 900 Β HE 1000 Β HE 100 M HE 120 M HE 140 M HE 160 M HE 180 M HE 200 M HE 220 M HE 240 M HE 260 M HE 280 M HE 300 M HE 320 M HE 340 M HE 360 M HE 400 M

HE450M HE 500 M HE 550 M HE 600 M HE 650 M HE 700 M HE 800 M HE 900 M HE 1000 M

10

12

11

Nsd/Np,.Rd=0,15 Nsd IkN]

MN.y [N.m]

Nsd

180 235 296 374 450 539 628 731

46 090 72 732 107689 156668 212 420 284134 364 721 465 310

240 313 395 499 600 718 838 975

817 906

567348

1090

678 948

1

1029 1 1

113 179

1246 1365 1504 1647 1753 1863 1976 2114 2306 2 562 2 760 367 458 556 670 781

906 1031

1377 1516 1657 2091 2153 2179 2200 2 248 2 315 2 376 2445

2509 2 579

2643 2 789 2 923

3065

··

953672 1073113 1200 715 1462 435 815 159 2 209 507 2 571679 2 955 562 3 367 145 3830 480 4 705 800 1

13

NSd/N pl.Rd=0,2

IkN]

209

1484 1572 1662 1820 2005 2 195

2 337

2484 2634 2 819 3 074

15

|

Nsd/NpI.Rd=0,25

16

17

|

NSd/N pl.Rd=0,3

18

Nsd/Np,.Rd=0.35

Nsd IkN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

43 379 68 453

299

40 668

64175

494 624 750 898

95 020 138 236

419 547 692 873 1051 1257

35 245 55 618 82 351

187429 250 707

359 469 593 749 900

37956

391

1047 1219 1362

321 813 410 568 500 601

1511

599 072

··

897573 1009 989 130 085 1376 409 1708 385 2 079 536 2 442119 2 837 162 1

·*

1078 1256 1463 1635 1813

841475 946864 1059 455 1290384

785377

2358

883 740 988 824 1204 358 1494 837 1819 594

2 493

2729 3008

1949 565

3 293

1466 1706 1907 2115

»·

2 227

601 611

1

383196 467227 559134

**

tttt

1855 1965 2 077 2 274 2 507 2 744 2 922 3105

59 897 88 685 129 021 174 934 233 993 300 358

*·

3509 3 842

1

3 184

·*

**

»»

**

**

**

**

**

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**

**

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a.

**

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**

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**

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**

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"

*·

295 226 386 312 497421 621 367 922130 1 100 588 1294 546 1 773 676 1 935 173 2069 406 2200 739 2483 752 2 858168 3 242361 3649 034 4 035159 4 442 201 4848 400 5 745 400 6 642 400 7622 200

893

1042 1208 1375 1836 2 021

2210 2 788 2 871

2906 2 933

2997 3086 3168 3260 3346 3438 3 524 3 719 3 897 4 087

689 623

tt*

**

611 741

729 278

820616 918194 1 118333 1388 063

2908

»»

490

119805 162439 217279 278904 355 825 433854 519196

··

2 598 2 751

··

102 650 152 583 214 844

19

MN.y [N.m]

101355 147452 199924 267421 343 267 437939 533974 639010

·*

14

96 612 143608 202 206 277860 363 588 468161 584 816

867887 1035 847 1218 396 1669 342 1821339 1947676 2 071284 2337649

612 764 926

90 574

134632 189 568

116

260 494

1302 1510 1719

340864 438901

1

2 295 2 526 2 762

2 690 041 3 051634 3453 096 3863 798 4 300125 4 744 735 5 745 400

3485 3589 3 632 3666 3 746 3858 3 959 4 075 4182 4298 4405 4649

**

**

aa

**

618

548 265 813 644

176 930

1339 1563 1812

243127 318140 409641 511714 759401

1

1707506 1825 946 1941829 2191546

4182 4306 4358 4400 4496

2 2 3 3

125657

112

1

2 062 2 754 3 031 3 314

971 107 142 247 1565 008

84 535

735 916

906 366

1066 097 1460 674

857 1069 1 297 1

563

**

78 497 116681 164 292 225 761

1823 2114 2406 3 213 3536 3 867 4880 5 024 5085

295415 380381 475163 705158 841626 989947 1356 340 1479 838 1582487

5 133 5 245 5 401

1

5286

593 672 704 217 1 812 374 2045 443 2353 785 2 670180 3 021 459 3 380 824 3 762 609 4151643

5 579

**

··

··

**

»·

»»

*·

*·

·*

**

**

**

*·

521 913

860 907 237 278 622 311

4 031367 4 448189 5 387 640

4 629 4 751

4890 5 018 5 158

1

1

5543 5 705 5 855 6 017 6 167

682 918

1899 340 2185 658 2 479 453 2 805 641 3139 336 3 493 851 *»

Plastic resistance of cross-sections

UB

S 460

Note; all values of columns 2 to 7 must be divided by γΜο Designation

UB 178x102x19 UB 203x102x23 UB 203 X 133 χ 25 UB 203x133x30 UB 254x102x22 UB 254x102x25 UB 254x102x28 UB 254 χ 146x31 UB 254 χ 146 χ 37 UB 254x146x43 UB 305x165x40 UB 305x165x46 UB 305x165x54 UB 356x171x45 UB 356x171x51 UB 356x171x57 UB 356 Χ 171x67 UB 406x178x54 UB 406x178x60 UB 406x178x67 UB 406x178x74 UB 457x152x52 UB 457x152x60 UB 457x152x67 UB 457x152x74 UB 457x152x82 UB 457x191x67 UB 457x191 χ 74 UB 457 Χ 191x82 UB 457x191x89 UB 457x191x98 UB 533x210x82 UB 533x210x92 UB 533x210x101 UB 533x210x109 UB 533x210x122 UB 610x229x101 UB 610x229x113 UB 610x229x125 UB 610x229x140 UB 610x305x149 UB 610x305x179 UB 610x305x238 UB 686x254x125 UB 686x254x140 UB 686x254x152 UB 686x254x170 UB 762x267x147 UB 762 Χ 267x173 UB 762x267x197 UB 838x292x176 UB 838 X 292 X 194 UB 838 χ 292x226 UB 914x305x201 UB 914x305x224 UB 914x305X253 UB 914x305x289 UB 914x419x343 UB 914x419x388

Ν,'Pi

'pl.z

'ply

M,

'pi-y

Mpl.Z

IkN]

[kN]

IkN]

[N.m]

[N.m]

262 329 340

440 522 572 707 387 476 566 689

78 778

19131 22 886

1758

387 415 444 472 435 467 537 533 598

162 291

25 231

1022

180810 222 286 260 512

918

54 912 64 895

286 615

1066 1250

331221 389195 356 327 412 162 464 665

1082 1228

836 949 884 919 1025 1

40 583 17146

914

761

112

1491

556 996

1065 1247 1403 1575

485133 551755 619173 690371 504179 592144

918 1

120

164

1271

1250 1372 1087

1445 1617 1323 1512 1679 1862

1

1

160

1278 1362 1485

2 073

1521

13 953

668 414 748 230 833 287 676671 760 254 842391 926 276 1026 908 946 964 1085 640 1 201 526 1 301 076 1 469 973 1 325 320

122 811

61320 75 004 85 873

98039 110 577 109 161

125166 139 793 155 653 174 295

2 113 041

2 551 780 3443 655

2 261

1837304

2651 2938

2 096 727 2 300 210

3335 2565

2 590151 2 371 542 2 850 932 3 296 627

293 559 326 686 373 257

I

3131545

4 695 5430

3 514 452 4 211215 3 841 570 4 385 876 5 032 400 5 782 200

7337

7120 800

8 433

8 128 200

4008 4 458 5 041 5 075 5 627

1509 172 1690 904 1905 354

619

257 288 322 343 360 335 351

399 406 452 530 555 586 639 715 695 712 791

853 766 818 907 966 1 054

857 907 995 1054 1 1

3964 5364

3 849

0,170 0,178 0.T75 0,164 0,250 0,233 0,217 0,183 0,162 0,158 0.172 0.167 0,167 0,210 0,196 0,191 0,182 0,219 0,202 0,201 0,196 0,250 0,233 0,230 0,222 0,219 0,218 0,208 0,207 0,201 0,198 0,237 0,222 0,216 0,215 0,210 0,239 0,226 0,218 0,215 0,184 0.182 0,178 0,243 0,230 0.225 0.220 0.250 0.239 0,228 0,250 0,243 0.227 0.250 0,246 0,236 0,233 0,194 0.189

96162 108813

2 462 2 792 3 278

3 177 3 749 3 031 3 497 4 327 3 391

Η

190 241

80132 91427 111766

2 042 2 256 2092 2 503

3064 3369

for Ν -My

IkN]

76 145 89 990

1795 2008 2180 2483 1854 2167

2 559 2 821

for Ν - My

21 164

1530 1644 1768 1945

3 291

(Nfimit /NpLRd)

118 556

144608 119147 140542

871

705 712

107672

Ν,limit

163 568 183 745

200461 229 867 184113 215 860

1

143 140 198

1279 1371

431 149 526 172

1498 1416 1495 1600 1758 1609 1912

724 012

2 479

249503

1783 1887

246 295

281265

2 005

297714

2198 2153

371 441

2 419

440 953

2634 2 576 2 758

387269 447949 557338 451710 535183 630450 736579 1329 200 1536 811

3 015

2943 3 227 3504

3 941 3898 4 288

Plastic resistance of cross-sections Note: all values of Nsj and Mjj y must be divided by γ^0 9

Designation

UB 178x102x19 UB 203x102x23 UB 203x133x25 UB 203x133x30 UB 254x102x22 UB 254x102 Χ 25 UB 254x102x28 UB 254 χ 146 Χ 31 UB 254 χ 146 χ 37 UB 254 Χ 146x43 UB 305x165x40 UB 305X165X46 UB 305x165x54 UB 356x171x45 UB 356x171x51 UB 356x171x57 UB 356x171x67 UB 406x178x54 UB 406x178x60 UB 406x178x67 UB 406x178x74 UB 457x152x52 UB 457x152x60 UB 457x152x67 UB 457x152x74 UB 457x152x82 UB 457x191x67 UB 457x191x74 UB 457x191x82 UB 457x191x89 UB 457x191x98 UB 533x210x82 UB 533x210x92 UB 533x210x101 UB 533x21 0x109 UB 533x210x122 UB 610x229x101 UB 610x229 Χ 113 UB 610x229x125 UB 610x229x140 UB 610x305x149 UB 610 Χ 305x179 UB 610x305 Χ 238 UB 686x254x125 UB 686x254x140 UB 686x254x152 UB 686x254x170 UB 762x267x147 UB 762x267x173 UB 762x267x197 UB 838x292x176 UB 838x292x194 UB 838x292 Χ 226 UB 914x305x201 UB 91 4x305x224 UB 91 4x305x253 UB 914x305x289 UB 914x419x343 UB 914x419 χ 388

10

|

12

11

NSd/NpI.Rd=0,15

13

|

Nsd/Np,.Rd=0,2

14

Nsd IkN]

MN.y [N.m]

Nsd IkN]

MN.y [N.m]

Nsd [kN]

167 203

78 778

107672

223 270

75 974 104 786

279 338

221

··

264 193

144 608

221

140 542

249 274 325 378 354 405 474 396 448

162291

501

590 476 528 590 652 460 526 590 652

119147

**

··

352 258 295 332

··

138314 **

140542 162291

··

|

15

Nsd/Npi.Rd=0,25 MN.y [N.m]

·*

**

17

Nsd IkN]

MN.y [N.m]

18

!

19

Nsd/Np,.Rd=0,35 NSd

[kN]

MN.y [N.m]

··

··

··

··

*"

**

·*

**

**

*·

5 27

121025

615

*

**

··

**

*·

**

**

**

**

··

·*

**

*

**

«·

368 415

|

··

129669

439

16

Nsd/Np,.Rd=0,3

*·

*

·*

112380

··

··

··

··

·"

··

*

··

··

*·

··

*

**

··

**

222 286 260 512

434 504

247652

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«

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389195

633

630

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464665 556996

668 787

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690371

544 688

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721

833 287

962

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590 653

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1047

785 864 722 810 888 958

926 276 1026 908

1072

1469 973

1430

890 993

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1099 1230 1311 1574 2 093 1 100 1231

1339 1496 1292 1521 1729 1546

1

152

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024 761

1441

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2 551780 3443 655

2098

**

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2 791

3 350 039

3 140 661

4 186

··

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703 1991

··

**

**

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**

**

**

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·*

k*

**

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1766 1971 2228 2 541 3 017 3 410

**

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4 547

620

Plastic resistance of cross-sections Note: all values of columns 2 to 7 must be divided by Ym0 1

Designation

UC 152x152x23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203 X 203 χ 52 UC 203x203x60 UC 203x203x71 UC 203x203x86 UC 254 χ 254 χ 73 UC 254x254x89 UC 254x254x107 UC 254x254x132 UC 254x254x167 UC 305x305x97 UC 305x305x1 18 UC 305x305x137 UC 305x305x158 UC 305x305x198 UC 305x305x240 UC 305x305x283 UC 356x368x129 UC 356x368x153 UC 356x368x177 UC 356x368x202 UC 356 χ 406x235 UC 356x406x287 UC 356 X 406 X 340 UC 356 X 406 X 393 UC 356x406x467 UC 356x406x551 UC 356 X 406 χ 634

2

3

4

5

6

7

8

Np,

Vp|.z

Vpl-y

Mpl.y

Mp|.z

Nfimit

{Nfimit /Np!.Rd)

for Ν -My IkN]

IkN]

IkN]

[N.m]

[N.m]

for Ν - My Η

++

265 307 379

570 788 976

451

1223 1394

1760 2167 2 702 3049 3 513 4 160

498 588 645 815 680 818

5043 4 283 5 212 6 274 7 734 9 791 5 679

6909 8 023

9263 11611 14 066 15 498 7 559 8 961 10 373

11832 13 774 16 823 18 621

21525

1012 1227 1563 946 1

146

1323 1522 1871 2 280 2 695 1

128

1335 1562 1794 2011 2491 2 967

30183

3455 4121 4 615

34 725

5 329

25 581

1601

1950 2354 1969 2418 2 905 3 625 4 628 2563 3140 3 670 4263 5 419 6609 7335 3 497 4164 4 823 5 513 6497

7962 8860 10 12 14 16

290 324 625 953

»·

113926 142030 228 822 261002 301795 367429 449 296 456352 562977 682838 859930 1 114 887 732352 900463 1 056 414 1233 003 1582 391 1953 654 2 195 235 1 140 465 1363 775 1 589 410 1826 999 2155 790 2 673 739 3009 603 3 535 650 4300 000 5 194 400 6 123 200

621

[kN]

**

·*

**

219 267

64 207 ·*

321

·*

350 412 437 550 478 567 696 826

140455 171 892 209 868 **

264642 320639 404052 523144 " **

484171 565847 727136 897270 1007129 ** »*

768571

882985 1096 284 1356 664 1523 789 786 101 2164 750 2 604 922 3 056 599 1

1029 677 810 925

1053 263 1511 1639 812 953 1

1

107

1262 1402 1712 1876 2152 2 510

2945 3 324

··

0,124 0,123 0,119 0,115 0.117 0,105 0,109 0,112 0.109 0.111

0,107 0.105 0,119 0,117 0,115 0,114 0,109 0,107 0,106 0,107 0,106 0,107 0.107 0,102 0,102 0,101

0,100 0,098 0,098 0,096

Plastic resistance of cross-sections Note: all values of Nsd and Mj^y must be divided by Ytø0 9

Designation

10

Nsd/Np,.Rd=0,15 Nsd IkN]

UC 152x152x23 UC 152x152x30 UC 152x152x37 UC 203x203x46 UC 203x203x52 UC 203x203x60 UC 203x203x71 UC 203x203x86 UC 254x254x73 UC 254x254x89 UC 254 X 254 χ 107 UC 254 x 254 x 132 UC 254x254x167 UC 305x305x97 UC 305x305x1 18 UC 305x305x137 UC 305x305x158 UC 305x305 x 198 UC 305x305x240 UC 305x305x283 UC 356x368x129 UC 356x368x153 UC 356x368x177 UC 356x368x202 UC 356x406x235 UC 356x406x287 UC 356x406x340 UC 356x406x393 UC 356x406x467 UC 356x406x551 UC 356x406x634

MN.y [N.m]

··

264 325 405

457 527 624 756 642 782 941 160

1

1469 852 1036 1203 1389 1742 2110 2 325

12

11

|

Nsd/Np,.Rd=0,2 Nsd IkN]

MN.y [N.m]

14

15

|

Nsd/Npl.Rd=0,25 NSd

MN.y [N.m]

IkN]

16

17

|

Nsd/Np,.Rd=0,3 Nsd IkN]

MN.y [N.m]

*·

*·

··

·*

··

··

**

**

*·

·*

tttt

··

»*

··

137674

433

129 576

121477

542

650

18

Nsd [kN]

113379

**

··

**

··

»·

**

aa

aa

··

·*

··

··

**

**

**

1230 1456

222249 266887 327777

349005

703 832

428 632

1009

··

536 884 652 875 818 347 1058 884

273537 328476 403418

**

1042

··

878

1040 1261

256441 307946 378 205

**

»*

1054 1248 1513

239345 287416 352991

··

**

1

765 **

**

3 427

410 559 499 257 625 795 809 735

**

**

1547 1958

1303 1568 1934 2448

**

**

**

**

a.

**

**

**

·*

»»

··

**

aa

**

a.

1014 988 182 419

1509183 1860 525 2086 660

1

255

1605 1853 2322 2 813 3100

473 721 576 066 722071

934310

1564 1882 2 320 2 937

955 282 1 112865 1420 407

2006

895 577

2 316

1751083 1963 915

3 517 3 875

1043 311 1331632 1641640

3483 4220

841 170

4 650

2903

1

2 407 2 779

442140 537662 673933

2 196 2 707

872 022

835872 973 757 1242856

1

1718 426

4 923 5 424

1422 755 1595681 *·

1

532 197

··

»»

··

**

**

**

··

**

«*

**

»»

5209

2 040 139 2 530183 2 844 795 3 339 121 4 052 694 4 892 539 5 755 618

2366

2 593 2 958

2 755

1

920 131

3443 4206 4655

3365

2381349 2677454

3 724 4 305

3 142 702

5116 6 037 6945

3 814 301 4604 743 5 417 052

5 381 6 395

7546 8 681

622

1334 530 1533 890 1800123

3 112 3 550

2 232 515 2 510 113 2 946 283

5 047

3 575 907 4 316 946 5 078 487

4132 5586 6 457

7674 9 055 10 417

904 203 154081

4064

**

1423 499 1636150

776167

3 242

··

2 075

**

2808

··

1512 468 1738 409

»»

1824

505 303 614471 770 209 996 597

*·

2 793 3 229 3 837 4 527

».

105280

··

**

2 523

-

759

134

2066

MN.y [N.m]

**

1344 1556 1775

1

19

Nsd/Np,.Rd=0,35

··

290 634

1

13

|

1

245 562

1431631 1680 114 2083 680 2 342 772 2 749 864 3 337 513 4 029 150 4 739 921

3 631 4141 4 821

5888 6 517 7 534 8 953 10 564

12154

** 1

156 593

1329372 1560106 1934 846 2175432 2 553445 3099119 3 741353 4 401355

European Commission

EUR 18366

Properties and in-service performance

Promotion of plastic design for steel and composite cross-sections: new required conditions in Eurocodes 3 and 4, practical tools for designers J. B. Schleich,

P.

Chantrain, B. Chabrolin,

Y.

Galea, A. Bureau, J. Anza,

F.

Espiga

Luxembourg: Office for Official Publications of the European Communities 1998

622 pp.

21

χ 29.7 cm

Technical steel research series ISBN 92-828-4894-9

Price {excluding VAT) in Luxembourg: ECU 114

At present in Eurocode 3 (design of steel structures) and in Eurocode 4 {design of composite steel and concrete structures), the plastic analysis is governed by two criteria. The first refers to the mechanical characteristics of steel and the second to the geometry of the profiles used. Previous research on the rotation capacity for plastic analysis, performed by ARBED, CRM and RWTH Aachen, has allowed us to understand the behaviour of plastic hinge formation for conventional steel. The results of this research have shown that the requirements of Eurocodes 3 and 4 {b/t ratio and its yield stress dependence, 1Jiy > 1.2, eu > 20. sy, Ar > 15 %) are very safe-sided and could be substantially reviewed especially for high-strength steels {S460).

A new single criterion called rotation capacity quantifies the capacity of a profile to develop a plastic hinge without discrimination for high-strength steels. This rotation capacity of plastic hinges includes a lot of parameters: geometrical and mechanical characteristics of the cross-sections, ductility; deformation range, local buckling problems, influence of welding procedures, influence of defects and the analysed structure {geometry, loads). Nowadays the more competitive plastic design is not often used on the market because guides and practical tools, which evaluate available rotation capacities of cross-sections and required rotation capacities for different structures, are missing. The first aim of this research is to determine the available rotation capacities for composite cross-sections (steel beam in conjunction with a concrete slab) for all the steel shapes included in sales programmes and as a function of different steel grades. The second aim is to determine the required rotation capacities for different types of structures.

These practical tools ensure a real promotion of plastic analysis because the designer will be able to know easily and quickly if a plastic analysis is possible by comparing the required rotation capacity for this structure with the available rotation capacities of the cross-sections used. The final aim of this research is to introduce these new rules of plastic analysis for steel and composite cross-sections into Eurocodes 3 and 4, with the support of expert analysis. In such a way the competitiveness of steel and composite cross-sections will be improved and with this advantage their market share will increase substantially.

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