Non-linear thermal post-buckling analysis of FGM ... · under non-uniform temperature rise across thickness ... Timoshenko beam theory is ... beam theory is that it considers uniform

Engineering Science and Technology, an International Journal 19 (2016) 1608–1625

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Non-linear thermal post-buckling analysis of FGM Timoshenko beamunder non-uniform temperature rise across thickness

http://dx.doi.org/10.1016/j.jestch.2016.05.0142215-0986/� 2016 Karabuk University. Publishing services by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author.E-mail addresses: [email protected], [email protected] (D. Das).

Peer review under responsibility of Karabuk University.

Amlan Paul a, Debabrata Das b,⇑aDepartment of Mechanical Engineering, Netaji Subhash Engineering College, Garia, Kolkata 700152, IndiabDepartment of Mechanical Engineering, Jadavpur University, Kolkata 700032, India

a r t i c l e i n f o

Article history:Received 8 April 2016Revised 10 May 2016Accepted 23 May 2016Available online 9 June 2016

Keywords:Functionally graded materialPost-bucklingGeometrically non-linearThermal loadTimoshenko beam

a b s t r a c t

The present work deals with geometrically non-linear post-buckling load–deflection behavior offunctionally graded material (FGM) Timoshenko beam under in-plane thermal loading. Thermal loadingis applied by providing non-uniform temperature rise across the beam thickness at steady-statecondition. FGM is modeled by considering continuous distribution of metal and ceramic constituentsacross the thickness using power law variation of volume fraction. The effect of geometricnon-linearity at large post-buckled configuration is incorporated using von Kármán type non-linearstrain–displacement relationship. The governing equations are obtained using the minimum potentialenergy principle. The system of non-linear algebraic equations is solved using Broyden’s algorithm.Four different FGMs are considered. A comparative study for post-buckling load–deflection behavior innon-dimensional form is presented for different volume fraction exponents and also for differentFGMs, each for different length–thickness ratios.� 2016 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The term FGM, i.e., functionally graded material was originatedby the researchers in Japan in mid-1980s (Yamanoushi et al. [1]).Since then it had gained great importance in the scientificcommunity because of its merits over conventional metals andcomposites. In FGM, the mixture of metal and ceramic withdesirable volume fractions provides smooth and continuousvariation of mechanical and physical properties in the desireddirection. And this makes it possible for the FGM components tohave the good qualities of both the metals and the ceramics inambient as well as in high temperature environment. Due to this,FGMs have found many applications in high temperature nuclearreactors and chemical plants, in high speed aircraft componentsand in many other branches of mechanical, electrical and civilengineering. With increasing applications, many theoreticalresearch works on mechanics of FGM components have beencarried out for the last few years. The present work deals withthe post-buckling load–deflection behavior of FGM Timoshenkobeams under non-linear thermal gradient across thickness.

In high temperature applications, often a structural componentlike beam is subjected to sustained temperature difference acrossits end surfaces. This facilitates conductive heat transfer throughthe beam thickness at steady-state condition and leads to thermalloading of the member. For such applications, a ceramic–metalFGM beams are often used, where the ceramic-rich layer is keptat the higher temperature side and the metal-rich surface is keptat the lower temperature side/ambient condition. This is due tothe fact that the ceramics have very good thermal resistance, andmetals have excellent strength and toughness. For designing FGMbeams in such thermal environments, its load–deflection behavioris very important. This particular problem is very significant as theFGM beams undergo bending in the presence of thermal compres-sive stress. The following paragraphs summarize some of therelated research works done by other notable researchers.

The investigation of thermal buckling load of third order sheardeformable FGM beam under uniform temperature rise is carriedout by Wattanasakulpong et al. [2]. Fu et al. [3] studied thethermo-piezoelectric buckling and dynamic stability for the piezo-electric functionally graded beams, subjected to one-dimensionalsteady heat conduction in the thickness direction employingEuler–Bernoulli beam theory. Kiani and Eslami [4] investigatedthe thermo-mechanical buckling problem of FGM Timoshenkobeams with temperature-dependent (TD) material properties forvarious temperature distributions across beam thickness.

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Fig. 1. Beam with dimensions and coordinate axes.

Table 1List of lowest order admissible functions for the displacement fields.

Function Boundary name Expression Boundary conditions

/1 SS and CS ðx=LÞf1� ðx=LÞg ujx¼0 ¼ 0; ujx¼L ¼ 0

b1 SS and CS ðx=LÞf1� ðx=LÞg wjx¼0 ¼ 0; wjx¼L ¼ 0

c1 SS cosðpx=LÞ wjx¼0–0; wjx¼L–0CS sinðpx=ð2LÞÞ wjx¼0 ¼ 0; wjx¼L–0

A. Paul, D. Das / Engineering Science and Technology, an International Journal 19 (2016) 1608–1625 1609

Fallah and Aghdam [5] studied the thermal buckling load foruniform temperature rise of FGM beams on elastic foundationemploying Euler–Bernoulli beam theory and consideringtemperature-independent (TID) material properties. Esfahaniet al. [6] studied the thermal buckling and post-buckling behaviorof FGM Timoshenko beams resting on non-linear hardening elasticfoundation. Ghiasian et al. [7] investigated the static and dynamicbuckling behavior of FGM Euler–Bernoulli beams subjected to uni-form temperature rise and resting on non-linear elastic foundation.The non-linear thermal post-buckling behavior of shear deformableStainless steel-Silicon Nitride beam resting on elastic foundation isstudied by Shen and Wang [8] under uniform and non-uniform

Table 2Temperature coefficients of FGM constituents.

Constituent Property P0 P�1

SUS304 E (Pa) 201.04 � 109 0a (1/K) 12.330 � 10�6 0t 0.3262 0K (W/mK) 15.379 0

Ti-6Al-4V E (Pa) 122.56 � 109 0a (1/K) 7.5788 � 10�6 0t 0.2884 0K (W/mK) 1.0000 0

Si3N4 E (Pa) 348.43 � 109 0a (1/K) 5.8723 � 10�6 0t 0.2400 0K (W/mK) 13.723 0

ZrO2 E (Pa) 244.27 � 109 0a (1/K) 12.766 � 10�6 0t 0.2882 0K (W/mK) 1.7000 0

Al2O3 E (Pa) 349.55 � 109 0a (1/K) 6.8269 � 10�6 0t 0.2600 0K (W/mK) �14.087 �1123.6

temperature rise. Ghiasian et al. [9] investigated the non-linearthermal dynamic buckling behavior of FGM Timoshenko beamssubjected to sudden uniform temperature rise. Free vibrationbehavior of a thermally or thermo-electrically buckled FGM beamswas investigated and reported in [10,11]. Thermal and thermo-electrical buckling and stability analysis of piezoelectric FGMbeams were investigated and discussed in [12–16]. Ghiasian et al.[17] carried out the non-linear vibration analysis of FGM beamsunder thermal loading considering TD material properties.

Zhao et al. [18] studied the thermal post buckling behavior ofsimply-supported thin FGM beams with temperature-independent (TID) material properties under uniform and somespecial cases of non-uniform temperature rise. The thermalpost-buckling load–deflection behavior of shear deformable FGMbeams under uniform temperature rise across thickness isinvestigated by Ma and Lee [19,20] both for temperature depen-dent and temperature independent material properties. Zhang[21] studied the non-linear bending behavior of shear deformableFGM beams under combined mechanical and steady-state thermalloading. Li [22] investigated the thermal post buckling behaviorof three-dimensional braided shear deformable beams withinitial geometrical imperfection subjected to uniform, linear and

P1 P2 P3

3.079 � 10�4 �6.534 � 10�7 08.086 � 10�4 0 0�2.002 � 10�4 3.797 � 10�7 0�1.264 � 10�3 2.092 � 10�6 �7.223 � 10�10

�4.586 � 10�4 0 06.638 � 10�4 �3.147 � 10�6 01.121 � 10�4 0 01.704 � 10�2 0 0

�3.070 � 10�4 2.160 � 10�7 �8.946 � 10�11

9.095 � 10�4 0 00 0 0�1.032 � 10�3 5.466 � 10�7 �7.876 � 10�11

�1.371 � 10�3 1.214 � 10�6 �3.681 � 10�10

�1.491 � 10�3 1.006 � 10�5 �6.778 � 10�11

1.133 � 10�4 0 01.276 � 10�4 6.648 � 10�8 0

�3.853 � 10�4 4.027 � 10�7 �1.673 � 10�10

1.838 � 10�4 0 00 0 0�6.227 � 10�3 0 0

Fig. 2. Variation of non-dimensional temperature rise (k�), normalized elastic modulus (E�) and normalized thermal expansion coefficient (a�) along the thickness directionfor L/h = 10 and k = 1.0: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.

1610 A. Paul, D. Das / Engineering Science and Technology, an International Journal 19 (2016) 1608–1625

non-linear temperature distribution through the thickness. Thenon-linear bending analysis of Euler–Bernoulli FGM beams wascarried out by Niknam et al. [23] under combined mechanicaland thermal loadings with uniform temperature rise across thick-ness. Kumar et al. [24] investigated the large amplitude free vibra-tion problem of an axially functionally graded (AFG) tapered beamsunder different boundary conditions. Paul and Das [25] investi-gated the free vibration behavior of pre-stressed FGM beams fordifferent boundary conditions using Ritz method.

For the present work, Timoshenko beam theory is used to con-sider the effect of shear deformation. The limitation of Timoshenkobeam theory is that it considers uniform distribution of shearstress across the beam thickness. For more realistic modeling oftransverse shear stress, higher order shear deformation theoriesmust be used. It must be mentioned that various improved higherorder shear deformation theories (HSDT) [26–39] had been devel-oped for analysis of various structural elements made of FGM. Insome of those research works, the effect of stretching deformationhas been considered in addition to the effect of shear deformation.Research works for static and dynamic analysis of micro and nonoscale beams and plates were reported in [40,41].

Based on the surface elasticity theory, post-buckling anddynamic analysis of functionally graded (FG) nanobeams underthermal environment were carried out by Ansari et al. [42,43].

Ansari et al. [44,45] investigated the size dependent thermal buck-ling and postbuckling behavior of FG micro beams and plates basedon modified strain gradient theory. The dynamic stability of func-tionally graded shear deformable thin-walled beams was studiedby Piovan and Machado [46]. Based on the nonlocal Timoshenkobeam theory, Ansari et al. [47] studied the thermal buckling behav-ior of single-walled carbon nanotubes embedded in an elastic med-ium. Mirzaei and Kiani [48] studied the snap-through phenomenonin a thermally postbuckled temperature dependent sandwichbeam. Akbas� and Kocatürk [49] investigated the post-bucklingbehavior of FG three-dimensional beams under thermal loadingusing total Lagrangian finite element model.

The buckling and vibration behavior of FGM sandwich beamhaving constrained viscoelastic layer was studied by Bhangaleand Ganesan [50] in thermal environment by using a finite elementformulation. Fallah and Aghdam [51] studied the large amplitudefree vibration and postbuckling problem of FGM beams restingon nonlinear elastic foundation. The thermal post-buckling analy-sis of FGM beams resting on nonlinear elastic foundations werereported in [52,53]. Atai et al. [54] studied the buckling and post-buckling behavior of semicircular FGM arches subjected to radialand tangential follower forces, and thermal loading. By using thetotal Lagrangian Timoshenko beam element approximation,Kocatürk and Akbas� [55,56] carried out thermal post-buckling

Fig. 3. Non-dimensional limit thermal load versus volume fraction exponent plots for different FGMs: (a) L/h = 5, (b) L/h = 10 and (c) L/h = 20.


analysis of FG beams. Using the same theory, Akbas� [57,58] studiedthe post-buckling behavior of AFG beams and edge cracked FGMbeams under compressive mechanical load. The effect ofmaterial-temperature dependence on the wave propagation of anFGM cantilever beam was investigated by Akbas� [59] under theapplication of an impact force.

From the literature review, it is clear that a detailed study onthe geometrically non-linear post-buckling load–deflection behav-ior of FGM Timoshenko beam under steady-state temperature dis-tribution across the beam thickness is scarce. So the present workis mainly carried out to study the thermal post-buckling equilib-rium path of various FGM beams with immovable simply-supported (SS) ends. A beam with one end clamped and the otherend simply supported (CS) is also considered for making a compar-ative study between the SS and CS beams. The temperature depen-dences of the mechanical properties (except thermal conductivity)are considered using a cubic polynomial relationship which in turnintroduces physical non-linearity into the problem. The

displacement based governing equations are derived using theenergy principle and the approximate solutions are obtained usingRitz method. Geometric non-linearity is incorporated into themathematical formulation using von Kármán type non-linearstrain–displacement relationship and the set of non-linear govern-ing equations is solved using Broyden’s algorithm. A comparativestudy of post-buckling equilibrium path in non-dimensional formis presented for different FGMs and also for different volume frac-tion exponents. The fact that the elastic modulus becomes negativeabove certain limit thermal load, depending on the material, is con-sidered for generation of results.

The significance of the present work is discussed here. Theliterature survey reveals that the present work has indirectand apparent similarity with some of the published workswhich also consider load–deflection behavior under non-uniform temperature rise. As an example, the work by Zhaoet al. [18] presented the thermal post-buckling load–deflectionbehavior of Titanium alloy-Zirconia FGM beam for some specific

Fig. 4. Comparison of post-buckling equilibrium path: (a) FGM 1 for uniform temperature rise with Ref. [6], (b) All the four FGM beams for non-uniform temperature rise withANSYS and (c) FGM 1 for uniform temperature rise with Ref. [19].


cases of non-uniform temperature distribution and by consider-ing temperature-independent (TID) material properties. Anotherexample is the work of Zhang [21] which deals with thermalload–deflection behavior of FGM beams under the presence ofmechanical loading. Hence, an exhaustive study on thermalpost-buckling load–deflection behavior in high temperatureenvironment with temperature dependent material propertiesis not available. The present work bridges this gap by consider-ing four different FGMs and by conducting a comparative studyfor different material compositions, length–thickness ratios andvolume fraction exponents. Apart from that, a comparativestudy is also presented for different through-thickness tempera-ture distributions, namely, uniform, linear and non-linear. Onthe other hand, the present method is very robust and quitegeneral in nature to investigate the static and dynamic behaviorof any structural elements for various boundary conditions. Asalready mentioned, Paul and Das [25] investigated the dynamicbehavior of pre-stressed FGM beams for various boundary

conditions using the same method. In the present work also,a comparative study between SS and CS beams is presented.

2. Mathematical formulation

The present work is envisaged to study the post-buckling load–deflection behavior of FGM Timoshenko beams under steady-statein-plane thermal loading incorporating geometric non-linearityinto the formulation. The thermal loading is applied by applyinga nonlinear temperature gradient across the thickness, which isobtained by solving the steady-state heat conduction equation.The material properties, namely, elastic modulus E, shear modulusG, thermal expansion coefficient a and Poisson’s ratio t are consid-ered to be temperature dependent. A beam with immovablesimply-supported (SS) ends is considered for the present work. AFGM beam having length L, thickness h and width b is shown inFig. 1, where, x, y and z denote the coordinate axes along thelength, width and thickness directions respectively.

Fig. 5. Comparison of non-dimensional post-buckling load–deflection behavior for different values of volume fraction exponent for L/h = 5: (a) FGM 1, (b) FGM 2, (c) FGM 3and (d) FGM 4.


For the present problem, the thermal gradient, applied alongthe thickness direction, gives rise to a thermal compressive forcealong the x axis and a thermal moment about the y axis due toimmovable end conditions. This is also true for a homogeneousbeam (like pure ceramic or pure metal) with any non-uniform tem-perature rise across the thickness. As the simply supported endscan not generate reaction moments, bending of the beam occurseven with a very small thermal gradient across the thickness andthe beam is said to acquire post-buckled configuration (loss of ini-tial configuration). Hence the buckling problem that involves bifur-cation does not arise for the present work.

The mathematical formulation is based on the minimumpotential energy principle in variational form and the effect of sheardeformation is included through Timoshenko beam theory (Shamesand Dym [60]). The effect of geometric non-linearity for largedeflection problems is included using von Kármán type non-linearstrain–displacement relationship. For FGM modeling, a continuousvariation of volume fraction of metal and ceramic across thethickness is assumed. It is to be mentioned that the subscriptsc and m refer to the ceramic and metal constituents respectively.Also the stress–strain relationship is assumed to be linear elastic

for the constituent materials. Although the physical neutral surfacefor a transversely deformed FGM beam does not coincide with thecentroidal surface, they are assumed to be coincident for the pre-sent work. The concept of physical neutral surface has been consid-ered by some of the researchers [3,19–21,40]. It must bementionedthat the consideration of the concept of physical neutral surfaceleads to a more practical solution.

2.1. FGM modeling

The continuous variation of volume fraction of ceramic across

the thickness is given by the power law relation as, Vc ¼ zh þ 1

2

� �k(Shen [61]) where, kð0 6 k 6 1Þ is the volume fraction exponentwhich governs the material variation profile across the thickness.The effective material property Pf of any FGM layer is determinedusing the Voigt model which gives Pf ¼ PcVc þ PmVm andVc þ Vm ¼ 1, where Pc and Pm are the material properties of theceramic and metal constituents respectively, and Vm is the volumefraction of the metal constituent. Accordingly, the top layer(z ¼ þh=2) is pure ceramic and the bottom layer is pure metallic



(z ¼ �h=2). The temperature dependence of the materialproperties (Pc or Pm) of the individual constituents is given by(Shen [61]),

Pc or; Pm ¼ P0½P�1T�1 þ 1þ P1T þ P2T

2 þ P3T3�; ð1Þ

where P0, P�1, P1, P2 and P3 are the coefficients of temperature and Tis the temperature in K. These coefficients are different for differentmaterial properties. Hence any effective material property given by,

Pf ðz; TÞ ¼ Pm þ ðPc � PmÞ zh þ 1

2

� �k is a function of both the coordinatealong the thickness direction and the temperature, where T is thetemperature at any value of z. It is evident that a zero value ofthe volume fraction exponent implies a pure ceramic beam. It isto be mentioned that the thermal conductivity (K) of the FGM con-stituents is assumed to be a function of z only.

2.2. Temperature distribution across thickness

The top surface, which is ceramic rich, is considered to be attemperature Tc and the bottom surface, which is metal rich, is con-sidered to be at temperature Tm (<Tc). With these temperatures at

the extreme layers, the variation of temperature across the thick-ness is obtained by solving the steady-state heat transfer equation,ddz KðzÞ dT

dz

� � ¼ 0. A polynomial series solution (Esfahani et al. [6],Javaheri and Eslami [62]) of this equation, retaining the first seventerms, is given as,

TðzÞ ¼ Tm þ DTC

zhþ 12

� �� Kcm

ðkþ 1ÞKm

zhþ 12

� �kþ1

þ K2cm

ð2kþ 1ÞK2m

"

� zhþ 12

� �2kþ1

� K3cm

ð3kþ 1ÞK3m

zhþ 12

� �3kþ1

þ K4cm

ð4kþ 1ÞK4m

� zhþ 12

� �4kþ1

� K5cm

ð5kþ 1ÞK5m

zhþ 12

� �5kþ1#

ð2Þ

where,

C ¼ 1� Kcm

ðkþ 1ÞKmþ K2

cm

ð2kþ 1ÞK2m

� K3cm

ð3kþ 1ÞK3m

þ K4cm

ð4kþ 1ÞK4m

� K5cm

ð5kþ 1ÞK5m

; DT ¼ Tc � Tm and Kcm ¼ Kc � Km:



2.3. Governing equation for post-buckling equilibrium path

The governing equation for stable post-buckling equilibriumpath is derived using the minimum potential energy principle,which in variational form (Shames and Dym [60]), is given as,

dðU þ VÞ ¼ 0 ð3Þwhere U is the total strain energy of the beam, V is the potentialenergy of the applied load and d is the variational operator. The totalstrain energy is composed of three parts, i.e., U ¼ Un þ Us þ Ur,where, Un and Us being the contributions from the normal strainand shear strain respectively, and Ur is the strain energy due tothermal pre-stress. As no other external load is applied, the valueof V is zero for the present problem. The present analysis is basedon three displacement fields, namely, the in-plane displacementfield u, the transverse displacement field w and the rotational fieldof beam cross section due to bending w. Here u, w and w are definedat the mid-plane of the beam.

The strain–displacement relationships for the normal strain exxat any layer z from the mid-plane and the shear strain exz are givenby,

exx ¼ dudx

þ 12

dwdx

� �2

� zdwdx

ð4Þ

and exz ¼ 12

dwdx

� w

� �: ð5Þ

Using linear stress–strain relationships, the expressions ofstrain energies Un and Us are as follows:

Un ¼ 12

An

Z L

0

dudx

� �2

dxþ An

Z L

0

14

dwdx

� �4

dxþ Cn

Z L

0

dwdx

� �2

dx

"

þ An

Z L

0

dwdx

� �2 dudx

� �dx� 2Bn

Z L

0

dwdx

� �dudx

� �dx

� Bn

Z L

0

dwdx

� �dwdx

� �2

dx

#ð6Þ

and Us ¼ qAs

2

Z L

0

dwdx

� �2

dxþZ L

0ðw2Þdx� 2

Z L

0

dwdx

� �ðwÞdx

" #;

ð7Þ

Fig. 8. Comparison of non-dimensional post-buckling load–deflection behavior for FGM 1, FGM 2, FGM 3 and FGM 4 beams for L/h = 5: (a) k = 0.0, (b) k = 0.2, (c) k = 2.0, (d)k = 10.0 and (e) k = 20.0.






Fig. 11. Non-dimensional post-buckling load–deflection behavior for different L/h ratio for k = 0.5: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.


where q is the shear correction factor, which is taken to be 5/6 forrectangular cross section.

With the presence of thermal gradient across the thickness andin-plane restraint at the ends, each layer is subjected to compres-sive thermal stress, which, at any layer z is given by,rthðzÞ ¼ �Efaf ½TðzÞ � T0�. Here, T0 is the stress-free temperaturefor thermal stress calculation.

The expression of Ur is thus given by,

Ur ¼ Ar2

Z L

0

dwdx

� �2

dxþ Ar

Z L

0

dudx

� �dx� Br

Z L

0

dwdx

� �dx

" #ð8Þ

In Eqs. (6)–(8), the following stiffness parameters are used:

ðAn;Bn;CnÞ ¼ bZ h=2

�h=2Ef ð1; z; z2Þdz; ð9Þ

As ¼ bZ h=2

�h=2Gf dz ð10Þ

and ðAr;BrÞ ¼ bZ h=2

�h=2rthð1; zÞdz ð11Þ

where the shear modulus is given by, Gf ¼ Ef2ð1þtf Þ.

Following Ritz method, the displacement fields u, w and w areassumed to be summation of the products of the correspondingadmissible functions and the unknown coefficients ci, as givenbelow:

u ¼Xnui¼1

ci/i; w ¼Xnwi¼1

cnuþibi and w ¼Xnsii¼1

cnuþnwþici; ð12Þ

where /i, bi and ci are the set of orthogonal admissiblefunctions for the field variables u, w and w respectively; andnu, nw and nsi are the number of functions for u, w and wrespectively. The lowest order admissible functions for each ofthese sets are selected so as to satisfy the boundary conditionsof the beam and the higher order functions are generatednumerically through Gram-Schmidt orthogonalization scheme.The lowest order admissible functions for each of thesedisplacement fields along with the boundary conditions aregiven in Table 1.

Putting the expression of various strain energies, given by Eqs.(6)–(8) in Eq. (3) and using the approximate displacement fields,given by Eq. (12), the governing equations are obtained in the fol-lowing form:

½Kji�fcig ¼ ff jg; ð13Þ

Fig. 12. Non-dimensional post-buckling load–deflection behavior for different temperature distributions for L/h = 10 and k = 0.75: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d)FGM 4.


where, ½Kji� and ff jg are the stiffness matrix and load vector respec-tively, each of dimension nuþ nwþ nsi. The stiffness matrix consistof two parts, i.e., ½Kji� = ½Kc

ji� + ½Krji �; where, ½Kc

ji� is the conventional

stiffness matrix and ½Krji � is the stress stiffness matrix. It is to be

mentioned that the first term of Eq. (8) contributes to ½Krji � and

the remaining two terms provides the load vector ff jg. The conven-tional stiffness matrix is a function of unknown coefficients fcig,due to the presence of non-linear strain–displacement relationship,given by Eq. (4) and this makes ½Kc

ji� as functions of fcig. The ele-

ments of ½Kcji�, ½Kr

ji � and ff jg are given in the Appendix.

2.4. Solution of the system of non-linear equations

Eq. (13) is a system of non-linear algebraic equations involvingthe unknown coefficients fcig. Initially it is tried to solve with asubstitution type iterative method using successive relaxation(Das et al. [63]). But this method failed to give post-buckling solu-tion after certain thermal loading level is achieved. This is becauseit could not obtain the solution once the post-buckling equilibrium

path changes its curvature. The exhibition of curvature change ofpost-buckling load–deflection behavior is evident from the pre-sented results in the next section. Hence a multi-dimensionalsecant method based on Broyden’s algorithm (Press et al. [64],Das et al. [65]) is used to solve Eq. (13). This algorithm is foundto be quite stable and provides solution even for change in curva-ture of the post-buckling equilibrium path. It must be mentionedthat the computational time for Broyden’s algorithm is more thanthe previously tried substitution method.

3. Results and discussion

In the present work, a study of thermal post-buckling load–de-flection behavior of FGM Timoshenko beam is carried out. Thepost-buckling path is represented in the form of non-dimensionalthermal load (k) versus normalized maximum transversedeflection (w�) plots. Four different functionally graded beammaterials are considered for the present work. These are StainlessSteel (SUS304)-Silicon Nitride (Si3N4), Stainless Steel-Zirconia(ZrO2), Stainless Steel-Alumina (Al2O3) and Titanium alloy(Ti-6Al-4V)-Zirconia and hereafter, these are referred as FGM 1,

Fig. 13. Non-dimensional post-buckling load–deflection behavior for temperature-dependent (TD) and temperature-independent (TID) material properties for L/h = 10: (a)FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.


FGM 2, FGM 3 and FGM 4 respectively. The temperature coeffi-cients (Reddy and Chin [66]) for various material properties ofthe constituents of these FGMs are given in Table 2. The resultsare generated for five different length–thickness (L/h) ratios whichare 5, 10, 15, 20 and 25. The results are generated for h = 0.01 mand b = 0.02 m.

The stress-free temperature T0 is taken as 300 K. Also the tem-perature of the bottom layer (Tm), which is metal rich, is alwaysconsidered to be equal to T0. By considering different temperaturevalues (Tc) for the top layer, starting from T0, various thermal gra-dients and in turn various thermal loads are applied. For each ofthese thermal loads, the value of the maximum post-bucklingdeflection wmax is obtained, which occurs at the mid-span(x = L/2) for SS beam. The non-dimensional thermal load, k andthe normalized maximum transverse deflection, w� are defined

as: k ¼ 12am0ðL=hÞ2ðTc � TmÞ and w� ¼ absðwmax=hÞ. Here, am0 isthe thermal expansion coefficient of the metal constituent at T0.Another parameter w� is defined as the normalized maximumpost-buckling deflection, considering the direction of the deflec-tion. It is defined as w� ¼ ðwmax=hÞ and is used for the validationstudy as presented in Fig. 4(c).

3.1. Variation of temperature and material properties

The variations along the beam thickness direction of non-dimensional temperature rise (k�) and the corresponding valuesof the effective elastic modulus (E�) and the effective thermalexpansion coefficient (a�) both in normalized forms are shown inFig. 2(a)–(d) for FGM 1, FGM 2, FGM 3 and FGM 4 respectively. Itis to be noted that Fig. 2 corresponds to L/h = 10 and k = 1.0. Thenon-dimensional temperature rise for any layer having tempera-

ture T is given by, k� ¼ 12am0ðL=hÞ2ðT � TmÞ. On the other hand,for any layer having effective elastic modulus, Ef and effectivethermal expansion coefficient, af ; the corresponding normalizedeffective properties are given by, E� ¼ Ef =Em0 and a� ¼ af=am0

respectively. Here, Em0 is the elastic modulus of the metal con-stituent at the stress-free temperature. Fig. 2(a)–(d) indicates thehighly non-linear nature of the temperature profile and the corre-sponding material properties across the beam thickness. It is seenthat E� increases from the bottom to the top layer for FGM 1 andFGM 4 (Fig. 2(a) and (d)), whereas it decreases across the thicknessfor FGM 2 (Fig. 2(b)). But for FGM 3, it increases initially and thendecreases, as seen in Fig. 2(c). On other hand, a� decreases for FGM

Fig. 14. Non-dimensional post-buckling load–deflection behavior for SS and CSbeams for L/h = 20 and k = 1.5.


1 (Fig. 2(a)) and increases for FGM 2 and FGM 4 (Fig. 2(b) and (d)).For FGM 3 (Fig. 2(c)), a� decrease along the beam thickness direc-tion but increases slightly as the top layer is approached.

As these plots are just the illustrations of non-linear variationsof temperature and the corresponding material properties, the val-ues of non-dimensional thermal load k for which Fig. 2 is pre-sented, are taken arbitrarily for different FGMs. It is to bementioned that Fig. 2(a)–(d) correspond to k = 15, 8, 30, 3 respec-tively and are presented for L/h = 10 and k = 1.0. These plots areexpected to be different for different volume fraction exponentsbut are not directly dependent on L/h ratios. Though L/h ratio doesnot affect the temperature distribution along the beam thicknessdirection but still needs to be mentioned as the non-dimensionalform of temperature rise k� is dependent on L/h. It is known thatthe present problem is strongly dependent on these non-linearvariations of elastic modulus, thermal expansion coefficient andtemperature.

It can be shown that the elastic modulus becomes negativeabove certain limiting values of k, which is denoted by k. As nega-tive elastic modulus does not have any physical meaning for thepresent problem, the results for the post-buckling behavior aregenerated only for its positive values. To identify the non-dimensional limit thermal load k, Fig. 3(a)–(d) are presented forL/h = 5, 10 and 20 respectively. These figures show the variationsof k with volume fraction exponent k. Fig. 3(a)–(d) show that kremains initially constant and then decreases with k for FGM 1,FGM 2 and FGM 4, whereas for FGM 3, it is constant initially andthen increases with k. It is to be noted that the dimensional valuesof limit thermal load are independent of L/h but it differs in non-dimensional form.

3.2. Validation study

The post-buckling equilibrium path of FGM 1 for uniform tem-perature rise (UTR) for L/h = 10 and k = 2.0 is compared with theresults presented in Ref. [6] in Fig. 4(a). The plot is presented fordimensional values of temperature rise and shows good agree-ment. The validation of the present work is also carried out withfinite element package ANSYS (version 10.0) for L/h = 10 andk = 1.0 for all the four FGM beams considered. The solution inANSYS is obtained using SHELL181 element and a layered beam

model has been used with 100 layers of equal thickness. The com-parison of post-buckling thermal load-maximum deflection plotsin non-dimensional plane for FGM 1, FGM 2, FGM 3 and FGM 4are shown in Fig. 4(b). The plots show good agreement of the pre-sent work with ANSYS and hence validate the present formulation.It must be noted that the finite element model used in ANSYS useslayered variation of material properties in contrast with continu-ous variation of the same in the present work. In Fig. 4(b), the unitvalue of w� is not reached for FGM 1 beam as the applied thermalload is not increased beyond the limit thermal load. On the otherhand, a maximum deflection level which is well below the unitvalue of w� is observed FGM 3 beam. The post-buckling equilib-rium path of FGM 1 for UTR for L/h = 20 and k = 1.0 is contrastedin Fig. 4(c) with the result presented in Ref. [19]. The comparisonshows matching, although not exactly, of the present result withRef. [19]. The deviation may be attributed to the fact that the resultin Ref. [19] is generated using the concept of physical neutral sur-face, which is not considered in the present work.

3.3. Post-buckling load–deflection behavior

Post-buckling load–deflection behavior in non-dimensionalplane (k�w�) for different values of k (k = 0.0, 0.1, 1.0, 5.0 and25.0) are shown in Fig. 5(a)–(d) for FGM 1, FGM 2, FGM 3 andFGM 4 respectively for L/h = 5. The results for non-dimensionalload–deflection behavior are attempted to be generated up to theunit value of w�. But this could not be achieved for all the casesdue to any one of the two important reasons. The first one, asalready mentioned, is due to the fact that the applied thermal loadis not increased beyond the limit thermal load, above which elasticmodulus becomes negative. The second reason being the fact thatthe load–deflection behavior reaches the zero-slope condition ink�w� plane indicating infinitely large tangent stiffness of thebeam. The post-buckling load–deflection beyond this point, whichshows decrease in deflection level with increase in thermal load, isnot considered to be the stable solution. This condition is hereafterreferred as the limit deflection condition.

Fig. 5(a) shows that FGM 1 beam, for all values of k, reaches thelimit deflection point which is much below the unit value of w�. Italso shows that the limit deflection level increases with k for itssmall values but decreases again with k for higher values of k.The post-buckling bending stiffness of FGM 1 beam decreases withincrease in k. The stiffness of FGM 2 beam increases with increas-ing values of k, as seen in Fig. 5(b). Also for higher value of k (=25),the maximum deflection level is restricted by the limit thermalload. FGM 3 beam exhibits (Fig. 5(c)) higher bending stiffnesslevels with increase in k values and also shows sharp decrease inmaximum deflection levels with increase in k. In all the cases ofFGM 3 beams, the maximum deflection level is controlled by limitdeflection condition. The stiffness of FGM 4 beam increases with kas evident from Fig. 5(d). The maximum deflection levels arereduced significantly with k for its higher values. It can also be seenthat the maximum deflection level is governed by limit deflectioncondition for higher values of k. From Fig. 5, it can be said that theeffect of volume fraction exponent is significant on post-bucklingequilibrium path. Besides, as will be seen in the next two para-graphs, the effect of L/h ratio is also important.

The comparison of non-dimensional post-buckling load–deflec-tion behavior, each for different values of k, are shown in Fig. 6(a)–(d) for L/h = 10 and in Fig. 7(a)–(d) for L/h = 20. Fig. 6(a)–(d) showsthat the stiffness decreases with increasing k values for FGM 1beam, whereas the trend is completely reverse for the other threeFGM beams. Fig. 6(a) shows that the maximum deflection levels forFGM 1 beam are restricted by the limit thermal loads. It also showsthat the maximum deflection level increases with k for its smallvalues but decreases again with k for higher values of k. For FGM


3 beam, w� is restricted by limit thermal load for lower values of k(i.e., k = 0.0, 0.1, 1.0) but is controlled by limit deflection conditionfor higher values of k (i.e., k = 5.0, 25.0). It also shows a sharpdecrease of w� values with increase in k.

The nature of variation of post-buckling bending stiffness with kfor different FGMs are similar for L/h = 20 as discussed for the otherL/h ratios. But the significant difference in post-buckling behavioris that the post-buckling deflection levels for FGM 1, FGM 2 andFGM 4 beams reach the unit value of w�. But for FGM 3 beam, unitvalue of w� is reached for lower values of k. On the other hand, forhigher k values, w� is governed by limit deflection condition as canbe seen from Fig. 7(c). The effect of the volume fraction exponent isfound to be highest for FGM 3 as the load–deflection plots arewidely scattered for different values of k. The variety of the thermalload–deflection behavior shown in Figs. 5–7 indicates that thematerial profile variation (effect of k) plays a crucial role in exhibit-ing the load–deflection behavior. This is mainly because the rela-tive variations in various mechanical properties at highertemperature are different for different constituent proportions.

Post-buckling load–deflection plots for four different FGMbeams in non-dimensional plane (k�w�) are shown in Fig. 8(a)–(e) for k = 0.0, 0.2, 2.0, 10.0 and 20.0 respectively for L/h = 5. Similarcomparative plots are presented in Fig. 9 for L/h = 10 and in Fig. 10for L/h = 20. Observing each of the figures of Figs. 8–10, the fourFGM beams can be arranged as FGM 4, FGM 2, FGM 1 and FGM 3in the order of increasing stiffness. This is true irrespective of thevalues of k and the values of L/h considered. The relative changein post-buckling deflection level among the four FGM beams con-sidered gradually increases with increase in applied thermal load.This is also true irrespective of the values of k and the values ofL/h considered. The stiffness exhibited by FGM 3 beam is the high-est as already mentioned. A careful examination of Figs. 8–10reveals that the gradual increase in post-buckling stiffness levelof FGM 3 beam with respect to the other three increases with kand this is more pronounced for higher L/h ratios. Figs. 8–10 indi-cate that the material-wise post-buckling behavior differs signifi-cantly for different L/h ratio. While selecting an FGM beam understeady-state heat conduction condition, the load–deflection behav-ior must be given due consideration. Because it strongly dependson the FGM composition, L/h ratio and material profile parameter k.

Post-buckling load–deflection plots for different length–thick-ness ratio are presented in Fig. 11(a)–(d) for FGM 1, FGM 2, FGM3 and FGM 4 respectively for k = 0.5. In each of the figures, L/h = 5, 10, 15, 20, 25 are considered. Some of the plots do not reachthe unit value of w� due to reaching of the limit thermal load. Asexpected, the stiffness level of the beam decreases with increasein the L/h ratio. This is true for all the FGMs considered exceptfor FGM 3. For FGM 3 (Fig. 11(c), where plots are shown with thinlines to have better visibility of the individual plots), the load–de-flection plots for different L/h ratio are found to be coincident, indi-cating invariance with respect to the L/h ratio in the non-dimensional plane.

Although the present study is meant for non-uniform tempera-ture rise, a comparative study of post buckling load–deflectionpath for uniform, linear and non-linear temperature rises are alsopresented. The comparative plots are shown in Fig. 12(a)–(d) forL/h = 10 and k = 0.75, each for different FGM compositions.Fig. 12 shows that some of the load–deflection plots do not reachthe unit value of w� due to the occurrence of limit thermal load.For low values of the thermal load, the behavior is seen to bealmost identical for the linear and non-linear temperature rises.For higher values of the thermal load, the plots for the linear andnon-linear temperature rises are found to be diverging, and thisis more pronounced for FGM 3 (Fig. 12(c)). Except for the low val-ues of the thermal load, the deflection levels for the case of uniformtemperature rise lie above that of the other two, indicating lowest

stiffness levels for this case with respect to the other two. For thethree cases considered, deflection levels are the lowest for the non-linear temperature rise, indicating highest post-buckling stiffness.

In order to illustrate the pronounced effect of temperaturedependence of the material properties on the post-buckling load–deflection path, Fig. 13(a)–(d) is presented for FGM 1, FGM 2,FGM 3 and FGM 4 respectively with L/h = 10. In each of the figures,the plots are shown for k = 0.1 and k = 1.0. Fig. 13 clearly indicatesthe significant difference in the post-buckling load–deflection pathfor the cases of TD and TID. It shows that the post-buckling stiff-ness levels are highly over estimated by considering TID. ForFGM 1, FGM 2 and FGM 4, the differences in the load–deflectionbehavior between TD and TID are seen to be significant. However,for FGM 4, it differs slightly relative to the other FGMs.

A comparison of the post-buckling equilibrium path between SSand CS beams is shown in Fig. 14 for L/h = 20 and k = 1.5. The com-parison plots are shown for all the FGMs considered. As expected,the CS beams for different FGM compositions exhibit higher stiff-ness levels compared to the SS beams. For FGM 3 beam, it is seenthat the difference in the post-buckling stiffness levels betweenthe SS and CS beams increase appreciably for higher values ofthe thermal load. This is unlike for the other three FGMs, for whichthe changes in the stiffness levels between the SS and CS beams areseen to be almost invariant with the thermal load.

4. Conclusions

Geometrically non-linear post-buckling behavior of FGMTimoshenko beams with immovable simply-supported ends isinvestigated. The analysis is carried out under thermal loadingwith steady-state thermal gradient across beam thickness andthe temperature dependence of the materials properties is consid-ered. The limit thermal load beyond which the elastic modulusbecomes negative is identified for various volume fraction expo-nents corresponding to four different FGM materials. The non-linear governing equations are solved using Broyden’s algorithmto obtain large deflection post-buckling behavior. Results forpost-buckling path are presented in the form of non-dimensionalthermal load-maximum deflection plots for four different FGMs.A comparative study for post-buckling behavior is presented fordifferent volume fraction exponents and also for different FGMs,each for different length–thickness ratios. Limit deflection pointsat which the post-buckling deflection becomes maximum withinfinitely large tangent stiffness are identified for some of theFGM beams. The results provide important information regardingthe nature of post-buckling bending stiffness of FGM Timoshenkobeam components for four different materials.

Appendix A

Elements of conventional stiffness matrix

½Kcji�j¼1;nu

i¼1;nu¼ An

Z L

0

d/j

dxd/i

dxdx;

½Kcji� j¼1; nu

i¼nuþ1; nuþnw¼ An

2

Z L

0

Xnwm¼1

cnuþmdbm

dx

!d/j

dxdbi�nu

dxdx;

½Kcji� j¼1;nu

i¼nuþnwþ1;nuþnwþnsi¼ �Bn

Z L

0

d/j

dxdci�nu�nw

dxdx;

½Kcji�j¼nuþ1;nuþnw

i¼1;nu¼ 0;



i¼nuþ1;nuþnw¼ qAs

Z L

0

dbj�nu

dxdbi�nu

dxdx

�

þAn

Z L

0

Xnum¼1

cmdam

dx

!dbj�nu

dxdbi�nu

dxdx

þAn

2

Z L

0

Xnwm¼1

cnuþmdbm

dx

!2dbj�nu

dxdbi�nu

dxdx

�Bn

Z L

0

Xnsim¼1

cnuþnwþmdcmdx

!dbj�nu

dxdbi�nu

dxdx

#;


i¼nuþnwþ1;nuþnwþnsi¼ �qAs

Z L

0

dbj�nu

dxci�nu�nwdx;

½Kcji�j¼nuþnwþ1;nuþnwþnsi

i¼1;nu¼ �Bn

Z L

0

dcj�nu�nw

dxd/i

dxdx;

½Kcji�j¼nuþnwþ1;nuþnwþnsi

i¼nuþ1;nuþnw¼ �Bn

2

Z L

0

Xnwm¼1

cnuþmdbm

dx

!"

� dcj�nu�nw

dxdbi�nu

dxdx�qAs

Z L

0cj�nu�nw

dbi�nu

dxdx

#

and ½Kcji�j¼nuþnwþ1;nuþnwþnsi

i¼nuþnwþ1;nuþnwþnsi¼ Cn

Z L

0

dcj�nu�nw

dxdci�nu�nw

dxdxþ qAs

Z L

0cj�nu�nw

ci�nu�nwdx.

Elements of stress stiffness matrix

½Krji �j¼nuþ1;nuþnw

i¼nuþ1;nuþnw¼Ar

Z L

0

dbj�nu

dxdbi�nu

dxdx; all other elements being zero:

Elements of load vector

ff jgj¼1;nu¼ �Ar

Z L

0

d/j

dxdx;

ff jgj¼nuþ1;nuþnw¼ 0

and ff jgj¼nuþnwþ1;nuþnwþnsi¼ Br

Z L

0

dcj�nu�nw

dxdx:

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http://dx.doi.org/10.1007/s11012-016-0396-0

http://dx.doi.org/10.1007/s11012-016-0396-0



















































http://dx.doi.org/10.1142/S0219455414500655

http://dx.doi.org/10.1142/S0219455414500655























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Non-linear thermal post-buckling analysis of FGM ... · under non-uniform temperature rise across thickness ... Timoshenko beam theory is ... beam theory is that it considers uniform