On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams

Abstract

This study examines the static bending behavior of functionally graded beams using a newly developed modified Timoshenko beam element. The mixed finite element formulation and Timoshenko beam theory serve as the foundation for the formulation of the proposed beam element. There are two nodes and two degrees of freedom in each node of the new beam element. The suggested element is free of shear locking, without the need for reduced or selective integrations, because of the mixed finite element formulation. Comparative results demonstrate high accuracy in computations, even with both regular and irregular meshes, as well as coarse and fine discretization. Because of its rapid convergence rate, the proposed element is an excellent tool for analyzing beam structures with complex geometries and load conditions. Several examples are provided to demonstrate the accuracy and high convergence of the proposed beam element. Additionally, the effects of various parameters, such as the power-law index and thickness-to-length ratio, on the bending behavior of functionally graded beams are investigated. The findings highlight the robustness and versatility of the developed beam element, which makes it a useful contribution to research into the computational mechanics of beam structures.

1. Introduction

Beams, plates, and shells are the most common structures that are widely used in many fields of engineering, such as aerospace engineering, civil construction, automobile engineering, and so on (Koizumi [1], Reddy et al. [2,3,4]). Therefore, the static and dynamic response of FG structures has attracted increasing research efforts in the past decade, using analytical methods (Wang et al. [5] and Tian et al. [6]) including the finite element method, isogeometric analysis, and the meshless method (Chen et al. [7]), etc. For example, Chen et al. [8] developed a mixed method for analysis of static bending and free vibration in beams resting on elastic bases. Pu et al. [9] used a modified Fourier series to analyse the bending of an FG sandwich beam with general boundary conditions. Ghazwani [10] developed a mixed beam element for the bending analysis of an FG beam with porosity effects. Kapuria et al. [11] analysed the bending and free vibration behaviors of FG beams using a theoretical model and experimental validation. Kadoli et al. [12] applied higher-order shear deformation theory to analyse the static bending of FG beams. Giunta et al. [13] used classical and advanced theories to analyse the mechanical behaviors of FG beams. Li et al. [14] proposed a new higher-order shear deformation theory to study the static and dynamic responses of FG beams. Ying et al. [15] obtained exact solutions based on two-dimensional elasticity to calculate the bending and free vibration of FG beams resting on elastic foundations. Using the Ritz method [16], Simsek analysed the static bending of FG beams subjected to a uniformly distributed load. Thai et al. [17] applied various theories of higher-order shear deformation to study the bending and free vibration of FG beams. Adiyaman et al. [18] studied the bending and buckling behaviors of two-dimensional FG beams and the effects of porosity. Nguyen et al. [19] studied the static bending and free vibration of axially loaded FG beams using new first-order shear deformation theory. Mohammadi et al. [20] used isogeometric analysis to study the nonlinear behavior of a Timoshenko beam reinforced with carbon nanotubes of FG with temperature-dependent material properties. Huang et al. [21] developed a new exact solution for mechanical analysis of two-directional FG Timoshenko beams. Tang et al. [22] studied the free vibration of non-uniform FG Timoshenko beams using an exact solution. Free and forced vibration of a bi-directional FG Timoshenko beam were studied by Şimşek [23]. Vo et al. [24] developed a quasi-3D theory for the static bending analysis of FG beams. Chaaban and Hadeji [25,26] gave analytical solutions for the bending and free vibration analysis of FG beams resting on elastic foundations. Chen et al. [27] studied the buckling and bending behaviors of FG beams and considered their porosity. Fouda et al. [28] presented a finite element model for the analysis of bending, buckling, and free vibrations in FG porous beams. Chakraborty et al. [2] developed a new beam element and applied it to study the bending and free vibration of FG beams. Vo et al. [29] developed a finite element method based on a refined shear deformation theory to investigate the free vibration and buckling behaviors of FG beams. Glabisz et al. [30] studied the stability of Timoshenko beams with nonlocal parameters dependent on frequency and initial stress. Based on a quasi-3D theory and isogeometric analysis, Fang et al. [31] examined the mechanical behaviors of thick porous beams. More details on the analysis of FG beams can found in previous works [32,33,34,35,36].

The purpose of the current work was to modify the Timoshenko beam element for static bending analysis. This element was established based on an idea discussed by Ghazwani [10]. The new beam element eliminates the requirement for reduced or selective integrations by providing two new parameters to account for the wrapping of rotations and the effects of boundary conditions. The number of degrees of freedom remains the same because these factors are removed at the element level. A comparative analysis was used to demonstrate the correctness of the suggested element and the rate of convergence. Next, the suggested beam element was used to examine the bending of FG beams. Careful consideration was also given to the implications of certain parameters, such as the slender ratio and the power-law index.

2. Functionally Graded Beams

A functionally graded (FG) beam with length $L$ and cross section $h\times b$ was considered, as presented in Figure 1. The material of the beam was a mixture of ceramic and metal, that the volume fraction of the materials varied continuously through the thickness of the beam according to the following formula (Santos et al. [37], Li et al. [38]):

$${\wp }_c={\left({\displaystyle \frac{z}{h}}+{\displaystyle \frac{1}{2}}\right)}^p, {\wp }_c+{\wp }_m=1$$

(1)

where ${\wp }_c, {\wp }_m$ are the volume fraction of the ceramic phase and metal phase, $h$ is the thickness of the beam, and $p$ is the power-law index. The Poisson’s ratio is constant, while the effective of Young’s modulus is calculated as follows:

$$\wp =E_m{\wp }_m+E_c{\wp }_c$$

(2)

Figure 1. Model of an FG beam.

3. Development of the Modified Timoshenko Beam Element

3.1. Displacement Field and Strain Energy

In Timoshenko beam theory, the axial displacement is neglected, and the displacements at a point of the beam are given by:

$$\left\{\begin{array}{l}u(x,z)=z\beta (x)\\ w(x,z)=w(x)\end{array}\right.$$

(3)

The strain field of the beam is obtained as follows:

$$\left\{\begin{array}{l}{\epsilon }_x=z{\beta }_{,x}\\ {\gamma }_{xz}=w_{,x}+\beta \end{array}\right.$$

(4)

According to Hook’s law, the stresses of the beam are calculated as follows:

$${\sigma }_x=E(z){\epsilon }_x; {\tau }_{xz}=kG(z){\gamma }_{xz}$$

(5)

where $E(z), G(z)=E(z)/\left[2(1+\nu )\right]$ are Young’s modulus and shear modulus, respectively, and $k$ is the shear correction factor, which usually equals 5/6. The variation of the strain energy of the beam can be obtained via the following formula:

$$\delta U={\displaystyle \underset{V}{\int }\left\{\delta {\epsilon }_x^0.z.{\sigma }_x+\delta {\gamma }_{xz}{\tau }_{xz}\right\}dV}={\displaystyle \underset{0}{\overset{L}{\int }}\left(\delta {\epsilon }_{x}M+\delta {\gamma }_{xz}Q\right)dx}$$

(6)

where $M$ and $Q$ are the resultant forces, which are calculated as follows:

$$M=b{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}z.{\sigma }_{x}dz} ; Q=kb{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\tau }_{xz}dz}$$

(7)

and $\delta {\epsilon }_x^0,\delta {\gamma }_{xz}$ are the virtual bending and shear strains:

$$\delta {\epsilon }_x^0=\delta {\beta }_{,x}, \delta {\gamma }_{xz}=\delta w_{,x}+\delta \beta$$

(8)

Substituting Equation (5) into Equation (7), we obtain:

$$M={\Im }_b{\epsilon }_x; Q={\Im }_s{\gamma }_{xz}$$

(9)

where the following applies:

$${\Im }_b=b{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}E(z)z^{2}dz} ; {\Im }_s=kb{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}G(z)dz}$$

(10)

3.2. A New Modified Timoshenko Beam Element

Consider a Timoshenko element as shown in Figure 2, with two nodes per element and three degrees of freedom per node.

Figure 2. The proposed Timoshenko beam element.

The vector of the nodal displacement of the beam element is given as follows:

$$d={\left\{\begin{array}{cccc}{w}_1& {\beta }_1& w_2& {\beta }_2\end{array}\right\}}^T$$

(11)

The displacements within the beam element approximate via nodal displacement according to the following formula:

$$w=H_1{w}_1+H_2{w}_2; \beta =H_1{\beta }_1+H_2{\beta }_2+H_m{\beta }_m$$

(12)

where ${\beta }_m$ is a parameter that is eliminated later, and $H_1, H_2, H_m$ are expressed as follows:

$$H_1={\displaystyle \frac{1}{2}}\left(1-\xi \right); H_2={\displaystyle \frac{1}{2}}\left(1+\xi \right); H_m={\displaystyle \frac{1}{4}}\left(1-{\xi }^2\right)$$

(13)

By substituting Equation (12) into Equation (8), we obtain:

$$\delta {\epsilon }_x=H\delta d+N_m\delta {\beta }_m$$

(14)

$$\delta {\gamma }_{xz}=H_s\delta d+H_m\delta {\beta }_m$$

(15)

where the following apply:

$$H=\left[\begin{array}{cccc}0& H_{1,x}& 0& H_{2,x}\end{array}\right]; N_m=H_{m,x}$$

(16)

$$H_s=\left[\begin{array}{cccc}{H}_{1,x}& H_1& H_{2,x}& H_2\end{array}\right]$$

(17)

The resultant shear force is rewritten as follows:

$$Q=Q_0-F ; F=F(x)={\displaystyle \underset{0}{\overset{x}{\int }}q(s)ds} ; \delta Q=\delta Q_0$$

(18)

Substituting Equation (18) into Equation (6), $\delta U$ is calculated as follows:

$$\delta U={\displaystyle \underset{0}{\overset{L}{\int }}\left(\delta {\epsilon }_x^{0}M+\delta {\gamma }_{xz}Q+\delta Q\left({\gamma }_{xz}-\frac{Q}{D_s}\right)\right)dx}$$

(19)

The expression of $\delta V$ is presented as follows:

$$\delta V=-{\displaystyle \underset{0}{\overset{L}{\int }}\delta d^T{H}_w^{T}qdx}$$

(20)

where the following applies:

$$H_w=\left[\begin{array}{cccc}{H}_1& 0& H_2& 0\end{array}\right]$$

(21)

Hamilton’s principle is employed to establish the discretised equations of motion of the beams as follows:

$${\displaystyle \underset{0}{\overset{t}{\int }}\left(\delta U+\delta V\right)dt}=0$$

(22)

Inserting $Q=Q_0-F, \delta Q=\delta Q_0$ into Equation (19), then into Equation (22), and rewritten in matrix form, we obtain:

$$\left[\begin{array}{ccc}\delta d^T& \delta Q_0& \delta {\beta }_m\end{array}\right]\left(\left[\begin{array}{ccc}{k}_{dd}& k_{dQ}& k_{d\beta }\\ k_{dT}^T& k_{QQ}& k_{Q\beta }\\ k_{d\beta }^T& k_{Q\beta }& k_{\beta \beta }\end{array}\right]\left[\begin{array}{c}d\\ Q_0\\ {\beta }_m\end{array}\right]-\left[\begin{array}{l}{f}_d\\ f_Q\\ f_{\beta }\end{array}\right]\right)=0$$

(23)

where the following apply:

$$k_{dd}={\displaystyle \underset{-1}{\overset{1}{\int }}{H}^T{\Im }_{b}H}{\displaystyle \frac{L}{2}}d\xi$$

(24)

$$k_{dQ}={\displaystyle \underset{-1}{\overset{1}{\int }}{H}_s^T\frac{L}{2}d\xi }$$

(25)

$$k_{d\beta }={\displaystyle \underset{-1}{\overset{1}{\int }}{H}^T{\Im }_b{H}_m\frac{L}{2}d\xi }$$

(26)

$$k_{QQ}=-{\displaystyle \underset{-1}{\overset{1}{\int }}\frac{1}{{\Im }_s}\frac{L}{2}d\xi }$$

(27)

$$k_{\beta \beta }={\displaystyle \underset{-1}{\overset{1}{\int }}{H}_m^T{\Im }_b{H}_m\frac{L}{2}d\xi }$$

(28)

$$k_{Q\beta }={\displaystyle \underset{-1}{\overset{1}{\int }}{H}_m\frac{L}{2}d\xi }$$

(29)

$$f_d={\displaystyle \underset{-1}{\overset{1}{\int }}{H}_w^{T}q\frac{L}{2}d\xi }+{\displaystyle \underset{-1}{\overset{1}{\int }}{H}_s^{T}F\frac{L}{2}d\xi }$$

(30)

$$f_Q=-{\displaystyle \frac{1}{{\Im }_s}}{\displaystyle \underset{-1}{\overset{1}{\int }}F\frac{L}{2}d\xi }$$

(31)

$$f_{\beta }={\displaystyle \underset{-1}{\overset{1}{\int }}{H}_{m}F\frac{L}{2}d\xi }$$

(32)

By eliminating the parameter ${\beta }_m$ from Equation (23), we obtain:

$$\left[\begin{array}{cc}\delta d^T& \delta Q_0\end{array}\right]\left(\left[\begin{array}{cc}{k}_{11}& k_{12}\\ k_{12}^T& k_{22}\end{array}\right]\left[\begin{array}{l}d\\ Q_0\end{array}\right]-\left[\begin{array}{l}{f}_1\\ f_2\end{array}\right]\right)=0$$

(33)

where the following apply:

$$k_{11}=k_{dd}-k_{d\beta }{\displaystyle \frac{1}{k_{\beta \beta }}}{k}_{d\beta }^T$$

(34)

$$k_{12}=k_{dQ}-k_{d\beta }{\displaystyle \frac{1}{k_{\beta \beta }}}{k}_{Q\beta }$$

(35)

$$k_{22}=k_{QQ}-k_{Q\beta }{\displaystyle \frac{1}{k_{\beta \beta }}}{k}_{Q\beta }$$

(36)

$$f_1=f_d-k_{d\beta }{\displaystyle \frac{1}{k_{\beta \beta }}}{f}_{\beta }$$

(37)

$$f_2=f_Q-k_{Q\beta }{\displaystyle \frac{1}{k_{\beta \beta }}}{f}_{\beta }$$

(38)

The parameter $Q_0$ is eliminated from Equation (33), and the static bending equations is obtained as the following formula:

$$Kd=f$$

(39)

where the following apply:

$$K=k_{11}-k_{12}{\displaystyle \frac{1}{k_{22}}}{k}_{12}^T$$

(40)

$$f=f_1-{\displaystyle \frac{f_2}{k_{22}}}{k}_{12}$$

(41)

4. Results and Discussion

4.1. Validation Study

In this subsection, the deflection of the non-dimensional center in the FG beam calculated using the present beam elements and well-established Timoshenko elements are compared with those reported by Li et al. [14] and Thai et al. [17], as given in Table 1. The material properties of $\mathrm{Al}$ are $E_m=70 \mathrm{GPa}$, ${\nu }_m=0.3$ and those of ${\mathrm{Al}}_2{\mathrm{O}}_3$ are $E_c=380 \mathrm{GPa}$, ${\nu }_c=0.3$. The deflection of the non-dimensional center of an FGM beam is computed as follows:

$$\overline{w}(L/2)=w\left(L/2\right){\displaystyle \frac{100E_m{h}^3}{q{L}^4}}$$

(42)

Table 1. Comparison of the deflection of the non-dimensional center $\overline{w}(L/2)$ of the FG beams.

$\mathit{L}/\mathit{h}$	$\mathit{p}$	Element Type	Number of Elements					Li et al. [14]	Thai et al. [17]
			2	12	18	24	32
5	0	Proposed	3.1657	3.1657	3.1657	3.1657	3.1657	3.1657	3.1654
		Timoshenko	0.8624	2.9836	3.0823	3.1183	3.1388	-	-
	1	Proposed	6.2599	6.2599	6.2599	6.2599	6.2599	6.2599	6.2594
		Timoshenko	1.5105	5.8417	6.0675	6.1503	6.1979	-	-
	5	Proposed	9.6483	9.6483	9.6483	9.6483	9.6483	9.7802	9.8281
		Timoshenko	3.0362	9.3314	9.5914	9.6854	9.7391	-	-
	10	Proposed	10.7195	10.7195	10.7195	10.7195	10.7195	10.8979	10.9381
		Timoshenko	3.6940	10.4678	10.7257	10.8187	10.8716	-	-
20	0	Proposed	2.8963	2.8963	2.8963	2.8963	2.8963	2.8962	2.8962
		Timoshenko	0.0702	1.5237	2.0700	2.3657	2.5720	-	-
	1	Proposed	5.8049	5.8049	5.8049	5.8049	5.8049	5.8049	5.8049
		Timoshenko	0.1190	2.8056	3.9391	4.5847	5.0495	-	-
	5	Proposed	8.8069	8.8069	8.8069	8.8069	8.8069	8.8151	8.8182
		Timoshenko	0.2573	5.0532	6.6292	7.4377	7.9846	-	-
	10	Proposed	9.6767	9.6767	9.6767	9.6767	9.6767	9.6879	9.6905
		Timoshenko	0.3229	5.8740	7.5244	8.3413	8.8829

For all cases of the ratio $L/h$ and the power-law index $p$, Table 1 demonstrates that the convergence of the modified Timoshenko beam elements is much higher than that of the classical Timoshenko beam elements with regular mesh.

The current results are consistent with those reported by Li et al. [14] and Thai et al. [17], despite the coarse mesh. Furthermore, Figure 3 shows the convergence of the modified Timoshenko beam element with irregular mesh. It can be seen that the numerical results for both the regular and irregular meshes are convergent. As a result, the suggested beam element is suitable for analysing the static bending behavior of FG beams.

Figure 3. Convergence rate of the modified Timoshenko beam element with irregular meshes.

4.2. Numerical Results

In this section, an FG beam of ${\mathrm{Ti}-6\mathrm{Al}-4\mathrm{V}/\mathrm{Si}}_3{\mathrm{N}}_4$ subjected to uniform distributed load is considered. The length of beams is $L$, the height is $h$, and the material properties are $E_c=323 \mathrm{GPa}$, $E_m=66.2 \mathrm{GPa}$, ${\nu }_c={\nu }_m=0.3$. The non-dimensional displacement of the FGM beam is calculated as follows:

$$w^{*}=w{\displaystyle \frac{100E_m{h}_0^3}{q{L}^4}}, h_0={\displaystyle \frac{L}{10}}$$

(43)

Table 2, Table 3 and Table 4 illustrate the non-dimensional maximum displacements of the FG beams under uniformly distributed load with various boundary conditions. It is clear that as the length-to-thickness ratio grows, the maximum deflections in the FG beams increase rapidly. Clamped beams have the smallest maximum deflections, whereas cantilever beams have the largest.

Table 2. The non-dimensional maximum displacement $w^{*}$ of the clamped FG beams.

$\mathit{L}/\mathit{h}$	$\mathit{p}=0$	$\mathit{p}=0.25$	$\mathit{p}=0.5$	$\mathit{p}=1$	$\mathit{p}=2$	$\mathit{p}=5$	$\mathit{p}=10$
4	0.0730	0.0899	0.1057	0.1327	0.1678	0.2110	0.2429
5	0.1200	0.1488	0.1759	0.2218	0.2799	0.3456	0.3933
8	0.3919	0.4908	0.5847	0.7428	0.9341	1.1199	1.2511
10	0.7204	0.9052	1.0809	1.3762	1.7288	2.0540	2.2813
15	2.2815	2.8769	3.4441	4.3960	5.5159	6.4881	7.1588
20	5.2837	6.6714	7.9947	10.2137	12.8103	15.0106	16.5204

Table 3. The non-dimensional maximum displacement $w^{*}$ of the simply supported FG beams.

$\mathit{L}/\mathit{h}$	$\mathit{p}=0$	$\mathit{p}=0.25$	$\mathit{p}=0.5$	$\mathit{p}=1$	$\mathit{p}=2$	$\mathit{p}=5$	$\mathit{p}=10$
4	0.2369	0.2973	0.3546	0.4510	0.5668	0.6762	0.7531
5	0.4403	0.5539	0.6620	0.8435	1.0593	1.2541	1.3898
8	1.7036	2.1501	2.5757	3.2896	4.1265	4.8413	5.3327
10	3.2823	4.1459	4.9695	6.3504	7.9639	9.3224	10.2532
15	10.9280	13.8141	16.5681	21.1837	26.5592	31.0191	34.0639
20	25.7791	32.5966	39.1035	50.0069	62.6906	73.1583	80.2955

Table 4. The non-dimensional maximum displacement $w^{*}$ of the cantilever FG beams.

$\mathit{L}/\mathit{h}$	$\mathit{p}=0$	$\mathit{p}=0.25$	$\mathit{p}=0.5$	$\mathit{p}=1$	$\mathit{p}=2$	$\mathit{p}=5$	$\mathit{p}=10$
4	2.0954	2.6409	3.1604	4.0324	5.0606	5.9612	6.5837
5	4.0027	5.0511	6.0504	7.7266	9.6927	11.3764	12.5345
8	15.9962	20.2147	24.2395	30.9858	38.8524	45.4154	49.9019
10	31.0628	39.2680	47.0982	60.2206	75.5009	88.1690	96.8159
15	104.2373	131.8167	158.1407	202.2495	253.5398	295.7938	324.5911
20	246.5837	311.8631	374.1753	478.5807	599.9251	699.6670	767.6072

4.2.1. The Effects of the Length-to-Thickness Ratio on the Bending of FG Beams

Figure 4 depicts how the length-to-thickness ratio $L/h$ affects the FG beams’ non-dimensional maximum displacements. It is clear that as the length-to-thickness ratio increases, so do the maximum deflections in the FG beams. Specifically, the deflections in the cantilever FG beams grow rapidly as $L/h$ increases. The maximum deflections in the cantilever FG beams were found to be forty times greater than those of the clamped beams. Furthermore, the increase in deflections in the pure-ceramic beams was slower than that in the FG beams. The reason for this is that pure ceramic beams consist of only the ceramic component, while FG beams consist of both the ceramic and metal components; therefore, the rigidity of pure ceramic beams is higher than that of FG beams.

Figure 4. The effects of L/h ratio on the deflection in FG beams: (a) CC beams; (b) CF beams.

4.2.2. The Effects of the Power-Law Index on the Bending of FG Beams

Figure 5 displays the fluctuation of the displacement of the FG beams’ non-dimensional center as a function of the power-law index $p$. The figure shows that the maximum deflections in the FG beams increased with an increase in the power-law index. The explanation is that as the value of the power-law index rises, the volume fraction of the metal increases and the volume fraction of the ceramic falls, resulting in a decrease in the stiffness of the FG beams. When the power-law index rises from 0 to 2, it significantly affects the deflection in the FG beams. Furthermore, the bending behavior of the FG beams is significantly influenced by the length-to-thickness ratio. Figure 6 presents the normal stress distribution through the thickness of an FG beam for two cases: CC and CF beams. It can be seen that the distribution of the normal stress is not symmetric and is non-linear through the thickness direction. The maximum normal stress appears at the upper surface and the minimum normal stress appears at the lower surface.

Figure 5. The effects of $p$ on the deflection in FG beams: (a) CC beams; (b) CF beams.

Figure 6. The effects of $p$ on the normal stress in FG beams: (a) CC beams; (b) CF beams.

5. Conclusions

In this study, a novel modified Timoshenko beam element was constructed to examine the static bending behavior of FG beams. There are only two nodes and two degrees of freedom in the suggested beam element. The current beam element has a straightforward formulation that does not make use of reduced or selective integration. Despite utilizing full Gaussian integration to produce the stiffness matrix and force vector, the suggested beam element can accurately forecast FG beam deflections with a limited number of elements. The FG beam’s static bending was examined using the current proposed beam element to show how several parameters affected its bending behavior. Several parameters were taken into consideration. The following conclusions can be drawn:

-

When the length-to-thickness ratio $L/h$ increases, the deflections in the FG beams increase rapidly.

-

When the power-law index $p$ increases, the deflections in the FG beams increase.

The modified Timoshenko beam elements were used for linear analysis only, and the application of the proposed elements for vibration, buckling, and nonlinear analysis should be studied in future works. The current findings demonstrate that the power-law index and the length-to-thickness ratio have a major impact on FG beams’ static bending behavior.