Dynamics of the Structures and Non Destructive Testing

105 Pages·2011·17.591 MB·105\105

by  Karama, Moussa

Dynamics of the
Structures
and Non Destructive
Testing
Edited by
Moussa Karama
Dynamics of the
Structures and
Non Destructive Testing
Special topic volume with invited peer reviewed papers only.
Edited by
Moussa Karama
Copyright  2011 Trans Tech Publications Ltd, Switzerland

All rights reserved. No part of the contents of this publication may be
reproduced or transmitted in any form or by any means without the
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Volume 482 of
Key Engineering Materials
ISSN 1013-9826
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Preface

Structural materials are fundamental in human history. Their use in airframe structures has steadily increased since the
1970s. Currently, the applications have expanded to include empennage, fuselage, wing and dynamic components of small
airplanes, transport aircraft and rotorcraft. Even though, the aircraft industry is very conservative in the adoption of new
designs and technologies and when new aircraft are introduced, they tend to build heavily upon past designs, introducing
only incremental updates in technology. However, the global world market has accelerated the race over the last two
centuries, with progress in structural materials pacing improvements in living. Advances in aluminum alloys for airframes and
super alloys for engines led to the current era of mass international travel by jet aircraft. Advances in high temperature
structural materials are leading the way to better engines and materials processing. Structural composites have
revolutionized pleasure boating and impacted many types of sports equipment. Advanced structural composites have been
essential to space vehicles and offer promises of lighter, more efficient materials for many applications.

With energy cost increase, the gain from weight-saving is most significant in the aerospace domain, but it is clearly also
advantageous for road transport and is starting to be recognised in shipbuilding. Consequently, improved reliability,
resistance to degradation and durability in severe environments are always current requirements. So the development of
composites, nano-composites and refractory alloys with specific properties becomes a key factor for industrial and
technological progress. Another challenge is the recyclability of advanced materials with the emergency of projects on
thermoplastic matrix composite fuselages. Moreover the incorporation of biopolymer extracted from different raw materials
can be an interesting alternative to reach the objective of 100% biodegradability. The purpose of these topics is to gather
researchers, specialists from universities and industries working on new composites and nano-composites formulation,
titanium alloys, materials and structure characterisation by destructive, non destructive testing, numerical analysis and
composite process

Prof. M. Karama
Guest Editor
Table of Contents
Preface
An Analytical Method for Free Vibration Analysis of Composite Beams Subjected to Initial
Thermal Stresses
A. Mahi, E.A. Adda-Bedia and A. Benkhedda 1
Polynomial Approach Modeling of Resonator Piezoelectric Disc
L. Elmaimouni, J.E. Lefebvre, F.E. Ratolojanahary, A. Raherison, B. Bahani and T. Gryba 11
Study of the Transmission of Ultrasonic Guided Wave at the Junction of Two Different
Elastic Plates with the Presence of a Defect
T. Belhoussine Drissi, B. Morvan, M. Predoi, J.L. Izbicki and P. Pareige 21
Identification of the Dynamic Coefficients of Hybrid Bearings
K. Atchonouglo, O. Bonneau, P. Jolly and C. Vallée 31
Impact Resistance of Composite Materials under Biaxial Preloading
C. Garnier, T. Djilali, R. Brault and S. Mistou 39
Characterization of Grains Size by Ultrasounds
A. Badidi Bouda, M.S. Aljohani, A. Mebtouche, R. Halimi and W. Djerir 49
Identification of Materials Properties Using Displacement Field Measurement
M. Fazzini, O. Dalverny and S. Mistou 57
Physical and Mechanical Properties of the Reinforced Polyester: Characterization -
Probabilistic Analysis
K. Ait Tahar and N. Harb 67
Seismic Vulnerability Assessment Using the Instrumentation of an Existing Building
F. Duco, J.P. Faye, S. Caperaa and E. Reubrez 79
High Temperature Tribological Behaviour of Metal Matrix Composites Produced by SPS
S. Orozco Gomez, K. Delbé, A. Benitez, J.Y. Paris and J. Denape 89
© (2011) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/KEM.482.1
An analytical method for free vibration analysis of composite beams
subjected to initial thermal stresses

A. Mahi1,a, E.A. Adda Bedia2,b, A. Benkhedda1,c
1Laboratoire des Sciences Aéronautiques, Université de Blida,
BP 270, 09000 Blida, Algeria

2Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbes,
BP 89 Ben M’hidi, 22000 Sidi Bel Abbes, Algeria.

[email protected], [email protected], [email protected]
Key words: Composite beams, free vibration, analytical method, first shear deformation theory.
Abstract. The purpose of this paper is to present exact solutions for the free vibration of
symmetrically laminated composite beams. The present analysis includes the first shear deformation
theory and the rotary inertia. The analytical solutions take into account the thermal effect on the free
vibration characteristics of the composite beams. In particular, the aim of this work is to derive the
exact closed-form characteristic equations for common boundary conditions. The different
parameters that could affect the natural frequencies are included as factors (aspect ratio, thermal
load-to-shear coefficient, ply orientation) to better perform dynamic analysis to have a good
understanding of dynamic behavior of composite beams. In order to derive the governing set of
equations of motion, the Hamilton’s principle is used. The system of ordinary differential equations
of the laminated beams is then solved and the natural frequencies’ equations are obtained
analytically for different boundary conditions. Numerical results are presented to show the
influence of temperature rise, aspect ratio, boundary conditions and ply orientation on the natural
frequencies of composite beams.
1. Introduction
More attention is accorded to composite materials in the last decades, especially to composite
beams. Nowadays, composite beams can be found as principal elements in primary structures in
many engineering applications, and mainly in aerospace and aeronautics. Thus, dynamic analysis
must be done accurately to predict the vibration characteristics of the beams. Only few literatures
are dealing with analytical free vibration analysis of beams [1]. To the author’s knowledge, no
results are found providing exact solutions which include initial thermal stresses.
In the present paper, the free vibration analysis is conducted using analytical method. The
equations of motion of laminated beam are derived using FSDBT theory including rotary inertia and
using Hamilton’s principle. To obtain the natural frequencies, the governing differential equations
are solved using an exact analytic method. The effects of temperature rise, aspect ratio, common
boundary conditions and ply orientation are investigated.
2 Dynamics of the Structures and Non Destructive Testing
2. Theoretical formulation
We consider a composite beam of length L, constant thickness h and width b, as shown on
Figure 1.

z
y

h x
b

L

Fig. 1: Laminated composite beam

The displacement field may be written as follows [1]:

u(x,z,t)=u(x,t)+zψ(x,t),

w(x,z,t)= w(x,t). (1)
where u and w are the displacements on the mid-plane and ψ is the rotation of the normal to the
mid-plane about y axis. The deformation field can then be derived easily to obtain:

∂u ∂ψ ∂w
ε = +z , γ =ψ+ .
x ∂x ∂x xz ∂x (2)

where ε is the deformation along the x axis and γ is the transverse shear strain.
x xz

The constitutive relations of the laminated beam can be expressed using the generalized Hook’s law
as follows [1]:

 ∂u 
N  A B  ∂x 
Mxx=B1111 D1111∂ψ, Qxz = A55γxz. (3)
 ∂x 
where:
n n n
N =b∑∫hk Q ε k dz, M =b∑∫hk Q ε k zdz, A =bK∑∫hk Q k dz.
x  11 x x  11 x 55  55 (4)
h h h
k=1 k−1 k=1 k−1 k=1 k−1

And K is the shear correction factor.

The transformed material constants are given by [2]:

Q =Q cos4θ+Q sin4θ+2(Q +2Q )sin2θcos2θ,
11 11 22 12 66
(5)
Q =G cos2θ+G sin2θ.
55 13 23

where:

E ν E ν E E
Q = 1 , Q = 12 2 = 21 1 , Q = 2 , Q =G .
11 1−νν 12 1−νν 1−νν 22 1−νν 66 12 (6)
12 21 12 21 12 21 12 21
Moussa Karama 3
The extensional, flexural-extensional coupling and flexural stiffness of the laminated beam are
[2]:

n
(A , B , D )=b∑∫hk Q k (1,z,z2)dz. (7)
11 11 11  11
h
k=1 k−1

3. Governing differential equations
In order to obtain the equations of motion, the Hamilton’s principle is used. The strain energy
due to mechanical stresses is [4]:

1
U = ∫σεdv. (8)
d 2 ij ij
v

The strain energy induced by the temperature rise is given by [3]:

2
1 ∂w
U = ∫σT   dv (9)
T 2 x  ∂x 
v

Where the initial thermal stress and axial load due to the temperature rise are given by [2]:

n
h
σT =−Q α∆T, NT =b∑∫ k [(Q α)∆T]kdz . (10)
x 11 x x 11 x
h
k=1 k−1

α is the thermal expansion coefficient and ΔT is the temperature rise.
x

The kinetic energy of the beam can be written in the following form [4]:

1 ∂u ∂u
E = ∫ρ i i dv. (11)
c 2 ∂t ∂t
v

The differential equations of motion are then derived using Hamilton’s principle [4]:

δ∫t[E −(U +U )] dt =0 . (12)
 0 c d T 

By collecting the terms of δu, δw and δψ separately and assuming that the in-plane displacement

u is negligible compared to the flexural-induced displacement w, and that the beam has a midplane
symmetry, the following equations of motion are obtained:

∂2w ∂ψ ∂2w ∂2w
−I + A  + −NT =0,
1 ∂t2 55 ∂x ∂x2  x ∂x2
(13)
∂2ψ ∂2ψ  ∂w
−I +D −A ψ+ =0.
 
3 ∂t2 11 ∂x2 55 ∂x 
where:
n
h
(I ,I ,I )=b∑∫ k ρk(1,z,z2)dz. (14)
1 2 3
h
k=1 k−1
4 Dynamics of the Structures and Non Destructive Testing
The harmonic solutions can be written as:

w W
 = Exp(iωt). (15)
ψ Ψ

where: ω is the circular frequency.

The system of equations is then expressed as:

∂2W ∂Ψ
+Lm +a2c2W =0,
∂ξ2 ∂ξ
(16)
∂2Ψ (1−a2b2c2) 1 ∂W
− Ψ− =0.
∂ξ2 c² Lc² ∂ξ

where:

x A I L4 I mD
ξ= , m= 55 , a2 = 1 ω2, b2 = 3 , c2 = 11 . (17)
L (A −NT) D I mL2 L2A
55 x 11 1 55

4. Analytical solution of governing equations
The equations (16) can be uncoupled to give:

∂4Ψ +m−1+a2(b2 +c2)∂2Ψ −a2(1−a2b2c2)Ψ =0,
 
∂ξ4  c2  ∂ξ2
(18)
∂4W +m−1+a2(b2 +c2)∂2W −a2(1−a2b2c2)W =0.
 
∂ξ4  c2  ∂ξ2

The solutions are found to be in the following form:

W = Achaαξ+ Ashaαξ+ A cosaβξ+ A sinaβξ,
1 2 3 4 (19)
Ψ = Bshaαξ+ B chaαξ+B sinaβξ+B cosaβξ.
1 2 3 4

where:

1/2
 2 
α= 2 −m−1+(b2 +c2)+ m−1+(b2 +c2) + 4 (1−a2b2c2) ,
2  a2c2  a2c2  a2 
 
(20)
1/2
 2 
β= 2 m−1+(b2 +c2)+ m−1+(b2 +c2) + 4 (1−a2b2c2) .
2 a2c2  a2c2  a2 
 