Elsevier

Composite Structures

Volume 92, Issue 6, May 2010, Pages 1308-1317
Composite Structures

Dynamic response of pre-stressed fibre metal laminate (FML) circular cylindrical shells subjected to lateral pressure pulse loads

https://doi.org/10.1016/j.compstruct.2009.11.012Get rights and content

Abstract

Dynamic response of fiber metal laminate cylindrical shells subjected to initial combined axial load and internal pressure were studied in this paper. First order shear deformation theory (FSDT) was utilized in the shell’s equilibrium equations and strain–displacement relations were based on Love’s first approximation theory. Equilibrium equations for buckling, free and forced vibration problems of the shell were solved using Galerkin method. The influences of FML parameters such as material properties lay up, Metal Volume Fraction (MVF), fiber orientation and initial stresses on dynamic response were investigated. The results were indicated that the FML lay up, has a significant effect on natural frequencies as well as transient dynamic response with respect to various values of MVF as well as pre-stress.

Introduction

Fibre metal laminate (FML) is a new advanced hybrid composite material combining thin metal layers with adhesive fibre prepreg as shown in Fig. 1. They combine good characteristics of metals such as ductility, impact and damage tolerances with the benefits of fiber composite materials such as high specific strength, high specific stiffness and good corrosion and fatigue resistance.

Due to the above advantages, cylindrical shells composed of FML materials are favourable to be used in the industries such as, automobile, aircraft, and space system. Most of composite cylindrical shells are used under impulse load and application of this loading may cause large deformation and damage and hence strength reduction. Therefore, considering dynamic response of the structure is necessary in the design process.

Some researches have been done on the buckling of composite cylindrical shells [1], [2]. Many researches have been done on the free vibration and dynamic response of composite cylindrical shells, for example Li and Chen [3] investigated transient dynamic response analysis of orthotropic circular cylindrical shell subjected to external hydrostatic pressure. They used classical shell theory and considered simply supported boundary conditions. Bardell et al. [4] investigated the free and forced vibration analysis of laminated cylindrical shell. Lam and Loy [5] studied the influence of boundary conditions for a thin laminated rotating cylindrical shell using first Love’s approximation and Galerkin method. Lee and Lee [6] studied free vibration and dynamic response for cross-ply composite circular cylindrical shell under radial impulse load and the boundary conditions were considered to be simply supported. They did not consider the pre-stress effect in the governing equations of motion. Khalili et al. [7] studied the free and forced vibration of simply supported composite circular cylindrical shells. The dynamic response was studied under transverse impulse, axial load and internal pressure by modal technique. However, according to the best knowledge of the authors, the dynamic response of FML cylindrical shells including initial stress effects has not been reported.

In this paper, free vibrations and transient response of hybrid FML circular cylindrical shells under transverse pressure pulse load, combined static axial loads and internal pressure is investigated based on first order shear deformation theory. The boundary conditions are considered to be simply supported. The pressure pulse load is in the form of triangle, which is applied on a small rectangular area. The effects of stacking sequence, fibre orientation, axial load, internal pressure and thickness to radius ratio on the time response of the shells were investigated. The new interesting results are presented which provide a helpful insight for aircraft fuselage skin designers.

A laminate coding system is used to comprehensively definition of FML laminates. The coding system of glass reinforced aluminum laminate is Al/G (1 + i)/i [0/90]S, for example, for i = 4, Al/G 5/4 [0/90]S, defines a laminate composed of five aluminum layers and four glass reinforced polymer perpregs with lay up [0/90]S alternatively stacked together. Similarly, if aramid reinforced polymer or carbon reinforced polymer prepregs are used instead of glass reinforced polymer prepregs, Al/A or Al/C is used instead of Al/G in the coding system, respectively. In a general case, FML is used instead of Al/G, Al/A or Al/C, i.e. FML 5/4[0/90]S. FMLs composed of glass, aramid and carbon reinforced polymer prepregs are named Glare, Arall and Care, respectively [8]. The thicknesses of all aluminum layers are assumed to be equal. Also, the thicknesses of all composite layers are assumed to be equal. The Metal Volume Fraction (MVF) is defined as the sum of the ratio of thicknesses of the individual aluminum layers to the total thickness of the laminate [8] as (MVF=1phal/h), where hal is the thickness of each separate aluminium layer and p is the number of aluminium layers. Hence, MVF = 0 and MVF = 1 represent pure composite and pure metal shells, respectively.

Section snippets

Governing equations

A circular cylindrical shell with mean radius R, thickness h and length L is shown in Fig. 2. u, v and w are the displacement components in the axial, tangential and radial directions, respectively. Based on the first order shear deformation theory (FSDT), the equilibrium equations for a shell under axial loads and internal pressure are as follows [3], [6]:Nx,x+Nxφ,φ/R+P(u,φφ/R+w,x)+qx(x,φ,t)=I1u,tt+I2βx,ttNxφ,x+Nφ,φ/R+Qφ/R+Nav,xx+P(v,φφ/R+w,φ)+qφ(x,φ,t)=(I1+2I2/R)v,tt+(I2+I3/R)βφ,ttQx,x+Qφ,φ/R-

Boundary conditions

The boundary conditions for cylindrical shell, which is simply supported along its curved edges at x=0 and x=L are considered as [13]:ν=w=Nx=Mx=βφ=0In order to solve the buckling and free vibration problems, the external excitations are taken to be zero. After substituting Eqs. (3), (5) into Eq. (1), the results can be simplified in the following form:L11L12L13L14L15L21L22L23L24L25L31L32L33L34L35L41L42L43L44L45L51L52L53L54L55u(x,φ,t)ν(x,φ,t)w(x,φ,t)βx(x,φ,t)βφ(x,φ,t)={0}

Lij are the differential

Buckling analysis

In order to obtain the critical axial buckling load, static solution is done, so the function of time in Eq. (8a), (8b) must be neglected. Substituting Eq. (8a), (8b) into Eq. (7) and using Galerkin method yields a set of five equations in the following form:02π0L(L11u+L12ν+L13w+L14βx+L15βφ)dηu(x)dxcosnφdxdφ=002π0L(L21u+L22ν+L23w+L24βx+L25βφ)ην(x)sinnφdxdφ=002π0L(L31u+L32ν+L33w+L34βx+L35βφ)ηw(x)cosnφdxdφ=002π0L(L41u+L42ν+L43w+L44βx+L45βφ)dηβx(x)dxcosnφdxdφ=002π0L(L51u+L52ν+L53w+L54βx+L

Free vibration analysis

To solve the governing equation, the function of time is treated as Tmn(t)=eiωmnt, where ωmn is the natural frequency. By considering axial compressive load equal to the fraction of critical buckling load and internal pressure, natural frequencies and mode shapes are obtained. By applying Galerkin method, similar to the buckling analysis, the following set of equations can be derived as follows:[Kij]-ωmn2[Mij]{AmnBmnCmnDmnEmn}T=0(i,j=1,,5)where Kij and Mij are the stiffness and mass matrices,

Dynamic response analysis

Fig. 3 shows the position of applied lateral pressure pulse load on a small rectangular area. The applied loads are defined as:qx(x,φ,t)=Qx(x,φ)f(t)=mnX¯mnf(t)=mnXmndηu(x)dxcosnφf(t)qφ(x,φ,t)=Qφ(x,φ)f(t)=mnY¯mnf(t)=mnYmnην(x)sinnφf(t)qr(x,φ,t)=Qr(x,φ)f(t)=mnP¯mnf(t)=mnPmnηw(x)cosnφf(t)mx(x,φ,t)=Mx(x,φ)f(t)=mnZ¯mnf(t)=mnZmndηβx(x)dxcosnφf(t)mφ(x,φ,t)=Mφ(x,φ)f(t)=mnW¯mnf(t)=mnWmnηβφ(x)sinnφf(t)

In Eq. (16), f(t) is function of time, Xmn,Ymn,Pmn,Zmn and Wmn are the constant

Definition of pressure pulse loads

Time duration of applying pressure pulse load begins from t = 0 to t=t1 (t1 is equal to the natural period of the shell). For triangular pulse load, the solution of convolution integral in Eq. (22) is obtained in the following form:f(t)=2f0(t/t1),0tt1/2f(t)=-4f0(t-t1/2)/t1,t1/2<tt1f(t)=2f0(t-t1)/t1,t>t10tf(τ)sinωmn(t-τ)dτ=2f0ωmntt1-sin(ωmnt)ωmnt1,0tt1/22f0ωmn1-tt1-sin(ωmnt)ωmnt1+2sinωmn(t-t1/2)ωmnt1,t1/2<tt12f0ωmn-sin(ωmnt)ωmnt1+2sinωmn(t-t1/2)ωmnt1-sinωmn(t-t1)ωmnt1,t>t1

Validation of the model

In order to investigate the validity of the results of the transient dynamic response, the history of radial displacement due to radial impulse load is compared with the results obtained by Ref. [6] as depicted in Fig. 4 for simply supported CRP shell with lay up [0/90/0/90/0/90] under triangle pulse type of peak load f0=300kPa, area of applied load 2l1×2l2=6.2×2cm2, and geometry R = 0.2 m, L = 0.2 m, h = 1.2 mm. No initial stresses are applied. As stated earlier, the FSDT is suitable for modeling the

Numerical results and discussions

A circular cylindrical shell of radius R = 3 m, length L = 3 m is considered. The material properties of the shell are defined in Table 2. In the study of dynamic response, triangular pulse load is used. The area of applied load is 2l1×2l2=0.2×0.2m2 and the value of maximum internal pressure is taken to be f0=5000 Pa. Transient response of the centre point of applied load area is studied. For sufficient convergence of dynamic response, the number of the considered modes is (m×n)=(45×55). Hereinafter,

Conclusions

The effect of stacking sequence, fibre orientation, axial load, internal pressure and the geometric parameter (R/h ratio) on the transient response of the FML shells under radial pressure pulse load has been studied based on first order shear deformation theory (FSDT) and using Galerkin method. New interesting results that hitherto not reported in the literature were obtained in this paper. Some of these new results are:

  • 1.

    Variation of MVF causes ωf as well as its corresponding mode shape to

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