An equivalent free-body analysis is performed if a static analysis (ANTYPE,STATIC) and inertia relief (IRLF,1) are used. This is a technique in which the applied forces and torques are balanced by inertial forces induced by an acceleration field. Consider the application of an acceleration field (to be determined) that precisely cancels or balances the applied loads:
(14–7) |
where:
= force components of the applied load vector |
= translational acceleration vector due to inertia relief (to be determined) |
ρ = density |
vol = volume of model |
= moment components of the applied load vector |
= rotational acceleration vector due to inertia relief (to be determined) |
x = vector cross product |
In the finite element implementation, the position vector {r} and the moment in the applied load vector are taken with respect to the center of mass. Equation 14–7 is rewritten in equivalent form as:
(14–8) |
where:
[M_{t}] = mass tensor for the entire finite element model |
[M_{r}] = mass moments and mass products of the inertia tensor for the entire finite element model |
When applicable, [M_{t}], [M_{r}], and other mass-related data are calculated from the total rigid body mass matrix. For more information, see Precise Calculation of Mass Related Information. When not applicable, these values are calculated using the equations in Mass-Related Information Calculation.
Once [M_{t}] and [M_{r}] are developed, then and in Equation 14–8 can be solved. The body forces that correspond to these accelerations are added to the user-imposed load vector, thereby making the net or resultant reaction forces zero. The output inertia relief summary includes (output as TRANSLATIONAL ACCELERATIONS) and (output as ROTATIONAL ACCELERATIONS). You may request only a mass summary for [M_{t}] and [M_{r}] (IRLF,-1).
The calculations for [M_{t}], [M_{r}], , and are made at every substep for every load step where they are requested, and reflect changes in material density and applied loads.
Several limitations apply:
Inertia relief is applicable only to the structural parts of linear analyses.
Element mass and/or density must be defined in the model.
In a model containing both 2-D and 3-D elements, only M_{t}(1,1) and M_{t}(2,2) in [M_{t}] and M_{r}(3,3) in [M_{r}] are correct in the precise mass summary. All other terms in [M_{t}] and [M_{r}] should be ignored. The acceleration balance is, however, correct.
Axisymmetric and generalized plane strain elements are not allowed.
If grounded gap elements are in the model, their status should not change from the original status. Otherwise, the exact kinematic constraints stated above might be violated.
The computation for [M_{t}] and [M_{r}] proceeds on an element-by-element basis:
(14–9) |
(14–10) |
in which [m_{e}] and [I_{e}] relate to individual elements, and the summations are for all elements in the model. The output `precision mass summary' includes components of [M_{t}] (labeled as TOTAL MASS) and [M_{r}] (MOMENTS AND PRODUCTS OF INERTIA TENSOR ABOUT ORIGIN).
The evaluation for components of [m_{e}] are simply obtained from a row-by-row summation applied to the element mass matrix over translational (x, y, z) degrees of freedom. It should be noted that [m_{e}] is a diagonal matrix (m_{xy} = 0, m_{xz} = 0, etc.). The computation for [I_{e}] is based on the following equation:
(14–11) |
where:
[M_{e}] = element mass matrix |
[b] = matrix which consists of nodal positions and unity components. It is a submatrix of the rigid body motion matrix [D], defined in Equation 14–288. |
[M_{e}] is dependent on the type of element under consideration. The description of the element mass matrices [M_{e}] is given in Derivation of Structural Matrices. The derivation for [b] comes about by comparing Equation 14–7 and Equation 14–8 on a per element basis, and eliminating to yield:
(14–12) |
where:
vol = element volume |
If the mass matrix in Equation 14–11 is derived in a consistent manner, the components in [I_{e}] are precise. This is demonstrated as follows. Consider the inertia tensor in standard form:
(14–13) |
which can be rewritten in product form:
(14–14) |
The matrix [Q] is the following skew-symmetric matrix:
(14–15) |
Next, the shape functions are introduced by way of their basic form,
(14–16) |
where:
[N] = matrix containing the shape functions |
Equation 14–15 and Equation 14–16 are combined to obtain:
(14–17) |
where:
(14–18) |
Inserting Equation 14–18 into Equation 14–14 leads to:
(14–19) |
Noting that the integral in Equation 14–19 is the consistent mass matrix for a solid element:
(14–20) |
It follows that Equation 14–11 is recovered from the combination of Equation 14–19 and Equation 14–20.
Equation 14–18 and Equation 14–20 apply to all solid elements (in 2-D, z = 0). For discrete elements, such as beams and shells, certain adjustments are made to [b] in order to account for moments of inertia corresponding to individual rotational degrees of freedom. For 3-D beams, for example, [b] takes the form:
(14–21) |
The [I_{e}] and [M_{r}] matrices are accurate when consistent mass matrices are used in Equation 14–11. However, the following limitation applies: