Energies are available in the solution output (by setting
Item = ALL, ESOL, or VENG on the OUTPR command) or
by choosing the appropriate items (SENE, TENE, KENE, ASENE, PSENE, AKENE, PKENE, DENE, and WEXT)
via postprocessing commands.
The following energy topics are available:
Also see Example: Energy Calculations in Transient and Harmonic Analyses in the Structural Analysis Guide.
For each element, 
is the potential energy (including strain energy), also called stiffness
energy, accessed by item SENE or TENE:
 | (14–332) |
where:
| NINT = number of integration points |
| {σ} = stress vector |
| {εel} = elastic strain vector |
| vol i = volume of integration point i |
 = plastic strain energy |
Es = stress stiffening energy 
|
| [Ke] = element stiffness/conductivity matrix |
| [Se] = element stress stiffness matrix |
| {ue} = element displacement vector |
Complex results (see POST1 and POST26 – Complex Results Postprocessing) are obtained following a harmonic analysis (ANTYPE,HARM) or a damped modal analysis (ANTYPE,MODAL with MODOPT,QRDAMP,,,,CPLX, or MODOPT,DAMP).
At each substep, by default, separate calculations are performed for:
When EngCalc = YES on the HROUT command
(harmonic analysis) or MXPAND command (modal analysis), the potential energy
is calculated as:
 | (14–333) |
It is the average potential energy over the deformation cycle with period 
, where:
| Ω = imposed circular frequency in harmonic analysis (rad/s), or imaginary part of the complex eigenvalue in damped modal analysis |
 = element real displacement vector |
 = element imaginary displacement vector |
The amplitude potential energy, accessed with item ASENE, is expressed as:
 | (14–334) |
The peak potential energy, accessed with item PSENE, can be written as:
 | (14–335) |
Average and amplitude potential energies define a cosine wave, which is the representation of the instantaneous potential energy. For the total system, the equation is:
 | (14–336) |
The average potential energy of the total system is:
 | (14–337) |
The amplitude potential energy of the total system is:
 | (14–338) |
where the residual quantity is:
 | (14–339) |
The phase of the cosine wave of the instantaneous potential energy of the total system is:
 | (14–340) |
The peak potential energy of the total system is:
 | (14–341) |
In the above equations:
| N = number of elements |
| i = superscript representing the ith element |
| j = superscript representing the jth element |
In the POST1 postprocessor, 
, 
, 
, and 
can be printed using the PRENERGY command. In the POST26
postprocessor, 
is accessed using the ENERSOL command.
After a prestressed full harmonic analysis using the PSTRES command with
NLGEOM,OFF (Prestressed Full-Harmonic Analysis Using the PSTRES Command (Legacy
Procedure) in the Structural Analysis Guide), 
is used instead of 
in Equation 14–332 to Equation 14–340.
Average, amplitude, and peak potential energies are calculated according to Equation 14–332 through Equation 14–340 for the element listed in Table 14.3: Element Types Supporting Additional Energy Calculations with the following restrictions:
When applicable, the plastic strain energy, artificial hourglass/drill stiffness energy, contact stabilization energy, or artificial stabilization energy do not contribute.
For linear perturbation full harmonic and linear perturbation full modal, the stress stiffness does not contribute.
For coupled-field elements used in a piezoelectric analysis or an electrostatic-structural analysis, those restrictions do not apply but only averaged potential energy is available. In the real part of complex results, this energy is the real part of Equation 10–87, the average stored elastic energy. In the imaginary part of complex results, this energy is the imaginary part of Equation 10–87, the average elastic loss.
The thermal strain vector (Equation 2–3) does not contribute.
In transient, harmonic, and modal analyses, 
is the kinetic energy accessed by item KENE:
 | (14–342) |
where:
 = time derivative of element displacement vector (that is, element velocity
vector) |
| [Me] = element mass matrix |
In harmonic and modal analyses, the kinetic energy can also be expressed as:
 | (14–343) |
In the case of complex results, the kinetic energy is calculated separately using the real part and the imaginary part by default (similar to potential energy; see Potential Energy (or Stiffness Energy) for Complex Results).
For the elements listed in Table 14.3: Element Types Supporting Additional Energy Calculations, if
EngCalc = YES on the HROUT or
MXPAND commands, the kinetic energy is calculated based on:
 | (14–344) |
This is the average kinetic energy over the deformation cycle of period 
. The amplitude kinetic energy, accessed with item AKENE, is:
 | (14–345) |
Therefore, the peak kinetic energy, accessed with item PSENE, is:
 | (14–346) |
Average and amplitude kinetic energies define a cosine wave, which is the representation of the instantaneous kinetic energy. For the total system, the equation is:
 | (14–347) |
The average kinetic energy of the total system is:
 | (14–348) |
The amplitude kinetic energy of the total system is:
 | (14–349) |
where the residual quantity is:
 | (14–350) |
The phase of the cosine wave of the instantaneous kinetic energy of the total system is:
 | (14–351) |
The peak kinetic energy of the total system is:
 | (14–352) |
In the POST1 postprocessor, 
, 
, and 
can be printed using the PRENERGY command. In the POST26
postprocessor, 
is accessed using the ENERSOL command.
For a damped modal analysis, calculations use the absolute value of
the imaginary part of the complex eigenvalue 
defined in Equation 14–243, instead
of 
.
The following limiations apply:
Average, amplitude, and peak kinetic energies are available only for the elements listed in Table 14.3: Element Types Supporting Additional Energy Calculations.
When postprocessing the expansion pass of a substructuring or component mode synthesis (CMS) analysis, kinetic energy is available only if the OUTRES command with option
DSUBres= ALL was issued in the first load step of the use pass.
The damping energy, 
, is the energy dissipated by viscoelastic damping forces over the deformation
path:
 | (14–353) |
where:
| [Ce] = element damping matrix |
In a transient analysis, if EngCalc = YES on the
TRNOPT command, Equation 14–353 is
approximated by:
 | (14–354) |
where:
| NSP = Number of substeps |
 = element velocity vector at substep i
|
| t i = time at substep i |
| t i-1 = time at substep i-1 |
Numerical integration done by Equation 14–354 is consistent only if solution data are written to the database for every substep (OUTRES,ALL,ALL, OUTRES,ESOL,ALL, or OUTRES,VENG, ALL).
In the case of complex results, if EngCalc = YES on the
HROUT or MXPAND command, the damping energy is calculated
based on:
 | (14–355) |
For a damped modal analysis, calculations use the absolute value of
the imaginary part of the complex eigenvalue defined in Equation 14–243, instead
of .
The following limitations apply:
Damping energy is available only for the element listed in Table 14.3: Element Types Supporting Additional Energy Calculations and the coupled-field elements.
Damping energy associated with the gyroscopic matrix is not available.
Damping energy is not available when postprocessing a mode-superposition transient analysis.
Damping energy is not available when postprocessing a mode-superposition harmonic analysis based on the QR damped eigensolver (MODOPT,QRDAMP).
Damping energy is not available when postprocessing an expansion pass following a transient analysis that uses the substructuring or component mode synthesis (CMS) method.
In the case of complex results, damping energy is available when postprocessing an expansion pass of the substructuring or component modes synthesis (CMS) method only if the OUTRES command with option
DSUBres= ALL was issued in the first load step of the use pass. In a component modes synthesis analysis withElcalc= NO on the CMSOPT command, damping energy is available only if damping was introduced in the use pass with the global damping commands: ALPHAD, BETAD, DMPRAT or DMPSTR. IfElcalc= YES on CMSOPT, damping energy is available only if damping was introduced in the generation pass andSEMATR= 3 on the SEOPT command.
When external loads are applied on nodes or elements, the general form of the work done by external loads over the deformation path is:
 | (14–356) |
where:
| {u(t)} = the displacement vector of the whole system if {F(t)} is the vector of nodal loads |
| or |
| {u(t)} = the element displacement vector if {F(t)} is the vector of loads on an element |
In a transient analysis, if Item = ALL,
ESOL, or VENG on the OUTRES command and EngCalc =
YES on the TRNOPT command, Equation 14–356 is approximated with:
 | (14–357) |
where:
| NSP = number of substeps |
| {F i } = system nodal or element load vector at substep i |
Numerical integration done by Equation 14–357 is consistent only if solution data are written to the database for every substep (OUTRES,ALL,ALL, OUTRES,ESOL,ALL, or OUTRES,VENG, ALL).
In the case of complex results, if Item =
ALL, ESOL, or VENG on the OUTRES command and
EngCalc = YES on the HROUT or
MXPAND command, at each substep the work done by external loads is expressed
as:
 | (14–358) |
where:
| {u1} = real part of the system nodal or element displacement vector |
| {u2} = imaginary part of the system nodal or element displacement vector |
| {F1} = real part of the system nodal or element load vector |
| {F2} = imaginary part of the system nodal or element load vector |
The following limitations apply:
Work done by external load is available only for the element listed in Table 14.3: Element Types Supporting Additional Energy Calculations.
Work due to constraint displacements is not available.
Work done by external loads is not available when postprocessing a mode-superposition transient analysis.
For complex results, work due to element pressure with an imaginary component is supported only for the SF command (
Lab= PRES and non-zeroVALUE2) and SFE command (Lab= PRES andKVAL= 2) applied on surface elements SURF153, SURF154 and SURF159.Work done by external loads is not available when postprocessing a mode-superposition harmonic analysis if
MSUPkey= YES on the MXPAND command. IfMSUPkey= NO, work due to element loads is available in the expansion pass of the real part, but not the work due to nodal loads.Work done by external loads is not available when postprocessing an expansion pass following a transient or harmonic analysis that uses the substructuring or component mode synthesis (CMS) method.
is the artificial energy associated with hourglass control, accessed by AENE
(for elements SOLID45, PLANE182,
SOLID185, SHELL181 only):
 | (14–359) |
where:
| NCS = total number of converged substeps |
 = hourglass strain energy defined in Flanagan and Belytschko([243]) due to one point integrations |
| [Q] = hourglass control stiffness defined in Flanagan and Belytschko([243]) |
Artificial energy has no physical meaning. It is used to control the hourglass mode
introduced by reduced integration. Use this rule to check whether or not the element is stable
due to the use of reduced integration: if 
< 5% is true, the element is considered stable (that is, it functions the
same way as a fully integrated element).
The following table lists element types that support additional
energy calculations (requested via Item = ALL, ESOL, or VENG on the
OUTRES command).
The following table list element limitations for energy computations.