Non-Linear Elastic Beam Deformations with Four-Parameter Timoshenko Beam Element Considering Through-the-Thickness Stretch Parameter and Reduced Integration

Abstract

In this work, non-linear elastic deformations of beams are investigated. The kinematics of the beam is derived based on an element with four-parameter containing a through-the-thickness stretch parameter to avoid Poisson locking. Moreover, the Kirchhoff-Saint Venant model is used to derive kinetic quantities. Next, a non-linear FE formula in Total Lagrangian form is obtained, and three-node beam element with two-node reduced integration is employed to avoid shear locking. Finally, to evaluate the performance of the derived formulations, some examples are provided. The results prove that the current formulation is in very good agreement with those available in the literature. More importantly, the formulation is capable of predicting the experimental results with high accuracy.

1. Introduction

1.1. Importance and Necessity of the Problem

Beams, fundamental structural elements, are very important piece across various engineering disciplines. Their applications span aerospace structures soaring through the atmosphere, high-rise buildings reaching for the sky, bridges connecting landscapes, and even the miniaturized marvels of micro- and nano-electromechanical systems (MEMS/NEMS) [1,2,3,4,5,6]. Biomechanics, too, finds application for beams in analyzing the intricate skeletal structures of living organisms [7,8]. Within this broad spectrum of applications, the analysis of beams subjected to large deformations has gained increasing importance. This growing focus stems from the recognition that real-world structures encounter a diverse range of loading scenarios. Static loads, dynamic forces, and even environmental factors can all induce significant deformations in beams [9,10]. Developing a more accurate framework to analyze these large deformations becomes crucial for designing safer and more reliable structures. While traditional beam theories, such as the Euler-Bernoulli model, have provided valuable tools for engineers, their limitations become apparent when dealing with scenarios involving substantial deformations. This necessitates the exploration of more sophisticated analytical approaches to accurately predict the behavior of beams under such conditions. Classical beam theories often assume linear material behavior. However, many engineering materials exhibit nonlinear stress-strain relationships. These limitations often stem from assumptions of small strains and linear material behavior, which can lead to inaccurate predictions in scenarios involving significant geometric changes. The Timoshenko beam theory, which accounts for shear deformation and rotational effects, provides a more comprehensive approach to modeling such scenarios [11,12].

The Timoshenko beam theory offers a more refined approach compared to the Euler-Bernoulli model that with additional considerations become particularly important for stockier beams or those experiencing substantial shear forces [13,14]. Within the domain of beam mechanics, achieving a nuanced understanding of the behavior exhibited by beams undergoing significant deflections necessitates the strategic deployment of two key elements: nonlinear constitutive models and geometrically exact formulations [15,16]. Hyperelasticity, a theoretical framework adept at describing the elastic response of materials even under substantial strains, emerges as a powerful tool for this endeavor [17]. This framework excels at capturing the nonlinear stress-strain relationships observed in a diverse array of materials, encompassing rubbers, polymers, and even the intricate biological tissues found within living organisms [18,19]. When applied to beam analysis, hyperelastic constitutive laws enable the accurate prediction of the complex interplay between material and geometric nonlinearities. However, directly solving the governing equations arising from the Timoshenko beam theory, especially when coupled with Hyperplastic and nonlinear material behavior, can be mathematically challenging.

The finite element method (FEM) offers a powerful tool for numerical analysis, enabling the exploration of complex deformation patterns in beams [20,21]. By discretizing the beam into smaller elements, the FEM transforms the complex problem into a system of algebraic equations that can be efficiently solved using numerical methods. This approach offers unparalleled flexibility in handling complex geometries, boundary conditions, and material properties, making it ideally suited for analyzing the large deformation response of beams even when considering nonlinear material and Hyperplastic material properties [22]. By incorporating the FEM into the framework, we can efficiently solve the governing equations arising from the Timoshenko beam theory with hyperplastic material considerations.

1.2. A Review on Literature

Several studies that have explored the finite element analysis of Timoshenko beams. Meier et al. [23] compared two geometrically exact finite element formulations for slender beams, namely the Kirchhoff-Love theory and the Simo-Reissner theory. Their study provided valuable insights into the accuracy and computational efficiency of these formulations for analyzing large deformations in beams. They highlighted the importance of considering the geometric nonlinearity in the finite element analysis of slender beams, which is particularly relevant to the study of Timoshenko hyperplastic beams. Cazzani et al. [24] performed an analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. Their study provided insights into the comparative performance of finite element and isogeometric analyses for modeling the behavior of Timoshenko beams under various loading conditions. The findings of this study are particularly relevant to the exploration of large deformations in Timoshenko hyperplastic beams, as they offer valuable information on the capabilities and limitations of different analysis methods. Sakai et al. [25] present a 3D elastic beam model for form-finding and analysis of elastic gridshells subjected to bending deformation at the self-equilibrium state. The accuracy of the proposed method utilizing the dynamic relaxation method is confirmed when compared to the results obtained by finite element analysis. In a study by Numanoğlu and his research group [26], a new eigenvalue problem solver was developed for the thermo-mechanical vibration of Timoshenko nanobeams using an innovative nonlocal finite element method. Finite element analyses of micro- and nanobeams were the topic of several studies [27,28,29,30]. Firouzi et al. [31] studied non-linear free vibration of Timoshenko beams using direct integration scheme. Moreover, Firouzi et al. [32] investigated large deformation of electro-visco-hyperelastic beams under electric charge. Wang et al. [33] used homotopy analysis method to analyze the deformation of a cantilever under a point load. Moreover, Beléndez et al. [34] performed an experiment on large deformation of a cantilever beam.

Cai et al. [35] proposed modeling of an active metabeam for broadband non-reciprocal wave suppression. Sahmani et al. [36] studied nonlinear in-plane buckling of FG porous microbeams by isogeometric analysis. Geometrically nonlinear stability of restrained nanobeams utilizing Stokes’ transformation was analyzed by Uzun et al. [37]. The role of homogenisation models in theoretically analysing the vibrations of three-layered microbeam was investigated by Yee et al. [38]. Settimi andLenci [39] analyzed free wave propagation of periodic flexural waves on an infinite elastic beam. Żur et al. [40] studied large deformations of hyperelastic beams using neo-Hookean material model. Bensaid et al. [41] studied free vibration and stability response of bidirectional graded material beams. Study of multiple-input multiple-output active vibration control of a composite sandwich beam by fractional order positive position feedback was done by Hameury et al. [42]. Liu et al. [43] studied free vibration of composite beams using Carrera unified formulation model. Pagani et al. [44] studied vibration of thin-walled slender structures by higher-order beam finite element.

While significant progress has been made in exploring such deformations through finite element analysis, knowledge gaps persist and necessitate further investigation. One potential future research direction is the development of a model that can accurately capture the behavior of Timoshenko beams under large deformations. Furthermore, the validation of finite element simulations through experimental testing of Timoshenko beams would enhance the confidence in the numerical results and contribute to the advancement of this research area.

This study demonstrates the progress and potential for further advancements in the finite element exploration of large deformations in Timoshenko beams. By adopting this framework, we establish a more accurate representation of the beam’s response under diverse loading conditions. These findings contribute significantly to our comprehension of the mechanical behavior exhibited by Timoshenko beams. Furthermore, they underscore the necessity for ongoing research endeavors to bridge the remaining knowledge gaps within this domain. Future investigations could strategically target free-form Timoshenko curved beams, with a particular emphasis on elucidating the nonlinear effects arising from large deformations. The nonlinear finite element analysis is developed, and through-the-thickness parameter is used to circumvent Poisson locking. Moreover, reduced integration is utilized to avoid shear locking. The experimental validation is also investigated.

The rest of the paper is organized as follows. In Section 2, the kinematics and kinetics of the four-parameter beam element are derived. Section 3 is allocated to derive FE formulation. In Section 4, some examples are provided to explore the applicability of the formulation. Finally, conclusion is discussed in Section 5.

2. Kinematics of the Four-Parameter Beam

An initially straight beam of which length, thickness and width are $L$, $h_0$ and $b$ is considered. A fixed Cartesian coordinate system $\{X,Z,Y\}$ with basis vectors $\{{\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3\}$ is located in the neutral surface of the beam, as displayed in Figure 1.

The reference position of each material point can be described as

$$\overrightarrow{X}=\mathit{X} {\overrightarrow{e}}_1+\mathit{Z} {\overrightarrow{e}}_3,$$

(1)

Moreover, the position of every arbitrary point after deformation can be expressed as [32]:

$$\overrightarrow{x}={\overrightarrow{x}}_0(\mathit{X})+[\mathit{Z}+{\mathit{Z}}^2\kappa (\mathit{X})]\overrightarrow{r}(\mathit{X}),$$

(2)

where ${\overrightarrow{x}}_0$, $\kappa$ and $\overrightarrow{r}$ are, respectively, the position in the centerline, the through-the-thickness stretch parameter, and the director vector. It should be mentioned that, the director vector is a unit vector in terms of rotation of the cross-section, $\phi$ is given by

$$\overrightarrow{r}=-\sin \phi {\overrightarrow{e}}_1+\cos \phi {\overrightarrow{e}}_2.$$

(3)

The deformation gradient is obtained via the following relation:

$$F=\mathrm{Grad}(x)={\displaystyle \frac{\mathsf{\partial }x}{\mathsf{\partial }X}}\approx F_0+Z{F}_1$$

(4)

Considering $\{u,v,\phi ,\kappa \}$ as unknown parameters, this expression is a four-parameter Timoshenko beam element. Moreover, the right Cauchy-Green tensor can be derived as follows:

$$C=C_0+Z{C}_1, C_0=F_0^T{F}_0, C_1=F_0^T{F}_1+ F_1^T{F}_0.$$

(5)

Finally, the Green-Lagrange tensor can be expressed via the following relation:

$$E={\displaystyle \frac{1}{2}}(C-I)\approx E_0+Z{E}_1, E_0={\displaystyle \frac{1}{2}}(C_0-I), E_1={\displaystyle \frac{1}{2}}{C}_1.$$

(6)

In this work, the well-known Kirchhoff-St Venant material model is used as follows:

$$\psi =\lambda {[\mathrm{tr}(E)]}^2+\mu \mathrm{tr}{E}^2,$$

(7)

where Lame’ constants, $\lambda$ and $\mu$ are defined by

$$\lambda ={\displaystyle \frac{E\upsilon }{(1+\upsilon )(1-2\upsilon )}}, \mu ={\displaystyle \frac{E}{2(1+\upsilon )}}.$$

(8)

In the latter relation, $E$ and $\upsilon$ are, respectively, the Young’s modulus and the Poisson’s ratio. In this case, the 2nd Piola-Kirchhoff stress tensor is expressed as:

$$\Pi =\lambda (\mathrm{tr}E)I+2\mu E.$$

(9)

In what follows, the stress is written in the linear format as follows:

$$\Pi ={\Pi }_0+Z{\Pi }_1,$$

(10)

in which ${\Pi }_0$ and ${\Pi }_1$ are calculated as

$${\Pi }_0=\lambda (\mathrm{tr}{E}_0)I+2\mu E_0, {\Pi }_1=\lambda (\mathrm{tr}{E}_1)I+2\mu E_1.$$

(11)

Using Equations (10) and (11), the fourth-order elasticity tensors are derived as

$$ℂ={ℂ}_0+Z{ℂ}_1,$$

(12)

$${ℂ}_0=\lambda I\otimes I+\mu (E_0\overline{\otimes }{E}_0+E_0\underset{\_}{\otimes }{E}_0), {ℂ}_1=\lambda I\otimes I+\mu (E_1\overline{\otimes }{E}_1+E_1\underset{\_}{\otimes }{E}_1),$$

(13)

where $\overline{\otimes }$ and $\underset{\_}{\otimes }$ are non-standard tensor product given by

$${[(•)\overline{\otimes }(\circ )]}_{ijkl}={(•)}_{ik}{(\circ )}_{jl}, {[(•)\underset{\_}{\otimes }(\circ )]}_{ijkl}={(•)}_{il}{(\circ )}_{jk}.$$

(14)

It should be mentioned that ${ℂ}_0$ and ${ℂ}_0$ are the linearized tensors of the total fourth-order elasticity tensor.

3. Finite Element Analysis

A finite element formulation for elastic deformation of 4-parameter beam in nonlinear regime is derived in this section. Based on principle of virtual work, the variation of internal energy is equal to the variation of work done by virtual external force

$$D{U}^{\mathrm{int}}=D{U}^{\mathrm{ext}},$$

(15)

where $D{U}^{\mathrm{int}}$ and $D{U}^{\mathrm{ext}}$ are the variations of internal energy and external work, respectively, and can be derived as follows:

$$D{U}^{\mathrm{int}}={\displaystyle {\int }_{V_0}DU} \mathrm{d}{V}_0={\displaystyle {\int }_L^{}{\displaystyle {\int }_A\Pi :DE}} \mathrm{d}{A}_0\mathrm{d}X,$$

(16)

$$D{U}^{\mathrm{ext}}={\displaystyle {\int }_L\left(q_1\right.}\cdot Du+q_2\cdot \left.Dv\right) dX+p_{1}Du+p_{2}Dv,$$

(17)

where $V_0$ and $A_0$ are the referential volume and cross-sectional area, respectively. Besides, $q_1$ and $q_2$ are the components of external traction acting on the length, and $p_1$ and $p_2$ are applied point forces in $X$- and $Z$-directions. Equation (16) can be re-written as

$$\begin{array}{l}D{U}^{\mathrm{int}}={\displaystyle {\int }_L^{}{\displaystyle {\int }_{A}D{\overrightarrow{E}}^T}}\overrightarrow{\Pi }\mathrm{d}{A}_0\mathrm{d}X={\displaystyle {\int }_L^{}{\displaystyle {\int }_A(D{\overrightarrow{E}}_0^T+ZD{\overrightarrow{E}}_1^T)(}}{\overrightarrow{\Pi }}_0+Z{\overrightarrow{\Pi }}_1)\mathrm{d}{A}_0\mathrm{d}X\\ ={\displaystyle {\int }_L^{}{\displaystyle {\int }_A(D{\overrightarrow{E}}_0^T{\overrightarrow{\Pi }}_0+ZD{\overrightarrow{E}}_0^T{\overrightarrow{\Pi }}_1+ZD{\overrightarrow{E}}_1^T{\overrightarrow{\Pi }}_0+Z^{2}D{\overrightarrow{E}}_1^T{\overrightarrow{\Pi }}_1)}}\mathrm{d}{A}_0\mathrm{d}X,\end{array}$$

(18)

The approximation of the generalized displacement field is defined as

$$\overrightarrow{Q}={\left\{u,v,\phi ,\kappa \right\}}^T={\displaystyle \sum _{I=1}^{NN}{H}_I\left(\xi \right)}{\left\{u_I,v_I,{\phi }_I,{\kappa }_I\right\}}^T,$$

(19)

in which $\overrightarrow{Q}$ is displacement vector, $NN$ is total nodes per element, and $H_I$ are the interpolation functions. In this paper, three-node element is used. Therefore, the interpolation functions are defined as follows:

$$H_1=-{\displaystyle \frac{1}{2}}\xi \left(1-\xi \right), H_2=\left(1-\xi \right)\left(1+\xi \right), H_3={\displaystyle \frac{1}{2}}\xi \left(1+\xi \right).$$

(20)

Using the latter relation, the virtual Green-Lagrange strain vector can be expressed via the following relation

$$\left\{\begin{array}{c}D{E}_r\\ \Delta E_r\end{array}\right\}={\displaystyle \sum _{I=1}^{NN}{B}_{rI}}\left\{\begin{array}{c}DQ\\ \Delta Q\end{array}\right\}, r=0,1$$

(21)

Substituting Equations (19) and (21) into (18) one can obtain:

$$D{U}^{\mathrm{int}}={𝒜}_{e=1}^{ne}D{Q}^T{f}^{\mathrm{int}}={𝒜}_{e=1}^{ne}{\displaystyle \sum _{I=1}^{NN}D{Q}_I^T{f}_I^{\mathrm{int}}},$$

(22)

in which ${𝒜}_{e=1}^{ne}$ is the operator for assemblage and $ne$ is total number of elements. Moreover, $f_I^{\mathrm{int}}$ is the vector of internal force expressed as

$$f_I^{\mathrm{int}}={\displaystyle {\int }_L(B_{0I}^T\overrightarrow{F}+B_{1I}^T\overrightarrow{m})} \mathrm{d}X.$$

(23)

Similarly, the variation of the external work is given by

$$D{U}^{\mathrm{ext}}={𝒜}_{e=1}^{ne}D{Q}^T{f}^{\mathrm{ext}}={𝒜}_{e=1}^{ne}{\displaystyle \sum _{I=1}^{NN}D{Q}_I^T{f}_I^{\mathrm{ext}}},$$

(24)

where $f_I^{\mathrm{ext}}$ is external force vector as follows:

$$f_I^{\mathrm{ext}}={\displaystyle {\int }_L{H}_I\overrightarrow{q}} \mathrm{d}X+H_I{\overrightarrow{p}}_1+H_I{\overrightarrow{p}}_2.$$

(25)

By back-substitution of Equations (22) and (24) in the principle of virtual work i.e., Equation (15), one can obtain that:

$${𝒜}_{e=1}^{ne}D{Q}^T(f^{\mathrm{int}}-f^{\mathrm{ext}})=0.$$

(26)

It is noted that, in the latter relation, $DQ$ can take arbitrary values, and therefore, the residual vector is defined as

$$ℛ=f^{\mathrm{int}}-f^{\mathrm{ext}}=0.$$

(27)

Equation (27) is a set of non-linear equations. Hence, an iterative Newton-Raphson method is employed to deal with it. To linearize the non-linear relation, one can write:

$$\begin{array}{ll}\Delta D{U}^{\mathrm{int}}& ={\displaystyle {\int }_L^{}{\displaystyle {\int }_{A_0}(D{E}^T}\Delta }\Pi +\Delta D{E}^{\mathrm{T}}\Pi )\mathrm{d}{A}_0\mathrm{d}X\\ & ={\displaystyle {\int }_L^{}{\displaystyle {\int }_{A_0}D{E}^T}\Delta }\Pi \mathrm{d}{A}_0\mathrm{d}X+{\displaystyle {\int }_L^{}{\displaystyle {\int }_{A_0}\Delta D{E}^{\mathrm{T}}\Pi \mathrm{d}{A}_0\mathrm{d}X=\mathrm{\Phi }+\mathrm{\Omega },}}\end{array}$$

(28)

where

$$\mathrm{\Phi }={\displaystyle {\int }_L^{}{\displaystyle {\int }_{A_0}D{E}^T}\Delta }\Pi \mathrm{d}{A}_0\mathrm{d}X, \mathrm{\Omega }={\displaystyle {\int }_L^{}{\displaystyle {\int }_{A_0}\Delta D{E}^{\mathrm{T}}\Pi \mathrm{d}{A}_0\mathrm{d}X.}}$$

(29)

Therefore, the linearization of variation of internal stored energy is derived as

$$\Delta D{U}^{\mathrm{int}}=\mathrm{\Phi }+\mathrm{\Omega }.$$

(30)

Finally, utilizing Equation (21), the Equation (30) can be re-written as

$$\Delta D{U}^{\mathrm{int}}={𝒜}_{e=1}^{ne}{\displaystyle \sum _{I=1}^{NN}{\displaystyle \sum _{J=1}^{NN}D{Q}_I^{\mathrm{T}}{K}_{IJ}\Delta Q_J}}=D{Q}^{\mathrm{T}}K\Delta Q,$$

(31)

in which $K_{IJ}$ is the stiffness matrix in the IJ’th block per element expressed via the following relation

$$K_{IJ}=K_{IJ}^{\mathrm{mat}}+K_{IJ}^{\mathrm{geo}}.$$

(32)

In the latter relation $K_{IJ}^{\mathrm{mat}}$ is the material stiffness matrix given by

$$K_{IJ}^{\mathrm{mat}}={\displaystyle {\int }_L^{}\{A{B}_{0I}^{\mathrm{T}}{ℂ}_0{B}_{0J}+\tilde{I}(B_{0I}^{\mathrm{T}}{ℂ}_1{B}_{1J}+B_{1I}^{\mathrm{T}}{ℂ}_0{B}_{1J}+B_{1I}^{\mathrm{T}}{ℂ}_1{B}_{0J})\}\mathrm{d}X}.$$

(33)

Moreover, $K_{IJ}^{\mathrm{geo}}$ is the geometric stiffness matrix expressed as

$$K_{IJ}^{\mathrm{geo}}={\displaystyle {\int }_L^{}(A\Delta D{E}_0^{\mathrm{T}}\Pi +\tilde{I}\Delta \delta E_1^{\mathrm{T}}\Pi )\mathrm{d}X}.$$

(34)

4. Demonstrative Examples

The kinematics and kinetics of the four-parameter Timoshenko beam are derived in the previous sections. Moreover, the finite element formulation for large deformation of Timoshenko beam is also derived. In this section, to evaluate the performance and ability of the formulations, some examples are provided. In all examples, three-node elements are utilized. Furthermore, two-node Gauss-Legendre reduced integration is employed to calculate all integrals to avoid shear locking phenomenon.

4.1. Analysis of a Cantilever Subjected to End Force

In this example, deflection of a clamped-free beam under a tip force is studied. As shown in the Figure 2a, a cantilever with length $L$ is applied to the point force $P$ at the free end. The load is so that the load parameter $P{L}^2/EI$ goes to 5, where $E$ is the elasticity modulus and $I$ is the moment of inertia. Our convergence analysis shows that a mesh of ten elements is enough to get converged results. The deformation-force curves are depicted in Figure 2a,b. It is deduced that, the result for vertical deflection ratio $-v/L$ and horizontal deflection ratio $-u/L$ in terms of load parameter $P{L}^2/EI$ are in

Very good agreement with the results reported in Wang et al. [33]. Moreover, the deformed shapes of the beam in ten load steps are displayed in Figure 3a. Besides, the convergence plot is shown in Figure 3b.

4.2. Analysis of a Clamped-Supported Beam under Distributed Load

In this example, finite elastic deformation of a clamped-supported beam is investigated. The beam is clamped in one end, and is supported in the other end so that it can move in x-direction. The beam is applied to uniform distributed load $q$ (see Figure 4b). Our convergence study reveals that a mesh of 20 elements is enough to get converged results. The curves for large vertical deformations in $X=L/4$, $X=L/2$ and $X=3L/4$ in terms of load parameter $p_{{}_{*}}=q{L}^2/EI$ are shown in Figure 4. It can be deduced from Figure 4a that the deflection ratio $v/L$ in $X=3L/4$ is larger than $X=L/4$, and obviously, the deformation is not symmetric. Furthermore, the deformed shapes of the clamped-supported beam are portrayed in Figure 5a in ten load step. Finally, the convergence diagrams of deformation in the middle of the beam in terms of number of elements are depicted in Figure 5b.

4.3. Validation with Experiment

In the final example, verification of the present model with experimental results is performed. To do so, the experiment done by Beléndez et al. [34] is considered. According to their experiment, a cantilever of which dimensions are $L=40 \mathrm{cm}$, $b=2.5 \mathrm{cm}$ and $h=0.04 \mathrm{cm}$ are considered. The cantilever is under its own weight $W=0.3032 N$ which is equal to distributed load $w=W/L=0.758 N/m$. Convergence analysis shows that a mesh of 5 elements is sufficient for solving this problem. The deformed shape in of the beam is depicted in Figure 6a. As can be observed, the final shape of the simulation in the present work is coincident with the final deformed shape of the experiment reported in [34]. This proves that the present model can predict the experimental results very good. Moreover, the curves of deformations in terms of loading are shown in Figure 6b.

5. Conclusions

In this paper, a non-linear finite element formulation for large elastic deflections of the Timoshenko beam is derived. The formulation is free of Poisson and shear locking, as it employs through-the-thickness stretch parameter and reduced integration. The demonstrative examples show very good agreement with the results reported in the literature. In Section 4.1, the present formulation can get the results for large deformations of a cantilever, and is coincident with the previous study in the literature. In Section 4.2, the deformation of a clamped-supported beam is investigated, and large deflections at three points of the beam are obtained. Besides, the formulations enable to capture the experimental results reported in the literature with completely convincing accuracy. In Section 4.3, the large defection of the beam under its own weight is investigated. A comparison to the experiment proves that the present model can predict the experiment very well.