homogenization procedure for the free and the steady state

Contemporary Engineering Sciences, Vol. 11, 2018, no. 3, 101 - 111

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ces.2018.817

Homogenization Procedure for the Free and

the Steady State Periodic Forced Response of

Laminated Composite Beams Resting on

Non-Linear Foundation

Merrimi El Bekkaye and El Bikri Khalid

Mohammed V University in Rabat. ENSET-Rabat, LaMIPI, B.P. 6207

Rabat Institute, Rabat, Morocco

Benamar Rhali

Mohammed V University in Rabat. Ecole Mohammadia d’Ingénieurs

Lersim, Av. Ibn Sina, Agdal, Rabat, Morocco

Copyright © 2018 Merrimi El Bekkaye et al. This article is distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstract

In the present paper, a theoretical model based on Hamilton's principle and

spectral analysis is used to study the non-linear free vibration of laminated

composite beams (LCB) resting on nonlinear Pasternak elastic Foundation with

immovable ends. A homogenization procedure is performed to reduce the

problem to that of isotropic homogeneous beams with effective bending stiffness

and axial stiffness. Iterative form solutions are presented to determine the

influence of the foundation parameters on the nonlinear frequency to the linear

frequency ratio of the LCB. Then, a multi-mode approach is developed to study

the nonlinear steady state periodic forced response of the beam. The results are

given for various foundation stiffness parameters, showing that an increase in the

value of linear elastic foundation stiffness or in modulus of Pasternak foundation

leads to a decrease in the nonlinear to linear frequency ratio, but this ratio

enhances with an increase in nonlinear elastic foundation stiffness. On the other hand, the beam is subjected to a concentrated force, an explicit formulation is used for

102 Merrimi El Bekkaye et al.

calculation of the non-linear steady state periodic forced response of beams.

Keywords: Non linear vibration, composite structure, foundation

1 Introduction

Many research works focus recently the non-linear dynamic response of

composite beams and plates [1]. A higher-order displacement-based theory has

been evaluated by construing the free vibration and the buckling behaviors of

laminated composite and sandwich beams [2]. Using trigonometric shear

deformation theory, a dynamic stiffness matrix of a uniform laminated composite

beam has been developed in [3]. The vibration analysis of the sandwich beams

with a flexible syntactic foam as the core material based on the high-order

sandwich panel theory has been investigated in [4].

Using Navier’s technique and solving the eigenvalue equations, analytical

solutions based on the global–local higher-order theory has been used by Zhen et

al [5], in their paper, authors have studied the effect of the order number of higher

order shear deformation as well as inter laminar continuity of transverse shear

stress on the global response of both laminated beams and sandwiches. Non-linear

dynamic behavior of an functionally graded beam with pinned–pinned ends

subjected to a moving harmonic load has been investigated in [6] by using

Timoshenko beam theory with the effect of the geometric nonlinearity. The

vibration of a simply supported laminated rectangular plate with non-ideal

boundary conditions on Pasternak foundation with in-plane loads has been

examined in [7] by using the Lindstedt–Poincare perturbation technique. In [8],

the differential quadrature method has been used to study 3-dimensional free

vibration analysis of continuous grading fiber reinforced (CGFR) plate on the

elastic foundations. The homotopy analysis method (HAM) has been used in [9]

to obtain an approximate analytical solution for geometrically non-linear

vibrations of thin laminated composite plates resting on nonlinear elastic

foundations. Baghani et al [10] presented an analytical expression for large free

vibration amplitude and post-buckling analysis of asymmetrically laminated

composite beams on elastic foundation. Geometric nonlinearity was considered

using Von Karman’s strain–displacement relations. Besides, the elastic foundation

has cubic nonlinearity with shearing layer. The nonlinear governing equation was

solved by employing the variational iteration method. The authors showed that the

third-order approximation of the VIM led to highly accurate solutions which are

valid for a wide range of vibration amplitudes and they have studied the effects of

different parameters on the ratio of nonlinear to linear natural frequency of beams

and the post-buckling load–deflection relation.

In the present work, using Hamilton’s principle and spectral analysis, the problem

of geometrically non linear free vibration of symmetrically and asymmetrically

laminated composite beams (LCB) resting on nonlinear and Pasternak elastic

foundations with immovable ends is investigated. The model is based on the

Homogenization procedure for the free and … 103

governing axial equation of the beam in which the axial inertia and damping are

ignored. A homogenization procedure is developed and used to reduce the

problem under consideration to that of an equivalent isotropic homogeneous

beam. Direct iterative method is employed first for solving the eigenvalue

equation for governing the frequency nonlinear vibration, in order to show the

effect foundation stiffness parameters, on dynamic response of laminated

composite beams. On the other hand, a multi-mode approach to the nonlinear

steady state periodic forced response of laminated composite beams is developed;

the method is based on an explicit formulation which allows an easy and

simplified calculation of the non-linear steady state periodic forced response of

beams.

2 Theoretical formulation

Consider in this section a laminated composite beam with length l, width b and

total thickness h as shown in Fig. 1. It assumed that the structure is composed of

perfectly bonded orthotropic layers and resting on a nonlinear Pasternak elastic

foundation.

Fig. 1 The geometry of the laminated composite beam.

Assuming that the beam has large amplitudes, the Von-Kármán type nonlinear

strain–displacement relationships as given below are used:

2

1

2

a aa

x

u W

x x

(1)

Where a

x is the non-linear membrane strain and the curvature of the beam is

defined by:

2

2.

a

x

WK

x

(2)

The total elastic strain energy TV of Euler-Bernoulli beams is:

0

1.

2

l

a

T x x y xV N M K dx (3)

h

b

z, w

x, u

l

Shearing layer Nonlinear layer

F c


In which xN and

yM are the axial internal force and the bending moment

respectively, which are related to strains as follows:

11 11

a

x x xN bA bB K (4)

11 11 .a

y x xM bB bD K (5)

Where 11 11,A B and 11D are extension-extension, bending-extension, bending-

bending coupling coefficients respectively, and can be evaluated using the

classical laminate plate theory.

The elastic strain energy VT of the beam is:

22

11 11 110

12

2

la a

T x x x xV A B K D K dx (6)

The total strain energy VT Can be written in terms of axial displacement ua and

transverse displacement Wa as:

2

2 2 22 2

11 11 112 2

0 0 0

1 1 1 1 12

2 2 2 2 2

l l la a a a a a

T

u W W u W WV bA dx bB dx bD dx

x x x x x x

(7)

T a c bV V V V

Where Va is the axial strain energy:

2

11

0

1

2

l

a

a xV bA dx (8)

Vc is the coupling axial-bending strain energy:

110

12

2

la

c x xV bB K dx (9)

Vb is the bending strain energy:

2

110

1

2

l

b xV bD K dx (10)

Note that in the case of isotropic and symmetric laminated composite beam the

Vc vanishes (B11 = 0).

The governing equation of motion related to axial free vibration for composite

beam is given by:


2

20

aa a xNu u

u ut t x

(11)

Once in plane inertia and damping are ignored, equation (8) leads to: 0xN

x

i.e 11 11 tana

x x xN bA bB K cons t (12)

Integrating (12) between the limits 0 and l and considering immovably axial end

conditions: (0) ( ) 0a au u l , leads to:

22

11 11

2

0 02 2

l la a

x

bA W bB WN dx dx

l x l x

(13)

Then equation (7) becomes:

2 2 22 2

11 11 11 11

2 2

0 0 0 0 0

1

4 2 2 2 2

l l l l la a a aa

T x x

bA W bB W W bD W bBV dx dx dx dx K dx

l x l x x x

(14)

In equation (14), 11

0

l

a

x xB K dx can be written only in term of W as follows:

2 22 2

11 1111 2 2

110 0 0 0

1

2 2 4

l l l la a aa

x x

B W B W WB K dx dx dx dx

A x l x x

(15)

Replacing (15) in (14) and rearranging, the potential energy TV can be written

in term of transverse displacement aW as:

2

2

08

l aeff

T

ES WV dx

l x

22

2

0

.2

l aeff

EI Wdx

x

(16)

Where 11effES bA and

2

1111

11

( )eff

BEI b D

A are the effective axial and

bending stiffness respectively. The expression for the total strain energy is

effective for replacing the laminated composite beam problem with an equivalent

classical isotropic beam problem.

The strain energy of the elastic foundation Vf and the kinetic energy of LCB, in

which axial and rotary inertia are ignored, are given by:

𝑉𝑓 =1

2∫ (𝐾𝐿𝑊2 +

1

2𝐾𝑁𝐿𝑊4 + 𝐾𝑆

2W

t) 𝑑𝑥

𝑙

0 (17)


2

0

.2

l aS WT dx

t

(18)

Where KL and KNL are the linear and the nonlinear Winkler foundation stiffness

respectively, KS is the shear modulus of Pasternak foundation.

For a general parametric study, we use the following non dimensional

formulation is used by putting:

* xx

l ,

*a

a WW

r , where

Ir

S

Applying Hamilton’s principle and expanding the displacement aW in the form

of a finite series, the following set of non linear amplitude equations is obtained:

2* * * *30

2i j r ijkr i ir i ira a a b a k a m , r = 1...n. (19)

From equation (19), 2* is given by:

2

* *

*

*

3

2i j ij i j k l ijkl

i j ij

a a k a a a a b

a a m

(20)

Where*

ijk , *

ijklb and *

ijm stand for the non dimensional classical rigidity tensor,

the non linear tensor and the mass tensor respectively, which are defined as:

2 * 12 *1

* * * * * *

2 2* *00

jiij L i j

wwk dx K w w dx

x x

**1

* *

* *0

jiS

wwK dx

x x

(21)

** * *1 1

* * *

* * * *0 0

ji k lijkl

ww w wb dx dx

x x x x

1

* * * * * *

0

NL i j k lK w w w w dx (22)

1* * * *

0ij i jm w w dx (23)

Where

2( )

,4( )

eff

eff

ESh

EI

4* ,

( )

LL

eff

K lK

EI

4 2* ,

2( )

NLNL

eff

K l hK

EI

4 2*

2( )

SS

eff

K l hK

EI ( 24, 25, 26,27)


3 The beam under a harmonic excitation force

An explicit formulation is used here to determine the effect of the excitation

frequency and level of the applied harmonic force, concentrated at the beam

middle span, on its dynamic response at large vibration amplitudes.

A multi-mode approach is developed with the objective to obtain a simplified

calculation of the non-linear steady state periodic forced response of laminated

composite beams.

The formulation obtained from equation (19) in this case of problem is:

23

1,...,2 ijkr i

c

i ir i j k e i ira k a a a b a m f i n (28)

Where

( )c c

i i ff F w x (29)

Fc is the concentrated force applied at xf . Then, the harmonic excitation force

is defined by:

( , ) ( )sinc c

f eF x t F δ x x t (30)

Where δ is the Dirac function and ωe the excitation frequency. The potential

energy of the force Fc(x,t) is given by:

2sin ( )c c

f i i eV a f t (31)

For first linear mode, basic function contribution a1 is defined in follows

equation, with the modal contributions 3, 5,…,11 obtained explicitly and

presented in equation (33).

2

3 *

1 1 1111

1 ** *

11 11

3

2 ,

c

e

f a b

ak m

2

3 *

1 111

** *

3

2 ( 3,5,...,11)

c

r r

r

rr e rr

f a b

rk m

(32, 33)

Also the amplitude of excitation force f1c* and e

are given by (32) as bellow:

2

21111 11

1 11 1 11

31

2

c

e

l

b fa

k a k

(34)


Here ω*1l the first linear frequency parameter defined by:

2

111

11

l

K

m

(35)

Then for a1, the non-linear amplitude Wa*nl1(x,a1) is given below:

3 * 3 * 3 *

1 1 1111 3 1 1113 11 1 11111* * * *

1 1 3 112 2 2* * *

11 11 33 33 1111 1111

3 3 3

2 2 2( , ) ( ) ( ) .. ( )

c c c

a

nl

e e e

f a b f a b f a b

W x a w x w x w xk m k m k m

(36)

4 Results and discussions

For symmetric and asymmetric lay-up orientations as shown in table 1, the

nonlinear frequency ratios nl/l are calculated for six layers of a uniform

thickness. We consider primary in this section that the laminated composite beam

is rested on foundations where parameters values are:

KL = KNL = KS = 0.

Table 1. Comparison of nonlinear frequency ratio nl/l for different lay-up

orientations of the laminated composite beams with max / 1.aW h

nl/l

Symmetric layup orientations

unidirectional 1.2397

(0/90/90)s 1.1353

(90/90/0)s 1.6724

Cross-ply 1.2397

Asymmetric layup orientations

Cross-ply* 1.2533

Balanced* 1.3128

General* 1.2847

It can be shown from Fig. 2 that an increase in the value of linear elastic

foundation stiffness or in modulus of Pasternak foundation leads to a decrease in

the nonlinear to linear frequency ratio. On the other hand, this ratio enhances with

an increase in nonlinear elastic foundation stiffness.


Fig. 2. The frequency ratio for various values of linear, nonlinear Winkler

foundation stiffness and the shear modulus of Pasternak foundation.

In Fig. 3, it can be seen the comparison of nonlinear frequency response

functions for different lay-up symmetric orientations of the laminated composite

beams, excited by harmonic force, the curves are plotted with KS=KL=KNL=0.

Fig. 3. Comparison of nonlinear frequency response functions for different lay-up

symmetric orientations of the laminated composite beams, excited by harmonic

force Fc*= 200, with KS=KL=KNL=0

0.8 1 1.2 1.4 1.6 1.8 2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(90/90/0)s

Cross ply

(0/90/90)s

Fc* = 200

W*max/h

ω*nl / ω

*l

W*Max

ωe*/ω*

l


5 Conclusions

Using Hamilton’s principle and spectral analysis, and neglecting axial inertia

and damping, a homogenisation procedure has been developed and used in this

work, to reduce the problem under consideration to that of an equivalent isotropic

homogeneous beam, with effective bending stiffness and axial stiffness

parameters. Iterative method has been employed to calculate the nonlinear

frequency ratios nl/l for symmetric and asymmetric lay-up orientations of the

laminated composite beam LCB resting on nonlinear Pasternak elastic Foundation

with immovable ends. The influence of the foundation parameters on the

nonlinear fundamental frequency has been studied. It shows that the effect of the

linear and the shearing foundation parameters is to decrease the nonlinear

frequency ratio of the LCB, whereas the effect of the nonlinear foundation

stiffness is to stiffen the nonlinear response. In the other hand, it can be shown

from the results that in the case of the beam excited by harmonic concentrated

force in the middle of its span, the nonlinear frequency change with lay-up

orientations, for fixed value of foundation. To complete the present study, it is

provided to analyze this effect on laminated composite plate and generalize this

method to 2D structures.

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Received: January 15, 2018; Published: February 5, 2018



https://doi.org/10.1016/j.ijengsci.2010.06.015

https://doi.org/10.1016/j.jfranklin.2010.12.002

https://doi.org/10.1016/j.apm.2010.05.012

homogenization procedure for the free and the steady state

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