**CHAPTER - 7**

**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**

** 7.4.1 Specially Orthotropic Plate**

** 7.4.2 Antisymmetric Cross-ply
Laminated Plate**

** 7.4.3 Antisymmetric Angle-Ply
Laminated Plate**

**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**

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:

- The material behaviour is linear and elastic.
- The plate is initially flat.
- The thickness of the plate is small compared to other dimensions.
- The translational displacements (and w) are small compared to the plate thickness, and the rotational displacements () are very small compared to unity.
- 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.
- 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 orientations
and 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 assumed
that 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 dx_{1 }dx_{2} (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, B_{ij} = 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 B_{ij} = 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 (D_{16 }
= D_{26} = 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 0^{0} or 90^{0}
fibre orientation, Eq. (7.13) is further simplified using Q_{16} = Q_{26}^{
}= 0.

For a homogeneous isotropic plate, Q_{11} = Q_{22}= E/(1-ν^{2}),
Q_{12} = ν Q_{1, }

Q_{66 }
= E/[2(1+ ν)] and Q_{16} = Q_{26} = 0 and hence Eq. 7.13 reduces
to

(7.14)

where
and expansion moments M^{e}’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 x_{1} and x_{2} 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 Rayleigh
Ritz method and Galerkin method.

** **

**7.3 BENDING OF LAMINATED PLATES**

**7.3.1 Specially Orthotropic Plate**

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 q_{mn} is determined for a specified distribution of
transverse load q (x_{1},x_{2} ) from the following integral:

(7.21)

It can be
shown that, for a uniformly distributed load q_{0},

(7.22)

where m, n are odd integers.

Hence for a specially orthotropic plate that is subjected to a transverse
uniformly distributed load q_{0}, the deflection w is given by

(7.23)

where m, n are odd integers.

The moments M_{1}, M_{2} and M_{6} and shear forces Q_{4}
and Q_{5} can be obtained from the following relations:

(7.24)

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 w_{p}(x_{1}), the
lateral load q(x_{1}) is assumed to have the same distribution in all
sections parallel to the x_{1}-axis and the plate is also considered
infinitely long the x_{2}-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
w_{h} (x_{1}, x_{2}) 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(x_{1}, x_{2})
to Eq. 7.15 can be expressed as follows:

(7.41)

The constant A_{m}, B_{m},Cm and Dm are determined
from the relevant boundry conditions at
. Note that q_{m }is determined from the loading
distribution q(x_{1}) using the following integration.

(7.42)

Hence for a uniformly distributed transverse load q_{0}

m = 1,3,5,… (7.43)

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

**7.3.2 Antisymmetric Cross-ply Laminated Plate**

** **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 A_{mn} , B_{mn }and W_{mn}. Solving these
equations, we obtain

(7.47)

where

and η = a/b

Using Eqs. 7.46 and 7.47, the stress and moment resultants (N_{1}, N_{2
}, N_{6 }, M_{1, }M_{2, }and M_{6} ) are
derived from Eqs. 6.52, and the shear forces Q_{4} and Q_{5} are
obtained from the last two relations of Eqs. 7.24.

Figure 7.3 exhibits the maximum nondimensional deflections (at x_{1}=a/2
and x_{12}=b/2) for simply supported antisymmetric cross-ply laminated
plates, which are plotted against the aspect ratio a/b. The deflections are
considerably higher in the case of a two-layered (n=1) plate because of the
extension-bending coupling (B_{11}). However, as the number of layers
increases, the coupling effect reduces and the results approach to that of an
orthotropic plate (n = ).

**7.3.3 Antisymmetric Angle-Ply Laminated Plate**

** **Next consider a rectangular angle-ply laminated composite
plate of stacking sequence [±Ø]_{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 and solving the resulting simultaneous algebraic equations we obtain

(7.51)

where

The values of maximum nondimensional deflections (at x_{1}=x_{2}=a/2)
for simply supported antisymmetric angle-ply laminated square plates are plotted
against the variation of Ø ranging from 0 to 45^{0}. A two-layered
laminate is found to exhibit much higher deflections due to higher values of
coupling terms B_{16} and B_{26} compared to the other cases.

**7.4 FREE VIBRATION AND BUCKLING**

**7.4.1 Specially Orthotropic Plate**

** **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 (x_{1},x_{2}, 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 critical combination 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.

For a laminated plate, where the compressive edge load
acts along the simply supported edges x_{1}=0,a and
the unloaded edges x_{2}=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 A_{m}, B_{m}, C_{m} and D_{m}
are determined from boundary conditions at x_{2}=0, b. For example, let
us consider the clamped edges along x_{2}=0, b i.e.,

X_{2}=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 A_{m}, B_{m}, C_{m}
and D_{m} 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.

**7.4.2 Antisymmetric Cross-ply Laminated Plate**

** **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 the following 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 k_{mn} 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, k_{mn}
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.

**7.4.3 Antisymmetric Angle-Ply Laminated Plate**

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. The displacement 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 supported antisymmetric 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.
Consider the plate to be infinite along the x_{1} direction and is
subjected to uniform shear along the edges x_{2} = ±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 x_{2}
= ±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.

** **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 static bending 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 *u _{1i}*(x

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 a_{i}, b_{i}, and c_{i. }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 application of conditions in Eq. 7.83 results a set of m+n+r algebraic
homogeneous equations in terms of m+n +r coefficients a_{i}, b_{i},
and c_{i}. 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 the time dependence as follows:

(7.84)

where the *U _{1}(x_{1}, x_{2})*,

** **The Galerkin method utilizes the governing differential
equations of the problem and the principle of virtual work to formulate the
variational problem. Here the virtual work of internal forces is obtained
directly from the differential equations without determining the strain energy.
The Galerkin method appears to be more general 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 laminated plates and so on.

Consider a general laminated plate (Fig. 7.1) to be in a state of
static equilibrium under loads q_{1,}q_{2 }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 the principle 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 *u _{1i}(x_{1},x_{2}),
u_{2i}(x_{1},x_{2}) *and

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. This provides m+n+r equations.

(7.89)

to determine m+n+r unknown coefficients a_{i}, b_{i}, and c_{i}.

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 are kinematically admissible and satisfy all the prescribed boundary conditions and compatibility conditions within the plate, the method leads to an exact solution.

Equations 7.85 can be used for the buckling analysis of a laminated
composite plate assuming q_{1}= q_{2}=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 a_{i}, b_{i}, and c_{i}. The
condition that for a non-trivial solution, the determinant of the coefficient
matrix should vanish yields the buckling 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 of buckling 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 a_{i}, b_{i}, and c_{i}. The
frequency equation is established from the condition that the determinant of the
coefficient matrix must vanish so 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 an
extension of Bernoulli's assumptions to a two-dimensional plate problem. Hence
the governing laminated plate equations 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 governing differential 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(x_{1}) only. Equations (7.92) assume
the form

(7.93)

Consider, for example, the following simply supported boundary conditions at x_{1}
= 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(x_{1}) 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 q_{m} for a particular distribution of load q(x_{1}) 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 q_{0,
}

_{ }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 U_{m} and W_{m}. 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, N_{cr}, 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 for which 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 k_{1}, k_{2 }and k_{6}
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 at the same location. In some cases, it is required to
determine the stresses in each ply, that correspond to the material axes x_{1}'
and x_{2}' (Fig. 6.12). These are obtained using the following relations
(see Eqs. A.11and A.19):

(7.102)

where m = cos Ø and n = sin Ø

In many practical design problems, the first ply failure is usually the design criterion. Once the stresses are 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'). The laminate failure, however, corresponds to the load at which the progressive failure of all plies takes place. The estimation of the laminate strength is more complex.

** **

- S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw Hill, NY, 1959.
- S. G. Lekhnitskii, Anisotropic Plates, Gordon and Breach, N.Y., 1968
- L.R.Calcote, Analysis of Laminated Composite Structure, Van Nostrand Rainfold, NY, 1969.
- J.E. Ashton and J.M Whitney, Theory of Laminated Plates, Technomic Publishing Co., Inc., Lancaster, 1984.
- J.C. Halpin, Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., Lancaster, 1987.
- J.M. Whitney, Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Co., Inc., 1987.
- R.M. Jones, Mechanics of Composite Materials, McGraw Hill, NY 1975.
- J.R. Vinson and T. –W, Chou, Composite Materials and their Use in Structures, Applied Science Publishers, London, 1975.
- K.T. Kedward and J.M. Whitney, Design Studies, Delware Composites Design Encyclopedia, Vol.5, Technomic Publishing Co., Inc., Lancaster, 1990.
- J.E. Ashton, Approximate Solutions for Unsymmetrically Laminated Plates, J. Composite Materials, 3, 1969, p. 189.

** **

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=q_{0} x_{1}/a.

3.
Determine the deflection equation for a square (axa) symmetric laminated
plate subjected to a transverse load q=q_{0} x_{1}/a when the
edges at x_{1} = 0, a are simply supported and those at x_{2} =
0, b are clamped.

4.
A simply supported antisymmetric cross-ply laminated (0^{0}/90^{0}/0^{0}/90^{0})
kelvar/epoxy composite square plate (0.5m x 0.5m x 5mm) is subjected to a
uniformly distributed load of 500N/m^{2}. Determine the deflection an
dply stresses at the centre of the plate. Use properties listed in Table 6.1.

5.
A simply supported antisymmetric angle-ply laminated (45^{0}/-45^{0}/45^{0}/-45^{0})
carbon/epoxy composite plate (0.75m x 0.5m x 5mm) is subjected to a uniformly
distributed transverse load q_{0}. 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 '_{11}^{t} =1450 MPa, X '_{11}^{c
}=1080 MPa, X '_{22}^{t }=60 MPa,

X'_{22}^{c} =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.