Assuming total displacement in a linear static analysis is:

(14–22)

where:

= total displacement

= deformable displacement

= rigid body motion

The corresponding acceleration vectors are: , and the general dynamic equation without damping is:

(14–23)

where:

= total mass matrix

= total stiffness matrix

= total external force

= total acceleration vector, normally

(14–24)

Substituting Equation 14–22 into Equation 14–23 results in

(14–25)

For a static analysis the following terms drop out:

- no deformable acceleration

- rigid body motion generates zero forces

and Equation 14–25 becomes

(14–26)

and can be calculated by solving Equation 14–26. It is worth noting a few facts in this static analysis. Since in a static analysis,Equation 14–24 can be viewed as:

(14–27)

This implies that the acceleration calculated from inertia relief only represents rigid body acceleration. If the structure is subject to rigid body motion only (an unconstrained or minimally constrained structure), deformable displacement is zero, . This condition leads to the computation of rigid body acceleration (Equation 14–8). When inertia relief loads are used with displacement boundary conditions (Equation 14–26), is no longer zero.

While inertia relief (IRLF) requires the use of supports to remove the rigid body modes from the model, automatic inertia relief (AIRL) allows the analysis of a genuinely free body without any support. The manual step of support definition and the influence of supports on the stress results are thus avoided. The results from IRLF and AIRL may exhibit different displacements, but the stress distribution should be very close, other than the side effects of the artificial supports.

Similar to IRLF, the purpose of AIRL is to balance the external forces applied on the structure by a free-body acceleration, to be determined. However the static balance equation cannot be solved directly given the absence of supports. As described here, an augmented linear system is formed, and Lagrangian multipliers are used to remove the free-body motions.

In the previous section, the total displacement was decomposed into a deformable component and a rigid-body motion component . Under the assumptions of no deformable acceleration and no strain energy induced by the rigid-body motion, it was shown that the general dynamic equation can be reformulated as a static problem:

(14–28)

The matrix [K] denotes a positive semi-definite stiffness matrix, and [M] denotes a positive definite mass matrix. By introducing [R], the matrix containing the n_rb rigid-body modes as column vectors, the free-body acceleration vector may be expressed with respect to the nodal free-body acceleration at a virtual reference point, denoted :

(14–29)

The static equilibrium balance is now expressed as:

(14–30)

Likewise, the free-body momentum balance can be expressed as:

(14–31)

According to the previously described decomposition of the total displacement , the deformable displacement does not contribute to the free-body motion of the structure. We choose to impose this property on the deformable displacement weighted by the mass:

(14–32)

Using Equation 14–30 and Equation 14–32, the following augmented system is obtained:

(14–33)

The introduction of off-diagonal blocks [M][R] and [R]^T[M] eliminates the rigid body modes from the model and lets us solve simultaneously for and using a sparse direct solver. A similar formulation is obtained when constraint equations exist in the model. Using the notations from Constraint Equations, the augmented system can be described as:

(14–34)

where:

and the constant unit time τ = 1 is introduced for the sake of dimensional homogeneity. The rigid body modes [R_I ] are extracted from the matrix [K*]. The new augmented system is obtained from the original Equation 14–30 and Equation 14–32 by applying the constraint equations according to the derivation made in Derivation of Matrix and Load Vector Operations, and by noticing that if and are respectively in the null space of [K] and [K*], then .

For a nonlinear static analysis, the first step is to calculate inertia relief acceleration (Equation 14–27) on the original geometry based on structural density at the beginning of the substep of a load step by Equation 14–8. The nonlinear equation to be solved (similar to Equation 14–16) is:

(14–35)

where

is the stress vector

is the finite element derivative matrix ()

The nonlinear Equation 14–35 is solved by Newton-Raphson iterative procedure as previously described. The nonlinearity can be material or large deformation (NLGEOM,ON) nonlinearity. In the case of NLGEOM,ON, the inertia force term is re-calculated at every substep to reflect the effects of updated geometry by:

(14–36)

while retains the same values as in the first substep, is recalculated by Equation 14–12 at every substep from the updated geometry to reflect the geometry change in NLGEOM,ON. Also, is calculated incrementally at every substep to account for follower force effects (if any) if NLGEOM,ON is issued.

The analysis process is the same as described in General Procedure for Linear Perturbation Analysis in the Structural Analysis Guide, but extra attention should be paid to the application of inertia relief loads.

You can include inertia relief in an eigenvalue buckling analysis by following the procedure described in Linear Perturbation Eigenvalue Buckling Analysis in the Structural Analysis Guide, paying special attention to the inclusion or exclusion of the inertia relief load (Equation 14–36).