Dynamic Analysis of Multilayered Composite Beams Considering Interlayer Slips

Abstract

This paper presents a new plane stress model for the dynamic analysis of multilayer composite beams with interlayer slip effects. In this model, the cross section of a multilayer composite beam is transformed into an equivalent plane stress cross section. Based on the equilibrium, constitutive and geometric equations of the plane stress problem, state equations are derived in terms of a set of state variables. The state variables are then expanded in Fourier series, and the state equations are solved using the state-space method. The proposed computational model makes it convenient to account for slip at each interface and can represent the entire transition of an interface from fully slipped to fully bonded. Interlayer slip and the corresponding interaction forces are incorporated naturally into the derivation of the governing equations, and the model gives accurate results. A steel–concrete–steel composite beam, a four-layer composite beam and a laminated timber beam are analyzed as examples of multilayer composite beams under both static and dynamic loading. The static analysis results are in good agreement with the literature results, with a maximum error of 0.63% for the maximum mid-span deflection and only 0.143% for the maximum interlayer slip value. Compared with finite element results, the natural frequencies and buckling loads obtained from the dynamic analysis exhibit maximum relative errors of 2.87% and 3.77%, respectively. The relationship between axial force and natural frequency is also presented, which verifies the accuracy and reliability of the proposed model and calculation method.

1. Introduction

Composite beams are structural members in which different materials and cross sections are connected by shear connectors so that they act together and make full use of the mechanical properties of each material. Compared with single component members, composite beams have higher load carrying capacity and stiffness and are widely used and studied. In general, connectors are required between the individual components so that they behave as an integral member. In practice, however, the components cannot be fully bonded. Tangential relative slip usually occurs along the interfaces between layers, and its magnitude depends on the stiffness of the connectors [1]. Interlayer slip causes the overall composite cross section to violate the plane section assumption, which complicates the structural response and has a significant influence on the mechanical performance of composite members.

Research on composite members with interlayer slip can be traced back to the 1950s. Newmark et al. [2]. developed a calculation theory for composite beams with interlayer slip based on Euler–Bernoulli beam theory. Viest [3,4] investigated the basic mechanical behavior of composite beams. Goodman [5] studied laminated timber structures in which interlayer slip was taken into account. Girhammar et al. [6,7,8,9] derived analytical expressions for composite beams with interlayer slip on the basis of Euler–Bernoulli beam theory. Xu and Wu [10] examined the static and dynamic behavior of composite beams with interlayer slip using Timoshenko beam theory. Based on a plane stress model, Xu and Wu [11,12] also derived the state-space equations for composite beams with interlayer slip, established a plane stress analysis model for two-layer composite beams, and carried out static and dynamic analyses. Shen et al. [13]. used the state-space method to derive the governing equations for the free vibration of composite beams with interlayer slip and studied the influence of axial force. Monetto [14] analyzed the bending equilibrium of three-layer beams with interlayer slip using Timoshenko beam theory. In addition to these theoretical formulations, many researchers have performed experimental studies [15,16,17,18], and finite element methods have also been applied effectively to the analysis of composite members with interlayer slip [19,20,21].

Most existing studies have focused on two-layer composite members, and research on the static and dynamic characteristics of three-layer and multilayer composite members remains relatively limited. In engineering practice, however, multilayer composite members are also widely used, such as steel–concrete–steel sandwich composite beams [22,23,24,25], laminated timber beams [26,27,28], sandwich beams [29,30,31], glass composite beams [32,33,34] and laminated beams strengthened with FRP [35,36]. For such multilayer composite members, some related investigations have been carried out. Ferdous et al. [37]. conducted experimental studies on the flexural shear behavior of laminated timber beams. Zhao et al. [38]. and Lacroix et al. [39]. experimentally investigated the mechanical response of steel–concrete–steel sandwich plates and FRP strengthened laminated timber beams under blast loading. Guo et al. [40]. studied experimentally the bending capacity of steel–concrete–steel composite beams. Chalak et al. [41]. analyzed the dynamic behavior of sandwich composite beams with a soft core based on zig zag theory. Magnucki [42] examined the bending response of sandwich beams using zig zag theory. Sayyad and Avhad [43] used higher order shear deformation theory to study the free vibration of curved sandwich beams. Li et al. [44] proposed a unified higher order shear deformation theory for the bending, free-vibration and buckling analysis of composite plates. However, studies based on Zig-zag theory or higher order shear deformation theory usually assume that the interfaces transmit only shear forces and that no slip occurs [45,46,47,48], which is not consistent with actual engineering practice.

Sousa and Silva [49,50] derived finite element formulations for multilayer composite members based on Euler–Bernoulli beam theory and Timoshenko beam theory. Monetto [14] obtained analytical solutions for three-layer composite beams with interlayer slip using Euler–Bernoulli beam theory. Keo et al. [51] developed finite element formulations for multilayer composite beams with interlayer slip on the basis of Timoshenko beam theory. These finite element formulations for multilayer composite members, derived from Euler–Bernoulli and Timoshenko beam theories, are prone to shear locking [52,53,54]. This indicates that further work is needed.

Dynamic analysis of composite members with interlayer slip has long been an important topic. Most existing studies have concentrated on two-layer composite beams [10,55,56,57,58,59], while relatively few have examined the dynamic behavior of three-layer and multilayer composite beams. Atashipour et al. [60] investigated the buckling stability of three-layer composite beams with interlayer slip using Timoshenko beam theory. Lin et al. [54] derived a finite element for three-layer composite beams with interlayer slip based on the variational principle and Timoshenko beam theory, and carried out dynamic analysis. For composite members with more than three layers, interlayer slip effects are usually not considered. In addition, the governing equations or finite element formulations in most existing work are derived from classical beam theory, so their applicability is restricted by the plane section assumption and the slenderness ratio.

The computational method in this paper is obtained by extending the dynamic analysis method of Xu and Wu [12] for steel concrete composite beams and a previously proposed plane stress static model [61] for multilayer composite beams. First, the dynamic plane stress state-space formulation is generalized from a two-layer configuration to a multilayer composite beam with an arbitrary number of layers. Second, slip compatibility and interaction-force conditions are introduced at every interface, so that different slip stiffness values can be assigned independently to different interfaces. Third, layer-wise density terms are incorporated into the state equations, which allows free-vibration analysis of multilayer composite beams with interlayer slip. Fourth, the initial axial stress is included in the dynamic formulation, leading to a characteristic equation that can be used for both natural-frequency and buckling-load analyses. Finally, a transfer-matrix-based computational procedure is implemented for multilayer systems, in which the layer transfer matrices and interface matrices are assembled sequentially.

In this study, a new plane stress model is used to analyze multilayer composite members with interlayer slip, and the state-space method is adopted to solve the resulting plane stress problem. First, the procedure for transforming the cross section of a multilayer composite beam with interlayer slip into an equivalent plane stress dynamic analysis model is described. The state-space equations of this plane stress dynamic model are then derived. By expanding the state variables in Fourier series, these equations are solved. Finally, typical examples reported in the literature are analyzed, and comparison with published results is used to verify the accuracy of the proposed computational method. This method does not rely on the plane section assumption and can describe cross-sectional deformation in an equivalent plane stress field. Moreover, since it does not employ low-order beam element interpolation functions, it avoids the problem of shear locking. It should be noted that the present model does not account for significant shear lag effects. However, when shear lag is significant, this transformation may not be suitable. The study of shear lag is out of the scope of this work.

2. Establishment of Plane Stress Model

For the multilayer composite beam cross section shown in Figure 1a, an equivalent transformation can be performed so that it becomes the unit width cross section shown in Figure 1b, and the problem can then be treated as a plane stress problem. To make these two planes equivalent, the axial stiffness EA and the bending stiffness EI of each layer must remain unchanged before and after the transformation. To achieve this, the original elastic modulus $E_i^{\prime }(i=1,2,\cdots ,n)$ is converted into an equivalent modulus $E_i=b_i{E}_i^{\prime }(i=1,2,\cdots ,n)$, while the height of each layer is kept unchanged so that each layer retains the same bending stiffness. In a similar way, the corresponding expression for $E_i{A}_i=b_i{E}_i^{\prime }{h}_i=E_i^{\prime }{A}_i^{\prime }$ and $E_i{I}_i=b_i{E}_i^{\prime }{h}_i^3/12=E_i^{\prime }{I}_i^{\prime }(i=1,2,\cdots ,n)$ can be obtained. For dynamic analysis, the density of the cross section must also be taken into account, and the actual material density ${\rho }_i^{\prime }(i=1,2,\cdots ,n)$ needs to be converted into an equivalent density ${\rho }_i=b_i{\rho }_i^{\prime }(i=1,2,\cdots ,n)$. In this way, the cross sections in Figure 1a,b are equivalent in the sense of one-dimensional beam theory. Such a transformation of the cross section may not be suitable when significant shear lag effects are present, but that case is beyond the scope of the present work.

The initial axial force $N_{x0}$ applied at both ends of the multilayer composite beam can be converted, according to Saint Venant’s principle, into equivalent initial axial stresses ${\sigma }_{x0}^{(i)}$ in each layer of the plane stress model. Finally, the dynamic problem of the multilayer composite beam subjected to end tensile forces is transformed into a two-dimensional plane stress dynamic analysis model, as shown in Figure 2. Where $N_{x0}>0$ represents tension and $N_{x0}<0$ represents compression. The model consists of n layers. The symbols $E_i$, $u_i$, $G_i$, ${\rho }_i$, and $h_i$ denote the elastic modulus, Poisson’s ratio, shear modulus, density and layer height of the i-th layer, respectively. Slip is allowed at every interface between adjacent layers, and the slip stiffness is denoted by $k_i(i=1,2,\cdots ,n-1)$. The relationship between the interlayer slip and the interaction force at interface i has been given in previous work [61]. In this study, the conventional assumption for steel concrete composite beams is adopted, namely that the interlayer interaction force is proportional to the interlayer slip.

3. Derivation and Solution of Equations

3.1. Establishment of State-Space Equations

The plane stress equilibrium equation for the i-th layer in the xy-plane is given by

$$\begin{array}{l}{\displaystyle \frac{𝜕{\sigma }_x^{(i)}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{xy}^{(i)}}{𝜕y}}={\rho }_i{\displaystyle \frac{{𝜕}^2{u}^{(i)}}{𝜕t^2}}+{\sigma }_{x_0}^{(i)}{\displaystyle \frac{{𝜕}^2{u}^{(i)}}{𝜕x^2}},\\ {\displaystyle \frac{𝜕{\tau }_{xy}^{(i)}}{𝜕x}}+{\displaystyle \frac{𝜕{\sigma }_y^{(i)}}{𝜕y}}={\rho }_i{\displaystyle \frac{{𝜕}^2{v}^{(i)}}{𝜕t^2}}+{\sigma }_{x_0}^{(i)}{\displaystyle \frac{{𝜕}^2{v}^{(i)}}{𝜕x^2}}\end{array}$$

(1)

where ${\sigma }_x^{(i)}$, ${\sigma }_y^{(i)}$ and ${\tau }_{xy}^{(i)}$ represent the normal stresses in the x- and y-directions and the shear stress in the i-th layer, respectively. ${\rho }_i$ is the density of the i-th layer, and t is time. The state equations derived in this section are all based on the assumption that each layer can be treated as an isotropic material in the analyzed plane. For isotropic materials, the constitutive equation is given by

$${\epsilon }_x^{(i)}={\displaystyle \frac{1}{E_i}}\left[{\sigma }_x^{(i)}-u_i{\sigma }_y^{(i)}\right], {\epsilon }_y^{(i)}={\displaystyle \frac{1}{E_i}}\left[-u_i{\sigma }_x^{(i)}+{\sigma }_y^{(i)}\right], {\gamma }_{xy}^{(i)}={\displaystyle \frac{1}{G_i}}{\tau }_{xy}^{(i)}$$

(2)

where ${\epsilon }_x^{(i)}$, ${\epsilon }_y^{(i)}$ and ${\gamma }_{xy}^{(i)}$ represent the normal strain and shear strain of the i-th layer, respectively. These strains can be expressed in terms of displacements through the geometric relations, i.e.,

$${\epsilon }_x^{(i)}={\displaystyle \frac{𝜕u^{(i)}}{𝜕x}}, {\epsilon }_y^{(i)}={\displaystyle \frac{𝜕v^{(i)}}{𝜕y}}, {\gamma }_{xy}^{(i)}={\displaystyle \frac{𝜕u^{(i)}}{𝜕y}}+{\displaystyle \frac{𝜕v^{(i)}}{𝜕x}}$$

(3)

where $u^{(i)}$ and $v^{(i)}$ represent the axial and vertical displacements of the i-th layer, respectively. By substituting Equations (2) and (3), the strain terms can be eliminated, yielding

$${\displaystyle \frac{𝜕u^{(i)}}{𝜕x}}={\displaystyle \frac{1}{E_i}}\left[{\sigma }_x^{(i)}-u_i{\sigma }_y^{(i)}\right], {\displaystyle \frac{𝜕v^{(i)}}{𝜕y}}={\displaystyle \frac{1}{E_i}}\left[-u_i{\sigma }_x^{(i)}+{\sigma }_y^{(i)}\right], {\displaystyle \frac{𝜕u^{(i)}}{𝜕y}}+{\displaystyle \frac{𝜕v^{(i)}}{𝜕x}}={\displaystyle \frac{1}{G_i}}{\tau }_{xy}^{(i)}$$

(4)

The governing Equations (1) and (4) now contain three stresses, ${\sigma }_x^{(i)}$, ${\sigma }_y^{(i)}$ and ${\tau }_{xy}^{(i)}$, as well as two displacements, $u^{(i)}$ and $v^{(i)}$. Next, Equations (1) and (4) are combined into a state-space form by substitution and simplification. First, the last term in Equation (4) and the second term in Equation (1) can be expressed as

$${\displaystyle \frac{𝜕u^{(i)}}{𝜕y}}=-{\displaystyle \frac{𝜕v^{(i)}}{𝜕x}}+{\displaystyle \frac{1}{G_i}}{\tau }_{xy}^{(i)}$$

(5)

$${\displaystyle \frac{𝜕{\sigma }_y^{(i)}}{𝜕y}}={\rho }_i{\displaystyle \frac{{𝜕}^2{v}^{(i)}}{𝜕t^2}}+{\sigma }_{x0}^{(i)}{\displaystyle \frac{{𝜕}^2{v}^{(i)}}{𝜕x^2}}-{\displaystyle \frac{𝜕{\tau }_{xy}^{(i)}}{𝜕x}}$$

(6)

By eliminating stress ${\sigma }_x^{(i)}$ from the first terms in Equations (1) and (4), we obtain

$${\displaystyle \frac{𝜕{\tau }_{xy}^{(i)}}{𝜕y}}={\rho }_i{\displaystyle \frac{{𝜕}^2{u}^{(i)}}{𝜕t^2}}+\left({\sigma }_{x0}^{(i)}-E_i\right){\displaystyle \frac{{𝜕}^2{u}^{(i)}}{𝜕x^2}}-u_i{\displaystyle \frac{𝜕{\sigma }_y^{(i)}}{𝜕x}}$$

(7)

Similarly, stress ${\sigma }_x^{(i)}$ is eliminated from the first two terms of Equation (4), and displacement $v^{(i)}$ is expressed in terms of $u^{(i)}$ and ${\sigma }_y^{(i)}$, i.e.,

$${\displaystyle \frac{𝜕v^{(i)}}{𝜕y}}=-u_i{\displaystyle \frac{𝜕u^{(i)}}{𝜕x}}+{\displaystyle \frac{1-u_i^2}{E_i}}{\sigma }_y^{(i)}$$

(8)

Finally, Equation (5) through Equation (8) can be written in matrix form as

$${\displaystyle \frac{𝜕}{𝜕y}}\left\{\begin{array}{c}{u}^{(i)}\\ {\sigma }_y^{(i)}\\ v^{(i)}\\ {\tau }_{xy}^{(i)}\end{array}\right\}=\left[\begin{array}{cccc}0& 0& -𝜕/𝜕x& 1/G_i\\ 0& 0& {\rho }_i{𝜕}^2/𝜕t^2+{\sigma }_{x0}^{(i)}{𝜕}^2/𝜕x^2& -𝜕/𝜕x\\ -u_i𝜕/𝜕x& (1-u_i^2)/E_i& 0& 0\\ {\rho }_i{𝜕}^2/𝜕t^2+\left({\sigma }_{x0}^{(i)}-E_i\right){𝜕}^2/𝜕x^2& -u_i𝜕/𝜕x& 0& 0\end{array}\right]\left\{\begin{array}{c}{u}^{(i)}\\ {\sigma }_y^{(i)}\\ v^{(i)}\\ {\tau }_{xy}^{(i)}\end{array}\right\}$$

(9)

Equation (9) represents the state-space equation for the plane stress dynamic problem of the i-th layer, where the variables $u^{(i)}$, ${\sigma }_y^{(i)}$, $v^{(i)}$ and ${\tau }_{xy}^{(i)}$ are referred to as state variables. It can be observed that if the density ${\rho }_i$ is set to zero, Equation (9) reduces to the static state equation [61], and for the stress ${\sigma }_x^{(i)}$, it can also be computed according to [61].

$${\sigma }_x^{(i)}=E_i{\displaystyle \frac{𝜕u^{(i)}}{𝜕x}}+u_i{\sigma }_y^{(i)}$$

(10)

Therefore, the state-space equation for the plane stress problem of multilayer composite beams is established.

3.2. Solution of the State Equation

Solving the state-space equation requires boundary conditions. If the beam is simply supported at both ends, the boundary conditions are:

$$v^{(i)}=0, {\sigma }_x^{(i)}=0 \mathrm{when} x=0 \mathrm{and} x=L$$

(11)

It should be noted that the simply supported boundary conditions in Equation (11) are imposed in the sense of the equivalent two-dimensional plane stress model. The condition $v^{(i)}$ constrains the vertical displacement along the end section of each layer, whereas ${\sigma }_x^{(i)}$ indicates that the axial normal traction at the ends is zero. Therefore, the beam is allowed to deform freely in the axial direction at the supports. The shear stress ${\tau }_{xy}^{(i)}$ is not prescribed to be zero at the ends and may represent the vertical reaction distributed along the supported end section. Thus, Equation (11) corresponds to an idealized line-supported boundary condition for the equivalent plane stress model, rather than a point support condition in classical one-dimensional beam theory. Consequently, the state variables can be expanded in Fourier series form:

$$\begin{array}{cc}{u}^{(i)}=H{U}^{(i)}(\zeta )\cos (m\pi \xi )\exp (\mathrm{j}\omega t),& {\sigma }_y^{(i)}=E_0{\sigma }^{(i)}(\zeta )\sin (m\pi \xi )\exp (\mathrm{j}\omega t)\\ v^{(i)}=H{V}^{(i)}(\zeta )\sin (m\pi \xi )\exp (\mathrm{j}\omega t),& {\tau }_{xy}^{(i)}=E_0{\tau }^{(i)}(\zeta )\cos (m\pi \xi )\exp (\mathrm{j}\omega t)\end{array}$$

(12)

where m represents the half-wave number in the x-direction, and $E_0$ is a parameter with stress dimensions used to emphasize the magnitude of stress. The symbols $\xi$ and $\zeta$ are dimensionless coordinates relative to the x and y directions, defined as follows:

$$\xi =x/L, \zeta =y/H$$

(13)

For the free-vibration analysis, the top and bottom surfaces are traction-free except for the interface interaction forces. For the static analysis under external transverse loading, the distributed load is introduced through the stress boundary condition on the top surface of the beam. A transverse load q(x) applied on the top surface is expanded into a Fourier sine series consistent with Equation (12), for a constant uniformly distributed load q, namely:

$$q(x)={\displaystyle \sum _{m=1}^{\infty }{q}_m\sin (\frac{m\pi x}{L})}, q_m=\left\{\begin{array}{ll}\frac{4q}{m\pi }\hfill & m=1,3,5\cdots \hfill \\ 0\hfill & m=2,4,6\cdots \hfill \end{array}\right.$$

(14)

q_m is the Fourier coefficient. The corresponding coefficients are imposed as the normal stress boundary condition on the top surface. The static solution is obtained by setting the inertia terms to zero and superposing the Fourier components.

Substituting Equation (12) into Equation (9) gives:

$$\begin{array}{l}{\displaystyle \frac{𝜕}{𝜕\zeta }}\left\{\begin{array}{c}{U}^{(i)}\\ {\sigma }^{(i)}\\ V^{(i)}\\ {\tau }^{(i)}\end{array}\right\}=\left[\begin{array}{cccc}0& 0& -{\alpha }_m& 1/{\overline{G}}_i\\ 0& 0& -{\overline{\rho }}_i{\mathrm{\Omega }}^2-{\overline{\sigma }}_{x0}^{(i)}{\alpha }_m^2& {\alpha }_m\\ u_i{\alpha }_m& (1-u_i^2)/{\overline{E}}_i& 0& 0\\ -{\overline{\rho }}_i{\mathrm{\Omega }}^2+\left({\overline{E}}_i-{\overline{\sigma }}_{x0}^{(i)}\right){\alpha }_m^2& -u_i{\alpha }_m& 0& 0\end{array}\right]\left\{\begin{array}{c}{U}^{(i)}\\ {\sigma }^{(i)}\\ V^{(i)}\\ {\tau }^{(i)}\end{array}\right\}\\ \begin{array}{cccccccc}& & & & & & & \end{array}(m=1,2,\cdots ,\infty )\end{array}$$

(15)

where

$$\begin{array}{l}{\alpha }_m=m\pi H/L, {\overline{G}}_i=G_i/E_0, {\overline{E}}_i=E_i/E_0,\\ {\overline{\sigma }}_{x0}^{(i)}={\sigma }_{x0}^{(i)}/E_0, {\overline{\rho }}_i={\rho }_i/{\rho }_0, {\mathrm{\Omega }}^2={\rho }_0{\omega }^2{H}^2/E_0\end{array}$$

(16)

stress ${\sigma }_x^{(i)}$ can be obtained by substituting Equation (12) into Equation (10):

$${\sigma }_x^{(i)}=E_0\left[-{\alpha }_m{\overline{E}}_i{U}^{(i)}(\zeta )+u_i{\sigma }^{(i)}(\zeta )\right]\sin (m\pi \xi )\exp \left(\mathrm{j}\omega t\right)$$

(17)

Thus, it can be seen that the boundary conditions at both ends are satisfied. Equation (15) is a system of ordinary differential equations, and its solution is:

$$\left\{\begin{array}{c}{U}_m^{(i)}(\zeta )\\ {\sigma }_m^{(i)}(\zeta )\\ V_m^{(i)}(\zeta )\\ {\tau }_m^{(i)}(\zeta )\end{array}\right\}=e^{\left[K_i\right](\zeta -{\zeta }_{i-1})}\left\{\begin{array}{c}{U}_m^{(i)}({\zeta }_{i-1})\\ {\sigma }_m^{(i)}({\zeta }_{i-1})\\ V_m^{(i)}({\zeta }_{i-1})\\ {\tau }_m^{(i)}({\zeta }_{i-1})\end{array}\right\} (\zeta \in \left[{\zeta }_{i-1},{\zeta }_i\right])$$

(18)

where ${\zeta }_0=0, {\zeta }_i=(h_1+h_2+\cdots +h_i)/H (i=1,2,\cdots ,n)$, and the matrix $\left[K_i\right]$ is the coefficient matrix of Equation (15), i.e.,

$$\left[K_i\right]=\left[\begin{array}{cccc}0& 0& -{\alpha }_m& 1/{\overline{G}}_i\\ 0& 0& -{\overline{\rho }}_i{\mathrm{\Omega }}^2-{\overline{\sigma }}_{x0}^{(i)}{\alpha }_m^2& {\alpha }_m\\ u_i{\alpha }_m& (1-u_i^2)/{\overline{E}}_i& 0& 0\\ -{\overline{\rho }}_i{\mathrm{\Omega }}^2+\left({\overline{E}}_i-{\overline{\sigma }}_{x0}^{(i)}\right){\alpha }_m^2& -u_i{\alpha }_m& 0& 0\end{array}\right]$$

(19)

Equation (18) establishes the relationship between the state variables at positions ${\zeta }_{i-1}$ and $\zeta$ in the i-th layer. The matrix exponential $\left[K_i\right]$ remains a matrix and serves as the transfer matrix for the state variables from ${\zeta }_{i-1}$ to $\zeta$. For the equilibrium equations and coordination conditions at the i-th slip interface in Figure 3, the same equations used in static analysis [61] can still be applied.

$$v^{(i+1)}=v^{(i)}, {\sigma }_y^{(i+1)}={\sigma }_y^{(i)}, k_i\left[u^{(i+1)}-u^{(i)}\right]=V_{si}={\tau }_{xy}^{(i+1)}={\tau }_{xy}^{(i)}$$

(20)

$$\left\{\begin{array}{c}{u}^{(i+1)}\\ {\sigma }_y^{(i+1)}\\ v^{(i+1)}\\ {\tau }_{xy}^{(i+1)}\end{array}\right\}=\left[\begin{array}{cccc}1& 0& 0& 1/k_i\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left\{\begin{array}{c}{u}^{(i)}\\ {\sigma }_y^{(i)}\\ v^{(i)}\\ {\tau }_{xy}^{(i)}\end{array}\right\}$$

(21)

Substituting Equation (12) into Equations (20) and (21) gives

$$\left\{\begin{array}{c}{U}^{(i+1)}({\zeta }_i)\\ {\sigma }^{(i+1)}({\zeta }_i)\\ V^{(i+1)}({\zeta }_i)\\ {\tau }^{(i+1)}({\zeta }_i)\end{array}\right\}=\left[\begin{array}{cccc}1& 0& 0& E_0/(k_{i}H)\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left\{\begin{array}{c}{U}^{(i)}({\zeta }_i)\\ {\sigma }^{(i)}({\zeta }_i)\\ V^{(i)}({\zeta }_i)\\ {\tau }^{(i)}({\zeta }_i)\end{array}\right\}$$

(22)

If the slip stiffness is infinitely large, i.e., $k_i\to \infty$, the interface is in a fully bonded state, and Equation (22) simplifies to the following form:

$$\left\{\begin{array}{c}{U}^{(i+1)}({\zeta }_i)\\ {\sigma }^{(i+1)}({\zeta }_i)\\ V^{(i+1)}({\zeta }_i)\\ {\tau }^{(i+1)}({\zeta }_i)\end{array}\right\}=\left\{\begin{array}{c}{U}^{(i)}({\zeta }_i)\\ {\sigma }^{(i)}({\zeta }_i)\\ V^{(i)}({\zeta }_i)\\ {\tau }^{(i)}({\zeta }_i)\end{array}\right\}$$

(23)

By applying the same method to Equation (17) $\zeta ={\zeta }_i$, the relationship between the state variables at the top and bottom interfaces of the i-th layer can be obtained, as follows:

$$\left\{\begin{array}{c}{U}_m^{(i)}({\zeta }_i)\\ {\sigma }_m^{(i)}({\zeta }_i)\\ V_m^{(i)}({\zeta }_i)\\ {\tau }_m^{(i)}({\zeta }_i)\end{array}\right\}=\left[T_i\right]\left\{\begin{array}{c}{U}_m^{(i)}({\zeta }_{i-1})\\ {\sigma }_m^{(i)}({\zeta }_{i-1})\\ V_m^{(i)}({\zeta }_{i-1})\\ {\tau }_m^{(i)}({\zeta }_{i-1})\end{array}\right\}$$

(24)

where the transfer matrix $\left[T_i\right]$ is

$$\left[T_i\right]=e^{\left[K_i\right]\Delta {\zeta }_i}$$

(25)

where $\Delta {\zeta }_i=h_i/H,(i=1,2,\cdots ,n)$. By using Equations (23)–(25), the following relationship between the state variables at the top and bottom of the plane stress model of the multilayer laminated beam can be obtained.

$$\left\{\begin{array}{c}{U}^{(n)}(1)\\ {\sigma }^{(n)}(1)\\ V^{(n)}(1)\\ {\tau }^{(n)}(1)\end{array}\right\}=\left[T_n\right]\left[C_{n-1}\right]\left[T_{n-1}\right]\cdots \left[C_2\right]\left[T_2\right]\left[C_1\right]\left[T_1\right]\left\{\begin{array}{c}{U}^{(1)}(0)\\ {\sigma }^{(1)}(0)\\ V^{(1)}(0)\\ {\tau }^{(1)}(0)\end{array}\right\}$$

(26)

where the matrix $\left[C_i\right](i=1,2,\cdots ,n-1)$ is the coefficient matrix of Equation (23). Equation (26) contains four equations and eight variables, requiring the addition of boundary conditions, specifically at the top and bottom of the beam. In the case of free vibration, the boundary conditions at both the top and bottom are zero, i.e.,

$${\sigma }_{y\mid y=0}=0, {\sigma }_{y\mid y=H}=0, {\tau }_{xy\mid y=0}=0, {\tau }_{xy\mid y=H}=0$$

(27)

Therefore, the stress conditions at the top and bottom of the beam become

$${\sigma }^{(1)}(0)=0, {\sigma }^{(n)}(1)=0, {\tau }^{(1)}(0)=0, {\tau }^{(n)}(1)=0$$

(28)

Substituting Equation (28) into Equation (26) gives

$$\left\{\begin{array}{c}{U}^{(n)}(1)\\ 0\\ V^{(n)}(1)\\ 0\end{array}\right\}=\left[T\right]\left\{\begin{array}{c}{U}^{(1)}(0)\\ 0\\ V^{(1)}(0)\\ 0\end{array}\right\}$$

(29)

where $\left[T\right]=\left[T_n\right]\left[C_{n-1}\right]\left[T_{n-1}\right]\cdots \left[C_2\right]\left[T_2\right]\left[C_1\right]\left[T_1\right]$ rearranges the second and fourth equations as follows:

$$\left[\begin{array}{cc}{T}_{21}& T_{23}\\ T_{41}& T_{43}\end{array}\right]\left\{\begin{array}{c}{U}^{(1)}(0)\\ V^{(1)}(0)\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}$$

(30)

where $T_{ij}$ represents the element in the i-th row and j-th column of matrix $\left[T\right]$. For $U^{(1)}(0)$ and $V^{(1)}(0)$ in Equation (30) to be non-zero, the determinant of the coefficient matrix in Equation (30) must be zero, i.e.,

$$\left|\begin{array}{cc}{T}_{21}& T_{23}\\ T_{41}& T_{43}\end{array}\right|=0$$

(31)

This is the characteristic equation for the free-vibration and buckling analysis of multilayer composite beams with interlayer slip and initial stress. The characteristic equation is solved using an interval scanning method combined with an iterative approach. The flowchart of the root-finding process for the characteristic equation is shown in Figure 3.

From the derivation process above, it can be seen that by adopting the plane stress model and the state-space method, interlayer slip is naturally incorporated. Furthermore, since the plane section assumption in classical beam theory is not used, the results obtained with the model and method in this paper align more closely with actual conditions.

4. Numerical Examples

4.1. Steel–Concrete–Steel Composite Beam

This section presents the static analysis example of a steel–concrete–steel three-layer composite beam from the work of Sousa and Silva [49]. Both static and dynamic analyses of this example will be carried out. The model and method for static analysis will follow previous research [61]. As shown in Figure 4, the elastic moduli of concrete and steel are 34,500 MPa and 200,000 MPa, respectively, and their Poisson’s ratios are 0.2 and 0.3. The interlayer slip stiffness at the top and bottom is assumed to be 40 MPa and 5 MPa, respectively.

4.1.1. Static Analysis

The beam is subjected to a uniform load of 10 kN/m at the top. The deflection and interlayer slip of the steel–concrete–steel composite beam are calculated.

Since the state variables are expanded using a Fourier series, as shown in Equation (12), the convergence of the series should first be examined. Figure 5 illustrates the convergence of the maximum deflection and the maximum interlayer slip of s₂. It can be seen that the solution converges when the number of terms exceeds 10, and when the number of terms reaches 15, the truncation error is within 0.0001%. Therefore, selecting 15 terms is sufficient to ensure the accuracy of the series solution. In the subsequent analysis, 15 terms are uniformly adopted to compute the final results.

Based on ABAQUS 2023 software, a finite element model of the steel–concrete–steel composite beam was established using B32 beam elements, as shown in Figure 6. The contact was simulated using Cartesian connectors, which were arranged along the beam length at a spacing of 5 mm. Large stiffness values were assigned in the X and Y directions, while the stiffness in the Z direction was set according to the slip stiffness in the example. The boundary conditions were set as simply supported.

By transforming the steel–concrete–steel composite beam into a plane stress model and solving it using the state-space method, both slip and deflection of the beam are obtained. Figure 7 shows the distribution of slip magnitude along the beam length for the two slip interfaces. The results indicate that the slip is maximum at both ends and zero at the mid-span, with the slip exhibiting an anti-symmetric distribution about the mid-span.

Sousa and Silva [49] derived the calculation method for multilayer composite beams based on Euler–Bernoulli beam theory. The results from this method are compared with those of Sousa and Silva [49] in

Table 1. Mid-span maximum deflection and maximum interlayer slip (mm).

	Present (a1)	Sousa [49] (a2)	Finite Element (a3)	Relative Error (%)
				$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$	$\left|{\mathit{a}}_1-{\mathit{a}}_3\right|/{\mathit{a}}_1\times 100\%$
vmax	10.94868	10.8796	10.92	0.63	0.26
s1	0.77847	0.77895	0.78	0.062	0.20
s2	1.003559	1.00212	1.00	0.143	0.35

. The maximum mid-span deflection v_max and maximum interlayer slip values (s₁, s₂) from both methods are in close agreement, with a relative error of 0.63% for v_max and only 0.143% for the maximum interlayer slip values, confirming the accuracy of both methods. It is also noted that the maximum deflection at mid-span v_max calculated by the method in this paper is slightly larger than the result from Sousa and Silva [49]. This difference is due to the fact that the model in this paper does not use the plane section assumption in classical beam theory, which results in a lower stiffness and, therefore, a slightly larger deflection. This is consistent with expectations. In addition, Table 1 compares the results obtained by the proposed method with those from finite element analysis, and good agreement is observed between them.

4.1.2. Dynamic Analysis

The density of the steel is 7850 kg/m³, and the density of the concrete is 2300 kg/m³. The natural frequency and buckling load of the steel–concrete–steel composite beam are calculated.

By converting this steel–concrete–steel composite beam into a plane stress dynamic analysis model and applying the state-space method, the natural frequencies and buckling loads of the composite beam can be calculated. The results are compared with those obtained from finite element simulations, as shown in

Table 2. First 10 natural frequencies of steel–concrete–steel composite beams (Hz).

Order	Present (a1)	Finite Element (a2)	Relative Error (%)$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$
1	19.44869	19.49	0.21
2	70.12896	70.94	1.16
3	151.8143	153.81	1.31
4	262.0415	263.99	0.74
5	397.5453	399.07	0.38
6	554.8248	555.45	0.11
7	730.4459	729.66	0.11
8	921.2254	918.53	0.29
9	1124.323	1119.30	0.45
10	1337.267	1329.70	0.57

and

Table 3. First 5 buckling loads of steel–concrete–steel composite beam (kN).

Order	Present (a1)	Finite Element (a2)	Relative Error (%)$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$
1	1874.3940	1845.9734	1.52
2	6099.1845	5942.8453	2.56
3	12,723.4883	13,055.6037	2.61
4	21,363.3596	22,169.6058	3.77
5	31,532.0797	30,440.2137	3.46

. The maximum relative errors between the calculated results and the finite element simulation results are 1.31% for the first ten natural frequencies and 3.77% for the first five buckling loads. The two methods validate each other and both yield accurate results. Figure 8 shows the first 10 mode shapes of the steel–concrete–steel composite beam.

Figure 9 shows the variation in the fundamental frequency with axial pressure. From the graph, it can be observed that when the axial pressure is zero, the fundamental frequency reaches its maximum value. As the axial pressure increases, the fundamental frequency decreases. When the axial pressure approaches the buckling load, the effective tangent stiffness of the structure resisting deformation continuously decreases until it approaches zero; since the natural frequency is determined by both stiffness and mass, as the stiffness approaches zero, the fundamental frequency also approaches zero.

4.2. Four-Layer Composite Beam

This example is also taken from Sousa and Silva [49], where both static [61] and dynamic analyses are performed. As shown in Figure 10, the length of the four-layer composite beam is 4 m, with an asymmetric cross section in the height direction. The elastic modulus of each layer is 50,000 MPa, and the interlayer slip stiffness is 5 MPa.

4.2.1. Static Analysis

The beam is subjected to a uniform load of 10 kN/m at the top. The deflection and interlayer slip of the four-layer composite beam are calculated.

Figure 11 shows the distribution of interlayer slip along the length of the four-layer composite beam. The pattern of slip distribution is similar to that of the three-layer composite beam, with maximum slip occurring at both ends and zero slip at the mid-span. The slip is symmetrically distributed around the mid-span.

Table 4. Mid-span maximum deflection and maximum interlayer slip (mm).

	Present (a1)	Sousa [49] (a2)	Finite Element (a3)	Relative Error (%)
				$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$	$\left|{\mathit{a}}_1-{\mathit{a}}_3\right|/{\mathit{a}}_1\times 100\%$
vmax	38.32448	38.27935	38.14	0.118	0.48
s1	2.709023	2.70896	2.71	0.0023	0.04
s2	2.144185	2.14333	2.14	0.0399	0.20
s3	1.494104	1.49301	1.49	0.0732	0.27

compares the maximum mid-span deflection and maximum interlayer slip values calculated by the present method with those obtained by Sousa and Silva [49]. The results from both methods are very close, with a relative error of 0.118% for v_max and only 0.0732% for the maximum interlayer slip values, confirming the consistency and accuracy of both methods. Similarly, Table 4 also compares the results obtained by the proposed method with those from finite element analysis, and good agreement is observed between them, which meets expectations.

4.2.2. Dynamic Analysis

The density of a four-layer composite beam is 2500 kg/m³. The natural frequency and buckling load of the four-layer composite beam are calculated.

Table 5. First 10 natural frequencies of four-layer composite beam (Hz).

Order	Present (a1)	Finite Element (a2)	Relative Error (%)$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$
1	10.93805	10.819	1.09
2	41.27305	41.957	1.66
3	91.48303	93.309	2.00
4	161.153	165.77	2.87
5	249.6951	254.87	2.07
6	356.3854	361.89	1.54
7	480.3877	488.99	1.79
8	620.7798	631.2	1.68
9	776.5766	791.51	1.92
10	946.7556	970.88	2.55

and

Table 6. First 5 buckling loads of four-layer composite beam (kN).

Order	Present (a1)	Finite Element (a2)	Relative Error (%)$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$
1	535.991436	520.763	2.84
2	1907.880047	1913.62	0.30
3	4165.974125	4204.8	0.93
4	7271.682873	7451.86	2.48
5	11,172.69835	11,413.04	2.15

present the comparisons between the calculated first 10 natural frequencies and first five buckling loads of the four-layer composite beam and the finite element results, respectively. The maximum relative errors between the calculated results and the finite element simulation results are 2.87% for the first ten natural frequencies and 2.84% for the first five buckling loads.

Figure 12 shows the variation in the fundamental frequency with axial pressure. From the graph, it is clear that when the axial pressure is zero, the fundamental frequency reaches its maximum value. As the axial pressure increases, the fundamental frequency decreases. When the axial pressure approaches the buckling load, the fundamental frequency approaches zero. The calculated results are consistent with expectations.

4.3. Laminated Timber Beam

This example is based on a previous study [62] of a five-layer laminated timber beam. As shown in Figure 13, the span of the laminated timber beam is 4 m, and its height is 0.3 m. The cross section is divided into five layers along the height direction, each with a height of 0.08 m. The elastic modulus of each layer is 8000 MPa, and the Poisson’s ratio is 0.3.

It should be noted that timber is generally an orthotropic material. In the present example, however, the laminated timber beam is modeled using equivalent isotropic material properties, with a single elastic modulus and Poisson’s ratio assigned to each layer. This treatment is adopted to demonstrate the proposed multilayer plane stress state-space procedure and to compare the limiting behavior with classical beam theory. Therefore, the results of this example should be interpreted as those of an equivalent isotropic laminated timber beam rather than a full orthotropic timber model.

4.3.1. Static Analysis

The laminated timber beam is subjected to a uniformly distributed load of 1 kN/m at the top. The deflection and interlayer slip of the laminated timber beam are calculated.

When the slip stiffness approaches zero, the composite beam tends toward complete slip, and the bending stiffness reaches the minimum bending stiffness EI_min. When the slip stiffness approaches infinity, the composite beam tends toward complete bonding, and the bending stiffness reaches the maximum bending stiffness EI_max. According to classical beam theory, the maximum deflection at mid-span when the slip stiffness is very small or very large is calculated to be 6.51 mm and 0.26 mm, respectively.

Figure 14 shows the variation in the maximum deflection at mid-span of the laminated timber beam with slip stiffness. From the graph, it is evident that the curves from both methods are nearly identical, confirming that the results are consistent. Furthermore, both methods converge to the classical beam theory value for the maximum deflection at mid-span when the slip stiffness is either very small or very large, which further verifies the accuracy of the present calculation results.

To investigate the variation in interlayer slip, the interlayer slip stiffness is assumed to be 50 MPa. Figure 15 shows the distribution of interlayer slip along the length of the beam when the slip stiffness is 50 MPa. The distribution pattern of slip in the five-layer laminated timber beam is similar to the other examples in this paper, with the maximum slip occurring at both ends of the beam and zero slip at the mid-span. The slip exhibits an anti-symmetric distribution around the mid-span. Additionally, for this example of a multilayer composite cross section with symmetry about the neutral axis and consistent slip stiffness at all interfaces, the farther the slip interface is from the neutral axis, the greater the slip.

4.3.2. Dynamic Analysis

The density of a laminated timber beam is 700 kg/m³. The natural frequency and buckling load of the laminated timber beam are calculated.

Since the slip stiffness of the laminated timber beam is unknown, it is assumed here that the slip stiffness is 50 MPa. Following the method described in Section 4.1.2, a finite element model of the laminated timber beam is established.

Table 7. First 10 natural frequencies of laminated timber beam (slip stiffness: 50 MPa) (Hz).

Order	Present (a1)	Finite Element (a2)	Relative Error (%)$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$
1	16.05434	16.24	1.16
2	42.78844	43.03	0.56
3	82.17152	83.34	1.42
4	135.4934	136.11	0.46
5	202.8563	204.88	1.00
6	283.959	285.12	0.41
7	378.3241	381.22	0.77
8	485.3783	486.52	0.24
9	604.4902	608.34	0.64
10	734.9943	736.96	0.27

and

Table 8. First 5 buckling loads of laminated timber beam (slip stiffness: 50 MPa) (kN).

Order	Present (a1)	Finite Element (a2)	Relative Error (%)$\left|{\mathit{a}}_1-{\mathit{a}}_2\right|/{\mathit{a}}_1\times 100\%$
1	1385.621	1364.38	1.53
2	2460.663	2453.37	0.30
3	4033.29	4109.00	1.88
4	6168.44	6337.02	2.73
5	8849.04	9086.76	2.69

compare the first 10 natural frequencies and the first five buckling loads of the laminated timber beam with the finite element results, respectively. The maximum relative errors between the calculated results and the finite element simulation results are 1.42% for the first ten natural frequencies and 2.73% for the first five buckling loads.

Figure 16 illustrates the variation in the fundamental frequency with axial pressure under the assumption of a 50 MPa slip stiffness. From the graph, it can be observed that when the axial pressure is zero, the fundamental frequency reaches its maximum. As the axial pressure increases, the fundamental frequency continuously decreases. As the axial pressure approaches the buckling load, the fundamental frequency also approaches zero, which aligns with expectations.

To investigate the effect of slip stiffness on the natural frequency of the laminated timber beam, the slip stiffness is varied from zero to infinity. When the slip stiffness approaches zero, the laminated timber beam exhibits the minimum bending stiffness EI_min, and the corresponding fundamental frequency, calculated using classical beam theory, is 7.66470 Hz. When the slip stiffness approaches infinity, the laminated timber beam exhibits the maximum bending stiffness EI_max, and the corresponding fundamental frequency is 38.32351 Hz.

Figure 17 shows the variation in the laminated timber beam’s fundamental frequency with slip stiffness. The graph demonstrates that the fundamental frequency increases as the slip stiffness increases. As the slip stiffness approaches zero, the fundamental frequency approaches the value corresponding to the minimum bending stiffness EI_min. As the slip stiffness approaches infinity, the fundamental frequency approaches the value corresponding to the maximum bending stiffness EI_max. These results are consistent with expectations, confirming the accuracy and reliability of the computational method proposed in this paper.

5. Conclusions

This paper investigates the free-vibration and buckling response of multilayer composite beams with interlayer slip. A new plane stress analysis model is proposed, and it is solved using the state-space method. The cross section of the multilayer composite beam is first transformed into an equivalent plane stress dynamic analysis model. Based on the dynamic equilibrium equations of the plane stress state, combined with the constitutive equations and geometric relations, the state equations are derived. The state variables are then expanded using Fourier series for the solution of these state equations. The static analysis results of typical multilayer composite beams from the literature are calculated, showing excellent agreement with previously published results, thus validating the accuracy of the previous work. This also enables further dynamic analysis, leading to the determination of natural frequencies, buckling loads, and the relationship between axial pressure and natural frequency. The main conclusions drawn from the above calculations and analysis are as follows:

• The proposed model can effectively analyze multilayer composite members considering interlayer slip. The interlayer slip and interaction forces can be naturally incorporated into the derivation of the equations, thereby enabling the description of the entire process from bending bond to full slip.
• The static analysis results are in good agreement with existing analytical solutions. For the steel–concrete–steel composite beam, the relative error of the maximum mid-span deflection v_max is only 0.143%, and the maximum relative errors of the maximum interlayer slip values s₁ and s₂ are 0.062% and 0.143%, respectively. For the four-layer composite beam, the relative error of v_max is 0.118% and that of the maximum interlayer slip values is only 0.0732%.
• The dynamic analysis results are in good agreement with the finite element results. For the steel–concrete–steel composite beam, the maximum relative error of the first ten natural frequencies is 1.31%, and that of the first five buckling loads is 3.77%; for the four-layer composite beam, the corresponding maximum errors are 2.87% and 2.84%, respectively; for the laminated timber beam, the corresponding maximum errors are 1.42% and 2.73%, respectively.
• The interlayer slip stiffness has a significant influence on the mechanical behavior of multilayer composite beams. As the slip stiffness increases, the natural frequencies also increase. For the laminated timber beam, when the slip stiffness approaches zero, its fundamental frequency approaches 7.66470 Hz, which corresponds to the minimum bending stiffness EI_min; when the slip stiffness continuously increases, its fundamental frequency approaches 38.32351 Hz, which corresponds to the maximum bending stiffness EI_max.