1 Introduction

Functionally graded materials (FGMs) are one kind of new composites that have a smoothand continuous spatial variation of material properties which can alleviate the stress concentrationfound in laminated composites. The gradient compositional variation in the constituentsfrom one surface to the other provides a good way to solve the problem of higher transverseshear stresses induced when two similar materials with a large difference in material propertiesare bonded. Therefore,FGMs have an enormous application potential in modern technologyand engineering,especially in the thermal environments where it can minimize the thermalstress concentration produced by the high temperature gradient. In recent years,many effortshave been made to investigate the mechanical behavior of the FGM structures. A survey in theliterature shows that comprehensive studies on static and dynamic behaviors on FGM beamstructures subjected to mechanical and thermal loadings based on different beam theories havebeen presented by different researchers.

On the basis of the Euler-Bernoulli beam theory (EBBT),the static and dynamic problemsof FGM beams were investigated^{[1, 2, 3, 4, 5, 6, 7, 8]}. Pradhan and Murmu^[1] studied thermo-mechanical vibrationof a simply supported sandwich beam with two FGM surface layers resting on an axiallyvariable elastic foundation by the differential quadrature method (DQM). Alshorbagy et al.^[2]examined free vibration of an FGM beam by the finite element method. Due to the deficiency inthe EBBT for consideration of the shear effect,the numerical results show that the slendernessof the beam has no influence on the frequencies and mode shapes. Simsek and Kocat¨urk^[3]analyzed free and forced vibration of an FGM simply supported beam under a concentratedmoving harmonic load,in which the effects of different material gradients,the velocity of themoving load,and the excitation frequency on the dynamic responses of the beam were examined.Free vibration and buckling of FGM beams with vertical edge cracks were studied byYang and Chen^[4]. In their numerical results,the effects of the material gradient,location,andnumbers of the cracks on the natural frequencies and the buckling loads were discussed. Li andLiu^[5] derived proportional relationships between the deflections,critical buckling loads,andnatural frequencies of FGM beams and those of the reference homogenous beams and provedthat these relationships are independent on the load and boundary conditions. Based on theexact geometrically nonlinear theory for the axially extensible beams,Li et al.^[6] performedinvestigation on small vibration of post-buckled FGM beams with surface-bonded piezoelectriclayers in thermal and electrical environments by a numerical shooting method. Yaghoobi andTorab^[7] presented large amplitude vibration and post-buckling of geometrically imperfect FGMbeams resting on nonlinear elastic foundation by the von Karman nonlinear strain-displacementrelation. The nonlinear thermal bending solution of a slender FGM beam with pined ends wasgiven by two coupled elliptic integral equations^[8],from which the numerical results for thebending response were presented in graphic and tabular forms.

In the framework of the first-order shear deformation theory or the Timoshenko beam theory(TBT),some researchers studied static and dynamic characteristics of FGM beams by analyticalas well as numerical methods^{[9, 10, 11, 12, 13, 14, 15, 16, 17]}. A new beam theory which had been developed fromlaminated plates was used to analyze free vibration of FGM beams by Sina et al.^[9]. By introducinga new unknown function to decouple the governing equations,Li^[10] presented analyticalsolutions for static bending and free vibration of FGM Timoshenko beams. Huang and Li^[11]also examined buckling of circular columns with variation of material properties in the radialdirection. Kiani and Eslami^[12] investigated thermomechanical buckling of FGM beams withtemperature-dependent material properties. They derived closed form solutions of the criticalbuckling loads under uniform,linearly,and nonlinearly temperature rise in the lateral directionfor different boundary conditions. By the equivalence condition between the shear strain energyof an FGM beam and that of a homogenous one,Murin et al.^[13] derived a new shear correctionfunction for the FGM beam with spatially continuous variations of material propertiesand used it in the modal analysis of FGM beams. Fu et al.^[14] investigated thermal bucklingof an FGM beam with a longitudinal through-width crack by dividing the cracked beam intofour sub-beams. Pradhan and Chakraverty^[15] presented an investigation on free vibration ofFGM beams by the Rayleigh-Ritz method based on both the classical and first-order shear deformationbeam theories considering the Poisson ratio effect in the axial stress-strain relation.Ansari et al.^[16] studied size-dependent bending,buckling,and free vibration of FGM beamsby the DQM. The size effects on the buckling loads and natural frequencies were analyzed by different strain gradient theories. Based on the von Karman nonlinear strain-displacement relation,some authors studied geometrically nonlinear deformation of FGM beams^{[17, 18, 19]}. Ma andLee investigated thermal post-buckling^[17] small free vibration^[18] of FGM beams with/withoutthermal post-buckling deformation subjected to an in-plane temperature by the analytical andnumerical shooting methods. Esfahani et al.^[19] studied thermal buckling and post-bucklingof FGM beams resting on nonlinear elastic foundation by the generalized DQM,in which theeffects of the boundary conditions,the thermal loading types,the foundation coefficients,thetemperature dependency of the material properties on the critical thermal loads and equilibriumpaths of the FGM beams were examined and discussed in detail.

Studies on bending and vibration of FGM beams based on the higher-order shear deformationtheory can also be found in the literature^{[20, 21, 22, 23, 24, 25, 26]}. Benatta et al.^[20] developed an analyticalsolution for static bending of simply supported functionally graded hybrid beams subjectedto a transverse uniform load. Sallai et al.^[21] further studied static bending of FGM beamsby adopting the parabolic,exponential,and Aydogdu’s model shape functions^[24] of the shearstrain distribution in the cross-section,in which the material properties were varied in the heightdirection as power-law,sigmoid,and exponential functions. Kadoli et al.^[22] carried out a numericalanalysis for static bending of FGM beams by the finite element method. Aydogdu andTaskin^[23] investigated free vibration of simply supported FGM beams based on first,parabolic,and exponential shear deformation beam theories,in which the natural frequencies were obtainedby the Navier type solution method. Simsek^[24] investigated free vibration response ofFGM beams by different beam theories,in which the system of equations of motion was derivedby Lagrange’s equations. Mahi et al.^[25] analyzed free vibration of FGM beams with thetemperature dependent material properties,in which the effects of the initial thermal stresson the natural frequencies were discussed. By the von Karman nonlinear strain-displacementrelation,Zhang^[26] studied nonlinear bending FGM beams with immovable simply supportedand clamped edges,in which the stretching-bending couplings in constitutive equations wereeliminated by introducing the physical neutral surface. More recently,Shen and Wang^[27] studiednonlinear bending,thermal post-buckling,and large amplitude vibration of FGM beamsresting on an elastic foundation. In their analysis,both the Voigt model and the Mori-Tanakamodel were used to predict the variation of the material properties which were assumed to betemperature-dependent.

In order to more accurately predict the behavior,the plane elasticity theory was also used toanalyze the thick FGM beams by other authors. For example,Sankar^[28] analyzed deformationsof simply supported FGM beams with Young’s modulus varying exponentially in the thicknessdirection and subjected to symmetrical sinusoidal transverse loads. Zhong and Yu^[29] studiedstatic bending deformation of a cantilever FGM beam with arbitrary through-the-thicknessvariation of material properties. In the sense of two-dimensional theory of elasticity,Ding etal.^[30] presented stress functions for anisotropic FGM beams in analytical forms and obtainedbending solutions under different boundary conditions.

Different from the above mentioned conventional studies on the static and dynamic responsesof functionally graded structures by analytical or numerical approaches,some researchers studiedthe relationship between the solutions of FGM structures and those of the correspondinghomogenous ones. By quantitatively investigating the numerical results about the deflections,the buckling loads,and the natural frequencies of FGM plates in the literature,Abrate^{[31, 32]}found that the values of these physical quantities of an FGM plate are proportional to those ofthe corresponding homogenous counterpart. However,further theoretical analysis showed thatthis proportional relation is exact only in the sense of the classical plate theory. By searching forthe similarity between the differential equations,Reddy et al.^[33] obtained axisymmetric bendingsolutions of FGM circular plates based on the first-order shear deformation plate theory interms of the deflection of the classical plate theory. Furthermore,Ma and Wang^[34] extendedthe above mentioned work^[33] to the case of the higher-order shear deformation theory or the Reddy-Bickford plate theory. By introducing a potential function,Cheng and Kitipornchai^[35] derived the bending solutions of non-homogenous polygonal plates with simply supported edges based on the first-order shear deformation theory in terms of the deflection of a homogenous thin plate.

The concept of the relationship among the Euler-Bernoulli and Timoshenko beams,theLevinson beam,and the Reddy-Bickford beam has been elaborated at length by Levinson^[36],Reddy et al.^{[37, 38]},Wang^[39],and Reddy et al.^[40]. A number of technical papers about therelationships between the solutions of shear deformable beams and plates and the correspondingclassical solutions have been published,and these abundant achievements have been integratedinto an academic book^[40]. However,in the above mentioned literature,the relationship of thebending solution of transversely inhomogenous Levinson beams with that of the homogenousEuler-Bernoulli beam (HEBB) has not been considered. Recently,Li et al.^[41] and Li andBatra^[42] investigated the relationships between bending and buckling solutions of the FGMTimoshenko beams and those of the corresponding HEBB. As a result,the solutions of the FGMTimoshenko beams are simplified as the calculation of the transition coefficients determinedeasily by the variation law of the gradient of the material properties and the geometry if thesolutions of the HEBB are known.

In this presentation,we further study the transition relation between the solutions of theFGM Levinson beams (FGMLBs) and those of the HEBB on the basis of the work in Ref. ^[41].The displacement field of the FGMLB is given by the higher-order shear deformation beamtheory^{[36, 40]}. Using the similarity between the differential equations and equivalence of theloadings,the generalized solutions of the deflection,the rotational angle,the resultant forces,and the bending moments of the FGMLB in terms of the deflection of the HEBB are derivedfor arbitrary load and boundary conditions. For some specified end constraints,the specificsolutions of the FGMLB are presented. Analytical expressions for the transition coefficientsbetween the two kinds of solutions are derived for continuous through-thickness variation of thematerial properties. The final object of this effort is to simplify the solution of the FGMLB asthe calculation of the transition coefficients including the higher-order shear deformation effectsbecause the solution of the reference HEBB can be found even in the textbooks.

2 Mathematical formulations of problem

2.1 Material properties

Consider an FGM beam (see Fig. 1) with a homogenous rectangular cross section,havingthe length l,the width b,and the depth h. We assume that the beam is made of two differentmaterials with the material properties varying continuously in the thickness direction fromthe top to the bottom surfaces. Furthermore,it is assumed that the material properties areindependent on the temperature change. The effective material properties,such as Young’smodulus E and the Poisson ratio υ,are continuous functions and are given in the followingform^[41]:

where z 2 [−h/2,h/2] is the lateral coordinate in the thickness direction being positive upward,P_b denotes the material property constant at the bottom surface of the beam,and ψP (z) is a known continuous function of the coordinate z. Especially,it gives ψP (−h/2) = 1 and ψP (h/2) = P_t/P_b,where Pt denotes the material property constant at the top surface of the beam. 2.2 Derivation of governing equations

Herein,we use the Levinson beam theory to establish the governing equations of FGMbeams. The Levinson beam theory is based on the same displacement field as the Reddy-Bickford beam theory. Therefore,the displacement field can be given as follows^{[36, 40]}:

where x is the axial axis coinciding with the geometrically central axis,u and w are the axial and transverse displacements at an arbitrary point in the beam,respectively,u₀ and w₀ are the displacements in the geometrically neutral surface of the beam in the directions of the x-and z-axes,respectively,φ(x) is the rotational angle of the cross section about the y-axis,and α = 4/(3h²). For a beam that is made of inhomogenous materials,its geometrically middle surface usually does not coincide with the physical neutral surface so that the axial displacement u₀(x) is not zero.

The linear strain-displacement relation gives the strain components as follows:

Using Hooke’s law,it yields

From Eq. (5),we obtain the resultant forces and the moment as follows:

where F_N and F_S are the axial force and the shear force,respectively,and M is the bending moment. The stiffnesses in Eqs. (6)-(8) are defined by

As opposed to using the variationally-derived equations of equilibrium in the Reddy-Bickford beam theory^[40],Levinson used the thickness-integrated equations of elasticity,which are exactly the same as those of the TBT^[36]

where q = q(x) is a distributed lateral force.

Substituting Eqs. (6)-(8) into Eq. (10) yields the equilibrium equations in terms of the dimensionless displacements as follows:

The dimensionless quantities in the above equations are defined by where E_b is Young’s modulus at the bottom surface of the beam,and A = bh and I = bh³/12 are the area and inertia moments of the cross-section,respectively.

From the definition of the stiffness coefficients,φ_i,in Eq. (15) together with Eq. (9),Eqs. (11)-(13) are applicable to arbitrary variation of the material properties as given by Eq. (1). If Young’s modulus of the FGM beam is assumed to vary as a power-law function in the thickness direction,and the Poisson ratio is assumed to be a constant,then we have^[37]

where p is the power of the material property variation function,and E_t is Young’s modulus at the top surface of the beam. Then,the analytical expressions of the dimensionless coefficients,∅_i,can be derived using Eqs. (1),(9),(15),and (16),which are presented in Appendix A. If we consider the variation of the Poisson ratio in the depth direction similar to Young’s modulus,we cannot get the analytical form of the coefficient,∅_xz. Instead,we get it by numerical integration.

Elimination of the axial displacement,U,in Eq. (12) using Eq. (11) yields

where three dimensionless parameters,c,c_s,and c_α,are defined by 3 General solutions in terms of those of HEBB

Substituting Eq. (13) into Eq. (17),we have the differential equation,which only includes the rotational angle,φ,as follows:

where λ_α = cc_αc_s represents the higher-order shear deformation effects.

Setting λ_α = 0 and φ = −dW_E/dξ in Eq. (19) yields the governing equation of the FGM Euler-Bernoulli beams as follows^[41]:

where WE is the dimensionless deflection of the FGM Euler-Bernoulli beam. Especially,for the reference homogenous beam (c = 1),we have^[41] where W_E^∗ is the dimensionless deflections of the reference HEBB. From the mathematical similarity between Eqs. (20) and (21),we obtain a linear relationship between the deflections^{[6, 41]}, It is shown that the relation (22) is independent on the load and boundary conditions^[6].

Using the equivalence of the loadings and substituting Eq. (21) into Eq. (19),we have

Integrating the above equation gives the general solution of rotational angle of FGMLB in terms of the HEBB as where β_i (i = 1,2,3) are integral constants. Here,it needs to explain that W_E^∗ is a particular solution of the reference HEBB under the same loadings and end constraints,which has satisfied its boundary conditions prior.

Furthermore,using Eq. (13),we can rewrite Eq. (17) in the form of

Substituting Eq. (24) into Eq. (25) and integrating it,we arrive at the deflection of FGMLB in terms of W_E^∗ as where β₄ is a constant,and β₁,β₂,and β₃ are determined by the specified boundary conditions of FGMLB. Equations (24) and (26) are a system of general solutions of FGMLB in terms of the deflection of the reference HEBB,which are valid for arbitrary load and boundary conditions. Herein,we have assumed that the solution W_E^∗ has satisfied the boundary conditions of the HEBB. Therefore,it should be pointed out that the load information or the load parameters have been included in the particular solution W_E^∗(ξ) so that the constants β_i (i = 1,2,3,4) are determined actuaactually only by the difference between the boundary conditions of the FGMLB and the HEBB.

Especially,if we set c_α = 0 in Eqs. (24) and (26),α = 0 in Eq. (9),and c_s = σ²/(10φ_xz),then they reduce to the solutions of FGM Timoshenko beams^[41]. Furthermore,if we let c_s = 0,then we have β_i ≡ 0. In other words,Eqs. (24) and (26) reduce to the solution of FGM Euler-Bernoulli beams^[6]. 4 Solutions for given load and boundary conditions

In this section,we find the particular solution for the FGMLB with specified loadings and end constraints. From Eq. (11),we can obtain the dimensionless axial displacement at the geometrically middle surface as

where β5 and β6 are integration constants which can be determined by the in-plane boundary conditions.

By Eqs. (6)-(7),(14),(24),(26),and (27),the dimensionless resultant forces and the bending moment can be expressed in terms of the deflection of reference HEBB as follows:

where f_SE^∗ = −W_′′′E^∗ and m_E^∗ = −W_′′E^∗ are the dimensionless shear force and the bending moment of the reference HEBB,respectively,and the prime denotes the derivative with respect to the coordinate ξ. The natural boundary conditions of the FGMLB can be given in terms of W_E^∗ using Eq. (28).

In the following,we derive the particular solutions of the FGMLB for some specified boundaryconditions with some combinations of clamped,simply supported,and free ends. The sixintegral constants in the general solutions (24)-(28) can be determined by the specified boundaryconditions of a specific problem. For the linear and static bending problem,we assumethat there is no axially external force acting on the beam,and the two ends are not restrictedin-plane at the same time (static determined axially) so that we have f_N ≡ 0,which gives β₅ = 0 in the light of Eq. (28). The constant β₆ can be determined by limiting the rigid bodydisplacement of the beam in the axial direction. For example,by setting U(0) = 0,the axialdisplacement can be finally expressed in terms of the deflection of the HEBB as follows:

If the beam is homogenous,then we have φ₁ = φ₃ = 0. Substitution of this result into Eq. (29) gives U(ξ) ≡ 0. This means that the physical neutral surface coincides with the geometrical middle plane for the homogenous beams^{[36, 37, 38, 39, 40]}. 4.1 Beams with two ends simply supported (S-S)

The boundary conditions of the FGMLB with two ends simply supported are written by

By substituting Eqs. (26) and (28c) into Eq. (30) and remembering that the deflection W_E^∗ is the particular solution of the HEBB,we find the integral constants as Then,the solution of the two-ends simply supported FGMLB under arbitrary loadings is given by where φ_E^∗ = −W_E^∗′ . 4.2 Beams with two ends clamped (C-C)

For an FGMLB with its two ends clamped,the boundary conditions are written by

Substituting Eqs. (24) and (26) into Eq. (33) and keeping in mind that W_E^∗ is a particular solution of the reference HEBB,the integral constants can be determined as follows: Substituting Eq. (34) into Eqs. (24)-(28),we obtain the bending solutions of FGMLB with two ends clamped subjected to arbitrary transverse loads in terms of W_E^∗. If the loads are symmetric about the beam center,then we have m_E^∗(0) = m_E^∗(1) and f_SE^∗(0) = −f_SE^∗(1),which leads to β₁ = 0. In this special case,we have 4.3 Beams with left end clamped and right end free (C-F)

For a beam clamped at the left end and free at the right end,the boundary conditions of the FGMLB are

Similarly,by the boundary conditions (36),the integration constants are determined as follows: Substitution of Eq. (37) into Eqs. (24) and (26) gives the deflection and the rotation angle of the FGMLB as follows: 4.4 Beams with left end clamped and right end simply supported (C-S)

If the beam is clamped at the left end and simply supported at the right one,the integral constants about the FGMLB are determined as follows:

Substituting these constants into Eqs. (24) and (26),we have

The solutions of the FGMLB for the above mentioned four kinds of boundary conditions are applicable to arbitrary load cases. As examples,for two kinds of loading cases,Q(ξ) = 1 and Q(ξ) = sin(πξ),we arrive at the particular solutions of the FGMLB with the above mentioned four kinds of boundary conditions,which are listed in Table 1.

For the special case of homogenous Levinson beams,we have φ₀ = φ₂ = c = 1,φ₁ = φ₃ = 0,and φ₄ = 1/80. Then,the solutions of the beams under uniform loadings in Table 1 reduce to the same results given by Levinson^[36] and Reddy et al.^{[38, 40]}.

The particular solutions of the axial displacement,the shear force,and the bending moment of the FGMLB for the above four kinds of boundary conditions can be obtained by Eqs. (28) and (29). Especially,for the statically determinate beams with S-S and C-F ends,we have

which are exactly the same as those of the HEBB.

From the above mentioned examples,it can be seen that the integral constants in the generalized solutions (24) and (26) are expressed in terms of the end bending moments m_E^∗(0),m_E^∗(1) and the end shearing forces f_SE^∗(0),f_SE^∗(1),including the coefficients c,c_s,and c_α. The deflection and the rotational angle of an FGMLB are composed of two parts. The first one is the dominant part being proportional to that of the HEBB. The second one represents the shear deformation correction part depending on the values of parameter c_s. From the definition (18),it can be seen that these two parameters are proportional to the square of the height-to-length ratio,σ₂,which means that the shear deformation can be neglected when the beam becomes very slender. Obviously,the terms with the parameter λ_α are the correction to the first-order shear deformation theory by the Levinson beam theory. 5 Numerical results and discussion

In this section,some examples with numerical results are presented to help the readers to assess accuracy and validity of the method and solutions in this paper. In numerical computation,we assume the material of the FGM beam to be composed of the ceramic (Alumina) with Young’s modulus E_c = 380 GPa and the metal (Aluminum) with E_m = 70 GPa. The Poisson ratios of the two constituents are υ_c = υ_m = 0.23. Young’s modulus at an arbitrary point of the FGM beam is given by Eqs. (1) and (16). We define Et = E_c and E_b = E_m. The Poisson ratio of the beam is assumed to be constant so that υ(z) ≡ 0.23.

Firstly,in Table 2 and 3, we list the values of dimensionless deflections at the beam centerof an S-S FGM beam subjected to the uniformly distributed load,Q(ξ) = 1,and a C-C FGMbeam under the non-distributed load,Q(ξ) = sin(πξ),respectively,for some specified valuesof the height-to-length ratio σ and the material gradient parameter p. For a given σ,the resultsin the first row are obtained by the numerical shooting method to solve the differentialequations (13) and (17) associated with the specific boundary conditions. Those in the secondrow are predicted by the present solution (26). In the third row,the values are calculatedby the Timoshenko beam solution reduced from Eq. (26). The values in the last row in thetables are given by Eq. (22) which is based on the EBBT. Obviously,exact agreement of thevalues in the first and the second rows shows that Eq. (26) can accurately predict the staticbending of the FGMLB. From the two tables,we can also find that the difference between thedeflections predicted by the TBT and those by the Levinson beam theory is not significant,which means that the first-order shear deformation beam theory can also accurately take intoaccount the shear deformation of the beams. Again,for the beams with larger height-to-lengthratios,the difference between the results predicted by the two shear deformation theories ismore significant. Furthermore,we can see that the effect of shear deformation of the C-C beam subjected to the non-distributed load is more significant than that of the S-S beam under the uniform load. From Eq. (19),we can know that if the load is linearly distributed,then the term with the coefficient,λ_α,vanishes,or the equation reduces to the same as that of the FGM Timoshenko beams.

For some given values of σ,Fig. 2 shows the free end deflection,W(1),of a C-F FGM beam subjected to the uniformly distributed unit load,Q = 1,varying continuously with the material gradient index,p. A comparison between the results predicted by Eq. (26) and the numerical shooting method is also given,which again shows very good accuracy and validity of the presented analytical approach.

In Fig. 3,we plot the curves of the deflection of S-S FGMLB versus the power index under the non-uniformly applied load,Q = sin(πξ),for a variety of values of σ. From the numerical results in the tables and Figs. 2 and 3,we note that the deflection increases with the increment in the values of p. This is due to the fact that the larger value of the power index means a smaller ceramic component of the beam,and hence the stiffness is reduced.

In Fig. 4,we present the curves of deflection distribution of a C-C FGMLB beam subjected to the non-uniform load,Q = sin(πξ),for different values of the power index p. In Fig. 5,the similar curves of a C-S FGMLB under the uniform load,Q = 1,are also illustrated for some given values of σ. Numerical values in Fig. 2 to Fig. 5 are obtained based on Eq. (26) incorporated with the specific load and boundary conditions. The numerical results both in the tables and figures show,as we know,that the deflections increase with the decrease in the value of the height-to length ratio because for the beams with the same height,the shear deformation effects decrease with the increase in their length of the cross-section. As a limit,we get the EBBT solution by letting σ be infinitesimal,or let the beam length become infinite,which means that the shear deformation of Euler-Bernoulli beams is totally neglected,in other words,the shearing stiffness is considered to be infinite in the classical beam theory.

6 Conclusions

Based on the Levinson beam theory,the general solutions for static bending of FGM beamswith arbitrary variation of material properties in the thickness direction are derived in terms ofthe deflection of the reference HEBB. The displacements and the resultant forces of the FGMLBare expressed in terms of the deflection of the HEBB with the same geometry,end constraints,and applied forces. Consequently,the bending solution of an FGMLB can be reduced into thatof an HEBB together with the calculation of three coefficients c,c_s,and c_α. Since the solutionsof the HEBB can be found even in the textbooks for a variety of boundary and load conditions,the solutions derived in this presentation are very convenient for engineers to get higher-ordershear deformation theory solutions of FGM beams directly without dealing with the complicatedbending-tension-shearing coupling problem. From Eq. (26) as well as Table 1,we can find thatthe deflection of an FGMLB consists of two parts. The first one is the dominant part whichis proportional to the deflection of the HEBB,and the second one is related to the higherordershare deformation which is a correction to the classical beam theory and depends on thevalues of parameters c_s and λ_α. The numerical results obtained by the presented analyticalsolutions and those by the shooting method to numerically solve the differential equations withboundary conditions show excellent agreement so that very good accuracy and validity of thepresented analytical approach are convinced. The analytical solutions in this paper can be usedas benchmarks to check both the analytical and numerical solutions of bending of FGM beamsbased on different shear deformation beam theories.