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Copyright © 2014 Tech Science Press CMES, vol.100, no.3, pp.201-222, 2014

Nonlinear Bending and Thermal Post-Buckling Analysis ofFGM Beams Resting on Nonlinear Elastic Foundations

Da-Guang Zhang1,2 and Hao-Miao Zhou1

Abstract: A model of FGM beams resting on nonlinear elastic foundations isput forward by physical neutral surface and high-order shear deformation theory.Material properties are assumed to be temperature dependent and von Kármánstrain-displacement relationships are adopted. Nonlinear bending and thermal post-buckling are given by multi-term Ritz method, and influences played by differentsupported boundaries, thermal environmental conditions, different elastic founda-tions, and volume fraction index are discussed in detail. It is worth noting that theeffect of nonlinear elastic foundation increases with increasing deflection.

Keywords: Functionally graded materials, Physical neutral surface, Nonlinearelastic foundations, Nonlinear bending, Thermal post-buckling.

1 Introduction

Functionally graded materials (FGMs) are a class of novel materials in which prop-erties vary continuously in a specific direction. To withstand the high-temperatureeffects, generally FGM structures made of ceramic and metal constituents are ad-dressed. Consequently, investigations on behaviors of FGM structures are identi-fied as an interesting field of study in recent years.

Many investigations on the linear bending and buckling of FGM or laminated orhomogeneous beams can be seen in [Sankar (2001); Venkataraman and Sankar(2003); Zhong and Yu (2007); Ying et al. (2008); Giunta et al. (2010); Aminbaghaiet al. (2012); Giunta et al. (2013); Dong et al. (2013); Dong et al. (2014); Elgoharyet al. (2014)]. And lots of studies have been also made on the nonlinear bending andpost-buckling of FGM beams based on Euler-Bernoulli beam theory, Timoshenkobeam theory and higher order shear deformation beam theory. However, relativelyfew have been made on the nonlinear analysis of FGM beams resting on elasticfoundations. Among those, Zhang (2013, 2014) put forward a model of the FGM

1 College of Information Engineering, China Jiliang University, Hangzhou, 310018, China.2 Corresponding author. Email: [email protected]

202 Copyright © 2014 Tech Science Press CMES, vol.100, no.3, pp.201-222, 2014

beams based on physical neutral surface and high order shear deformation theory,and analyzed nonlinear bending, thermal post-buckling and nonlinear vibration be-haviors by Ritz method. Ma and Lee (2011) derived governing equations for boththe static behavior and the dynamic response of FGM beams subjected to uniformin-plane thermal loading, based on the physical neutral surface and the first ordershear deformation beam theory, then Ma and Lee (2012) gave exact solutions fornonlinear static responses of a shear deformable FGM beam under an in-plane ther-mal loading. Li et al. (2006) studied thermal post-buckling of functionally gradedmaterial Timoshenko beams. Almeida et al. (2011) presented geometric nonlinearanalysis formulation for beams of functionally graded cross-sections by means ofa total Lagrangian formulation. Rahimi and Davoodinik (2010) discussed the largedeflection of a functionally graded cantilever beam under inclined end loading byfully accounting for geometric nonlinearities using analytical and Adomian decom-position methods, then Davoodinik and Rahimi (2011) extended their works to aflexible tapered functionally graded cantilever beam. Rahimi et al. (2013) investi-gated the post-buckling behavior of functionally graded beams by means of an exactsolution method. Zhao et al. (2007) derived the non-linear differential equationsof post-buckling for FGM rod subjected to thermal load. They considered rodswith both ends pinned and used the shooting method to solve the equations. Fuet al. (2012) discussed nonlinear analysis of buckling, free vibration and dynamicstability for the piezoelectric functionally graded beams in thermal environment.Ke et al. (2009) studied post-buckling behavior of functionally graded materialbeams including an edge crack effect based on Timoshenko beam theory and von-Kármán’s strain–displacement relations. They applied the Ritz method to obtainthe non-linear governing equations and used the Newton-Raphson method to ob-tain the postbuckling load-end shortening curves and post-buckling deflection-endshortening curves. Anandrao et al. (2010) investigated the buckling and thermalpost-buckling behavior of uniform slender FGM beams. The von-Kármán strain–displacement relationship is used to obtain the equations. Single-term Ritz methodand finite elements method are applied to obtain the response of the beam. ThenAnandrao et al. (2012) studied large amplitude free vibration and thermal post-buckling of shear flexible FGM beams using finite element formulation based onfirst order Timoshenko beam theory. Ansari et al. (2013) investigated thermalpost-buckling characteristics of FGM microbeams undergoing thermal loads basedon the modified strain gradient theory. Shegokar and Lal (2013) provided thestochastic nonlinear bending response of FGM beam with surface bonded piezo-electric layers subjected to thermo-electro-mechanical loadings with uncertain ma-terial properties. Shen and Wang (2014) dealt with the large amplitude vibration,nonlinear bending and thermal postbuckling of FGM beams resting on an elasticfoundation in thermal environments. Fallah and Aghdam (2011, 2012) investigated

Nonlinear Bending and Thermal Post-Buckling Analysis 203

nonlinear free vibration and post-buckling analysis of functionally graded beams onnonlinear elastic foundation. Esfahani et al. (2013) examined thermal buckling andpost-buckling analysis of FGM Timoshenko beams resting on a non-linear elasticfoundation. Ghiasian et al. (2013) studied static and dynamic buckling of an FGMbeam resting on a non-linear elastic foundation subjected to uniform temperaturerise loading and uniform compression.

The present paper extends the previous works [Zhang (2013); Zhang (2014)] tononlinear bending and thermal post-buckling analysis of FGM beams resting onnonlinear elastic foundations based on physical neutral surface and high order sheardeformation theory. Temperature dependent material properties and von Kármánstrain-displacement relationship are taken into consideration. The material proper-ties of FGMs are assumed to be graded in thickness direction according to a volumefraction power law distribution and are expressed as a nonlinear function of tem-perature. Approximate solutions of FGM beams are obtained by Ritz method.

2 Temperature dependent material properties of FGM beams

Figure 1: Geometry and coordinates of a FGM beam resting on nonlinear elasticfoundations.

Consider a FGM beam (thickness h and length Lx) with rectangular cross-section,which is made from a mixture of metals and ceramics. The coordinate system isillustrated in Fig. 1. It can be assumed that the effective material properties P ofFGMs, such as Young’s modulus E, Poisson’s ratio ν , thermal conductivity κ and


thermal expansion coefficient α , can be expressed as

P = PcVc +PmVm (1)

in which Vm and Vc are the metal and ceramic volume fractions and are relatedby Vm +Vc = 1, Pm and Pc denote the temperature-dependent properties of metaland ceramic beam, respectively, and may be expressed as a nonlinear function oftemperature [Touloukian (1967)]

P = P0(P−1T−1 +1+P1T +P2T 2 +P3T 3) (2)

in which T = T0+∆T and T0=300 K (room temperature), P−1, P0, P1, P2 and P3 arethe coefficients of temperature T (K) and are unique to the constituent materials.

3 Modeling of FGM beams resting on nonlinear elastic foundations based onphysical neutral surface and high order shear deformation theory

As is customary, the foundation is assumed to be a compliant foundation, meaningno part of the beam lifts off the foundation in the deformed region. The load-displacement relationship of the foundation is assumed to be p=K1w−K2d2w/dx2

+K3w3, where p is the force per unit area, K1 is the Winkler foundation stiffness,K2 is a constant showing the effect of the shear interactions of the vertical elementsand K3 is nonlinear elastic foundation coefficients.

According to model of FGM beams [Zhang (2013); Zhang (2014)] based on physi-cal neutral surface and high order shear deformation theory, the displacements, thestrains and the stresses have the same form as the previous works [Zhang (2013);Zhang (2014)], see Appendix A.1-8. For the sake of brevity, the deducing processof the formulae is omitted, and the governing equations can be derived accordingto energy variational principle.

D11d2ψx

dx2 − 4F11

3h2d3wdx3 − A44

(ψx +

dwdx

)− dMT

dx= 0 (3a)

A44

(dψx

dx+

d2wdx2

)+

4F11

3h2d3ψx

dx3 − 16H11

9h4d4wdx4 − 4

3h2d2PT

dx2 +Nxd2wdx2

−K1w+K2d2wdx2 −K3w3 +q = 0

(3b)

And if the boundaries have no in-plane displacements on the geometric middleplane, i.e. prevented from moving in the x-direction, Nx can then be written inintegral form as

Nx =1Lx

∫ Lx

0

A11

[z0

dψx

dx− 4c0

3h2

(d2wdx2 +

dψx

dx

)+

12

(dwdx

)2]−NT

dx (3c)


All symbols used in Eq. (3) are defined by Zhang (2013, 2014), see Appendix A.9.In the following analysis, two cases of boundaries will be considered.

w = 0, Mx =−z0Nx, Px =−c0Nx, (for immovable simply supported ends) (4a)

w =dwdx

= ψx = 0, (for immovable clamped ends) (4b)

4 Ritz method for approximate solutions of nonlinear problems of FGMbeams

Ritz method is adopted in this section to obtain approximate solutions of FGMbeams, and in symmetrical problems about the beams with immovable simply sup-ported ends, it can be assumed that

w =M

∑i=1,2···

ai sin(2i−1)πx

Lx(5a)

where M is total number of series, and where ai are undetermined coefficients.We assume that the temperature variation is uniform and occurs in the thicknessdirection only, i.e. NT , MT and PT are constants for FGM beams, so substitutingEq. (5a) into Eq. (3a), ψx can be determined as

ψx =M

∑i=1,2···

ci cos(2i−1)πx

Lx(5b)

where

ci =

4F113h2

[(2i−1)π

Lx

]3− A44

(2i−1)πLx

A44 + D11

[(2i−1)π

Lx

]2 ai (6)

For symmetrical problems about the beam with immovable clamped ends, it can beassumed that

w =M

∑i=1,2···

ai

(1− cos

2iπxLx

)(7a)

Similarly, ψx can be determined as

ψx =M

∑i=1,2···

ci sin2iπxLx

(7b)


where

ci =

4F113h2

(2iπLx

)3− A44

2iπLx

A44 + D11

(2iπLx

)2 ai (8)

Cubic algebraic equations about ai can be obtained by substituting w and ψx intothe following expression.

∂Π

∂ai= 0 (9)

in which Π =U +V , in which the strain energy U is

U =12

∫Ω

(σxεx + τxzγxz −σxα∆T )dΩ+12

∫ Lx

0

[K1w2 +K2

(dwdx

)2

+12

K3w4

]dx

(10a)

where Ω denotes domain of FGM beams. Work done by applied forces V is

V =−∫ Lx

0qwdx (10b)

As for FGM beams with given loads (like transverse uniformly distributed loadsq0 and thermal loads NT , MT , PT ) and other known coefficients, ai can be solvedby Newton-Raphson iterative method or other equivalent methods. For the sake ofbrevity, nonlinear algebraic equations and the solving process are omitted. Substi-tuting these coefficients back into Eqs. (5) and (7), w and ψx may then be com-pletely determined. In addition, if critical thermal buckling loads exist, criticalvalue can be easily obtained by making solutions of deflection coefficients ai ap-proach to zero.

In addition, the present paper extends convergence studies of the previous works[Zhang (2013)] to nonlinear bending and thermal post-buckling analysis of FGMbeams resting on nonlinear elastic foundations, so M = 3 is also used in all thefollowing calculations.

5 Results and discussions

Numerical results are presented in this section for nonlinear bending and thermalpost-buckling of FGM beams. A Si3N4/SUS304 is selected as an example. Typ-ical values for Young’s modulus E (in Pa), Poisson’s ratio ν , thermal expansioncoefficient α (in K−1) and thermal conductivity κ(in W/mK) are listed in Table 1


from Reddy and Chin (1998). One dimensional temperature field is assumed to beconstant in the x− y plane of the layer. In such a case, the temperature distributionalong the thickness can be obtained by solving a steady-state heat transfer equation

− ddz

[κ(z,T )

dTdz

]= 0 (11)

This equation can be solved by imposing boundary condition of T = Tt at the topsurface (z = −h/2) and T = Tb at bottom surface (z = h/2). The solution of thisequation is

T = Tt − (Tt −Tb)

∫ z− h

2

1κ(z,T )dz∫ h

2− h

2

1κ(z,T )dz

(12)

Note that the temperature field is uniform when Tt = Tb.

Table 1: Temperature-dependent coefficients for ceramic and metals [Reddy andChin (1998)].

Material Properties P−1 P0 P1 P2 P3

Si3N4

E (Pa) 0 348.43e+9 -3.070e-4 2.160e-7 -8.946e-11ν 0 0.24 0 0 0

α(1/K) 0 5.8723e-6 9.095e-4 0 0κ(W/mK) 0 13.723 -1.032e-3 5.466e-7 -7.876e-11

SUS304

E (Pa) 0 201.04e+9 3.079e-4 -6.534e-7 0ν 0 0.3263 -2.002e-4 3.797e-7 0

α(1/K) 0 12.330e-6 8.086e-4 0 0κ(W/mK) 0 15.379 -1.264e-3 2.092e-6 -7.223e-10

5.1 Comparison studies

To ensure the accuracy and effectiveness of the present method, two examples aresolved for nonlinear bending and thermal post-buckling analysis of isotropic andFGM beams.

Example 1. The central deflection-load curves for a beam with immovable simplysupported ends subjected to a transverse uniform distributed load q0L3

x/D11 are cal-culated and compared in Table 2 with results of Horibe and Asano (2001) using theboundary integral equation method. In this example, r =

√D11/A11 is the radius

of gyration, and Lx/r = 100, Pasternak type (k1, k2)=(100, 50) for the Pasternak


Table 2: Comparisons of nonlinear bending for isotropic beams resting on differentelastic foundations.

q0L3x/D11

wcenter/Lx

Horibe and Asano (2001) Present(100, 0) (100, 50) (100, 0) (100, 50)

40 0.053751 0.048414 0.05370 0.0429580 0.069528 0.060816 0.06948 0.06076120 0.080385 0.072745 0.08034 0.07269160 0.088954 0.082007 0.08891 0.08196200 0.096154 0.089705 0.09611 0.08966

elastic foundation and (k1, k2)=(100, 0) for the Winkler elastic foundation, wherek1 = K1L4

x/D11 and k2 = K2L2x/D11. The present results agree well with results of

Horibe and Asano (2001) except at one point, and the result of Horibe and Asano(2001) may be incorrect in the case of (k1, k2)=(100, 50) with a dimensionlessuniform distributed load of 40.

Figure 2: Comparisons of thermal post-buckling for Si3N4/SUS304 beams with im-movable clamped ends resting on different elastic foundations subjected to uniformtemperature rise.


Figure 3: Comparisons of thermal post-buckling for Si3N4/SUS304 beams withimmovable simply supported ends resting on different elastic foundations subjectedto heat conduction.

Example 2. Thermal post-buckling for Si3N4/SUS304 beams with different sup-ported ends resting on different elastic foundations are calculated and comparedin Figs. 2 and 3 with results of Esfahani et al. (2013) using generalized differ-ential quadrature method. In this example, length to thickness ratio Lx/h = 25,volume fraction index N = 1, and the dimensionless foundation stiffnesses are(k1,k2,k3)= (100,0,0) for the Winkler elastic foundation, (k1,k2,k3)= (100,10,0)for the Pasternak elastic foundation, (k1,k2,k3) = (100,0,50) and (k1,k2,k3) =(100,10,50) for the nonlinear elastic foundation, where k1 = 12K1Lx

4/E0h3, k2 =12K2Lx

2/E0h3 and k3 = 12K3Lx4/E0h, E0 is Young’s modulus of Si3N4 at reference

temperature. Excellent agreements can be seen from Figs. 2 and 3.

5.2 Parametric studies

A parametric study was undertaken for nonlinear bending and thermal post-bucklingof Si3N4/SUS304 beams with Lx/h= 50. The volume fraction Vc is defined by Vc =(1/2− z/h)N , and the dimensionless foundation stiffnesses are (k1, k2, k3)=(50, 0,0) for the Winkler elastic foundation, (k1, k2, k3)=(50, 5, 0) for the Pasternak elasticfoundation, (k1, k2, k3)=(50, 0, 10) and (k1, k2, k3)=(50, 5, 10) for the nonlinearelastic foundation, where k1 = K1L4

x/E0h3, k2 = K2L2x/E0h3 and k3 = K3L4

x/E0h,E0 is Young’s modulus of SUS304 at reference temperature. The top surface is


ceramic-rich, whereas the bottom surface is metal-rich, hence Tt = Tc and Tb = Tm

for heat conduction.

Figure 4: Effect of volume fractions on nonlinear bending of Si3N4/SUS304 beamswith immovable simply supported ends resting on nonlinear elastic foundations.

The central dimensionless deflections of Si3N4/SUS304 FGM beams with immov-able simply supported ends and clamped ends subjected to transverse uniformlydistributed loads in different temperature fields are calculated, see Figs. 4-9. Theeffect of volume fractions on nonlinear bending of Si3N4/SUS304 beams with dif-ferent supported ends resting on nonlinear elastic foundations can be seen in Figs.4, 5, it can be observed that the deflections increase with increasing value of volumefraction index N. The effect of different elastic foundations on nonlinear bendingof Si3N4/SUS304 beams with different supported ends and volume fraction indexN = 2 can be seen in Figs. 6, 7, it can be observed that the effect of nonlinear elasticfoundation increase with increasing deflection. The effect of different temperaturefields on nonlinear bending of Si3N4/SUS304 beams with different supported endsand volume fraction index N = 2 resting on nonlinear elastic foundations can beseen in Figs. 8, 9, downward initial deflections under uniform temperature risefields and the upward initial deflections under heat conduction fields can be ob-served for the beams with immovable simply supported ends, while no initial de-flections under both uniform temperature rise and heat conduction fields can beobserved for the beams with immovable clamped ends.


Figure 5: Effect of volume fractions on nonlinear bending of Si3N4/SUS304 beamswith immovable clamped ends resting on nonlinear elastic foundations.

Figure 6: Effect of elastic foundations on nonlinear bending of Si3N4/SUS304beams with immovable simply supported ends.


Figure 7: Effect of elastic foundations on nonlinear bending of Si3N4/SUS304beams with immovable clamped ends.

Figure 8: Effect of different temperature fields on nonlinear bending ofSi3N4/SUS304 beams with immovable simply supported ends.


Figure 9: Effect of different temperature fields on nonlinear bending ofSi3N4/SUS304 beams with immovable clamped ends.

Figure 10: Thermal post-buckling behaviors for Si3N4/SUS304 beams with im-movable simply supported ends resting on nonlinear elastic foundations subjectedto uniform temperature rise.


Figure 11: Thermal post-buckling behaviors for Si3N4/SUS304 beams with im-movable simply supported ends resting on nonlinear elastic foundations subjectedto heat conduction.

Figure 12: Thermal post-buckling behaviors for Si3N4/SUS304 beams with im-movable clamped ends resting on nonlinear elastic foundations subjected to uni-form temperature rise.


Figure 13: Thermal post-buckling behaviors for Si3N4/SUS304 beams with im-movable clamped ends resting on nonlinear elastic foundations subjected to heatconduction.

Figure 14: Effect of elastic foundations on thermal post-buckling of Si3N4/SUS304beams with immovable simply supported ends subjected to uniform temperaturerise.


Figure 15: Effect of elastic foundations on thermal post-buckling of Si3N4/SUS304beams with immovable simply supported ends subjected to heat conduction.

Figure 16: Effect of elastic foundations on thermal post-buckling of Si3N4/SUS304beams with immovable clamped ends subjected to uniform temperature rise.


Figure 17: Effect of elastic foundations on thermal post-buckling of Si3N4/SUS304beams with immovable clamped ends subjected to heat conduction.

Thermal post-buckling behaviors for Si3N4/SUS304 FGM beams with differentsupported ends resting on nonlinear elastic foundations subjected to uniform tem-perature rise and heat conduction are calculated, see. Figs. 10-13. Bifurcation ofbuckling can occur for FGM beams with simply supported ends due to effect ofuniform temperature rise or heat conduction, while bifurcation of buckling can notoccur for FGM beams with clamped ends due to effect of uniform temperature riseand heat conduction. The effect of elastic foundations on thermal post-bucklingof Si3N4/SUS304 beams with different supported ends and volume fraction indexN = 2 resting on nonlinear elastic foundations subjected to uniform temperaturerise and heat conduction are calculated, see. Figs. 14-17, and it can be observedthat the effect of nonlinear elastic foundation increase with increasing deflection.

6 Conclusions

A model of the FGM beams resting on nonlinear elastic foundations is successfullyestablished by physical neutral surface and high-order shear deformation theory. Innonlinear bending and post-buckling analysis, influences played by different sup-ported boundaries, thermal environmental conditions, different elastic foundationand volume fraction index are discussed in detail. In nonlinear bending analysis,downward initial deflections under uniform temperature rise fields and the upward


initial deflections under heat conduction fields can be observed for the FGM beamswith immovable simply supported ends, while no initial deflections under both uni-form temperature rise and heat conduction fields can be observed for the FGMbeams with immovable clamped ends. In thermal post-buckling analysis, bifurca-tion of buckling can occur for FGM beams with simply supported ends due to effectof uniform temperature rise or heat conduction, while bifurcation of buckling cannot occur for FGM beams with clamped ends due to effect of uniform tempera-ture rise and heat conduction. It is worth noting that the effect of nonlinear elasticfoundation is significant with increasing deflection.

Acknowledgement: This research was supported by the Fund of the NationalNatural Science Foundation of China under Grant No. 11172285, and ZhejiangProvincial Natural Science Foundation of China under Grant No. LR13A020002.The authors would like to express their sincere appreciation to these supports.

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Appendix A

The displacement fields

u = u0 +(z− z0)ψx −4

3h2

(z3 − c0

)(dwdx

+ψx

)(A.1a)

w = w(x) (A.1b)

in which z0 and c0 are defined by

z0 =

∫ h2− h

2zE(z,T )dz∫ h

2− h

2E(z,T )dz

, c0 =

∫ h2− h

2z3E(z,T )dz∫ h

2− h

2E(z,T )dz

(A.2)

Considering nonlinear von Kármán strain-displacement relationships, the strainscan be expressed by

εx = ε(0)x +(z− z0)ε

(1)x +

(z3 − c0

)ε(3)x , γxz = γ

(0)xz + z2

γ(2)xz (A.3)

in which

ε(0)x =

du0

dx+

12

(dwdx

)2

, ε(1)x =

dψx

dx, ε

(3)x =− 4

3h2

(dψx

dx+

d2wdx2

)(A.4a)

γ(0)xz = ψx +

dwdx

, γ(2)xz =− 4

h2

(ψx +

dwdx

)(A.4b)

According to Hooke’s law, the stresses can be determined as

σx = E(z,T ) [εx −α(z,T )∆T ] , τxz =E(z,T )

2 [1+ν(z)]γxz (A.5)

The constitutive equations can be deduced by proper integration.

Nx

Mx

Px

=

A11 0 00 D11 F110 F11 H11

ε

(0)x

ε(1)x

ε(3)x

− NT

MT

PT

,


[Qx

Rx

]=

[A44 D44D44 F44

][γ(0)xz

γ(2)xz

](A.6)

In which the beam stiffnesses are defined by

(A11,D11,F11,H11) =∫ h

2

− h2

E(z,T )(

1,(z− z0)2,(z− z0)

(z3 − c0

),(z3 − c0

)2)

dz

(A.7a)

(A44,D44,F44) =∫ h

2

− h2

E(z,T )2 [1+ν(z)]

(1,z2,z4)dz (A.7b)

The axial forces, bending moments and higher order bending moment caused byelevated temperature are defined by

(NT ,MT ,PT ) =∫ h

2

− h2

E(z,T )α(z,T )∆T(1,z− z0,z3 − c0

)dz (A.8)

where ∆T = T −T0 is temperature rise from some reference temperature T0 at whichthere are no thermal strains.

The symbols used in the governing equations are defined by

D11 = D11 −4F11

3h2 , F11 = F11 −4H11

3h2 , D11 = D11 −4F11

3h2 ,

A44 =A44−4D44

h2 , D44 =D44−4F44

h2 , A44 = A44−4D44

h2 , MT =MT −4

3h2 PT ,

(A.9)

nonlinear bending and thermal post-buckling analysis of ... · post-buckling analysis of fgm...

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