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A steel-concrete composite beam element with material nonlinearities and partial shear interaction
ARTICLE IN PRESS Finite Elements in Analysis and Design 45 (2009) 966–972
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A steel-concrete composite beam element with material nonlinearities and partial shear interaction Hamid R. Valipour , Mark A. Bradford Centre for Infrastructure Engineering and Safety (CIES), School of Civil and Environmental Engineering, The University of New South Wales, Sydney, 2052, Australia
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 June 2009 Received in revised form 21 September 2009 Accepted 28 September 2009
This paper presents the formulation of a novel force-based 1D steel-concrete composite element that captures material nonlinearities and partial shear interaction between the steel profile and the reinforced concrete slab. By decomposing the material total strain into its elastic and inelastic components, a total secant solution strategy based on a direct iterative scheme is introduced and the corresponding solution strategy is outlined. A composite Simpson integration scheme, together with piecewise interpolation of the slip strain along the element axis, is employed to calculate the slip forces along the element axis consistently. The accuracy and efficiency of the formulation are verified by some numerical examples reported elsewhere in the literature, and it is shown that the formulation with just one element can lead to virtually closed form analytical results as long as the integrals in the formulation are calculated accurately. & 2009 Elsevier B.V. All rights reserved.
Keywords: Flexibility formulation Material nonlinearity Secant stiffness Steel-concrete composite
1. Introduction Over the last two decades, the analysis of steel-concrete composite structures has followed two specific themes. For the first of these, researchers have tried to derive analytical solutions of the governing differential equation for generic cases by adopting simple assumptions such as linear elastic material behaviour [1], which have provided simple equations for engineering design purposes that can also be used as benchmark solutions for evaluating numerical methods. For the second, the focus has been on developing finite element models and numerical procedures, which offer better versatility compared with the generic models and which can be used for analysing large steel-concrete composite framed structures including material and geometrical nonlinearities [2–7]. The present paper falls into the second category of these themes. The finite element formulation can be cast within the framework of displacement-based, force-based or mixed (hybrid) methods [8]. Although the displacement and force-based formulations have the same degree of approximation, for frame elements which are the focus of this paper, the force-based formulations lead to superior accuracy when compared with displacement-based formulations because of the exact fulfilment of the equilibrium equations [9,10]. The formulation of displacement-based elements, however, is quite simple and straightfor-
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ward and is usually favoured in comparison with force-based elements. The first composite elements which were developed generally took advantage of transverse cubic displacement shape functions that require mesh refinement to capture the nonlinear response with reasonable accuracy [11]. Formulation of this type, when implemented with linear longitudinal shape functions, can lead to inaccurate results because of curvature locking. Salari et al. proposed a force-based element [3], where cubic interpolation function are used to approximate the bond slip shear forces along the element [3]. The formulation of the element, however, is rather complex and it does not satisfy the kinematic condition of shear slip continuity [3]. Ayoub [12] developed a forced-based composite element, which takes advantage of a linear function for shear slip interpolation. The mixed formulation concept was adopted by Ayoub and Filipou [13], in order to develop an efficient inelastic composite element. In addition, some researchers have tried to solve the simplified form of the governing differential equations, and to cast the results in matrix form, rather than using a conventional method based on predefined force or displacement interpolation functions [2]. A full discourse on different available approaches for formulating 1D steel-concrete composite elements can be found in Spacone and El-Tawil [5]. In the present study, the concept of a total secant approach based on decomposing the strain at cross-section fibres into their elastic and inelastic components is adopted to derive the secant stiffness of the cross-section [14]. Exact force interpolation functions are employed to derive the element secant stiffness, and a direct iteration scheme consistent with this secant
ARTICLE IN PRESS H.R. Valipour, M.A. Bradford / Finite Elements in Analysis and Design 45 (2009) 966–972
formulation is then presented. A composite Simpson integration scheme, accompanied with a parabolic piecewise interpolation of the slip (shear) strains along the element, is used to estimate consistently the shear slip forces along the element. Comparison studies illustrate the efficiency of the formulation.
967
from the right hand side of the Eq. (1) does not violate the generality of the formulation, which is adopted to simplify the derivation of the expressions [15]. Equilibrium of the cross-section requires that Z T Z Z Z sx1 dA sx2 dA ðy1 þ HÞsx1 dA y2 sx2 dA ; DðxÞ ¼ O1
O2
O1
O2
ð4Þ
2. Element formulation 2.1. Equilibrium equations Fig. 1a shows a 4-node 1D plane frame element AB, with four degrees of freedom at each end. It is noteworthy that adopting partial horizontal and full vertical interaction between the reinforced concrete slab and the steel profile leads to different horizontal translations for the slab end nodes with respect to steel profile Fig. 1a). Further, the rotational degrees of freedom are assigned to the steel profile end nodes (Fig. 1a). The generalised displacement and force vectors for the unrestrained system are denoted by q and Q, respectively (Fig. 1a), and the corresponding vectors for the restrained system are denoted by q and Q respectively (Fig. 1b). The equilibrium of the configuration Ax, shown in Fig. 1c, yields DðxÞ ¼ b1 ðxÞQ þDb ðxÞ þD ðxÞ; 2
1
6 b1 ðxÞ ¼ 4 0 H
0
0
1
0
0
x=l 1
ð1Þ 0
ð2Þ
and Db ðxÞ ¼
Z
x 0
Db ðsÞ ds
Z 0
x
T Db ðsÞ ds 0 ;
2.2. Compatibility equations Assuming partial shear interaction and adopting the NavierBernoulli hypothesis for the slab and steel profile separately, the compatibility requirement yields
exa ¼ era ya k
ða ¼ 1; 2Þ
ð5Þ
and
eb ¼ ðer2 er1 Þ Hk;
3
0 7 5 x=l
where O1 and O2 denote the reinforced concrete slab and steel profile domains, respectively, y1 and y2 are the distances of the fibres (transverse integration points) from the mid-plane of the concrete slab and steel components respectively (Fig. 2) and sx1 and sx2 are the total longitudinal stress components at the monitoring points in the concrete and steel respectively.
ð3Þ
where DðxÞ ¼ ½Nc ðxÞ Ns ðxÞ MðxÞT is the vector of the section generalised forces, b1 ðxÞ is the force interpolation matrix, Q ¼ ½Q 1 Q 2 Q 3 Q 4 T is the nodal generalised force vector for the restrained configuration, Db ðxÞ is the vector of section generalised force due to shear slip forces, H denotes the distance between reference axis of the reinforced concrete slab and the steel profile and D ðxÞ is a vector of the sectional internal forces and is solely produced by member loads. Removing the term D ðxÞ
ð6Þ
where exa denotes the total longitudinal strain component at the monitoring points (fibres), era is the axial strain at the mid-plane of the material domain Oa , k denotes the total curvature of the cross-section, eb is the slip strain and subscript a denotes the component of a composite section; (a = 1 for the concrete slab and a = 2 for the steel profile). 2.3. Constitutive law in total strain form Using a total strain constitutive law and decomposing the total strain, exa into its elastic ðeexa Þ and inelastic, ðepxa Þ components, the total stress and strain in the longitudinal direction are related by
sxa ¼ Eea ðexa epxa Þ
ða ¼ 1; 2Þ;
ð7Þ
where Eea is the elastic secant modulus of the theoretical unloading curve and is a function of the stress and strain components at the integration point under consideration. If a uni-axial constitutive law is adopted, the material total secant modulus ðEe Þ and inelastic strain component, ðepx Þ can be calculated directly (Fig. 3), whereas for a multi-axial constitutive law an iterative procedure is required to determine the material state [14]. Introducing the identities Z Eea dA; ð8aÞ ka11 ¼ Oa
y1 hc
RC slab mid-plane H
y2 hs
εr1
Steel profile mid-plane
εb ε b : Slip strain ε r2 κ
Fig. 1. (a) Unrestrained 2-node frame element AB in x–y plane (b) restrained simply supported configuration and (c) free body diagram of Ax.
Fig. 2. Steel-concrete composite section and strain distribution with partial shear interaction.
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σ
and
εe qp ¼
L oa d Unl ing o ad i ng
σ
E e=
εp
ε
σ ( ε ε p)
ka22 ¼
Z
ε
ya Eea dA;
ð8bÞ
Oa
Z Oa
Z
ð8cÞ
Z
ð17Þ
e
Kq ¼ Q þQ b Q p
ð18Þ
Q p ¼ e Kq p
ð19Þ
and
Eea epxa dA;
ð8dÞ
Oa
Mpa ðxÞ ¼
T
b ðxÞe f s ðxÞDp ðxÞ dx;
0
in which
y2a Eea dA;
Npa ðxÞ ¼
l
where q is the generalised deformation vector for the restrained beam, e F is the flexibility matrix of the restrained element, q p represents the vector of the nodal generalised plastic deformations and q b is a vector of nodal generalised deformations due to section shear slip between slab and steel profile along the element for the restrained beam. If Q p and Q b represent the generalised force vectors conjugate to the deformation vectors q p and q b , respectively, Eq. (14) degenerates to
Fig. 3. Total secant concept within a uni-axial plastic-damage model.
ka12 ¼
Z
ya Eea epxa dA;
ð8eÞ
Oa
Q b ¼ eK qb
ð20Þ
where e K is the stiffness matrix of the restrained element and is obtained by inverting e F.
and then substituting Eqs. (5) and (7) into Eq. (4) gives DðxÞ ¼ e ks ðxÞdðxÞ þDp ðxÞ; in which 2 e
6 ks ðxÞ ¼ 4
ð9Þ
k111
0
k112
0
k211
k212
k112 Hk111
k212
k222 þ k122 Hk112
3 7 5
ð10Þ
and Dp ðxÞ ¼ ½Np1 ðxÞ Np2 ðxÞ Mp1 ðxÞ þ Mp2 ðxÞ HNp1 ðxÞT ;
ð11Þ
where e ks ðxÞ is the secant stiffness matrix of the cross-section, Dp ðxÞ is the residual plastic force vector for the section and dðxÞ ¼ ½er1 ðxÞ er2 ðxÞ kðxÞT is the section generalized strain vector. The flexibility matrix of the section e f s ðxÞ is obtained by inverting the section stiffness matrix, and so Eq. (9) can be rearranged as dðxÞ ¼ e f s ðxÞfDðxÞ Dp ðxÞg:
ð12Þ
Adopting the small strain assumption within Navier-Bernoulli beam theory and applying the principle of virtual work for the simply supported configuration AB (Fig. 1b), leads to the following compatibility equation Z l T q¼ b ðxÞ dðxÞ dx: ð13aÞ 0
2
1 6 0 bðxÞ ¼ 4 0
0
0
1
0
0
x=l 1
0
3
0 7 5
ð13bÞ
x=l
3. Rigid body motion and corresponding transformation The flexibility formulation presented in the previous section was derived in the element reference system which was restrained. Thus, a transformation is required to relate the deformations and corresponding force vectors in the restrained system (without rigid body modes) to the unrestrained system (with rigid body modes). By adopting the small strain-displacement theory, neglecting second order effects and by satisfying equilibrium, the transformation between the force vectors of the restrained and unrestrained systems is established as (Fig. 1) Q ¼ TT Q þQ s ; in which 2 1 0 6 60 1 T¼6 60 0 4 0 0
ð21Þ
0 0
0 0
1 0
0 1
0 0
1=l
1
0
0
1=l
1=l
0
0
0
1=l
ð22Þ
and Q s ¼ ½0 0 ðH=lÞDb ðlÞ 0 Db ðlÞ Db ðlÞ ðH=lÞDb ðlÞ 0T ;
ð23Þ
where Q s is a vector containing the total nodal forces due to shear Rl slip forces and Db ðlÞ ¼ 0 Db ðsÞ ds. The displacement compatibility requirements lead to the transformation between the displacements vectors of the restrained and unrestrained systems as q ¼ Tq:
If Eqs. (1) and (12) are substituted into Eq. (13a), then
3 0 7 07 7 07 5 1
ð24Þ T
q ¼ e FQ þ q b q p ; in which Z l T e F¼ b ðxÞe f s ðxÞb1 ðxÞ dx;
ð14Þ
TTe KTq ¼ TT Q þ TT Q b TT Q p :
Z
l 0
T
b ðxÞe f s ðxÞDb ðxÞ dx
ð25Þ
T
ð15Þ
By eliminating T Q from Eqs. (21) and (25), the element equilibrium equation including rigid body modes (for the unrestrained beam) is obtained as
ð16Þ
TTe KTq ¼ Q Q s þTT Q b TT Q p :
0
qb ¼
Pre-multiplying Eq. (18) by T and substituting q from Eq. (24) gives
ð26Þ
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4. Shear-slip interpolation along the element axis
6. Numerical examples
With reference to Eq. (1), it can be observed that incorporating partial shear interaction into the formulation necessitates the horizontal slip shear forces, Db ðsÞ, to be prescribed along the element. In existing models, a predefined slip displacement or slip shear force function is adopted for calculating the slip shear stresses, which of course can lead to some level of approximation within the formulation, depending on the adopted predefined function [3,13]. In this paper, a different approach based on piecewise interpolation of the slip strains together with the composite Simpson integration scheme is employed to estimate the slip and corresponding shear forces, which is more straightforward and accurate when compared with other available methods [16]. By adopting the total secant concept, the force-strain relationship of the shear connection can be simplified to Z x eb ðsÞ ds ; s0 ¼ ðq2 q1 Þ Hq4 ð27Þ Db ðxÞ ¼ kb s0 þ
6.1. Analysis of a simply supported beam and propped cantilever
0
where kb is the secant stiffness of the shear connection which is usually known from empirical data [17], and s0 is the slip at the initial node of the element. Using the composite Simpson scheme, the integrals on the right hand side of Eqs. (3), (23) and (27) can be estimated easily for the odd numbered longitudinal sections. For the even numbered sections, a piecewise parabolic interpolation of the slip strain eb ðsÞ (or slip force Db ðsÞ) is used (Fig. 4). A composite Simpson scheme with (2n +1) integration points along the element axis divides the element into n equal sub-elements. It is assumed that the slip strain (slip shear) varies parabolically along each sub-element (Fig. 4). Having the value of slip (slip shear) for the mid and end nodes of the sub-element, a second order function (parabola) is used to interpolate the slip (slip shear) along the sub-element and using this parabola, the integrals in Eqs. (3), (23) and (27) can be calculated analytically for the even numbered sections.
5. Direct iteration solution scheme If the results at the end of the kth load step are available, then values of the strain eðkÞ and the corresponding stress sðkÞ at each integration point (fibre), the secant section flexibility matrix e f sðkÞ ðxÞ together with the vectors DpðkÞ ðxÞ, DbðkÞ ðxÞ, Q pðkÞ , Q bðkÞ and Q sðkÞ , the element secant stiffness matrix e K ðKÞ , and the structure stiffness matrix e KSt: ðkÞ , are available to start the solution for the next load step. The flowchart shown in Fig. 5 describes the solution strategy for the (k+ 1)th load step.
2nd order Parabola εb
Sub-element
3
1
...
5
A εb
εb
εb
2
1
4
εb
3
εb
εb
3
5
4
2n-1
2n+1
Integration B point εb
5
εb
εb
2n-1
2n+1
Fig. 4. Slip strains and composite Simpson integration points along the element.
In the first part of this example, a simply supported beam subjected to a uniformly distributed load of w ¼ 1 kN=m and an axial force of P=50 kN is analysed. The geometry of the member, section details and material properties are given in Fig. 6. Girhammar and Gopu [18] derived an analytical solution for this beam by solving the governing differential equation and adopting linear elastic material behaviour. One half of the beam is modelled by a single flexibility-based element. A composite Simpson scheme with 5 integration points through the depth of each component (i.e. flange and web) and 11 integration points along the element axis is used. The variation of the deflection along the element is shown in Fig. 7, which shows perfect agreement between the closed form analytical solution and the results obtained from the total secant flexibility-based element. Furthermore, the beam is modelled with 4 displacement-based elements which provide deflection response comparable with analytical results and just 1 flexibility-based element. The displacement-based elements take advantage of Hermitian shape functions and linear shape function for slip. In the second part of this example, a propped cantilever with span length of 10 m and subjected to a uniform distributed load of w= 10 kN/m is analysed. The section consists of two rectangular elements. The top element has a width of 600 mm, depth of 300 mm and an elastic modulus E1 =20 GPa while the bottom element has a width of 60 mm, depth of 300 mm and an elastic modulus E2 =200 GPa. The linear modulus for the shear connection along the beam is taken as k= 4.5 MPa. The entire beam is modelled by just a single element with 21 Simpson integration points along the element within the total secant flexibility formulation. The variation of the slip along the element obtained from the flexibility formulation is compared with the analytical results of Ranzi et al. [19] in Fig. 8, which again shows perfect agreement. Furthermore, the result obtained from a displacement-based formulation using 20 elements along the beam is given in Fig. 8 which provides accuracy comparable with analytical results and just 1 flexibility-based element. The displacement-based element employs Hermitian shape functions together with linear shape function for slip.
6.2. Continuous composite beam The 2-bay continuous composite beam (CTB4) from the experiments reported by Ansourian [20] is analysed in this section. The geometry of the beam, loading and section details are shown in Fig. 9a. The material properties are: fyp ¼ 237 MPa (yield stress of steel profile), fyr ¼ 430 MPa (yield stress of slab reinforcement), Es ¼ 200 GPa (steel initial modulus of elasticity), fcp ¼ 27:5 MPa (concrete compressive strength) and ft ¼ 1:7 MPa (concrete tensile strength). For the tensile concrete, a linear elastic brittle failure model was assumed and for concrete under compression, the CEB-FIP Model Code 1990 [21], with a linear softening branch, is adopted. For the shear studs, an exponential shear-slip model is employed [22]. A damage model for unloading/reloading cycles for the concrete is adopted (Fig. 9b). The strain corresponding to the concrete compressive strength is ec0 ¼ 0:002 and the ultimate strain of the concrete is taken as ecu ¼ 0:01. The steel behaviour is assumed to be linear elastic-perfectly plastic-hardening, with a strain hardening modulus of Esh ¼ 2000 MPa and an unloading/ reloading modulus equal to initial elastic modulus.
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Fig. 5. Flowchart of the direct iterative solution scheme for the sample (k+ 1)th load step within the total secant stiffness approach.
l= 4 m
300 mm 50 mm 150 mm
Elastic Modulus (MPa) Flange Web Shear connection 12000 8000 50
50 mm
Fig. 6. Geometry, section details and material properties of the composite beam analysed by Girhammar and Gopu [18].
One half of the beam is modelled by a single flexibility-based element with 15 integration points along the element. The slab depth is divided to 11 layers and the steel profile flange and web are divided to 3 and 11 layers, respectively. The load versus mid-span deflection and the load versus curvature of the beam are shown in Fig. 10. It can be seen that the flexibility formulation developed in this paper can capture the response reasonably well with just one element, whereas available stiffness-based methods require many more elements to capture the experimental response with comparable accuracy [2]. The ultimate loading capacity
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971
Coordinate along the beam (m) 0
0.8
1.6
2.4
3.2
600
4
-4
-8
Load (kN)
Deflection (mm)
0
Analytical solution [18] Present flexibility formulation Displacement-based formulation
-12
400 Experiment [20] Present flexibility formulation
200
Fig. 7. Variation of the deflection along the simply supported composite beam subjected to uniformly distributed load [18].
0 0
10
20
1
30 40 Deflection (mm)
50
60
0 0
2
4
6
8
10
Analytical solution [19] Present flexibility formulation Displacement-based formulation
-1
Load (kN)
Slip (mm)
600
Coordinate along the beam (m)
-2
400 Experiment- mid span [20] Present flexibility formulation- mid span
200
Experiment- middle support [20]
Fig. 8. Variation of slip along the composite propped cantilever subjected to uniformly distributed load [19].
Present flexibility formulation- middle support 0
P/2
P/2
RC Slab
0
20
40
60
80
100
Curvature x 10-6 (1/mm) Steel Profile l= 4 m
Fig. 10. (a) Load versus mid-span deflection and (b) load versus curvature of beam CTB4 from Ansourian’s test [20].
l= 4 m 800 mm 100 mm
φ10 @100 mm
longitudinal reinforcement (mm2) 10 mm Middle support Mid-span
Top 804
Bot 767
Top 160
19 mm x 75 mm Nelson studs-84 mm
6.5 mm
Bot 160
IPBL 200 200 mm Beam Section σ ft
ε cu
ε c0
ε
ing ng ad Lo loadi Un Linear Softening
CEB-FIP Model Code 1990 [21] fcp
Fig. 9. (a) Geometry, loading and section details of continuous beam CTB4 from Ansourian’s test [20] and (b) outline of the unloading/reloading within the adopted damage model for concrete.
of the structure predicted by the numerical analysis is P= 531 kN, which is quite close to P= 518 kN reported in the experimental work.
6.3. Behaviour of a composite beam in the hogging region The simply supported composite beam (CB2) from the experiments reported by Loh et al. [23] is analysed in this section. The geometry of the beam, loading and section details are shown in Fig. 11 and more details of test set up can be found in Loh et al. [23]. The material properties are: fyf ¼ 345 MPa (yield stress of the steel profile flange), fyw ¼ 400 MPa (yield stress of the steel profile web), fyr ¼ 500 MPa (yield stress of the slab reinforcement), Es ¼ 200 GPa (steel initial modulus of elasticity), fcp ¼ 26:2 MPa and ft ¼ 2:0 MPa. For the tensile concrete, a linear elastic brittle failure model was assumed and for concrete under compression, the CEB-FIP Model Code 1990 [21] with a linear softening branch is adopted. For the shear studs, an exponential shear-slip model with a yield load of 100 kN/shear stud is employed [22]. A damage model for unloading/reloading cycles for the concrete is adopted. The strain corresponding to the concrete compressive strength is ec0 ¼ 0:002, and the ultimate strain of the concrete is taken as ecu ¼ 0:01. The steel behaviour is assumed to be linear elastic-perfectly plastichardening, with a strain hardening modulus of Esh ¼ 750 MPa and an unloading/reloading modulus equal to initial elastic modulus. One half of the beam is modelled by a single flexibility-based element with 5 integration points along the element. The slab depth is divided to 12 layers and the steel profile flange and web are divided to 3 and 11 layers, respectively. The load versus deflection and the bending moment versus curvature of beam at mid-span are shown in Fig. 12. It can be seen
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P
19 mm x 100 mm shear stud@480 mm
250UB 25.7
Steel Profile 120 mm
RC Slab l= 2.5 m
515 mm
0.125 m
0.125 m
6 φ 16
φ 12
Beam Section
Fig. 11. Geometry, loading and section details of simple beam CB2 from test of Loh [23].
which demonstrates the superior efficiency of the formulation. Further, the ultimate capacity of members can be captured reasonably well with the approach.
Load (kN)
600
400 References
200
Experiment [23] Present flexibility formulation
0 0
20
40 60 80 Deflection (mm)
100
120
Bending Moment (kN.m)
400 300 200 Experiment-mid span [23]
100
Present flexibility formulation-mid span 0 0
20
40
60
80
100
Curvature x 10-6 (1/mm) Fig. 12. (a) Load versus deflection and (b) bending moment versus curvature at mid-span of beam CB2 from test of Loh [23].
that the global response as well as the ultimate loading capacity of the beam predicted by the flexibility formulation, correlate reasonably well with the experimental data.
7. Conclusions A novel flexibility-based element in the framework of the total secant approach has been derived for 1D composite elements with partial shear interaction, and the corresponding direct iterative solution scheme was presented. The formulation takes account of material nonlinearities and preserves the continuity of shear slip, without using a predefined force or displacement shape function. A composite Simpson integration scheme with a piecewise parabolic interpolation of the slip strains was employed to calculate the slip shear forces along the element. The new formulation takes advantage of a total secant stiffness approach, which typically offers better numerical stability compared with methods based on tangent stiffness approaches, and it can easily handle material and shear connection constitutive models with horizontal plateaux. It was shown that the new formulation with only one element can provide accurate results for elements under service load,
[1] A. Heidarpour, M.A. Bradford, Generic non-linear modelling of a bi-material composite beam with partial shear interaction, International Journal of NonLinear Mechanics 44 (3) (2009) 290–297. [2] G. Ranzi, M.A. Bradford, Direct stiffness analysis of a composite beam-column element with partial interaction, Computers and Structures 85 (15–16) (2007) 1206–1214. [3] M.R. Salari, E. Spacone, P.B. Shing, D.M. Frangopol, Nonlinear analysis of composite beams with deformable shear connectors, Journal of Structural Engineering 124 (10) (1998) 1148–1158. [4] C. Faella, E. Martinelli, E. Nigro, Shear connection nonlinearity and deflections of steel-concrete composite beams: a simplified method, Journal of Structural Engineering 129 (1) (2003) 12–20. [5] E. Spacone, S. El-Tawil, Nonlinear analysis of steel-concrete composite structures: state of the art, Journal of Structural Engineering 130 (2) (2004) 159–168. [6] A. Dall’asta, A. Zona, Non-linear analysis of composite beams by a displacement approach, Computers and Structures 80 (27–30) (2002) 2217–2228. [7] Y.-L. Pi, M.A. Bradford, B. Uy, Second order nonlinear inelastic analysis of composite steel-concrete members. I: theory, Journal of Structural Engineering 132 (5) (2006) 751–761. [8] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, Vol. 1: The BasisButterworth-Heinemann, Oxford, London, 2000. [9] I. Carol, J. Murcia, Nonlinear time-dependent analysis of planar frames using an ‘exact’ formulation. I. Theory, Computers and Structures 33 (1) (1989) 79–87. [10] M.R. Salari, E. Spacone, Analysis of steel-concrete composite frames with bond-slip, Journal of Structural Engineering 127 (11) (2001) 1243–1250. [11] B.J. Daniels, M. Crisinel, Composite slab behavior and strength analysis. Part I: calculation procedure, ASCE, Journal of Structural Engineering 119 (1) (1993) 16–35. [12] A. Ayoub, A force-based model for composite steel-Concrete beams with partial interaction, Journal of Constructional Steel Research 61 (3) (2005) 387–414. [13] A. Ayoub, F.C. Filippou, Mixed formulation of nonlinear steel-concrete composite beam element, Journal of Structural Engineering 126 (3) (2000) 371–381. [14] H.R. Valipour, S.J. Foster, Non-local damage formulation for a flexibility-based frame element, ASCE, Journal of Structural Engineering 135 (10) (2009) 1213–1221. [15] H.R. Valipour, S.J. Foster, Nonlinear dynamic analysis of reinforced concrete frames subjected to ground motion, 20th Australasian Conference on Mechanics of Structures and Materials, ACMSM 2008, 2–5 Dec., Toowoomba, Queensland, 2008, 153–159. [16] H.R. Valipour, S.J. Foster, A total secant flexibility-based formulation for frame elements with physical and geometrical nonlinearities. Finite Elements in Analysis and Design 2008; (submitted for publication). [17] D.J. Oehlers, M.A. Bradford, Steel and Concrete Composite Structural Members: Fundamental Behaviour, Pergamon Press, Oxford, 1995. [18] U.A. Girhammar, V.K.A. Gopu, Composite beam-columns with interlayer slipexact analysis, Journal of Structural Engineering New York, N.Y 119 (4) (1993) 1265–1282. [19] G. Ranzi, M.A. Bradford, B. Uy, A direct stiffness analysis of a composite beam with partial interaction, International Journal for Numerical Methods in Engineering 61 (5) (2004) 657–672. [20] P. Ansourian, Experiments on continuous composite beams, part 2, Proc Ins. Eng., 1981. [21] CEB-FIP model code 1990: Design code (1993). Thomas Telford, London. [22] J.G. Ollgaard, R.G. Slutter, J.W. Fisher, Shear strength of stud connectors in lightweight and normalweight concrete, 8 (2) (1971) 55–64. [23] H.Y. Loh, B. Uy, M.A. Bradford, The effects of partial shear connection in te hogging moment regions of composite beams; Part I: Experimental study, Journal of Constructional Steel Research 60 (2004) 897–919.
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.de/locate/finel
A steel-concrete composite beam element with material nonlinearities and partial shear interaction Hamid R. Valipour , Mark A. Bradford Centre for Infrastructure Engineering and Safety (CIES), School of Civil and Environmental Engineering, The University of New South Wales, Sydney, 2052, Australia
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 June 2009 Received in revised form 21 September 2009 Accepted 28 September 2009
This paper presents the formulation of a novel force-based 1D steel-concrete composite element that captures material nonlinearities and partial shear interaction between the steel profile and the reinforced concrete slab. By decomposing the material total strain into its elastic and inelastic components, a total secant solution strategy based on a direct iterative scheme is introduced and the corresponding solution strategy is outlined. A composite Simpson integration scheme, together with piecewise interpolation of the slip strain along the element axis, is employed to calculate the slip forces along the element axis consistently. The accuracy and efficiency of the formulation are verified by some numerical examples reported elsewhere in the literature, and it is shown that the formulation with just one element can lead to virtually closed form analytical results as long as the integrals in the formulation are calculated accurately. & 2009 Elsevier B.V. All rights reserved.
Keywords: Flexibility formulation Material nonlinearity Secant stiffness Steel-concrete composite
1. Introduction Over the last two decades, the analysis of steel-concrete composite structures has followed two specific themes. For the first of these, researchers have tried to derive analytical solutions of the governing differential equation for generic cases by adopting simple assumptions such as linear elastic material behaviour [1], which have provided simple equations for engineering design purposes that can also be used as benchmark solutions for evaluating numerical methods. For the second, the focus has been on developing finite element models and numerical procedures, which offer better versatility compared with the generic models and which can be used for analysing large steel-concrete composite framed structures including material and geometrical nonlinearities [2–7]. The present paper falls into the second category of these themes. The finite element formulation can be cast within the framework of displacement-based, force-based or mixed (hybrid) methods [8]. Although the displacement and force-based formulations have the same degree of approximation, for frame elements which are the focus of this paper, the force-based formulations lead to superior accuracy when compared with displacement-based formulations because of the exact fulfilment of the equilibrium equations [9,10]. The formulation of displacement-based elements, however, is quite simple and straightfor-
Corresponding author. Tel.: + 61 2 9385 5022; fax: + 61 2 9313 8341.
E-mail address: [email protected] (H.R. Valipour). 0168-874X/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2009.09.011
ward and is usually favoured in comparison with force-based elements. The first composite elements which were developed generally took advantage of transverse cubic displacement shape functions that require mesh refinement to capture the nonlinear response with reasonable accuracy [11]. Formulation of this type, when implemented with linear longitudinal shape functions, can lead to inaccurate results because of curvature locking. Salari et al. proposed a force-based element [3], where cubic interpolation function are used to approximate the bond slip shear forces along the element [3]. The formulation of the element, however, is rather complex and it does not satisfy the kinematic condition of shear slip continuity [3]. Ayoub [12] developed a forced-based composite element, which takes advantage of a linear function for shear slip interpolation. The mixed formulation concept was adopted by Ayoub and Filipou [13], in order to develop an efficient inelastic composite element. In addition, some researchers have tried to solve the simplified form of the governing differential equations, and to cast the results in matrix form, rather than using a conventional method based on predefined force or displacement interpolation functions [2]. A full discourse on different available approaches for formulating 1D steel-concrete composite elements can be found in Spacone and El-Tawil [5]. In the present study, the concept of a total secant approach based on decomposing the strain at cross-section fibres into their elastic and inelastic components is adopted to derive the secant stiffness of the cross-section [14]. Exact force interpolation functions are employed to derive the element secant stiffness, and a direct iteration scheme consistent with this secant
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formulation is then presented. A composite Simpson integration scheme, accompanied with a parabolic piecewise interpolation of the slip (shear) strains along the element, is used to estimate consistently the shear slip forces along the element. Comparison studies illustrate the efficiency of the formulation.
967
from the right hand side of the Eq. (1) does not violate the generality of the formulation, which is adopted to simplify the derivation of the expressions [15]. Equilibrium of the cross-section requires that Z T Z Z Z sx1 dA sx2 dA ðy1 þ HÞsx1 dA y2 sx2 dA ; DðxÞ ¼ O1
O2
O1
O2
ð4Þ
2. Element formulation 2.1. Equilibrium equations Fig. 1a shows a 4-node 1D plane frame element AB, with four degrees of freedom at each end. It is noteworthy that adopting partial horizontal and full vertical interaction between the reinforced concrete slab and the steel profile leads to different horizontal translations for the slab end nodes with respect to steel profile Fig. 1a). Further, the rotational degrees of freedom are assigned to the steel profile end nodes (Fig. 1a). The generalised displacement and force vectors for the unrestrained system are denoted by q and Q, respectively (Fig. 1a), and the corresponding vectors for the restrained system are denoted by q and Q respectively (Fig. 1b). The equilibrium of the configuration Ax, shown in Fig. 1c, yields DðxÞ ¼ b1 ðxÞQ þDb ðxÞ þD ðxÞ; 2
1
6 b1 ðxÞ ¼ 4 0 H
0
0
1
0
0
x=l 1
ð1Þ 0
ð2Þ
and Db ðxÞ ¼
Z
x 0
Db ðsÞ ds
Z 0
x
T Db ðsÞ ds 0 ;
2.2. Compatibility equations Assuming partial shear interaction and adopting the NavierBernoulli hypothesis for the slab and steel profile separately, the compatibility requirement yields
exa ¼ era ya k
ða ¼ 1; 2Þ
ð5Þ
and
eb ¼ ðer2 er1 Þ Hk;
3
0 7 5 x=l
where O1 and O2 denote the reinforced concrete slab and steel profile domains, respectively, y1 and y2 are the distances of the fibres (transverse integration points) from the mid-plane of the concrete slab and steel components respectively (Fig. 2) and sx1 and sx2 are the total longitudinal stress components at the monitoring points in the concrete and steel respectively.
ð3Þ
where DðxÞ ¼ ½Nc ðxÞ Ns ðxÞ MðxÞT is the vector of the section generalised forces, b1 ðxÞ is the force interpolation matrix, Q ¼ ½Q 1 Q 2 Q 3 Q 4 T is the nodal generalised force vector for the restrained configuration, Db ðxÞ is the vector of section generalised force due to shear slip forces, H denotes the distance between reference axis of the reinforced concrete slab and the steel profile and D ðxÞ is a vector of the sectional internal forces and is solely produced by member loads. Removing the term D ðxÞ
ð6Þ
where exa denotes the total longitudinal strain component at the monitoring points (fibres), era is the axial strain at the mid-plane of the material domain Oa , k denotes the total curvature of the cross-section, eb is the slip strain and subscript a denotes the component of a composite section; (a = 1 for the concrete slab and a = 2 for the steel profile). 2.3. Constitutive law in total strain form Using a total strain constitutive law and decomposing the total strain, exa into its elastic ðeexa Þ and inelastic, ðepxa Þ components, the total stress and strain in the longitudinal direction are related by
sxa ¼ Eea ðexa epxa Þ
ða ¼ 1; 2Þ;
ð7Þ
where Eea is the elastic secant modulus of the theoretical unloading curve and is a function of the stress and strain components at the integration point under consideration. If a uni-axial constitutive law is adopted, the material total secant modulus ðEe Þ and inelastic strain component, ðepx Þ can be calculated directly (Fig. 3), whereas for a multi-axial constitutive law an iterative procedure is required to determine the material state [14]. Introducing the identities Z Eea dA; ð8aÞ ka11 ¼ Oa
y1 hc
RC slab mid-plane H
y2 hs
εr1
Steel profile mid-plane
εb ε b : Slip strain ε r2 κ
Fig. 1. (a) Unrestrained 2-node frame element AB in x–y plane (b) restrained simply supported configuration and (c) free body diagram of Ax.
Fig. 2. Steel-concrete composite section and strain distribution with partial shear interaction.
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σ
and
εe qp ¼
L oa d Unl ing o ad i ng
σ
E e=
εp
ε
σ ( ε ε p)
ka22 ¼
Z
ε
ya Eea dA;
ð8bÞ
Oa
Z Oa
Z
ð8cÞ
Z
ð17Þ
e
Kq ¼ Q þQ b Q p
ð18Þ
Q p ¼ e Kq p
ð19Þ
and
Eea epxa dA;
ð8dÞ
Oa
Mpa ðxÞ ¼
T
b ðxÞe f s ðxÞDp ðxÞ dx;
0
in which
y2a Eea dA;
Npa ðxÞ ¼
l
where q is the generalised deformation vector for the restrained beam, e F is the flexibility matrix of the restrained element, q p represents the vector of the nodal generalised plastic deformations and q b is a vector of nodal generalised deformations due to section shear slip between slab and steel profile along the element for the restrained beam. If Q p and Q b represent the generalised force vectors conjugate to the deformation vectors q p and q b , respectively, Eq. (14) degenerates to
Fig. 3. Total secant concept within a uni-axial plastic-damage model.
ka12 ¼
Z
ya Eea epxa dA;
ð8eÞ
Oa
Q b ¼ eK qb
ð20Þ
where e K is the stiffness matrix of the restrained element and is obtained by inverting e F.
and then substituting Eqs. (5) and (7) into Eq. (4) gives DðxÞ ¼ e ks ðxÞdðxÞ þDp ðxÞ; in which 2 e
6 ks ðxÞ ¼ 4
ð9Þ
k111
0
k112
0
k211
k212
k112 Hk111
k212
k222 þ k122 Hk112
3 7 5
ð10Þ
and Dp ðxÞ ¼ ½Np1 ðxÞ Np2 ðxÞ Mp1 ðxÞ þ Mp2 ðxÞ HNp1 ðxÞT ;
ð11Þ
where e ks ðxÞ is the secant stiffness matrix of the cross-section, Dp ðxÞ is the residual plastic force vector for the section and dðxÞ ¼ ½er1 ðxÞ er2 ðxÞ kðxÞT is the section generalized strain vector. The flexibility matrix of the section e f s ðxÞ is obtained by inverting the section stiffness matrix, and so Eq. (9) can be rearranged as dðxÞ ¼ e f s ðxÞfDðxÞ Dp ðxÞg:
ð12Þ
Adopting the small strain assumption within Navier-Bernoulli beam theory and applying the principle of virtual work for the simply supported configuration AB (Fig. 1b), leads to the following compatibility equation Z l T q¼ b ðxÞ dðxÞ dx: ð13aÞ 0
2
1 6 0 bðxÞ ¼ 4 0
0
0
1
0
0
x=l 1
0
3
0 7 5
ð13bÞ
x=l
3. Rigid body motion and corresponding transformation The flexibility formulation presented in the previous section was derived in the element reference system which was restrained. Thus, a transformation is required to relate the deformations and corresponding force vectors in the restrained system (without rigid body modes) to the unrestrained system (with rigid body modes). By adopting the small strain-displacement theory, neglecting second order effects and by satisfying equilibrium, the transformation between the force vectors of the restrained and unrestrained systems is established as (Fig. 1) Q ¼ TT Q þQ s ; in which 2 1 0 6 60 1 T¼6 60 0 4 0 0
ð21Þ
0 0
0 0
1 0
0 1
0 0
1=l
1
0
0
1=l
1=l
0
0
0
1=l
ð22Þ
and Q s ¼ ½0 0 ðH=lÞDb ðlÞ 0 Db ðlÞ Db ðlÞ ðH=lÞDb ðlÞ 0T ;
ð23Þ
where Q s is a vector containing the total nodal forces due to shear Rl slip forces and Db ðlÞ ¼ 0 Db ðsÞ ds. The displacement compatibility requirements lead to the transformation between the displacements vectors of the restrained and unrestrained systems as q ¼ Tq:
If Eqs. (1) and (12) are substituted into Eq. (13a), then
3 0 7 07 7 07 5 1
ð24Þ T
q ¼ e FQ þ q b q p ; in which Z l T e F¼ b ðxÞe f s ðxÞb1 ðxÞ dx;
ð14Þ
TTe KTq ¼ TT Q þ TT Q b TT Q p :
Z
l 0
T
b ðxÞe f s ðxÞDb ðxÞ dx
ð25Þ
T
ð15Þ
By eliminating T Q from Eqs. (21) and (25), the element equilibrium equation including rigid body modes (for the unrestrained beam) is obtained as
ð16Þ
TTe KTq ¼ Q Q s þTT Q b TT Q p :
0
qb ¼
Pre-multiplying Eq. (18) by T and substituting q from Eq. (24) gives
ð26Þ
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969
4. Shear-slip interpolation along the element axis
6. Numerical examples
With reference to Eq. (1), it can be observed that incorporating partial shear interaction into the formulation necessitates the horizontal slip shear forces, Db ðsÞ, to be prescribed along the element. In existing models, a predefined slip displacement or slip shear force function is adopted for calculating the slip shear stresses, which of course can lead to some level of approximation within the formulation, depending on the adopted predefined function [3,13]. In this paper, a different approach based on piecewise interpolation of the slip strains together with the composite Simpson integration scheme is employed to estimate the slip and corresponding shear forces, which is more straightforward and accurate when compared with other available methods [16]. By adopting the total secant concept, the force-strain relationship of the shear connection can be simplified to Z x eb ðsÞ ds ; s0 ¼ ðq2 q1 Þ Hq4 ð27Þ Db ðxÞ ¼ kb s0 þ
6.1. Analysis of a simply supported beam and propped cantilever
0
where kb is the secant stiffness of the shear connection which is usually known from empirical data [17], and s0 is the slip at the initial node of the element. Using the composite Simpson scheme, the integrals on the right hand side of Eqs. (3), (23) and (27) can be estimated easily for the odd numbered longitudinal sections. For the even numbered sections, a piecewise parabolic interpolation of the slip strain eb ðsÞ (or slip force Db ðsÞ) is used (Fig. 4). A composite Simpson scheme with (2n +1) integration points along the element axis divides the element into n equal sub-elements. It is assumed that the slip strain (slip shear) varies parabolically along each sub-element (Fig. 4). Having the value of slip (slip shear) for the mid and end nodes of the sub-element, a second order function (parabola) is used to interpolate the slip (slip shear) along the sub-element and using this parabola, the integrals in Eqs. (3), (23) and (27) can be calculated analytically for the even numbered sections.
5. Direct iteration solution scheme If the results at the end of the kth load step are available, then values of the strain eðkÞ and the corresponding stress sðkÞ at each integration point (fibre), the secant section flexibility matrix e f sðkÞ ðxÞ together with the vectors DpðkÞ ðxÞ, DbðkÞ ðxÞ, Q pðkÞ , Q bðkÞ and Q sðkÞ , the element secant stiffness matrix e K ðKÞ , and the structure stiffness matrix e KSt: ðkÞ , are available to start the solution for the next load step. The flowchart shown in Fig. 5 describes the solution strategy for the (k+ 1)th load step.
2nd order Parabola εb
Sub-element
3
1
...
5
A εb
εb
εb
2
1
4
εb
3
εb
εb
3
5
4
2n-1
2n+1
Integration B point εb
5
εb
εb
2n-1
2n+1
Fig. 4. Slip strains and composite Simpson integration points along the element.
In the first part of this example, a simply supported beam subjected to a uniformly distributed load of w ¼ 1 kN=m and an axial force of P=50 kN is analysed. The geometry of the member, section details and material properties are given in Fig. 6. Girhammar and Gopu [18] derived an analytical solution for this beam by solving the governing differential equation and adopting linear elastic material behaviour. One half of the beam is modelled by a single flexibility-based element. A composite Simpson scheme with 5 integration points through the depth of each component (i.e. flange and web) and 11 integration points along the element axis is used. The variation of the deflection along the element is shown in Fig. 7, which shows perfect agreement between the closed form analytical solution and the results obtained from the total secant flexibility-based element. Furthermore, the beam is modelled with 4 displacement-based elements which provide deflection response comparable with analytical results and just 1 flexibility-based element. The displacement-based elements take advantage of Hermitian shape functions and linear shape function for slip. In the second part of this example, a propped cantilever with span length of 10 m and subjected to a uniform distributed load of w= 10 kN/m is analysed. The section consists of two rectangular elements. The top element has a width of 600 mm, depth of 300 mm and an elastic modulus E1 =20 GPa while the bottom element has a width of 60 mm, depth of 300 mm and an elastic modulus E2 =200 GPa. The linear modulus for the shear connection along the beam is taken as k= 4.5 MPa. The entire beam is modelled by just a single element with 21 Simpson integration points along the element within the total secant flexibility formulation. The variation of the slip along the element obtained from the flexibility formulation is compared with the analytical results of Ranzi et al. [19] in Fig. 8, which again shows perfect agreement. Furthermore, the result obtained from a displacement-based formulation using 20 elements along the beam is given in Fig. 8 which provides accuracy comparable with analytical results and just 1 flexibility-based element. The displacement-based element employs Hermitian shape functions together with linear shape function for slip.
6.2. Continuous composite beam The 2-bay continuous composite beam (CTB4) from the experiments reported by Ansourian [20] is analysed in this section. The geometry of the beam, loading and section details are shown in Fig. 9a. The material properties are: fyp ¼ 237 MPa (yield stress of steel profile), fyr ¼ 430 MPa (yield stress of slab reinforcement), Es ¼ 200 GPa (steel initial modulus of elasticity), fcp ¼ 27:5 MPa (concrete compressive strength) and ft ¼ 1:7 MPa (concrete tensile strength). For the tensile concrete, a linear elastic brittle failure model was assumed and for concrete under compression, the CEB-FIP Model Code 1990 [21], with a linear softening branch, is adopted. For the shear studs, an exponential shear-slip model is employed [22]. A damage model for unloading/reloading cycles for the concrete is adopted (Fig. 9b). The strain corresponding to the concrete compressive strength is ec0 ¼ 0:002 and the ultimate strain of the concrete is taken as ecu ¼ 0:01. The steel behaviour is assumed to be linear elastic-perfectly plastic-hardening, with a strain hardening modulus of Esh ¼ 2000 MPa and an unloading/ reloading modulus equal to initial elastic modulus.
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Fig. 5. Flowchart of the direct iterative solution scheme for the sample (k+ 1)th load step within the total secant stiffness approach.
l= 4 m
300 mm 50 mm 150 mm
Elastic Modulus (MPa) Flange Web Shear connection 12000 8000 50
50 mm
Fig. 6. Geometry, section details and material properties of the composite beam analysed by Girhammar and Gopu [18].
One half of the beam is modelled by a single flexibility-based element with 15 integration points along the element. The slab depth is divided to 11 layers and the steel profile flange and web are divided to 3 and 11 layers, respectively. The load versus mid-span deflection and the load versus curvature of the beam are shown in Fig. 10. It can be seen that the flexibility formulation developed in this paper can capture the response reasonably well with just one element, whereas available stiffness-based methods require many more elements to capture the experimental response with comparable accuracy [2]. The ultimate loading capacity
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971
Coordinate along the beam (m) 0
0.8
1.6
2.4
3.2
600
4
-4
-8
Load (kN)
Deflection (mm)
0
Analytical solution [18] Present flexibility formulation Displacement-based formulation
-12
400 Experiment [20] Present flexibility formulation
200
Fig. 7. Variation of the deflection along the simply supported composite beam subjected to uniformly distributed load [18].
0 0
10
20
1
30 40 Deflection (mm)
50
60
0 0
2
4
6
8
10
Analytical solution [19] Present flexibility formulation Displacement-based formulation
-1
Load (kN)
Slip (mm)
600
Coordinate along the beam (m)
-2
400 Experiment- mid span [20] Present flexibility formulation- mid span
200
Experiment- middle support [20]
Fig. 8. Variation of slip along the composite propped cantilever subjected to uniformly distributed load [19].
Present flexibility formulation- middle support 0
P/2
P/2
RC Slab
0
20
40
60
80
100
Curvature x 10-6 (1/mm) Steel Profile l= 4 m
Fig. 10. (a) Load versus mid-span deflection and (b) load versus curvature of beam CTB4 from Ansourian’s test [20].
l= 4 m 800 mm 100 mm
φ10 @100 mm
longitudinal reinforcement (mm2) 10 mm Middle support Mid-span
Top 804
Bot 767
Top 160
19 mm x 75 mm Nelson studs-84 mm
6.5 mm
Bot 160
IPBL 200 200 mm Beam Section σ ft
ε cu
ε c0
ε
ing ng ad Lo loadi Un Linear Softening
CEB-FIP Model Code 1990 [21] fcp
Fig. 9. (a) Geometry, loading and section details of continuous beam CTB4 from Ansourian’s test [20] and (b) outline of the unloading/reloading within the adopted damage model for concrete.
of the structure predicted by the numerical analysis is P= 531 kN, which is quite close to P= 518 kN reported in the experimental work.
6.3. Behaviour of a composite beam in the hogging region The simply supported composite beam (CB2) from the experiments reported by Loh et al. [23] is analysed in this section. The geometry of the beam, loading and section details are shown in Fig. 11 and more details of test set up can be found in Loh et al. [23]. The material properties are: fyf ¼ 345 MPa (yield stress of the steel profile flange), fyw ¼ 400 MPa (yield stress of the steel profile web), fyr ¼ 500 MPa (yield stress of the slab reinforcement), Es ¼ 200 GPa (steel initial modulus of elasticity), fcp ¼ 26:2 MPa and ft ¼ 2:0 MPa. For the tensile concrete, a linear elastic brittle failure model was assumed and for concrete under compression, the CEB-FIP Model Code 1990 [21] with a linear softening branch is adopted. For the shear studs, an exponential shear-slip model with a yield load of 100 kN/shear stud is employed [22]. A damage model for unloading/reloading cycles for the concrete is adopted. The strain corresponding to the concrete compressive strength is ec0 ¼ 0:002, and the ultimate strain of the concrete is taken as ecu ¼ 0:01. The steel behaviour is assumed to be linear elastic-perfectly plastichardening, with a strain hardening modulus of Esh ¼ 750 MPa and an unloading/reloading modulus equal to initial elastic modulus. One half of the beam is modelled by a single flexibility-based element with 5 integration points along the element. The slab depth is divided to 12 layers and the steel profile flange and web are divided to 3 and 11 layers, respectively. The load versus deflection and the bending moment versus curvature of beam at mid-span are shown in Fig. 12. It can be seen
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P
19 mm x 100 mm shear stud@480 mm
250UB 25.7
Steel Profile 120 mm
RC Slab l= 2.5 m
515 mm
0.125 m
0.125 m
6 φ 16
φ 12
Beam Section
Fig. 11. Geometry, loading and section details of simple beam CB2 from test of Loh [23].
which demonstrates the superior efficiency of the formulation. Further, the ultimate capacity of members can be captured reasonably well with the approach.
Load (kN)
600
400 References
200
Experiment [23] Present flexibility formulation
0 0
20
40 60 80 Deflection (mm)
100
120
Bending Moment (kN.m)
400 300 200 Experiment-mid span [23]
100
Present flexibility formulation-mid span 0 0
20
40
60
80
100
Curvature x 10-6 (1/mm) Fig. 12. (a) Load versus deflection and (b) bending moment versus curvature at mid-span of beam CB2 from test of Loh [23].
that the global response as well as the ultimate loading capacity of the beam predicted by the flexibility formulation, correlate reasonably well with the experimental data.
7. Conclusions A novel flexibility-based element in the framework of the total secant approach has been derived for 1D composite elements with partial shear interaction, and the corresponding direct iterative solution scheme was presented. The formulation takes account of material nonlinearities and preserves the continuity of shear slip, without using a predefined force or displacement shape function. A composite Simpson integration scheme with a piecewise parabolic interpolation of the slip strains was employed to calculate the slip shear forces along the element. The new formulation takes advantage of a total secant stiffness approach, which typically offers better numerical stability compared with methods based on tangent stiffness approaches, and it can easily handle material and shear connection constitutive models with horizontal plateaux. It was shown that the new formulation with only one element can provide accurate results for elements under service load,
[1] A. Heidarpour, M.A. Bradford, Generic non-linear modelling of a bi-material composite beam with partial shear interaction, International Journal of NonLinear Mechanics 44 (3) (2009) 290–297. [2] G. Ranzi, M.A. Bradford, Direct stiffness analysis of a composite beam-column element with partial interaction, Computers and Structures 85 (15–16) (2007) 1206–1214. [3] M.R. Salari, E. Spacone, P.B. Shing, D.M. Frangopol, Nonlinear analysis of composite beams with deformable shear connectors, Journal of Structural Engineering 124 (10) (1998) 1148–1158. [4] C. Faella, E. Martinelli, E. Nigro, Shear connection nonlinearity and deflections of steel-concrete composite beams: a simplified method, Journal of Structural Engineering 129 (1) (2003) 12–20. [5] E. Spacone, S. El-Tawil, Nonlinear analysis of steel-concrete composite structures: state of the art, Journal of Structural Engineering 130 (2) (2004) 159–168. [6] A. Dall’asta, A. Zona, Non-linear analysis of composite beams by a displacement approach, Computers and Structures 80 (27–30) (2002) 2217–2228. [7] Y.-L. Pi, M.A. Bradford, B. Uy, Second order nonlinear inelastic analysis of composite steel-concrete members. I: theory, Journal of Structural Engineering 132 (5) (2006) 751–761. [8] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, Vol. 1: The BasisButterworth-Heinemann, Oxford, London, 2000. [9] I. Carol, J. Murcia, Nonlinear time-dependent analysis of planar frames using an ‘exact’ formulation. I. Theory, Computers and Structures 33 (1) (1989) 79–87. [10] M.R. Salari, E. Spacone, Analysis of steel-concrete composite frames with bond-slip, Journal of Structural Engineering 127 (11) (2001) 1243–1250. [11] B.J. Daniels, M. Crisinel, Composite slab behavior and strength analysis. Part I: calculation procedure, ASCE, Journal of Structural Engineering 119 (1) (1993) 16–35. [12] A. Ayoub, A force-based model for composite steel-Concrete beams with partial interaction, Journal of Constructional Steel Research 61 (3) (2005) 387–414. [13] A. Ayoub, F.C. Filippou, Mixed formulation of nonlinear steel-concrete composite beam element, Journal of Structural Engineering 126 (3) (2000) 371–381. [14] H.R. Valipour, S.J. Foster, Non-local damage formulation for a flexibility-based frame element, ASCE, Journal of Structural Engineering 135 (10) (2009) 1213–1221. [15] H.R. Valipour, S.J. Foster, Nonlinear dynamic analysis of reinforced concrete frames subjected to ground motion, 20th Australasian Conference on Mechanics of Structures and Materials, ACMSM 2008, 2–5 Dec., Toowoomba, Queensland, 2008, 153–159. [16] H.R. Valipour, S.J. Foster, A total secant flexibility-based formulation for frame elements with physical and geometrical nonlinearities. Finite Elements in Analysis and Design 2008; (submitted for publication). [17] D.J. Oehlers, M.A. Bradford, Steel and Concrete Composite Structural Members: Fundamental Behaviour, Pergamon Press, Oxford, 1995. [18] U.A. Girhammar, V.K.A. Gopu, Composite beam-columns with interlayer slipexact analysis, Journal of Structural Engineering New York, N.Y 119 (4) (1993) 1265–1282. [19] G. Ranzi, M.A. Bradford, B. Uy, A direct stiffness analysis of a composite beam with partial interaction, International Journal for Numerical Methods in Engineering 61 (5) (2004) 657–672. [20] P. Ansourian, Experiments on continuous composite beams, part 2, Proc Ins. Eng., 1981. [21] CEB-FIP model code 1990: Design code (1993). Thomas Telford, London. [22] J.G. Ollgaard, R.G. Slutter, J.W. Fisher, Shear strength of stud connectors in lightweight and normalweight concrete, 8 (2) (1971) 55–64. [23] H.Y. Loh, B. Uy, M.A. Bradford, The effects of partial shear connection in te hogging moment regions of composite beams; Part I: Experimental study, Journal of Constructional Steel Research 60 (2004) 897–919.