Derivation of Thick Plate Equations

Differential Form

Introduction

This notebook derives the governing differential equations for a thick, anisotropic plate. First-order shear deformation theory is used. This formulation is used in all of the MSC sandwich analyses.

The derivation is based on developing a general expression for the strain energy density. The govenering equations can then be derived using the variational approach.

Kinematic Equations

The assumed displacement field will include rotations about the midplane that are independent of the slope of the displacement w. The rotations are designated and . The assumed displacements are functions of x and y. The full displacement field is then

For small displacements, we use the conventional definitions of strain. The following vector is in the order .

strain =
{D[u[x,y,z],x],
D[v[x,y,z],y],
D[w[x,y],z],
D[w[x,y],y]+D[v[x,y,z],z],
D[w[x,y],x]+D[u[x,y,z],z],
D[u[x,y,z],y]+D[v[x,y,z],x]}

where an expression such as implies the first derivative of with respect to the first argument (x).

Strain Energy Density

Assume a plane stress stiffness matrix for an anisotropic (monoclinic) material

stiff = {{q[1,1],q[1,2],q[1,3],0,0,q[1,6]},
         {q[1,2],q[2,2],q[2,3],0,0,q[2,6]},
         {q[1,3],q[2,3],q[3,3],0,0,q[3,6]},
         {0,0,0,q[4,4],q[4,5],0},
         {0,0,0,q[4,5],q[5,5],0},
         {q[1,6],q[2,6],q[3,6],0,0,q[6,6]}};

Next, form the strain energy density in 3D. The substitutions that follow implicitely integrate through the thickness, so that A, B, and D have conventional lamination theory definitions. Note that the nature of the assumed displacement field precludes the coupling matrix B from ever appearing any of the final equations. The terms are effective transverse shear stiffnesses. Although the following operation implies direct of the properties to obtain these quantities, it is know that better accuracy will be obtained if effective properties are used that account for the true distribution of the interlaminar stress.

strainEnergy = Expand[strain . stiff . strain] /2/.
   {z^2*q[i_, j_] :→Subscript[D,ToString[i]<>ToString[j]],
    z*q[i_, j_] :→Subscript[B,ToString[i]<>ToString[j]],
    q[i_, j_] :→Subscript[A,ToString[i]<>ToString[j]]};

Work Contribution Due to Pressure Load

External work will be performed by an applied, pressure distribution P[x,y]. This can be expressed as follows.

In[28]:=

Out[28]=

Work Contribution Due to Interaction with Membrane Loads

Direct membrane forces result in work when they interact with finite out-of-plane rotations. The work can be conveniently derived as follows.

strain2 =
{D[w[x,y],x],
D[w[x,y],y]};

The applied membrane load intensities are .

Work Contribution Due to Kinetic Energy

For dynamic analysis, there is a kinetic energy term. The following is a vector of  derivatives of the displacements with respect to time (velocity).

ρ is the areal density of the plate (lb/in) while is the first moment of the density distributon through the thickness, =.

The kinetic energy density is then.

For a vibration problem, one can assume that displacments have the form w[x,y,t]=w[x,y] Sin[ω t]. In that case, the energy can be expressed as

where it is implied that all displacements are multiplied by Sin[ω t]

Equations for Deflection of a Plate Under an Applied Pressure Load

Variational approach for deriving the governing equations.

The operation results in the following three, coupled differential equations.

Equations for Linear Buckling

Variational approach for deriving the governing equations.

Linear buckling will occur when the following three coupled differential equations are satisfied for non-trivial (zero) displacements.

Equations for Vibration

Variational approach for deriving the governing equations.

The equations for steady-state vibration follow. The natural frequency is the value of ω that satisfies the coupled equations for non-trivial displacmements.

Created by Mathematica  (July 9, 2004)