EFFECTIVE LENGTH OF T-STUB OF RHS COLUMN BASE PLATES František Wald, Valéry Bouguin*, ZdenČk Sokol and Jean-Pierre Muzeau* Czech Technical University in Prague * University of Blaise Pascal, Clermont Ferrand ABSTRACT
The paper presents the application of the component method to column bases of the RHS columns. The decomposition of the connection into the components is described. An analytical model is assembled to determine the moment resistance and the rotational stiffness of the column base under different axial loads. The effective length of T-stub in tension is derived analytically and checked by the FE simulation as a main contribution. The prediction model is verified by comparison with the published test results. 1 INTRODUCTION The RHS columns are connected to the foundation by base plates and/or by embeddings. In seismic areas are both fixing combined with infilling of bottom part of column by cocnrete. The base plates are designed thick to transfer primarily compression forces into concrete block and are restrained by stiffener. The anchor bolts are used longer compare to the bolts between plates due to the washer plates, thicker base plate, grout, and enbeddement in concrete, which allows deformation and separation during the loading. The difference to beamto–column connection shell be introduced into the prediction of strength, stiffness and rotational capacity of the base plate in tension. RHS column
Side view
Component base plate in bending and anchor bolts in tension
Base plate
Grout
An example of anchor bolt Foundation
Packs
Component column web and flange in shear and compression Component base plate and concrete block in copression
Embedded anchor bolt Conical sleeve Anchoring plate
Component anchor bolt, key, in shear
Top view
b)
a)
Fig. 1 Example of base plate a) and description main components b)
393
The compression part of the base plate designed for resistance of the concrete in crushing under the flexible base plate. The model of effective area under the base plates is commonly accepted and applied in design recommendations, see Annex L in (1). The behaviour of the tension part of the base plate is mostly guiding the column base resistance and stiffness (2) in case of loading by bending moment. The knowledge of behaviour of end plates in beam to column connection ware redefined in (1) using models developed in last years by applying the component method. The connection is disintegrated into components, which behaviour is described, and composed back to model connection characteristics.
2 COMPONENT METHOD FOR BASE PLATE The column base with base plate is in component method divided into components, see at Fig. 1, (2). In the base plate can be recognised the component the base plate in bending and anchor bolt in tension, the component column web and flange in shear and compression, the component the anchor bolt, shear key, in shear and the component column web and flange in shear and in compression. The design procedure is summarised on flow chart on Fig. 2. Geometrical and material properties
Discterisation to component
Base plate in bending and anchor bolt in tension
Concrete in compression and base plate in bending
"eff
fj
T-stub effective length
Concrete bearing strength in joint
Stiffness of component
Resistance of component
FRd
kp ,k b and k t
Ft.Rd
kc Resistance and stiffness of component
Check of prying Assembly for resistance Resistance
M Rd
Assembly for stiffness
Shape factor of curve
Sj.ini
Stiffness
P
Moment rotation curve
Fig. 2 The design procedure for the base plate of the RHS column If the anchor bolts are activated in tension, the base plate is subjected to tensile forces and deforms in bending while the anchor bolts elongate. The failure of the tensile zone may
394
result from the yielding of the plate, from the failure of the anchor bolts, or from a combination of both phenomena. In Eurocode 3 (1) is the design resistance of a T-stub of flange in tension of effective length Ɛeff is derived for three possible plastic collapse mechanisms of T-stub in tension follow for three failure modes. In the particular case of base plates the elongation of the anchor bolts in tension is mostly such, in comparison to the flexural deformability of the base plate, that no prying forces develop at the extremities of the T-stub flange. In this case, the failure results either from that of the anchor bolts in tension (Mode 3) or from the yielding of the plate in bending see Figure 3, where a two hinges mechanism develops in the T-stub flange. This failure is not likely to appear in beam-to-column joints and splices because of the limited elongation of the bolts in tension. This particular failure mode can be named Mode 1*. The corresponding resistance writes
F1* .Rd
2 " eff m pl .Rd m
.
(1)
F / 6 B t.Rd 1
Mode 2
Mode 3
0,8
F1*.Rd
Mode 1
0,6
Mode 1*
0,4 0,2
B
0 0
0,5
1
1,5
B
2 4 "eff m pl.Rd /6 B t.Rd
Fig. 3 Failure mode 1*, typical for base plates due to long deformed length of the anchor bolts When the Mode 1* mechanism forms, large base plate deformations develop; they may result finally in contacts between the concrete block and the extremities of the T-stub plage, i.e. in prying forces. Further loads may therefore be applied to the T-stub until failure is obtained through Mode 1 or Mode 2, see Fig. 3. But to reach this level of resistance, so large deformations of the T-stub are necessary, which are not acceptable in design conditions. The extra-strength which separates Mode 1* from Mode 1 or Mode 2 in this case is therefore disregarded. As a result, in cases where no prying forces develop, the design resistance of the T-stub is taken as equal to
FRd
min F1* .Rd , F3.Rd , when F3.Rd
6 Bt .Rd .
(2)
The influence on Mode 1* failure under washer plate, cover plate, aimed at strengthening the base plate, which can be considered (1) as
F1* .Rd
2 " eff m pl .Rd m bp.Rd
m
,
(3)
for
395
m bp.Rd
¦t
0,25
2 bp
f y .bp / J M 0 ,
(4)
where fy.bp is the yield stress of the cover plate, tbp is the thickness of the cover plate. 1 ' d
a
c
ac b
" eff.1
D
bc
lb
y
x yield line
D
la
a)
" eff.2 " eff.5
eb
" eff.3
ea
"
eff.4
b)
Fig. 4 The base plate geometry a), assumption of the range of effective length of T-stub for base plate b) The effective length of the base plate T-stub can be determined by the yield line method (3). The yield line is a straight line, and this line is perpendicular to a line, that pass through the bolt and the corner of the plate. D represents the angle of the yield line with the edge and c the minimal distance between the corner of the plate and the yield line. The following relations can be obtained (4) x , y
tan D
(5)
where x and y are the variable coordinates of the bolt. For the design of the parameter c, we use the work method of the yield line theory. The internal work Wi
¦ >T
j
; muj ; " j
n
@
§1 1 · m pl ¨¨ x y ¸¸ . x ¹ ©y
(6)
The external work
We
Pu '
Fpl ' .
(7)
' represents the deformation of the plate in the bolt position, see Fig. 4.
' 1
d c
x2 y 2 , c
x2 y 2 Fpl c
(8)
§x y· m pl ¨¨ ¸¸ , ©y x¹
(9)
396
cm 4
" eff
x2 y 2 , xy
(10)
For the resistance can be derived
Fpl
c m pl
x2 y 2 , xy
(11)
x2 y 2 cst (12) wc xy Five cases may be observed for the yield lines round by the corner of the column, see Tab. 1 from (4), if are taken into account the modes without the contact of the edge of base plate to the concrete surfaces, e.j. in no prying cases. w Fpl
m pl
Tab. 1 The calculation of the effective length of a T-stub per bolt, Case 1 to 3 Case 1
W ext
W int
Case 2
Fpl G
4 S m pl G
Fpl
4 S m pl
m
a ac ea 2
" eff .1
S m
G
Case 3
a ac 2 e a a ac
W ext
Fpl G
W int
m pl
Fpl " eff .2
m pl
G W ext
b a ac
W int
b a ac 2 ea
b 4
Fpl " eff .3
b bc 2 a ac 2 2 ea 2 eb 2 b bc 2 a ac 2 Fpl G
§e e · m pl ¨¨ a b ¸¸ © e b ea ¹ mpl § ea eb · ¨ ¸ G ¨© eb ea ¸¹
a ac 2 b bc 2 8
§ ea eb · ¸¸ ¨¨ © e b ea ¹
The Case 4 and Case 5 are similar to 2 and 1 respectively. The results of prediction of effective lengths per anchor bolt are summarised in Tab. 2. The prediction of the base plate stiffness and resistance depends on the prying or noprying mode. In case of prying the calculation can be based on standard Annex J procedure. The boundaries between modes ware developed in (6). The simplified boundaries, for derivation see (2), where prying do not occur are
As " eff .ini t 3 d Lb 8,82 m 3
(13)
The anchor bolts into the concrete is fixed by hooked bars for light anchoring, cast-inplace headed anchors and bounded anchors to drilled holes. Models for the anchoring design resistance compatible with Eurocodes based on the ultimate limit state concept have been prepared (5). The anchor bolt effective free length Lb = Lbf + Lbe consist of physical free length Lbf and embedded free length (3). In case of embedded anchor bolts, which can be estimated as Lbe = 8 d. The stiffness of the plate as an independent component of no prying is
397
Tab. 2 Effective length of a T-stub " eff for base plate of RHS columns for a bolt a
b m
m
m
m
"eff .1 S m
Sm § b bc · eb ¸ S .¨ © 2 ¹ " eff .5
§ a ac · ea ¸ © 2 ¹
" eff .2
S .¨
b/ 4
" eff .4
a/ 4
a b
ac bc eb
" eff .3
m
8
ea
" eff k p.*
a ac 2 b bc 2
§ ea eb ¨¨ © e b ea
· ¸¸ ¹
min ( " eff .1; " eff .2 ; " eff .3 ; " eff .4 ; " eff .5 )
Fp
" eff .ini t 3
0,425 " eff t 3
E Gp
2 m3
m3
(14)
and the contribution by the bolt elongation is
k b.*
Fb E Gb
2,0
As . Lb
(15)
The stiffness of the component of base plate in bending and bolts in tension can be summarised from above simplified predictions as 1 / kt = 1 / kb.i + 1 / kp.i .
(16)
The positive influence of washer plate is limited till 5% of deformation and can be for practical design neglected. The resistance of the component in compression based on effective area under the base plate is described in Eurocode L (1). This component contributes only a bit to the base plate stiffness, but is taken into account for consistency of prediction (2). The effective area under the flexible base plate is modelled round the column. The position of the neutral axes is calculated from the force equilibrium for resistance. For the stiffness calculation is taken into account the effective area under the flanges only, see Fig. 2 (2). The position of compression force is located at the centre of compression flange. The tensile force is located at the anchor bolts. The force represents the resistances in tension, Ft.l.Rd and in compression, Fc.l.Rd, Fc.r.Rd. The stiffness is calculated based on component stiffness for the spring in position described at Fig. 8. Two cases are observed, with tension in anchor bolts and without activation of the anchor bolt.
398
a) Model for resistance
b) Model for stiffness Active part of equivalent plate Active part of equivalent plate
Equivalent rigid plate MRd
Centre of compressed part
NSd
NSd
Neutral axis
Fc.Rd
Ft.Rd zt
Equivalent rigid plate MRd
Centre of compressed part Neutral axis Fc.Rd
Ft.Rd zt
zc z
zc z
I
Model for anchoring in action
G t.l Model for no tension in anchoring
G c.r
I G c.l
G c.r
Fig. 5 The force equilibrium of base plate, for the full effective area used for the resistance calculation a), with the effective area under the flanges only applied for the stiffness prediction b), the assembling for large eccentricity and small eccentricity without tension in the anchor bolt
3 EFFECTIVE LENGTH OF T-STUB BY FE SIMULATION¨ The boundary between the failure modes ware observed using the FE model. Three layers brick model of base plate was applied by code Ansys. The step by step procedure was incorporated with for multi-linear model of material (7). The position of the bolt was changed round the column corner. The yielding to the base plate, highlights the expected failure mode, see Fig. 6, by reaching the plateau of the material diagram. The size of the bolt nut / washer plate and the relative stiffness of the anchor bolt was studied numerically. For the rigid anchor bolt as a limited case can be observed the prolongation of the yielding in the base plate and column corner on Fig. 7.
399
Fig. 6 The FE mesh of simulation, the different yield patterns under the moving of the anchorage round the base plate corner (4)
a)
b)
c)
Fig. 7 The influence of the size of the bolt nut, (concentrated load a), nut of diameter 25 mm b), 50 mm, washer plate of 80 mm c)) for the indefinitely stiff bolt
400
M RHS 200 x 200 x 12 4 x M 36
M RHS 152,4 x 152,4 x 12,7 4 x O 19
P 32 420 x 165 Grout 30 x 310 x 200 1000 x 550 x 550
experiment
M, kNm
P 20 -300 x 190 Grout 30 x 310 x 200 1270 x 610 x 460
M, kNm
60
60
50 40
experiment
50
analytical
40
30
30
20 10
analytical
20
Picard and Beaulieu 12 F (9)
Nakashima 20 - 12 (8)
10
0
0 0
0,01
0,02
0,03
0,04
0
I rad
0,01
0,02
0,03
0,04
I rad
Fig. 8 Comparison of the predicted moment rotation curve to the experimental on of Picard and Beaulieu (9) as well as Nakashima (8) 4 VERIFICATION TO EXPERIMENTS Two test setups, (8) and (9), with fully described working diagram of the test by M - I curve validates the design model (4). The material and geometry is introduced in measured / reported values. The geometrical basic characteristics are shown on the Fig. 8. Both tests were loaded by bending only. Fro steel the nominal value of material properties was available only. Comparing the initial stiffness, the experimental and analytical results are closed for tests with low influence of axial force. This comes from the steel grade. In the analytical model we did not use measured material property but. The results exhibit a good agreement of proposed prediction model to presented tests. The behavior of base plates loaded by normal forces display the accuracy of prediction of the resistance of the concrete in compression that is more limited. 5 CONCLUSION The Eurocode 3 procedure based on Annex J and Annex L procedures can determine the resistance and stiffness of base plates of RHS columns. The presented study shows a good prediction of the behavior by the simple engineering model. ACKNOWLEDGMENT This work has been supported by the grant of Czech Ministry of Education No. MŠM 21 000 000 1. Within the framework of the activities of the COST C1 European Project (Semi-rigid behaviour of civil engineering structural connections) and the Technical Committee 10 of ECCS (European Steel Fabricator Association) an ad-hoc working group was established to prepare a background document for European standardisation. The authors would like to thanks of all the members for help explicitly to Mr. C. M. Steenhuis, TU Eindhoven and Mr. J. P. Jaspart, University of Liège. NOTATION
401
d fy k m t x, y z B E F Lb M N S W G,'
I P J Ɛ
diameter of the bolt yield stress of steel component stiffness distance from the bolt axes to the weld edge, bending resistance of base plate thickness of the base plate axes, coordinates lever arm bolt force Young’s modulus of steel force free length of the anchor bolt bending moment axial force stiffness work component deformation connection rotation shape factor partial safety factor length of the T-stub
Subscripts effective free length of bolt b physical free length of bolt bf embedded free length of bolt bp bp c eff ini j l p r t Rd Sd pl int ext
cover plate compressed effective initial joint left plate right tension, T-stub design resistance acting plastic internal external
REFERENCES
(1) Eurocode 3, ENV - 1993-1-1, Design of Steel Structures - General rules and roles for buildings, CEN, Brussels 1992, including Part A2: Design of Steel Structures - General rules and roles for buildings, Annex J, European Prenorm, CEN, Brussels 1998. (2) Column Bases in Steel Building Frames, COST C1, ed. K. Weynand, Brussels, 1999, p.116. (3) Wald F.: Column Bases, ýVUT, Praha, 1995, p. 137, ISBN 80-01-01337-5. (4) Bougin V.: Column Bases of the Rectangular Hollow Sections Columns, Diploma theses, Clermont Ferrand, 2000, p. 129. (5) Fastenings to Concrete and Masonry Structures, State of the Art Report, CEB, Thomas Telford Services Ltd, London 1994, p. 248, ISBN 0 7277 1937 8. (6) Wald F., Obata M., Matsuura S., Goto Y.: Prying of Anchor Bolts, Nagoya University Report, Nagoya 1993, pp. 241-249. (7) Wald F., Baniotopoulos Ch. C.: Numerical modelling of column base connection, in COST C1 Conference, Liege, 1998, p. 2-7. (8) Nakashima S.: Experimental Behavior of Encased Steel Square Tubular Column-Base Connections, in Proceedings of the First Word Conference on Constructional Steel Design, Word Developments, Elsevier Applied Science, ed. Dowling P., Harding J. E., Bjorhovde R., Martinez-Romeo E., Acapulco1992, pp. 240-249. (9) Picard A., Beaulieu D.: Behaviour of a simple column base connection, Canadian Journal of Civil Engineering, Vol. 12, No. 1, 1985, pp. 126-136.
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