postbuckling analysis of a nonlinear beam with axial...

J Eng Math (2014) 88:121–136DOI 10.1007/s10665-013-9682-1

Postbuckling analysis of a nonlinear beam with axialfunctionally graded material

Kun Cai · David Y. Gao · Qing H. Qin

Received: 25 July 2013 / Accepted: 12 November 2013 / Published online: 19 April 2014© Springer Science+Business Media Dordrecht 2014

Abstract The postbuckling analysis of a modified nonlinear beam composed of axial functionally graded material(FGM) is investigated by a canonical dual finite element method (CD-FEM). The governing equation of the axialFGM nonlinear beam is derived through a variational method. The CD-FEM is adopted to find the nonconvexpostbuckling configurations of the beam according to Gao’s triality theory. Using duality transition, the originalpotential energy functional becomes a functional of deformation and dual stress fields. By variation of the mixedcomplementary energy, the coupling equations are derived to find deformation and dual stress fields. In FEM,matrices of a beam element depend on the gradient of material property (elastic modulus). To obtain general formsof matrices of a beam element, the graded elastic modulus is approximated by piecewise linear functions with respectto axial position. Numerical examples are presented to show the effects of graded elasticity on the postbucklingconfigurations of the beam.

Keywords Canonical duality · FEM · Functionally graded material · Postbuckling · Triality theory

1 Introduction

A functionally graded material (FGM) is a type of advanced material in which the compositions change continuouslythrough the whole body. For a FGM with two or more constituent phases, the gradient of each phase can be designedfor the purpose of achieving specified physical properties, e.g., thermal or corrosive resistance, low density, etc.[1]. Due to their excellent material properties and ease of design and production, FGMs have been widely adoptedas important components of structures in many fields, including microelectromechanical systems (MEMSs), spacestructures, turbine rotors, and gears. In the literature, FGM components have been modeled either as plate/shell [2]or as two-dimensional elasticity or beam/columns [3–5] for numerical and performance analysis.

K. CaiCollege of Water Resources and Architectural Engineering, Northwest A&F University, Yangling 712100, China

D. Y. GaoSchool of Science, Information Technology and Engineering, University of Ballarat, Mt Helen, VIC 3353, Australia

Q. H. Qin (B) · D. Y. GaoResearch School of Engineering, Australia National University, Canberra, ACT 0200, Australiae-mail: [email protected]

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122 K. Cai et al.

For FGM beam modeling, much effort has been focused on the following five aspects: (1) static deformation [6];(2) dynamic analysis [7–9]; (3) coupling stress theory [10,11]; (4) crack and fracture [8,12]; and (5) buckling andpostbuckling [8,12,13].

Although thorough buckling/postbuckling analyses of homogeneous beam structures have been conducted, onlya few reports on the buckling/postbuckling analysis of a FGM beam can be found in the literature [8,12]. Basedon Bernoulli–Euler beam theory and the rotational spring model, Yang and Chen [8] investigated the free vibrationand elastic buckling behavior of FGM beams with open edge cracks and presented the related natural frequencies,critical buckling load, and corresponding mode shapes. Singh and Li [13] presented a low-dimensional nonprismaticmodel to calculate the buckling loads of an axial FGM column. Ke et al. [12] studied the postbuckling responseof a FGM Timoshenko beam with an edge crack. In these works, the authors considered von Kármán nonlinearkinematics, and the Ritz method was adopted for numerical analysis. In these analyses, the gradient variation ofmaterial properties was assumed to exist along the thickness of the beam only. The FGM beams involved in theaforementioned studies were classified into two major types in terms of the gradient direction of their materialproperties. The first type has material properties varying along the thickness of the beam [6,9,14,15]. The othertype has material properties changing along the beam axis [10,13,16,17]. It is noted that, among the previouslymentioned reports, only the work by Singh and Li [13] and Shahba et al. [17] involved buckling analysis. That isthe motivation for us in this work: postbuckling analysis of FGM beams by a canonical dual finite element method(CD-FEM).

In this work, governing equations for nonlinear beams with axial FGM [18] are developed. Based on Gao’s trialitytheory [19], the effects of FGM properties on the postbuckling configurations of nonlinear beams are investigatedusing a CD-FEM.

The article is organized as follows. In Sect. 2, the Gao beam model is introduced and the governing equation forbeams with axial variation of FGMs is derived. Based on the governing equation obtained, a complementary dualprinciple is then constructed. In Sect. 3, the CD-FEM is presented in which material properties are approximatedthrough linear interpolation. In Sect. 4, a flowchart showing the global and local extrema is presented. Numericalexamples are presented to show the effects of material gradient on the postbuckling configurations of beams inSect. 5.

2 Gao Beam model

Consider a cantilever beam subjected to a distributed lateral load q(x) and axial compressive force F at the rightend, as shown in Fig. 1. The gradient of elastic modulus of the beam changes along the beam axial direction. Thespan of the beam is L , the height is 2h, and the thickness is b = 1.0. To solve this problem using a finite elementmethod (FEM), the beam is discretized into a number of beam elements. Taking the ith element as an example,the two ends of the element are marked A and B. Its nodal variables are denoted by (wAi , θAi , σAi , wBi , θBi , σBi )

T

(Fig. 1).

Fig. 1 FGM beam model (cantilever type)

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Postbuckling analysis 123

If the thickness of the beam is very large, e.g., 2h/L ∈ [0.1, 0.2], lateral deformation of the beam cannot beneglected. Considering this, in 1996 Gao [18] developed a nonlinear beam model with three assumptions, i.e., (1)the cross sections of the beam are initially uniform along the beam axis and have a symmetry axis about whichbending occurs; (2) the cross sections remain perpendicular to the beam axis before and after deformation, andshear deformation is ignored (Kirchhoff-Love hypothesis); and (3) the beam undergoes moderately large elasticdeformation, i.e., w(x) ∼ h/L and θ ∼ w,x (x) [18]. The control equation is

E Iw,xxxx − αEw2,xw,xx + Eλw,xx − f (x) = 0 ∀x ∈ [0, L] . (1)

The related axial displacement field can be given by the equation

ux = −1

2(1 + ν)w2

,x − λ

2h (1 + ν), (2)

where α = 3h(1 − v2) > 0, I = 2h3/3, λ = (1 + ν)(1 − v2)F/E > 0, and f (x) = (1 − v2)q(x). The coefficientE is the elastic modulus of the material, and v is Poisson’s ratio.

In this nonlinear beam model, both the lateral stress and the lateral deformation of cross section are considered,which is different from the Euler–Bernoulli beam model. Simultaneously, the axial deformation could be relativelylarge, considering the third assumption. Therefore, this nonlinear beam model can be used for studying both pre-and postbuckling analysis of beams in practical engineering [20].

Clearly, in this model, the elasticity of the material is assumed to be constant. If the beam is made of FGM, e.g.,the elastic modulus of the material changes along either the lateral or the axial direction, the governing equationgiven previously is invalid. In the present work, the beam model is modified to suit a beam with axial FGM. In otherwords, in the current beam model the elastic modulus of the material is assumed to change along the axial directiononly, while Poisson’s ratio remains constant [12,21].

First, if we assume the beam to be in a plane stress state, its deformation reads

u =(

u(x) − yθ(x)

w(x)

), (3)

where the rotation angle is θ = tan−1 (∂w/∂x).The Green strain tensor can be written as

E = 1

2

(∇u + ∇uT + ∇uT · ∇u

)=

(εxx εxy

εyx εyy

), (4)

where ∇ is the gradient operator, and the components of the Green strain tensor are

εxx = u,x (x) − yθ,x (x) + 1

2

(u,x (x) − yθ,x (x)

)2 + 1

2w2

,x ,

εyy = 1

2θ2, (5)

εxy = 1

2

(w,x (x) − θ(x)

) − u,x (x)θ(x).

Since we consider the beam material to be isotropic, the components of the Piola–Kirchhoff stress are found to be

σx = E(x)

1 − v2 (εxx + νεyy), σy = E(x)

1 − v2 (νεxx + εyy). (6)

Therefore, the potential energy of the beam is obtained using the following equation:

Π p(u, w) =∫

E(x)

2(1 − ν2)

(ε2

xx + ε2yy + 2νεxxεyy

)d −

L∫0

q(x)w dx ± Fu(x)

∣∣∣x=0,L

. (7)

For true deformation, the vanishing variation of potential energy

δΠ p(u, w, δu, δw) = 0. (8)

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124 K. Cai et al.

leads to the equilibrium equations⎧⎪⎪⎨⎪⎪⎩

∫ h−h σx,x dy = 0 ∀x ∈ [0, L] ,

∫ h

−h

[yσx,xx + (

σx + σy),x w,x + (

σx + σy)w,xx

]dy + q(x) = 0.

(9)

and boundary conditions⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∫ h−h(σx |x=0,L)δu dy = ∓F δu|x=0,L ,

∫ h−h y (σx |x=0,L)δw,x dy = 0,

∫ h−h

[yσx,x + (

σx + σy)w,x

]x=0,L δw dy = 0.

(10)

If the modulus is not a constant but a function of x within the solution domain, then the governing equations canbe obtained by the variation of potential energy as

E,x

(u,x + 1 + ν

2w2

,x

)+ E

[u,xx + (1 + ν)w,xw,xx

] = 0, (11)

d

dx

[E(x)

(u,x + 1 + ν

2w2

,x

)]= 0 ∀x ∈ [0, L] . (12)

Through integration by parts, the new governing equations for the beam with gradient-modulus materials areobtained as

I[E(x)w,xx

],xx − h(1 − ν2)

[E(x)w3

,x

],x

+ λEw,xx − f (x) = 0, (13)

u,x = −1

2(1 + ν)w2

,x − λ

2h(1 + ν)∀x ∈ [0, L] . (14)

Equation (13) can be reduced to Eq. (1) when the elastic modulus is a constant.The total potential energy of the beam associated with Eq. (13) is given by

Πp(w) =L∫

0

(1

2E Iw2

,xx + 1

12Eαw4

,x − 1

2Eλw2

,x

)dx −

L∫0

f (x)w dx, (15)

where w ∈ Ua → R, the admissible deformation space of the beam considering certain necessary boundaryconditions. The Euler buckling load in classic beam theory can be defined as

Fcr = infw∈Ua

∫ L0 E Iw2

,xx dx∫ L0 w2

,x dx. (16)

If the end compressive force F < Fcr, then the beam is in an unbuckled state. The total potential energy Πp isconvex, and the nonlinear differential equation (1) or (13) has only one solution. However, if F > Fcr, then thebeam is in a postbuckling state, the total potential energy Πp is nonconvex, and Eq. (1) or (13) may have at mostthree (strong) solutions at each material point x ∈ [0, L]: two minimizers corresponding to the two stable bucklingstates, and one local maximizer corresponding to an unstable buckling state. Using numerical methods to solve thenonconvex variational equation

δΠp (w, δw) = δ

L∫0

(1

2E Iw2

,xx + 1

12Eαw4

,x − 1

2Eλw2

,x

)dx −

L∫0

f (x)w dx = 0, (17)

we must encounter the nonuniqueness in a finite-dimensional space. However, to find the global optimal solution of anonconvex problem is usually NP-hard due to the lack of a global optimality condition [19]. For nonconvex problems,

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the extremal property of the generalized Hellinger–Reissner principle and the existence of a purely stress-basedcomplementary variational principle are two well-known debates that have existed in the field of nonlinear elasticityfor over 40 years [22–24]. The first open problem was partially solved by Gao and Strang [25] by introducing aso-called complementary gap function. They recovered a broken symmetry in nonlinear governing equations oflarge deformation problems and proved that this gap function provided a global optimality condition. The secondproblem was solved by Gao [26], and a pure complementary energy variational principle was first proposed in bothnonlinear beam theory and general nonlinear elasticity [27]. A general review of this history was given in [28].

In [18] (by Gao), a canonical dual transformation was presented, i.e.,

σ = Eα

3w2

,x − Eλ. (18)

Using this canonical dual stress in the present work, the generalized total complementary energy of the beamΞ : Ua × Sa → R can be expressed as

Ξ (w, σ) =L∫

0

(1

2E Iw2

,xx + 1

2σw2

,x − 3

4Eα(σ + Eλ)2 − f (x)w

)dx, (19)

where Sa is the admissible space of the dual stress σ .The Gao–Strang complementary gap function [25] for this beam model is defined as

G(w, σ ) =L∫

0

(E I

2w2

,xx + σ

2w2

,x

)dx . (20)

This positive gap function provides a global stability criterion for general large deformation problems [25]. In1997, Gao found that the negative gap function could be used to identify the largest local extrema in postbucklinganalysis. Therefore, the so-called triality theory was proposed. Furthermore, a pure complementary energy principlefor finite elasticity theory was established in 1999 [26]. Since then, the canonical duality theory has been graduallydeveloped [27]. This theory consists mainly of (1) a canonical dual transformation, (2) a complementary dualvariational principle, and (3) the triality theory. Detailed information concerning this theory and its extensiveapplications in nonconvex mechanics and global optimization is available in the monograph [19].

Based on the Gao–Strang generalized complementary energy [25], the pure complementary energy of thisnonlinear beam is obtained as

Πd(σ ) = {Ξ(w, σ ) |δwΞ(w, σ ) = 0 } , (21)

which is defined on a statically admissible space Sa. Then we have the following theorem.

Theorem 1: Complementary dual principle The complementary energy Πd(σ ) is canonical dual to the totalpotential energy Πp(w) in the sense that if (w, σ ) ∈ Ua × Sa is a critical point of Ξ (w, σ ), then w ∈ Ua is a criticalpoint of Πp(w), σ ∈ S is a critical point of Πd(σ ), and Πp(w) = Ξ(w, σ ) = Πd(σ ).

In computational mechanics, it is well known that the traditional FEM is based on the potential variationalprinciple, which produces upper bound approaches to related boundary-value problems. The dual FEM is basedon the complementary energy principle, which was originally studied by Pian [29] and Hodge and Belytschko[30] mainly for infinitesimal deformation problems. Hybrid FEMs are based on the generalized Hellinger–Reissnercomplementary energy principle and have certain advantages both for solving elastic deformation problems [31–33]and conducting thick plate analysis [34]. From a mathematical point of view, numerical discretization for nonconvexvariational problems should lead to global optimization problems that may have many local extrema. It is well knownin computational science that traditional direct approaches to solving nonconvex minimization problems in globaloptimization are fundamentally difficult or even impossible. Therefore, most nonconvex optimization problems areconsidered NP-hard [19].

The purpose of the present work is to undertake the numerical investigation of a large deformed beam with axialFGM using CD-FEM based on the canonical duality theory developed by Gao in 2000 [19] and the triality theoryproved recently [35].

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126 K. Cai et al.

Fig. 2 Nodal modulus ofbeam with axiallygradient-modulus material.(Color figure online)

3 Canonical dual FEM

3.1 Linearization of elemental modulus

For a FGM, the gradient modulus is a function of x (x is the position of a material point along the beam axis)in , and different materials have different functions of x . It is difficult, therefore, to provide a general formulaof an element stiffness matrix in FEM. To overcome this difficulty, the original curve of modulus vs. position xis approximated with a number of piecewise lines. The modulus E becomes, then, piecewise linear. As in FEM,accurate deformation of the material properties can be obtained as the number of finite elements in the beam issufficiently large. To improve the accuracy without increasing the number of elements, we can use several straightlines to approximate the original curve within each element. For the sake of simplicity, however, in the present workthe curve is approximated by only one straight line within each element.

Figure 2 shows the definition of the nodal Young’s modulus of a beam that is discretized into m ele-ments. E1 is the modulus of the material at the first node and Em+1 is the modulus at the last node. Theexact function of E(x) in each element is represented by the blue curve. In this method, the blue curve isapproximated with a straight line (black) in each element. That black line can be defined using the followingfunction:

E(x ∈ ei ) = 1

2((Ei+1 + Ei ) + (Ei+1 − Ei ) ξ) = ai + biξ (∀ξ ∈ [−1, 1]) . (22)

If the real modulus of the material in the beam changes linearly, no error occurs with the current piecewise lin-earization method. However, from Fig. 2 we can determine that the modulus (black straight line) of the first element(e1) is higher than the real modulus (blue curve). Conversely, the approximate modulus of the last element (em)

is smaller than the real modulus. To reduce the error between the approximate and real moduli, the real materialproperty curve of each element (e.g., e1 or em) can be approximated with more line segments.

3.2 Shape functions of deformation and dual stress fields

Suppose the ith elemental domain is Ωi , which is the subdomain of the real region Ω , and the element has twonodes, e.g., A and B (Fig. 1). Each node has three unknown parameters. At a given element, say element i, the threeparameters are deflection (wAi ), rotating angle (θAi ), and dual stress (σAi ) at node A and wBi , θBi , and σBi at nodeB. The deformation field and the dual stress field are approximated separately.

The deformation field and the dual stress field are approximated using

whi (x) = NT

w · wi , (23)

σ hi (x) = NT

σ · σ i , (24)

where wi = (wAi , θAi , wBi , θBi )T is the nodal displacement vector of the ith element, and σT

i = (σAi , σBi ) is thenodal dual stress element. Their shape functions are as follows:

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Nw =

⎛⎜⎜⎜⎜⎜⎝

14 (1 − ξ)2(2 + ξ)

Li8 (1 − ξ)2(1 + ξ)

14 (1 + ξ)2(2 − ξ)

Li8 (1 + ξ)2(ξ − 1)

⎞⎟⎟⎟⎟⎟⎠ , (25)

N Tσ = 1

2[(1 − ξ) (1 + ξ)] , (26)

where Li is the length of the ith element.

3.3 Discretized generalized complementary energy

Assuming the beam is discretized with m elements, the total energy in Eq. (19) can be expressed in discretized formas follows:

Ξ h (wi , σ i ) =m∑

i=1

(1

2wT

i · Gi (σ i ) · wi − 1

2σT

i · Ki · σ i − λTi · σ i − f T

i · wi − ci

)

= 1

2wT · G(σ ) · w − 1

2σT · K · σ − λT · σ − f T · w − c, (27)

where w ∈ R2(m+1), G(σ ) ∈ R2(m+1) × R2(m+1), K ∈ R(m+1) × R(m+1), σ ∈ R(m+1), λ ∈ R(m+1),

f ∈ R2(m+1), and c ∈ R.In the following equations, E(x) is replaced by a piecewise linear function of x [see Eq. (22)] in the integral

process:

Gi (σ i ) =∫i

(E I N′′

w · (N′′

w

)T +((

N′σ

)T · σ i

)· N′

w · (N′

w

)T)

dx, (28)

where

Gi =

⎛⎜⎜⎜⎝

g11 g12 g13 g14

g22 g23 g24

g33 g34

(sym) g44

⎞⎟⎟⎟⎠ , (29)

with “sym” denoting “symmetric” and the elements gi j being⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

g11 = 32 t0 + 3

5 t2, g12 = 3Li4 t0 − Li

4 t1 + Li20 (t2 + t3),

g13 = − 32 t0 − 3

5 t2, g14 = 3Li4 t0 + Li

4 t1 + Li20 (t2 − t3),

g22 = L2i

2 t0 − L2i

4 t1 + L2i

30 (2t2 − t3), g23 = −g12,

g24 = L2i

4 t0 − L2i

60 t2, g33 = g11, g34 = −g14,

g44 = L2i

2 t0 + L2i

4 t1 + L2i

30 (2t2 + t3),

(30)

and

t0 = 8I

L3e

ai , t1 = 8

L3e

bi , t2 = σBe + σAe

2, t3 = σBe − σAe

2. (31)

Ki =∫i

(3

2EαNσ · NT

σ

)dx = 3Li

16α

(k11 k12

k21 k22

). (32)

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128 K. Cai et al.

Here, the ki j are given by⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

k11 = 1b2

i

[2ai − 4 (bi + ai ) + (bi +ai )

2

biln

(bi +aibi −ai

)],

k22 = 1b2

i

[2ai + 4 (bi + ai ) + (bi +ai )

2

biln

(bi +aibi −ai

)],

k12 = k21 = 1b2

i

[2ai +

(b2

i −a2i

)bi

ln(

bi +aibi −ai

)] (33)

if bi = 0,

k11 = k22 = 3

2ai, k12 = k21 = 3

4ai(34)

if bi = 0, and

λi =∫i

3

2αλNσ (x) dx = 3λLi

4α

(11

), (35)

fi =∫i

f (x)Nw dx =1∫

−1

f

(L

2(ξ + 1)

)Nw

L

2dξ (36)

ci =∫i

3

4αλ2 E dx = 3

4αλ2Li ai . (37)

The vanishing variation of Eq. (27) with respect to its arguments,

δΞ h (w, σ ) = δ

(1

2wT · G(σ ) · w − 1

2σT · K · σ − λT · σ − f T · w − c

)

= (G(σ ) · w − f ) · δw +(

1

2wT · G,σ (σ ) · w − K · σ − λ

)· δσ = 0, (38)

yields the following two equations:

G(σ ) · w − f = 0, (39)1

2wT · G,σ (σ ) · w − K · σ − λ = 0, (40)

where G,σ denotes the gradient of G with respect to σ . Let Sha be a discretized feasible stress space such that G is

invertible for any given σ ∈ Sha . Then, on the discretized feasible deformation space U h

a , the displacement vectorw can be obtained as

w = G−1(σ ) · f . (41)

If this is substituted into the generalized complementary energy equation (27), the discretized pure complementaryenergy can be explicitly given by

Πd (σ ) = −1

2f T · G−1(σ ) · f − 1

2σT · K · σ − λT · σ − c. (42)

The following discretized subspaces are defined to identify both the global and local extrema of Πd:

Sh+a = {

σ ∈ Rnσ | G(σ ) be positive and definite}, (43)

Sh−a = {

σ ∈ Rnσ | G(σ ) be negative and definite}, (44)

where nσ is the dimension of the discretized stress field, and, correspondingly, nw is the dimension of the discretizeddeformation field.

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Theorem 2: Triality theory Suppose (w, σ ) is a critical point of Ξ h (w, σ ).

(1) Global solution: the critical point w ∈ U ha is a global minimizer of Πp(w) if and only if the critical point

σ ∈ Sh+a is a global maximizer of Πd (σ ), i.e.,

Πp (w) = minw∈U h

a

Πp (w) ⇔ maxσ∈Sh+

a

Πd (σ ) = Πd (σ ) . (45)

(2) Local stable solution: the critical point w ∈ U ha is a local maximizer of Πp(w) if and only if σ ∈ Sh−

a is a localmaximizer of Πd (σ ):

Πp (w) = maxw∈U h

a

Πp (w) ⇔ maxσ∈Sh−

a

Πd (σ ) = Πd (σ ) . (46)

(3) Local unstable solution: if σ ∈ Sh−a and nw = nσ , then the critical point w ∈ U h

a is a local minimizer ofΠp(w) if and only if σ is a local minimizer of Πd (σ ). If σ ∈ Sh−

a but nw > nσ (this is the case studied in thepresent work), the vector w = G−1(σ ) · f is a saddle point of Πp(w) on U h

a , which is a local minimum only ona subspace of U h

a such that the dimension of US equals nσ , i.e.,

Πp (w) = minw∈US

Πp (w) ⇔ minσ∈Sh−

a

Πd (σ ) = Πd (σ ) . (47)

In what follows, we use this triality theorem to obtain both global and local extrema of a postbuckling beam.Figure 3 shows a flowchart of the canonical dual algorithm for finding a beam’s global and local extrema in the

postbuckling state. In the current work, the initial dual stress is set at zero. The maximum iteration k* is set at 20in this algorithm.

4 Numerical examples

In the following examples, the beams considered have the same span, L = 1.0 m, and height, 2h = 0.1 m. ThePoisson ratio of the material is 0.3. The modulus of the left end is E1, that of the right is ENNode, and the parameterR = 1 − ENNode/E1 is used for the linear modulus function. “NNode” is the total number of nodes in the beam.

4.1 Mesh-dependency study

The cantilever beam used in this example is shown in Fig. 4. Uniform pressure q is applied laterally on the beam,and f (x) = (1 − v2) q(x) = 0.01 N/m. The left end is fixed. A compressive force F is applied on the right endof beam, and λ = (1 + ν)(1 − v2)F/E1 = 0.02 m2, where E1 = 1,000 Pa. The modulus decreases linearly, andR = 0.5, i.e., E(x) = E1(1 − Rx/L). Five mesh cases are considered in the mesh-dependency study, i.e., ENum,the number of elements in the beam, is equal to 20, 30, 40, 50, and 60, respectively.

Because the elastic modulus changes linearly, the solutions of a beam with a FGM are accurate with respect tothe matrices of the beam elements.

Figure 5 presents the global stable configurations of beams under postbuckling states. It indicates that the globalsolution has a slight mesh dependency. Because the number of elements is over 40, the differences among theforegoing solutions can be neglected.

In Fig. 6, the deflections of the beams are also different from each other, just as the mesh schemes are different.This indicates that the local stable solutions cannot be captured using the present method with a linear stressinterpolation scheme. Thus, this type of solution will not be discussed in the following examples.

Figure 7 shows the unstable postbuckling states of beams composed of FGM. One also encounters a meshdependency when searching for an unstable solution. But compared with local stable solutions, the differencesamong the local unstable states are very slight. In particular, the differences among the solutions with respect toENum = 20, 40, and 60 are very small. When ENum = 30 and 50, the curves are obviously different from eachother. Thus, in the following examples, mesh scheme-based unstable postbuckling solutions are shown.

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130 K. Cai et al.

Fig. 3 Flowchart of present algorithm Fig. 4 Cantilever beam model

-2.0E-03

-1.0E-03

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

0.0 0.2 0.4 0.6 0.8 1.0

Beam axis/m

Bea

m D

efle

ctio

n/m

E_Num=20

E_Num=30

E_Num=40

E_Num=50

E_Num=60

-1.0E-02

-8.0E-03

-6.0E-03

-4.0E-03

-2.0E-03

0.0E+00

2.0E-03

0.0 0.2 0.4 0.6 0.8 1.0

Beam axis/m

Bea

m D

efle

ctio

n/m

E_Num=20

E_Num=30

E_Num=40

E_Num=50

E_Num=60

Fig. 5 Stable postbuckling configuration of beams with globalmaximum of Ξh

Fig. 6 Local stable postbuckling configuration of beams withlocal maximum of Ξh

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-4.0E-03

-3.0E-03

-2.0E-03

-1.0E-03

0.0E+00

1.0E-03

2.0E-03

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Bea

m D

efle

ctio

n/m

E_Num=20

E_Num=30

E_Num=40

E_Num=50

E_Num=60

Fig. 7 Unstable postbuckling configuration of beams with localminimum of Ξh

Fig. 8 Clamped/clamped beam model

-1.0E-03

-8.0E-04

-6.0E-04

-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

0.0 0.2 0.4 0.6 0.8 1.0

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m D

efle

ctio

n/m

R=0

R=0.1

R=0.2

R=0.8

R=0.95

-6.0E-03

-4.0E-03

-2.0E-03

0.0E+00

2.0E-03

4.0E-03

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Bea

m D

efle

ctio

n/m

R=0

R=0.1

R=0.2

R=0.8

R=0.95

Fig. 9 Stable postbuckling configurations of beams with differ-ent gradient-modulus materials

Fig. 10 Unstable postbuckling configurations of beams with dif-ferent gradient-modulus materials

4.2 Constant versus linear FGMs

Figure 8 shows a clamped/clamped beam with 40 elements. The lateral load is a piecewise uniform pressure q thatyields f (x) = (1 − v2)q(x) = 0.05 N/m. The left end is fixed. On the right end of the beam is a compressive forceF that leads to λ = (1+ν)(1−v2)F/E1 = 0.02 m2. The material modulus decreases from E1 = 1,000 Pa linearly[E(x) = E1(1 − Rx/L)] to E41. Five cases are considered: (1) R = 1 − E41/E1 = 0; (2) R = 0.1; (3) R = 0.2;(4) R = 0.8; (5) R = 0.95. The most and least stable postbuckling configurations are presented separately to showthe effects of modulus change on postbuckling behavior.

Because the elastic modulus changes linearly, the element matrices are obtained without any approximation, andthe solutions of beams with a FGM are accurate.

From Fig. 9 we can see that the deflection curves are different for beams with different types of material. Ahigher downward slope of the modulus leads to a greater deflection of most beams. However, the shapes of thecurves are similar, in that they have the same number of peaks.

Figure 10 shows the unstable configurations of beams with different gradient-modulus materials. The deflectioncurve corresponding to R = 0 is the traditional solution of postbuckling with a global maximum of the mixed energy

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132 K. Cai et al.

-4.0E-05

-2.0E-05

0.0E+00

2.0E-05

4.0E-05

Beam Axis/m

Bea

m D

efle

ctio

n/m

(a) Constant

(b) Parabola

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Bea

m D

efle

ctio

n/m

(a) Constant

(b) Parabola

(b)(a)

Fig. 11 Stable postbuckling configurations (with a global maximum of Ξ h) for two load cases, i.e., with E being constant or E versusx being a parabolic curve. a f (x) = 0.01N/m, b f (x) = 0.05N/m

functional. The shape of a deflection curve with R = 0.1 is different from that of a curve with R = 0. However,when R = 0.2, the shape of the deflection curve is similar to that of a curve with R = 0. When R = 0.8 or 0.95,different curves are obtained. The difference between the curves reflects the association of unstable configurationswith modulus variation.

4.3 Parabolic curve of modulus

The beam model is the same as that shown in Fig. 8, a clamped/clamped beam. The modulus function of the material

can be expressed as a parabolic curve, i.e., E(x) = 12 E0

(( 2x−LL

)2 + 1), x ∈ [0, L], and E0 = 1,000 Pa. The

function implies a symmetrical distribution of material. The lateral load is the piecewise uniform pressure q, whichmakes f (x) = (1 − v2)q(x) equal 0.01 and 0.05 N/m for different loading cases, and λ = (1 + ν)(1 − v2)F/E0 =0.02 m2. The most and least stable postbuckling configurations of the beam are presented to show the sensitivity ofpostbuckling behavior to modulus distribution. The traditional solution, in which the material modulus is a constant,is also given for comparison.

Figure 11 shows deformation curves with the same shape in two different loading cases, where the deflection isproportional to the lateral load, which implies that the stable deformation configurations of the beam are fixed withregard to the lateral pressure, increasing by the same magnitude.

For unstable postbuckling configurations, Fig. 12 shows that a higher lateral load leads to a smaller deflectionwhen the modulus is a parabola function of the position. Therefore, the deformation of a beam in an unstable statedoes not obey the rule that a higher lateral load leads to higher deflection. That rule is usually observed in a stablestate.

4.4 Sine curve of modulus

The beam model in this example is the same as that shown in Fig. 8, a clamped/clamped beam. The function forthe modulus of the material can be expressed as sine curves E(x) = E0 (1 + 0.8 sin (kπx/L)), x ∈ [0, L] , andE0 = 1,000 Pa. The lateral load is the piecewise uniform pressure q, which makes f (x) = (1 − v2)q(x) equal0.01 N/m for different loading cases. Force F on the right end yields λ = (1 + ν)(1 − v2)F/E0 = 0.02 m2.The most and least stable postbuckling configurations of the beam are presented to demonstrate the sensitivity ofpostbuckling behavior to modulus distribution. The traditional solution, in which the material modulus is a constant,

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-3.0E-03

-2.0E-03

-1.0E-03

0.0E+00

1.0E-03

2.0E-03

3.0E-03

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Bea

m D

efle

ctio

n/m

(a) Constant

(b) Parabola

-5.0E-03

-3.0E-03

-1.0E-03

1.0E-03

3.0E-03

5.0E-03

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Bea

m D

efle

ctio

n/m

(a) Constant

(b) Parabola

(a) (b)

Fig. 12 Unstable postbuckling configurations (with local minimum of Ξ h) for different lateral load cases. a f (x) = 0.01N/m, bf (x) = 0.05N/m

0.0E+00

2.0E+02

4.0E+02

6.0E+02

8.0E+02

1.0E+03

1.2E+03

1.4E+03

1.6E+03

1.8E+03

2.0E+03

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Bea

m m

odul

us/P

a

k = 0k = 0.5 k = 1 k = 2k = 5

-6.0E-04

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0.0E+00

2.0E-04

4.0E-04

6.0E-04

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Beam Axis/m

Bea

m D

efle

ctio

n/m

k = 0k = 0.5 k = 1 k = 2k = 5

Fig. 13 Modulus curves of material in beam Fig. 14 Stable postbuckling configurations (with global maxi-mum of Ξh)

is also given for comparison. Five cases are considered in this analysis: (1) k = 0 (traditional solution); (2) k = 0.5;(3) k = 1; (4) k = 2; (5) k = 5. The modulus curves are shown in Fig. 13.

In Fig. 14, three curves show an approximate rotating symmetry, namely, the k = 0 curve, the k = 1 curve, andthe k = 5 curve, as expected (because the modulus curves of the material in the beam show a symmetry, and thelateral load is distributed in rotating symmetry; see Fig. 8). The two remaining curves (for k = 0.5 and k = 2) haveno such symmetry. The deflection of the right part of the blue curve is clearly lower than that of the left part becausethe right part of the beam has a higher-modulus material.

Figure 15 demonstrates that the deflections of the beam for the curves associated with k = 1 or k = 5 are largerthan those of the other curves. Meanwhile, the curves show no symmetry, although the material modulus curves(the k = 1 curve and the k = 5 curve) are symmetric. This again implies the association of unstable postbucklingconfigurations with the modulus distribution.

4.5 Exponential curve of modulus

In Fig. 16a, the clamped/simply supported beam used in this example is discretized into 40 elements with thesame length. A uniform pressure q is applied laterally on the beam, and f (x) = (1 − v2)q(x) = 0.01 N/m. A

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134 K. Cai et al.

-1.5E-02

-1.0E-02

-5.0E-03

0.0E+00

5.0E-03

1.0E-02

1.5E-02

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Bea

m D

efle

ctio

n/m

k = 0k = 0.5 k = 1 k = 2k = 5

Fig. 15 Unstable postbuckling configurations (with local mini-mum of Ξh)

Fig. 16 Clamped/simply supported beam model

0.0E+00

5.0E+02

1.0E+03

1.5E+03

2.0E+03

2.5E+03

3.0E+03

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Bea

m m

odul

us/P

a

k = 0

k = 0.2

k = 0.4

k = 0.6

k = 0.8

-1.0E-03

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0.0E+00

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Beam Axis/m

Bea

m D

efle

ctio

n/m

d=0

d=0.2

d=0.4

d=0.6

d=0.8

Fig. 17 Modulus curves of material in beam Fig. 18 Stable postbuckling configurations of beam with axialexponential FGM

compressive force F is applied on the right end of beam, and λ = (1 + ν)(1 − v2)F/E1 = 0.02 m2, whereE1 = 1,000 exp(1.0) Pa. The modulus function along the beam axis is E(x) = E1

[exp(1 − x/L) − d

]. Five cases,

i.e., d = 0, 0.2, 0.4, 0.6, and 0.8, are considered in finding the global stable states.Figure 18 gives the stable postbuckling configurations of a beam with an axial exponential FGM. The deflection

of the beam with d = 0 is the smallest one among the five solutions. The reason for this is simple: the elasticmodulus of the material is the highest one at the same location along the axis among the five cases (Fig. 17).

5 Conclusions

The postbuckling configurations of large deformed beams with an axial FGM were investigated using CD-FEM.To find all the solutions, the triality theory was adopted. The numerical results yield the following findings:

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(I) The most stable postbuckling configuration of a beam is reactive to the elastic modulus distribution but not tothe lateral load. A symmetrical structure with a symmetrical material distribution still results in symmetricaldeflection. The most stable postbuckling solution has no mesh dependency with enough elements in the beamand can be captured easily.

(II) Local (stable or unstable) solutions are sensitive to both the material distribution and the lateral load. Asymmetrical structure with a symmetrical material distribution commonly leads to asymmetrical deflections.The two solutions have an obvious mesh dependency.

Acknowledgments The financial support of the US Air Force Office of Scientific Research (Grant No. FA9550-10-1-0487), theNational Natural Science Foundation of China (Grant No. 50908190), the Youth Talents Foundation of Shaanxi Province (Grant No.2011kjxx02), and the Research Foundation (GZ1205) of the State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian University of Technology, Dalian, China, is fully acknowledged.

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