A Unified Numerical Approach to the Dynamics of Beams with Longitudinally Varying Cross-Sections, Materials, Foundations, and Loads Using Chebyshev Spectral Approximation

Abstract

Structures with inhomogeneous materials, non-uniform cross-sections, non-uniform supports, and subject to non-uniform loads are increasingly common in aerospace applications. This paper presents a simple and unified numerical dynamics model for all beams with arbitrarily axially varying cross-sections, materials, foundations, loads, and general boundary conditions. These spatially varying properties are all approximated by high-order Chebyshev expansions, and discretized by Gauss–Lobatto sampling. The discrete governing equation of non-uniform axially functionally graded beams resting on variable Winkler–Pasternak foundations subjected to non-uniformly distributed loads is derived based on the Euler–Bernoulli beam theory. A projection matrix method is employed to simultaneously assemble spectral elements and impose general boundary conditions. Numerical experiments are performed to validate the proposed method, considering different inhomogeneous materials, boundary conditions, foundations, cross-sections, and loads. The results are compared with those reported in the literature and obtained by the finite element method, and excellent agreement is observed. The convergence, accuracy, and efficiency of the proposed method are demonstrated.

1. Introduction

Inhomogeneous materials, non-uniform cross-sections, variable elastic foundations, and non-uniformly distributed loads are commonplace in the static and dynamic analysis of structures, making the problems significantly more complex. In particular, in the last few decades, functionally graded materials (FGMs) and structures have been booming [1], and the associated structural optimization methods are rapidly developing [2]. By optimizing material gradient, geometrical shape, and size simultaneously, FGM structures can precisely meet different requirements on mechanical performance in different parts of the same structure [3,4,5,6,7,8]. With the development of additive manufacturing technology, more of such complex structures are being produced and used in engineering [9,10]. However, sophisticated material distributions and structure shapes, along with complicated foundations and loads, result in significant difficulties in the mechanical analysis of these structures. Figure 1 shows a rare but possible case, where the beam is designed to be non-uniform and inhomogeneous in order to work better on a variable elastic foundation when subjected to a non-uniformly distributed load. Similar but simpler beams can be found in the literature [7,11,12]. Numerous works have been separately dedicated to the mechanics of FGM beams [13,14], non-uniform beams [15,16], beams on variable elastic foundations [17,18], and beams subjected to non-uniformly distributed loads, but have rarely considered these factors simultaneously.

There are many works on axially functionally graded (AFG) beams with variable cross-sections, almost focusing on solving their natural frequencies and mode shapes using various analytical [14,19] or numerical methods [12,20,21], with only a few exceptions that considered dynamic responses. Calim [22] investigated the dynamic response of tapered AFG beams using a complementary functions method. Han [23] provided an analytical solution of the steady-state dynamic responses of non-uniform AFG beams. Chen [24] studied the vibration response of a double-FG porous beam system acted on by a moving load. However, these works did not involve foundations, and the cross-sections and loads are simple, not arbitrary. Moreover, works on the dynamics of non-uniform FGM beams on elastic foundations are rare, especially AFG beams. Li et al. [25] proposed an analytical solution for the free vibration of non-uniform FGM beams resting on Pasternak elastic foundations, but the material properties vary along the thickness direction and the Pasternak elastic parameters are constants. Kumar [26] studied the free vibration of an AFG beam on a uniform Pasternak foundation, where only clamped–clamped and simply-supported boundary conditions were considered. Ta et al. [27] provided an analytical formulation for the eigen solutions of FGM beams with variable cross-sections supported by springs at both ends. Robinson and Adali [28] investigated the buckling of non-uniform AFG beams on Winkler–Pasternak foundations, where foundation parameters are still constants. In general, the above studies consider, at most, two cases of inhomogeneity or non-uniformity, and there are only a few studies on forced vibration.

In addition, the spectral method [29,30,31,32,33,34] is used in beam section optimization and vibration analysis because of its rapid convergence and high accuracy. Fang et al. [29,30] studied the free vibration of rotating tapered beams made of axially functional graded materials by using the Chebyshev–Ritz method. Soltani et al. [31] presented a new hybrid approach for stability and free vibration analyses of axially functionally graded non-uniform beams resting on a constant Winkler–Pasternak elastic foundation. Liu et al. [32] studied the random vibration characteristics of functionally graded porous (FGP) curved beams with elastically restrained ends by using the Chebyshev spectral Method. Abdalla [33] presented the pseudo-spectral method for the minimization of the volume of elastic straight beams undergoing plane deformation and subject to buckling loads. Wang et al. [34] developed an improved weak-form quadrature element (IWQE) method based on Chebyshev interpolation and Gauss–Lobatto quadrature together.

A review of the literature shows that previous works partially, but not fully, considered the non-homogeneity and non-uniformity in materials, geometry, foundations, and loads. Analytical and numerical methods for specified material gradients, cross-sections, foundations, and boundary conditions gradually emerged [35]. But a general and consistent numerical method that can tackle all these non-homogeneities and non-uniformities is still rare. To the authors’ knowledge, the dynamic response of non-uniform and axially functionally graded beams resting on the variable elastic foundations and subjected to non-uniformly distributed loads has also not been studied. From the standpoint of structural optimization, the more complex loads and foundations are, the more sophisticated material distributions and structural geometry after optimizations. Therefore, a general numerical method for the statics and dynamics of this kind of complex beam is essential for structural optimization.

Thus, the objective of this paper is to present a novel Chebyshev spectral method that can cope with inhomogeneities or non-uniformities of materials, geometries, foundations, and loads simultaneously, for the statics and dynamics of sophisticated beams. This method combines the characteristics of the finite element method (FEM), the spectral method, and the mesh-free method, independent of material gradients, laws of variation of cross-sections and foundations, and can handle all classical boundary conditions and arbitrarily distributed loads. Most importantly, it treats the four kinds of non-homogeneity and non-uniformity, i.e., materials, geometry, foundations, and loads, in a consistent manner.

2. Problem Description

Consider the beam shown in Figure 1, which is made of a functionally graded material; it has an arbitrarily varying cross-section, rests on an inhomogeneous Winkler–Pasternak foundation, and is subjected to a non-uniformly distributed load. This means that all material, geometric, foundation, and load parameters vary with the coordinate x, i.e.,

$$E=E\left(x\right),\rho =\rho \left(x\right),A=A\left(x\right),I=I\left(x\right),k_w=k_w\left(x\right),k_p=k_p\left(x\right),q=q(x,t)$$

(1)

where E is the Young’s modulus, $\rho$ is the material density, A is the area of the cross-section, I is the moment of inertia of the cross-section. $k_w$ and $k_p$ are the Winkler–Pasternak elastic parameters of the foundation, $q(x,t)$ is a non-uniformly distributed load. Without loss of generality, all these functions are arbitrary. When some of them become constants, the problem reduces to a simpler classical problem, as shown in Figure 2. This paper aims to provide a general numerical approach to solve the static and dynamic responses of all kinds of beams shown in Figure 1 and Figure 2, and to validate the convergence, accuracy, and efficiency.

3. Theoretical Formulation

In this section, firstly, a Chebyshev spectral approximation proposed by Yagci et al. [36] and modified by the authors [21] is briefly reviewed, which is employed to approximate all the spatially varying parameters. Then the discrete equations governing the bending and vibration of the beams are derived from Lagrange’s equation. Finally, a projection matrix method is introduced to assemble spectral elements and apply boundary conditions.

3.1. Chebyshev Spectral Approximation

Discretizing and approximating functions are the core problems in computational mechanics. Exploiting the superior properties of the Chebyshev polynomials, Yagci et al. [36] proposed a spectral Tchebyshev technique for solving linear and nonlinear beam equations. This technique uses Chebyshev polynomials as spatial basis functions and applies Galerkin’s method to obtain the spatially discretized equations of motion. The authors improved and extended this technique in recent works [21,37,38,39], and re-derived the inner-product matrix and weighted inner-product matrix, making them diagonal in order to improve the efficiency of the Chebyshev spectral approximation. Herein, this method is briefly reviewed and then employed to establish the Chebyshev spectral element of non-uniform and inhomogeneous beams.

The Chebyshev polynomials are a group of recursive orthogonal polynomials defined by

$$T_0\left(x\right)=1,T_1\left(x\right)=x,T_{k+1}=2x{T}_k\left(x\right)-T_{k-1}\left(x\right),k\ge 1$$

(2)

where k is an integer and $x\in [-1,1]$. For the sake of approximating functions defined on an arbitrary interval $[{\ell }_1,{\ell }_2]$, define the scaled Chebyshev polynomials as

$${\mathcal{T}}_k\left(x\right)=T_k\left(\xi \left(x\right)\right),\xi \left(x\right)=\frac{2}{{\ell }_2-{\ell }_1}x-\frac{{\ell }_2+{\ell }_1}{{\ell }_2-{\ell }_1}$$

(3)

Then an infinitely differentiable and square-integrable function $y(x,t)$ in the interval $[{\ell }_1,{\ell }_2]$ can be approximated by a truncated Chebyshev series expansion as

$$y(x,t)=\sum _{k=0}^{N-1}{a}_k\left(t\right){\mathcal{T}}_k\left(x\right)$$

(4)

Adopting the Gauss–Lobatto sampling

$$x_k=\frac{{\ell }_1+{\ell }_2}{2}-\frac{{\ell }_2-{\ell }_1}{2}cos[k\pi /(N-1)],y_k=y(x_k,t)$$

(5)

where $k=0,1,\dots ,N-1$. There exists a one-to-one mapping between the expansion coefficients vector $\mathbf{a}=\left\{a_k\right\}$ and the sampling value vector $\mathbf{y}=\left\{y_k\right\}$, which can be expressed as

$$\mathbf{a}={\mathsf{\Gamma }}_F\mathbf{y},\mathbf{y}={\mathsf{\Gamma }}_B\mathbf{a}$$

(6)

where ${\mathsf{\Gamma }}_F$ is an $N\times N$ forward transformation matrix, and ${\mathsf{\Gamma }}_B$ is its inverse matrix.

The n-th spatial derivative of $y(x,t)$ can be obtained by [36]

$${\mathbf{y}}^{\left(n\right)}={\mathbf{Q}}_n\mathbf{y}$$

(7)

where ${\mathbf{Q}}_n$ is the n-th derivative matrix, a $N\times N$ sparse matrix with constant elements determined by the interval $[{\ell }_1,{\ell }_2]$ and the number of Chebyshev polynomials used, i.e., N.

Regarding the inner product of two functions, $f\left(x\right)$ and $g\left(x\right)$, both of them are infinitely differentiable and square-integrable, and can be calculated by [21]

$${\int }_{l_1}^{l_2}f\left(x\right)g\left(x\right)\mathrm{d}x={\mathbf{f}}^{\mathrm{T}}\mathbf{V}\mathbf{g}$$

(8)

where $\mathbf{V}$ is the inner product matrix, a diagonal constant matrix determined by N, $\mathbf{f}=\left\{\mathbf{f}\left({\mathbf{x}}_{\mathbf{k}}\right)\right\}$, and $\mathbf{g}=\left\{\mathbf{g}\left({\mathbf{x}}_{\mathbf{k}}\right)\right\}$ is the sampling value vector of $f\left(x\right)$ and $g\left(x\right)$, respectively.

Likewise, the weighted inner product between $f\left(x\right)$ and $g\left(x\right)$ about a weight function $h\left(x\right)$ can be given by [21]

$${\int }_{{\ell }_1}^{{\ell }_2}f\left(x\right)h\left(x\right)g\left(x\right)\mathrm{d}x={\mathbf{f}}^{\mathrm{T}}{\mathbf{V}}_h\mathbf{g}$$

(9)

where ${\mathbf{V}}_h$ is the weighted inner product matrix about the weight function $h\left(x\right)$, which is also a diagonal constant matrix determined by N and the values of $h\left(x\right)$ at the sampling points.

3.2. Spectral Element of Non-Uniform AFG Beams Resting on Variable Foundation

Considering the beam in Figure 1, if its material, cross-section, and foundation parameters vary smoothly, one Chebyshev spectral element is adequate for the static and dynamic analysis of the beam. Utilizing the aforementioned Chebyshev spectral approximation and the Euler–Bernoulli beam theory, the formula of a novel Chebyshev spectral element for non-uniform AFG beams resting on variable Winkler–Pasternak foundations is presented in this section.

Denote the transverse displacement of the neutral axis of the Euler–Bernoulli beam as $w(x,t)$, the vertical foundation reaction is presumed to be

$$p(x,t)=-k_w\left(x\right)w(x,t)+k_p\left(x\right)\frac{{\partial }^{2}w}{\partial x^2}$$

(10)

where $k_w$ is the sub-grade reaction modulus, and $k_p$ is the shear foundation modulus.

The kinetic energy and the strain energy of the beam are given by

$$T=\frac{1}{2}{\int }_0^l\rho \left(x\right)A\left(x\right){\left(\frac{\partial w}{\partial t}\right)}^2\mathrm{d}x,U_s=\frac{1}{2}{\int }_0^{l}E\left(x\right)I\left(x\right){\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2\mathrm{d}x$$

(11)

The elastic potential energy of the elastic foundation is [40]

$$U_f=\frac{1}{2}{\int }_0^l\left[k_w\left(x\right)w^2+k_p\left(x\right){\left(\frac{\partial w}{\partial x}\right)}^2\right]\mathrm{d}x$$

(12)

The work conducted by the external load can be calculated by

$$W={\int }_0^{l}q(x,t)w\mathrm{d}x$$

(13)

The transverse displacement $w(x,t)$ can be approximated by a truncated Chebyshev expansion,

$$w(x,t)\approx \sum _{k=0}^{N-1}{a}_k{\mathcal{T}}_k\left(x\right)$$

(14)

Adopting the Gauss–Lobatto sampling to discrete the displacement field, then the aforementioned Chebyshev spectral approximation can be applied. Define

$$h_1\left(x\right)=\rho \left(x\right)A\left(x\right),h_2\left(x\right)=E\left(x\right)I\left(x\right),h_3\left(x\right)=k_w\left(x\right),h_4\left(x\right)=k_p\left(x\right)$$

(15)

According to Equations (7) to (9), the energies and the work can be computed as

$$T=\frac{1}{2}{\dot{\mathbf{w}}}^{\mathrm{T}}{\mathbf{V}}_{h_1}\dot{\mathbf{w}},U_s=\frac{1}{2}{\mathbf{w}}^{\mathrm{T}}\left({{\mathbf{Q}}_2}^{\mathrm{T}}{\mathbf{V}}_{h_2}{\mathbf{Q}}_2\right)\mathbf{w}$$

(16)

$$U_f=\frac{1}{2}{\mathbf{w}}^{\mathrm{T}}{\mathbf{V}}_{h_3}\mathbf{w}+\frac{1}{2}{\mathbf{w}}^{\mathrm{T}}\left({{\mathbf{Q}}_1}^{\mathrm{T}}{\mathbf{V}}_{h_4}{\mathbf{Q}}_1\right)\mathbf{w},W={\mathbf{q}}^{\mathrm{T}}\mathbf{V}\mathbf{w}={{\mathbf{F}}_{\mathrm{equal}}}^{\mathrm{T}}\mathbf{w}$$

(17)

where $\dot{\mathbf{w}}$ is the sampled vector of the velocity $\dot{w}$ at the Gauss–Lobatto points, ${\mathbf{V}}_{h_i}$ is the weighted inner product matrix of the weight function $h_i,i=1,2,3,4$, and ${\mathbf{Q}}_1$ and ${\mathbf{Q}}_2$ are, respectively, the 1-th and 2-th derivative matrices, as shown in Equation (7). $\mathbf{q}$ is the sampled vector of the distributed load $q\left(x\right)$, and ${\mathbf{F}}_{\mathrm{equal}}={\mathbf{V}}^{\mathrm{T}}\mathbf{q}$ is the equivalent concentrated loads applied at the Gauss–Lobatto points.

The Lagrangian, denoted as $L=T-U_s-U_f$, serves as the basis for deriving the governing equation of motion by employing Lagrange’s equation

$$\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{w_i}}-\frac{\partial L}{\partial w_i}={{\mathbf{F}}_{\mathrm{equal}}}_i,i=0,1,\dots ,N-1$$

(18)

The obtained discrete governing equation is given by

$${\mathbf{V}}_{h_1}\ddot{\mathbf{w}}+\left({{\mathbf{Q}}_2}^{\mathrm{T}}{\mathbf{V}}_{h_2}{\mathbf{Q}}_2+{\mathbf{V}}_{h_3}+{{\mathbf{Q}}_1}^{\mathrm{T}}{\mathbf{V}}_{h_4}{\mathbf{Q}}_1\right)\mathbf{w}={\mathbf{F}}_{\mathrm{equal}}$$

(19)

For convenience, it can be rewritten in a concise form as follows:

$$\mathcal{M}\ddot{\mathcal{X}}+\mathcal{K}\mathcal{X}=\mathcal{F}$$

(20)

where

$$\mathcal{X}=\mathbf{w},\mathcal{M}={\mathbf{V}}_{h_1},\mathcal{K}={{\mathbf{Q}}_2}^{\mathrm{T}}{\mathbf{V}}_{h_2}{\mathbf{Q}}_2+{\mathbf{V}}_{h_3}+{{\mathbf{Q}}_1}^{\mathrm{T}}{\mathbf{V}}_{h_4}{\mathbf{Q}}_1,\mathcal{F}={\mathbf{F}}_{\mathrm{equal}}$$

(21)

It should be noted that boundary conditions are not imposed yet. They will be handled along with assembling spectral elements by a projection matrix method, which will be explained in detail in the next subsection.

3.3. Assembling Spectral Elements and Imposing Boundary Conditions

When discontinuities or abrupt changes in material properties, sections, foundations, or loads exist, more than one Chebyshev spectral element should be employed. Figure 3 illustrates some cases of these kinds. In this section, projection matrices are introduced to assemble Chebyshev spectral elements and handle general boundary conditions and compatibility between elements.

For a beam divided into $\mathcal{N}$ spectral elements, there are $\mathcal{N}$ governing equations of elements

$${\mathcal{M}}^{\left(i\right)}{\ddot{\mathcal{X}}}^{\left(i\right)}+{\mathcal{K}}^{\left(i\right)}{\mathcal{X}}^{\left(i\right)}={\mathcal{F}}^{\left(i\right)},i=1,2,\dots ,\mathcal{N}$$

(22)

where the superscript $\left(i\right)$ indicates element i.

The first step of the assembly procedure is just writing them together in a matrix form.

$$\left[\begin{array}{cccc}{\mathcal{M}}^{\left(1\right)}& & & \\ & {\mathcal{M}}^{\left(2\right)}& & \\ & & \ddots & \\ & & & {\mathcal{M}}^{\left(\mathcal{N}\right)}\end{array}\right]\left[\begin{array}{c}{\ddot{\mathcal{X}}}^{\left(1\right)}\\ {\ddot{\mathcal{X}}}^{\left(2\right)}\\ ⋮\\ {\ddot{\mathcal{X}}}^{\left(\mathcal{N}\right)}\end{array}\right]+\left[\begin{array}{cccc}{\mathcal{K}}^{\left(1\right)}& & & \\ & {\mathcal{K}}^{\left(2\right)}& & \\ & & \ddots & \\ & & & {\mathcal{K}}^{\left(\mathcal{N}\right)}\end{array}\right]\left[\begin{array}{c}{\mathcal{X}}^{\left(1\right)}\\ {\mathcal{X}}^{\left(2\right)}\\ ⋮\\ {\mathcal{X}}^{\left(\mathcal{N}\right)}\end{array}\right]=\left[\begin{array}{c}{\mathcal{F}}^{\left(1\right)}\\ {\mathcal{F}}^{\left(2\right)}\\ ⋮\\ {\mathcal{F}}^{\left(\mathcal{N}\right)}\end{array}\right]$$

(23)

or a more concise form, ${\mathcal{M}}_{\mathrm{g}}{\mathcal{X}}_{\mathrm{g}}+{\mathcal{K}}_{\mathrm{g}}{\mathcal{X}}_{\mathrm{g}}={\mathcal{F}}_{\mathrm{g}}$, where subscript $\mathrm{g}$ means global.

All boundary conditions and continuity requirements between elements can be written as

$${\mathit{\beta }}^{\mathrm{T}}{\mathcal{X}}_{\mathrm{g}}=0$$

(24)

where $\mathit{\beta }$ is an $N·\mathcal{N}$-dimensional vector. For example, for a beam clamped at the left end, both the displacement and the slope must be zero, i.e.,

$${\left.w\right|}_{x=0}=0,{\left.\partial w/\partial x\right|}_{x=0}=0$$

(25)

This can be written in a discrete form as

$${{\mathit{e}}_1}^{\mathrm{T}}{\mathcal{X}}^{\left(1\right)}=0,{{\mathit{e}}_1}^{\mathrm{T}}{\mathit{Q}}_1^{\left(1\right)}{\mathcal{X}}^{\left(1\right)}=0$$

(26)

where ${\mathit{e}}_1$ is an N-dimensional vector whose 1-th element is unity and all other elements are zero.

Another example is the continuity requirements at the common node of two adjacent elements i and $i+1$. For the Euler–Bernoulli beam, the displacement, slope, and internal moment at the common node should be the same for two adjacent elements. That means w, $\partial w/\partial x$, and ${\partial }^{2}w/\partial x^2$ are continuous at the common node, i.e.,

$${{\mathbf{e}}_N}^{\mathrm{T}}{\mathcal{X}}^{\left(i\right)}={{\mathbf{e}}_1}^{\mathrm{T}}{\mathcal{X}}^{(i+1)},{{\mathbf{e}}_N}^{\mathrm{T}}{\mathbf{Q}}_1^{\left(i\right)}{\mathcal{X}}^{\left(i\right)}={{\mathbf{e}}_1}^{\mathrm{T}}{\mathbf{Q}}_1^{(i+1)}{\mathcal{X}}^{(i+1)},{{\mathbf{e}}_N}^{\mathrm{T}}{\mathbf{Q}}_2^{\left(i\right)}{\mathcal{X}}^{\left(i\right)}={{\mathbf{e}}_1}^{\mathrm{T}}{\mathbf{Q}}_2^{(i+1)}{\mathcal{X}}^{(i+1)}$$

(27)

where ${\mathbf{e}}_i,i=1,N$ represents N-dimensional vectors, whose i-th element is unity and all other elements are zero. These three constraints can be written in the following form:

$$\left[\begin{array}{cccccc}0& \dots & {{\mathbf{e}}_N}^{\mathrm{T}}& -{{\mathbf{e}}_1}^{\mathrm{T}}& \dots & 0\\ 0& \dots & {{\mathbf{e}}_N}^{\mathrm{T}}{\mathbf{Q}}_1^{\left(i\right)}& -{{\mathbf{e}}_1}^{\mathrm{T}}{\mathbf{Q}}_1^{(i+1)}& \dots & 0\\ 0& \dots & {{\mathbf{e}}_N}^{\mathrm{T}}{\mathbf{Q}}_2^{\left(i\right)}& -{{\mathbf{e}}_1}^{\mathrm{T}}{\mathbf{Q}}_2^{(i+1)}& \dots & 0\end{array}\right]\left[\begin{array}{c}{\mathcal{X}}^{\left(1\right)}\\ ⋮\\ {\mathcal{X}}^{\left(i\right)}\\ {\mathcal{X}}^{(i+1)}\\ ⋮\\ {\mathcal{X}}^{\mathcal{N}}\end{array}\right]=\left[\begin{array}{c}0\\ ⋮\\ 0\\ 0\\ ⋮\\ 0\end{array}\right]$$

(28)

or a compact form

$$\left[\begin{array}{c}{\mathit{\beta }}_1^{\mathrm{T}}\\ {\mathit{\beta }}_2^{\mathrm{T}}\\ {\mathit{\beta }}_3^{\mathrm{T}}\end{array}\right]{\mathcal{X}}_{\mathrm{g}}=\mathcal{B}{\mathcal{X}}_{\mathrm{g}}=\mathbf{0}$$

(29)

Every row of the matrix $\mathcal{B}$, i.e., ${\mathit{\beta }}_i,i=1,2,3$, represents a boundary condition or continuity requirement.

Now, the problem evolves into solving the equation

$$\left\{\begin{array}{c}{\mathcal{M}}_{\mathrm{g}}{\mathcal{X}}_{\mathrm{g}}+{\mathcal{K}}_{\mathrm{g}}{\mathcal{X}}_{\mathrm{g}}={\mathcal{F}}_{\mathrm{g}}\hfill \\ \mathcal{B}{\mathcal{X}}_{\mathrm{g}}=\mathbf{0}\hfill \end{array}\right.$$

(30)

The projection matrix method introduced in Ref. [36] is an effective way to impose the boundary conditions and continuity requirements. The solution can be expressed as

$${\mathcal{X}}_{\mathrm{g}}=\mathcal{P}{\hat{\mathcal{X}}}_{\mathrm{g}}$$

(31)

where $\mathcal{P}$ is the $(N·\mathcal{N})\times (N·\mathcal{N}-n)$ projection matrix, n is the number of boundary conditions and continuity requirements, i.e., the number of rows of matrix $\mathcal{B}$.

Multiply both sides of Equation (23) by ${\mathcal{P}}^{\mathrm{T}}$, the global governing equation becomes

$$\widehat{{\mathcal{M}}_{\mathrm{g}}}\widehat{{\mathcal{X}}_{\mathrm{g}}}+\widehat{{\mathcal{K}}_{\mathrm{g}}}\widehat{{\mathcal{X}}_{\mathrm{g}}}=\widehat{{\mathcal{F}}_{\mathrm{g}}}$$

(32)

where

$$\widehat{{\mathcal{M}}_{\mathrm{g}}}={\mathcal{P}}^{\mathrm{T}}{\mathcal{M}}_{\mathrm{g}}\mathcal{P},\widehat{{\mathcal{K}}_{\mathrm{g}}}={\mathcal{P}}^{\mathrm{T}}{\mathcal{K}}_{\mathrm{g}}\mathcal{P},\widehat{{\mathcal{F}}_{\mathrm{g}}}={\mathcal{P}}^{\mathrm{T}}{\mathcal{F}}_{\mathrm{g}}$$

(33)

Solving the above equation, the static and dynamic responses of non-uniform AFG beams resting on variable foundations and subject to non-uniformly distributed loads can be obtained.

4. Numerical Examples

To validate the proposed methods, a series of numerical experiments are carried out, involving both static and dynamic problems. Various types of FGMs, foundations, loads, and sections are considered, including

• Case 1: Uniform beams on the homogeneous foundation;
• Case 2: Stepped beams made of axially functionally graded materials;
• Case 3: Non-uniform AFG beams resting on the variable foundation.

The results calculated by the proposed method are compared with analytical and numerical results found in the literature or obtained by FEM.

4.1. Uniform Beams on Homogeneous Foundations

First, consider the simplest case, a uniform Euler–Bernoulli beam made of homogeneous material resting on a homogeneous elastic foundation. The parameters of the rectangular beam include a length of 2 m, a width of 0.05 m, and a thickness of 0.05 m. The material density is $\rho =7850\mathrm{kg}/{\mathrm{m}}^3$; Young’s modulus is $E=210\mathrm{GPa}$. The static deflections under uniformly distributed loads, natural frequencies, and dynamic responses are all computed by the proposed method.

For static problems, define the dimensionless mid-span deflection and foundation parameters as follows:

$$\overline{w}=\frac{wEI}{q{l}^4},{\overline{k}}_w=\frac{k_w{l}^4}{EI},{\overline{k}}_p=\frac{k_p{l}^2}{EI}$$

(34)

where q is the uniformly distributed load with unit $\mathrm{N}/\mathrm{m}$. The relative error is defined as

$${Error}_N=|{\overline{w}}_N-{\overline{w}}_{exact}|/{\overline{w}}_{exact}$$

(35)

where ${\overline{w}}_N$ is the value of $\overline{w}$ calculated by the proposed method using N Chebyshev polynomials, and ${\overline{w}}_{exact}$ is the exact value of $\overline{w}$ in the literature [41].

Figure 4 presents the convergence of the relative errors. Two kinds of boundary conditions, i.e., clamped–clamped (CC) and simply-supported (SS), and various values of foundation parameters are considered. It can be found that the relative errors converge rapidly as the number of Chebyshev polynomials increases. When 12 polynomials are used in one Chebyshev spectral element, the relative errors ${Error}_N$ in all cases are smaller than ${10}^{-4}$, i.e., $0.01\%$.

Table 1. Mid-span dimensionless deflection $\overline{w}\times {10}^{-3}$ of a homogeneous beam under the uniformly distributed load.

B.C.	Foundation Parameters		DQM [42]	Exact [41]	Present
	${\overline{\mathit{k}}}_{\mathit{w}}$	${\overline{\mathit{k}}}_{\mathit{p}}$
CC	0	0	2.6064	2.60417	2.60417
		10	2.0862	2.08454	2.08455
		25	1.6081	1.60687	1.60688
	10	0	2.5547	2.55256	2.55256
		10	2.0528	2.05116	2.05116
		25	1.5880	1.58681	1.58683
	100	0	2.1670	2.16547	2.16547
		10	1.7935	1.79229	1.79229
		25	1.4273	1.42633	1.42635
SS	0	0	13.02290	13.02083	13.02083
		10	6.44827	6.44771	6.44771
		25	3.66111	3.66091	3.66092
	10	0	11.80567	11.80396	11.80396
		10	6.13325	6.13275	6.13275
		25	3.55668	3.55649	3.55649
	100	0	6.40074	6.40020	6.40020
		10	4.25582	4.25557	4.25557
		25	2.82846	2.82834	2.82835

tabulates the values of mid-span deflection when 16 polynomials are used. The closed-form solutions given by Doeva [41] and the differential quadrature method (DQM) solution presented by Chen et al. [42] are also given for comparison. All the results show that the solutions obtained by the proposed method agree well with the exact analytical solutions, superior to the DQM solutions. Moreover,

Table 2. Maximum dimensionless deflection $\overline{w}\times {10}^{-3}$ of a homogeneous beam under the uniformly distributed load, ${\overline{k}}_p=10$.

	${\overline{\mathit{k}}}_{\mathit{w}}=10$			${\overline{\mathit{k}}}_{\mathit{w}}=25$			${\overline{\mathit{k}}}_{\mathit{w}}=100$
	SC	SF	CF	SC	SF	CF	SC	SF	CF
FEM [43]	3.54113	36.29669	23.01367	3.39874	25.59052	18.34552	2.82864	10.01332	8.92297
Present	3.54142	36.29669	23.01367	3.39903	25.59052	18.34552	2.82896	10.01332	8.92297
$Erro{r}_{16}$	0.008%	0%	0%	0.009%	0%	0%	0.011%	0%	0%

presents the values of the maximum non-dimensional deflections for the other three kinds of boundary conditions, including simply-supported-clamped (SC), simply-supported-free (SF), and clamped-free (CF). The maximum difference between the present results and those in Ref. [43] is a mere $0.011\%$. The above results demonstrate the accuracy and convergence of the proposed method.

For free vibration problems, define a non-dimensional frequency parameter

$${\omega }_{nor}=\sqrt{\omega l^2\sqrt{\rho A/\left(EI\right)}}$$

(36)

where $\omega$ denotes the natural frequency ($\mathrm{rad}/\mathrm{s}$). The convergence of the first three non-dimensional frequency parameters is shown in Figure 5, where three kinds of boundary conditions are considered, including simply-supported (SS), clamped-simply-supported (CS), and clamped–clamped (CC), and one Chebyshev spectral element is used. It is seen from Figure 5 that a rapid convergence is observed, the values stabilize in a very small range when more than eight polynomials are used. The values of non-dimensional frequency parameters are listed in

Table 3. ${\omega }_{nor}$ of a uniform and homogeneous Euler–Bernoulli beam on the Winkler–Pasternak foundation.

B.C.	${\overline{\mathit{k}}}_{\mathit{w}}$	${\overline{\mathit{k}}}_{\mathit{p}}$	Model Number
			1			2			3
			[44]	FEM	Present	[44]	FEM	Present	[44]	FEM	Present
SS	0	0	3.142	3.1416	3.14159	6.283	6.2832	6.28318	9.425	9.4248	9.42478
	$0.6{\pi }^4$	0	3.533	3.5333	3.53329	6.341	6.3413	6.34128	9.442	6.4422	9.44218
	$0.6{\pi }^4$	1	3.588	3.5880	3.58795	6.380	6.3796	6.37963	9.468	9.4685	9.46845
CS	0	0	3.927	3.9266	3.92660	7.069	7.0686	7.06858	10.210	10.210	10.2102
	$0.6{\pi }^4$	0	4.148	4.1484	4.14843	7.109	7.1096	7.10959	10.224	10.224	10.2239
	$0.6{\pi }^4$	1	4.188	4.1881	4.18814	7.139	7.1392	7.13924	10.246	10.246	10.2458
CC	0	0	4.730	4.7300	4.73004	7.853	7.8532	7.85320	10.996	10.996	10.9956
	$0.6{\pi }^4$	0	4.862	4.8624	4.86245	7.883	7.8832	7.88320	11.007	11.007	11.0066
	$0.6{\pi }^4$	1	4.889	4.8890	4.88896	7.907	7.9066	7.90659	11.025	11.025	11.0251

. The results in the literature [44] and those obtained by the finite element method are also given for comparison. The finite element analysis is carried out via COMSOL Multiphysics 6.0 simulation software, using 100 Euler–Bernoulli beam elements. Excellent agreement between the results of the three methods is observed, demonstrating the validity of the proposed method.

The dynamic responses of the beam under two types of point loads at the center are also investigated.

$$q_1\left(t\right)=q_{0}sin\left(\frac{11}{12}{\omega }_{1}t\right)$$

(37a)

$$q_2\left(t\right)=\left\{\begin{array}{cc}\hfill q_0\left[1+sin\left(40\pi t-\pi /2\right)\right],\hfill & t\le 0.05\mathrm{s}\\ \hfill 0,\hfill & t>0.05\mathrm{s}\end{array}\right.$$

(37b)

where $q_0=1000\mathrm{N}$, ${\omega }_1$ is the first natural frequency of the beam. The two ends of the beam are clamped, and the parameters of the foundation are assumed to be ${\overline{k}}_w=0.6{\pi }^4$, ${\overline{k}}_p=1$. Figure 6 shows the trace of the displacements of the midpoint of the beam. The black lines and the red dashed lines are the results obtained by the Chebyshev spectral element method and the FEM, respectively. The former uses a Chebyshev spectral element of the 16-order, and the latter uses 100 finite elements. It can be seen that two sets of lines almost overlap, indicating excellent agreement. The accuracy of the proposed method is demonstrated.

4.2. AFG Stepped Beams

Consider a Euler–Bernoulli beam of a rectangular cross-section and three step changes in the cross-section, as shown in Figure 7. The geometric characteristics of the beam are the same as those used in Ref. [45], including $l_1/l=0.25$, $l_2/l=0.30$, $l_3/l=0.25$, $l_4/l=0.2$, $A_2/A_1=0.8$, $A_3/A_1=0.65$, $A_4/A_1=0.25$, $I_2/I_1={\left(0.8\right)}^3$, $I_3/I_1={\left(0.65\right)}^3$, $I_4/I_1={\left(0.25\right)}^3$. The length of the beam, $l=1\mathrm{m}$, and the width and thickness of the first section are both 0.01 m. The beam is made of an axially functionally graded material consisting of two constituents, namely Zirconia ZrO₂ and aluminum Al. Their properties $E_{zr}=200\mathrm{GPa}$, ${\rho }_{zr}=5700\mathrm{kg}/{\mathrm{m}}^3$, and $E_{al}=70\mathrm{GPa}$, ${\rho }_{al}=2702\mathrm{kg}/{\mathrm{m}}^3$. The variations of the properties of the beam follow the below laws.

$$E\left(x\right)=E_{zr}+\left(E_{al}-E_{zr}\right){(x/l)}^2,\rho \left(x\right)={\rho }_{zr}+\left({\rho }_{al}-{\rho }_{zr}\right){(x/l)}^2$$

(38)

Four Chebyshev spectral elements are adopted for the dynamic analysis of this beam, as shown in Figure 7. Each element employs eight Chebyshev polynomials. The dimensionless natural frequency is defined to be

$${\omega }_{nor}=\sqrt{\omega l^2\sqrt{{\rho }_{zr}{A}_1/\left(E_{zr}{I}_1\right)}}$$

(39)

Considering four types of boundary conditions, the calculated values of the first three modes are tabulated in

Table 4. The first three dimensionless natural frequencies, ${\omega }_{nor}$, of the stepped beam made of axially functionally graded material.

B.C.	Mode 1		Mode 2		Mode 3
	[45]	Present	[45]	Present	[45]	Present
CC	3.4077	3.407699	5.72338	5.723375	8.39335	8.393376
CS	3.00028	3.000280	5.43323	5.433228	8.04824	8.047760
SS	2.08694	2.086939	4.62467	4.624669	7.38797	7.387968
CF	2.3605	2.360501	4.32307	4.323069	5.903	5.902994

. Results in Ref. [45], which were computed by the symbolic-numeric method of initial parameters, are also given for comparison. An excellent agreement between the two sets of results is observed.

Assume that the beam is cantilevered, and subjected to a non-uniformly distributed load described by

$$q\left(x,t\right)=\left\{\begin{array}{cc}\hfill 0,& x<l_1+l_2+l_3\hfill \\ \hfill sin\left(360\pi t\right),& x\ge l_1+l_2+l_3\hfill \end{array}\right.$$

(40)

The dynamic response is investigated by the proposed method and the FEM, respectively. Displacement traces of four points, respectively, located at $x=0.25l$, $x=0.55l$, $x=0.8l$, and $x=l$, are plotted in Figure 8. As expected, the two sets of lines nearly are identical. It should be noted that only four Chebyshev spectral elements are used, each of them employing eight Chebyshev polynomials. But an equivalent precision, compared with 200 finite elements, is gained.

4.3. Non-Uniform AFG Beams on Variable Foundations

Finally, the most complex situation, as shown in Figure 1, is considered, where the materials, cross-sections, foundations, and load are all non-uniform or inhomogeneous. The beam has a length of $l=2\mathrm{m}$, and its cross-section varies along the longitudinal direction.

$$b\left(x\right)=b_0{e}^{-{\left(x-l/2\right)}^2/2}h\left(x\right)=h_0{e}^{-{\left(x-l/2\right)}^2/2}$$

(41)

where the width is $b_0=0.05\mathrm{m}$ and the height is $h_0=0.05\mathrm{m}$, The material parameters also varies axially, following the laws

$$\rho \left(x\right)={\rho }_1+\left({\rho }_2-{\rho }_1\right){\left(x/l\right)}^2,E\left(x\right)=E_1+\left(E_2-E_1\right){\left(x/l\right)}^2$$

(42)

where ${\rho }_1=5700\mathrm{kg}/{\mathrm{m}}^3$, ${\rho }_2=2702\mathrm{kg}/{\mathrm{m}}^3$, $E_1=200\mathrm{GPa}$, and $E_2=70\mathrm{GPa}$. The varying foundation parameters can be given by:

$$k_w\left(x\right)=k_{w1}+\left(k_{w2}-k_{w1}\right)\left(x/l\right),k_p\left(x\right)=k_{p1}+\left(k_{p2}-k_{p1}\right)\left(x/l\right)$$

(43)

$$k_{w1}=0.6{\pi }^4{E}_1{I}_0/l^4,k_{w2}=1.2{\pi }^4{E}_1{I}_0/l^4,k_{p1}=E_1{I}_0/l^2,k_{p2}=2E_1{I}_0/l^2$$

(44)

where $I_0=b_0{h}_0^3/12$.

A series of static problems are first investigated to assess the proposed method. Four kinds of distributed loads are taken into account, as follows.

$$\begin{array}{cc}\hfill \mathrm{Uniform}:& q\left(x\right)=q_0\hfill \end{array}$$

(45a)

$$\begin{array}{cc}\hfill \mathrm{Sinusoidal}:& q\left(x\right)=q_{0}sin(x\pi /l),\hfill \end{array}$$

(45b)

$$\begin{array}{cc}\hfill \mathrm{Exponential}:& q\left(x\right)=q_0{e}^{(x/l)}\hfill \end{array}$$

(45c)

where $q_0=1000\mathrm{N}/\mathrm{m}$. Define the dimensionless deflection $\overline{w}={10}^{3}w{E}_1{I}_0/q_0{l}^4$, and the relative error

$$err=\left|\overline{w}-{\overline{w}}_{\mathrm{FEM}}\right|/{\overline{w}}_{\mathrm{FEM}}$$

(46)

Table 5. Static deflection $\overline{w}$ of the non-uniform AFG beam on the variable foundation subjected to non-uniform distributed loads.

B.C.	Load	$\mathit{x}=0.2\mathit{l}$		$\mathit{x}=0.4\mathit{l}$		$\mathit{x}=0.6\mathit{l}$		${\left\{\mathit{err}\right\}}_{\mathbf{max}}$
		FEM	Present	FEM	Present	FEM	Present
SS	Uniform	4.97926	4.99585	7.31450	7.32638	7.53748	7.52987	0.333%
	Sinusoidal	3.78410	3.79682	5.71830	5.72696	5.87506	5.86841	0.336%
	Exponential	7.78435	7.81942	12.0451	12.0756	13.1182	13.1159	0.451%
SC	Uniform	4.58116	4.59948	6.42648	6.43455	6.07975	6.07271	0.400%
	Sinusoidal	3.51090	3.51966	5.09786	5.10337	4.85422	4.84952	0.250%
	Exponential	7.0490	7.02686	10.3421	10.3248	10.2835	10.2878	0.314%
CC	Uniform	2.43342	2.43564	4.52038	4.52419	4.86512	4.86252	0.091%
	Sinusoidal	1.93322	1.93491	3.70152	3.70425	3.96564	3.96320	0.087%
	Exponential	3.83054	3.83662	7.49591	7.50600	8.48754	8.48691	0.159%

shows the dimensionless deflection of the beam under various boundary conditions and loads, calculated by the proposed Chebyshev spectral element method and the FEM. It can be observed that the two kinds of results agree very well, and the biggest relative error is merely $0.451\%$. Figure 9 depicts the deflection of the beam under different boundary conditions and loads. The dashed lines representing the FEM results and the black lines representing the Chebyshev spectral element results are almost identical, with very small differences. The accuracy of the proposed method is validated again.

The dynamic characteristics of the beam are also investigated and shown in

Table 6. Natural frequencies (Hz) of the non-uniform AFG beam on the variable foundation.

B.C.	Mode 1		Mode 2		Mode 3		${\left\{\mathit{err}\right\}}_{\mathbf{max}}$
	Present	FEM	Present	FEM	Present	FEM
SS	49.7113	49.7130	121.554	121.554	259.659	259.659	0.003%
CC	61.9321	61.9331	157.299	157.299	319.936	319.936	0.002%
CS	56.7793	56.7806	140.869	140.869	291.068	291.068	0.002%
SC	53.3359	53.3372	135.827	135.827	285.790	285.790	0.002%

. Two 12th-order Chebyshev spectral elements and 100 finite elements are employed, respectively. The largest relative error between the two sets of solutions is a mere 0.003%.

Finally, the dynamic responses of the non-uniform AFG beam under three types of distributed loads are studied. The loads are formulated as

$$\begin{array}{cc}\hfill \mathrm{Uniform}:& q(x,t)=\left\{\begin{array}{c}{q}_0,t<60\mathrm{ms}\hfill \\ 0,t\ge 60\mathrm{ms}\hfill \end{array}\right.\hfill \end{array}$$

(47a)

$$\begin{array}{cc}\hfill \mathrm{Sinusoidal}:& q(x,t)=q_{0}sin\left(x\pi /l\right)sin\left(\sqrt{\frac{E_1}{{\rho }_1}}\frac{\pi }{10l}t\right)\hfill \end{array}$$

(47b)

$$\begin{array}{cc}\hfill \mathrm{Exponential}:& q(x,t)=q_0\frac{1}{\sqrt{2\pi \sigma }}{e}^{\left[x/l-{\left(t-t_1\right)}^2/\left(2{\sigma }^2\right)\right]}\hfill \end{array}$$

(47c)

where $t_1=5\mathrm{ms}$, $\sigma =0.5$. The displacement history of three points in 100 ms, respectively, at $x=l/4$, $x=l/2$, and $x=3l/4$, is plotted in Figure 10. The Chebyshev spectral element program developed by the Julia language uses two 12-order spectral elements and a velocity Verlet algorithm. Julia is a new open-source, high-level, dynamic programming language, designed to give users the speed of C/C++ while remaining as easy to use as Python, developed specifically for scientific computing [46]. The finite element analysis carried out by COMSOL adopts a generalized alpha algorithm and uses 100 elements. In both two methods, the absolute and relative tolerances are set to ${10}^{-6}$, and a fixed step of 0.01 ms is adopted. It can be found that their results agree with each other very well. The computation time of the proposed method is 450 ms, whereas the FEM takes 25 s when running on the same computer. This example shows that even when the materials, cross-sections, foundations, and loads are non-uniform, the proposed method still exhibits high accuracy and efficiency.

5. Conclusions

This paper presents an accurate and effective numerical method for the static and dynamic analysis of beams with inhomogeneous material, variable spring foundations, non-uniform cross-sections, and loads. Their variations in space are all discretized by Gauss–Lobatto sampling and then approximated by high-order Chebyshev expansions. The discrete governing equation of beam spectral elements is derived by using the Lagrange’s equation and Euler–Bernoulli beam theory. Projection matrices are introduced to assemble the Chebyshev spectral elements and handle boundary conditions and compatibility between elements. The numerical method is programmed in the Julia language and validated by a series of numerical experiments. Various material gradients, non-uniform cross-sections, variable foundations, distributed loads, and boundary conditions are considered. The convergence and accuracy of the proposed method have been validated by comparing its results with solutions found in the literature and obtained by the FEM. It has been demonstrated that the proposed method can tackle inhomogeneities and non-uniformities in materials, cross-sections, foundations, and loads very well. Even when all of them are present, high accuracy and efficiency can still be achieved.

In fact, the proposed method can handle AFG beams, beams with material gradients along the thickness direction, and is adaptable for various foundation types, including Winkler–Pasternak and viscoelastic foundations. Furthermore, it is suitable for both Euler-Bernoulli and Timoshenko beam models. Due to space limitations, these situations are not discussed in this article. The authors will demonstrate them in subsequent papers.