RBEs and MPCs in MSC Nastran A Rip-Roarin

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 2

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

Motion at one GRID drives another • Simple Rotation of Green Grid drives X translation and Z rotation of Red Grid 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) 5

Small Displacement Theory & Rotations • Small displacement theory: sin( ) = tan( ) = cos( ) = 1 • For Rz @ A Y Tx. B B Rz. B = Rz. A= Tx. B = (- )*LAB Ty. B = 0 - A 6 X

Typical “Rigid” Elements in MSC. Nastran • Geometry-based – RBAR – RBE 2 } Really-rigid “rigid” elements • Geometry- & User-input based – RBE 3 • User-input based – MPC 7

Common Geometry-Based Rigid Elements • RBAR – Rigid Bar with six DOF at each end • RBE 2 – Rigid body with independent DOF at one GRID, and dependent DOF at an arbitrary number of GRIDs. 8

The RBAR • The RBAR is a rigid link between two GRID points 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 EID GA GB CNA RBAR 535 1 2 123456 10 CNB CMA CMB 123456

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

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

The RBE 2 • One independent GRID (all 6 DOF) • Multiple dependent GRID/DOFs 13

RBE 2 Example • Rigidly “weld” multiple GRIDs to one other GRID: RBE 2 EID RBE 2 99 GN CM GM 1 GM 2 GM 3 GM 4 GM 5 101 123456 1 1 3 4 2 101 14 2 3 4

RBE 2 Example RBE 2 EID RBE 2 99 GN CM GM 1 GM 2 GM 3 GM 4 GM 5 101 123456 1 2 3 4 • Note: No relative motion between GRIDs 1 -4 ! 3 – No deformation of element(s) between these GRIDs 101 15 1 4 2

Common RBE 2/RBAR Uses • RBE 2 or RBAR between 2 GRIDs – “Weld” 2 different parts together • 6 DOF connection – “Bolt” 2 different parts together • 3 DOF connection • RBE 2 – “Spider” or “wagon wheel” connections – Large mass/base-drive connection 16

RBE 3 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) 17

RBE 3 Description 18

RBE 3 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 19

RBE 3 Description • UM fields can be used to move the dependent DOF away from the reference grid – For Example (in 1 -D): U 99 = (U 1 + U 2 + U 3) / 3 3 * U 99 = U 1 + U 2 + U 3 -U 1 = + U 2 + U 3 - 3 * U 99 20

RBE 3 Is Not Rigid! • RBE 3 vs. RBE 2 – RBE 3 allows warping and 3 D effects – In this example, RBE 2 enforces beam theory (plane sections remain planar) RBE 2 RBE 3 21

RBE 3: 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 22

RBE 3: 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 23

RBE 3: 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 24

RBE 3: How it Works? • Step 2: Continued… F 1 m FCG CG MCG F 2 m 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 = {Mcg. Wiri/(W 1 r 12+W 2 r 22+W 3 r 32)} 25 F 3 m

RBE 3: 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. 26

Example 1 • RBE 3 distribution of loads when force at reference grid at CG passes through CG of master grids 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 28

Example 1: Force Through CG • Load through CG with uniform weighting factors results in uniform load distribution 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 RBE 3 would be singular 30

Example 2 • How does the RBE 3 distribute loads when force on reference grid does not pass through CG of master grids? 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. 32

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

Example 3: Transverse Load on Beam • If uniform weighting factors are used, the load is equally distributed to all grids. 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 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 36

Example 3: Transverse Load on Beam • Displacements with quadratic weighting factors virtually equivalent to those from RBE 2 (Beam Theory), but do not impose “plane sections remain planar” as does RBE 2. 37

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

Example 3: Transverse Load on Beam • RBE 2 Displacement contour – Max Y disp=. 00685 39

Example 4 • Use RBE 3 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? 40

Example 4: Use RBE 3 for Unconstrained Motion • Solution: – Use RBE 3 – Move dependent DOF from reference grid to selected master grids with UM option on RBE 3 (otherwise, reference grid cannot be SPC’d) – Apply SPC to reference grid 41

Example 4: Use RBE 3 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 RBE 3 – Note: “M” is the NASTRAN DOF set name for dependent DOF 42

Example 4: Use RBE 3 for Unconstrained Motion “UM” Grids 43

Example 4: Use RBE 3 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 RBE 3 rigid body motion 44

Example 4: Use RBE 3 for Unconstrained Motion • Result is free expansion due to internal pressure. (note: poisson effect causes shortening) 45

Example 4: Use RBE 3 for Unconstrained Motion • Resulting MPC Forces are numeric zeroes verifying that no stiffness has been added. 46

Example 5 • Connect 3 D model to stick model • 3 D model with 7 psi internal pressure • Use RBE 3 instead of RBE 2 so that 3 D model can expand naturally at interface. – RBE 3 will also allow warping and other 3 D effects at the interface. 47

Example 5: 3 D to Stick Model Connection • 120” diameter cylinder • 7 psi internal pressure • 10000 Lb. transverse load on stick model • RBE 3: Reference grid at center with 6 DOF, Master Grids with 3 translations 48

Example 5: 3 D to Stick Model Connection 49

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

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

Example 5: 3 D to Stick Model Connection • Shell outer fiber stresses at interface slightly higher than beam bending stresses – 3 D effects – Shell model under internal pressure and not bound by beam theory assumptions 52

Example 6 • Use RBE 3 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 RBE 3. – Connect centerline grids with PLOTELs 53

Example 6: Using RBE 3 to Visualize “Beam” Modes • Generic engine courtesy of Pratt & Whitney 54

Example 6: Using RBE 3 to Visualize “Beam” Modes • RBE 3’s used to connect various components to centerline. • Each component’s centerline grids connected by it’s own set of PLOTELs 55

Example 6: Using RBE 3 to Visualize “Beam” Modes • Complex Mode Animation 56

Example 6: Using RBE 3 to Visualize “Beam” Modes • Animation of the PLOTEL segments shows that this is a whirl mode • Relative motion of various components more clearly seen 57

Example 7 • Use RBE 3 to connect incompatible elements – Beam to plate – Beam to solid – Plate to solid • Alternative to RSSCON 58

Example 7: RBE 3 Connection of Incompatible Elements 59

Example 7: RBE 3 Connection of Incompatible Elements • Use RBE 3 to connect beams to plates at two corners • Use RBE 3 to connect beams to solids at two corners • Use RBE 3 to connect plates to solid – Plate thickness is same as solid thickness in this example 60

Example 7: RBE 3 Connection of Incompatible Elements • RBE 3 connection of beams to plates – Map 6 DOF of beam into plate translation DOF – For best results, beam “footprint” should be similar to RBE 3 “footprint”, otherwise joint will be too stiff 61

Example 7: RBE 3 Connection of Incompatible Elements • RBE 3 connection of beams to solids – Map 6 DOF of beam into solid translation DOF – For best results, beam “footprint” should be similar to RBE 3 “footprint”, otherwise joint will be too stiff 62

Example 7: RBE 3 Connection of Incompatible Elements • RBE 3 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) 63

Example 7: RBE 3 Connection of Incompatible Elements • Deformation contours show continuity at RBE 3 interfaces 64

Example 7: RBE 3 Connection of Incompatible Elements • Bending stress contours consistent across RBE 3 interface 65

RBE 3 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) 66

RBE 3 Usage Guidelines • Example: What can happen if master rotations included? – Modified RBE 3 from Example 5 – Displacements clearly incorrect when all 6 DOF listed for master grids (next page) 67

RBE 3 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) 68

RBE 3 Usage Guidelines • Make check run with PARAM, CHECKOUT, YES – Section 9. 4. 1 of MSC. Nastran Reference Manual (V 68) – 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 – V 2001: Case control command • GROUNDCHECK (SET=(G, N))=YES – V 70. 7 and earlier: • Use CHECKA alters from SSSALTER library 69

RBE 3: Additional Reading • Much RBE 3 information has been posted on MSC’s Knowledge Base – http: //www. mechsolutions. com/support/knowbase/index. html 70

RBE 3: Additional Reading • Recommended TANs – TAN#: 2402 RBE 3 - The Interpolation Element. – TAN#: 3280 RBE 3 ELEMENT CHANGES IN VERSION 70. 5, improved diagnostics – TAN#: 4155 RBE 3 ELEMENT CHANGES IN VERSION 70. 7 – TAN#: 4494 Mathematical Specification of the Modern RBE 3 Element – TAN#: 4497 AN ECONOMICAL METHOD TO EVALUATE RBE 3 ELEMENTS IN LARGE-SIZE MODELS 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 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) aj. Xi = a 1 X 1 + a 2 X 2 + a 3 X 3+…+ an. Xn 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 Ux 2 = Ux 1 1 • Re-arrange so left-hand side is zero • List dependent term first 0 = - Ux 2 + Ux 1 74

MPC Format • For example: 2 – Set X motion of GRID 2 = X motion of GRID 1 1 0 = - UX 2 + UX 1 = (-1. )UX 2 + (+1. )UX 1 UX 2 = UX 1 MPC SID G 1 C 1 A 1 G 2 C 2 A 2 MPC 535 2 1 -1. 0 1 1 +1. 0 75

General Approach to MPCs • Write down relationship you want to impose on a per DOF level: aj. Xi = a 1 X 1 + a 2 X 2 +…+ an. Xn • Move dependent term to 1 st term on right hand side: 0 = -ai. Xi + a 1 X 1 + a 2 X 2+…+ an. Xn 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 77

Use of MPC to tie GRIDs together • Write down relationship you want to impose on a per DOF level: UX 2 = UX 1 2 UY 2 = UY 2 1 UZ 3 = UZ 3 q. X 2 = q. X 1 q. Y 2 = q. Y 1 q. Z 2 = q. Z 1 78

Use of MPC to tie GRIDs together • Move dependent term to 1 st term on right hand side: 0 = -UX 2 + UX 1 MPC, 535, 2, 1, -1. 0, 1, 1, +1. 0 0 = -UY 2 + UY 2 MPC, 535, 2, 2, -1. 0, 1, 2, +1. 0 0 = -UZ 3 + UZ 3 MPC, 535, 2, 3, -1. 0, 1, 3, +1. 0 0 = -q. X 2 + q. X 1 MPC, 535, 2, 4, -1. 0, 1, 4, +1. 0 0 = -q. Y 2 + q. Y 1 MPC, 535, 2, 5, -1. 0, 1, 5, +1. 0 0 = -q. Z 2 + q. Z 1 MPC, 535, 2, 6, -1. 0, 1, 6, +1. 0 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 UX 2 = UX 1 q. Z 2 = q. Z 1 80

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

MPCs for Relative Motion • Introduce “placeholder” variable – Good use for SPOINTs • Write out desired relationship as before U 1000 = UX 2 – UX 1 • Move dependent term to RHS 0 = - U 1000 + UX 2 – UX 1 82 1 ? 2

MPCs for Relative Motion • Write out MPCs 1 0 = -U 1000 + UX 2 – UX 1 ? 2 SPOINT 1000 MPC + 535 1000 1 -1. 0 1 1 -1. 0 83 2 1 +1. 0

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

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

MPCs for Relative GAP • Set initial gap value via SPC! 1 2 0 = -U 1000 + U 1001 + UX 2 – UX 1 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 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 87

MPC used to Maintain Separation 1 1. 00 0. 25 2 U 1 = U 2 + (desired – initial) 0 = -U 1 + U 2 + U 1000 SPOINT, 1000 MPC, 535, 1, 2, -1. 0, +, , 1000, 1, +1. 0 SPC, 2002, 1000, 1, -. 75 88 2, 2, +1. 0

Use of MPCs for AVERAGE Motion • Determine average motion of DOFs U 1000 = (U 1+ U 2 + U 3 + U 4 +U 5 +U 6)/6 4 5 Z 6 3 0 = -6*U 1000 + U 1+ U 2 + U 3 + U 4 +U 5 +U 6 2 1 89

MPCs as Bell-crank or Control System • Output of 1 DOF scales another 1 U 2 = U 1/1. 65 2 1. 1. 65 00 à 0 = -1. 65*U 2 + U 1 MPC SID G 1 C 1 A 1 G 2 C 2 A 2 MPC 535 2 1 -1. 65 1 1 +1. 0 90

Units Conversion • Somewhat frivolous application, but why not? – Convert radians to degrees – Convert inches to meters 91 q 2 = q 1 * 57. 29578 39. 37 * X 2 = X 1

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. 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 93

MPCs and RBEs • Off the shelf Add them to your modeling arsenal today! – RBAR – RBE 2 • Customizable – RBE 3 • Handmade – MPC 94