MPC Equations Rigid Elements-5!13!2010

Introduction to MPC Equations and Rigid

Elements

Glenn Grassi, MSC Software

May 2010

Agenda

• MSC.Nastran Set Definitions

• What are MPC Equations

• Forming and using MPC Equations

• Rigid Elements in MSC.Nastran

• Q & A

3

Special thanks to

Lance Proctor

Jim Swan

Jack Castro

for their contributions to this presentation.

MSC/MD Nastran Set Definitions

• Each degree of freedom of an MSC/MD Nastran analysis

model is defined as being a member of a “user-set”

• These set notations, such a G-set, M-set, S-set, etc are

identifications and classifications of how each degree of

freedom (dof) in the analysis participates in the solution

sequence

4

MSC/MD Nastran Set Definitions

• When a GRID entry is place in the bulk data section there

are (6) dof’s added into the model

• Other entries such as SPOINT or EPOINT, for example, will

add (1) dof in the model

• The collection of all dof’s that are entered into the model will

initially be label as belonging to the G-set

5

MD Nastran Set Definitions

• The dof’s that are developed are initially placed into the

G-set (Global set) in numerical order according to their ID

number

• If the follow bulk data entries were defined the G-set would

be defined as containing 13 dof’s

– GRID,1,,0.,0.,0.

– GRID,20,,1.,0.,0.

– SPOINT,2

6


• As the solution proceeds these dof’s may be relabeled and

transferred to other sets.

• If dof’s are constrained using multipoint constraint (MPC)

entries or rigid elements (RBAR, RBE2, RBE3, etc) then

some dof’s are relabeled as now belonging to the M-set

(Dependent Set)

7


• The remaining dof’s that are not defined in the M-set would

then be relabeled as belonging to the N-set (Independent

Set)

8

• Once a dof is labeled a belong to the M-set it cannot be

relabeled again. This is know as a “mutually exclusive” set

definition. In this example dof’s 7 and 9 belong to the M-set

and cannot be redefined.


• The dof’s in the N-set can be further reduced by adding

single point constraints (SPC, AUTOSPC, PS) into the

model.

9

• Dof’s 3 and 8 now joint dof’s 7 and 9 as being “mutually

exclusive”


• If a dof was incorrectly specified in the M-set and the S-set

then a Fatal Message (2101) would occur.

10


• The dof’s that are not constrained will be relabeled as

belonging to the F-set (Free).

11


• As the analysis progresses, further set reductions and

relabeling are possible

12

• The original 13 dof problem has been trimmed down to a 5

dof problem


• A complete description of user sets can be found in the

MSC/MD Nastran Reference Manual and Dynamic User

Guide.

13


14

• A printout of the user set definitions can be obtained by

including PARAM,USETPRT,2 in the analysis.

MPC Equations

15

• A MultiplePointConstraint Equation (MPC) is a linear

relationship between two or more degrees of freedom that

are expressed in the form

Σj Rj uj = 0

Where

• uj = any degree of freedom defined by a grid point

or an spoint

• Rj = user-defined scale factor

MPC Equations

16

• Multipoint constraints have many important practical

applications and can be used to Tie GRIDs together

• Determine relative motion between GRIDs

• Maintain separation between GRIDs

• Determine average motion between GRIDs

• Model bell-crank or control system

• Units conversion

UY6 = UY7

MPC Equations

17

Σj Rj uj = 0

= 0+

+

M-set

N-set

MPC Equations

18

• Simple example of an MPC Equation.

Y6 = Y7

MPC Equations

19

• Y6 = Y7

• 0 = Y7 - Y6 or 0 = - Y6 + Y7

Independent dof

Dependent dof (first one listed)

• Add coefficients

0 = 1.0 * Y7 – 1.0 * Y6

MPC 1 7 2 1.0 6 2 -1.0

MPC Equations

20

MPC Equations

21

MPC Equations

22

• By including MPCFORCE=ALL the MPC forces acting on the

grids can be printed.

MPC Equations

23

Angle ?

MPC Equations

24

• SPOINT has an ID for a single dof

• ID must be unique (cannot conflict with grids, epoints, etc)

• There are no directions associated with the SPOINT

– dof 0 is usually used in MPC equations

– dof 1 can also be used (cannot use 2 thru 6)

SPOINT 100

SPOINT 200 300

SPOINT 401 THRU 430

MPC Equations

25

• Calculate the relative angle in radians between GRID 6 and GRID 7 by

introducing an SPOINT 100

• Calculate the relative angle in degrees between GRID 6 and GRID 7 by

introducing an SPOINT 200

• MPC equation: SPOINT100 = RZ7 - RZ6 {SPOINT100 - RZ7 + RZ6 = 0 }

• MPC equation: SPOINT200 = SPOINT100 x 57.2958

{SPOINT200 - SPOINT100 x 57.2958 = 0 }

MPC 1 100 0 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

MPC Equations

26

MPC Equations

27

• SPOINT 100 = 7.238917E-03 (radians)

• SPOINT 200 = 4.147595E-01 (degrees)

How to enforce a 5.0 degree angle

MPC Equations

28

MPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

MPC Equations

29

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

30

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

31

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

32

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 100 0 -57.2958 200 0 1.0

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

33

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 100 0 -57.2958 200 0 1.0

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

34

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

35

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 7 6 -1.0 100 0 1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

36

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 7 6 -1.0 100 0 1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

37

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 7 6 -1.0 100 0 1.0

+ 6 6 1.0

MPC 1 100 0 -57.2958 200 0 1.0

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

38

MPC = 1

SPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 7 6 -1.0 100 0 1.0

+ 6 6 1.0

MPC 1 100 0 -57.2958 200 0 1.0

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPC 1 200 0 5.0

MPC Equations

39

Enforced a 5.0 degree angle

MPC Equations

40

MPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

MPC Equations

41

MPC = 400

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPCADD 400 1 2 3

MPC 1 7 2 1. 6 2 -1.

MPC 2 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 3 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

MPC Equations

42

Dependent DOF coefficient = -1.0

(pre-defined in MSC.Patran)

0 = 1.0 * Y7 - 1.0 * Y6 ( original )

0 = -1.0 * Y7 + 1.0 * Y6 ( modified x -1.0)

Y6 = Y7

MPC Equations

43

MSC.Patran does not define SPOINT’s

– Use Create/Node/Edit

Constrain all dof’s except the dof = 1

MPC Equations

44

MPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

SPOINT 100 200

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 0 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 0 1.0 100 0 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

MPC = 1

Begin Bulk

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

GRID 100 23456

GRID 200 23456

MPC 1 7 2 1. 6 2 -1.

MPC 1 100 1 1.0 7 6 -1.0

+ 6 6 1.0

MPC 1 200 1 1.0 100 1 -57.2958

$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------

MPC Equations

45

MPC Equations

46

MPC, 535, 2, 1, -1.0, 1, 1, +1.0

MPC, 535, 2, 2, -1.0, 1, 2, +1.0

MPC, 535, 2, 3, -1.0, 1, 3, +1.0

MPC, 535, 2, 4, -1.0, 1, 4, +1.0

MPC, 535, 2, 5, -1.0, 1, 5, +1.0

MPC, 535, 2, 6, -1.0, 1, 6, +1.0

0 = -UX2 + UX1

0 = -UY2 + UY2

0 = -UZ2 + UZ2

0 = - X2 + X1

0 = - Y2 + Y1

0 = - Z2 + Z1

Use of MPC to tie GRIDs together

1

2

MPC Equations

47

MPC used to Maintain Separation

• Enforce a separation between GRIDs

– Similar to using a gap

– Changes which DOF are dependent/independent

– Example:

– Initially 1” apart

– Keep separation = 0.25”

1

2

0.25”1.0”

48

MPC used to Maintain Separation

1

2

0.25

U1 = U2 + (desired – initial)

0 = -U1 + U2 + U1000

SPOINT,1000

MPC, 535, 1, 2, -1.0, 2, 2, +1.0

+, , 1000, 1, +1.0

SPC, 2002, 1000, 1, -0.75

1.00 desired

initial

total shrink =

2.0 x -0.375 = -0.75

Relative motion:

U1000 = U1 – U2

MPC Equations

49

Use of MPCs for AVERAGE Motion

• Determine average motion of DOFs

U1000 = (U1+ U2 + U3 + U4 +U5 +U6)/6

0 = -6*U1000 + U1+ U2 + U3 + U4

+U5 +U6

4

5

2

3

6

1

MPC Equations

50

MPCs as Bell-crank or Control System

• Output of 1 DOF scales another

U2 = U1/1.65

0 = -1.65*U2 + U12

1

1 +1.0-1.65 12 1MPC 535

C2 A2A1 G2G1 C1MPC SID

1.6

5

Rigid Elements

51

• The multipoint constraint, or MPC entry, provides the capability to model rigid

bodies and to represent other relationships which can be treated as rigid

constraints.

• The MPC entry provides considerable generality but lacks user convenience

since the user must supply all of the coefficients in the equations of constraint

• To enhance user convenience, nine rigid body elements (R-Type) are available

in MSC.Nastran.

• These elements require only the specification of the degrees-of-freedom that

are involved in the equations of constraint. All coefficients in these equations of

constraint are calculated internally in MSC.Nastran.

Rigid Elements

52

Not Exactly Rigid

- Averaging element

RBEs and MPCs

• Not necessarily “rigid” elements

– Working Definition:

The motion of a DOF is dependent on

the motion of at least one other DOF

Motion at one GRID drives another

• Simple Translation

X motion of Green Grid drives X motion

of Red Grid

Motion at one GRID drives another

• Simple Rotation

Rotation of Green Grid drives X translation

and Z rotation of Red Grid

Linear RBEs and MPCs

The motion of a DOF is dependent on

the motion of at least one other DOF

– Displacement, not elastic relationship

– Not dictated by stiffness, mass, or force

– Linear relationship

– Small displacement theory

– Dependent v. Independent DOFs

– Stiffness/mass/loads at dependent DOF

transferred to independent DOF(s)

Small Displacement Theory & Rotations

• Small displacement theory:

sin( ) ≈ tan( ) ≈

cos( ) ≈ 1

• For Rz @ A

RzB = RzA=

TxB = ( )*LAB

TyB = 0

X

Y

A

B

TxB

• Geometry-based– RBAR

– RBE2

• Geometry- & User-input based– RBE3

• User-input based– MPC

• Less Common “Rigid” elements (not covered today)– RBAR1, RJOINT, RROD, RTRPLT, RTRPLT1, RBE1,

RSSCON, RSPLINE

Commonly used “Rigid” Elements in MSC.Nastran

}Really-rigid “rigid” elements

Common Geometry-Based Rigid Elements

• RBAR

– Rigid Bar with six DOF at each

end

– RBE2

– Rigid body with

independent DOF at one

GRID, and dependent DOF

at an arbitrary number of

GRIDs.

The RBAR

• The RBAR is a rigid link between two GRID points

– Proper rigid body motion is preserved

The RBAR

– Can mix/match dependent DOF between the GRIDs, but this is rare

– The independent DOFs must be capable of describing the rigid body motion of the element

1234561234561 2RBAR 535

CMA CMBCNA CNBGA GBRBAR EID

– Most common to have all the

dependent DOFs at one GRID,

and all the independent DOFs at

the other

B

A

RBAR Example: Fastener

• Use of RBAR to “weld” two parts of a model

together:

1234561234561 2RBAR 535


B

A

RBAR Example: Pin-Joint

• Use of RBAR to form pin-jointed attachment

1231234561 2RBAR 535


B

A

RBAR definition in Patran

The RBE2

• One independent GRID (all 6 DOF)

• Multiple dependent GRID/DOFs

RBE2 Example

• Rigidly “weld” multiple GRIDs to one other GRID:

32RBE2 4110199 123456

GM5GM3GM2RBE2 GM4GM1GNEID CM

13

2

101

4

RBE2 Example

• Note: No relative motion between GRIDs 1-4 !

– No deformation of element(s) between these GRIDs

32RBE2 4110199 123456

GM5GM3GM2RBE2 GM4GM1GNEID CM

13

2

101

4

Common RBE2/RBAR Uses

• RBE2 or RBAR between 2 GRIDs

– “Weld” 2 different parts together

• 6DOF connection

– “Bolt” 2 different parts together

• 3DOF connection

• RBE2

– “Spider” or “wagon wheel” connections

– Large mass/base-drive connection

RBE2 definition in Patran

RBE3 Elements

• NOT a “rigid” element

• IS an interpolation element

• Does not add stiffness to the structure (if used

correctly)

• Motion at a dependent

GRID is the weighted

average of the motion(s) at

a set of master

(independent) GRIDs

RBE3 Description

RBE3 Description

• By default, the reference grid DOF will be the

dependent DOF

• Number of dependent DOF is equal to the number of

DOF on the REFC field

• Dependent DOF cannot be SPC’d, OMITted,

SUPORTed or be dependent on other RBE/MPC

elements

– PARAM,AUTOMSET,YES can resolve conflicts

RBE3 Is Not Rigid!

• RBE3 vs. RBE2– RBE3 allows warping

and 3D effects

– In this example, RBE2 enforces beam

theory (plane sections remain planar)

RBE3 RBE2

• Forces/moments applied at reference grid are distributed to the master grids in same manner as classical bolt pattern analysis

• Step 1: Applied loads are transferred to the CG of the weighted grid group using an equivalent Force/Moment

• Step 2: Applied loads at CG transferred to master grids according to each grid’s weighting factor

RBE3: How it Works? – Applied Forces


• Step 1: Transform force/moment at reference

grid to equivalent force/moment at the weighted

CG of master grids.

MCG=MA+FA*e

FCG=FA

CG

FCG

MCG

FA

MA

Reference Grid

e

CG


• Step 2: Move loads at CG to master grids

according to their weighting values.

• Force at CG divided amongst master grids according

to weighting factors Wi

• Moment at CG mapped as equivalent force couples on

master grids according to weighting factors Wi


• Step 2: Continued…

CG

FCG

MCG

Total force at each master node is sum of...

Forces derived from force at CG: Fif = FCG{Wi/ Wi}

F1m

F3mF2m

Plus Forces derived from moment at CG:

Fim = {McgWiri/(W1r12+W2r2

2+W3r32)}

RBE3: How it Works? – Mass Distribution

• Masses smeared to the master grids similar to

forces distribution• Mass is distributed to the master grids with weighting factors

• Rotational inertia is transferred to master grids

• Reference node inertial force is distributed in same manner as when

static force is applied to the reference grid.

Example 1

• RBE3 distribution of loads when force at reference grid at

CG passes through CG of master grids

Example 1: Force Through CG

• Simply supported beam

• 10 elements, 11 nodes numbered 1 through 11

• 100 LB. Force in negative Y on reference grid 99


• Load through CG with uniform weighting factors results in

uniform load distribution


• Comments…

• RBE3 Require 6 RBMODES

• x rotation DOF is added to satisfy equilibrium

Example 2

• Force does not pass thru CG of master grids

Example 2: Load not through CG

• The resulting force distribution is not intuitively

obvious

• Note forces in the opposite direction on the left side of the beam.

Upward loads on left

side of beam result

from moment caused

by movement of

applied load to the CG

of master grids.

Example 3

• Use of weighting factors to generate realistic

load distribution: 100 LB. transverse load on 3D

beam.

Example 3: Transverse Load on Beam

• If uniform weighting

factors are used, the

load is equally

distributed to all

grids.


Displacement Contour

• The uniform load distribution results in too much

transverse load in flanges causing them to droop.


• Assume quadratic distribution of

load in web

• Assume thin flanges carry zero

transverse load

• Master DOF 1235. DOF 5 added to

make RY rigid body motion

determinate

• Displacements with quadratic weighting factors

virtually equivalent to those from RBE2 (Beam

Theory), but do not impose “plane sections

remain planar” as does RBE2.



• RBE3 Displacement Contour

• Max Y disp=.00685


• RBE2 Displacement contour

• Max Y disp=.00685

Example 4

• Use RBE3 to get

“unconstrained” motion

• Cylinder under pressure

• Which Grid(s) do you

pick to constrain out

Rigid body motion, but

still allow for free

expansion due to

pressure?

Example 4: Use RBE3 for Unconstrained Motion

•Solution:• Use RBE3

• Move dependent DOF from reference grid to selected master

grids with UM option on RBE3 (otherwise, reference grid cannot

be SPC’d)

• Apply SPC to reference grid

• Since reference grid has 6 DOF, we must assign 6 “UM” DOF to a set of master grids

• Pick 3 points, forming a nice triangle for best numerical conditioning

• Select a total of 6 DOF over the three UM grids to determine the 6 rigid body motions of the RBE3

• Note: “M” is the NASTRAN DOF set name for dependent DOF


How Do I create UM set in Patran?

UX, UY, UZ onlyUy, Uz in cyl coord

sys is determinate

Reassign

Dependant

termsPick 3 nodes @

approx 120


• For circular geometry, it’s convenient to use

a cylindrical coordinate system for the

master grids.• Put THETA and Z DOF in UM set for each of the three UM grids

to determine RBE3 rigid body motion

Equation (consider avg x disp of grid 99)

Avg motion: U99x = (U1x + U2x + U3x) / 3

Default MPC: -3.*U99x + U1x + U2x + U3x =0

Rearrange UM: U1 + U2 + U3 - 3 * U99 =0

What is the UM?

• UM fields can be used to move the dependent DOF away from the reference grid

• For Example (in 1-D):

First term in MPC equation is dependent; Same equation, different order

99

1 2 3

99

1 2 3

Example 4: Use RBE3 for Unconstrained

Motion

“UM” Grids


• Result is free expansion due to internal pressure. (note: poisson effect causes shortening)

Example 4: Use RBE3 for Unconstrained

Motion

• Resulting MPC

Forces are

numeric zeroes

verifying that no

stiffness has

been added.

–PARAM,AUTOMSET,YES can also be

used in many instances instead of UM

RBE3 – Non Uniform Distribution – CHEXA(8)

Coefficients all 1.0

Coefficients 1.0, 0.5 and 0.25


Coefficients 1.0, 0.5 and 0.25


Stress and Deflection

Correct Stress = 2,500

Correct Disp = 2.5e-3

Max Stress 5,830

Max Disp = 2.86e-3



Coefficients: 1.0 -2.0 0.5 -1.0 0.25


Stress and Deflection

Correct Stress = 2,500

Correct Disp = 2.5e-3

Max Stress 11,600

Max Disp = 3.21e-3Coefficients all 1.0

Coefficients:

1.0 -2.0 0.5 -1.0 0.25

RBE3: Additional Reading

• Recommended TANs– TAN#: 2402 RBE3 - The Interpolation Element.

– TAN#: 3280 RBE3 ELEMENT CHANGES IN VERSION 70.5, improved diagnostics

– TAN#: 4155 RBE3 ELEMENT CHANGES IN VERSION 70.7

– TAN#: 4494 Mathematical Specification of the Modern RBE3 Element

– TAN#: 4497 AN ECONOMICAL METHOD TO EVALUATE RBE3 ELEMENTS IN LARGE-SIZE

MODELS

http://simcompanion.mscsoftware.com)

–Visit SimCompanion

http://www.mechsolutions.com/support/knowbase/NASTRAN/tan/tan2402.html





http://simcompanion.mscsoftware.com/

Thank You

MPC Equations Rigid Elements-5!13!2010

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