A New Method of Resolution of the Bending of Thick FGM Beams Based on Refined Higher Order Shear Deformation Theory

The aim of this work is to study the static bending of functionally graded beams accounting higher order of shear deformat ion theory. The governing equations, derived from the virtual work principle, are a set of ordinary d ifferential equations describing a static bending of a thick beam. Thus, this paper presents the differential transform method used to solve the previous system of equations. The results obtained lay the foundation to determine the exact analytical solution for different boundary conditions and external loadings. The axial displacement and the bending and shear displacements, in the exact analytical form, of a thick clamped-clamped beam with functionally graded material under a unifo rm load will be fully developed. Moreover, normal and shear stresses will be analyzed. To confirm the efficiency of this work, a comparison with the numerical results provided by literature is performed. Through this work, the given analytical results provide engineers with an accurate tool to determine the analytical solution for the bending of plates and shells. In addition, the geometric and material parameters that appear clearly in the analytical results allow for a more optimized design of functional ly graded material beams. This type of beams is frequently used in mechanical engineering fields such as aerospace engineering.


Introduction
The bending of thick functionally graded (FG) beams continues to occupy an important spot in mechanical engineering. Therefore, it is necessary to provide engineers with analytical results, enabling them to predict the bending behavior of FG thick beamsin a parametric way and thus improve the performance of systems in a competitive manner.
Using a fin ite element method, R. Kadoli et al [1] studied the static bending behavior of FG beams with higher order shear deformation beam theory (HOSDT).
Simsek [2] developed a numerical solution for Timoshenko beam theory (TBT) and the HOSDT using Rit z method. Giunta G and al [3] adopted Navier closed form solution to solve classical beam theories. Zhong [4] and Li et al [5] presented analytical solution for cantilever FG beams. Vo, Thuc and al [6] presented finite element solution with Hermite interpolation based on refined shear deformation theory. Farhatnia and al (2019) [18] developed a fin ite elementary approach to study the bending and buckling FG beam based on refined Zigzag Theory. Razouki and al (2019) [19] applied differential transform method(DTM) and gave the exact analytical solution to the bending FG beam based on higher order shear deformat ion theory (simply supportedsimply supported beam case).
In this paper, DTM is applied to the governing equations, which are obtained fro m the principle of virtual work, to give a general solution [19]. The general exact solution form for a bending FG beam with various higher-order shear deformation beam theories is fully developed. The exact solution is given for clamped-clamped beam(C-C) subjected to a uniformly d istributed load (UDL). The results are compared with the existing numerical ones to validate the obtained solution. In addition, the analytical expression for transverse deflection is given to show the effect of shear displacement.

Constitutive Relations
Consider a FG beam with length L and rectangular  FGMs are co mposite materials made of ceramic and metal. The materialś properties of FG beams are assumed to vary continuously through the depth of the beam according to a power law as [1,2,5,13]

Displacement Fields and Strains
The displacement fields of various higher-order shear deformation beam theories are given in a general form as [6,14,15,17].
where u is the axial d isplacement of a point on the midplane o f the beam; w b and w s are the bending and shear components of transverse displacement of a point on the midplane of the beam. f(y) is a shape function indicating the distribution of the transverse shear strain and shear stress through the depth of the beam [6]. The non-zero strains are given by: The axial strain The shear strain By assuming that the material of FG beam obeys Hooke's law, the stresses in the beam become σ x = E ( y ) ε x Axial normal stress (4a) τ xy = G ( y ) γ xy Shear stress (4b) where G ( y ) is the Shear modulus related to the Young's modulus E ( y ) by:

Strain Energy and External Load Work
The virtual of the strain energy of the FG beam is given by [6,16]: δ = ∫(σ x δε x + τ xy δγ )dsdx (5) where is the variational symbol, S the cross -sectional area of the uniform beam. By giving the virtual forms 11 and 12 fro m Eq. (3) and substituting the subsequent results into Eq. (5) we obtain [6,19] The virtual potential energy of the applied transverse load q(x) is given by [6,19]:

Equilibrium Equations
The princip le of virtual work states that if a body is in equilibriu m then the total virtual work done is zero [15,16].
Substituting the expressions of δ and δ fro m Eq. (6) and Eq. (8) into Eq. (9) and integrating by parts space variables, and collecting the coefficients of δu, δw b , and δw s , the follo wing equations of equilibriu m of the functionally graded beam are obtained [6,19]: The boundary conditions are of the form: specify By substituting Eqs. (3) and (4) into Eqs. (7), the constitutive equations for the stress resultants are obtained as follows [6,19]: The coefficients of the Eqs. (13) are given in the appendix A.
Then, Eqs (10) can be expressed in terms of the displacements u, w b and w s as follows [18]:

Differential Transform Method
The differential transformation is defined as follows [7,8,9,10,11,12]: in which ( ) the original function, ( ) = is the transformed function. The inverse differential transformation is defined as:

DTM and the Equilibrium System
Assuming that the solutions of the shear w s ( x ) , bending w b ( x ) , axial u ( x ) displacements and external load q ( x ) are a polyno mial forms (power series) as follows [19]: And applying the DTM (for x 0 = 0 ) to Eqs. (17) and Eqs. (14) leads to a recurrent system as follows: For k=0, 1, 2, 3, … And where A 1 , A 2 , B 1 , B 2 , B 3 , C 1 , C 2 and C 3 are given by the appendix B

Coefficients of the Displacement Series
In which

Shear Displacement Series
Fro m Eqs. (24c) and Eq.(24e) we obtain by using the Taylorś series expansion: Hence [19] w s ( x ) = s 0 + s 1 x + Where ( ) is expressed once the load form is defined

Bending Displacement Solution
Fro m Eq. (25c) we obtain by using the Taylorś series expansion: Hence, the final general form of the bending displacement expression is given by [19]: In which is expressed once the load form is defined.

Axial Displacement Solution
Fro m Eqs. (26b) and Eq.(26d) we obtain using the Taylorś series expansion: Where ( ) is expressed once the load form is defined.

Load Form Series
The final exact forms of r u ( x ) given by Eq.(30d), r b ( x ) given by Eq.(29d) and r s ( x ) given by Eq.(28d ) are to be expressed for each external load. In this section we develop the uniformly distributed load.
For uniform load we have

Boundary Conditions and Relevant Exact Analytical Solution: Clamped-Clamped Beam
In this section, the exact analytical solution is given for the clamped-clamped FG beam bending under uniform load fig. 2:

Conclusions
The analytical solution to the static bending problem of FGM beams with HOSDT was presented and applied to the case of a constant cross-section recessed-embedded beam under the action of a UDL. The results obtained clearly show the possibility of analyzing the shear effects as well as taking into account the effect of the different parameters related to the FGM law. The comparison of the results with those obtained by a numerical resolution confirms the effectiveness of the DTM method. This approach, which is successfully used to solve this type of ODE equations, naturally coupled, can be applied to solve physics or engineering problems governed by the same type of differential equation systems. It should also be noted that numerical solving is, in this case, not necessary. For TBT based on Reddy [16] f ( y ) = For full ceramic(n=0) or fu ll metallic (n ⇢ ∞) and according to the appendix A, we obtain.