RBEs and MPCs in MSC.Nastran
A Rip-Roarin’ Review of
Rigid Elements
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
Slide 2
Motion at one GRID drives another
• Simple Translation
X motion of Green Grid drives X motion
of Red Grid
Slide 3
Motion at one GRID drives another
• Simple Rotation
Rotation of Green Grid drives X translation
and Z rotation of Red Grid
Slide 4
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)
Slide 5
Small Displacement Theory & Rotations
• Small displacement theory:
sin() = tan() = 
cos() = 1
• For Rz @ A
Y
TxB
B
RzB = RzA= 
TxB = (-)*LAB
TyB = 0
-
A
Slide 6
X
Typical “Rigid” Elements in MSC.Nastran
• Geometry-based
– RBAR
– RBE2
} Really-rigid “rigid” elements
• Geometry- & User-input based
– RBE3
• User-input based
– MPC
Slide 7
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.
Slide 8
The RBAR
• The RBAR is a rigid link between two
GRID points
Slide 9
The RBAR
B
– Most common to have all the
dependent DOFs at one GRID,
and all the independent DOFs at
A
the other
– 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
RBAR
RBAR
EID
535
GA
1
GB
2
Slide 10
CNA CNB
123456
CMA
CMB
123456
RBAR Example: Fastener
• Use of RBAR to “weld” two parts of a
model together:
RBAR
RBAR
EID
535
GA
1
GB
2
CNA CNB
123456
B
A
Slide 11
CMA
CMB
123456
RBAR Example: Pin-Joint
• Use of RBAR to form pin-jointed
attachment
RBAR
RBAR
EID
535
GA
1
GB
2
CNA CNB
123456
B
A
Slide 12
CMA
CMB
123
The RBE2
• One independent GRID (all 6 DOF)
• Multiple dependent GRID/DOFs
Slide 13
RBE2 Example
• Rigidly “weld” multiple GRIDs to one
other GRID:
RBE2
RBE2
EID
99
GN
CM GM1 GM2 GM3 GM4 GM5
101 123456 1
2
3
4
1
3
4
2
101
Slide 14
RBE2 Example
RBE2 EID
RBE2 99
GN CM GM1 GM2 GM3 GM4 GM5
101 123456 1
2
3
4
• Note: No relative motion between
GRIDs 1-4 !
3
– No deformation of element(s)
between these GRIDs
101
Slide 15
1
4
2
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
Slide 16
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)
Slide 17
RBE3 Description
Slide 18
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
Slide 19
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 + U3 - 3 * U99
Slide 20
RBE3 Is Not Rigid!
• RBE3 vs. RBE2
– RBE3 allows warping
and 3D effects
– In this example, RBE2 enforces beam
theory (plane sections remain planar)
RBE2
RBE3
Slide 21
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
Slide 22
RBE3: How it Works?
• Step 1: Transform force/moment at
reference grid to equivalent force/moment
at weighted CG of master grids.
FA
MA
CG
FCG
Reference Grid
CG
e
MCG
FCG=FA
MCG=MA+FA*e
Slide 23
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
Slide 24
RBE3: How it Works?
• Step 2: Continued…
F1m
FCG
CG
MCG
F2m
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)}
Slide 25
F3m
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.
Slide 26
Example 1
• RBE3 distribution of loads when force at
reference grid at CG passes through
CG of master grids
Slide 27
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
Slide 28
Example 1: Force Through CG
• Load through CG with uniform weighting
factors results in uniform load distribution
Slide 29
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
Slide 30
Example 2
• How does the RBE3 distribute loads
when force on reference grid does not
pass through CG of master grids?
Slide 31
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.
Slide 32
Example 3
• Use of weighting factors to generate
realistic load distribution: 100 LB.
transverse load on 3D beam.
Slide 33
Example 3: Transverse Load on Beam
• If uniform
weighting
factors are
used, the load
is equally
distributed to all
grids.
Slide 34
Example 3: Transverse Load on Beam
• The uniform load distribution results in
too much transverse load in flanges
causing them to droop.
Displacement Contour
Slide 35
Example 3: Transverse Load on Beam
• 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
Slide 36
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.
Slide 37
Example 3: Transverse Load on Beam
• RBE3 Displacement Contour
– Max Y disp=.00685
Slide 38
Example 3: Transverse Load on Beam
• RBE2 Displacement contour
– Max Y disp=.00685
Slide 39
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?
Slide 40
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
Slide 41
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
Slide 42
Example 4: Use RBE3 for
Unconstrained Motion
“UM” Grids
Slide 43
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
Slide 44
Example 4: Use RBE3 for
Unconstrained Motion
• Result is free expansion due to internal
pressure. (note: poisson effect causes shortening)
Slide 45
Example 4: Use RBE3 for
Unconstrained Motion
• Resulting
MPC Forces
are numeric
zeroes
verifying that
no stiffness
has been
added.
Slide 46
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.
Slide 47
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
Slide 48
Example 5: 3D to Stick Model
Connection
Slide 49
Example 5: 3D to Stick Model
Connection
• Undeformed/Deformed plot shows
continuity in motion of 3D and Beam
model
Slide 50
Example 5: 3D to Stick Model
Connection
• MPC forces at
interface show
effect of both the
tip shear and
interface
moment.
Slide 51
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
Slide 52
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
Slide 53
Example 6: Using RBE3 to Visualize
“Beam” Modes
• Generic engine courtesy of Pratt &
Whitney
Slide 54
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
Slide 55
Example 6: Using RBE3 to Visualize
“Beam” Modes
• Complex
Mode
Animation
Slide 56
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
Slide 57
Example 7
• Use RBE3 to connect incompatible
elements
– Beam to plate
– Beam to solid
– Plate to solid
• Alternative to RSSCON
Slide 58
Example 7: RBE3 Connection of
Incompatible Elements
Slide 59
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
Slide 60
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
Slide 61
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
Slide 62
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)
Slide 63
Example 7: RBE3 Connection of
Incompatible Elements
• Deformation contours show continuity at
RBE3 interfaces
Slide 64
Example 7: RBE3 Connection of
Incompatible Elements
• Bending stress contours consistent
across RBE3 interface
Slide 65
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)
Slide 66
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)
Slide 67
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)
Slide 68
RBE3 Usage Guidelines
• 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
Slide 69
RBE3: Additional Reading
• Much RBE3 information has been posted on
MSC’s Knowledge Base
– http://www.mechsolutions.com/support/knowbase/index.html
Slide 70
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
Slide 71
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
Slide 72
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
Slide 73
General Approach For Use of MPCs
• Write out desired displacement equality
relationship on a per DOF level
– Dependent motion = (your equation goes here)
2
Ux2 = Ux1
1
• Re-arrange so left-hand side is zero
• List dependent term first
0 = - Ux2 + Ux1
Slide 74
MPC Format
• For example:
2
– Set X motion of GRID 2
= X motion of GRID 1
1
0 = - UX2 + UX1
= (-1.)UX2 + (+1.)UX1
UX2 = UX1
MPC
SID
G1
C1
A1
G2
C2
A2
MPC
535
2
1
-1.0
1
1
+1.0
Slide 75
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
Slide 76
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
Slide 77
Use of MPC to tie GRIDs together
• Write down relationship you want to
impose on a per DOF level:
UX2 = UX1
2
UY2 = UY2
1
UZ3 = UZ3
qX2 = qX1
qY2 = qY1
qZ2 = qZ1
Slide 78
Use of MPC to tie GRIDs together
• Move dependent term to 1st term on
right hand side:
0 = -UX2 + UX1
MPC, 535, 2, 1, -1.0, 1, 1, +1.0
0 = -UY2 + UY2
MPC, 535, 2, 2, -1.0, 1, 2, +1.0
0 = -UZ3 + UZ3
MPC, 535, 2, 3, -1.0, 1, 3, +1.0
0 = -qX2 + qX1
MPC, 535, 2, 4, -1.0, 1, 4, +1.0
0 = -qY2 + qY1
MPC, 535, 2, 5, -1.0, 1, 5, +1.0
0 = -qZ2 + qZ1
MPC, 535, 2, 6, -1.0, 1, 6, +1.0
Slide 79
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
Slide 80
MPCs for Relative Motion
• What’s the relative motion between
GRIDs 1 and 2?
1
?
2
Slide 81
MPCs for Relative Motion
• Introduce “placeholder” variable
– Good use for SPOINTs
• Write out desired
relationship as before
U1000 = UX2 – UX1
• Move dependent term to RHS
0 = - U1000 + UX2 – UX1
Slide 82
1
?2
MPCs for Relative Motion
• Write out MPCs
1
0 = -U1000 + UX2 – UX1
?2
SPOINT 1000
MPC
+
535
1000
1
-1.0
1
1
-1.0
Slide 83
2
1
+1.0
MPCs for Relative GAP
• What is the gap between GRIDs 1 and 2?
1
2
Initial
gap
Slide 84
MPCs for Relative GAP
• Write equation:
– Introduce new placeholder
variable for initial gap
UGAP = UINIT + UX2 – UX1
0 = -UGAP + UINIT + UX2 – UX1
Slide 85
1
2
MPCs for Relative GAP
• Set initial gap value via SPC!
1
2
0 = -U1000 + U1001 + UX2 – UX1
SPOINT, 1000
$ Gap value
SPOINT, 1001
$ Initial Gap
MPC,
+,
SPC,
535, 1000, 1, -1., 1001, 1, +1.
,
2, 1, +1.,
1, 1, -1.
2002, 1001,1,0.5 $ Set initial gap
Slide 86
MPC used to Maintain Separation
• Enforce a separation between GRIDs
– Similar to using a gap
– Changes which DOF are
dependent/independent
• Example:
1
– Initially 1” apart
– Keep separation = 0.25”
0.25
2
Slide 87
MPC used to Maintain Separation
1
1.00
0.25
2
U1 = U2 + (desired – initial)

0 = -U1 + U2 + U1000
SPOINT,1000
MPC,
535,
1, 2, -1.0,
+,
, 1000, 1, +1.0
SPC,
2002, 1000, 1, -.75
Slide 88
2, 2, +1.0
Use of MPCs for AVERAGE Motion
• Determine average motion of DOFs
U1000 = (U1+ U2 + U3 + U4 +U5 +U6)/6
4

5
3
0 = -6*U1000 + U1+ U2 + U3 + U4
+U5 +U6
6
2
1
Slide 89
MPCs as Bell-crank or Control System
• Output of 1 DOF scales another
1
U2 = U1/1.65
0 = -1.65*U2 + U1
1.65
2
MPC
MPC
SID
535
G1
2
C1
1
Slide 90
A1
-1.65
G2
1
C2
1
A2
+1.0
Units Conversion
• Somewhat frivolous application, but why
not?
– Convert radians
to degrees
q2 = q1 * 57.29578
– Convert inches
to meters
39.37 * X2 = X1
Slide 91
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.
Slide 92
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
Slide 93
MPCs and RBEs
• Off the shelf
Add them to
your
modeling
arsenal
today!
– RBAR
– RBE2
• Customizable
– RBE3
• Handmade
– MPC
Slide 94