Accepted Manuscript

Elastic Buckling and Static Bending of Shear Deformable Functionally Graded

Porous Beam

D. Chen, J. Yang, S. Kitipornchai

PII: S0263-8223(15)00597-8

DOI: http://dx.doi.org/10.1016/j.compstruct.2015.07.052

Reference: COST 6632

To appear in: Composite Structures

Please cite this article as: Chen, D., Yang, J., Kitipornchai, S., Elastic Buckling and Static Bending of Shear

Deformable Functionally Graded Porous Beam, Composite Structures (2015), doi: http://dx.doi.org/10.1016/

j.compstruct.2015.07.052

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers

we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and

review of the resulting proof before it is published in its final form. Please note that during the production process

errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Elastic Buckling and Static Bending of Shear Deformable

Functionally Graded Porous Beam

D. Chen1, J. Yang

2*, S. Kitipornchai

1

1 School of Civil Engineering, the University of Queensland, Brisbane, St Lucia 4072, Australia

2 School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO

Box 71, Bundoora, VIC 3083 Australia

2

Elastic Buckling and Static Bending of Shear Deformable

Functionally Graded Porous Beam

D. Chen1, J. Yang

2*, S. Kitipornchai

1

1 School of Civil Engineering, the University of Queensland, Brisbane, St Lucia 4072, Australia

2 School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO

Box 71, Bundoora, VIC 3083 Australia

___________________________________________________________________________

Abstract

This paper presents the elastic buckling and static bending analysis of shear deformable

functionally graded (FG) porous beams based on the Timoshenko beam theory. The elasticity

moduli and mass density of porous composites are assumed to be graded in the thickness

direction according to two different distribution patterns. The open-cell metal foam provides

a typical mechanical feature for this study to determine the relationship between coefficients

of density and porosity. The partial differential equation system governing the buckling and

bending behavior of porous beams is derived based on the Hamilton’s principle. The Ritz

method is employed to obtain the critical buckling loads and transverse bending deflections,

where the trial functions take the form of simple algebraic polynomials. Four different

boundary conditions are considered in the paper. A parametric study is carried out to

investigate the effects of porosity coefficient and slenderness ratio on the buckling and

bending characteristics of porous beams. The influence of varying porosity distributions on

the structural performance is highlighted to shed important insights into the porosity design to

achieve improved buckling resistance and bending behavior.

Keywords:

Functionally graded porous beam; Timoshenko beam theory; elastic buckling; static bending;

Ritz method; porosity distribution.

1. Introduction

Structures made of functionally graded materials (FGMs) have attracted tremendous

attention from research and engineering communities due to their unique advantages offered

*Corresponding author. Tel.: 61-03-99256169; Fax: 61-03-99256108

E-mail address: j.yang@rmit.edu.au (J. Yang)

3

by smooth and continuously graded distribution of material composition along one or more

directions. Numerous studies have been conducted on FGM beams, plates and shells in the

past few decades. One of the latest developments in FGMs is structures with graded porosity

which introduces pores into the microstructure to meet the desired structural performance by

tailoring the local density of the structure. This represents an important opportunity in a wide

range of engineering applications. For example, the graded metal foam offers unique

potential for lightweight structures in aerospace, automotive and civil engineering [1-12]. The

excellent energy-absorbing capability of porous FGMs also makes them the perfect

candidates for structures under dynamic or impact loading [13-15]. Despite of its practical

importance, research in this emerging area is only at the beginning stage and is very limited

with most of the previous studies devoted to the compression behavior [15-17] and energy

dissipation performance [18-20] only.

The stability and bending behavior are important mechanical characteristics of FGM

structures and have received increasing research attention. Yang et al. [21] studied the

buckling and post-buckling performance of the shear deformable composite plate with initial

geometric imperfection based on the higher order shear deformation theory. Liew et al. [22]

investigated the thermal post-buckling behaviour of FG laminated plates with temperature-

dependent material properties. Ke and Yang [23] analyzed the bending, buckling and free

vibration of FGM annular microplates with the modified couple stress theory and Mindlin

plate theory. Simsek and Yurtcu [24] used the nonlocal Timoshenko and Euler-Bernoulli

beam theory to examine the analytical solutions for the static bending and buckling of a FG

nanobeam.

Studies on the stability and bending behavior of porous FGM structures, especially for

beams, are still limited in number. Magnucki and Stasiewicz [25] proposed an explicit

solution for the critical buckling load of a compressed porous beam based on the broken-line

hypothesis and the principle of stationarity for the total potential energy. Jasion and

Magnucka-Blandzi [26] presented the analytical, numerical, and experimental studies of

three-layered sandwich beams and circular plates with metal foam core to compare the values

of critical buckling loads obtained by different methods. The effective design of a sandwich

metal foam beam was conducted by Magnucka-Blandzi and Magnucki [27] to calculate the

optimal dimensionless parameters, considering the mass and critical force of a simply

supported beam simultaneously.

As for porous plates and shells, Magnucka-Blandzi [28] studied the dynamic stability for a

circular porous plate to determine the critical loads. The influence of unstable regions for the

Mathieu equation was described. She [29, 30] also examined the non-linear dynamic stability

and axi-symmetrical deflection and buckling of circular porous plates. Belica and Magnucki

[31, 32] investigated the dynamic stability of a porous cylindrical shell under different

loading conditions and obtained the critical loads for a family of porous shells. In addition,

the optimization analysis of porosity parameter for a porous-cellular shell is conducted by

4

Magnucki and Malinowski [33] to achieve the maximized critical load as well sticas the

minimized mass.

As one of the pioneering studies on porous structures, Biot [34] proposed the

poroelasticity theory by introducing the bulk dynamic and kinematic variables [35]. Based on

this theory, Detournay and Cheng [36] expressed the stress-strain relationship for saturated

porous structures, which was used by Jabbari and Mojahedin [35] to conduct the buckling

analysis of thin circular FG plates made of saturated porous materials. The general

equilibrium and stability equations were derived from the energy method and classical plate

theory, associated with the method of calculus of variations. They [37] also presented the

buckling analysis of soft ferromagnetic FG circular saturated porous plates in transverse

magnetic fields. The thermal buckling behavior of FG thin circular plate made of saturated

porous materials was analysed by Jabbari and Hashemitaheri [38]. The critical buckling

temperature was discussed for different thermal distributions. Jabbari and Joubaneh [39]

considered the porous circular plate integrated with piezoelectric actuator layers and obtained

the buckling load of such a plate. Joubaneh and Mojahedin [40] further presented the thermal

buckling analysis of porous circular plate bounded with piezoelectric sensor-actuator patches.

It should be noted that in the above-mentioned studies, only one specific porosity

distribution was considered and no detailed discussion concerning the effects of different

porosity parameters on the structural performance of porous beams was given. This paper

focuses on the elastic buckling and bending performance of porous beams with two different

porosity distributions. Timoshenko beam theory is employed to account for the effect of

transverse shear deformation. The algebraic governing equation system is derived then solved

by Ritz method. The critical buckling loads and static bending deflections for beams of two

porosity distributions under different boundary conditions are presented with changing

porosity coefficient and slenderness ratio. A detailed parametric study is carried out to

highlight the influence of porosity parameters on the structural performance of porous beams.

2. Functionally graded porous beam

A FG porous Timoshenko beam of thickness h and length L with two different porosity

distributions along the thickness direction is shown in Fig. 1a for porosity distribution 1 [25]

and Fig. 1b for porosity distribution 2 [35]. This beam is depicted in a rectangular coordinate

system with z-axis in the thickness direction and x-axis in the length direction. Due to the

non-uniform porosity distribution, Young’s modulus, shear modulus and mass density vary

smoothly based on Eqs. (1a)-(1c) and Eqs. (2a)-(2c) corresponding to porosity distribution 1

and 2, respectively. As can be seen, both porosity distributions share the same maximum and

minimum values of elasticity moduli and mass density. In distribution 1, the minimum values

are on the midplane of the beam with the largest size and density of internal pores whereas

the maximum values exist on the top and bottom surfaces which are equal to the values of

homogeneous beams made of pure materials. While in distribution 2, elasticity moduli and

mass density are the maximum on the top surface then generally decrease to the minimum

values on the bottom surface.

5

(a) Porosity distribution 1

(b) Porosity distribution 2

Fig. 1. Two porosity distribution patterns.

Porosity distribution 1:

1 0( ) 1 cosE z E e (1a)

1 0( ) 1 cosG z G e (1b)

1( ) 1 cosmz e (1c)

Porosity distribution 2:

1 0( ) 1 cos2 4

E z E e

(2a)

1 0( ) 1 cos2 4

G z G e

(2b)

1( ) 1 cos2 4

mz e

(2c)

( )E z

1E

1E

/ 2h

/ 2h

1

1

0E ( )z

0

z

1E

( )E z

0E

x

z

/ 2h

/ 2h

1

0

( )z

x

6

where /z h , the porosity coefficient 0 0 1 0 11 / 1 /e E E G G (

00 1e ), the minimum

and maximum values of Young’s modulus 0E and

1E are related to the minimum and

maximum values of shear modulus 0G and

1G by / 2 1i iG E v ( 0,1i ) where v is the

Poisson’s ratio which is a constant across the beam thickness. The porosity coefficient for

mass density is defined as 0 11 /me (0 1me ) in which 0 and

1 are the minimum

and maximum values of mass density, respectively.

The relationship between relative density and relative Young’s modulus for an open-cell

metal foam [41, 42] is

2

0 0

1 1

E

E

(3)

This can be used to obtain the relationship between 0e and me as below

01 1me e (4)

It should be noted from the above equation that 0me e , which indicates that the difference

between the maximum and minimum mass density values is always smaller than that between

the maximum and minimum values of elasticity moduli, as can also be seen from Fig. 1.

3. Theoretical formulation

3.1 Basic equations

According to Timoshenko beam theory, the following displacement field is used to include

the effect of transverse shear strain which is assumed to be constant in the thickness direction.

0( , , ) ( , ) ( , )xu x z t u x t z x t (5a)

0( , , ) ( , )w x z t w x t (5b)

where 0u and 0w denote the axial and transverse displacements of a point on the midplane of

the beam, and x is the rotation of the cross section of the beam. The linear strain-

displacement relationships are

0 xxx

uz

x x

(6a)

0xz x

w

x

(6b)

where xx and xz are the normal and transverse shear strains that are related to the normal

stress xx and shear stress xz based on the linear elastic constitutive law as

7

11( )xx xxQ z (7a)

55( )xz xzQ z (7b)

where

11 2

( )( )

1

E zQ z

v

(8a)

55

( )( ) ( )

2(1 )

E zQ z G z

v

(8b)

Based on the Eqs. (5a)-(8b), the strain energy U of the porous beam can be expressed as

/2

0 /2

1( )d d

2

L h

xx xx xz xzh

U z x

2 2 2

20 0 0 011 11 11 55 55 55

0

12 2 d

2

Lx x

x x

u u w wA B D A A A x

x x x x x x

(9)

where the stiffness components are calculated by

/2

2

11 11 11 11/2

{ , , } ( ) 1, , dh

h

A B D Q z z z z

(10a)

/2

55 55/2

( )dh

h

A kQ z z

(10b)

where 5 / 6k is the shear correction factor.

The work V done by external loads is written as

2

00 0 0

0

1d ( ')

2

L

x

wV N Qw x Fw x

x

(11)

where 0xN is the axial force along the beam length, Q and F are the transverse distributed

load and point load applied at the top surface of the beam ( / 2z h ), respectively, and 'x

denotes the position of the point load.

By introducing the following dimensionless quantities

0 0,

,u w

u wh

, x , 5511 11 1111 11 11 55 2

110 110 110 110

, , , , , ,AA B D

a b d aA A h A h A

, x

L ,

L

h , 0

110

xNP

A ,

110

Qq L

A ,

110

Ff

A ,

''

x

L (12)

8

where 110A is the value of

11A of the homogeneous beam made of pure materials, the

dimensionless strain energy *U and dimensionless work *V done by the applied loads can be

obtained as

2 2 21

* 2 2

11 11 11 55 55 5500

12 2 d

2

U u u w wU a b d a a a

(13a)

21

*

00

1d ( ')

2

V wV P qw fw

(13b)

where

2

0 110

hA

L (14)

Based on the Hamilton’s principle, the total energy of the porous beam is expressed in the

dimensionless form as

* * *U V (15)

3.2 Ritz method

The Ritz method is employed to obtain the critical buckling load and transverse bending

deflection of the porous beam with hinged-hinged (H-H), clamped-clamped (C-C), clamped-

hinged (C-H), and clamped-free (C-F) end supports. The trial displacement functions for

beams with different boundary conditions take the form of the following simple algebraic

polynomials [43].

H-H beam:

1

1

1

1

( ) (1 )

( ) (1 )

( )

N

j

j

j

N

j

j

j

N

j

j

j

u A

w B

C

(16)

9

C-C beam:

1

1

1

( ) (1 )

( ) (1 )

( ) (1 )

N

j

j

j

N

j

j

j

N

j

j

j

u A

w B

C

(17)

C-H beam:

1

1

1

( ) (1 )

( ) (1 )

( )

N

j

j

j

N

j

j

j

N

j

j

j

u A

w B

C

(18)

C-F beam:

1

1

1

( )

( )

( )

N

j

j

j

N

j

j

j

N

j

j

j

u A

w B

C

(19)

where N is the total number of polynomial terms, jA , jB and jC are the unknown

coefficients. The above trial functions satisfy the geometric boundary conditions of the beams.

Following the standard procedure of Ritz method, substituting the trial functions into Eq. (15)

then minimizing the obtained total energy with respect to the unknown coefficients

*

*

*

0

0

0

j

j

j

A

B

C

(20)

the following algebraic governing equation system can be obtained

10

L p q f( ) = 0P q f K V d V V (21)

where d is the unknown coefficient vector, T T T T{{ } { } { } }j j jA B Cd , j = 1, 2, …, N, LK is

the linear stiffness matrix, pV ,

qV and fV are the geometric stiffness matrices associated

with buckling load P, distributed load q and point load f.

4. Solution procedure

4.1 Elastic buckling analysis

By neglecting the distributed load and point load, Eq. (21) reduces to a homogeneous

equation system below from which the dimensionless buckling load crP can be obtained as its

lowest eigenvalue.

L p( ) 0P K V d (22)

4.2 Static bending analysis

In the absence of the buckling load, the algebraic equation system below can be used to

solve the static bending deflection of the porous beam under either a distributed load q or a

point load f.

L q 0q K d V (23)

L f 0f K d V (24)

5. Numerical results

A parametric study is carried out with comprehensive numerical results presented in both

tabular and graphical forms to investigate the influences of porosity coefficient and

slenderness ratio on the buckling and bending characteristics of functionally graded porous

beams under different boundary conditions. The material of the porous beam is assumed to be

steel foam with 1 200E GPa, 1/ 3v , ρ1 = 7850kg/m3. The cross section of beam is 0.1h

m, 0.1b m (b is the width of the beam).

5.1 Convergence and validation

Table 1 compares the dimensionless critical buckling loads cr 0 110/xP N A for H-H, C-C,

C-H and C-F beams with two different porosity distributions and varying number N of

polynomial items in the trial functions. It is observed that increasing the number of

11

polynomial items N improves the accuracy of results and leads to convergent solutions at

10N . Hence 10N is used in the following numerical calculations.

The validation analysis is done through direct comparison between the present results and

finite element results obtained by using commercial finite element software package ANSYS.

To simulate the graded distribution of elasticity moduli and mass density along the thickness

direction, the beam is divided into a number of layers of same thickness. Table 2 shows the

dimensionless critical buckling load, dimensionless bending deflections 0 /w w h under a

distributed load 41 10Q N/m, and a point load 41 10F N at the tip of the beam with

varying slenderness ratio. As can be seen, convergent results can be achieved with a total

number of 10 layers for porosity distributions 1 or 20 layers for porosity distributions 2,

respectively. Our results are in excellent agreement with the finite element results.

Table 1

Dimensionless critical buckling load of porous beams under different end supports ( / 20L h ,

00.5e ).

Porosity distribution 1

N H-H C-C C-H C-F

2 0.0040517 0.1232831 0.0099497 0.0010114

4 0.0033406 0.0136335 0.0067870 0.0008403 6 0.0033387 0.0130155 0.0067560 0.0008402

8 0.0033387 0.0130109 0.0067559 0.0008402

10 0.0033387 0.0130109 0.0067559 0.0008402 12 0.0033387 0.0130109 0.0067559 0.0008402

Porosity distribution 2

N H-H C-C C-H C-F

2 0.0035146 0.1216477 0.0083545 0.0008367

4 0.0028706 0.0113467 0.0056651 0.0006940

6 0.0028688 0.0108101 0.0056384 0.0006939

8 0.0028688 0.0108060 0.0056383 0.0006939

10 0.0028688 0.0108060 0.0056383 0.0006939 12 0.0028688 0.0108060 0.0056383 0.0006939

12

Table 2

Validation analysis for C-F porous beams (0

0.5e ).

Dimensionless critical buckling load

Porosity distribution 1 Porosity distribution 2

/L h Present ANSYS Present ANSYS

10 0.00334 0.00333 0.00276 0.00275

20 0.00084 0.00084 0.00069 0.00069

50 0.00013 0.00013 0.00011 0.00011

Dimensionless bending deflection under a distributed load

Porosity distribution 1 Porosity distribution 2

/L h Present ANSYS Present ANSYS

10 0.00083 0.00083 0.00100 0.00099

20 0.01307 0.01311 0.01582 0.01572

50 0.50898 0.51007 0.61646 0.61182

Dimensionless bending deflection under a point load

Porosity distribution 1 Porosity distribution 2

/L h Present ANSYS Present ANSYS

10 0.00219 0.00221 0.00265 0.00264

20 0.01741 0.01746 0.02108 0.02093

50 0.27142 0.27193 0.32874 0.32620

5.2 Elastic buckling

Fig. 2 plots the dimensionless critical buckling load versus porosity coefficient curves for

both porosity distributions. Results show that an increase in the porosity coefficient leads to

the deceasing of critical buckling load, indicating that the internal pores decrease the effective

stiffness of beams. It can also be seen that the varying porosity coefficient has a more

remarkable effect on the stiffness of the beams with porosity distribution 2. Fig. 3 highlights

the significant influence of slenderness ratio on the buckling characteristics of functionally

graded porous beams. As expected, a beam with larger slenderness ratio has a smaller critical

buckling load. Among the four boundary conditions considered, the C-C beam has the

highest while the C-F beam has the lowest critical buckling load.

13

0.0 0.2 0.4 0.6 0.80.000

0.004

0.008

0.012

0.016

C-F

C-C

C-H

H-H

Dim

en

sio

nle

ss c

riti

cal

bu

ck

lin

g l

oad

Porosity coefficient (e0)

Porosity distribution 1:

Porosity distribution 2:

Fig. 2. Dimensionless critical buckling load of functionally graded porous beams: Effect of porosity

coefficient ( / 20L h )

10 20 30 40 500.00

0.01

0.02

0.03

0.04

0.05Porosity distribution 1:

Porosity distribution 2:

C-F

C-C

C-H

H-H

Dim

en

sio

nle

ss c

riti

cal

bu

ck

lin

g l

oad

Slenderness ratio (L / h)

Fig. 3. Dimensionless critical buckling load of functionally graded porous beams: Effect of

slenderness ratio (0

0.5e )

5.3 Static bending

It is assumed in this section that the uniformly distributed load is Q = 1×104 N/m and the

point load on the mid-span of the beam is 41 10F N, unless otherwise stated.

Fig. 4 and Fig. 5 illustrate the effects of the porosity coefficient and slenderness ratio on

the dimensionless maximum deflection of functionally graded porous beams under a

uniformly distributed load. As can be observed, increasing the porosity coefficient and

14

slenderness ratio lead to larger deflections. It is evident that the beams with porosity

distribution 1 have higher effective stiffness than beams with distribution 2. Among the three

boundary conditions (C-C, C-H, H-H) considered in this figure, the deflection is the largest

for the H-H beam whereas it is the smallest for the C-C beam. Similar results can be obtained

when the beam is subjected to a point load at the mid-span of the beam, as shown in Fig. 6

and Fig. 7, respectively.

0.0 0.2 0.4 0.6 0.80.0000

0.0005

0.0010

0.0015

0.0020

0.0025Porosity distribution 1:

Porosity distribution 2:

C-C

C-H

H-H

Dim

en

sio

nle

ss m

ax

imu

m d

efl

ecti

on

Porosity coefficient (e0)

Fig. 4. Dimensionless maximum deflection under a uniformly distributed load: Effect of porosity

coefficient ( / 20L h ).

10 20 30 40 500.00

0.01

0.02

0.03

0.04

0.05

0.06Porosity distribution 1:

Porosity distribution 2:

C-C

C-H

H-H

Dim

en

sio

nle

ss m

ax

imu

m d

efl

ecti

on

Slenderness ratio (L / h)

Fig. 5. Dimensionless maximum deflection under a uniformly distributed load: Effect of slenderness

ratio (0

0.5e ).

15

0.0 0.2 0.4 0.6 0.80.0000

0.0005

0.0010

0.0015

0.0020Porosity distribution 1:

Porosity distribution 2:

C-C

C-H

H-H

Dim

en

sio

nle

ss m

ax

imu

m d

efl

ecti

on

Porosity coefficient (e0)

Fig. 6. Dimensionless maximum deflection under a point load: Effect of porosity coefficient

( / 20L h ).

10 20 30 40 500.000

0.005

0.010

0.015

0.020Porosity distribution 1:

Porosity distribution 2:

C-C

C-H

H-H

Dim

en

sio

nle

ss m

ax

imu

m d

efl

ecti

on

Slenderness ratio (L / h)

Fig. 7. Dimensionless maximum deflection under a point load: Effect of slenderness ratio (0

0.5e ).

In what follows, the normal stress distribution is also given in dimensionless form as

Q F

Q F, ,A A

QL F

(25)

16

where Q

and F

refer to the dimensionless normal bending stresses when the beam is

under the action of a uniformly distributed load and a point load, respectively, and A is the

cross section area of the beam.

-0.50 -0.25 0.00 0.25 0.50

-2.0

-1.0

0.0

1.0

2.0

D

imen

sio

nle

ss n

orm

al

stre

ss

z / h

e0 = 0.00:

e0 = 0.25:

e0 = 0.50:

e0 = 0.75:

e0 = 0.95:

(a) Porosity distribution 1

-0.50 -0.25 0.00 0.25 0.50

-1.0

0.0

1.0

2.0

3.0

e0 = 0.00:

e0 = 0.25:

e0 = 0.50:

e0 = 0.75:

e0 = 0.95:

Dim

en

sio

nle

ss n

orm

al

stre

ss

z / h

(b) Porosity distribution 2

Fig. 8. Effect of porosity coefficient on the variation of dimensionless normal stress through the

thickness for H-H beam under a distributed load ( / 20L h , / 0.5x L ).

17

Fig. 8 compares the effect of porosity coefficient on the dimensionless normal stress along

the thickness of the H-H beam under a uniformly distributed load. It is worthy of noting that

the normal bending stress varies linearly along the thickness direction for a non-porous beam

(0

0e ) but changes nonlinearly for functionally graded beams. This is due to the nonlinear

gradient in material properties caused by the non-uniform porosity distribution. For porosity

distribution 1, a bigger porosity coefficient represents a larger pore size and higher internal

pore intensity hence lower local stiffness around the midplane of the beam, which

consequently leads to lower normal stress in this region and higher stress on both the top and

bottom surfaces. The normal bending stress is symmetric about the mid-plane for porosity

distribution 1 due to its symmetric pore distribution but is unsymmetric for porosity

distribution 2 where the pore size gradually increases from the top surface to the bottom

surface. As a result, the maximum normal bending stress which is on the top surface is much

bigger than that at the bottom and the difference between them tends to be much bigger as 0e

increases. Fig. 9 presents the effect of slenderness ratio on the dimensionless normal bending

stress. As expected, for both porosity distributions, a slender beam would have bigger normal

bending stress due to its weaker bending stiffness and larger deflections.

It should be mentioned that the above observations on the normal bending stress, although

for an H-H beam under a distributed load, are also valid for beams with other boundary and

loading conditions which are not presented herein for brevity.

-0.50 -0.25 0.00 0.25 0.50-5.0

-2.5

0.0

2.5

5.0

Dim

en

sio

nle

ss n

orm

al

stre

ss

z / h

L / h = 10:

L / h = 20:

L / h = 30:

L / h = 40:

L / h = 50:

(a) Porosity distribution 1

18

-0.50 -0.25 0.00 0.25 0.50-3.0

-1.5

0.0

1.5

3.0

4.5 L / h = 10:

L / h = 20:

L / h = 30:

L / h = 40:

L / h = 50:

Dim

en

sio

nle

ss n

orm

al

stre

ss

z / h (b) Porosity distribution 2

Fig. 9. Effect of slenderness ratio on the variation of dimensionless normal stress through the

thickness for H-H beam under a distributed load (0

0.5e , / 0.5x L ).

6. Conclusions

The elastic buckling and static bending of FG porous beams with various boundary

conditions and two different porosity distributions have been investigated. Theoretical

formulations are within the framework of Timoshenko beam theory. Ritz method is employed

to obtain the critical buckling load, transverse bending deflection, and normal bending stress.

The effects of porosity coefficient and slenderness ratio on the critical buckling load,

maximum deflection and associated stress distribution are discussed. Numerical results show

that:

(1) An increase in the porosity coefficient and slenderness ratio leads to lower critical

buckling loads of functionally graded porous beams;

(2) The maximum deflections for the porous beams increase with an increase in the porosity

coefficient and slenderness ratio;

(3) The through-thickness normal stress distribution changes from linear to nonlinear with

the increasing porosity coefficient, and varied more dramatically with the increasing

slenderness ratio;

(4) The porosity distribution has a significant influence on the buckling and static bending

behaviour of the beam. Compared with the unsymmetric distribution pattern, the

symmetric distribution offers better buckling capacity and improved bending resistance.

19

Acknowledgments

The work described in this paper was fully funded by two research grants from the

Australian Research Council under Discovery Project scheme (DP130104358,

DP140102132). The authors are grateful for their financial support.

References

[1] Smith B, Szyniszewski S, Hajjar J, Schafer B, Arwade S. Steel foam for structures: A review of

applications, manufacturing and material properties. Journal of Constructional Steel Research

2012; 71: 1-10. [2] Ashby MF, Evans T, Fleck NA, Hutchinson J, Wadley H, Gibson L. Metal Foams: A Design

Guide. Elsevier; 2000.

[3] Rabiei A, Vendra L. A comparison of composite metal foam's properties and other comparable

metal foams. Materials Letters 2009; 63(5): 533-536. [4] Lim TJ, Smith B, McDowell D. Behavior of a random hollow sphere metal foam. Acta

Materialia 2002; 50(11): 2867-2879.

[5] Raj S, Ghosn L, Lerch B, Hebsur M, Cosgriff L, Fedor J. Mechanical properties of 17-4PH stainless steel foam panels. Materials Science and Engineering: A 2007; 456(1): 305-316.

[6] Park C, Nutt S. Anisotropy and strain localization in steel foam. Materials Science and

Engineering: A 2001; 299(1): 68-74. [7] Badiche X, Forest S, Guibert T, Bienvenu Y, Bartout JD, Ienny P, Croset M, Bernet H.

Mechanical properties and non-homogeneous deformation of open-cell nickel foams:

application of the mechanics of cellular solids and of porous materials. Materials Science and

Engineering: A 2000; 289(1): 276-288. [8] Gibson L. Mechanical behavior of metallic foams. Annual review of materials science 2000;

30(1): 191-227.

[9] Kwon Y, Cooke R, Park C. Representative unit-cell models for open-cell metal foams with or without elastic filler. Materials Science and Engineering: A 2003; 343(1): 63-70.

[10] Sanders W, Gibson L. Mechanics of hollow sphere foams. Materials Science and Engineering:

A 2003; 347(1): 70-85. [11] Warren W, Kraynik A. The linear elastic properties of open-cell foams. Journal of Applied

Mechanics 1988; 55(2): 341-346.

[12] Banhart J. Manufacture, characterisation and application of cellular metals and metal foams.

Progress in materials science 2001; 46(6): 559-632. [13] Avalle M, Belingardi G, Montanini R. Characterization of polymeric structural foams under

compressive impact loading by means of energy-absorption diagram. International Journal of

Impact Engineering 2001; 25(5): 455-472. [14] Rajendran R, Sai KP, Chandrasekar B, Gokhale A, Basu S. Preliminary investigation of

aluminium foam as an energy absorber for nuclear transportation cask. Materials & Design

2008; 29(9): 1732-1739.

[15] Avalle M, Belingardi G, Ibba A. Mechanical models of cellular solids: parameters identification from experimental tests. International Journal of Impact Engineering 2007; 34(1):

3-27.

[16] Zhang Y, Tang Y, Zhou G, Wei J, Han F. Dynamic compression properties of porous aluminum. Materials Letters 2002; 56(5): 728-731.

[17] Zhao Y, Taya M, Kang Y, Kawasaki A. Compression behavior of porous NiTi shape memory

alloy. Acta materialia 2005; 53(2): 337-343. [18] Al-Hadhrami A, Elliott L, Ingham D. A new model for viscous dissipation in porous media

across a range of permeability values. Transport in Porous Media 2003; 53(1): 117-122.

20

[19] Vural M, Ravichandran G. Dynamic response and energy dissipation characteristics of balsa

wood: experiment and analysis. International Journal of Solids and structures 2003; 40(9): 2147-2170.

[20] Nield D. The modeling of viscous dissipation in a saturated porous medium. Journal of Heat

Transfer 2007; 129(10): 1459-1463.

[21] Yang J, Liew KM, Kitipornchai S. Imperfection sensitivity of post-buckling behavior of higher-order shear deformable functionally graded plates. International Journal of Solids and

Structures 2006; 43(17): 5247-5266.

[22] Liew KM, Yang J, Kitipornchai S. Thermal postbuckling of laminated plates comprising FGM with temperature-dependent material properties. Journal of Applied Mechanics ASME 2004;

71(6): 839-850.

[23] Ke LL, Yang J, Kitipornchai S, Bradford MA. Bending, buckling and vibration of size-dependent functionally graded annular microplates. Composite structures 2012; 94(11): 3250-

3257.

[24] Şimşek M, Yurtcu H. Analytical solutions for bending and buckling of functionally graded

nanobeams based on the nonlocal Timoshenko beam theory. Composite Structures 2013; 97: 378-386.

[25] Magnucki K, Stasiewicz P. Elastic buckling of a porous beam. Journal of Theoretical and

Applied Mechanics 2004; 42(4): 859-868. [26] Jasion P, Magnucka-Blandzi E, Szyc W, Magnucki K. Global and local buckling of sandwich

circular and beam-rectangular plates with metal foam core. Thin-Walled Structures 2012; 61:

154-161. [27] Magnucka-Blandzi E, Magnucki K. Effective design of a sandwich beam with a metal foam

core. Thin-Walled Structures 2007; 45(4): 432-438.

[28] Magnucka-Blandzi E. Dynamic stability of a metal foam circular plate. Journal of Theoretical

and Applied Mechanics 2009; 47: 421-433. [29] Magnucka-Blandzi E. Non-linear analysis of dynamic stability of metal foam circular plate.

Journal of Theoretical and Applied Mechanics 2010; 48(1): 207-217.

[30] Magnucka-Blandzi E. Axi-symmetrical deflection and buckling of circular porous-cellular plate. Thin-Walled Structures 2008; 46(3): 333-337.

[31] Belica T, Magnucki K. Dynamic stability of a porous cylindrical shell. PAMM 2006; 6(1): 207-

208.

[32] Belica T, Magnucki K. Stability of a porous-cellular cylindrical shell subjected to combined loads. Journal of Theoretical and Applied Mechanics 2013; 51(4): 927-936.

[33] Magnucki K, Malinowski M, Lewinski J. Optimal design of an isotropic porous-cellular

cylindrical shell. In: ASME 2006 Pressure Vessels and Piping/ICPVT-11 Conference, Vancoucer, Canada, 2006.

[34] Biot M. Theory of buckling of a porous slab and its thermoelastic analogy. Journal of Applied

Mechanics 1964; 31(2): 194-198. [35] Jabbari M, Mojahedin A, Khorshidvand A, Eslami M. Buckling analysis of a functionally

graded thin circular plate made of saturated porous materials. Journal of Engineering

Mechanics 2014; 140(2): 287-295.

[36] Detournay E, Cheng AHD. Fundamentals of poroelasticity, Chapter 5, Comprehensive Rock Engineering: Principles, Practice and Projects, Vol. 2. Oxford (UK): Pergamon Press; 1993.

[37] Jabbari M, Mojahedin A, Haghi M. Buckling analysis of thin circular FG plates made of

saturated porous-soft ferromagnetic materials in transverse magnetic field. Thin-Walled Structures 2014; 85: 50-56.

[38] Jabbari M, Hashemitaheri M, Mojahedin A, Eslami M. Thermal buckling analysis of

functionally graded thin circular plate made of saturated porous materials. Journal of Thermal Stresses 2014; 37(2): 202-220.

[39] Jabbari M, Joubaneh EF, Khorshidvand A, Eslami M. Buckling analysis of porous circular

plate with piezoelectric actuator layers under uniform radial compression. International Journal

of Mechanical Sciences 2013; 70: 50-56.

21

[40] Joubaneh EF, Mojahedin A, Khorshidvand A, Jabbari M. Thermal buckling analysis of porous

circular plate with piezoelectric sensor-actuator layers under uniform thermal load. Journal of Sandwich Structures and Materials 2014; 0(00): 1-23.

[41] Gibson LJ, Ashby M. The mechanics of three-dimensional cellular materials. Proceedings of

the Royal Society of London A: Mathematical, Physical and Engineering Sciences 1982;

382(1782): 43-59. [42] Choi J, Lakes R. Analysis of elastic modulus of conventional foams and of re-entrant foam

materials with a negative Poisson's ratio. International Journal of Mechanical Sciences 1995;

37(1): 51-59. [43] Ke LL, Yang J, Kitipornchai S. Nonlinear free vibration of functionally graded carbon

nanotube-reinforced composite beams. Composite Structures 2010; 92(3): 676-683.