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Article

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

by
Mofareh Hassan Ghazwani
1 and
Pham V. V. Vinh
2,*
1
Department of Mechanical Engineering, College of Engineering and Computer Sciences, Jazan University, P.O. Box 45124, Jazan 82917, Saudi Arabia
2
Institute of Energy & Mining Mechanical Engineering, Nha Trang 57100, Vietnam
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(1), 73; https://doi.org/10.3390/math13010073
Submission received: 8 October 2024 / Revised: 20 November 2024 / Accepted: 27 December 2024 / Published: 28 December 2024

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 × 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]):
c = z h + 1 2 p ,   c + m = 1
where c ,   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:
= E m m + E c c

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:
u ( x , z ) = z β ( x ) w ( x , z ) = w ( x )
The strain field of the beam is obtained as follows:
ε x = z β , x γ x z = w , x + β
According to Hook’s law, the stresses of the beam are calculated as follows:
σ x = E ( z ) ε x ;       τ x z = k G ( z ) γ x z
where E ( z ) ,   G ( z ) = E ( z ) / 2 ( 1 + ν ) 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:
δ U = V δ ε x 0 . z . σ x + δ γ x z τ x z d V = 0 L δ ε x M + δ γ x z Q d x
where M and Q are the resultant forces, which are calculated as follows:
M = b h / 2 h / 2 z . σ x d z       ;       Q = k b h / 2 h / 2 τ x z d z
and δ ε x 0 , δ γ x z are the virtual bending and shear strains:
δ ε x 0 = δ β , x ,   δ γ x z = δ w , x + δ β
Substituting Equation (5) into Equation (7), we obtain:
M = b ε x ;       Q = s γ x z
where the following applies:
b = b h / 2 h / 2 E ( z ) z 2 d z     ;     s = k b h / 2 h / 2 G ( z ) d z

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.
The vector of the nodal displacement of the beam element is given as follows:
d = w 1 β 1 w 2 β 2 T
The displacements within the beam element approximate via nodal displacement according to the following formula:
w = H 1 w 1 + H 2 w 2 ;       β = H 1 β 1 + H 2 β 2 + H m β m
where β m is a parameter that is eliminated later, and H 1 ,   H 2 ,   H m are expressed as follows:
H 1 = 1 2 1 ξ ;   H 2 = 1 2 1 + ξ ;   H m = 1 4 1 ξ 2
By substituting Equation (12) into Equation (8), we obtain:
δ ε x = H δ d + N m δ β m
δ γ x z = H s δ d + H m δ β m
where the following apply:
H = 0 H 1 , x 0 H 2 , x ;   N m = H m , x
H s = H 1 , x H 1 H 2 , x H 2
The resultant shear force is rewritten as follows:
Q = Q 0 F   ;       F = F ( x ) = 0 x q ( s ) d s   ;       δ Q = δ Q 0
Substituting Equation (18) into Equation (6), δ U is calculated as follows:
δ U = 0 L δ ε x 0 M + δ γ x z Q + δ Q γ x z Q D s d x
The expression of δ V is presented as follows:
δ V = 0 L δ d T H w T q d x
where the following applies:
H w = H 1 0 H 2 0
Hamilton’s principle is employed to establish the discretised equations of motion of the beams as follows:
0 t δ U + δ V d t = 0
Inserting Q = Q 0 F ,   δ Q = δ Q 0 into Equation (19), then into Equation (22), and rewritten in matrix form, we obtain:
δ d T δ Q 0 δ β m k d d k d Q k d β k d T T k Q Q k Q β k d β T k Q β k β β d Q 0 β m f d f Q f β = 0
where the following apply:
k d d = 1 1 H T b H L 2 d ξ
k d Q = 1 1 H s T L 2 d ξ
k d β = 1 1 H T b H m L 2 d ξ
k Q Q = 1 1 1 s L 2 d ξ
k β β = 1 1 H m T b H m L 2 d ξ
k Q β = 1 1 H m L 2 d ξ
f d = 1 1 H w T q L 2 d ξ + 1 1 H s T F L 2 d ξ
f Q = 1 s 1 1 F L 2 d ξ
f β = 1 1 H m F L 2 d ξ
By eliminating the parameter β m from Equation (23), we obtain:
δ d T δ Q 0 k 11 k 12 k 12 T k 22 d Q 0 f 1 f 2 = 0
where the following apply:
k 11 = k d d k d β 1 k β β k d β T
k 12 = k d Q k d β 1 k β β k Q β
k 22 = k Q Q k Q β 1 k β β k Q β
f 1 = f d k d β 1 k β β f β
f 2 = f Q k Q β 1 k β β f β
The parameter Q 0 is eliminated from Equation (33), and the static bending equations is obtained as the following formula:
K d = f
where the following apply:
K = k 11 k 12 1 k 22 k 12 T
f = f 1 f 2 k 22 k 12

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 Al are E m = 70   GPa , ν m = 0.3 and those of Al 2 O 3 are E c = 380   GPa , ν c = 0.3 . The deflection of the non-dimensional center of an FGM beam is computed as follows:
w ¯ ( L / 2 ) = w L / 2 100 E m h 3 q L 4
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.

4.2. Numerical Results

In this section, an FG beam of Ti - 6 Al - 4 V / Si 3 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   GPa , E m = 66.2   GPa , ν c = ν m = 0.3 . The non-dimensional displacement of the FGM beam is calculated as follows:
w * = w 100 E m h 0 3 q L 4 ,   h 0 = L 10
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.

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.

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.

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.

Author Contributions

Conceptualization, methodology, software, formal analysis, investigation, resources, validation, visualization, writing—original draft preparation, and writing—review and editing, M.H.G.; supervision and project administration, P.V.V.V. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

No data were used in this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Model of an FG beam.
Figure 1. Model of an FG beam.
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Figure 2. The proposed Timoshenko beam element.
Figure 2. The proposed Timoshenko beam element.
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Figure 3. Convergence rate of the modified Timoshenko beam element with irregular meshes.
Figure 3. Convergence rate of the modified Timoshenko beam element with irregular meshes.
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Figure 4. The effects of L/h ratio on the deflection in FG beams: (a) CC beams; (b) CF beams.
Figure 4. The effects of L/h ratio on the deflection in FG beams: (a) CC beams; (b) CF beams.
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Figure 5. The effects of p on the deflection in FG beams: (a) CC beams; (b) CF beams.
Figure 5. The effects of p on the deflection in FG beams: (a) CC beams; (b) CF beams.
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Figure 6. The effects of p on the normal stress 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.
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Table 1. Comparison of the deflection of the non-dimensional center w ¯ ( L / 2 ) of the FG beams.
Table 1. Comparison of the deflection of the non-dimensional center w ¯ ( L / 2 ) of the FG beams.
L / h p Element TypeNumber of ElementsLi et al. [14]Thai et al. [17]
212182432
50Proposed3.16573.16573.16573.16573.16573.16573.1654
Timoshenko0.86242.98363.08233.11833.1388--
1Proposed6.25996.25996.25996.25996.25996.25996.2594
Timoshenko1.51055.84176.06756.15036.1979--
5Proposed9.64839.64839.64839.64839.64839.78029.8281
Timoshenko3.03629.33149.59149.68549.7391--
10Proposed10.719510.719510.719510.719510.719510.897910.9381
Timoshenko3.694010.467810.725710.818710.8716--
200Proposed2.89632.89632.89632.89632.89632.89622.8962
Timoshenko0.07021.52372.07002.36572.5720--
1Proposed5.80495.80495.80495.80495.80495.80495.8049
Timoshenko0.11902.80563.93914.58475.0495--
5Proposed8.80698.80698.80698.80698.80698.81518.8182
Timoshenko0.25735.05326.62927.43777.9846--
10Proposed9.67679.67679.67679.67679.67679.68799.6905
Timoshenko0.32295.87407.52448.34138.8829
Table 2. The non-dimensional maximum displacement w * of the clamped FG beams.
Table 2. The non-dimensional maximum displacement w * of the clamped FG beams.
L / h p = 0 p = 0.25 p = 0.5 p = 1 p = 2 p = 5 p = 10
40.07300.08990.10570.13270.16780.21100.2429
50.12000.14880.17590.22180.27990.34560.3933
80.39190.49080.58470.74280.93411.11991.2511
100.72040.90521.08091.37621.72882.05402.2813
152.28152.87693.44414.39605.51596.48817.1588
205.28376.67147.994710.213712.810315.010616.5204
Table 3. The non-dimensional maximum displacement w * of the simply supported FG beams.
Table 3. The non-dimensional maximum displacement w * of the simply supported FG beams.
L / h p = 0 p = 0.25 p = 0.5 p = 1 p = 2 p = 5 p = 10
40.23690.29730.35460.45100.56680.67620.7531
50.44030.55390.66200.84351.05931.25411.3898
81.70362.15012.57573.28964.12654.84135.3327
103.28234.14594.96956.35047.96399.322410.2532
1510.928013.814116.568121.183726.559231.019134.0639
2025.779132.596639.103550.006962.690673.158380.2955
Table 4. The non-dimensional maximum displacement w * of the cantilever FG beams.
Table 4. The non-dimensional maximum displacement w * of the cantilever FG beams.
L / h p = 0 p = 0.25 p = 0.5 p = 1 p = 2 p = 5 p = 10
42.09542.64093.16044.03245.06065.96126.5837
54.00275.05116.05047.72669.692711.376412.5345
815.996220.214724.239530.985838.852445.415449.9019
1031.062839.268047.098260.220675.500988.169096.8159
15104.2373131.8167158.1407202.2495253.5398295.7938324.5911
20246.5837311.8631374.1753478.5807599.9251699.6670767.6072
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Ghazwani, M.H.; Vinh, P.V.V. On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams. Mathematics 2025, 13, 73. https://doi.org/10.3390/math13010073

AMA Style

Ghazwani MH, Vinh PVV. On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams. Mathematics. 2025; 13(1):73. https://doi.org/10.3390/math13010073

Chicago/Turabian Style

Ghazwani, Mofareh Hassan, and Pham V. V. Vinh. 2025. "On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams" Mathematics 13, no. 1: 73. https://doi.org/10.3390/math13010073

APA Style

Ghazwani, M. H., & Vinh, P. V. V. (2025). On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams. Mathematics, 13(1), 73. https://doi.org/10.3390/math13010073

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