In general, the equations that are solved for static linear analyses are:
 | (14–72) |
or
 | (14–73) |
where:
[K] = total stiffness or conductivity matrix =
 |
| {u} = nodal degree of freedom (DOF) vector |
| N = number of elements |
| [Ke] = element stiffness or conductivity matrix |
| {Fr} = nodal reaction load vector |
{Fa}, the total applied load vector, is defined by:
 | (14–74) |
where:
| {Fnd} = applied nodal load vector |
| {Fe} = total of all element load vector effects (pressure, acceleration, thermal, gravity) |
Equation 14–72 thru Equation 14–74 are similar to Equation 15–1 thru Equation 15–4.
If sufficient boundary conditions are specified on {u} to guarantee a unique solution, Equation 14–72 can be solved to obtain the node DOF values at each node in the model.
Rewriting Equation 14–73 for linear analyses by separating out the matrix and vectors into those DOFs with and without imposed values,
 | (14–75) |
where:
| s = subscript representing DOFs with imposed values (specified DOFs) |
| c = subscript representing DOFs without imposed values (computed DOFs) |
Note that {us} is known, but not necessarily equal to {0}. Since the reactions at DOFs without imposed values must be zero, Equation 14–75 can be written as:
 | (14–76) |
The top part of Equation 14–76 may be solved for {uc}:
 | (14–77) |
The actual numerical solution process is not as indicated here but is done more efficiently using one of the various equation solvers discussed in Equation Solvers.
The reaction vector
, may be developed for linear models
from the bottom part of Equation 14–76:
 | (14–78) |
where:
 = reaction forces (output using either OUTPR,RSOL or PRRSOL command) |
Alternatively, the nodal reaction load vector may be considered over all DOFs by combining Equation 14–73 and Equation 14–74 to get:
 | (14–79) |
where only the loads at imposed DOF are output. Where applicable, the transient/dynamic effects are added:
 | (14–80) |
where:
| [M] = total mass matrix |
| [C] = total damping or conductivity matrix |
 ,
 = defined below |
The element static nodal loads are:
 | (14–81) |
where:
 = element nodal loads (output using OUTPR,NLOAD, or PRESOL commands) |
| e = subscript for element matrices and load vectors |
The element damping and inertial loads are:
 | (14–82) |
 | (14–83) |
where:
 = element damping nodal
load (output using OUTPR,NLOAD, or PRESOL commands) |
 = element inertial nodal
load (output using OUTPR,NLOAD, or PRESOL commands) |
Thus,
 | (14–84) |
The derivatives of the nodal DOF with respect to time are:
 = first derivative of the nodal DOF with
respect to time, e.g., velocity |
 = second derivative of the nodal DOF with
respect to time, e.g., acceleration |
Transient Analysis and Harmonic Analysis discuss the transient and harmonic damping and inertia loads.
If an imposed DOF value is part of a constraint equation, the nodal reaction load vector is further modified using the appropriate terms of the right hand side of Equation 14–194; that is, the forces on the non-unique DOFs are summed into the unique DOF (the one with the imposed DOF value) to give the total reaction force acting on that DOF.
The following circumstances could cause a disequilibrium, usually a moment disequilibrium:
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The following circumstances could cause an apparent disequilibrium:
All nodal coordinate systems are not parallel to the global Cartesian coordinate system. However, if all nodal forces are rotated to the global Cartesian coordinate system, equilibrium should be seen to be satisfied.
The solution is not converged. This applies to the potential discrepancy between applied and internal element forces in a nonlinear analysis.
The mesh is too coarse. This may manifest itself for elements where there is an element force printout at the nodes, such as SHELL61 (axisymmetric-harmonic structural shell).
The “TOTAL” of the moments (MX, MY, MZ) given with the reaction forces does not necessarily represent equilibrium. It only represents the sum of all applicable moments. Moment equilibrium would also need the effects of forces taken about an arbitrary point.
Axisymmetric models are used with forces or pressures with a radial component. These loads will often be partially equilibrated by hoop stresses, which do not show up in the reaction forces.
Shell elements have an elastic foundation described. The load carried by the elastic foundation is not seen in the reaction forces.
In substructure expansion pass with the resolve method used, the reaction forces at the master degree of freedom are different from that given by the backsubstitution method (see Substructuring Analysis).