composite materials and structures

4/14/12 COMPOSITE MATERIALS AND STRUCTURES

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CHAPTER - 7

LAMINATED COMPOSITE BEAMS AND PLATES

7.1 INTRODUCTION

7.2 THIN LAMINATED PLATE THEORY

7.3 BENDING OF LAMINATED PLATES

7.3.1 Specially Orthotropic Plate

7.3.2 Antisymmetric Cross-ply Laminated Plate

7.3.3 Antisymmetric Angle-Ply Laminated Plate

7.4 FREE VIBRATION AND BUCKLING




7.5 SHEAR BUCKLING OF COMPOSITE PLATE

7.6 RITZ METHOD

7.7 GALERKIN METHOD

7.8 THIN LAMINATED BEAM THEORY

7.9 PLY STRAIN, PLY STRESS AND FIRST PLY FAILURE

7.10 BIBLIOGRAPHY

7.11 EXERCISES

7.1 INTRODUCTION

The formulae presented in this chapter are based on the classical bending theory of thin composite plates.

The small deflection bending theory for a thin laminate composite beam is developed based on Bernoulli's

assumptions for bending of an isotropic thin beam. The development of the classical bending theory for a thin

laminated composite plate follows Kirchhoff's assumptions for the bending of an isotropic plate. Kirchhoff's main

suppositions are as follows:

1. The material behaviour is linear and elastic.

2. The plate is initially flat.

3. The thickness of the plate is small compared to other dimensions.

4. The translational displacements ( and w) are small compared to the plate thickness, and the



rotational displacements ( ) are very small compared to unity.

5. The normals to the undeformed middle plane are assumed to remain straight, normal and inextensional

during the deformation so that transverse normal and shear strains ( ) are neglected in

deriving the plate kinematic relations.

6. The transverse normal stresses are assumed to be small compared with other normal stress

components . So that they may be neglected in the constitutive relations.

The relations developed earlier in sections 6.12 and 6.13 are essentially based on the above Kirchhoff's

basic assumptions. Some of these relations will be utilized in the present chapter to derive the governing

equations for thin composite plates. It may be noted that Kirchhoff's assumptions are merely an extension of

Bernoulli's from one-dimensional beam to two-dimensional plate problems. Hence a classical plate bending

theory so developed can be reduced to a classical beam bending theory. Here, also, the governing plate

equations are derived first, and the beam equations are subsequently obtained from the plate equations.

7.2 THIN LAMINATED PLATE THEORY

Consider, a rectangular, thin laminated composite plate of length a, width b and thickness h as shown in

Fig.7.1. The plate consists of a laminate having n number of laminae with different materials, fibre orientationsand thicknesses. The plate is subjected to surface loads q's and m's per unit area of the plate as well as edge

loads per unit length. The expansional strains, which may be caused due to moisture and

temperature, are also included. Note that and w are mid-plane displacement components. It is assumedthat Kirchhoff's assumptions for the small deflection bending theory of a thin plate are valid in the present case.

One of these assumptions is related to transverse strains which are

neglected in derivation of plate kinematic relations i.e., stress-strain relations.

Considering the dynamic equilibrium of an infinitesimally small element dx1 dx2 (Fig. 7.2) the following

equations of motion are obtained

(7.1)

(7.2)

(7.3)

(7.4)



(7.5)

where commas are used to denote partial differentiation, and dots relate to differentiation with respect to time, t.Combining Eqs. 7.3 through 7.5, we obtain

(7.6)

where

, and

and is the density of the laminate at a distance z.

Substituting Eqs. 6.60 in Eqs. 7.1, 7.2 and 7.6 and noting Eqs. 6.47 and 6.48,we obtain the following

governing differential equations in terms of mid-plane displacements and w.

(7.7)

(7.8)

(7.9)



Note that the rotary inertia effects are usually neglected in development of a thin plate theory.

The proper boundary conditions are chosen from the following combinations:

(7.10)

where the subscript n is the direction normal to the edge of the plate, and relates to the tangential direction.

Equations 7.7 through 7.9 can be further simplified using the stiffnesses listed in Table 6.3 for various

kinds of laminates.

For symmetric laminates, Bij = 0 and R = 0. Hence Eqs.7.7, 7.8 and 7.9 can be accordingly modified.

Note that these equations become uncoupled. The modified forms of Eqs. 7.7 and 7.8 (with Bij = 0 and R = 0)

represent the stretching behaviour of a symmetric laminated plate. The bending behaviour of a symmetric

laminated plate is represented by the equation

(7.11)

However, for a specially orthotropic plate with symmetric cross-ply lamination (D16 = D26 = 0), Eq. 7.11 is

further reduced to

(7.12)

For a homogeneous anisotropic parallel-ply laminate, where all plies have the same fibre orientation θ,

noting that we obtain from Eq. 7.11.



(7.13)

In a similar way, for an orthotropic plate with either 00 or 900 fibre orientation, Eq. (7.13) is further

simplified using Q16 = Q26 = 0.

For a homogeneous isotropic plate, Q11 = Q22= E/(1-ν2), Q12 = ν Q1,

Q66 = E/[2(1+ ν)] and Q16 = Q26 = 0 and hence Eq. 7.13 reduces to

(7.14)

where and expansion moments Me�s are accordingly derived.

The solution of Eqs. 7.8 through 7.10 is difficult to achieve due to the presence of bending-extensional

and other coupling terms as well as mixed order of differentiation with respect to x1 and x2 in each relation. The

closed form solutions are restricted to a few simple laminate configurations, loading conditions, plate geometry

and boundary conditions. The other analytical methods are based on the variational approaches such RayleighRitz method and Galerkin method.

7.3 BENDING OF LAMINATED PLATES


Consider a rectangular, symmetric cross-ply laminated composite plate (Fig. 7.1) which is subjected to

transverse load q only. Equation 7.12 reduces to

(7.15)

All Edges Simply Supported

Consider the simply supported conditions as given below



(7.16)

We assume the Navier-type of solution. Let

(7.17)

that satisfies the boundary conditions vide Eq. 7.16. Further we assume that

(7.18)

Substitution of Eqs. 7.17 and 7.18 in Eq. 7.15 yields

(7.19)

Note that, for an isotropic plate,

(7.20)

The loading coefficient qmn is determined for a specified distribution of transverse load q (x1,x2 ) from the

following integral:

(7.21)

It can be shown that, for a uniformly distributed load q0,

(7.22)

where m, n are odd integers.



Hence for a specially orthotropic plate that is subjected to a transverse uniformly distributed load q0, the

deflection w is given by

(7.23)

where m, n are odd integers.

The moments M1, M2 and M6 and shear forces Q4 and Q5 can be obtained from the following relations:

(7.24)

Two Opposite Edges Simply Supported

We now consider the simply supported conditions at and either or both of the conditions at

(Fig. 7.1) may be simply supported, clamped or free. We can proceed with the Levy's type of solution.

The solution of Eq.7.15 consists of two parts: homogeneous solution and particular solution. Thus,

(7.25)

For this particular solution wp(x1), the lateral load q(x1) is assumed to have the same distribution in all

sections parallel to the x1-axis and the plate is also considered infinitely long the x2-direction. Equation 7.15

takes the following form:



(7.26)

Assume

(7.27)

and

(7.28)

Substituting Eqs. 7.27 and 7.28 in Eq. 7.26, we obtain

(7.29)

Hence the particular solution is given by

(7.30)

The homogeneous solution is obtained from the following form of Eq.7.15

(7.31)

Let us express wh (x1, x2) by

(7.32)

Eq. 7.32 satisfies simply supported boundry conditions at (Eq. 7.16). Substitution of Eq. 7.32 in Eq.

7.31 yields

(7.33)



(7.34)

Let the solution be

(7.35)

Substituting Eq. 7.35 in Eq.7.34, the characteristic equation is obtained as follows:

(7.36)

the solution of which is given by

(7.37)

The value of , in general, is complex. Hence, the roots of Eq. 7.36 can be expressed in the form and

, where α and β are real and positive and are given as

(7.38)

(7.39)

Hence, the solution is

(7.40)

Hence referring to Eqs. 7.25, 7.30 and 7.40, the final solution w(x1, x2) to Eq. 7.15 can be expressed as

follows:



(7.41)

The constant Am, Bm,Cm and Dm are determined from the relevant boundry conditions at .

Note that qm is determined from the loading distribution q(x1) using the following integration.

(7.42)

Hence for a uniformly distributed transverse load q0

m = 1,3,5,� (7.43)

The moments and shear forces are computed using Eqs. 7.24.


Now consider the rectangular plate, shown in Fig. 7.1, to be made up of cross-ply laminations of

stacking sequence [0/90]n (refer case 6 of Table 6.3). Equations 7.7 through 7.9 then reduce to

(7.44)

The simply supported boundary conditions considered here are



(7.45)

Assume the following displacement components

(7.46)

that satisfy the simply supported boundary conditions vide Eqs. 7.45. The transverse load q is represented by the

double Fourier series in Eq.7.18. Now substitution of Eqs. 7.18 and 7.46 in Eqs 7.44 results in three

simultaneous algebraic equations in terms of Amn , Bmn and Wmn. Solving these equations, we obtain

(7.47)

where



and η = a/b

Using Eqs. 7.46 and 7.47, the stress and moment resultants (N1, N2 , N6 , M1, M2, and M6 ) are derived from

Eqs. 6.52, and the shear forces Q4 and Q5 are obtained from the last two relations of Eqs. 7.24.

Figure 7.3 exhibits the maximum nondimensional deflections (at x1=a/2 and x12=b/2) for simply

supported antisymmetric cross-ply laminated plates, which are plotted against the aspect ratio a/b. Thedeflections are considerably higher in the case of a two-layered (n=1) plate because of the extension-bendingcoupling (B11). However, as the number of layers increases, the coupling effect reduces and the results approach

to that of an orthotropic plate (n = ).


Next consider a rectangular angle-ply laminated composite plate of stacking sequence [��]n (refer

case 7 of Table 6.3) that is subjected to transverse load q. Equations 7.7 through 7.9 become

(7.48)

The following simply supported boundary conditions are assumed

(7.49)

The transverse load q is given by Eq. 7.18. The following displacement field



(7.50)

satisfies simply supported boundary conditions in Eqs. 7.49. Substituting Eqs. 7.18 and 7.50 in Eqs 7.48 andsolving the resulting simultaneous algebraic equations we obtain

(7.51)

where

The values of maximum nondimensional deflections (at x1=x2=a/2) for simply supported antisymmetric

angle-ply laminated square plates are plotted against the variation of � ranging from 0 to 450. A two-layered



laminate is found to exhibit much higher deflections due to higher values of coupling terms B16 and B26 compared

to the other cases.

7.4 FREE VIBRATION AND BUCKLING


Consider a rectangular specially orthotropic plate (Fig. 7.5) which is subjected to compressive loads

and per unit length along the edges. The plate is also assumed to be vibrating freely in the

transverse direction. Equation 7.12 then becomes (note that )

(7.52)

All Edges Simply Supported

The deflected shape w (x1,x2, t) is assumed in the following form:

(7.53)

that satisfies the simply supported boundary conditions in Eqs. 7.16. Substitution of Eq. 7.53 in Eq. 7.52 yields

the frequency equation as follows

(7.54)

where

,

and . Note that is the circular frequency.

The non-dimensional frequency parameter can be computed for a particular mode shape m and n for

various values of aspect ratio (a/b), stiffness ratios and inplane loads. From Eq. 7.54, it is evident that a criticalcombination of compressive inplane biaxial loads can reduce the frequency to zero. When the frequency is zero,the inplane loads correspond to the buckling loads of the plate. Further, it may be noted that the tensile inplane



loads will increase the frequency of the plate.

Two Opposite Loaded Edges Simply Supported

For a laminated plate, where the compressive edge load acts along the simply supported

edges x1=0,a and the unloaded edges x2=0, b may have any arbitrary boundary condition, a solution to Eq .

7.52 (note that ) can be assumed to be in the form

(7.55)

Substituting Eq. 7.55 in Eq. 7.52 yields

(7.56)

A solution to Eq. 7.56 can be obtained as follows

(7.57)

Substituting Eq. 7.57 in Eq. 7.56 we obtain

(7.58)

where

and



with

Equation 7.58 has four roots i.e., with

and

Thus the solution is

(7.59)

The coefficients Am, Bm, Cm and Dm are determined from boundary conditions at x2=0, b. For example, let us

consider the clamped edges along x2=0, b i.e.,

X2=0,b: w=w, 2=0 (7.60)

Combining Eqs. 7.59 and using Eqs. 7.60, we obtain the following homogeneous algebraic equations

(7.61)

The frequency equation is obtained from the condition that the determinant of the coefficients of Am, Bm,

Cm and Dm must vanish. This leads to

(7.62)

The critical value of N is computed by satisfying Eq. 7.62 for a particular m when the frequency

becomes zero.




Consider the transverse force vibration of a simply supported rectangular antisymmetrically laminated

cross-ply plate (see section 7.3.2), when subjected to compressive loads , and .

Equations 7.44 hold good except the third equation where 'q' is replaced by � �.The following displacement field

(7.63)

satisfy the boundary conditions in Eqs. 7.45. These, on substitution in Eqs. 7.44 modified as above, result in thefollowing homogeneous algebraic equations:

(7.64)

where



and .

The frequency relation is derived by vanishing the determinant of the coefficient matrix of Eq. 7.64 and is

given by

(7.65)

wher and kmn are defined in Eq. 7.54.

The critical buckling load corresponds to the lowest value of k that satisfies Eq. 7.65 when the frequency

is zero. The load parameters, kmn and nondimensional frequency parameters, are plotted against the aspect

ratio, a/b for simply supported antisymmetric cross-ply laminate as shown in Figs. 7.6 and 7.7, respectively.


Now consider the transverse free vibration of a simply supported rectangular antisymmetric angle-ply

laminated plate (vide section 7.3.3) which is subjected to compressive loads , and

. The third equation in Eqs. 7.48 is modified substituting - in place of q. Thedisplacement field that satisfies the boundary conditions in Eqs. 7.49 is assumed as

(7.66)

These displacement relations, when substituted in the modified Eqs. 7.48 yield the following homogeneous



algebraic equations:

(7.67)

where

The condition that the determinant of the coefficient matrix in Eq.7.67 vanishes, determines the frequency

equation as follows:

(7.68)

where



Figures 7.8 and 7.9 depict the variation of bucklin

g load parameters and Figure 7.10 shows that of the frequency parameter for simply supportedantisymmetric angle-ply laminated square plates.

7.5 SHEAR BUCKLING OF COMPOSITE PLATE

A closed form solution for the shear buckling of a finite composite plate does not exist. This is true also

for an isotropic panel. Here the solution of a long specially orthotropic composite plate is considered. Considerthe plate to be infinite along the x1 direction and is subjected to uniform shear along the edges x2 = �b/2 (Fig.

7.11). In absence of inertia and other loads except the edge shear , Eq. 7.12 becomes

(7.69)

Assuming the solution to be of the form

(7.70)

where k is a longitudinal wave parameter and . Substituting Eq. 7.70 in Eq.7.69 we obtain

(7.71)

A solution to Eq.7.71 is assumed to be of the form

(7.72)

which on substitution in Eq. 7.71 yields the following characteristic equation

(7.73)

Equation 7.73 has four roots and the solution to Eq. 7.69 is written as follows:



(7.74)

Combining Eqs. 7.70 and 7.74, the solution is obtained as

(7.75)

The substitution of Eq. 7.75 in the specified boundary conditions at the edges x2 = �b/2 will result four

homogeneous algebraic equations in terms of the four coefficients A, B, C and D. For a non-trivial solution, the

determinant of this coefficient matrix must vanish. This condition yields the equation for the shear buckling

problems. The critical buckling load corresponds to the minimum value of at particular value of k.

7.6 RITZ METHOD

The Ritz method (also known as the Rayleigh-Ritz method), in most cases, leads to an approximate

analytical solution unless the chosen displacement configurations satisfy all the kinematic boundary conditions and

compatability conditions within the body. It is developed minimizing energy functional on the

basis of energy principles. The principle of minimum potential energy can be used for formulation of staticbending and buckling problems. The free vibration problem is formulated using Hamilton 's principle.

The total strain energy of a general laminated plate is given by (Fig. 7.1)

(7.76)

Substituting Eq. 6.59 in Eq . 7.76, one obtains

(7.77)

Substituting Eqs.6.47 through 6.49 in Eq. 7.77, and using Eqs. 6.53, 6.61 and 6.62 we obtain



(7.78)

The potential energy of external surface tractions and edge loads due to the deflections of the plate is given by

(7.79)

The kinetic energy of the laminated plate can be expressed as

(7.80)

where P, R, and I are defined in Eqs. 7.6.

The Ritz method can be utilized for seeking solution to a particular problem. The approximate

displacement functions are chosen to be in the following form

(7.81)



where the shape functions u1i(x1, x2), u2i(x1, x2) and wi(x1, x2) are capable of defining the actual deflection

surface and satisfy individually atleast the geometric boundary conditions, and a1, b1, and c1 are undetermined

constants.

For the bending of a general laminated plate, the energy functions, is defined as

(7.82)

where U and V are represented in Eqs 7.78 and 7.79, respectively.

The principle of minimum potential energy leads to the following conditions.

(7.83)

that provide m + n + r simultaneous algebraic equations for the computation of m+n+r unknown coefficients ai,

bi, and ci. The approximate solution is thus obtained by substituting these coefficients in the assumed

displacement functions in Eqs 7.81.

For the solution of plate buckling problem, only the edge loads are retained assuming ,

, in the expression for the potential energy V in Eqs. 7.79 and 7.82. The applicationof conditions in Eq. 7.83 results a set of m+n+r algebraic homogeneous equations in terms of m+n +r coefficients

ai, bi, and ci. The vanishing of the determinant of the coefficient matrix yields the buckling equation from which

critical buckling loads are determined.

For the free vibration problem, the displacement functions in Eqs. 7.81 can be modified to include thetime dependence as follows:

(7.84)

where the U1(x1, x2), U2(x1, x2), and W(x1, x2), correspond to the right-hand-side expressions in Eqs. 7.81.

The energy functional П* includes the strain energy U and kinetic energy T. Hence П* = U+T. In the absence of



surface forces and moments, edge loads and expansional stress resultants and moments, and substituting Eq.

7.84 in the above energy functional П* and carrying out the derivation of П*with respect to ai, bi, and ciand

equating them to zero i.e. , and , we obtain a set of m+n+r coefficients ai, bi, and

ci. The frequency equation is derived from the condition that the determinant of the coefficient matrix must vanish.

7.7 GALERKIN METHOD

The Galerkin method utilizes the governing differential equations of the problem and the principle ofvirtual work to formulate the variational problem. Here the virtual work of internal forces is obtained directlyfrom the differential equations without determining the strain energy. The Galerkin method appears to be moregeneral than the Ritz method and can be very effectively utilized to solve diverse general laminated plate bending

problems involving small and large deflection theories, linear and nonlinear vibration and stability of laminatedplates and so on.

Consider a general laminated plate (Fig. 7.1) to be in a state of static equilibrium under loads q1,q2 and q

only Then the governing differential equations in Eqs. 7.7 through 7.9 can be expressed as follows:

(7.85)

The equilibrium of the plate is obtained by integrating Eqs. 7.85 over the entire area of the plate. Note that, if

required, the edge loads and expansional force resultants and moments can also be included in Eqs.

7.85.

Assuming small arbitrary variations of the displacement functions and applying theprinciple of virtual work, we obtain the variational equations as follows:

(7.86)



As in the Ritz method, we assume the approximate displacement functions in Eqs. 7.81, where the shape

functions u1i(x1,x2), u2i(x1,x2) and wi(x1,x2), satisfy the displacement boundary conditions but not necessarily

the forced boundary conditions, in which case the method leads to an approximate solution.

Now,

(7.87)

Substitution of Eq. 7.87 in Eq. 7.86 yields

(7.88)

The variations of expansion coefficients are arbitary and not inter-related. Thisprovides m+n+r equations.



(7.89)

to determine m+n+r unknown coefficients ai, bi, and ci.

Note that, in a rigorous sense, the variational relations in Eqs. 7.86 are valid only, if the assumed

displacement functions are the exact solutions of the problem. Thus, when these displacements arekinematically admissible and satisfy all the prescribed boundary conditions and compatibility conditions within theplate, the method leads to an exact solution.

Equations 7.85 can be used for the buckling analysis of a laminated composite plate assuming q1= q2=0

and replacing �q� with � � where , and .Following Eqs. 7.86 through 7.89 we obtain the variational relations of the following form:

(7.90)

Equations 7.90 are a set of m+n+r homogeneous algebraic equations in terms of m+n+r coefficients ai, bi, and ci.

The condition that for a non-trivial solution, the determinant of the coefficient matrix should vanish yields thebuckling equation.

For the free vibration problem, the displacement functions are assumed to be of the form given in Eqs.7.84. Considering only the transverse inertia in Eqs. 7.7 through 7.9 and neglecting surface forces and moments,edge loads and expansional stress resultants and moments, we obtain, following the procedure as in the case ofbuckling above, the variational relations of the form



(7.91)

These are, again, a set of m+n+r homogeneous algebraic equations in terms of m+n+r coefficients ai, bi, and ci.

The frequency equation is established from the condition that the determinant of the coefficient matrix must vanishso as to obtain a non-trivial solution.

7.8 THIN LAMINATED BEAM THEORY

The small deflection bending theory of thin laminated composite beams can be developed based on

Bernoulli's assumptions for bending of an isotropic beam. Note that Kirchhoff's assumptions are essentially anextension of Bernoulli's assumptions to a two-dimensional plate problem. Hence the governing laminated plateequations as developed in earlier sections can be reduced to one-dimensional laminated beam equations.

Consider a thin laminated narrow beam of length L, unit width and thickness h (Fig. 7.12). The governingdifferential equations defined in Eqs. 77 through 7.9 reduce to the following two one-dimensional forms:

(7.92)

Consider the bending of a laminated composite shown in Fig. 7.12 under actions of transverse load q(x1)

only. Equations (7.92) assume the form

(7.93)

Consider, for example, the following simply supported boundary conditions at x1 = 0, L:

and (7.94)

Assume the displacement functions to be of the forms



(7.95)

that satisfy the boundary conditions in Eqs. 7.94. Assume the transverse load q(x1) as

(7.96)

Substituting Eqs. 7.95 and 7.96 in Eqs. 7.93 and carrying out the algebraic manipulation, we obtain

and (7.97)

where qm for a particular distribution of load q(x1) is obtained from

(7.98)

Equations 7.95 in conjunction with Eqs. 7.97 provide solution to the above beam bending problem. Note, that

for a uniformly distributed transverse load q0,

Next consider the free transverse vibration and buckling of a simply supported laminated beam. The

following governing differential equations are considered ( ; see Eqs. 7.92):

(7.99)



The displacement functions chosen are

(7.100)

that satisfy the boundary conditions defined by Eqs. 7.94. Substituting Eqs. 7.100 in Eqs. 7.99, we obtain two

algebraic homogeneous equations in terms of coefficients Um and Wm. For a non-trivial solution, the determinant

of the coefficient matrix must vanish. This yields the frequency equation to be in the form

(7.101)

Note that the critical buckling load, Ncr, corresponds to the minimum value of compressive force N for a specific

mode shape m, when the frequency is zero.

It is to be mentioned that the approximate analysis methods such as the Ritz method and Galerkin

method can be used to obtain solutions for laminated composite beams with various other support conditions forwhich closed form solutions may not be easily obtainable.

7.9 PLY STRAIN, PLY STRESS AND FIRST PLY FAILURE

Once the mid-plane displacement are determined, as discussed in the previous sections,

the mid-plane strains and curvatures k1, k2 and k6 are determined using Eqs. 6.47 and 6.48.

Next the strains for any ply located at a distance z from the mid-plane (see Fig. 6.16) are

computed utilizing Eqs. 6.49. Equations 6.50 are then employed to determine the ply stresses atthe same location. In some cases, it is required to determine the stresses in each ply, that correspond to thematerial axes x1' and x2' (Fig. 6.12). These are obtained using the following relations (see Eqs. A.11and A.19):



(7.102)

where m = cos � and n = sin �

In many practical design problems, the first ply failure is usually the design criterion. Once the stressesare determined in each ply of a laminate, one of the failure theories presented in section 6.14 is employed to

determine the load at which any one of the laminae in the laminated structure fails first ('first ply failure'). Thelaminate failure, however, corresponds to the load at which the progressive failure of all plies takes place. Theestimation of the laminate strength is more complex.

7.10 BIBLIOGRAPHY

1. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw Hill, NY, 1959.

2. S. G. Lekhnitskii, Anisotropic Plates, Gordon and Breach, N.Y., 1968

3. L.R.Calcote, Analysis of Laminated Composite Structure, Van Nostrand Rainfold, NY, 1969.

4. J.E. Ashton and J.M Whitney, Theory of Laminated Plates, Technomic Publishing Co., Inc., Lancaster,1984.

5. J.C. Halpin, Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., Lancaster, 1987.

6. J.M. Whitney, Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Co., Inc.,1987.

7. R.M. Jones, Mechanics of Composite Materials, McGraw Hill, NY 1975.

8. J.R. Vinson and T. �W, Chou, Composite Materials and their Use in Structures, Applied Science

Publishers, London, 1975.

9. K.T. Kedward and J.M. Whitney, Design Studies, Delware Composites Design Encyclopedia, Vol.5,

Technomic Publishing Co., Inc., Lancaster, 1990.

10. J.E. Ashton, Approximate Solutions for Unsymmetrically Laminated Plates, J. Composite Materials, 3,1969, p. 189.

7.11 EXERCISES

1. Derive the governing differential equations as defined in Eqs.7.7, 7.8 and 7.9.

2. Determine the deflection equation for a simply supported square (axa) symmetric laminated plate

subjected to a transverse load q=q0 x1/a.

3. Determine the deflection equation for a square (axa) symmetric laminated plate subjected to a

transverse load q=q0 x1/a when the edges at x1 = 0, a are simply supported and those at x2 = 0, b



are clamped.

4. A simply supported antisymmetric cross-ply laminated (00/900/00/900) kelvar/epoxy composite

square plate (0.5m x 0.5m x 5mm) is subjected to a uniformly distributed load of 500N/m2.Determine the deflection an dply stresses at the centre of the plate. Use properties listed in Table6.1.

5. A simply supported antisymmetric angle-ply laminated (450/-450/450/-450) carbon/epoxy composite

plate (0.75m x 0.5m x 5mm) is subjected to a uniformly distributed transverse load q0. Determine

the load at which the first ply failure occurs. Use the Tsai-Hill or Tsai-Wu strength criterion. See

Table 6.1 and also assume X '11t =1450 MPa, X '11

c =1080 MPa, X '22t =60 MPa,

X'22c =200 MPa and X '12= 80 MPa.

6. Determine the transverse natural frequencies for the plates defined in Problems 4 and 5 above.

Neglect the transverse load.

7. Determine the uniaxial compressive buckling loads for the plates defined in problem 4 and 5 above.

Neglect the transverse load.

8. Make a comparative assessment between the Ritz method and the Galerkin method.