

Chapter 10
Approximate theories for solids with special shapes: Â rods, beams, membranes, plates and shells
10.6 Simplified versions of general shell theory Â flat plates and membranes
In many practical cases of interest the general equations of shell theory can be vastly simplified.Â In this section, we summarize the governing equations for a number of special solids, including equations governing behavior of flat plates, and membranes.
10.6.1 Flat plates with small outofplane deflections and negligible inplane loading
This is the simplest version of plate theory, and is used in most practical applications. The figure illustrates the problem to be solved.Â An initially flat plate, which has uniform (small) thickness h, Youngâ€™s modulus Â Poissonâ€™s ratio Â and mass density ,Â is subjected to a distributed force Â per unit area (acting vertically upwards).Â The edge of the plate can be constrained in various ways, as discussed in more detail below.Â Â We wish to determine its deformed shape, as well as the internal forces and moments in the plate.
All vector and tensor quantities can be expressed in a fixed Cartesian basis Â illustrated in the figure.Â The covariant and reciprocal basis vectors are identical so there is no need to use the system of raised and lowered indices that was needed in general shell theory. In addition, since the basis vectors are independent of position, the Christoffel symbols are all zero.Â We continue to use the convention that Greek subscripts can have values 1 or 2, while Latin subscripts may have values 1,2,3.
We make the following assumptions
The following (approximate) results can then be extracted from the general shell equations:
Kinematics: Â The curvature change tensor is , while the inplane strain tensor is . Â The normal vector to the deformed plate can be approximated as Â The displacement field in the plate is approximated as Â The (infinitesimal) strain field in the plate is approximated as .Â
Kinetics Â The external force is characterized by the force per unit area Â acting on the surface of the plate; Â The inplane stress tensor , so the internal forces are completely characterized by the internal moment tensor Â and transverse force tensor ; Â The components Â represent the vertical force per unit length acting on an internal plane perpendicular to the Â direction; Â The physical significance of Â is illustrated in the figure: Â characterizes the moment per unit length acting on planes inside the shell that are normal to the Â direction, while Â characterizes the moment per unit length acting on planes that are normal to .Â Note that Â represents a moment about the Â axis, while Â is a moment acting about the Â axis.
MomentCurvature relation reduces to
Equations of MotionÂ are approximated by Â (rotational inertia has been neglected). These equations can be combined to eliminate V
and can also be expressed in terms of the displacements as
Edge boundary conditions.Â The edge of the plate is characterized by a curve C that lies in the midplane of the shell, encircling Â in a counterclockwise sense.Â Â We let Â denote arclength measured around C from some convenient origin, and use Â and Â denote unit vectors tangent and normal to C.Â Elementary plate theory offers the following choices of boundary condition for each point on C:
More general boundary conditions, in which the edge of the plate is subjected to prescribed forces and moments, can also be derived from the results given in Section 10.5.8 if this is of interest.
Strain energy and kinetic energy of a flat plate: The formula for the strain energy and kinetic energy of the plate can be expressed in terms of displacements as
The second term in the integral for the kinetic energy represents the energy associated with the plateâ€™s outofplane rotation, and can be ignored in most practical applications.
10.6.2 Flat plates with small outofplane deflections and significant inplane loading
This version of plate theory is used to model plates that are subjected to substantial loading parallel to the plane of the plate (usually due to loads applied at its boundaries).Â The theory assumes that displacements are small enough to use linearized measures of strain, but includes nonlinear terms associated with the inplane loading in the equilibrium equations.Â The theory can be used to calculate buckling loads for plates, but does not accurately model their deformation if the buckling loads are exceeded.
The problem to be solved is stated in Sect 10.6.1.Â Â The majority of the governing equations are identical to those of standard plate theory, given in 10.6.1 Â the equations which need to be modified to account for transverse forces are listed below.
Kinematics: Â The inplane strain tensor is approximated as Â The displacement field in the plate is approximated as Â The (infinitesimal) strain field in the plate is approximated as , where the components of the curvature change tensor are given in 10.6.1.Â
Kinetics Â The external force consists of a force per unit area Â acting on the surface of the plate. Â The internal forces are characterized as described in 10.6.1, except that the stress resultant tensor Â is nonzero. The components of the stress resultant tensor can be interpreted as illustrated in the figure: Â represents the force per unit length, acting in the Â direction, on an internal plane perpendicular to the Â direction within the plate.Â Â
Stress resultantstrain and MomentCurvature relations Â Â Â Â Â
Equations of motion Â reduce to Â Â Â Â Â Â Â The second two equations can be combined to eliminate V
This result can also be expressed in terms of displacement as
Edge boundary conditions. The edge of the plate is characterized as described in Section 10.6.1.Â Boundary conditions for the transverse displacement , transverse force Â and internal moment Â are identical to those listed in 10.6.2.Â Â In addition, the inplane displacements or forces must satisfy the following boundary conditions:
10.6.3 Flat plates with small inplane and large transverse deflections (von Karman theory)
This version of plate theory is used to model plates that are subjected to substantial loading parallel to the plane of the plate (usually due to loads applied at its boundaries), and also experience substantial outofplane displacement.Â The theory uses a nonlinear strain measure to account for the inplane stretching that results from finite transverse displacement and rotation, and includes nonlinear terms associated with the inplane loading in the equilibrium equations.Â The theory can be used to estimate the shape of a buckled plate if the buckling loads are exceeded.Â Â
We make the following assumptions
Most of the governing equations of VonKarman plate theory are identical to those listed in the preceding section.Â Nevertheless, the full set of governing equations is give below for convenience.
Kinematics: Â The inplane strain tensor is approximated by .Â The additional, nonlinear, term in this expression is the main feature of VonKarman theory. Â The curvature change tensor has components Â The normal vector to the deformed plate can be approximated as Â The displacement field in the plate is approximated as Â The (infinitesimal) strain field in the plate is approximated as .Â
Kinetics Â The external force is characterized by the force per unit area Â acting on the surface of the plate. Â The transverse stress tensor , so the Internal forces are characterized by the inplane stress resultant tensor , the transverse force tensor Â and internal moment tensor .Â The physical significance of the components of these tenors is discussed in Sections 10.6.1 and 10.6.2
Stress resultantstrain and MomentCurvature relations Â Â Â Â Â
Equations of motion Â reduce to Â Â Â Â Â Â Â where the rotational inertia term has been neglected in the last equation.
Edge boundary conditions:
Alternative forms for the VonKarman equations: If the plate is in static equilibrium (so the velocity and acceleration of the plate is zero), the VonKarman equations can be written in a compact form by expressing the inplane forces Â in terms of an Airy stress function , following the procedure outlined for plane elasticity problems in Section 5.2.Â Â Â As a result, the governing equations can be reduced to a pair of coupled, nonlinear partial differential equations forÂ Â and .Â These formulas will not be given here, but can be found, e.g. in Timoshenko and WoinowskyKrieger, `Theory of Plates and Shells,â€™Â McGrawHill (1964).
10.6.4 Stretched, flat membrane with small outofplane deflections
This is a simplified version of the stretched plate theory outlined in 10.5, which can be used if the plate is so thin that internal bending moments can be neglected.Â Â The problem to be solved is illustrated in the figure.Â A `membraneâ€™ with Youngâ€™s modulus , Poissonâ€™s ratio , mass density , and thickness h is initially planar and lies in the plane perpendicular to the Â direction. The edge of the membrane is subjected to a load per unit length Â and prevented from moving transverse to the membrane.Â A force per unit area Â acts on the membrane, inducing a small, time dependent, transverse deflection .
Kinematics: Â The inplane strain tensor is approximated as Â The displacement field in the plate is approximated as .Â We assume that . Â The curvature of the membrane has components Â The (infinitesimal) strain field in the membrane is approximated as
Kinetics Â The external force consists of a force per unit area Â acting on the surface of the plate. Â The internal forces are characterized by the stress resultant tensor Â (the internal moments are neglected).
Stress resultantstrain relations Â Â Â Â Â Â Equations of motion are approximated as Â Â Â Â Â Â Â The second equation can also be expressed in terms of displacement as
Edge boundary conditions On the edge of the membrane
10.6.5 Membrane equations in cylindricalpolar coordinates
In this section, we rewrite the governing equations for a stretched membrane in a cylindricalpolar coordinate system, to provide a simple example of the use of general curvilinear coordinates.Â We reconsider the membrane described in the preceding section, but now assume that the membrane is circular, with radius R.
HEALTH WARNING: We use polar coordinates Â as the choice of curvilinear coordinates.Â However, all vector and tensor quantities will be expressed as covariant or contravariant components in the natural basis for this coordinate system, not as components in a cylindricalpolar basis of unit vectors .Â Â Â Â Â
Coordinate system and kinematic relations
Here the reciprocal basis has simply been written down by inspection Â you can readily verify that .Â Note that neither the natural basis vectors nor the reciprocal basis vectors are unit vectors.
where terms of order , etc have been neglected in the expression for .Â The reciprocal basis vectors can also be calculated, but are not required in the analysis to follow.
Notice that the curvature components all have different units Â this is because the basis vectors themselves have units. It is easy to check that the terms in the dyadic product Â all have correct units.
Equations of Motion: The general equations of motion for a shell are Â We proceed to simplify these for a flat membrane.
Boundary conditions:
Special case: membrane subjected to uniform biaxial inplane loading: If the membrane is subjected to a uniform radial force per unit length Â acting on its edge at r=R, the first two equations of motion and the boundary conditions are satisfied by .Â This corresponds to a state of uniform biaxial tension in the membrane.Â The equation of motion for the transverse deflection reduces toÂ


(c) A.F. Bower, 2008 