Energies are available in the solution printout (by setting Item = VENG on the OUTPR command) or in postprocessing (by choosing items SENE, TENE, KENE, and AENE on the ETABLE command or using the PRENERGY command). For each element, is the potential energy (including strain energy), accessed with SENE or TENE on ETABLE:

(14–313)

is the kinematic energy, accessed with KENE on ETABLE (computed only for transient, harmonic and modal analyses):

(14–314)

is the artificial energy associated with hourglass control, accessed with AENE on ETABLE (SOLID45, PLANE182, SOLID185, SHELL181 only):

(14–315)

where:

NINT = number of integration points

{σ} = stress vector

{εel} = elastic strain vector

voli = volume of integration point i

= plastic strain energy

Es = stress stiffening energy

[Ke] = element stiffness/conductivity matrix

[Se] = element stress stiffness matrix

{ue} = element DOF vector

= time derivative of element DOF vector

[Me] = element mass matrix

NCS = total number of converged substeps

{γ} = hourglass strain energy defined in Flanagan and Belytschko([243]) due to one point integrations.

[Q] = hourglass control stiffness defined in Flanagan and Belytschko([243]).

As may be seen from the bottom part of Equation 14–313 as well as Equation 14–314, all types of DOFs are combined, e.g., SOLID5 using both UX, UY, UZ, TEMP, VOLT, and MAG DOF. An exception to this is the piezoelectric elements, described in Piezoelectrics, which do report energies by separate types of DOFs in the NMISC record of element results. See Eigenvalue and Eigenvector Extraction when complex frequencies are used. Also, if the bottom part of Equation 14–313 is used, any nonlinearities are ignored. Elements with other incomplete aspects with respect to energy are reported in Table 14.3: Exceptions for Element Energies.

Artificial energy has no physical meaning. It is used to control the hourglass mode introduced by reduced integration. The rule-of-thumb to check if the element is stable or not due to the use of reduced integration is if < 5% is true. When this inequality is true, the element using reduced integration is considered stable (i.e., functions the same way as fully integrated element).

Element type limitations for energy computation are given in Table 14.3: Exceptions for Element Energies.