RBEs and MPCs in MSC.Nastran

RBEs and MPCs in MSC.NastranRBEs and MPCs in MSC.Nastran

A Rip-Roarin’ Review of

Rigid Elements

RBEs and MPCsRBEs 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 anotherMotion at one GRID drives another

• Simple Translation

X motion of Green Grid drives X motion

of Red Grid

Motion at one GRID drives anotherMotion at one GRID drives another

• Simple Rotation

Rotation of Green Grid drives X translation

and Z rotation of Red Grid

RBEs and MPCsRBEs 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 & RotationsSmall Displacement Theory & Rotations

• Small displacement theory:sin() = tan() = cos() = 1

• For Rz @ ARzB = RzA=

TxB = (-)*LAB

TyB = 0 X

Y

A

B

-

TxB

• Geometry-based– RBAR– RBE2

• Geometry- & User-input based– RBE3

• User-input based– MPC

Typical “Rigid” Elements in MSC.NastranTypical “Rigid” Elements in MSC.Nastran

} Really-rigid “rigid” elements

Common Geometry-Based Rigid ElementsCommon 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 RBARThe RBAR

• The RBAR is a rigid link between two GRID points

The RBARThe 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: FastenerRBAR Example: Fastener

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

1234561234561 2RBAR 535


B

A

RBAR Example: Pin-JointRBAR Example: Pin-Joint

• Use of RBAR to form pin-jointed attachment

1231234561 2RBAR 535


B

A

The RBE2The RBE2

• One independent GRID (all 6 DOF)

• Multiple dependent GRID/DOFs

RBE2 ExampleRBE2 Example

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

32RBE2 4110199 123456

GM5GM3GM2RBE2 GM4GM1GNEID CM

13

2

101

4

RBE2 ExampleRBE2 Example

• Note: No relative motion between GRIDs 1-4 !– No deformation of element(s)

between these GRIDs

32RBE2 4110199 123456GM5GM3GM2RBE2 GM4GM1GNEID CM

13

2

101

4

Common RBE2/RBAR UsesCommon 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

RBE3 ElementsRBE3 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 DescriptionRBE3 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

U99 = (U1 + U2 + U3) / 3

3 * U99 = U1 + U2 + U3

-U1 = + U2 + U3 - 3 * U99


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

– For Example (in 1-D):

RBE3 Is Not Rigid!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

RBE3: How it Works?RBE3: How it Works?

• 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


• Step 1: Transform force/moment at reference grid to equivalent force/moment at 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/(W1r1

2+W2r22+W3r3

2)}


• Masses on reference grid are smeared to the master grids similar to how forces are distributed– Mass is distributed to the master grids according

to their weighting factors– Motion of reference mass results in inertial force

that gets transferred to master grids

– Reference node inertial force is distributed in same manner as when static force is applied to the reference grid.

Example 1Example 1

• RBE3 distribution of loads when force at reference grid at CG passes through CG of master grids

Example 1: Force Through CGExample 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

Example 1: Force Through CGExample 1: Force Through CG

• Load through CG with uniform weighting factors results in uniform load distribution

Example 1: Force Through CG Example 1: Force Through CG

• Comments…– Since master grids are co-linear, the x

rotation DOF is added so that master grids can determine all 6 rigid body motions, otherwise RBE3 would be singular

Example 2Example 2

• How does the RBE3 distribute loads when force on reference grid does not pass through CG of master grids?

Example 2: Load not through CGExample 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 3Example 3

• Use of weighting factors to generate realistic load distribution: 100 LB. transverse load on 3D beam.

Example 3: Transverse Load on BeamExample 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 4Example 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 Example 4: Use RBE3 for Unconstrained MotionUnconstrained 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


“UM” Grids


• 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


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


• Resulting MPC Forces are numeric zeroes verifying that no stiffness has been added.

Example 5Example 5

• Connect 3D model to stick model

• 3D model with 7 psi internal pressure

• Use RBE3 instead of RBE2 so that 3D model can expand naturally at interface.– RBE3 will also allow warping and other 3D

effects at the interface.

Example 5: 3D to Stick Model Example 5: 3D to Stick Model ConnectionConnection

• 120” diameter cylinder

• 7 psi internal pressure

• 10000 Lb. transverse load on stick model

• RBE3: Reference grid at center with 6 DOF, Master Grids with 3 translations


• Undeformed/Deformed plot shows continuity in motion of 3D and Beam model


• MPC forces at interface show effect of both the tip shear and interface moment.


• Shell outer fiber stresses at interface slightly higher than beam bending stresses– 3D effects – Shell model under

internal pressure and not bound by beam theory assumptions

Example 6Example 6

• Use RBE3 to see “beam” type modes from a complex model

• Sometimes it’s difficult to identify and describe modes of complex structures

• Solution: – Connect complex structure down to

centerline grids with RBE3. – Connect centerline grids with PLOTELs

Example 6: Using RBE3 to Visualize Example 6: Using RBE3 to Visualize “Beam” Modes“Beam” Modes

• Generic engine courtesy of Pratt & Whitney


• RBE3’s used to connect various components to centerline.

• Each component’s centerline grids connected by it’s own set of PLOTELs


• Complex Mode Animation


• Animation of the PLOTEL segments shows that this is a whirl mode

• Relative motion of various components more clearly seen

Example 7Example 7

• Use RBE3 to connect incompatible elements– Beam to plate– Beam to solid– Plate to solid

• Alternative to RSSCON

Example 7: RBE3 Connection of Example 7: RBE3 Connection of Incompatible Elements Incompatible Elements

Example 7: RBE3 Connection of Example 7: RBE3 Connection of Incompatible ElementsIncompatible Elements

• Use RBE3 to connect beams to plates at two corners

• Use RBE3 to connect beams to solids at two corners

• Use RBE3 to connect plates to solid– Plate thickness is same as solid thickness

in this example


• RBE3 connection of beams to plates– Map 6 DOF of beam into plate translation DOF– For best results, beam “footprint” should be similar to

RBE3 “footprint”, otherwise joint will be too stiff


• RBE3 connection of beams to solids– Map 6 DOF of beam into

solid translation DOF– For best results, beam

“footprint” should be similar to RBE3 “footprint”, otherwise joint will be too stiff


• RBE3 connection of plates to solids– Coupling of plate

drilling rotation to solid not recommended

– Plate and solid grids can be equivalent, coincident, or disjoint (as shown)


• Deformation contours show continuity at RBE3 interfaces


• Bending stress contours consistent across RBE3 interface

RBE3 Usage GuidelinesRBE3 Usage Guidelines

• Do not specify rotational DOF for master grids except when necessary to avoid singularity caused by a linear set of master grids

• Using rotational DOF on master grids can result in implausible results (see next two slides)


• Example: What can happen if master rotations included?– Modified RBE3 from Example 5– Displacements clearly incorrect when all 6

DOF listed for master grids (next page)


• Deformation with all 6 DOF specified for master grids at interface

• Deformation with 3 translation DOF specified for master grids (same loads/BC’s)


• Make check run with PARAM,CHECKOUT,YES – Section 9.4.1 of MSC.Nastran Reference Manual (V68)– EMH printout should be numeric zeroes (no grounding)

– No MAXRATIO error messages from decomposition of Rgmm

and Rmmm matrices (numerically stable)

• Perform grounding check of at least KGG and KNN matrix– V2001: Case control command

• GROUNDCHECK (SET=(G,N))=YES

– V70.7 and earlier: • Use CHECKA alters from SSSALTER library

RBE3: Additional ReadingRBE3: Additional Reading

• Much RBE3 information has been posted on MSC’s Knowledge Base– http://www.mechsolutions.com/support/knowbase/index.html

http://www.mechsolutions.com/support/knowbase/index.html





RBE3: Additional ReadingRBE3: 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://www.mechsolutions.com/support/knowbase/NASTRAN/tan/tan2402.html





User-Input based “Rigid” ElementsUser-Input based “Rigid” Elements

• MPCs– Most general-purpose way to define

motion-based relationships– Could be used in place of ALL other RBEi

• Lack of geometry makes this impractical

– Can be changed between SUBCASEs

MPC DefinitionMPC Definition

• “Rigid” elements– Definition: The motion of a DOF dependent

on the motion of (at least one) other DOF • Linear Relationship• One (1) dependent DOF• “n” independent DOF (n >= 1)

ajXi = a1X1 + a2X2 + a3X3+…+ anXn

General Approach For Use of MPCsGeneral Approach For Use of MPCs

• Write out desired displacement equality relationship on a per DOF level– Dependent motion = (your equation goes here)

0 = - Ux2 + Ux1

• Re-arrange so left-hand side is zero

• List dependent term first

Ux2 = Ux1

2

1

MPC FormatMPC Format

• For example:– Set X motion of GRID 2

= X motion of GRID 1

UX2 = UX1 0 = - UX2 + UX1

= (-1.)UX2 + (+1.)UX1

1 +1.0-1.0 12 1MPC 535

C2 A2A1 G2G1 C1MPC SID

2

1

General Approach to MPCsGeneral Approach to MPCs

• Write down relationship you want to impose on a per DOF level:

ajXi = a1X1 + a2X2 +…+ anXn

0 = -aiXi + a1X1 + a2X2+…+ anXn

• Move dependent term to 1st term on right hand side:

Why would I want to use an MPC?Why would I want to use an MPC?

• Tie GRIDs together (RBEi)

• Determine relative motion between GRIDs

• Maintain separation between GRIDs

• Determine average motion between GRIDs

• Model bell-crank or control system

• Units conversion

Use of MPC to tie GRIDs togetherUse of MPC to tie GRIDs together

• Write down relationship you want to impose on a per DOF level:

UX2 = UX1

UY2 = UY2

UZ3 = UZ3

X2 = X1

Y2 = Y1

Z2 = Z1

12

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


• Move dependent term to 1st term on right hand side:

0 = -UX2 + UX1

0 = -UY2 + UY2

0 = -UZ3 + UZ3

0 = -X2 + X1

0 = -Y2 + Y1

0 = -Z2 + Z1


• Use CAUTION when tying non-coincident GRIDs together!

• Watch for how those rotations and translations couple!2

1 UX2 = UX1

Z2 = Z1

MPCs for MPCs for RelativeRelative Motion Motion

• What’s the relative motion between GRIDs 1 and 2?

1 2?


• Introduce “placeholder” variable– Good use for SPOINTs

1 2?

• Move dependent term to RHS

0 = - U1000 + UX2 – UX1

• Write out desired relationship as before

U1000 = UX2 – UX1


• Write out MPCs1 2?

0 = -U1000 + UX2 – UX1

SPOINT 1000

MPC 535 1000 1 -1.0 2 1 +1.0

+ 1 1 -1.0

Initial gap

MPCs for Relative MPCs for Relative GAPGAP

• What is the gap between GRIDs 1 and 2?

1 2


1 2

UGAP = UINIT + UX2 – UX1

0 = -UGAP + UINIT + UX2 – UX1

• Write equation:– Introduce new placeholder

variable for initial gap


• Set initial gap value via SPC!1 2

SPOINT, 1000 $ Gap value

SPOINT, 1001 $ Initial Gap

MPC, 535, 1000, 1, -1., 1001, 1, +1.

+, , 2, 1, +1., 1, 1, -1.

SPC, 2002, 1001,1,0.5 $ Set initial gap

0 = -U1000 + U1001 + UX2 – UX1

MPC used to Maintain SeparationMPC 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

MPC used to Maintain SeparationMPC used to Maintain Separation1

20.25

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, -.75

1.00

Use of MPCs for AVERAGE MotionUse 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

Z

4

5

2

3

6

1

MPCs as Bell-crank or Control SystemMPCs as Bell-crank or Control System

• Output of 1 DOF scales another

U2 = U1/1.65

0 = -1.65*U2 + U1

2

1

1 +1.0-1.65 12 1MPC 535

C2 A2A1 G2G1 C1MPC SID

1.6

5

1.00

Units ConversionUnits Conversion

• Somewhat frivolous application, but why not?

– Convert radians to degrees

2 = 1 * 57.29578

– Convert inches to meters

39.37 * X2 = X1

Rigid Element OutputRigid Element Output

• Since Rigid elements are a specialized input of MPC equations, the output is requested by MPCFORCE case control command.– COMMON ERROR

• The MPCFORCEs are associated with GRID IDs, not Element IDs. So when selecting a SET for output, be sure the set is for GRID IDs, not Element IDs.

Guidelines for “Rigid” ElementsGuidelines for “Rigid” Elements

• Linear ONLY– Relationships calculated based on initial

geometry

• Can cause internal constraints for thermal conditions

• Be careful that independent GRID has 6 DOF

MPCs and RBEsMPCs and RBEs

• Off the shelf– RBAR– RBE2

• Customizable– RBE3

• Handmade– MPC

Add them to your

modeling arsenal today!

RBEs and MPCs in MSC.Nastran

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