Post-processing

Using shell elements, I asked for strain Time History output (RunnameT01) and Animation files, but the values remain equal to zero; why?

The strain tensor is not computed by default; it must be asked for in the Radioss input file (Runname_0000.rad) by setting flag Istrain (flag to compute strains for post-processing) to 1 in option /DEF_SHELL or in the shell property set.

On the contrary, the strain tensor is always computed and available for /MAT/LAW25 (COMPSH) and /MAT/LAW27 (PLAS_BRIT).

/ANIM/ELEM, BRICK or /SHELL/EPSD & Variable EPSD for a group of shells or 3-node shells for Time History: I asked for strain rate output but the values remain equal to zero into post-processors; why?

Strain rate filtering needs to be activated (Fsmooth=1), but it is available for most material laws but not for all.

It is not possible to get these outputs if the material law does not allow filtering (or “smooth”) the strain rate. On the contrary, using Fsmooth =1 and Fcut =1.E+30 will allow for all these laws to get these outputs without filtering the strain rate (indeed, filtering is activated but the cut-off frequency is so high that no filtering happens at all).

In certain cases, the outputs are also available even if strain rate filtering had not been asked for (Fsmooth =0).

This variable EPSD is available for both Animations and Time History in case of shell elements; it is only available for Animations in case of solid elements.

Output to ANIM or Time History files of strain rate for shell elements.
Table 1. 3- and 4-Node SHELL Elements
Material Law Available with V51? Available with V90?
LAW2 In any case In any case
LAW15 If Fsmooth=1 If Fsmooth=1
LAW25 If Fsmooth=1 If Fsmooth=1
LAW27 If Fsmooth=1 If Fsmooth=1
LAW36 In any case In any case
LAW44 If Fsmooth=1 If Fsmooth=1
LAW48 If Fsmooth=1 If Fsmooth=1
Output to ANIM files of strain rate for solid elements.
Table 2. SOLID Elements
Material Law Available with V51? Available with V90?
LAW2 In any case In any case
LAW36 In any case In any case
LAW44 If Fsmooth=1 If Fsmooth=1
LAW48 If Fsmooth=1 If Fsmooth=1
LAW50 If Fsmooth=1 If Fsmooth=1

What are the stresses SIGX, SIGY, and VONM in Animation files if I use integration points for the shells?

The stresses SIGX, SIGY… in Animation files represent the mean stresses through the thickness of the shell element. The VONM stress represents the von Mises criteria applied to these mean stresses SIGX, SIGY… In the same way the stresses F1, F2, F12, Q1 and Q2 given in Time History correspond to these mean stresses.

These mean stresses are computed by summation of the stresses at each integration point, averaged by the integration weights (refer to Integration Points Throughout the Thickness in the Radioss Theory Manual). They are used for the internal forces calculation.(1)
σ = k = 1 n w k σ k

Which value is output when using /ANIM/ELEM/EPSP when using different element type?

/ANIM/ELEM/EPSP output the plastic strain of element.
  • For Bricks

    It is the mean value calculated using relative volumes of the different integration points.

  • For Quads

    /ANIM/ELEM/EPSP is not available for Quad element. No value will be output.

  • For Shells
    • The plastic strain at the middle integration point is output. When an even number of integration points are requested, then the N/2 + 1 integration point is output.
    • For 4-node shell element with Ishell=12 (QBAT) element formulation, the mean value of EPSP of the 4 in-plane gauss points of the middle integration point is output.
    • It is recommended to use /ANIM/SHELL/EPSP/Keyword4, to get the plastic strain results at the upper and lower integration points. Especially in bending, the plastic strain in mid-layer will be less than the outer integration points.
  • For Beams

    It is the mean value calculated using the relative areas of the different integration points.

What is the output to Animation files with /ANIM/ … /ENER?

The specific energy per mass unit.

What is the output to Animation files with /ANIM/ … /HOUR?

The Hourglass energy per mass unit.

Using shell elements with QEPH formulation (Ishell=24), the hourglass energy of the part and the subset are not equal to zero in Time History; why?

When looking to the SUBSET or the PART in Time History, the hourglass energy is not zero.

This is because energy absorbed due to the numerical damping is output there. This means, in output the place of hourglass energy has been used to present this viscous energy.

The viscous energy is related to coefficient dn for shell property which using QEPH (Ishell =24) and QBAT and DKT18 (Ishell =12 or Ish3n =30).

The energy corresponding to the physical stabilization of hourglass is counted as internal energy for this formulation.

Using /ANIM/GZIP the Animation files are not readable; why?

This option uses the Gnu tool: GZIP which is normally available on Linux. Verify that it is installed correctly on the machine Radioss is running on. On Windows, GZIP is included with the HyperWorks installation.

What is the difference between /ANIM and /OUTP for EPSP output?

Runname_nnnn.sty files contain both membrane and max (over the integration points through the thickness) values; whereas Annn files contain only membrane value.

Is it possible to get more (or less) Animation files while a computation is running?

Yes it is possible to write an Animation file by writing a control file in the data directory.

For the run number nn (/RUN/Runname/nn in the Radioss Engine input file), you have to write the file Runname_nn_0000_[C].rst with the process /ANIM in it.

Radioss Engine writes an Animation file at this time.

The other options available with control files are described in the Control File (C-File) file.

In order to change the Animation files writing frequency, you have to stop your Radioss computation while writing a RESTART file, by using a control file (option /STOP). Then you can chain a second run with a different frequency for the Animation files writing.

How can I plot deleted elements to understand the propagation of a fracture?

Select Preferences > Option > Visualization menu in HyperView to display Eroded Elements.

This will help you to understand the propagation of a fracture.

How is the generalized stress tensor /ANIM/SHELL/TENS/MEMB and /ANIM/SHELL/TENS/BEND computed?

The generalized membrane and bending stress tensor is computed for each plane (layer) according the deformation, the bending behavior of the shell elements, and the material law

For the shell property /PROP/TYPE1 (SHELL) or /PROP/TYPE9 (SH_ORTH):
  • For global integration, (N=0)
    The exact computation is done from the generalized strain tensor and the result will correspond to:(2)
    σ g = { M e m b r a n e s t r e s s = 1 t t / 2 + t / 2 σ d z B e n d i n g s t r e s s = 1 t 2 t / 2 + t / 2 σ z d z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaC4Wd8aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacqGH9aqpdaGa daWdaeaafaqabeGabaaabaWdbiaad2eacaWGLbGaamyBaiaadkgaca WGYbGaamyyaiaad6gacaWGLbqbaeqabeqaaaqaaaaacaWGZbGaamiD aiaadkhacaWGLbGaam4CaiaadohacqGH9aqpdaWcaaWdaeaapeGaaG ymaaWdaeaapeGaamiDaaaadaGfWbqabSWdaeaapeGaeyOeI0IaamiD aiaac+cacaaIYaaapaqaa8qacqGHRaWkcaWG0bGaai4laiaaikdaa0 WdaeaapeGaey4kIipaaOGaaC4WdiaadsgacaWG6baapaqaa8qacaWG cbGaamyzaiaad6gacaWGKbGaamyAaiaad6gacaWGNbqbaeqabeqaaa qaaaaacaWGZbGaamiDaiaadkhacaWGLbGaam4CaiaadohacqGH9aqp daWcaaWdaeaapeGaaGymaaWdaeaapeGaamiDa8aadaahaaWcbeqaa8 qacaaIYaaaaaaakmaawahabeWcpaqaa8qacqGHsislcaWG0bGaai4l aiaaikdaa8aabaWdbiabgUcaRiaadshacaGGVaGaaGOmaaqdpaqaa8 qacqGHRiI8aaGccaWHdpGaamOEaiaadsgacaWG6baaaaGaay5Eaiaa w2haaaaa@76FD@
  • For multiple integration points through the thickness (N > 0)
    The generalized stress ( σ g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaC4Wd8aadaWgaaWcbaWdbiaadEgaa8aabeaaaaa@38A1@ ) is computed for each plane (layer) and integrated according to the defined weights, which includes the position through the thickness and the relative thickness.(3)
    σ g = { M e m b r a n e s t r e s s = i = 1 N w i N σ i B e n d i n g s t r e s s = i = 1 N w i M σ i } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaC4Wd8aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacqGH9aqpdaGa daWdaeaafaqabeGabaaabaWdbiaad2eacaWGLbGaamyBaiaadkgaca WGYbGaamyyaiaad6gacaWGLbqbaeqabeqaaaqaaaaacaWGZbGaamiD aiaadkhacaWGLbGaam4CaiaadohacqGH9aqpdaGfWbqabSWdaeaape GaamyAaiabg2da9iaaigdaa8aabaWdbiaad6eaa0WdaeaapeGaeyye IuoaaOGaam4DamaaDaaaleaacaWGPbaabaGaamOtaaaakiabgwSixl abeo8aZnaaBaaaleaacaWGPbaabeaaaOWdaeaapeGaamOqaiaadwga caWGUbGaamizaiaadMgacaWGUbGaam4zauaabeqabeaaaeaaaaGaam 4CaiaadshacaWGYbGaamyzaiaadohacaWGZbGaeyypa0ZaaybCaeqa l8aabaWdbiaadMgacqGH9aqpcaaIXaaapaqaa8qacaWGobaan8aaba WdbiabggHiLdaakiaadEhadaqhaaWcbaGaamyAaaqaaiaad2eaaaGc cqGHflY1cqaHdpWCdaWgaaWcbaGaamyAaaqabaaaaaGccaGL7bGaay zFaaaaaa@743B@
    Weight for membrane stress tensor ( w i N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4DamaaDaaaleaacaWGPbaabaGaamOtaaaaaaa@38F6@ ) computation.
    #point(s) w i N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4DamaaDaaaleaacaWGPbaabaGaamOtaaaaaaa@38F6@
    1 1.0000
    2 0.5000 0.5000
    3 0.2500 0.5000 0.2500
    4 0.1667 0.3333 0.3333 0.1667
    5 0.1250 0.2500 0.2500 0.2500 0.1250
    6 0.1000 0.2000 0.2000 0.2000 0.2000 0.1000
    7 0.0833 0.1667 0.1667 0.1667 0.1667 0.1667 0.0833
    8 0.0714 0.1429 0.1429 0.1429 0.1429 0.1429 0.1429 0.0714
    9 0.0625 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.0625
    10 0.0556 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.0556
    Weight for bending stress tensor ( w i M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4DamaaDaaaleaacaWGPbaabaGaamOtaaaaaaa@38F6@ ) computation.
    #point(s) w i M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4DamaaDaaaleaacaWGPbaabaGaamOtaaaaaaa@38F6@
    1 0.0000
    2 -0.0833 0.0833
    3 -0.0833 0.0000 0.0833
    4 -0.0648 -0.0556 0.0556 0.0648
    5 -0.0521 -0.0625 0.0000 0.0625 0.0521
    6 -0.0433 -0.0600 -0.0200 0.0200 0.0600 0.0433
    7 -0.0370 -0.0556 -0.0278 0.0000 0.0278 0.0556 0.03
    8 -0.0323 -0.0510 -0.0306 -0.0102 0.0102 0.0306 0.0510 0.0323
    9 -0.0286 -0.0469 -0.0313 -0.0156 0.0000 0.0156 0.0313 0.0469 0.0286
    10 -0.0257 -0.0432 -0.0309 -0.0185 -0.0062 0.0062 0.0185 0.0309 0.0432 0.0257

For the shell property defined by layers, /PROP/TYPE10 (SH_COMP), /PROP/TYPE11 (SH_SANDW), /PROP/TYPE17 (STACK).

The generalized stress ( σ g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaC4Wd8aadaWgaaWcbaWdbiaadEgaa8aabeaaaaa@38A1@ ) is computed for each layer and integrated according to the relative layer thickness (layer thickness/total thickness) and the position on the layer z (isotropic value of z: -0.5 < z < 0.5).(4)
σ g = { M e m b r a n e s t r e s s = i = 1 N t h i c k i t h i c k σ i B e n d i n g s t r e s s = i = 1 N t h i c k i t h i c k z i σ i } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaC4Wd8aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacqGH9aqpdaGa daWdaeaafaqabeGabaaabaWdbiaad2eacaWGLbGaamyBaiaadkgaca WGYbGaamyyaiaad6gacaWGLbqbaeqabeqaaaqaaaaacaWGZbGaamiD aiaadkhacaWGLbGaam4CaiaadohacqGH9aqpdaGfWbqabSWdaeaape GaamyAaiabg2da9iaaigdaa8aabaWdbiaad6eaa0WdaeaapeGaeyye IuoaaOWaaSaaaeaacaWG0bGaamiAaiaadMgacaWGJbGaam4AamaaBa aaleaacaWGPbaabeaaaOqaaiaadshacaWGObGaamyAaiaadogacaWG RbaaaiabgwSixlabeo8aZnaaBaaaleaacaWGPbaabeaaaOWdaeaape GaamOqaiaadwgacaWGUbGaamizaiaadMgacaWGUbGaam4zauaabeqa beaaaeaaaaGaam4CaiaadshacaWGYbGaamyzaiaadohacaWGZbGaey ypa0ZaaybCaeqal8aabaWdbiaadMgacqGH9aqpcaaIXaaapaqaa8qa caWGobaan8aabaWdbiabggHiLdaakmaalaaabaGaamiDaiaadIgaca WGPbGaam4yaiaadUgadaWgaaWcbaGaamyAaaqabaaakeaacaWG0bGa amiAaiaadMgacaWGJbGaam4AaaaacqGHflY1caWG6bWaaSbaaSqaai aadMgaaeqaaOGaeyyXICTaeq4Wdm3aaSbaaSqaaiaadMgaaeqaaaaa aOGaay5Eaiaaw2haaaaa@87D9@
Note:
  1. N=1 defines a membrane element. The bending stress tensor is zero.
  2. For fully integrated element shells (Ishell=12), the stress tensor output for each plane (layer) is the average value of the 4 Gauss points.
  3. For the property /PROP/TYPE51, several integration points through the thickness can be defined for each layer. The generalized stress computation will be done according to the shell property defined by the layers with different height through the thickness (z) using the layer position and the selected distribution through the layer formulation (/PROP/TYPE51).
  4. Integration points through shell thickness:
    Number of Integration Points Distribution of Integration Points Number of Layer
    /PROP/TYPE1 N

    ( 0 N 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaad6eacqGHKjYOcaaIXaGaaGimaiaaicdaaaa@3D1D@ )

    Lobatto integration scheme -
    /PROP/TYPE9 N

    ( 1 N 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaad6eacqGHKjYOcaaIXaGaaGimaiaaicdaaaa@3D1D@ )

    Lobatto integration scheme -
    /PROP/TYPE10 1 per layer Middle of layer N

    ( 0 N 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaad6eacqGHKjYOcaaIXaGaaGimaiaaicdaaaa@3D1D@ )

    /PROP/TYPE11 1 per layer Middle of layer N

    ( 0 N 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaad6eacqGHKjYOcaaIXaGaaGimaiaaicdaaaa@3D1D@ )

    /PROP/TYPE16 1 per layer Middle of layer N

    ( 0 N 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaad6eacqGHKjYOcaaIXaGaaGimaiaaicdaaaa@3D1D@ )

    /PROP/TYPE17 Npt_ply in /PROP/TYPE19

    ( 1 N p t _ p l y 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaad6eacaWGWbGaamiDaiaac+facaWGWbGaamiBaiaadMhacqGH KjYOcaaI5aaaaa@4167@ )

    Uniform integration scheme Pply_IDi

    ( 1 P p l y _ I D i n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaadcfacaWGWbGaamiBaiaadMhacaGGFbGaamysaiaadseacaWG PbGaeyizImQaamOBaaaa@4230@ )

    /PROP/TYPE51 Npt_ply in /PROP/TYPE19

    ( 1 N p t _ p l y 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaad6eacaWGWbGaamiDaiaac+facaWGWbGaamiBaiaadMhacqGH KjYOcaaI5aaaaa@4167@

    Iint=1:

    Uniform integration scheme

    Iint=2:

    Gauss integration scheme

    Pply_IDi

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    For the position and weight of the Gauss integration scheme and Lobatto integration scheme, refer to “Integration points throughout the thickness” in the Theory Manual.