J. Cent. South Univ. (2019) 26: 3087-3102

DOI: https://doi.org/10.1007/s11771-019-4238-9

Flexural and eigen-buckling analysis of steel-concrete partially composite plates using weak form quadrature element method

XIA Jun(夏军)¹, SHEN Zhi-qiang(申志强)¹, LIU Kun(刘昆)², SUN Cheng-ming(孙成名)³

1. Undergraduate School, National University of Defense Technology, Changsha 410072, China;

2. School of Aeronautics and Astronautics, Sun Yat-Sen University, Shenzhen 518001, China;

3. School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Flexural and eigen-buckling analyses for rectangular steel-concrete partially composite plates (PCPs) with interlayer slip under simply supported and clamped boundary conditions are conducted using the weak form quadrature element method (QEM). Both of the derivatives and integrals in the variational description of a problem to be solved are directly evaluated by the aid of identical numerical interpolation points in the weak form QEM. The effectiveness of the presented numerical model is validated by comparing numerical results of the weak form QEM with those from FEM or analytic solution. It can be observed that only one quadrature element is fully competent for flexural and eigen-buckling analysis of a rectangular partially composite plate with shear connection stiffness commonly used. The numerical integration order of quadrature element can be adjusted neatly to meet the convergence requirement. The quadrature element model presented here is an effective and promising tool for further analysis of steel-concrete PCPs under more general circumstances. Parametric studies on the shear connection stiffness and length-width ratio of the plate are also presented. It is shown that the flexural deflections and the critical buckling loads of PCPs are significantly affected by the shear connection stiffness when its value is within a certain range.

Key words: weak form quadrature element method; partially composite plates; interlayer slip; flexural analysis; eigen-buckling analysis

Cite this article as: XIA Jun, SHEN Zhi-qiang, LIU Kun, SUN Cheng-ming. Flexural and eigen-buckling analysis of steel-concrete partially composite plates using weak form quadrature element method [J]. Journal of Central South University, 2019, 26(11): 3087-3102. DOI: https://doi.org/10.1007/s11771-019-4238-9.

1 Introduction

The composite structure exhibits enhanced mechanical performance compared to the dedication of its sub-components working independently. Steel-concrete partially composite plates (PCPs) structure, such as composite bridge deck, composite box girder, composite floor system or shear walls in buildings and reinforcement for overpass recently reported by NIE et al [1] and WU et al [2], is a typical composite structure and has been extensively used in structural engineering area. The representative steel-concrete partially composite plate is consisted of a steel plate carrying a reinforced concrete slab through distributed shear connectors. An interlayer slip arises undoubtedly between the interface of the steel plates and concrete slab due to the incomplete-rigidness of the shear connectors in common practice. This phenomenon is also described as partial interaction and has remarkable influence on mechanical properties of the steel-concrete partially composite plates in most instances. Thus, the effect of interlayer slip must be carefully considered in the analysis model and design code for steel-concrete partially composite plate structures.

A great deal of studies considering the steel-concrete partially composite beams with interlayer slips have been conducted over the past decades and still in progress, covering linear and nonlinear analysis of static, dynamic and buckling problem using experimental, analytic and numerical method [3-8]. Recently, a stiffness matrix method based on the shear deformable beam theory of partially composite beams was proposed by LIN et al [6] and used for flexural analysis. FANG et al [7] examined the dynamic characteristics of steel- concrete partially composite beams with a simplified analytical model, in which the so called mode stiffness matrix was formulated. ECSEDI et al [8] offered analytical solutions for 2-layer partially composite beams considering the effect of cross-sectional shear deformation and its influence on flexural deflection was parametrically studied.

The aforementioned research works are valuable for partially composite beams but not completely applicative for partially composite plates. However, there are still little literatures available for composite plates with partial interaction so far. The initial analytical model for partially composite plates was most likely presented by CLARKE et al [9]. The steel plate was assumed to be very thin compared to the reinforced concrete slab, even was a negligible order of level. This analytical model presented by CLARKE et al would not be suitable when the steel plates become thicker. Then, SATO [10, 11] given a set of partial differential equations for linear flexural and buckling analysis of partially composite plates. Since introducing excessive assumptions, the critical loads of simply supported partially composite were underestimated when the shear connection stiffness is relatively small [1]. Recently, analytical solutions for eigen-buckling analysis of steel-concrete composite plates with interlayer slip acted by uniaxial compression and pure shear were developed by NIE et al [1] and WU et al [12] respectively. In their model, the effect of interlayer slip was considered by assuming shear deformable infinitely thin layer between the bottom steel plate and the reinforced concrete slab.

Moreover, these analytical solutions mentioned above are restricted to PCPs with certain shape, simple loads and boundary conditions. The enormous demand for nonlinear and linear analysis of composite plates in general boundary conditions leads to the development of numerical methods among which the finite element analysis, particularly using commercial FEM code, appear to be more popular. The so-called “slip locking” problems [13] encountered in finite element analysis for one dimension partially composite beams with relative larger interface connection stiffness using classical low-order beam element will undoubtedly be recurrent for PCPs if the similar elements are adopted. As for an alternative solution, more sophisticated three dimension finite element model was presented in Ref. [1] for flexural and buckling analysis of PCPs utilizing the ANSYS software. The concrete slab and steel plate were both meshed with solid finite element, and the shear connectors on the interface were represented by spring elements. Another similar finite element model was established in Ref. [2] using the ABAQUS software. These finite element formulation strategies proceeding unexceptionally increase the expense of analysis. Thousands (even tens of thousands) of degrees of freedom (DOF) is needed for three dimension finite element formulation to meet the convergence requirement of the problem.

The weak form quadrature element method (hereinafter referred to as QEM) recently proposed by ZHONG et al [14] has been utilized in many types of structural problems [15-19], also including linear and nonlinear analysis for partially composite beams performed by the authors [20-23]. Meanwhile, the excellent performance of the weak form QEM in increasing computational efficiency was observed in these works. In present work, a numerical model considering interlayer slip for rectangular steel-concrete PCPs was established using the weak form QEM. The flexural and eigen-buckling analysis were conducted for steel-concrete PCPs with shear connection stiffness commonly used in practice. The QEM numerical results were compared with those from analytic and finite element methods. The characteristics of fast convergence and high computational efficiency of QEM were observed again. In addition, parametric studies on the shear connection stiffness and length-width ratio of the plate are also conducted. It is found that the shear connection stiffness significantly affects the flexural deflections and the critical buckling loads of the partially composite plates when its value is within a certain range.

2 Theoretical formulation

The kinematical model and QEM element formulation for steel-concrete partially composite plates will be derived based on thin plate theory (Kirchhoff plate theory) in this section.

2.1 Kinematical model and constitutive relations

Figure 1 shows a commonly used rectangular steel-concrete partially composite plate in an orthogonal coordinate system (O-XYZ) which was applied to marking an arbitrary material point of the partially composite plate before deformation. The rectangular steel-concrete partially composite plate possessing uniform thickness (h^t+h^b) is formed using steel plate (bottom-component) connected to a concrete slab (top-component) via distributed welded stud shear connectors. Without losing generality and as shown in Figure 1, the x-axis and y-axis of the orthogonal coordinate system are parallel to the adjacent two sides of the rectangular partially composite plate, respectively. And the z-axis is taken positive downward from the XOY plane. The length of the two adjacent sides of the partially composite plate are denoted by a and b, respectively. An arbitrary reference XOY plane is located at distances of z^t(z^b) from the centroid of the top (bottom)-component with uniform thickness h^t (h^b) and z^s from the interface between the two sub components.

Figure 1 A rectangular partially composite plate in Cartesian coordinate system

For simplicity, the steel-concrete composite plate with interlayer slip is modeled by two sub-thin plates connected by flexible shear connectors which allow for biaxial relative displacement, interlayer slip, between the two sub-components interface while the transverse uplift is prohibited. The two sub-components are both assumed to be experiencing small rotations, displacements and strains. That is to say the two sub-components possess their own longitudinal deformation independently but have the same bending deflection, rotation and curvature.

The displacement field (Y direction view) of the partially composite plates is illustrated in Figure 2. The displacement field consists of transverse displacement w along the Z coordinate axis, biaxial displacements u^t(u^b) and v^t(v^b) at the centroids of the top (bottom)-component along the X and Y coordinate axes respectively. The rotation angles of the cross-sectional plane about the X, Y axes are denoted by θ_x and θ_y.

Figure 2 Displacement field for a rectangular partially composite plate (Y direction view)

The Kirchhoff hypothesis implies, as discussed above, the displacement field of the partially composite plates as follows:

   (1)

and

                      (2)

Equation (1) indicates the first order partial derivatives of the flexural deflection to the x-axis and y-axis, respectively.

So, for small strain and moderate rotation (e.g., 10°-15°) problems as we concerned in this work, the strain vector can be listed as

                                (3)

The strain vector descripted in Eq. (3) is comprised of linear term and nonlinear term. The terms and in Eq. (3) are the (mixed-) second order derivatives of the deflection w to the x-axis and y-axis, respectively, also known as curvatures.

The biaxial interlayer slip u_s, v_s between the two sub-components can be obtained through

                         (4)

where denotes the distance of the two sub-component centroids.

The variables correlated with the linear term in Eq. (3) and Eq. (4) can be arranged in a vector

          (5)

So, the generalized stresses σ can be related to the generalized linear strains ε with the help of the following constitutive equation,

σ=Eε                                   (6)

The generalized elasticity matrix E in Eq. (6) is given by

              (7)

with

                         (8)

where E^t (E^b) is the elastic modulus and v^t (v^b) is the Poisson ratio of top(bottom)-component. Moreover, K_x and K_y are the shear connection stiffness along X and Y coordinate axes. And the generalized stresses vector σ is,

            (9)

where

                 (10)

f_s-x and f_s-y are the intensities of shear force offered by the shear connector along X and Y coordinate axes.

2.2 Geometric mapping

Different from finite element method, both of the derivatives and integrals in the variational description, or weak form, for a problem to be solved is directly evaluated by the aid of identical numerical interpolation points for one quadrature element. More than that, these numerical interpolation points are also the element nodes. In order to solve the two-dimension problem occupying arbitrarily-shaped physical domain utilizing the weak form QEM, meshing the physical domain with several relative large quadrilaterals is easy-implemented and necessary.

Then, these relative large quadrilaterals need to be transformed from physical domain shown in Figure 3(a) using geometric mapping formulation to normalized domain depicted in Figure 3(b) for the purpose of expediently using numerical integration and differentiation tool. For a rectangular partially composite plate studied in this paper, using only one quadrature element is fully competent and convergence of the solutions can be achieved by increasing the number of interpolation points.

Figure 3 Geometric mapping for one partially composite plate quadrature element when Nl=Nm=5

As for a straight-sided rectangular quadrature element, the following straightforward linear geometric mapping formulation is generally utilized

          (11)

with

    (12)

In Eqs. (11) and (12), x and y are the coordinates of an arbitrary point in quadrature element under physical domain, and ξ and η are the coordinates of it in normalized domain. x_k and y_k (k from 1 to 4) is the coordinates of the quadrature element corner nodes in physical domain.

The first order derivatives in physical domain depicted in Eq. (13) could be converted into normalized domain as follows:

          (13)

Simultaneously, the inverse transformation of above process can be obtained by means of the inverse of Jacobian matrix J,

        (14)

Then, the first order and second order differentials of the coordinates ξ, η to the coordinates x, y which are essential for developing the quadrature element of composite plate with interlayer slip can be acquired through

             (15)

and

    (16)

In Eqs. (15) and (16), the value of the determinant for J can be calculated through

                     (17)

2.3 Quadrature element formulation

The strain energy of the partially composite plate quadrature element can be expressed as

                       (18)

where represents the strain energy related to the linear terms in Eq. (3), and represents the strain energy related to the nonlinear ones.

The strain energy related to the linear terms can be succinctly written as

                     (19)

With the help of the geometric mapping presented in Subsection 2.2, could be further written as

  (20)

In Eq. (20), the generalized linear strains ε is converted into normalized domain denoted as with the help of the transformation matrix D. And the nonzero elements of the 11×17 matrix D are given below,

 (21)

Meanwhile, the normalized strain vector is

   (22)

An efficient numerical integration scheme is orthogonally utilized in the quadrature element to evaluate the strain energy. Then, Eq. (20) can be calculated as

      (23)

Nl (Nm) is the numerical integration points number in ξ-(η-) axis. Accordingly, H_i (H_j) are the numerical integration weighting coefficients on the correlative interpolation points in Eq. (23). The variable values at interpolation points are denoted by the variable name and right subscript label. For instance,In general, Gauss- Lobatto numerical integration scheme [24] is often employed for QEM and its integration points distributes in the normalized standard domain along ξ coordinate axis as

 (24)

In Eq. (24), ξ_i is the (i-1)-th roots of the polynomialwhere is the (N-1)-order Legendre polynomial.

The node displacement vectorof the quadrature element for partially composite plate in normalized domain is

     (25)

where

 (26)

It should be noted that there are 5 DOFs, the related to an inner node of the quadrature element; at each corner node, there is 7 DOFs, w_ij and two slopes w_,ξij, w_,ηij (when i=1 or Nl and j=1 or Nm); and 6 DOFs, w_ij and the normal slopeare assigned to all element edge nodes besides the corner ones. All these DOFs of quadrature element are depicted in Figure 3.

With the help of the classical differential quadrature method (DQM), the first order partial derivatives for the longitudinal displacements u^t(u^b) and v^t(v^b) under normalized domain in Eq. (23) are then rewritten as a linear weighted sum of the related node displacements. For instance,

      (27)

whereis the first order differentiation weighting coefficient of the classical differential quadrature method. Considering the C¹ continuous requirement of the Kirchhoff hypothesis, we choose the generalized differential quadrature method (GDQM) in association with the classical DQM to computing the transverse deflection w, i.e.,

           (28)

In Eq. (28),is the m-th order differentiation weighting coefficient of the generalized differential quadrature method. More detailed introduction of DQM and GDQM, along with procedures for calculating numerical differentiation weighting coefficient and can be found in relevant literatures Refs. [25, 26].

Then, the normalized strain field vector in Eq. (23) can be rewritten as

                            (29)

where B_ij are matrices of size 17×(5NmNl+2(Nm+ Nl)), involving the weighting coefficients from Eqs. (27) and (28).

The node displacement vector d^(e) of the quadrature element for partially composite plate in physical domain is

     (30)

where

       (31)

with n being the outward normal of element edge. In detail, there are 5 DOFs, w_ij for 2≤i≤Nl-1 and 2≤j≤Nm-1, related to an inner node of the quadrature element; at each corner node, there exists 7 DOFs, w_ij and two slopes w_,xij, w_,yij (when i=1 or Nl and j=1 or Nm); and 6 DOFs, w_ij and the directional derivative , are assigned to all element edge nodes without the corner nodes.

The relationship between normalized node displacement vector and its counterpart d^(e) in the physical domain can be written as

                           (32)

According to Eqs. (25) and (30), most elements in physical displacement vector d^(e) and normalized displacement vector are the same. One can readily obtain the transformation matrix T in Eq. (32) through minor modification from an identity matrix possessing the same dimension with d^(e). As for the DOFs belong to corner nodes, the correlative elements of the identity matrix need to be changed by means of the following equation,

                 (33)

And for the DOFs belong to edge nodes besides the corner ones, the correlative elements in the identity matrix need to be modified utilizing the following directional derivative equations,

      (34)

where n_x and n_y are the direction cosines of the outward normal n. Similarly, the first order differentials, w_,η and w_,ξ, in the right-hand side of Eq. (34) are calculated using the classical differential quadrature method. Substituting Eq. (29) and Eq. (30) into Eq. (23) yields

                       (35)

and the element stiffness matrix is,

    (36)

Similarly, the strain energy related to the nonlinear term in Eq. (18) can be given by

                 (37)

where

     (38)

and

            (39)

Likewise, using the same numerical tool as in Eq. (35), Eq. (37) turns into

                        (40)

with the element geometric stiffness matrix,

 (41)

For the partially composite plate undergoing transverse distributed force q(x, y), the energy expression of external forces can be computed in the same way,

      (42)

The total potential energy Π^(e) of the quadrature element is composed of two parts,

                       (43)

The first order variation of the total potential energy equals zero when the element in an equilibrium state, i.e.

                 (44)

As for Eq. (44), it should be noted that the nonlinear strain energy is often neglected when performing a linear flexural analysis. Substituting Eq. (35) and Eq. (42) into Eq. (44) yields

           (45)

According to the arbitrariness of δd^(e), Eq. (45) yields the following equilibrium criterion of the element,

                         (46)

For linear buckling analysis, a limit or critical state of stability exists when the second order variation of the total potential energy equals zero [27], i.e.,

       (47)

Substituting Eq. (35) and Eq. (40) into Eq. (47) yields

 (48)

Similarly, the following linear stability criterion of the element can be acquired,

                    (49)

Applying Eqs. (45) and (48) to the whole partially composite plate leads to the global equilibrium criterion and linear stability criterion of the system,

                 (50)

The assemblage process of the global stiffness matrix K, the global geometric matrix K_σ, the global force vector F and the global displacement vector d are consistent with the classical finite element formulation.

The deflection of partially composite plates can be solved from the linear algebraic equation Eq. (50) after introducing reasonable boundary condition. As for the eigen-buckling problem, the internal forces in Eq. (39) are generally assumed to be proportional to the axial external applied load. Thus, with a proportional factor λ, Eq. (50) can be turned into a generalized eigenvalue problem,

                       (51)

where K_σ0 is the global geometric matrix caused by a unit applied external axial load. The minimal eigenvalues of Eq. (51), denoted as λ_cr, determines the linear buckling load of the partially composite plate. More details of the weak form QEM could be found in our previous works [14-21].

3 Numerical examples

Some representative examples are discussed in this section for linear flexural and buckling problems of PCPs in order to validate the effectiveness of the QEM.

3.1 Flexural analysis

3.1.1 Example one

To investigate the convergence of QEM, a rectangular steel-concrete partially composite plate with sides of a=b=3 m acted by uniformly distributed lateral load q₀=1 MN/m² is analyzed for flexural problem. The thickness of steel plate and concrete slab are h^b=0.01 m and h^t=0.1 m, respectively. The modulus and Poisson ratio for the steel plate are E^b=210000 MPa and ν^b=0.3 while these two material parameters for the concrete slab are E^t=30000 MPa and ν^t=0.2. The partially composite plate with continuous levels of shear connection, i.e., K=K_x=K_y varying from 0.1 to 100 GPa, is taken into consideration to evaluate the proposed QEM formulation.

Two kinds of representative boundary conditions for partially composite plates, clamped end (C) and simply supported end (S), are considered. For a clamped end, it is required that

            (52)

And the following Eq. (53) is enforced in the case of simply supported end.

                        (53)

For brevity, two capital letters S and C are chosen to represent the boundary conditions for the 4 edges of the rectangular partially composite plate. Thus, two typical boundary conditions of the plate: SSSS and CCCC are considered in the present investigation. In order to examine the capability of the proposed QEM formulation for PCPs, only one QEM element is used to mesh the entire rectangular partially composite plate. And the flexural deflection is non-dimensionalized through Eq. (54),

     (54)

The numerical results for the maximum normalized deflection and X-direction interlayer slip u_s in point A (located at ξ=-1, η=0 in normalized coordinate system) for various shear connection stiffness under the SSSS boundary conditions are listed in Tables 1 and 2. The maximum normalized deflection and X-direction interlayer slip u_s in point B (located at ξ=-1/2, η=0 in normalized coordinate system) under the CCCC boundary conditions are listed in Tables 3 and 4. The accuracy of the QEM formulation is enhanced by increasing the number of integration points.

From Tables 1-4, it is seen that both the deflection and interlayer slip solutions converge rapidly with the number of the numerical integration points increase. Moreover, the number of integration points for convergence increases with the shear connection stiffness enlargement. When the shear connection stiffness is relatively small,i.e., K≤1 GPa, convergence of the quadrature element solution requires no more than 9×9 integration points in one element, i.e., the total number of the DOFs is only 441. However, this number may increase to 12×12, i.e., the total number of the DOFs increases to 768, when the shear connection stiffness is relatively large, i.e., K>10 GPa. The so called “slip-locking” problem appearing in conventional low-order finite element elements [13] is not observed in QEM formulations.

Table 1 Normalized maximum deflection (SSSS)

Table 2 u_s in point A of PCPs (SSSS) (Unit: mm)

Table 3 Normalized maximum deflection (CCCC)

Table 4 u_s in point B of PCPs (CCCC) (Unit: mm)

The distribution of interlayer slips u_s and v_s along the interface for partially composite plates with SSSS and CCCC boundary condition are plotted in Figures 4 and 5 for K=0.1 GPa, K=10 GPa and K=100 GPa, respectively.

In general, the interlayer slip reduces significantly with the increase of the shear connection stiffness. As for the SSSS boundary condition, the position of maximum of interlayer slip moves gradually from the edge-midpoints of plate to its corner with the shear connection stiffness enlargement. And for the CCCC boundary condition, the position of maximum for interlayer slip located at the mid-span cross section of the composite plate moves from the 1/4 span to the edge. In addition, the distribution of interlayer slip is smoother when the shear connection stiffness is relatively small and less integration points is needed for convergence, while the distribution become steepened especially around the position of the extremum when the shear connection stiffness is relatively large and more integration points are needed.

In order to further explore the flexural behavior of steel-concrete partially composite plates, the influences of length-width ratio of the plate and shear connection stiffness on the deflection of the plate are assessed. One quadrature element with 13×13 integration points is adopted to satisfy the precision requirement. As shown in Figure 6, the maximum deflections increase with the increase of length-width ratio for both boundary conditions and the magnitude change inversely with the shear connection stiffness. Moreover, it is also noted that the maximum deflections decrease remarkably with the shear connection stiffness increasing in a certain range, i.e., 0.1 GPa≤K≤10 GPa. Thus, the flexural deflection of PCPs is more susceptible to the effect of shear connection stiffness in this range.

3.1.2 Example two

In this example, a simply supported (SSSS) rectangular partially steel-concrete composite plate with sides of a=b=3 m under uniformly distributed lateral load q₀=10 MN/m² is examined. The thickness of steel plate and concrete slab are h^b=0.006 m and h^t=0.13 m, respectively. The elastic modulus and Poisson ratio of the bottom steel plate are E^b=206000 MPa and ν^b=0.3 while these two material parameters for the concrete slab are E^t=29420 MPa and ν^t=0.2.

Figure 4 Distribution of interlayer slip along interface of a partially composite SSSS plate:

The same partially steel-concrete composite plate has been ever analyzed by NIE et al [1] and SATO [10]. A 3D finite element model for PCPs was constructed in the commercial finite element software ANSYS by NIE et al [1]. Both concrete slab and steel plate were meshed with solid elements (solid 45). The transverse displacements of interface nodes were coupled and the welded stud shear connectors distributing along the interface were molded by spring elements (Combine 39). The analytic model was also obtained in Ref. [1] in which the interlayer slip effect was modeled as a thin shear-layer. The maximum deflections calculated from QEM for different shear connection stiffness are compared with those from FEM and analytic solution offered by NIE et al [1] in Table 5. One quadrature element with 13×13 integration points is adopted to satisfy the precision requirement. The definition of the symbols, D_v, K and h_v, in Table 5 can be found in Ref. [1].

From Table 5, one can see that the numerical results of QEM are less than those from FEM without exception since the three-dimensional FEM model is more flexible than our two dimensional plate model. The maximum relative deviation of QEM from FEM is -10.63% when the partially composite plates have a smaller shear connection stiffness, i.e., On the other hand, the numerical results of QEM are larger than those of analytic solution when the shear connection stiffness is relatively small and less than that when the shear connection stiffness increase. The maximum relative deviation of QEM from analytic solution is -7.42% when Although both of the QEM formulation and analytic solution are conducted based on the plate theory, the deviation of computation result between them cannot be explained directly owing to the essential assumption and simplification in the derivation of the analytic solution [1]. Considering the theoretical basis and solution process of these methods are different, the relative deviation of QEM from FEM and analytic solution is still acceptable.

Figure 5 Distribution of interlayer slip along interface of a partially composite CCCC plate:

Figure 6 Maximum deflection of PCPs with SSSS (a) and CCCC (b) boundary conditions

Table 5 Maximum deflection of PCPs (SSSS unit: m)

3.2 Eigen-buckling analysis

The effectiveness of quadrature element for PCPs in eigen-buckling analysis is checked in this subsection. The same steel-concrete partially composite plate as in the subsection 3.1.2 is chosen for uniaxial compressive eigen-buckling analysis. And another partially composite plate reported in Ref. [12] is re-examined for pure shear eigen- buckling analysis. This rectangular partially composite plate with sides of a=b=3 m is simply supported (SSSS), and the thickness of concrete slab and steel plate are h^t=0.10 m and h^b=0.01 m, respectively. The elastic modulus and Poisson ratio of the bottom steel plate is E^b=206000 MPa and ν^b=0.3 while these two material parameters for the concrete slab is E^t=32500 MPa and ν^t=0.2.

The whole steel-concrete partially composite plate is also meshed using only one quadrature element. The uniaxial compressive and pure shear critical buckling load of the PCPs, denoted as N_xx-cr and N_xy-cr, are calculated from the weak form QEM. It is found that both of the solutions for N_xx-cr and N_xy-cr converge rapidly with the increasing of the integration points. And the number of integration points for convergence increases with the shear connection stiffness enlargement. However, this number is no more than 15×15 when the shear connection stiffness is relatively large, i.e., K>10 GPa.

The uniaxial compressive and pure shear critical buckling loads calculated from QEM for different shear connection stiffness are compared with those from FEM and analytic solution offered by NIE et al [1], SATO [11] and WU et al [12] in Figure 7. One quadrature element with 15×15 integration points is adopted to satisfy the precision requirement. A brief introduction for the finite element and analytic model can be found in the subsection 3.1.2.

As shown in Figure 7, the computation results from QEM formulation agree well with those from FEM and analytic solution. The critical buckling loads from QEM formulation are closer to analytic solution when the shear connection stiffness is relatively small while they are closer to those from FEM when the shear connection stiffness is relatively large. For the uniaxial compressive critical buckling analysis, the maximum relative deviation of QEM from FEM is no more than 2.74%, and the critical buckling loads were underestimated by SATO [11] especially when moderate shear connection stiffness are considered, i.e., 1 GPa

To further explore the eigen-buckling behavior of PCPs, the effect of length-width ratio of the rectangular partially composite plate on critical buckling load is assessed. The partially composite plate reported in Ref. [12] under simply supported boundary condition is chosen as benchmark. As shown in Figure 8, the critical buckling loads increase with the shear connection stiffness increasing, more obviously in a certain range, i.e., K>1 GPa. For the uniaxial compressive critical buckling analysis, the critical loads for PCPs with a/b=1.5 are slightly greater than those for PCPs with a/b =1 and a/b =2.5. This fluctuant feature was also identified in uniaxial compressive critical buckling analysis for thin plates [28] since the critical load was determined not only by the length-width ratio but also the buckling mode. As for the pure shear critical buckling analysis, the critical buckling loads decrease with the length-width ratio increasing.

Figure 7 Critical buckling load of PCPs under uniaxial compressive (a) and pure shear loads (b)

4 Conclusions

A numerical model for steel-concrete partially composite plate with interlayer slip is established utilizing the weak form QEM. And the flexural and eigen-buckling analysis are conducted for steel-concrete PCPs with typical continuous levels of shear connection stiffness.

Figure 8 Critical buckling load of PCPs with respect to length-width ratio:

1) The weak form quadrature element method shows excellent performance in flexural and eigen-buckling analysis of rectangular steel- concrete PCPs with common levels of shear connection stiffness. Only one quadrature element is usually sufficient for a rectangular partially composite plate and the numerical integration order of quadrature element can be adjusted neatly to meet the convergence requirement.

2) The shear connection stiffness significantly affects the flexural deflection and the critical buckling load of the steel-concrete PCPs when its value is within a certain range, i.e., 0.1 GPa≤K≤10 GPa for flexural problem and K>1 GPa for eigen-buckling problem.

3) The maximum flexural deflections increase with the enlargement of length-width ratio while the magnitude changes inversely with the shear connection stiffness.

4) The uniaxial compressive critical loads present slight fluctuation with the increase of length-width ratio, while the pure shear critical buckling loads decrease with the length-width ratio increasing.

Finally, it is worth putting out, the programming implementation of the quadrature element formulation for steel-concrete partially composite plate established in this work is carried out using the MATLAB software environment. Furthermore, developing user-defined element based on the quadrature element formulation in the present typical commercial finite element software environment, such as ANSYS, ABAQUS, will be a more direct and effective way to expand the application scope of the novel numerical method. The finite element software ABAQUS, for instance, has already provided the user subroutine interface ‘UEL’, ‘UELMAT’ to define a new element and relate it to ABAQUS materials, and these development work will be conducted in our subsequent research.

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(Edited by HE Yun-bin)

中文导读

界面滑移钢-混凝土组合板弹性弯曲及稳定性的弱形式求积元分析

摘要：利用抗剪连接件将钢板与钢筋混凝土板进行组合形成的钢-混凝土组合板，能充分发挥混凝土和钢材优越的材料性能，在建筑、桥梁结构的新建和加固中得到了广泛应用。由于抗剪连接件的非完全刚性，在对钢-混凝土组合板进行力学分析时，应对界面滑移效应进行充分考虑。本文采用弱形式求积元法对矩形钢-混凝土组合板在简支和固支边界条件下的弯曲和特征值屈曲问题进行了分析。弱形式求积元法的显著特点是利用同一组插值点直接计算待求解问题弱形式描述中的积分和导数，通过调整单元的阶次来满足待求解问题的收敛要求。通过与现有文献中有限元及解析解的数值结果进行比较，验证了本文数值模型的高效性。对于工程中常见抗剪连接刚度的矩形钢-混凝土组合板，仅用一个求积元单元就能得到满意的计算结果。进一步的参数化研究表明，当抗剪连接件的连接刚度在特定范围内变化时，其对钢-混凝土组合板的弯曲变形和临界屈曲荷载有显著影响。

关键词：弱形式求积元法；部分组合板；界面滑移；弯曲分析；特征值屈曲分析

Foundation item: Project(51508562) supported by the National Natural Science Foundation of China; Project(ZK18-03-49) supported by the Scientific Research Program of National University of Defense Technology, China

Received date: 2018-10-10; Accepted date: 2019-08-22

Corresponding author: SHEN Zhi-qiang, PhD, Associate Professor; Tel: +86-731-87022972; E-mail: zq_shen@outlook.com; ORCID: 0000-0002-8824-8996