The principle of virtual work states that a virtual (very small) change of the internal strain energy must be offset by an identical change in external work due to the applied loads, or:
 | (2–41) |
where:
| U = strain energy (internal work) = U1 + U2 |
| V = external work = V1 + V2 + V3 |
| δ = virtual operator |
The virtual strain energy is:
 | (2–42) |
where:
| {ε} = strain vector |
| {σ} = stress vector |
| vol = volume of element |
Continuing the derivation assuming linear materials and geometry, Equation 2–41 and Equation 2–42 are combined to give:
 | (2–43) |
The strains may be related to the nodal displacements by:
 | (2–44) |
where:
| [B] = strain-displacement matrix, based on the element shape functions |
| {u} = nodal displacement vector |
It will be assumed that all effects are in the global Cartesian system. Combining Equation 2–44 with Equation 2–43, and noting that {u} does not vary over the volume:
 | (2–45) |
Another form of virtual strain energy is when a surface moves against a distributed resistance, as in a foundation stiffness. This may be written as:
 | (2–46) |
where:
| {wn} = motion normal to the surface |
| {σ} = stress (or pressure) carried by the surface |
| areaf = area of the distributed resistance |
Both {wn} and {σ} will usually have only one nonzero component. The point-wise normal displacement is related to the nodal displacements by:
 | (2–47) |
where:
| [Nn] = matrix of shape functions for normal motions at the surface |
The stress, {σ}, is
 | (2–48) |
where:
| k = the foundation stiffness in units of force per length per unit area |
Combining Equation 2–46 through Equation 2–48, and assuming that k is constant over the area,
 | (2–49) |
Next, the external virtual work will be considered. The inertial effects will be studied first:
 | (2–50) |
where:
| {w} = vector of displacements of a general point |
| {Fa} = acceleration (D'Alembert) force vector |
According to Newton's second law:
 | (2–51) |
where:
| ρ = density (input as DENS on MP command) |
| t = time |
The displacements within the element are related to the nodal displacements by:
 | (2–52) |
where [N] = matrix of shape functions. Combining Equation 2–50, Equation 2–51, and Equation 2–52, and assuming that ρ is constant over the volume,
 | (2–53) |
The pressure force vector formulation starts with:
 | (2–54) |
where:
| {P} = the applied pressure vector (normally contains only one nonzero component) |
| area p = area over which pressure acts |
Combining equations Equation 2–52 and Equation 2–54,
 | (2–55) |
Unless otherwise noted, pressures are applied to the outside surface of each element and are normal to curved surfaces, if applicable.
Nodal forces applied to the element can be accounted for by:
 | (2–56) |
where:
|
Finally, Equation 2–41, Equation 2–45, Equation 2–49, Equation 2–53, Equation 2–55, and Equation 2–56 may be combined to give:
 | (2–57) |
Noting that the {δu}T vector is a set of arbitrary virtual displacements common in all of the above terms, the condition required to satisfy equation Equation 2–57 reduces to:
 | (2–58) |
where:
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Equation 2–58 represents the equilibrium equation on a one element basis.
The above matrices and load vectors were developed as “consistent”. Other formulations are possible. For example, if only diagonal terms for the mass matrix are requested (LUMPM,ON), the matrix is called “lumped” (see Lumped Matrices). For most lumped mass matrices, the rotational degrees of freedom (DOFs) are removed. If the rotational DOFs are requested to be removed (KEYOPT commands with certain elements), the matrix or load vector is called “reduced”. Thus, use of the reduced pressure load vector does not generate moments as part of the pressure load vector. Use of the consistent pressure load vector can cause erroneous internal moments in a structure. An example of this would be a thin circular cylinder under internal pressure modelled with irregular shaped shell elements. As suggested by Figure 2.3: Effects of Consistent Pressure Loading, the consistent pressure loading generates an erroneous moment for two adjacent elements of dissimilar size.
