1. RBEs and MPCs in MSC.Nastran
A Rip-Roarin’ Review of
Rigid Elements

2. The motion of a DOF is dependent on
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

3. Motion at one GRID drives another
Simple Translation
X motion of Green Grid drives X motion
of Red Grid

4. Motion at one GRID drives another
Simple Rotation
Rotation of Green Grid drives X translation
and Z rotation of Red Grid

5. The motion of a DOF is dependent on
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)

6. Small Displacement Theory & Rotations
sin() = tan() = 
cos() = 1
For A
RzB = RzA= 
TxB = (-)*LAB
TyB = 0 Y
TxB B - A X

7. Typical “Rigid” Elements in MSC.Nastran
Geometry-based
RBAR
RBE2
Geometry- & User-input based
RBE3
User-input based
MPC }
Really-rigid “rigid” elements

8. 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.

9. The RBAR
The RBAR is a rigid link between two GRID points

10. The RBAR B
Most common to have all the dependent DOFs at one GRID, and all the independent DOFs at the other A
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
123456 1 2
RBAR
535
CMA
CMB
CNA
CNB GA GB
EID

11. RBAR Example: Fastener
Use of RBAR to “weld” two parts of a model together:
123456 1 2
RBAR
535
CMA
CMB
CNA
CNB GA GB
EID B A

12. RBAR Example: Pin-Joint
Use of RBAR to form pin-jointed attachment
123
123456 1 2
RBAR
535
CMA
CMB
CNA
CNB GA GB
EID B A

13. The RBE2
One independent GRID (all 6 DOF)
Multiple dependent GRID/DOFs

14. Rigidly “weld” multiple GRIDs to one other GRID:
RBE2 Example
Rigidly “weld” multiple GRIDs to one other GRID: 3 2
RBE2 4 1
101 99
123456
GM5
GM3
GM2
GM4
GM1 GN
EID CM 1 3 2
101 4

15. Note: No relative motion between GRIDs 1-4 !
RBE2 Example 3 2
RBE2 4 1
101 99
123456
GM5
GM3
GM2
GM4
GM1 GN
EID CM
Note: No relative motion between GRIDs 1-4 !
No deformation of element(s) between these GRIDs 1 3 2
101 4

16. RBE2 or RBAR between 2 GRIDs
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

17. RBE3 Elements
Motion at a dependent GRID is the weighted average of the motion(s) at a set of master (independent) GRIDs
NOT a “rigid” element
IS an interpolation element
Does not add stiffness to the structure (if used correctly)

18. RBE3 Description

19. 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

20. RBE3 Description
UM fields can be used to move the dependent DOF away from the reference grid
For Example (in 1-D):
U99 = (U1 + U2 + U3) / 3
3 * U99 = U1 + U2 + U3
-U1 = + U2 + U * U99

21. RBE3 Is Not Rigid! RBE3 vs. RBE2 RBE3 RBE2
RBE3 allows warping and 3D effects
In this example, RBE2 enforces beam theory (plane sections remain planar)
RBE3
RBE2

22. 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

23. RBE3: How it Works?
Step 1: Transform force/moment at reference grid to equivalent force/moment at weighted CG of master grids. CG
FCG
MCG FA MA
Reference Grid e CG
FCG=FA
MCG=MA+FA*e

24. RBE3: How it Works?
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

25. RBE3: How it Works? Step 2: Continued…
F1m
F3m
F2m CG
FCG
MCG
Total force at each master node is sum of...
Forces derived from force at CG: Fif = FCG{Wi/Wi}
Plus Forces derived from moment at CG:
Fim = {McgWiri/(W1r12+W2r22+W3r32)}

26. RBE3: How it Works?
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.

27. Example 1
RBE3 distribution of loads when force at reference grid at CG passes through CG of master grids

28. 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

29. Example 1: Force Through CG
Load through CG with uniform weighting factors results in uniform load distribution

30. 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

31. Example 2
How does the RBE3 distribute loads when force on reference grid does not pass through CG of master grids?

32. 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.

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

34. Example 3: Transverse Load on Beam
If uniform weighting factors are used, the load is equally distributed to all grids.

35. Example 3: Transverse Load on Beam
The uniform load distribution results in too much transverse load in flanges causing them to droop.
Displacement Contour

36. Example 3: Transverse Load on Beam
Assume quadratic distribution of load in web
Assume thin flanges carry zero transverse load
Master DOF DOF 5 added to make RY rigid body motion determinate

37. Example 3: Transverse Load on Beam
Displacements with quadratic weighting factors virtually equivalent to those from RBE2 (Beam Theory), but do not impose “plane sections remain planar” as does RBE2.

38. Example 3: Transverse Load on Beam
RBE3 Displacement Contour
Max Y disp=.00685

39. Example 3: Transverse Load on Beam
RBE2 Displacement contour
Max Y disp=.00685

40. 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?

41. 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

42. Example 4: Use RBE3 for Unconstrained Motion
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

43. Example 4: Use RBE3 for Unconstrained Motion
“UM” Grids

44. Example 4: Use RBE3 for Unconstrained Motion
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

45. Example 4: Use RBE3 for Unconstrained Motion
Result is free expansion due to internal pressure. (note: poisson effect causes shortening)

46. Example 4: Use RBE3 for Unconstrained Motion
Resulting MPC Forces are numeric zeroes verifying that no stiffness has been added.

47. Connect 3D model to stick model 3D model with 7 psi internal pressure
Example 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.

48. Example 5: 3D to Stick Model Connection
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

49. Example 5: 3D to Stick Model Connection

50. Example 5: 3D to Stick Model Connection
Undeformed/Deformed plot shows continuity in motion of 3D and Beam model

51. Example 5: 3D to Stick Model Connection
MPC forces at interface show effect of both the tip shear and interface moment.

52. Example 5: 3D to Stick Model Connection
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

53. Use RBE3 to see “beam” type modes from a complex model
Example 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

54. Example 6: Using RBE3 to Visualize “Beam” Modes
Generic engine courtesy of Pratt & Whitney

55. Example 6: Using RBE3 to Visualize “Beam” Modes
RBE3’s used to connect various components to centerline.
Each component’s centerline grids connected by it’s own set of PLOTELs

56. Example 6: Using RBE3 to Visualize “Beam” Modes
Complex Mode Animation

57. Example 6: Using RBE3 to Visualize “Beam” Modes
Animation of the PLOTEL segments shows that this is a whirl mode
Relative motion of various components more clearly seen

58. Use RBE3 to connect incompatible elements
Example 7
Use RBE3 to connect incompatible elements
Beam to plate
Beam to solid
Plate to solid
Alternative to RSSCON

59. Example 7: RBE3 Connection of Incompatible Elements

60. Example 7: RBE3 Connection of Incompatible 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

61. Example 7: RBE3 Connection of Incompatible Elements
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

62. Example 7: RBE3 Connection of Incompatible Elements
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

63. Example 7: RBE3 Connection of Incompatible Elements
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)

64. Example 7: RBE3 Connection of Incompatible Elements
Deformation contours show continuity at RBE3 interfaces

65. Example 7: RBE3 Connection of Incompatible Elements
Bending stress contours consistent across RBE3 interface

66. RBE3 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)

67. Example: What can happen if master rotations included?
RBE3 Usage Guidelines
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)

68. RBE3 Usage Guidelines
Deformation with all 6 DOF specified for master grids at interface
Deformation with 3 translation DOF specified for master grids (same loads/BC’s)

69. Make check run with PARAM,CHECKOUT,YES
RBE3 Usage Guidelines
Make check run with PARAM,CHECKOUT,YES
Section 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

70. RBE3: Additional Reading
Much RBE3 information has been posted on MSC’s Knowledge Base

71. RBE3: Additional Reading
Recommended TANs
TAN#: RBE3 - The Interpolation Element.
TAN#: RBE3 ELEMENT CHANGES IN VERSION 70.5, improved diagnostics
TAN#: RBE3 ELEMENT CHANGES IN VERSION 70.7
TAN#: Mathematical Specification of the Modern RBE3 Element
TAN#: AN ECONOMICAL METHOD TO EVALUATE RBE3 ELEMENTS IN LARGE-SIZE MODELS

72. User-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

73. ajXi = a1X1 + a2X2 + a3X3+…+ anXn
MPC 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

74. General Approach For Use of MPCs
Write out desired displacement equality relationship on a per DOF level
Dependent motion = (your equation goes here) 2 1
Ux2 = Ux1
Re-arrange so left-hand side is zero
List dependent term first
0 = - Ux2 + Ux1

75. MPC Format For example: UX2 = UX1 0 = - UX2 + UX1
Set X motion of GRID 2
= X motion of GRID 1 2 1
UX2 = UX1
0 = - UX2 + UX1
= (-1.)UX2 + (+1.)UX1 1
+1.0
-1.0 2
MPC
535 C2 A2 A1 G2 G1 C1
SID

76. General Approach to MPCs
Write down relationship you want to impose on a per DOF level:
ajXi = a1X1 + a2X2 +…+ anXn
Move dependent term to 1st term on right hand side:
0 = -aiXi + a1X1 + a2X2+…+ anXn

77. 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

78. Use of MPC to tie GRIDs together
Write down relationship you want to impose on a per DOF level:
UX2 = UX1
UY2 = UY2
UZ3 = UZ3
qX2 = qX1
qY2 = qY1
qZ2 = qZ1 2 1

79. Use of MPC to tie GRIDs together
Move dependent term to 1st term on right hand side:
0 = -UX2 + UX1
0 = -UY2 + UY2
0 = -UZ3 + UZ3
0 = -qX2 + qX1
0 = -qY2 + qY1
0 = -qZ2 + qZ1
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

80. Use of MPC to tie GRIDs together
Use CAUTION when tying non-coincident GRIDs together!
Watch for how those rotations and translations couple! 2 1
UX2 = UX1
qZ2 = qZ1

81. MPCs for Relative Motion
What’s the relative motion between GRIDs 1 and 2? 1 ? 2

82. MPCs for Relative Motion
Introduce “placeholder” variable
Good use for SPOINTs
Write out desired relationship as before ? 1 2
U1000 = UX2 – UX1
Move dependent term to RHS
0 = - U UX2 – UX1

83. MPCs for Relative Motion
Write out MPCs ? 1 2
0 = -U UX2 – UX1
SPOINT 1000
MPC

84. What is the gap between GRIDs 1 and 2?
MPCs for Relative GAP
What is the gap between GRIDs 1 and 2? 1 2
Initial gap

85. MPCs for Relative GAP Write equation: 0 = -UGAP + UINIT + UX2 – UX1
Introduce new placeholder variable for initial gap 1 2
UGAP = UINIT + UX2 – UX1
0 = -UGAP + UINIT + UX2 – UX1

86. Set initial gap value via SPC!
MPCs for Relative GAP
Set initial gap value via SPC! 1 2
0 = -U U UX2 – UX1
SPOINT, $ Gap value
SPOINT, $ Initial Gap
MPC, , 1000, 1, -1., 1001, 1, +1.
+, , 2, 1, +1., 1, 1, -1.
SPC, , 1001,1,0.5 $ Set initial gap

87. 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
0.25 2

88. MPC used to Maintain Separation1
1.00
0.25 2
U1 = U2 + (desired – initial)
 0 = -U1 + U2 + U1000
SPOINT,1000
MPC, , 1, 2, -1.0, 2, 2, +1.0
+, , 1000, 1, +1.0
SPC, , 1000, 1, -.75

89. Use of MPCs for AVERAGE Motion
Determine average motion of DOFs
U1000 = (U1+ U2 + U3 + U4 +U5 +U6)/6
 0 = -6*U U1+ U2 + U3 + U4 +U5 +U6 4 5 3 Z 6 2 1

90. MPCs as Bell-crank or Control System
Output of 1 DOF scales another 1
U2 = U1/1.65
0 = *U2 + U1 2
1.65
1.00 1
+1.0
-1.65 2
MPC
535 C2 A2 A1 G2 G1 C1
SID

91. Somewhat frivolous application, but why not?
Units Conversion
Somewhat frivolous application, but why not?
Convert radians to degrees
q2 = q1 *
Convert inches to meters
39.37 * X2 = X1

92. Rigid 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.

93. Guidelines 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

94. Add them to your modeling arsenal today!
MPCs and RBEs
Off the shelf
RBAR
RBE2
Customizable
RBE3
Handmade
MPC
Add them to your modeling arsenal today!