Continuous Beams
Hogging Moment Region
Sagging Moment Region`
Continuous Composite Beam System
Continuous Composite Beam System
Continuous Composite Beam Ad A d v an antt ag ages es
greater load greater load capaci capacity ty due du e to redistri redist ribut bution ion of moments greater grea ter stif st iffness fness and there therefor fore e reduc reduce e defl de fle ect ction ion and vibra vibr atio tion. n.
Disadvantages
inc rease increa se com complexity plexity in i n desi design gn susce susc eptibl ptible e to buckling buck ling in the nega negativ tive e moment region ove ov er inte i nterna rnall suppor supports. ts. Two for ms of buckl buckling ing ma m ay occur: occ ur: (i) local buckli buc kling ng of the t he web and/or and/or bottom bot tom flange f lange (ii) late la tera rall tors torsional ional bucklin buc kling. g.
Optimum Span/Depth Ratio of Composite Beam Simply Supported Beam: L/D = 18 to 22 Continuous Beam L/D = 25 to 28 L = Span Length D= Overall depth, including the concrete or composite slab
Introduction
Continuous beams may be more economical than simply supported beams.
However, special phenomena may However, m ay occur which must be taken into account in design, such as: – local buckling of compressed plate elements – lateral-torsional buckling – cracking of concrete due to tensile stresses
These all occur in the hogging moment regions.
In the sagging moment regions, design checks are similar to those of simply supported beams.
Hogging Moment Resistance depends on the reinforcements within the effective width
hs
P.N.A. t f
f
b f compression
Effective Width of the Concrete Flange (1) The effective width of the concrete flange depends on the distance between the zero-moment points, approximated by Le.
Le 2 Le 1
0.85 L1
L1
b eff = b o + b e1 + b e 2
0.25 L1
L2 Le 3
Le 4
2 L3
0.7 L2
L2
be1 or be2 = Le/8 < actual width Le = length between points of zero moment
L3
Effective Width of the Concrete Flange (2) Le 2 Le 1
0.25 L1
L2 Le 3
0.85 L1
L1
Le 4
2 L3
0.7 L2
L3
L2
At mid-span or an internal support: 2
beff Le / 8
bei i 1
Defining bei
b0
bi
At an end support: 2
beff
b0
b
i ei i 1
i
0.55 0.025 Le / bei
b0: distance between the centers of the outstand shear connectors
1.0
Hogging Moment Resistance effective width For internal beam, the effective width of concrete slab in the negative (hogging) bending region is given by Le = 0.5 L and corresponds to an effective width of L/8, where L is the clear span (rather than L/4 for the positive (sagging) moment in a simply supported beam). This means that the bar reinforcement is concentrated in a relatively narrow width over the internal supports. Le 2 Le 1
0.85 L1
L1
beff = L/8
0.25 L1
L2 Le 3
Le 4
2 L3
0.7 L2
L2
L3
beff = 0.7L/4
if bo = 0, and L=L1= L2
Example : Effective width of edge beam A continuous beam of uniform section consists of two spans and a cantilever, as shown in the figure below. Calculate the effective width for the mid-span regions AB and CD, for the support regions BC and DE. 6m
8m
2m
N
A
N’
B
C
E
D 0.3 0.15
2.0
Units: m Section N-N’
Example : Effective width of edge beam For segment AB, Le
0.85 6
be 1
Le / 8
be 1
0.3 m
0.3 0.15
5.1 m
2.0
0.638 m > 0.3 m
Section N-N’ be 2
Le / 8
0.638 m < 2 m 2
beff
b0
bei i 1
beff
0.15 0.638 0.3
1.088 m
Example 6.1
Perform similar calculation for the rest of the locations, All units: m AB
BC
CD
DE
Le
5.1
3.5
5.6
4
Le /8
0.638
0.438
0.7
0.5
b0
0.15
0.15
0.15
0.15
be1
0.3
0.3
0.3
0.3
-
-
-
-
0.638
0.438
0.7
0.5
-
-
-
-
1.088
0.888
1.15
0.95
1
be2 2
beff
Classification of Composite Cross-Section EN 1993-1-1 clause 5.5.2: Classification should be according to the less favorable class of elements in compression. A steel component restrained by attaching it to a reinforced concrete element may be placed in a more favorable class. Simply supported composite beams (sagging moment) are almost always in Class 1 or 2, because: 1. the depth of web in compression (if any) is small 2. the connection to the concrete slab prevents local buckling of the steel flange.
Classification of Cross-Section under hogging moment (1) 1) Composite sections without concrete encasement Compression outstand flange unrestrained from buckling follow EN 1993-1-1 Table 5.2 Web follow EN 1993-1-1 Table 5.2
c
c
t
t
Cross-section classification (5) Classification boundaries for webs in pure bending and uniform compression (EC3) Class
Pure bending
Uniform compression
1
72ε
33ε
2
83ε
38ε
3
124ε
42ε
t
rolled welded d / t rolled d/t welded
t
ε=
235 N/mm f y
2
Classification for web (negative bending) Class
Web subj ect to bending
Web sub ject to compression
Web subj ect to bending and compression
Stress distri bution (compression positive)
1
2
+ +
d/t 72
d/t 83
d/t 33
d/t 38
when >0,5: d/t 396 when <0,5: d/t 36 when >0,5: d/t 456 when <0,5: d/t 41,5
Stress distri bution (compression positive) 3
d/t 124
d/t 42
when d/t 42
>-1:
when
-1:
Classification of encased beam under hogging moment (2) 2) Composite sections with concrete encasement
b bc 0.8
b c
b bc
1.0
b
Web encased in concrete can be assumed to be Class 1 or Class 2 provided: the concrete that encases the steel section should be reinforced, mechanically connected to the steel section, and capable of preventing buckling of the web and of any part of the compression flange towards the web (clause 5.5.3(2) of BS EN 1994-1-1).
Classification of encased beam under hogging moment (3)
Classification of outstand flanges in uniform compression Class
Type
Web uncased
Web encased
(EC3) 1 2 3
c
(EC4)
Rolled
9ε
10ε
Welded
9ε
9ε
Rolled
10ε
15ε
Welded
10ε
14ε
Rolled
14ε
21ε
Welded
14ε
20ε
welded t
rolled
c/t
ε= c t
235 N/mm f y
2
Cross-section classification
Class of the section is defined as class of the element with the less favourable behaviour (e.g.: class 1 web and class 2 flange = class 2 section)
Exception: if compression flange is at least class 2 and web is class 3, then the section can be considered class 2: – with the same cross-section, if the web is encased – with an effective web, if the web is not encased
b
eff
hc hp 20t w ε
d tw
20t w ε
Classification of encased beam under hogging moment (4) Additional requirements on Class 1 and 2 sections if the reinforcement is in tension: 1) Ductility requirement on steel reinforcement ductility class B and C steels 2) The minimum area of reinforcement for Class 1 and 2 sections: f y f ctm A s A kc s c s 235 f sk Ac= effective area of the concrete flange f y = nominal yield strength of the structural steel, MPa f ctm = mean tensile strength of the concrete (EN 1992-1-1, Table 3.1 for normal weight concrete or 11.3.1 for lightweight concrete, see next slide) f sk = characteristic yield strength of the reinforcement k c = coefficient accounting for the stress distribution prior to concrete cracking δ : = 1 for Class 2 and 1.1 for Class 1 (more reinforcing steels for Class 1)
Mechanical properties of concrete Extract from Table 3.1 in EN 1992-1-1 (for normal weight concrete) f ck
(MPa)
25
30
35
40
45
50
55
60
f ctm (MPa)
2.6
2.9
3.2
3.5
3.8
4.1
5.2
4.4
Ecm
31
33
34
35
36
37
38
39
(GPa)
For light weight concrete, the f ctm and E cm values for a given grade are reduced by the following coefficient: flctm
f ctm 0.4 0.6 / 2200
Elctm
Ectm
/ 2200
2
ρ refers to the density of lightweight concrete in kg/m 3
Classification of Cross-Section (8)
k c
1 1
h c / 2 z0
0.3
hc N.A.
hc: thickness of the concrete flange, excluding ribs and haunches. z0: distance between the centroids of the uncracked concrete flange and the uncracked composite section, calculated using n0 for short term loading
n0
E a / Ecm
E a: modulus of elasticity for structural steel
z0
Plastic cross-section resistance
Basic assumptions: – full connection between steel and concrete – steel & reinforcement are full yielded – resistance of concrete in tension is zero
Moment Resistance
beff
Fs d
R w = dtf y
Fa
t
Maximum Tensile force in reinforcement
F s = As f sk / γ s Maximum Compressive force on steel section
Fa
= Aa f y / γ a
γs = 1.15
γa = 1.0
Moment Resistance at Hogging Moment region
Two main cases for which formulae are developed – case 1: plastic neutral axis is in the flange of the steel section – case 2: plastic neutral axis is in the web of the steel section
and
Plastic cross-section resistance
Case 1: Fa > Fs ≥ R w plastic neutral axis is in the steel flange R w = Fa – 2bf tf f y
hs
P.N.A. t f
hs
f
Fa b f
PNA from top of steel beam
z f =
Fa − F s
compression
2 f y b f
Moment about top of the steel flange
Mpl,Rd = 0.5Faha + Fshs - (Fa -
(Fa-Fs)0.5Zf
Fs)2/(4bf f y)
Simplified Moment Mpl,Rd ≈ 0.5F ha + F hs
hs = position of rebars from the beam flange
F s = As f sk / γ s Fa
= Aa f y / γ a
and
Plastic cross-section resistance Case 2: Fa > Fs < R w plastic neutral axis is in the web of the steel section
b
tension f / γ s sk
eff
F
hc
s
hp Fa
P.N.A. zw
ha
tw
Fa
h a /2
f y / γa f y / γa
PNA from centriod of steel beam
zw
=
F s
2tw f y
Moment about t he center of th e steel beam
M pl− . Rd −
M pl Rd
= M apl . Rd + Fs (0.5ha + hs ) − 0.5 Fs zw = M apl Rd + F (0.5h + h ) − 0.25F 2 / (t
Fs f )
F
= Asf sk / γ s
=A
f / γ
Analysis of Continuous Beam In BS EN 1994-1-1, the design of continuous composite beams may be based on two approaches to determine the design bending moments in negative (hogging) and positive (sagging) bending: • Clause 5.4.4 states that linear elastic analysis may be used for composite beams with all section classifications using maximum permitted moment redistributions • Clause 5.4.5 states that rigid plastic analysis may be used for Class 1 sections
Methods of Analysis •Simplified table of moment coefficients (BS5950:Part 1: 3-1) •Elastic analysis – uncracked section •Elastic analysis – cracked section •Plastic hinge analysis (class 1 section only)
Simplified table of moment coefficients (BS5950 Part 1: 3-1 Table 2.)
** * * * *
This method accounts for pattern loading, cracking of concrete and yielding of steel. 30
Examples (BS5950 Part 1: 3-1 Table 2) 2 spans Non reinforced plastic section 0.50wl2/8
0.79wl2/8 3 spans Reinforced compact section 0.67wl2/8
2
0.80wl /8
0.52wl2/8
Restrictions on the use of Simplified moment coefficients Some restrictions are placed on this method: uniform section with equal flanges and without haunches. Steel beam should be the same in each span. The loading should be uniformly distributed. Unfactored imposed load should not exceed 2.5 times the unfactored dead load. No span should be less than 75% of the longest. End span should not exceed 115% of the length of the adjacent span. There should not be any cantilevers.
Elastic Analysis of Continuous Beam
The elastic analysis method depends on whether the composite cross-section is considered to be uncracked, or cracked in negative bending. For uncracked analysis, the stiffness of the beam is treated as being constant along its length. For cracked analysis, the stiffness of the beam is reduced in the negative (hogging) moment region and hence lower percentage redistributions of moment are permitted in comparison to the uncracked case.
Cracked and un-cracked analysis EIg
Uncracked analysis: use EIg through out Cracked analysis: Use EIn near the support and EIg outside the 15% length of the support
EIg
(b) Crack section
EIn
EIg
Analysis of Continuous Distribution of bending moment Composite Beam
Elastic Analysis Uncracked Section (positive moment) Ig
= n r
I ay
+ 2h p + h a )2 beff h 3c + + Iay 4(1 + nr ) 12n
Aa ( h c
is the ratio of the elastic moduli of steel to concrete taking into account the creep of the concrete (may assume n = 13) is the ratio of the cross-sectional area of the steel section relative to the concrete section , Aa/(beff hc). is the second moment of area of the steel section
Cracked Section, Negative Moment
hc
hr
h p
I n
= I ay +
Aa Ar ( ha
+ 2( hp + hc − hr )) 2 4( Aa + Ar )
ha
Ar = area of reinforcement hr = distance of reinforcement from the top of concrete slab
Elastic analysis – moment redistributions Maximum moment redistributions for elastic global analysis of continuous composite beams (per cent of the initial value of the bending moment to be reduced) Analysis Method for Composit e Section
Section Class to SS EN 1993-1-1 1
2
3
4
Un-cracked Section
40%
30%
20%
10%
Cracked Section
25%
15%
10%
0%
Resistance against combined bending and shear
Interaction diagram
Low bending – shear capacity not reduced
High bending and shear – interaction formula
2V 2 − M v −.Rd = M −f .Rd + ( M Rd − M −f .Rd ) ⋅ 1 − Sd − 1 V pl .Rd
V Sd V pl.Rd
C
B
0,5 V pl.Rd
A
M
_ f.Rd
M
_ Rd
Moment capacity of flanges only
M
_ V.Rd
Low shear – moment capacity not reduced
Problems for continuous beam design High shear Lateral-torsional buckling at hogging moment region Shear connection design Loss of serviceability due to concrete cracking
Lateral-torsional buckling
The theory of lateral-torsional buckling of continuous beams over supports is rather complex.
In reality, lateral-torsional buckling is affected by: – beam distortion / lateral deflection of compressed flange – torsional rigidity of section
In design, two types of simplified approach may be followed: – simplified calculation of lateral-torsional buckling resistance according to analogy to steel beams (EC3 approach) – application of certain detailing rules that prevent lateral-torsional buckling
Lateral-torsional buckling
EC3 approach
−
M b .Rd
EC3 LT buckling curves
−
= χ LT M Rd
λ LT =
− M pl − M cr
In this approach, the elastic critical moment is determined using the so-called “inverted U-frame model”. The use of this model is subject to certain conditions. This model is not discussed here in detail No lateral-torsional buckling if the lateral-torsional buckling slenderness ratio < 0.4,
λ LT
≤ 0.4
Lateral Torsional Buckling at Hogging Moment Region
≤
Prevention of lateral-torsional buckling by bracing
0,4.
λ LT < 0.4
No reduction in capacity due to lateral torsional buckling
Lateral torsional buckling can also be prevented by limiting the depth of the steel member at the hogging moment region
Maximum depth h (mm) of uncased steel member to avoid lateral-torsional buckling checks (EC 4 Table 6.1 ) Member
Steel grade S235
Steel grade S275
Steel grade S355
Steel grade S420 or S460
IPE/UB
600
550
400
270
HE/UC
800
700
650
500
Shear connection design
Basic rules •
Connectors should be ductile
•
Plastic design of shear connection is possible even if global analysis is elastic, provided that the end cross-sections of the critical length to be designed are at least Class 2
•
In hogging moment regions, use of full shear connection is recommended
•
In sagging moment regions, partial shear connection may be applied
Shear Connections in Negative Moment Regions Nn = Fs /PRd F s =
As f sk
γ s = tension resistance of reinforcement
PRd = design capacity of a shear connector in negative moment regions considering concrete cracking Total no. of shear connectors = N p + Nn needed between the point of maximum moment and each adjacent support
Fa
N p
= ⋅ min Aa f y ; P Rd 1
Fc
0.85beff hc f ck 1.5
Fs
N n
=
1
⋅
As f sk
P Rd 1.15
N p + N n = ⋅ min Aa f y ; P Rd 1
0.85beff hc f ck 1.5
+
As f sk
1.15
Deflection Deflection is affected by
Pattern loading
Cracking of concrete
Yielding of rebars
But yielding of rebars and cracking of concrete have less influence on deflections in services than they do on analyses for ultimate limit states. Simplified method is developed for uniform beam in which deflection is estimated based on uniformly distributed load that the hogging end moments M1 and M2 reduce the md-span moment deflection from δo to δc
Serviceability deflection
Deflection of a continuous beam (simplified)
c =
o{1-0.6(M1+M2)/Mo}
δo = Deflection of a simply supported beam Mo = maximum sagging moment in the beam when it is simply supported
M1 and M2 are moments after redistribution for pattern loads, etc. 49
Influence of Pattern loading To account for pattern loading as shown in the figure, reduce the uncracked moments at the internal supports by 40% M1 and M2 are moments after redistribution for pattern loads, etc. 50
Serviceability – Cracking of concrete
In continuous beams, concrete cracking is mainly due to tensile stresses in the hogging moment regions
This cracking is prevented by limiting bar spacing or bar diameters in the reinforcement
Serviceability – Cracking of concrete
Limiting bar spacing (for high bond bars only) to avoid cracking over supports stress in reinforcement 2 σs, N/mm 160 200 240 280 320 360
maximum bar spacing for wk = 0,4 mm 300 300 250 250 150 100
this stress is calculated considering tension stiffening
maximum bar spacing for wk = 0,3 mm 300 250 200 150 100 50
maximum bar spacing for wk = 0,2 mm 200 150 100 50 – –
unless using a more precise method: σ s = σ s 0 + ∆σ s
…with... ∆σ s =
0.4 f ctm
σ s = σ s 0 + ∆σ s σso = tensile stress in the reinforcement
∆σ s =
0.4 f ctm α st ρ s
f ctm is the mean tensile strength of concrete; ρs is the “reinforcement ratio” expressed as αst = As / Act Act is the area of concrete flange in tension within the effective width As is the total area of reinforcement within the area Act αst is the ratio AI Aa I a where A and I are the area and second moment of area of the composite section neglecting concrete in tension and any sheeting, and Aa and Ia are the same properties for the bare steel section.
Conclusions
Continuous beams offer advantages over simply supported beams, but special phenomena need particular attention during design in the hogging moment regions
In the case of both elastic and plastic design, cross-section classification and resistance calculation are key issues
Lateral-torsional buckling at the hogging moment regions must be prevented by appropriate detailing or by direct check
In shear connection design, hogging moment regions require full shear connection
In the hogging moment regions, the serviceability limit state of cracking of concrete may be relevant
