Direct stiffness analysis of a composite beam-column element with partial interaction

Computers and Structures 85 (2007) 1206–1214 www.elsevier.com/locate/compstruc

Direct stiffness analysis of a composite beam-column element with partial interaction G. Ranzi b

a,*

, M.A. Bradford

b

a School of Civil Engineering, The University of Sydney, Building J05, NSW 2006, Australia Australian Government Federation Fellow, The University of New South Wales, UNSW, Sydney, Australia

Received 23 February 2006; accepted 21 November 2006 Available online 30 January 2007

Abstract This paper presents a stiffness formulation for the analysis of composite steel–concrete beam-columns with partial shear interaction (PI). This formulation is based on the direct stiffness method (DSM). The advantage of the proposed method is that no approximated displacement and/or force fields are introduced in the element derivation, unlike other modelling techniques available in the literature. Some simple structural systems, such as simply supported beams and propped cantilevers, subjected to a point load and to a uniformly distributed load are then considered to validate the accuracy of the results obtained using the proposed formulation against results derived based on closed form solutions; for continuous beams, the results have been validated against those calculated using highly refined mesh of high order finite elements. This has been carried out for different levels of shear connection stiffness to highlight the ability of the proposed method to overcome the curvature locking problems observed in some conventional displacement formulations. The generic applicability of this technique to the analysis of continuous beams is then illustrated, in particular, highlighting its ability to account for material nonlinearities at both service and ultimate conditions. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Beam-column; Composite beams; Direct stiffness method; Partial shear interaction; Stiffness method

1. Introduction Over the last few decades, several researchers have investigated the behaviour of composite steel–concrete beams with partial shear interaction (PI). The seminal work by Newmark et al. [1] represents one of the earliest contributions in the understanding of the partial interaction behaviour of composite beams, and their model is usually referred to as Newmark’s model. Since then, several modelling techniques have been presented which usually require some sort of discretisation in the spatial domain (i.e. along the beam length) to be introduced, such as that for finite element methods and for finite difference methods. It is beyond the scope of this paper to provide a lengthy dis-

*

Corresponding author. Tel.: +61 2 9351 5215; fax: +61 2 9351 3343. E-mail address: [email protected] (G. Ranzi).

0045-7949/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.11.031

course of the current state of the art, and for this purpose reference should be made, among the others, to Spacone and El-Tawil [2] and to Leon and Viest [3]. Despite the fact that the partial interaction behaviour of composite beams has been studied over the last 50 years, there have been some recent interesting numerical contributions still focussing on their linear-elastic behaviour. Despite the problem appearing quite simple at face value, the analysis of steel–concrete composite members with PI is quite complicated. Worthy of mention is the contribution by Faella et al. [4] who, in 2002, presented a stiffness element with 6dof, viz. the vertical displacement, the rotation and the slip at both element ends, where the governing differential equation of the PI problem was expressed and solved with respect to the curvature, while the slip expression was defined in terms of the hyperbolic functions. Ranzi et al. [5] presented another formulation based on the direct stiffness approach for an element with 6dof, i.e.

G. Ranzi, M.A. Bradford / Computers and Structures 85 (2007) 1206–1214

the vertical displacement, the rotation and the slip at both ends and observed that numerical instabilities occur in the calculation of some stiffness coefficients for low values of the dimensionless stiffness parameter aL, as defined by Girhammar and Pan [6], when these coefficients are derived using the exponential functions (or the hyperbolic functions) in the expression for the slip, and they proposed a modelling procedure to avoid such instabilities. The modelling technique proposed in this paper intends to derive an 8dof stiffness element which represents an extension of the 6dof element mentioned previously, in which the freedoms consist of the axial displacement at the level of the reference axis, the vertical displacement, the rotation and the slip at both element ends, and these are depicted in Fig. 1. The main advantage of this technique is that no interpolation in the displacement fields and/or discretisations are introduced along the element length. Hitherto, the inclusion of axial force to produce a robust algorithm incorporating PI has not been reported. Applications of this technique are then demonstrated for simply supported beams and for propped cantilevers subject to an uniformly distributed load and to an axial loading. These cases are also used to validate the accuracy of

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the proposed stiffness formulation against closed form solutions derived by the authors. This has been carried out for different levels of shear connection stiffness. By doing so, it will be demonstrated how the proposed element is capable of overcoming the curvature locking problems that has been observed in some conventional displacement formulations for high connection stiffness, which have been reported by Dall’Asta and Zona [7] to occur for values of the dimensionless stiffness parameter aL > 10: A 2-span continuous composite beam is analysed to demonstrate the ease of use of this method and its results are validated against those obtained by means of a highly refined finite element with 16dof proposed by Dall’Asta and Zona [7]. Finally, the suitability of the proposed technique to describe the behaviour of indeterminate and of continuous beams is highlighted; in particular, it is shown how material nonlinearities can be easily implemented in the modelling taking advantage of the highly refined functions at the basis of the proposed stiffness element. 2. Partial interaction analysis 2.1. General

Element 1 N10

Arbitrary reference axis

N1L

M0

ML N0

NL R0

Element 2

RL

z

y0

unL un 0 vL

v0

v′0 v′L

s0 sL

Fig. 1. Nodal displacements and actions of the 8dof stiffness element.

The generic composite beam considered by the proposed stiffness formulation is composed of a concrete slab, steel reinforcement, a steel joist and a shear connection as shown in Fig. 2. The top and bottom elements are referred to as elements 1 and 2, respectively. The composite cross-section is thus represented as A ¼ A1 [ A2 , where A1 and A2 are the cross-sections of elements 1 and 2, respectively. The area A1 represents the slab and is further sub-divided into Ac and Ar which represent the areas of the concrete component and of the reinforcement respectively ðA1 ¼ Ac [ Ar Þ, while A2 represents the cross-section of the steel joist only and it is denoted as As. The strain diagram is defined uniquely by the strain in the top fibre of the cross-section u00 , the curvature v00 and the slip strain s 0 , where the prime denotes a derivative with respect to the coordinate along the beam strain in the top fibre, u0′

y0

y

arbitrary reference axis (a) composite cross-section

slip strain,

s′

curvature,

v′′

strain at the level of the reference axis, u n′ (b) strain diagram

Fig. 2. Composite cross-section and strain diagram.

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G. Ranzi, M.A. Bradford / Computers and Structures 85 (2007) 1206–1214

z. Implicit in this is the validity of the Euler–Bernoulli hypothesis that plane sections remain plane (except at the interface) and a linearisation of the curvature for which v 0 2 is deemed to vanish [1]. The composite beam is assumed to occupy the prismatic spatial region V ¼ A  ½0; L, where A represents the composite cross-section which is an arbitrary cross-section that is symmetric about the plane of bending, while [0, L] is defined along the beam coordinate z (which is perpendicular to the cross-section at any location along the beam length, and with z 2 ½0; LÞ. For generality, the model is derived with reference to an arbitrary axis located at a distance y0 below the top fibre of the cross-section from which the cross-sectional properties of the beam are defined. As the axial displacement is controlled at the level of the reference axis, it will be assumed, without any loss of generality, that the reference axis is located in the steel joist (i.e. bottom element) as would occur in real beams. All materials are assumed to behave in a linear-elastic fashion. In particular, for the steel joist, the reinforcement and the concrete slab, their linear-elastic material properties can be expressed generically as rc ¼ Er ec ¼ Ec ½u00 þ ðy þ y 0 Þv00 þ s0 dcs 

ð1Þ

where c = c, r, s for the concrete slab, the reinforcement and the steel beam respectively, dcs = 0, dcr = 0 and dss = 1, y is the vertical coordinate from the reference axis, rc, ec and Ec are the generic stress, strain and elastic modulus in the material of domain c. The shear connection is also assumed to behave in a linear-elastic fashion so that q ¼ ks

ð2Þ

where q is the shear flow per unit length (shear flow force), k the shear connection stiffness (with units of force per length2) and s is the slip. 2.2. Analytical model The analytical model utilised to derive the 8dof stiffness element is constructed based on an unknown strain diagram, which requires three parameters to be fully defined. As noted, these parameters are the strain in the top fibre of the cross-section u00 , the curvature v00 and the slip strain s 0 . The three equations utilised to solve the problem are those for horizontal equilibrium at the composite cross-section, rotational equilibrium at the composite cross-section δz

q (z) Fig. 3. Free body diagram of the top element.

ML

z

NL L

R0

RL

Fig. 4. General single span beam.

and horizontal equilibrium of a free body diagram of the top element as shown in Fig. 3. For simplicity and again without any loss of generality, a single span beam is considered as shown in Fig. 4 subjected to a pattern of loading which produces a variation of the bending moment M(z) and of the axial force N(z) (referred to as M and N for simplicity), whose variations are not necessarily known initially if the beam is statically indeterminate. Similarly to Newmark’s model, no vertical separation is assumed to occur between elements 1 and 2 (i.e. the top and bottom elements), so that the curvature is the same in both elements. 2.2.1. Horizontal and rotational equilibrium Horizontal and rotational equilibrium are established by equating the internal actions to the external ones, which are referred to as N and M, as Z Z r dA ¼ N ; M i ¼ yr dA ¼ M ð3a; bÞ Ni ¼ A

A

and Ni and Mi are the internal axial force and moment resisted by the composite cross-section, and r is the generic stress in the composite cross-section. Based on Eq. (3), the unknown curvature and strain in the top fibre of the composite cross-section are expressed in terms of the slip strain. 2.2.2. Horizontal equilibrium of a free body diagram of the top element The slip strain is then obtained by enforcing horizontal equilibrium of a free body diagram of the top element as shown in Fig. 3, which can be written as N 01 þ q ¼ N 01 þ ks ¼ 0

ð4Þ

where N1 is the axial force resisted by the top element. Eq. (4) represents the governing differential equation of the PI problem which can be re-arranged in the following compact form as ~as00  ks ¼ aM 0 þ a1 N 0

N1 +N′1δ z

N1

w(z)

M0 N0

ð5Þ

where all notation is defined in Appendix. Solving Eq. (5) for the slip and slip strain yields the following solutions, which are defined as the sum of a general solution corresponding to the homogeneous differential equation sH and the particular solution sP, as

G. Ranzi, M.A. Bradford / Computers and Structures 85 (2007) 1206–1214

s ¼ sH ðC 1 ; C 2 Þ þ sP s0 ¼ s0H ðC 1 ; C 2 Þ þ s0P

ð6aÞ ð6bÞ

in which C1 and C2 represent the two constants of integration. The actual expression for sP depends upon the applied loading conditions as specified on the right-hand side of Eq. (5). 2.2.3. Modelling The other variables defining the strain diagram, which are the curvature and the strain in the top fibre of the cross-section, can be determined substituting the expressions for the slip and slip strain obtained from Eqs. (6) into Eq. (3) as u00 ¼ b1 M þ b2 N þ b3 s0 00

v ¼ r1 M þ r 2 N þ r3 s

0

ð7Þ ð8Þ

The expressions for the rotation and deflection can then be obtained by integrating the curvature about the coordinate along the beam length z. Hence, Z Z Z b1 v0 ¼ r1 M dz þ r2 N dz þ r3 s0 dz þ C ð9Þ Z Z Z Z M dz dz þ r2 N dz dz v ¼ r1 Z Z ^ 1z þ C b2 s0 dz dz þ C þ r3 ð10Þ and the strain at the level of the reference axis is determined as u0n ¼ l1 M þ l2 N þ l3 s0

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M0

ML

w

N0

z

NL L

R0

RL

Fig. 5. General single span beam subjected to an uniformly distributed load w.

loads to account for member loading are obtained for an uniformly distributed load, as shown in Fig. 5. The same procedure can be utilised to calculate the nodal equivalent loads related to other loading conditions. The analytical formulation utilised is based in Eqs. (6)–(13). The system of equations to be solved based on the DSM method (considering one stiffness element only) can then be expressed as 3 2 k 11 k 12 k 13 k 14 k 15 k 16 k 17 k 18 6 k 22 k 23 k 24 k 25 k 26 k 27 k 28 7 7 6 7 6 6 k 33 k 34 k 35 k 36 k 37 k 38 7 7 6 6 k 44 k 45 k 46 k 47 k 48 7 * 7~ 6 qeq ¼ 6 d q þ~ 7d ¼ K~ 6 k 55 k 56 k 57 k 58 7 7 6 6 k 66 k 67 k 68 7 7 6 7 6 4 k 77 k 78 5 k 88

ð11Þ

By integrating u0n over the beam length, the axial displacement at the level of the reference axis un can be determined as Z Z Z un ¼ l1 M dz þ l2 N dz þ l3 s0 dz þ C u ð12Þ The constants of integration can then be determined utilising the appropriate static and/or kinematic boundary conditions of the composite beam-column analysed. Denoting the generic stress in element 1 as r1, the axial force resisted by element 1 (i.e. the reinforced concrete slab), which is also needed for the direct stiffness technique, is determined as Z N1 ¼ r1 dA ¼ q1 M þ q2 N þ q3 s0 ð13Þ A1

3. The 8dof stiffness element 3.1. General The derivation of the 8dof stiffness element is carried out in this section by means of the direct stiffness method (DSM). This procedure is well described in structural analysis textbooks [8,9]. The stiffness coefficients are derived based on an unloaded element, while the equivalent nodal

ð14Þ in which *

q ¼ ½N 0 ; R0 ; M 0 ; N 10 ; N L ; RL ; M L ; N 1L 

*

d ¼ ½un0 ; v0 ; v00 ; s0 ; unL ; vL ; v0L ; sL  *

T

T

ð15Þ ð16Þ

*

where q and d are the vectors of nodal actions and dis* placements shown in Fig. 1, q eq is the vector of equivalent nodal loads due to an uniformly distributed load w, kij are the stiffness coefficients (i = 1, . . . , 8; j = 1, . . . , 8), and K is the element stiffness matrix. In this definition, the slip may be thought of being conjugate in an abstract sense to the axial force within element 1. All terms (i.e. stiffness coefficients and equivalent nodal loads) are lengthy and are given in [10], while the complete description of their derivation is outlined below. 3.2. Derivation of the stiffness matrix and equivalent nodal loads The stiffness coefficients are derived by means of the direct stiffness approach based on an unloaded stiffness element, for which the variations of the moment and the axial force along the beam can be expressed as M ¼ M 0 þ R0 z;

N ¼ N 0

ð17a; bÞ

G. Ranzi, M.A. Bradford / Computers and Structures 85 (2007) 1206–1214

in which l2 ¼

k ~ a

ð19Þ

where C1 and C2 represent the constant of integrations for the slip expression. In implementing the DSM method, each column of the stiffness matrix is derived by restraining all freedoms except the one related to the column considered, for which a unit displacement is imposed. For example, the first column of the stiffness matrix is obtained enforcing the kinematic state described by *

d ¼ ½1; 0; 0; 0; 0; 0; 0; 0

T

ð20Þ

and which when substituted into Eqs. (6)–(13) allows the constants of integration C1 and C2 (by the use of b 1 and C b 2 (by the use of v0 ¼ vL ¼ 0Þ, Cu s0 ¼ sL ¼ 0Þ, C (by the use of un0 = 1), and the actions N0, R0 and M0 (by the use of unL ¼ v00 ¼ v0L ¼ 0Þ to be obtained, while NL, RL and ML can then be obtained from elementary statics. The values of the axial forces in the top element N10 and N1L at z = 0, L, respectively, are determined from Eq. (13) by the use of N 10 ¼ N 1 ðz ¼ 0Þ and N 1L ¼ N 1 ðz ¼ LÞ. The other columns of the stiffness matrix can be derived by * sequentially imposing the kinematic state defined by T d ¼ ½0; d2j ; d3j ; d4j ; d5j ; d6j ; d7j ; d8j  , where dij is the Kronecker delta, from j = 2 through to j = 8, which allow the five constants of integration and the nodal actions to be obtained at each step, and hence the corresponding entries in the jth column of the stiffness matrix to be determined. The equivalent nodal loads required to model an uniformly distributed load w are derived applying the kine* matic state defined by d ¼ ½0; 0; 0; 0; 0; 0; 0; 0T while the expressions required for the moment and axial force along the element become wz2 M ¼ M 0 þ R0 z  ; 2

N ¼ N 0

ð21a; bÞ

This then produces the following expressions for the slip based on Eq. (6a): a aw z ð22Þ s ¼ C 1 elL þ C 2 elL  R0 þ k k 3.3. Numerical instabilities It was observed that for low values of the dimensionless shear connection stiffness lL, which is equivalent to the dimensionless stiffness aL introduced by Girhammar and Pan [6] and where L is the element length, the stiffness coef-

ficients calculated based on the procedure previously described exhibit some numerical instabilities. The exponential terms in Eq. (18) (viz. in the expression for the slip) are a potential source of numerical instability in the calculation of the stiffness coefficients for low values of the dimensionless stiffness parameter lL. This is caused by the lack of numerical precision introduced by the large terms which arise when the exponentials in lL are inverted within the stiffness relationships and are multiplied with other exponentials in lL, therefore leading to a loss of significant figures in the calculated number stored by the computer. This behaviour was also observed by Ranzi et al. [5]. The normalised slip freedom ks/ks1 on the diagonal of the stiffness matrix is plotted against the dimensionless stiffness lL in Fig. 6 to highlight this behaviour, where ks is the stiffness related to the slip freedom, i.e. K(4, 4) of Eq. (14), while ks1 is equal to the stiffness coefficient K(4, 4) calculated for lL ¼ 1 (the latter coefficient being obtained modifying the value of the shear connection stiffness). It can be seen in Fig. 6, that numerical instabilities appear when lL reaches a small (indeed infinitesimal) value, and reliable results are obtained when lL > 0:05. For lower values of lL (i.e. for lL < 0:05Þ, a modified expression for the slip has been utilised to avoid these numerical instabilities; this has been carried out by replacing the exponentials appearing in the expression for the slip

600

Ratio K (4,4) over [ K (4,4) calculated for μ L =1]

Based on the loading condition of Eq. (17), the solution of the governing differential equation outlined in Eq. (5) and expressed in a generic form in Eq. (6a) can be re-arranged as a s ¼ C 1 elL þ C 2 elL  R0 ð18Þ k

500 400 300 200 100 0 0

100

200

300

400

500

600

700

800

-100

900

1000 μL

-200 -300 -400

6

Ratio K (4,4) over [ K (4,4) calculated for μ L =1]

1210

5 4

lack of numerical precision

3 2 1 0 0 -1

0.0005

0.001

0.0015

0.002

0.0025

0.003

μL

-2 -3

Fig. 6. Behaviour of stiffness coefficient K(4, 4) derived based on the exponential expressions for the slip for varying lL.

G. Ranzi, M.A. Bradford / Computers and Structures 85 (2007) 1206–1214

s ¼ C1

j j 7 7 X X ðlzÞ a j ðlzÞ þ C2  R0 ð1Þ k j! j! j¼0 j¼0

ð23Þ

The proposed modelling procedure relies on two stiffness matrices; one derived expressing the slip with the exponentials as in Eq. (18) and one with their truncated Taylor series expansion as defined in Eq. (23). Therefore, the stiffness coefficients calculated for an element whose dimensionless stiffness coefficient lL is greater than 0.05 are based on the exponential expression for the slip, while for lL less than 0.05 these are based on their truncated Taylor expansion. The dimensionless stiffness lL is not just dependent upon the structural system considered, but also upon the discretisation utilised as L represents the length of the stiffness element. The robustness of the modelling technique is guaranteed by verifying that the stiffness matrices, i.e. the one based on the exponentials and the one based on their truncated Taylor series, can be interchanged. This is carried out ensuring that the coefficients of both stiffness matrices are identical for lL equal to 0.05 regardless of the actual value of lL for the element. It has been observed in the applications performed that only two stiffness coefficients, viz. K(4, 4) and K(4, 3) of Eq. (14), need to be compared for lL equal to 0.05 to ensure the validity of this approach. A similar approach has been utilised in the derivation of the equivalent nodal loads due to an uniformly distributed load. 4. Applications 4.1. Introduction The accuracy of the results obtained using the modelling technique presented in this paper is now validated against those calculated based on closed form solutions available in the literature. For this purpose, the case of a simply supported beam subjected to an uniformly distributed load and to axial loading reported by Girhammar and Gopu [11] using a first order analysis and the case of a propped cantilever subjected to a uniformly distributed load presented in [12] have been utilised. A 2-span continuous composite beam is then analysed to highlight the ease of use of the proposed modelling tech-

u11

u14

u13

u15

u12

nique and the results for this case are compared against those obtained by means of a highly refined finite element recently proposed by Dall’Asta and Zona [7] which consists of 16dof (Fig. 7). In this case, the occurrence of concrete cracking in the hogging moment regions has also been considered; which represents a likely situation at service condition. The results have been plotted for different levels of the dimensionless stiffness parameter lL to highlight the ability of the proposed approach to overcome the curvature locking problems which occur for high values of shear connection stiffness as reported by Dall’Asta and Zona [7]. The suitability of the proposed modelling technique to describe the nonlinear behaviour of determinate and indeterminate members at ultimate loads has been highlighted in the modelling of one continuous beam whose experimental results have been reported by Ansourian [13]. 4.2. Simply supported beam subjected to a UDL and to axial loading A simply supported beam subjected to an uniformly distributed load and to axial loading has been modelled using one element only. The cross-sectional properties specified are those used in the worked example by Girhammar and Gopu [11], which comprises a top rectangular concrete element having a width of 300 mm, a depth of 50 mm and an elastic modulus E1 ¼ 12; 000 MPa, and the corresponding values for the bottom rectangular timber element are 50 mm width, 150 mm depth and E2 ¼ 8000 MPa. The beam is 4 m long and it is subjected to an axial load of 50 kN and an uniformly distributed load of 1 kN/m. The proposed element is capable of applying the longitudinal axial force at various levels of the composite cross-section. Fig. 8 outlines how the deflections calculated along the beam length using the DSM method and based on the closed form solutions match perfectly for various levels of the dimensionless stiffness lL, including the one used in the worked example by Girhammar and Gopu [11] for which lL equals 8.43 (as the shear connection stiffness equals 50 MPa). Coordinate along the beam (m) 0

0.5

1

1.5

2

2.5

3

3.5

4

0 -0.005 Deflection (m)

in Eq. (18) with their Taylor series expansion truncated at the eighth term as

1211

-0.01 -0.015 -0.02 -0.025

u21

θ1 v1

μL = 1 - Girhammar & Gopu [11]

θ3 u24

u23

u25

v3 Fig. 7. 16dof finite element.

u22 v2

θ2

μL = 1 - DSM

μL = 8.43 - Girhammar & Gopu [11]

μL = 8.43 - DSM

μL = 100 - Girhammar & Gopu [11]

μL = 100 - DSM

Fig. 8. Variation of the deflection along a simply supported beam presented as a worked example in [11] for various levels of lL (DSM = Direct Stiffness Method).

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G. Ranzi, M.A. Bradford / Computers and Structures 85 (2007) 1206–1214

0

1

Coordinate along the beam (m) 2 3 4

1500mm x 130mm 5

6

0

Ec = 28 600 MPa Es = 200 000 MPa Ereo = 200 000 MPa yreo = 35mm Areo (0.6%) = 1170 mm2

Deflection (m)

-0.01 -0.02

170mm x 12mm

-0.03 -0.04

300mm x 7mm -0.05 μL = 1 - CFS

μL = 10 - CFS

μL = 100 - CFS

μL = 1 - DSM

μL = 10 - DSM

μL = 100 - DSM

170mm x 12mm

Fig. 9. Variation of the deflection along a propped cantilever subjected to an uniformly distributed load for various levels of lL (CFS = Closed Form Solution – DSM, Direct Stiffness Method).

Fig. 11. Cross-sectional and material properties for the application on a continuous composite beam at service condition [14].

Coordinate along the beam (m)

-0.0005

0.003 0.002

0

0.001

0.0005

0 0 -0.001

1 2 3 Coordinate along the beam (m)

4

5

6

-0.002

Deflection (m)

Slip (m)

0

0.002

0.003

-0.004 μL = 10 - CFS

μL = 100 - CFS

μL = 10 - DSM

μL = 100 - DSM

Fig. 10. Variation of the slip along a propped cantilever subjected to a uniformly distributed load for various levels of lL (CFS, Closed Form Solution; DSM, Direct Stiffness Method).

4.3. Propped cantilever subjected to a UDL Similarly to the simply supported case, the modelling of the propped cantilever requires only one element. A propped cantilever 6 m long is analysed subjected to an uniformly distributed load of 1 kN/m for different level of the shear connection stiffness. The results obtained using the DSM method and the closed form solutions for the deflection along the beam length are in perfect agreement, as it can be seen from Figs. 9 and 10. In these comparisons, linear-elastic material properties have been specified to emphasise the robustness and accuracy of the proposed modelling technique; these limitations have been released in the following applications. 4.4. Continuous beam – service condition The modelling of a 2-span continuous composite beam subjected to a uniformly distributed load of 10 kN/m has been carried out using two stiffness elements, whose cross-sectional properties are those presented in a worked example by Oehlers and Bradford [14] and are illustrated in Fig. 11.

3

4

5

6

7

8

9

10

0.001

-0.003

μL = 1 - DSM

2

0.0015

0.0025

μL = 1 - CFS

1

0.0035

μL μL μL μL μL μL

= 1 - DSM (concrete uncracked) = 1 - 16dof FEM (concrete uncracked) = 1 - DSM (concrete cracked) = 100 - DSM (concrete uncracked) = 100 - 16dof FEM (concrete uncracked) = 100 - DSM (concrete cracked)

Fig. 12. Variation of the deflection along a continuous beam subjected to an uniformly distributed load for various levels of lL (DSM, Direct Stiffness Method; 16dof FEM, 16dof Finite Element Method).

The left-hand support of the beam is assumed to be pinned while the other two supports are roller supports. The left and right spans have lengths of 6 m and 4 m, respectively. Fig. 12 plots the variation of the deflection along the beam length for various levels of the shear connection dimensionless stiffness lL (calculated based on the average length of the two spans, i.e. 5 m). In particular, the calculated results have been compared in the linear-elastic range against those obtained by means of the 16dof finite element (Fig. 7) adopting a very fine mesh; the results are shown to perfectly match. The use of the proposed modelling technique has been extended in the nonlinear range to account for the cracking of the concrete in the hogging moment regions which is typical of the structural response at service condition. For this purpose, four stiffness elements have been utilised in the analysis, i.e. two modelling the composite member in the sagging moment regions and two in the hogging moment regions. An iterative procedure has been implemented to reach the final solution. 4.5. Continuous beam – ultimate behaviour The experimental test carried out on a 2-span continuous beam reported by Ansourian [13] are considered in this

G. Ranzi, M.A. Bradford / Computers and Structures 85 (2007) 1206–1214 Load-deflection curves (Left span) - Beam CTB1

250

Load (kN)

200

150

100 Experiment

50

Modelling - 30 elements Modelling - 60 elements

0

0

10

20

30 Deflection (mm)

40

50

60

Fig. 13. Load-deflection curves (left span) from beam CTB1 [13]: comparison of experimental results with modelling results with 30 and 60 elements, respectively.

Load-deflection curves (Right span) - Beam CTB1

250

Load (kN)

200

150

100 Experiment

50

Modelling - 30 elements Modelling - 60 elements

0

0

2

4

6

8 10 12 Deflection (mm)

14

16

18

20

Fig. 14. Load-deflection curves (right span) from beam CTB1 [13]: comparison of experimental results with modelling results with 30 and 60 elements, respectively.

section. In particular, the proposed modelling technique has been utilised to model the nonlinear response of the beam labelled as CTB1 and reference should be made to [13] for more details on the specimens and testing set-up. The nonlinear behaviour of all materials forming the cross-sections have been included in the analysis based on the properties reported in [13]. The nonlinear simulations have been carried out by means of the secant stiffness method [15]. Figs. 13 and 14 plot the experimental values of the deflection along the beam length as well as those calculated adopting different degrees of mesh discretisations; this highlights the ability of the proposed approach to predict the nonlinear response of composite members which has the advantage to rely on highly refined functions, i.e. exponential functions, to describe the structural response.

5. Conclusions This paper has presented a stiffness formulation for the analysis of composite beam-columns with partial shear interaction (PI). A 8dof stiffness element has been derived using the direct stiffness method (DSM), whose freedoms

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are the axial displacement at the level of the reference axis, the vertical displacement, the rotation and the slip at the element ends. No approximated displacement and/or force fields are introduced in the proposed modelling technique, as are required by other techniques available in literature. The results obtained using the DSM method have been validated against, and shown to perfectly match, those obtained based on closed form solutions available in literature. The validation has been carried out for a simply supported beam subjected to an uniformly distributed load and to an axial load, and for a propped cantilever subjected to a uniformly distributed load. In these comparisons, several values of the dimensionless stiffness coefficient lL have been considered to highlight the ability of the derived elements to overcome curvature locking problems, which have been observed to occur in instantaneous analyses in some conventional displacement formulations. Moreover, the DSM element does not require discretisation lengthwise between supports and is able to handle axial laods applied at different levels of the cross-section. The case of a 2-span continuous beam has then been considered at service conditions. The calculated results have been validated against those obtained using a refined mesh of a high order finite element which has been recently proposed in the literature. The results have been shown to perfectly match. The occurrence of concrete cracking in the hogging moment regions has been included in the analysis. Finally, the experimental results reported in the literature of a 2-span continuous beam have been used as a benchmark to test the ability of the proposed approach to model the ultimate behaviour accounting for the nonlinearities of all materials forming the cross-section. It has been shown how it is capable of well predicting the structural response thanks to the highly refined functions, i.e. exponential functions, at the basis of the proposed modelling technique. Appendix Ac, Ar, As = area of the concrete component, of the reinforcement and of the steel joist, respectively e 1 ¼ A c E c þ Ar E r ; AE

e 2 ¼ As E s ; AE

e ¼ AE e 1 þ AE e2 AE

Bc, Br, Bs = first moment of area of the concrete component, of the reinforcement and of the steel joist, respectively, calculated about the arbitrary reference axis e 1 ¼ Bc E c þ Br E r ; B E e 2 ¼ Bs E s ; B E e ¼ BE e 1 þ BE e 2; BE e þ y0AE e e þ IE e BE y BE ; b2 ¼ 0 ; b1 ¼  e E e  BE e2 e E e  BE e2 A EI A EI   e E e 2 þ y0 BE e 2AE e 1  BE e 1AE e 2  AE e e 2I E B EB b3 ¼ ; e E e  BE e2 A EI Ec, Er, Es are the elastic modulus of the concrete component, of the reinforcement and of the steel joist, respectively.

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Ic, Ir, Is are the second moment of area of the concrete component, of the reinforcement and of the steel joist, respectively, calculated about the arbitrary reference axis e 1 ¼ I c Ec þ I r Er ; IE

e 2 ¼ I s Es ; IE

e ¼ IE e1 þ I E e 2; IE

l1 ¼ b1 þ y 0 r1 ; l2 ¼ b2 þ y 0 r2 ; l3 ¼ b3 þ y 0 r3 þ 1; e e AE B E ; r2 ¼ ; r3 ¼ a r1 ¼ 2 e e e e e e2 A EI E  B E A EI E  B E e 2AE e 2 þ BE e 2AE e 1  I EA e E e 1AE e2 BE 1 2 ~ ; a¼ 2 e E e  BE e A EI e 1AE e 2  BE e 2AE e1 BE ; a¼ e E e  BE e2 A EI e 1I E e  BE e 1BE e AE : a1 ¼ e E e  BE e2 A EI References [1] Newmark NM, Siess CP, Viest IM. Tests and analysis of composite beams with incomplete interaction. Proc Soc Exp Stress Anal 1951;9(1):75–92. [2] Spacone E, El-Tawil S. Nonlinear analysis of steel–concrete composite structures: state of the art. J Struct Eng ASCE 2004;30(2):159–68. [3] Leon RT, Viest IM. Theories of incomplete interaction in composite beams. In: Proceedings of the composite construction in steel and concrete III, Irsee, Germany, 9–14 June. p. 858–70.

[4] Faella C, Martinelli E, Nigro E. Steel and concrete composite beams with flexible shear connection: ‘‘exact’’ analytical expression of the stiffness matrix and applications. Comput Struct 2002;80:1001–9. [5] Ranzi G, Bradford MA, Uy B. A direct stiffness analysis of a composite beam with partial interaction. Int J Numer Meth Eng 2004;61:657–72. [6] Girhammar UA, Pan D. Dynamic analysis of composite members with interlayer slip. Int J Solids Struct 1993;30(6):797–823. [7] Dall’Asta A, Zona A. A Finite elements for the analysis of composite members with interlayer slip. In: Proceedings of CTA XVIII (Italian Workshop on Steel); 2001. [8] Griffiths DW, Nethercot DA, Rockey KC, Evans HR. Finite element method: a basic introduction. 2nd ed. Halsted Press; 1975. [9] Weaver W, Gere JM. Matrix analysis of framed structures. 3rd ed. Chapman & Hall; 1990. [10] Ranzi G, Bradford MA. Time-dependent analysis of composite beams with partial interaction using the direct stiffness approach. UNICIV Report R-423, School of Civil and Environmental Engineering, The University of New South Wales, Australia; 2004. [11] Girhammar UA, Gopu VKA. Composite beam-columns with interlayer slip-exact analysis. J Struct Eng ASCE 1993;119(4):1265–81. [12] Ranzi G, Bradford MA. Analytical solutions for the time-dependent behaviour of composite beams with partial interaction. Int J Solids Struct 2006;43:3770–93. [13] Ansourian P. Experiments on continuous composite beams. Proc Inst Eng 1981;71(Part 2):25–51. [14] Oehlers DJ, Bradford MA. Composite steel and concrete structural members: fundamental behaviour. Oxford: Pergamon Press; 1995. [15] Ranzi G. Partial interaction analysis of composite beams using the direct stiffness method. PhD thesis, UNSW; 2003.