Dynamic stiffness matrix of partial-interaction composite beams

Abstract

Composite beams have a wide application in building and bridge engineering because of their advantages of mechanical properties, constructability and economic performance. Unlike static characteristics, the methods of studying the dynamic characteristics of partial-interaction composite beams were limited, especially dynamic stiffness matrix method. In this article, the dynamic stiffness matrix of partial-interaction composite beams was derived based on the assumption of the Euler–Bernoulli beam theory, and then it was used to predict the frequencies of the free vibration of the single-span composite beams with various boundary conditions or different axial forces. The corresponding vibration modes and buckling loads were also obtained. From the comparison with the existing results, the numerical results obtained by the proposed method agreed reasonably with those in the literatures. The dynamic stiffness matrix method is an accurate method which can determine natural vibration frequencies and vibration mode shapes in any precision theoretically. As a result, when the higher precision or natural frequencies of higher order are required, the dynamic stiffness matrix method is superior when compared to other approximate and numerical methods. The dynamic stiffness matrix method can also be combined with the finite-element method to calculate the free vibration frequencies and natural mode shapes of composite beams in complex conditions.

Introduction

Composite beams have a wide application in building and bridge engineering because of their advantages of mechanical properties, constructability and economic performance. They often have to withstand all kinds of dynamic loads, so it is necessary to have an in-depth research on the dynamic characteristics of composite beams. The dynamic stiffness matrix (DSM) method is an effective method for solving the problem of dynamic characteristics. It was proposed in the early 1940s. After decades of development, it has been a powerful tool for solving engineering problems in structural vibration, particularly when higher natural frequencies and better accuracies are required. ¹ The DSM method is usually referred to as an accurate method because it precisely solves the differential equations of the structure and is different from the traditional finite-element method or other approximate methods which are required to assume the displacement or other physical quantities beforehand. Theoretically, the DSM method can determine natural vibration frequencies and vibration mode shapes in any precision, and it also can obtain accurate results without considering the number of the elements. The DSM has the property of element mass and element stiffness at the same time. It is composed of the analytical solution of the differential equations. DSM is the transcendental function of frequency. Therefore, this kind of problem is often referred to as transcendental eigenproblem.

More and more researches pay attention on the DSM method since it was proposed in the early 1940s. Around 1970, Williams and Wittrick^2,3 proposed the well-known Williams–Wittrick algorithm to solve natural vibration frequencies of the structure using the DSM method. In 1977, Hopper and Williams ⁴ proposed a method to calculate natural mode shapes. To get the natural mode shapes, a method in which subspace iteration was utilized in conjunction with a frequency-dependent mass and stiffness formulation was described and applied to framed structures by Richards and Leung. ⁵ In 1993, Leung ⁶ proposed the DSM method of the non-conservative loads to analyze thin-walled structures under the influence of a follower axial force and an in-plane moment. Shortly afterwards, a computer-assisted analytical method was introduced for any structural members, the differential governing equations of which were expressible in matrix polynomial form by Leung and Zeng. ⁷ In 1994, Ye and Williams ⁸ discussed the range of eigenvalues. A general theory of development of the DSM of a structural element was outlined by Banerjee ⁹ in 1997. In 2000, Chan and Williams ¹⁰ analyzed the orthonormalization conditions for the modes of piecewise continuous structures obtained by the transcendental stiffness matrix method and showed how they can be extended to orthonormalize any set of coincident modes. In 2002, Williams et al. ¹¹ had a thorough discussion on the mathematical and physical concepts in the eigenproblem of the DSM method.

Since 1990, there have been more and more researches on laminated beams with the DSM method. The DSM method was developed to predict the free vibration characteristics of composite beams for which the bending and torsional displacements were (materially) coupled by Banerjee and Williams ¹² in 1995. Then, Banerjee and Williams ¹³ considered the effects of shear deformation and rotation inertia. In 1998, Banerjee ¹⁴ considered the effect of axial forces based on the results. In 2005, Banerjee and Sobey ¹⁵ examined the free vibration characteristics of a three-layered sandwich beam with the DSM method.

However, the recently developed DSM of Banerjee and Sobey ¹⁵ did not account for the interlayer slips of the three-layered sandwich beam. In 2007, Xu and Wu ¹⁶ examined the two-dimensional analytical solution of simply supported composite beams with interlayer slips. They then discussed the static, dynamic, and buckling characteristics of partial-interaction composite members using Timoshenko’s beam theory, in which the analytical solutions were derived under different boundary conditions. ¹⁷ On the basis, Wu et al. ¹⁸ examined the dynamic characteristics of composite beam under axial force and gave the exact formula for calculating the free vibration frequency. In 2008, Xu and Wu ¹⁹ examined free vibration and buckling of composite beams with interlayer slip by two-dimensional theory. Recently, Nguyen et al. ²⁰ developed an exact dynamic analysis procedure for the free vibration of shear-deformable two-layer beams with interlayer slip, and the effect of transverse shear deformation of two layers was taken into account. Accordingly, the main improvement of the proposed model, when compared to Wu et al. ¹⁸ model, allows the layers to have independent shear strains which depend on their shear modulus. In 2013, Chakrabarti et al. ²¹ adopted one-dimensional finite-element model based on a higher order beam theory to examine the dynamic response of composite beams with partial interaction and considered the effect of partial shear interaction between the adjacent layers, as well as the transverse shear deformation of the beam. However, according to the authors’ knowledge, the DSM of composite beam with interlayer slip has not been reported.

Based on the assumption of Euler–Bernoulli beam theory, this article deduced the DSM of the composite beams with interlayer slips with axial force. An analytical expression of the DSM of the composite beams was also given by the help of MATLAB. Finally, according to DSM method, for the composite beam without axial force, the free vibration frequencies of the composite beams under four boundary conditions were demonstrated; for the composite beam with axial force, the vibration frequencies of simply supported composite beams under different axial forces were obtained. The corresponding vibration modes and buckling loads were also obtained. The numerical results agreed with those in the literatures very well.

DSM

Xu and Chen ²² proposed the differential equations of the free vibration of composite beams with interlayer slips under the effect with axial force $P$ as follows:

$$\begin{array}{cc}\multicolumn{1}{c}{}& \frac{{\partial }^2}{\partial x^2}\left(D\frac{{\partial }^{2}w}{\partial x^2}\right)-\frac{\partial }{\partial x}\left\{\frac{\partial }{\partial x}\left[\overline{\mathrm{EA}}\frac{\partial }{\partial x}(u_s-\frac{\partial w}{\partial x}h)\right]h\right\}\hfill \\ \multicolumn{1}{c}{}& -P\frac{{\partial }^{2}w}{\partial x^2}+\rho A_0\frac{{\partial }^{2}w}{\partial t^2}=0\hfill \end{array}$$

(1)

$$-\frac{\partial }{\partial x}\left[\overline{\mathrm{EA}}\frac{\partial }{\partial x}(u_s-\frac{\partial w}{\partial x}h)\right]+k_s{u}_s=0$$

(2)

where $w$ denotes the deflection, $u_s$ denotes the relative slip at the interface, $k_s$ denotes the rigidity of the shear connector, $t$ denotes time, $h$ is the distance between the centroids of components 1 and 2, and

$$\begin{array}{cc}\multicolumn{1}{c}{}& \overline{\mathrm{EA}}=\frac{E_1{A}_1\times E_2{A}_2}{E_{1}A+E_2{A}_2},D=E_1{I}_1+E_2{I}_2,\hfill \\ \multicolumn{1}{c}{}& \overline{D}=D+\overline{\mathrm{EA}}{h}^2,\rho A_0={\rho }_1{A}_1+{\rho }_2{A}_2\hfill \end{array}$$

(3)

where $E_i,A_i,I_i$ , and ${\rho }_i(i=1,2)$ denote Young’s modulus, moment of inertia, cross-sectional area, and the density of two materials, respectively. To solve equations (1) and (2), the displacement $w$ and $u_s$ are expressed as

$$w=W{e}^{i\omega t},u_s=U_S{e}^{i\omega t}$$

(4)

where $W$ and $U_S$ represent amplitudes of $w$ and $u_s$ , $\omega$ denotes frequency, while $t$ denotes time and $i=\sqrt{-1}$ . Substitution of equation (4) into equations (1) and (2) gives

$$\overline{D}\frac{{\text{d}}^{4}W}{\text{d}{x}^4}-\overline{\mathrm{EA}}h\frac{{\text{d}}^3{U}_S}{\text{d}{x}^3}-P\frac{{\text{d}}^{2}W}{\text{d}{x}^2}-\rho A_0{\omega }^{2}W=0$$

(5)

$$\overline{\mathrm{EA}}h\frac{{\text{d}}^{3}W}{\text{d}{x}^3}-\overline{\mathrm{EA}}\frac{{\text{d}}^2{U}_S}{\text{d}{x}^2}+k_s{U}_S=0$$

(6)

Eliminating $W$ or $U_S$ using equations (5) and (6) gives

$$a\frac{{\text{d}}^{6}R}{\text{d}{\xi }^6}+b\frac{{\text{d}}^{4}R}{\text{d}{\xi }^4}+c\frac{{\text{d}}^{2}R}{\text{d}{\xi }^2}+\mathrm{dR}=0$$

(7)

where

$$R=W\mathrm{or}{U}_S,\xi =\frac{x}{L}$$

(8)

$$\begin{array}{cc}\multicolumn{1}{c}{}& a=\frac{\overline{\mathrm{EA}}D}{L^6},b=-\frac{\overline{\mathrm{EA}}P+\overline{D}{k}_s}{L^4},\hfill \\ \multicolumn{1}{c}{}& c=-\frac{\overline{\mathrm{EA}}\rho A_0{\omega }^2-P{k}_s}{L^2},d=k_s\rho A_0{\omega }^2\hfill \end{array}$$

(9)

and $L$ is the length of the element of composite beams.

The general solutions of equation (7) are based on the root characteristic of its eigen-equation

$$a{r}^6+b{r}^4+c{r}^2+d=0$$

(10)

The six roots of equation (10) are $\pm \alpha$ , $\pm \beta$ , $\pm \gamma i$ , where $\alpha$ , $\beta$ , and $\gamma$ are all real. ¹⁸ Consequently, the solution of equation (7) can be written as

$$\begin{array}{cc}\multicolumn{1}{c}{}& W\left(\xi \right)=A_1\cosh \alpha \xi +A_2\sinh \alpha \xi +A_3\cosh \beta \xi \hfill \\ \multicolumn{1}{c}{}& +A_4\sinh \beta \xi +A_5\cos \gamma \xi +A_6\sin \gamma \xi \hfill \end{array}$$

(11)

$$\begin{array}{c}\multicolumn{1}{c}{U_S\left(\xi \right)=B_1\cosh \alpha \xi +B_2\sinh \alpha \xi +B_3\cosh \beta \xi }\\ \multicolumn{1}{c}{+B_4\sinh \beta \xi +B_5\cos \gamma \xi +B_6\sin \gamma \xi }\end{array}$$

(12)

where $A_j$ and $B_j(j=1,2,\dots ,6)$ are two different sets of constants. Substitution of equations (11) and (12) into equation (6) shows

$$\begin{array}{cc}\multicolumn{1}{c}{}& B_1=k_{\alpha }{A}_2,B_2=k_{\alpha }{A}_1,B_3=k_{\beta }{A}_4,\hfill \\ \multicolumn{1}{c}{}& B_4=k_{\beta }{A}_3,B_5=k_{\gamma }{A}_6,B_6=-k_{\gamma }{A}_5\hfill \end{array}$$

(13)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& k_{\alpha }=\frac{\overline{\mathrm{EA}}h{\alpha }^3}{\overline{\mathrm{EA}}{\alpha }^{2}L-k_s{L}^3},k_{\beta }=\frac{\overline{\mathrm{EA}}h{\beta }^3}{\overline{\mathrm{EA}}{\beta }^{2}L-k_s{L}^3},\hfill \\ \multicolumn{1}{c}{}& k_{\gamma }=\frac{\overline{\mathrm{EA}}h{\gamma }^3}{\overline{\mathrm{EA}}{\gamma }^{2}L+k_s{L}^3}\hfill \end{array}$$

(14)

The expressions for the bending moment $M(\xi )$ , shear force $Q\left(\xi \right)$ , and axial force $N\left(\xi \right)$ are obtained from equations (11) and (12) as follows

$$\begin{array}{cc}\multicolumn{1}{c}{}& M\left(\xi \right)={\lambda }_{\alpha }{A}_1\cosh \alpha \xi +{\lambda }_{\alpha }{A}_2\sinh \alpha \xi +{\lambda }_{\beta }{A}_3\cosh \beta \xi \hfill \\ \multicolumn{1}{c}{}& +{\lambda }_{\beta }{A}_4\sinh \beta \xi +{\lambda }_{\gamma }{A}_5\cos \gamma \xi +{\lambda }_{\gamma }{A}_6\sin \gamma \xi \hfill \end{array}$$

(15)

$$\begin{array}{cc}\multicolumn{1}{c}{}& Q\left(\xi \right)={\mu }_{\alpha }{A}_1\sinh \alpha \xi +{\mu }_{\alpha }{A}_2\cosh \alpha \xi +{\mu }_{\beta }{A}_3\sinh \beta \xi \hfill \\ \multicolumn{1}{c}{}& +{\mu }_{\beta }{A}_4\cosh \beta \xi -{\mu }_{\gamma }{A}_5\sin \gamma \xi +{\mu }_{\gamma }{A}_6\cos \gamma \xi \hfill \end{array}$$

(16)

$$\begin{array}{cc}\multicolumn{1}{c}{}& N\left(\xi \right)={\eta }_{\alpha }{A}_1\cosh \alpha \xi +{\eta }_{\alpha }{A}_2\sinh \alpha \xi +{\eta }_{\beta }{A}_3\cosh \beta \xi \hfill \\ \multicolumn{1}{c}{}& +{\eta }_{\beta }{A}_4\sinh \beta \xi +{\eta }_{\gamma }{A}_5\cos \gamma \xi +{\eta }_{\gamma }{A}_6\sin \gamma \xi \hfill \end{array}$$

(17)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\lambda }_{\alpha }=\frac{\overline{\mathrm{EA}}\mathrm{hL}{k}_{\alpha }\alpha -\overline{D}{\alpha }^2}{L^2},{\lambda }_{\beta }=\frac{\overline{\mathrm{EA}}\mathrm{hL}{k}_{\beta }\beta -\overline{D}{\beta }^2}{L^2},\hfill \\ \multicolumn{1}{c}{}& {\lambda }_{\gamma }=\frac{\overline{D}{\gamma }^2-\overline{\mathrm{EA}}\mathrm{hL}{k}_{\gamma }\gamma }{L^2}\hfill & \hfill \end{array}$$

(18)

$${\mu }_{\alpha }=\frac{{\lambda }_{\alpha }\alpha }{L},{\mu }_{\beta }=\frac{{\lambda }_{\beta }\beta }{L},{\mu }_{\gamma }=\frac{{\lambda }_{\gamma }\gamma }{L}$$

(19)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\eta }_{\alpha }=\frac{\overline{\mathrm{EA}}h{\alpha }^2-\overline{\mathrm{EA}}L{k}_{\alpha }\alpha }{L^2},{\eta }_{\beta }=\frac{\overline{\mathrm{EA}}h{\beta }^2-\overline{\mathrm{EA}}L{k}_{\beta }\beta }{L^2},\hfill \\ \multicolumn{1}{c}{}& {\eta }_{\gamma }=\frac{\overline{\mathrm{EA}}L{k}_{\gamma }\gamma -\overline{\mathrm{EA}}h{\gamma }^2}{L^2}\hfill & \hfill \end{array}$$

(20)

The positive direction is shown in Figure 1

Figure 1.

Sign convention for positive bending moment (M), shear force (Q), and axial force (N).

To obtain the DSM of an element of composite beams, the internal force for both ends of the element can be expressed in terms of the displacements $(W_i,{\theta }_i,U_{\mathrm{Si}},W_j,{\theta }_j,U_{\mathrm{Sj}})$ at the ends as shown in Figure 2, in which the rotational angle $\theta$ is defined as

$$\theta =\frac{1}{L}\frac{\text{d}W}{\text{d}\xi }$$

(21)

Figure 2.

End conditions for forces and displacements of the beam element.

From equations (11)–(14) and equation (21), the displacements of an element of composite beams at the two ends can be rendered in terms of the parameters $A_j$ , namely

$$\mathbf{U}\mathbf{=}\mathbf{BA}$$

(22)

where

$$\begin{array}{cc}\multicolumn{1}{c}{}& \mathbf{U}\mathbf{=}{\left[\begin{array}{cccccc}\hfill W_i\hfill & \hfill {\theta }_i\hfill & \hfill U_{\mathrm{Si}}\hfill & \hfill W_j\hfill & \hfill {\theta }_j\hfill & \hfill U_{\mathrm{Sj}}\hfill \end{array}\right]}^{\text{T}},\hfill \\ \multicolumn{1}{c}{}& \mathbf{A}\mathbf{=}{\left[\begin{array}{cccccc}\hfill A_1\hfill & \hfill A_2\hfill & \hfill A_3\hfill & \hfill A_4\hfill & \hfill A_5\hfill & \hfill A_6\hfill \end{array}\right]}^{\text{T}}\hfill \end{array}$$

(23)

and

$$\mathbf{B}=\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill \alpha /L\hfill & \hfill 0\hfill & \hfill \beta /L\hfill & \hfill 0\hfill & \hfill \gamma /L\hfill \\ \multicolumn{1}{c}{0}& \hfill k_{\alpha }\hfill & \hfill 0\hfill & \hfill k_{\beta }\hfill & \hfill 0\hfill & \hfill k_{\gamma }\hfill \\ \multicolumn{1}{c}{\cosh \alpha }& \hfill \sinh \alpha \hfill & \hfill \cosh \beta \hfill & \hfill \sinh \beta \hfill & \hfill \cos \gamma \hfill & \hfill \sin \gamma \hfill \\ \multicolumn{1}{c}{\alpha \sinh \alpha /L}& \hfill \alpha \cosh \alpha /L\hfill & \hfill \beta \sinh \beta /L\hfill & \hfill \beta \cosh \beta /L\hfill & \hfill -\gamma \sin \gamma /L\hfill & \hfill \gamma \cos \gamma /L\hfill \\ \multicolumn{1}{c}{k_{\alpha }\sinh \alpha }& \hfill k_{\alpha }\cosh \alpha \hfill & \hfill k_{\beta }\sinh \beta \hfill & \hfill k_{\beta }\cosh \beta \hfill & \hfill -k_{\gamma }\sin \gamma \hfill & \hfill k_{\gamma }\cos \gamma \hfill \end{array}\right]$$

(24)

Similarly, the internal forces of an element of composite beams at the two ends can also be represented by the parameters $A_j$ with the help of equations (15)–(20) as follows

$$\mathbf{F}\mathbf{=}\mathbf{DA}$$

(25)

where

$$\mathbf{F}\mathbf{=}{\left[\begin{array}{cccccc}\hfill Q_i\hfill & \hfill M_i\hfill & \hfill N_i\hfill & \hfill Q_j\hfill & \hfill M_j\hfill & \hfill N_j\hfill \end{array}\right]}^{\text{T}}$$

(26)

$$\mathbf{D}=\left[\begin{array}{cccccc}\hfill 0\hfill & \hfill -{\mu }_{\alpha }\hfill & \hfill 0\hfill & \hfill -{\mu }_{\beta }\hfill & \hfill 0\hfill & \hfill -{\mu }_{\gamma }\hfill \\ \multicolumn{1}{c}{{\lambda }_{\alpha }}& \hfill 0\hfill & \hfill {\lambda }_{\beta }\hfill & \hfill 0\hfill & \hfill {\lambda }_{\gamma }\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{-{\eta }_{\alpha }}& \hfill 0\hfill & \hfill -{\eta }_{\beta }\hfill & \hfill 0\hfill & \hfill -{\eta }_{\gamma }\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{{\mu }_{\alpha }\sinh \alpha }& \hfill {\mu }_{\alpha }\cosh \alpha \hfill & \hfill {\mu }_{\beta }\sinh \beta \hfill & \hfill {\mu }_{\beta }\cosh \beta \hfill & \hfill -{\mu }_{\gamma }\sin \gamma \hfill & \hfill {\mu }_{\gamma }\cos \gamma \hfill \\ \multicolumn{1}{c}{-{\lambda }_{\alpha }\cosh \alpha }& \hfill -{\lambda }_{\alpha }\sinh \alpha \hfill & \hfill -{\lambda }_{\beta }\cosh \beta \hfill & \hfill -{\lambda }_{\beta }\sinh \beta \hfill & \hfill -{\lambda }_{\gamma }\cos \gamma \hfill & \hfill -{\lambda }_{\gamma }\sin \gamma \hfill \\ \multicolumn{1}{c}{{\eta }_{\alpha }\cosh \alpha }& \hfill {\eta }_{\alpha }\sinh \alpha \hfill & \hfill {\eta }_{\beta }\cosh \beta \hfill & \hfill {\eta }_{\beta }\sinh \beta \hfill & \hfill {\eta }_{\gamma }\cos \gamma \hfill & \hfill {\eta }_{\gamma }\sin \gamma \hfill \end{array}\right]$$

(27)

From equations (22) and (25), the parameters $\mathbf{A}$ can be eliminated, and the relationship between the internal forces and displacements at the two ends of the composite beam can be obtained as

$$\mathbf{F}\mathbf{=}\mathbf{KU}$$

(28)

where the matrix $\mathbf{K}$ is the desired DSM and determined by

$$\mathbf{K}\mathbf{=}\mathbf{D}{\mathbf{B}}^{-1}$$

(29)

Applications

Free vibration frequencies

In general, each of the two ends of the beam has three different kinds of boundary conditions, that is, simply supported (S), clamped (C), and free (F) cases. They are combined to make four cases in practice, that is, SS, CC, CF, and CS, in which SS, for instance, means both ends are simply supported. ²³

If a composite beam is simply supported at the two ends, the boundary conditions are

$$w_1=w_2=0,M_1=M_2=0,N_1=N_2=0$$

(30)

Substituting equation (30) into equation (28) gives

$$\left[\begin{array}{cccc}\hfill k_{22}\hfill & \hfill k_{23}\hfill & \hfill k_{25}\hfill & \hfill k_{26}\hfill \\ \multicolumn{1}{c}{k_{32}}& \hfill k_{33}\hfill & \hfill k_{35}\hfill & \hfill k_{36}\hfill \\ \multicolumn{1}{c}{k_{52}}& \hfill k_{53}\hfill & \hfill k_{55}\hfill & \hfill k_{56}\hfill \\ \multicolumn{1}{c}{k_{62}}& \hfill k_{63}\hfill & \hfill k_{65}\hfill & \hfill k_{66}\hfill \end{array}\right]\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill {\theta }_1\hfill \\ \multicolumn{1}{c}{u_{s1}}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}\hfill {\theta }_2\hfill \\ \multicolumn{1}{c}{u_{s2}}\end{array}}\end{array}\right\}=\left\{\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right\}$$

(31)

where $k_{\mathrm{ij}}$ is the element at the ith row and jth column of the matrix $\mathbf{K}$ . Equation (31) is a homogeneous linear algebraic equation set, and the non-trivial solutions can be obtained only if the determinant of its coefficient matrix vanishes, that is

$$\left|\begin{array}{cccc}\hfill k_{22}\hfill & \hfill k_{23}\hfill & \hfill k_{25}\hfill & \hfill k_{26}\hfill \\ \multicolumn{1}{c}{k_{32}}& \hfill k_{33}\hfill & \hfill k_{35}\hfill & \hfill k_{36}\hfill \\ \multicolumn{1}{c}{k_{52}}& \hfill k_{53}\hfill & \hfill k_{55}\hfill & \hfill k_{56}\hfill \\ \multicolumn{1}{c}{k_{62}}& \hfill k_{63}\hfill & \hfill k_{65}\hfill & \hfill k_{66}\hfill \end{array}\right|=0$$

(32)

This is the characteristic equation of the free vibration of partial-interaction composite beams, from which the natural frequencies $\omega$ of the composite beams with two simply supported ends can be solved. In fact, equation (32) is transcendental with respect to the frequency $\omega$ since the elements $k_{\mathrm{ij}}$ are related to the frequency $\omega$ through the eigen-equation in equation (10). The transcendental equation has an infinite number of its roots, that is, the frequencies of the beam. To find the solutions of the characteristic equations, the frequency $\omega$ is chosen from an initial value and stepped through a sequence of small increments, and the value of the determinant in equation (32) is computed for each stepped $\omega$ and recorded. After a sufficient number of sign crossings have been identified, the values for $\omega$ that yield a zero determinant can be isolated and refined using bisection. The sign change is monitored by computing the values of the determinants using the Gauss elimination method given by the software MATLAB.

Modal shapes

After the frequencies are obtained, the corresponding modal shapes can be computed by the following procedure. First, substituting equation (30) into equations (11), (15), and (17) gives

$$\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{\cosh \alpha }& \hfill \sinh \alpha \hfill & \hfill \cosh \beta \hfill & \hfill \sinh \beta \hfill & \hfill \cos \gamma \hfill & \hfill \sin \gamma \hfill \\ \multicolumn{1}{c}{{\lambda }_{\alpha }}& \hfill 0\hfill & \hfill {\lambda }_{\beta }\hfill & \hfill 0\hfill & \hfill {\lambda }_{\gamma }\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{{\lambda }_{\alpha }\cosh \alpha }& \hfill {\lambda }_{\alpha }\sinh \alpha \hfill & \hfill {\lambda }_{\beta }\cosh \beta \hfill & \hfill {\lambda }_{\beta }\sinh \beta \hfill & \hfill {\lambda }_{\gamma }\cos \gamma \hfill & \hfill {\lambda }_{\gamma }\sin \gamma \hfill \\ \multicolumn{1}{c}{{\eta }_{\alpha }}& \hfill 0\hfill & \hfill {\eta }_{\beta }\hfill & \hfill 0\hfill & \hfill {\eta }_{\gamma }\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{{\eta }_{\alpha }\cosh \alpha }& \hfill {\eta }_{\alpha }\sinh \alpha \hfill & \hfill {\eta }_{\beta }\cosh \beta \hfill & \hfill {\eta }_{\beta }\sinh \beta \hfill & \hfill {\eta }_{\gamma }\cos \gamma \hfill & \hfill {\eta }_{\gamma }\sin \gamma \hfill \end{array}\right]\left\{\begin{array}{c}\hfill A_1\hfill \\ \multicolumn{1}{c}{A_2}\\ \multicolumn{1}{c}{A_3}\\ \multicolumn{1}{c}{A_4}\\ \multicolumn{1}{c}{A_5}\\ \multicolumn{1}{c}{A_6}\end{array}\right\}=\left\{\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right\}$$

(33)

Then, the solved frequencies are substituted into equation (33), and non-trivial solution of the coefficients $A_i(i=1,2,\dots ,6)$ except an arbitrary constant factor can be solved. These coefficients are finally substituted into equations (11)–(21), and the corresponding modal shapes can be obtained.

Critical loads of buckling

equation (32) is also the characteristic equation of the buckling of partial-interaction composite beams if the frequency $\omega$ is taken as zero. Since the elements $k_{\mathrm{ij}}$ have the implicit relationships with the axial force $P$ through equation (10), the characteristic equation is transcendental with respect to the axial force $P$ . In a similar manner with the frequency, the critical loads of the buckling of the composite beams are obtained using the bisection method as well as the corresponding modal shapes of the buckling.

Numerical examples

In order to validate the proposed DSM, the composite beams used by Xu and Wu ¹⁷ and Shen et al. ²⁴ were used. For composite beams without axial force, frequencies of free vibration of those composite beams under different boundary conditions were calculated. The results were compared with those of Xu and Wu ¹⁷ and Shen et al. ²⁴ For the composite beams with axial force, vibration frequencies of simply supported composite beams with different axial forces were calculated, and the results were compared with those of Wu et al. ¹⁸ and Shen et al. ²⁵ The corresponding vibration modes and buckling loads were also obtained and compared with those of Xu and Wu. ¹⁷ Table 1 shows the free vibration frequencies of composite beams under four boundary conditions. Table 2 shows the vibration frequencies of the composite beams under four axial forces. Table 3 shows the first five buckling loads under four boundary conditions. Figure 3 shows the first four deflection mode shapes of the SS beam. Figure 4 shows the first four relative slip mode shapes of the SS beam. Figure 5 shows the first four deflection mode shapes of the SC beam. Figure 6 shows the first four relative slip mode shapes of the SC beam. It is shown that the results of this article are in line with the existing results, which indicates the feasibility of the proposed DSM method.

Table 1.

The first 10 flexural frequencies without axial force (rad/s).

Order	SS			SC			CC			CF
	Xu and Wu 17	Shen et al. 24	This study	Xu and Wu 17	Shen et al. 24	This study	Xu and Wu 17	Shen et al. 24	This study	Xu and Wu 17	Shen et al. 24	This study
1	64.85	64.85	64.85	89.57	89.56	89.56	118.16	118.50	118.16	25.12	25.12	25.12
2	210.65	210.65	210.65	248.45	248.45	248.45	289.81	290.00	289.81	126.84	126.83	126.83
3	417.72	417.72	417.72	472.46	472.45	472.45	532.01	532.50	532.00	308.91	308.91	308.91
4	692.22	692.22	692.22	765.59	765.59	765.59	843.87	844.00	843.87	550.42	550.42	550.42
5	1038.44	1038.44	1038.44	1130.99	1130.99	1130.99	1228.54	1228.50	1228.53	861.75	861.75	861.75
6	1458.39	1458.39	1458.39	1570.25	1570.24	1570.24	1687.06	1687.00	1687.05	1244.70	1244.69	1244.69
7	1953.01	1953.01	1953.01	2084.18	2084.17	2084.17	2220.29	2220.50	2220.28	1701.93	1701.93	1701.93
8	2522.76	2522.76	2522.76	2673.21	2673.20	2673.21	2828.56	2829.00	2828.56	2233.77	2233.76	2233.76
9	3167.89	3167.93	3167.89	3337.59	3337.63	3337.58	3512.16	3511.50	3512.16	2840.97	2840.99	2840.96
10	3888.55	3887.40	3888.55	4077.45	4078.75	4077.45	4271.21	4265.50	4271.20	3523.54	3523.45	3523.53

Table 2.

The first 10 flexural frequencies with different axial forces (rad/s).

Order	P = −50 kN			P = 0 kN			P = 100 kN			P = 1000 kN
	Wu et al. 18	Shen et al. 25	This study	Wu et al. 18	Shen et al. 25	This study	Wu et al. 18	Shen et al. 25	This study	Wu et al. 18	Shen et al. 25	This study
1	58.56	58.56	58.56	64.85	64.85	64.85	75.88	75.88	75.88	140.44	140.44	140.44
2	203.15	203.15	203.15	210.65	210.65	210.65	224.90	224.90	224.90	326.26	326.26	326.26
3	409.28	409.28	409.28	417.72	417.72	417.72	434.12	434.12	434.12	560.50	560.50	560.50
4	683.19	683.19	683.19	692.22	692.22	692.22	709.93	709.93	709.93	852.91	852.91	852.91
5	1029.06	1029.06	1029.06	1038.44	1038.44	1038.44	1056.95	1056.95	1056.95	1210.91	1210.91	1210.91
6	1448.78	1448.78	1448.78	1458.39	1458.39	1458.39	1477.42	1477.42	1477.42	1638.77	1638.77	1638.77
7	1943.25	1943.25	1943.25	1953.01	1953.01	1953.01	1972.38	1972.38	1972.38	2138.84	2138.84	2138.84
8	2512.89	2512.89	2512.89	2522.76	2522.76	2522.76	2542.36	2542.36	2542.36	2712.46	2712.46	2712.46
9	3157.95	3157.96	3157.95	3167.89	3167.88	3167.89	3187.67	3187.68	3187.67	3360.43	3360.48	3360.43
10	3878.56	3878.41	3878.56	3888.55	3888.70	3888.55	3908.45	3908.65	3908.45	4083.22	4080.87	4083.22

Table 3.

The first five buckling loads under four boundary conditions (kN).

Order	SS		SC		CC		CF
	Xu and Wu 17	This study	Xu and Wu 17	This study	Xu and Wu 17	This study	Xu and Wu 17	This study
1	271.0222	271.0183	444.2441	444.2355	714.8772	714.8625	84.0703	84.0699
2	714.8772	714.8625	937.2378	937.2198	1148.8943	1148.8748	485.8875	485.8780
3	1249.3871	1249.3657	1544.8915	1544.8685	1929.8718	1929.8469	966.4534	966.4349
4	1929.8718	1929.8469	2310.5175	2310.4918	2695.5925	2695.5664	1569.3896	1569.3662
5	2779.6112	2779.5842	3249.6805	3249.6532	3807.1939	3807.1660	2332.8660	2332.8400

Figure 3.

The first four deflection mode shapes of the SS beam.

Figure 4.

The first four relative slip mode shapes of the SS beam.

Figure 5.

The first four deflection mode shapes of the SC beam.

Figure 6.

The first four relative slip mode shapes of the SC beam.

Conclusion

Based on the assumption of Euler–Bernoulli beam theory, the DSM of composite beams with interlayer slips with axial force was determined. Furthermore, an accurate solution for free vibration frequency of composite beams without axial force under different boundary conditions and vibration frequency of simply supported composite beams under different axial forces were obtained. The corresponding vibration modes and buckling loads were also obtained. The results in this article may provide a reference for other approximate methods or numerical methods.