S2-1NAS105, Section 2, May 2005 SECTION 2 ELEMENT TECHNIQUES

S2-2NAS105, Section 2, May 2005

S2-3NAS105, Section 2, May 2005 TABLE OF CONTENTS SectionPage THE MOST POPULAR MSC.NASTRAN ELASTIC ELEMENTS…………………………………2-7 EVOLUTION OF PLATE FINITE ELEMENTS…………………………………………………… HIGHER ORDER VERSUS LOWER ORDER ELEMENTS …………………………………… MODERN FINITE ELEMENTS……………………………………………………………………….2-13 FEATURES USED BY THE CURRENT ELEMENTS…………………………………………… FINITE ELEMENT VOLUME INTEGRATION………………………………………………………2-15 ADDED STRAIN FUNCTIONS……………………………………………………………………….2-16 HEXA…………………………………………………………………………………………………….2-17 ELEMENT PERFORMANCE…………………………………………………………………………2-18 PATCH TEST – CLASSICAL…………………………………………………………………………2-19 PATCH TEST FOR SOLIDS………………………………………………………………………….2-20 STRAIGHT CANTILEVER BEAM…………………………………………………………………….2-21 TWISTED BEAM……………………………………………………………………………………….2-22 TYPES OF GEOMETRIC DISTORTION FROM A SQUARE PLATE……………………………2-23 RECTANGULAR PLATE………………………………………………………………………………2-24 RESULTS OF CLAMPED RECTANGULAR PLATE WITH PRESSURE LOAD … … … … … 2-25 SCORDELIS-LO ROOF……………………………………………………………………………… RESULTS FOR SCORDELIS-LO ROOF……………………………………………………………2-27 SUMMARY OF SOME SAMPLE TEST PROBLEMS………………………………………………2-28

S2-4NAS105, Section 2, May 2005 TABLE OF CONTENTS SectionPage RECOMMENDATIONS………………………………………………………………………… MPCS AND R-TYPE ELEMENTS…………………………………………………………….2-31 MPC – BULK DATA ENTRY…………………………………………………………………….2-36 MULTIPOINT CONSTRAINT EXAMPLES…………………………………………………….2-38 R-TYPE ELEMENTS……………………………………………………………………………2-46 THE R ELEMENTS…………………………………………………………………………… SAMPLE USES OF R ELEMENTS……………………………………………………………2-48 COMMONLY USED R ELEMENTS………………………………………………………… RBE2 AND RBAR……………………………………………………………………………… RBAR: THE CONSTRAINT EQUATIONS…………………………………………………….2-51 RBAR – BULK DATA ENTRY………………………………………………………………….2-52 RBE2 – BULK DATA ENTRY……………………………………………………………………2-54 THE RBE2…………………………………………………………………………………………2-56 RBE2 EXAMPLE………………………………………………………………………………….2-57 COMMON RBE2/RBAR USES………………………………………………………………….2-58 RBE3 – THE WIFFLETREE……………………………………………………………………2-59 RBE3 – BULK DATA ENTRY……………………………………………………………………2-60 RBE3 DESCRIPTION…………………………………………………………………………….2-64 RBE3 IS NOT RIGID…………………………………………………………………………… RBE3: HOW IT WORKS………………………………………………………………………….2-66

S2-5NAS105, Section 2, May 2005 SectionPage EXAMPLE 1…………………………………………………………………………………………2-71 EXAMPLE 1: FORCE THROUGH CG………………………………………………………… EXAMPLE 2…………………………………………………………………………………………2-75 EXAMPLE 2: LOAD NOT THROUGH CG………………………………………………………2-76 RSPLINE – THE LINEAR SPLINE………………………………………………………………2-77 RSPLINE – BULK DATA ENTRY……………………………………………………………… RSSCON – BULK DATA ENTRY……………………………………………………………… TABLE OF CONTENTS

S2-6NAS105, Section 2, May 2005

S2-7NAS105, Section 2, May 2005 THE MOST POPULAR MSC.NASTRAN ELASTIC ELEMENTS

S2-8NAS105, Section 2, May 2005 EVOLUTION OF PLATE ELEMENTS n It was years before good plate elements were available in finite element programs. n One of the first approximations was to use a beam simulation. n Triangles – the first plate elements were 3–noded triangles using polynomials to approximate the deformations. n The constants were evaluated in terms of the displacements of the corners. Unfortunately, there were not enough DOF available to allow the evaluation of a complete second–order polynomial and people had to improvise to try to correct for this.

S2-9NAS105, Section 2, May 2005 EVOLUTION OF PLATE ELEMENTS (Cont.) n Many of the early plate elements were made up by using a combination of triangles. n Finally Irons came up with the idea of Isoparametric elements. n These elements work using polynomials also, but use Gauss Integration so that the integrals may be evaluated exactly, even for elements with curved edges, and faces. n Each isoparametric element is based on a perfect element. n These perfect elements have dimensions of 2 units (non– dimensional) and are mapped onto the physical geometry of the element. Assumed polynomials

S2-10NAS105, Section 2, May 2005 EVOLUTION OF PLATE ELEMENTS (Cont.) n The following are examples of perfect elements n For each of these elements, the dimensions and vary from -1 to +1. n The reason for this is that it is easier to come up with shape functions to interpolate the deformations in terms of the perfect element. Gaussian integration is used to evaluate the stiffness and other element matrices. Assuming Isoparametric Shapes (Mapping to Square)

S2-11NAS105, Section 2, May 2005 EVOLUTION OF PLATE ELEMENTS (Cont.) n The order of an element is determined by the order of the polynomial used to represent the deformed shape.

S2-12NAS105, Section 2, May 2005 HIGHER ORDER VERSUS LOWER ORDER ELEMENTS n According to Zienkiewicz 1,...A dramatic improvement of accuracy arises with the same number of degrees of freedom when complex elements are used. --- A considerable cost saving occurs. (Not necessarily true today; good lower order plate, shell, and solid elements exist.) n Further, the data preparation is considerably reduced with complex elements. (Not true, particularly if a mesh generator is used.) n On the other side of the picture, it will be sometimes seen that the very much reduced number of complex elements may not be adequate to represent all of the local geometries of the real problem with the minimum number of elements. (Therefore, the lower order elements are needed for special regions.) n With the use of higher order elements, progressively, the departure from an easily conceived physical idealization occurs. (True.) n Probably the most serious economic problem of complex curvilinear elements is the computer time necessary for performing the numerical integration. (True, but not important for large problems.) 1. O.C. Zienkiewicz, The Finite Element Method in Engineering Science, (Third Edition), McGraw-Hill, London, 1977.

S2-13NAS105, Section 2, May 2005 MODERN FINITE MODELING n Finite elements approximate engineering theory by using polynomial displacement functions. n Based on experience, people have found that using standard isoparametric elements can give incorrect answers in many situations. Nastran elements are customized to provide the best performance possible. n The following pages show some of the methods used in customizing the elements. n REMEMBER – FINITE ELEMENTS ARE AN APPROXIMATION TO ENGINEERING THEORY AND ARE NOT EXACT!!!!!

S2-14NAS105, Section 2, May 2005 FEATURES USED BY THE CURRENT ELEMENTS 1. Assumed strains at integration points 2. HEXA now has internal strain functions 3. Reduced integration

S2-15NAS105, Section 2, May 2005 FINITE ELEMENT VOLUME INTEGRATION n Full integral n Reduced Integration

S2-16NAS105, Section 2, May 2005 ADDED STRAIN FUNCTIONS n (Some people call them Bubble Functions.)

S2-17NAS105, Section 2, May 2005 HEXA n Assumed strain functions n Lumped mass for higher order HEXA INTOPT = 0 The theory gives negative mass at corners

S2-18NAS105, Section 2, May 2005 ELEMENT PERFORMANCE n In 1985 Dr. R. MacNeal and R. Harder published (Finite Elements in Analysis and Design, North Holland Publishing Co., pp. 3-20) a series of proposed models to test the quality of finite elements. n These problems were designed to punish the elements based on their understanding of the elements and how they performed. n The following shows some of the models and results using the current elements in MSC.NASTRAN (V2001). n

S2-19NAS105, Section 2, May 2005 PATCH TEST CLASSICAL n Use regular shape, irregular pattern n Non-redundant constraints n Apply boundary forces or displacements for constant stress. The patch tests check n Convergence n Compatibility

S2-20NAS105, Section 2, May 2005 PATCH TEST FOR SOLIDS Location of Inner Nodes Outer Dimensions: unit cube E = 1.0 x 10 6 ; v = 0.25

S2-21NAS105, Section 2, May 2005 STRAIGHT CANTILEVER BEAM Length = 6.0 E = 1.0 x 10 7 Width = 0.2v = 0.30 Depth = 0.1Mesh = 6 x 1 Loading: Unit forces at free end Note: All elements have equal volume.

S2-22NAS105, Section 2, May 2005 TWISTED BEAM Undeformed Geometry Length = 12.0 E = 29.0 x 10 6 Width = 1.1v = 0.22 Depth = 0.32Mesh = 12 x 2 Twist = 90 o (root to tip) Loading: unit forces at tip

S2-23NAS105, Section 2, May 2005 TYPES OF GEOMETRIC DISTORTION FROM A SQUARE PLATE n Aspect ratio n Skew n Taper (2 directions) n Warp Reasonable Limits Up to 10:1 Normally < 4:1 Up to ~5% is acceptable normally No real limit, but element does not include warpage T a /Q a = 0.5 – 0.75 T a = Largest of the Areas of Triangles formed at each corner grids. Q a = Area of the Quadrilateral.

S2-24NAS105, Section 2, May 2005 RECTANGULAR PLATE

S2-25NAS105, Section 2, May 2005 RESULTS OF CLAMPED RECTANGULAR PLATE WITH PRESSURE LOAD Aspect Ratio = 1.0 Aspect Ratio = 5.0 Reduced Shear Integration

S2-26NAS105, Section 2, May 2005 SCORDELIS-LO ROOF

S2-27NAS105, Section 2, May 2005 RESULTS FOR SCORDELIS-LO ROOF

S2-28NAS105, Section 2, May 2005 SUMMARY OF SOME SAMPLE TEST PROBLEMS

S2-29NAS105, Section 2, May 2005 RECOMMENDATIONS n Plates u QUAD4Is not very sensitive to aspect ratios u QUAD4Membrane behavior can be sensitive to irregular irregular shapes u TRIA3Is nearly as good but has some aspect ratio sensitivity u TRIA3Tends to be stiff for membrane loads u TRIA3May have some local stress perturbations when mixed with QUAD4 u QUAD8Has best results for curved shapes u QUAD8Should not span more than 20 – 30 degrees u QUAD8Should have midsize nodes placed accurately. u QUADR/TRIAR (New in V2004) - Try

S2-30NAS105, Section 2, May 2005 RECOMMENDATIONS (Cont.) n Solids u HEXARecommendation is eight nodes and strain function integration (0) u HEXAMay be used as plates with strain function integration u HEXAGives poor results with irregular shapes u HEX20Recommended for irregular shapes and curved surfaces (use reduced integration) u PENTARecommended for irregular shapes u PENTAMay exhibit local irregular when mixed with HEXA

S2-31NAS105, Section 2, May 2005 MPCs AND R- TYPE ELEMENTS

S2-32NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINTS (MPC) n Each MPC entry is used to specify one displacement (U m ) as a linear combination of one or more other displacements(U n ). n Nastran divides the G-set into 2 sets. M = dependent DOFs, N = independent DOFs n Then performs the reduction from the G to N set.

S2-33NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINTS (MPC) (Cont.) n General form for MPC equations: where n The equations for all MPCs and R-type elements are assembled to form the constraint equations: Scaling coefficient for the dependent DOFs Scaling coefficient for the independent DOFs Displacement of dependent DOFs Displacement of independent DOFs

S2-34NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINTS (MPC) (Cont.) u This can be written as n The G-set matrices are rewritten as follows: n As explained in The MSC.NASTRAN REFERENCE GUIDE, Section 9.4.3, the equation Therefore, R M must not be singular

S2-35NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINTS (MPC) (Cont.) Becomes where

S2-36NAS105, Section 2, May 2005 MPC – BULK DATA ENTRY Defines a multipoint constraint equation of the form where u j represents degree of freedom Cj at grid scalar point Gj

S2-37NAS105, Section 2, May 2005 MPC – BULK DATA ENTRY (Cont.) Remarks: 1. Multipoint constraint sets must be selected with the Case Control command MPC = SID. 2. The first degree of freedom (G1, C1) in the sequence is defined to be the dependent degree of freedom assigned by one MPC entry cannot be assigned dependent by another MPC entry or by a rigid element. 3. Forces of multipoint constraint may be recovered in the linear structured solution sequences (101 – 200) with the MPCFORCE Case Control command and in SOL 24 with RF Alter RF23D24 (see the MSC.NASTRAN Reference Manual, Chapter 15). 4. The m-set degrees of freedom specified on this entry may not be specified on other entries that define mutually exclusive sets. See the MSC.NASTRAN Quick Reference Guide, Appendix B for a list of these entries. 5. By default, the grid point connectivity created by the MPC, MPCADD, and MPCAX entries is not considered during resequencing, (see the PARAM, NEWSEQ description in the MSC.NASTRAN quick Reference Guide, Section 6). In order to consider the connectivity during resequencing, SID must be specified on the PARAM, MPCX entry. Using the example above, specify PARAM, MPCX, 3.

S2-38NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINT EXAMPLE 1. Thick plate with bars attached. Using plate theory assumption (plane sections remain plane), we can write the equations for the in- plane motion of Grid Points 2 and 3 as function of the motion of Grid point 1.

S2-39NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINT EXAMPLE (Cont.) Notes: n Select MPC = 1 in the Case Control to use these entries. n The MPC equations are written using the displacement coordinate system of the GRID points. If GRID points involved in MPC have different coordinate system, be careful, to avoid grounding the structure. MPC, 1, 2, 1, 1., 1, 1, -1.,,+MPC1A +MPC1A,, 1, 6,.5 MPC, 1, 3, 1, 1., 1, 1, -1.,,+MPC1B +MPC1B,, 1, 6, -.5 MPC, 1, 2, 2, 1., 1, 2, -1. MPC, 1, 3, 2, 1., 1, 2, -1. MPC, 1, 2, 6, 1., 1, 6, -1. MPC, 1, 3, 6, 1., 1, 6, -1. MPC entries are:

S2-40NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINT EXAMPLE (Cont.) n 1A. If GRID 1 has the following displacement coordinate system (CID =1), and GRID 2 and 3 have the basic displacement coordinate system, the MPC equations would look like: X Y CORD2R, 1,, 0., 0., 0., 0., 0., 1.,, 0., 1., 0. GRID, 1,, 4.,.5, 0., 1 $ CD = 1 MPC, 1, 2, 1, 1., 1, 2, 1.,,&MPC1A +MPC1A,, 1, 6, 0.5 MPC, 1, 3, 1, 1., 1, 2, 1.,,&MPC1B +MPC1B,, 1, 6, -0.5 $ MPC, 1, 2, 2, 1., 1, 1, -1. MPC, 1, 3, 2, 1., 1, 1, -1. $ MPC, 1, 2, 6, 1., 1, 6, -1. MPC, 1, 3, 6, 1., 1, 6, -1.

S2-41NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINT EXAMPLE (Cont.) 2. Calculate the distance between two points: Example: There is a tolerance requirement on a structure. We want to know the clear distance between two specified points. o = 10.0 = Initial clearance

S2-42NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINT EXAMPLE (Cont.) Total distance = o + Use a scalar point to represent the distance. * Call out MPC = 10 in the Case Control Section ** Call out SPC = 1 in the Case Control Section $ Component 1 of node 1001 is SPC'ed to 10.0 (the $ original distance between node 1 and node 2) GRID, 1001 SPC, 1, 1001, 1, 10.0 $ Create a scalar variable to hold (final) distance $ between node 1 and node 2 SPOINT, 1000 $ Set MPC: Ux1000 = Ux Ux2 - Ux1 MPC, 10, 1000, 0, 1., 1001, 1, -1.,,+MPC10A +MPC10A,, 2, 1, -1., 1, 1, 1.

S2-43NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINT EXAMPLE (Cont.) 3. Average displacement To get the average displacement of Grid Points 1 and 2 of the previous example, add the following: $ Create a scalar variable (1002) to hold the average $ distance between node 1 and node 2 SPOINT, 1002 $ MPC, 10, 1002, 0, 1., 1, 1, -0.5,,+MPC10B +MPC10B,, 2, 1, -0.5

S2-44NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINT EXAMPLE (Cont.) 4 Enforce a relative gap. n In order to do this, we wish to constrain the SPOINT to have the desired gap. Therefore, the SPOINT must be independent on the MPC and we need to re-write the MPC entries. * Call out MPC = 10 in the Case Control Section ** Call out SPC = 1 in the Case Control Section $ Create a scalar variable (1000) to hold the final $ distance (clearance) between node 1 and node 2 SPOINT, 1000 $ Relative gap (final) = (Ux2 - Ux1) + gap (initial) $ MPC Eq.: Ux1000 = Ux2 - Ux1 + Ux1001 input as: $ -Ux1 +Ux2 +Ux1001 -Ux1000 = 0. MPC, 10, 1, 1, -1., 2, 1, 1.,,+MPC10A +MPC10A,, 1001, 1, 1., 1000, 0, -1. $ SPC, 1, 1001, 1, 0.02 $Initial gap SPC, 1, 1000, 0, $Final gap to be allowed

S2-45NAS105, Section 2, May 2005 MULTIPOINT CONSTRAINTS EXAMPLE (Cont.) n MPC at Selective Mesh Refinement n U 8 = 0.5*(U 7 + U 9 ) ; U 17 = 0.5*(U 9 + U 18 )

S2-46NAS105, Section 2, May 2005 R – TYPE ELEMENTS

S2-47NAS105, Section 2, May 2005 THE R ELEMENTS n Generate internal MPC equations that eliminate dependent degrees of freedom. (The user selects the dependent points.) n Are automatically included in the solution (the MPC = command does not effect R–elements) n R elements do not account for nonlinear, mass, thermal expansion, or heat transfer effects. n Internal forces and grid point forces are calculated for output.(MPCFORCE case control request – output is by GRID point, not by R–element. The output at each GRID point is the summation from all R–elements and MPCs connected to it. The MPCForces should sum to zero, if no structural element is attached at the grid.)

S2-48NAS105, Section 2, May 2005 SAMPLE USES OF R ELEMENTS n When very stiff structure sections are inconvenient to model or numerically troublesome n When different pieces of the model are mismatched and the grid points cannot be connected with conventional elements n If connecting joints are free to slide and/or rotate in specific directions n When elements are offset from grid points n To distribute input loading or enforced motions n To connect incompatible elements

S2-49NAS105, Section 2, May 2005 COMMONLY USED R ELEMENTS n RBARRigid bar connecting two grid points with 6 independent and 1-6 dependent DOFs n RBE2Rigid element with six independent DOFs at one grid point and any number of dependent DOFs n RBE3Interpolation element with 1 – 6 dependent DOFs and any number of independent DOFs. Used for distributing loads or obtaining average displacement. No stiffness is introduced by RBE3. n RSPLINE Interpolation element with any number of independent and dependent DOFs. Uses the displacement pattern of a beam element based on the independent DOFs to obtain displacements of the dependents DOFs. n RSSCONInterpolation element used to connect shell elements to solid elements (added in v69).

S2-50NAS105, Section 2, May 2005 RBE2 and RBAR n These are rigid elements n Simpler versions of RBE1 n RBAR connects to only two grid points with a total of six independent degrees of freedom. The independent DOF are selected by you, but must be able to define any motion is space. n RBE2 has one independent grid point (all six degrees of freedom) and any number of dependent grid points.

S2-51NAS105, Section 2, May 2005 RBAR: The CONSTRAINT EQUATIONS n Equations for RBAR A-SA Vector Notation: Component Notation :

S2-52NAS105, Section 2, May 2005 RBAR – BULK DATA ENTRY Defines a rigid bar with six degrees of freedom at each end CMA, CMB Component numbers of dependent degrees of freedom in the global coordinate system assigned by the element at grid points GA and GB. See Remarks 2 and 3. (Integers 1 through 6 with no embedded blanks, or zero or blank).

S2-53NAS105, Section 2, May 2005 RBAR – BULK DATA ENTRY (Cont.) Remarks: 1. The total number of components in CNA and CNB must equal six; for example, CNA = 1236, CNB = 34. Furthermore, they must jointly be capable of representing any general rigid body motion of the element. 2. If both CMA and CMB are zero or blank, all of the degrees of freedom not in CNA and CNB will be made dependent; i.e, they will be made members of the m-set. 3. The m-set coordinates specified on this entry may not be specified on other entries that define mutually exclusive sets. Se the MSC.NASTRAN Quick Reference Guide, Appendix B for a list of these entries. 4. Element identification numbers must be unique. 5. Rigid elements, unlike MPCs, are not selected through the Case Control Section. 6. Forces of multipoint constraint may be recovered in the linear structured solution sequences (101 – 200), with the MPCFORCE Case Control command. 7. Rigid elements are ignored in heat transfer problems. 8. See the MSC.NASTRAN Reference Manual, Section 5.5, for a discussion of rigid elements.

S2-54NAS105, Section 2, May 2005 RBE2 – BULK DATA ENTRY Defines a rigid body whose independent degrees of freedom are specified at a single grid point and whose dependent degrees of freedom are specified at an arbitrary number of grid points.

S2-55NAS105, Section 2, May 2005 RBE2 – BULK DATA ENTRY (Cont.) Remarks: 1. The components indicated by CM are made dependent (members of the m- set) at all grid points GMI. 2. Dependent degrees of freedom assigned by one rigid element may not also be assigned dependent by another rigid element or by a multipoint constraint. 3. Element identification numbers must be unique. 4. Rigid elements, unlike MPCs are not selected through the Case Control Section. 5. Forces of multipoint constraints may be recovered in the linear structured solution sequences (101 – 200) with MPCFORCE Case Control command and in SOL 24 with Alter RF24D24 (see the MSC.NASTRAN Reference Manual, Chapter 15). 6. Rigid elements are ignored in heat transfer problems. 7. See the MSC.NASTRAN Reference Manual,Section 5.5, for a discussion of rigid elements. 8. The m-set coordinates specified on this entry may not be specified on other entries that define mutually exclusive sets. See the MSC.NASTRAN Quick Reference Guide, Appendix B for a list of these entries.

S2-56NAS105, Section 2, May 2005 The RBE2 n One independent GRID (all 6 DOF) n Multiple dependent GRID/DOFs Note: Small Rotations assumed

S2-57NAS105, Section 2, May 2005 RBE2 Example n Note: No relative motion between GRIDs 1-4 ! u No deformation of element(s) between these GRIDs 32RBE GM5GM3GM2RBE2GM4GM1GNEIDCM

S2-58NAS105, Section 2, May 2005 Common RBE2/RBAR Uses n RBE2 or RBAR between 2 GRIDs u Weld two different parts together l 6 DOF Connection u Ball Joint two different parts together l 3 DOF connection n RBE2 u Spider or wagon wheel connections u Large mass/base-drive connection

S2-59NAS105, Section 2, May 2005 RBE3 – THE WIFFLETREE n The RBE3 is an interpolation element. Example: n Basic equations n Select one to six dependents from any DOF. Number of dependents = number of U ref. > >

S2-60NAS105, Section 2, May 2005 RBE3 – BULK DATA ENTRY Defines the motion at a reference grid point as the weighted average of the motions at a set of other grid points. Note: RBE3 degenerates into RBAR if there is only 1 Gij entry..

S2-61NAS105, Section 2, May 2005 RBE3 – BULK DATA ENTRY (Cont.)

S2-62NAS105, Section 2, May 2005 RBE3 – BULK DATA ENTRY (Cont.) Remarks: 1. It is recommended that for most applications only the translation components 123 be used for Ci. An exception is the case where the Gi,j are collinear. A rotation component may then be added to one grid point to stabilize its associated rigid body mode for the element. 2. Blank spaces may be left at the end of a Gi, j sequences. 3. The default for UM should be used except in cases where the user wishes to include some or all REFC components in displacement sets exclusive from the m-set. If the default is not used for UM. a. The total number of components in the m-set (I.e.., the total number of dependent degrees of freedom defined by the element) must be equal to the number of components in REFC (four components in the example). b. The components specified after UM must be a subset of the component specified under REFC and (Gi, j, Ci). c. The coefficient matrix [R m ] described in the MSC.NASTRAN Reference Manual, Section must be nonsingular. SOL 60 or PARAM, CHECKOUT in SOLs 101 – 200 may be used to check for this condition. 4. Dependent degrees of freedom assigned by one rigid element may not also be assigned dependent by another rigid element or by a multipoint constraint.

S2-63NAS105, Section 2, May 2005 RBE3 – BULK DATA ENTRY (Cont.) Remarks: 5. Rigid elements, unlike MPCs, are not selected through the Case Control section. 6. Forces of multipoint constraint may be recovered in the linear structured solution sequences (101 – 200) with the MPCFORCE Case Control command and SOL 24 with RF Alter RF24D24 (see the MSC.NASTRAN Reference Manual, Chapter 15). 7. Rigid elements are ignored in heat transfer problems. 8. The m-set coordinates specified on this entry may not be specified on other entries that define mutually exclusive sets. See the MSC.NASTRAN Quick Reference Guide, Appendix B for a list of these entries. 9. The formulation for RBE3 was change in V70.7. It now gives unit independent consistent answers. Only models that connected rotation DOF for Ci (ignoring the recommendation in Remark 1) are affected. The formulation prior to V70.7 may be obtained by setting SYSTEM(310)=1.

S2-64NAS105, Section 2, May 2005 RBE3 Description n By default, the reference grid DOF will be the dependent DOF. n Number of dependent DOF is equal to the number of DOF on the REFC field. n Dependent DOF cannot be SPCd, OMITted, SUPORTed or be dependent on other RBE/MPC elements.

S2-65NAS105, Section 2, May 2005 RBE3 Is Not Rigid! n RBE3 vs. RBE2 u RBE3 allows warping and 3D effects u In this example, RBE2 enforces beam theory (plane sections remain planar) RBE3 RBE2

S2-66NAS105, Section 2, May 2005 RBE3: How it Works? n Forces/Moments applied at reference grid are distributed to the master grids in same manner as classical bolt pattern analysis. u Step 1: Applied loads are transferred to the CG of the weighted grid group using an equivalent Force/Moment u Step 2: Applied loads at CG transferred to master grids according to each grids weighting factor Note: If independent DOFs contain rotations, RBE3 does not work like classical bolt pattern analysis.

S2-67NAS105, Section 2, May 2005 RBE3: How it Works? n If independent DOFs include rotations, moments at CG are mapped as equivalent force couples, and concentrated moments. n Step 1: Transform force/moment at reference grid to equivalent force/moment at weighted CG of master grids. M CG =M A +F A * e F CG =F A CG F CG M CG FAFA MAMA Reference Grid e CG

S2-68NAS105, Section 2, May 2005 RBE3: How it Works? n Step 2: Move loads at CG to master grids according to their weighting values. u Force at CG divided amongst master grids according to weighting factors W i u Moment at CG mapped as equivalent force couples on master grids according to weighting factors W i

S2-69NAS105, Section 2, May 2005 RBE3: How it Works? n Step 2: Continued… CG F CG M CG Total force at each master node is sum of... Forces derived from force at CG: F if = F CG {W i / W i } F 1m F 3m F 2m Plus Forces derived from moment at CG: F im = {M cg W i r i /(W 1 r 1 2 +W 2 r 2 2 +W 3 r 3 2 )}

S2-70NAS105, Section 2, May 2005 RBE3: How it Works? n Masses on reference grid are smeared to the master grids similar to how forces are distributed u Mass is distributed to the master grids according to their weighting factors u Motion of reference mass results in inertial force that gets transferred to master grids u Reference node inertial force is distributed in same manner as when static force is applied to the reference grid.

S2-71NAS105, Section 2, May 2005 Example 1 n RBE3 distribution of loads when force at reference grid at CG passes through CG of master grids

S2-72NAS105, Section 2, May 2005 Example 1: Force Through CG n Simply supported beam u 10 elements, 11 nodes numbered 1 through 11 n 100 LB. Force in negative Y on reference grid 99

S2-73NAS105, Section 2, May 2005 Example 1: Force Through CG n Load through CG with uniform weighting factors results in uniform load distribution

S2-74NAS105, Section 2, May 2005 Example 1: Force Through CG n Comments… u 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 RBE3, 11,,99, , 1., 123, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 *** USER FATAL MESSAGE 2038 (RBE3D) USER ACTION: ADD MORE DOFS TO THE CONNECTED POINTS TO INSURE THAT THEY CAN CONSTRAIN ALL 6 RIGID BODY MODES OF THE ELEMENT. Corrected RBE3 data (add DOF4 to one or more Master Grids): RBE3, 11,,99, , 1., 1234, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ~

S2-75NAS105, Section 2, May 2005 Example 2 n How does the RBE3 distribute loads when force on reference grid does not pass through CG of master grids?

S2-76NAS105, Section 2, May 2005 Example 2: Load not through CG n The resulting force distribution is not intuitively obvious u 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.

S2-77NAS105, Section 2, May 2005 RSPLINE – THE LINEAR SPLINE n The RSPLINE is an interpolation element using the equations of a beam – Example

S2-78NAS105, Section 2, May 2005 RSPLINE – THE LINEAR SPLINE (Cont.) n Basic equations n In matrix notation match at grid points

S2-79NAS105, Section 2, May 2005 RSPLINE – BULK DATA ENTRY Defines multipoint constraints for the interpolation of displacements at grid points.

S2-80NAS105, Section 2, May 2005 RSPLINE – BULK DATA ENTRY (Cont.) Remarks: 1. Displacements are interpolated from the equations of an elastic beam passing through grid points. 2. A blank field for Ci indicates that all six degrees of freedom of Gi are independent. Since G1 must be independent, no field is provided for C1. Since the last grid point must also be independent, the last field must be a Gi, not a Ci. For the example shown G1, G3, and G6 are independent. G2 has six constrained degrees of freedom while G4 and G5 each have three. 3. Dependent (I.e., constrained) degrees of freedom assigned by one rigid element may not be assigned dependent by another rigid element or by a multipoint constraint. 4. Degrees of freedom declared to be independent by one rigid body element can be made dependent by another rigid body element or by a multipoint constraint. 5. EIDs must be unique. 6. Rigid elements (including RSPLINE), unlike MPCs, are not selected through the Case Control Section.

S2-81NAS105, Section 2, May 2005 RSPLINE – BULK DATA ENTRY (Cont.) Remarks: (Cont.) 7. Forces of multipoint constraint may be recovered in the linear structured solution sequences (101 –200) with the MPCFORCE Case Control command and in SOL 24 with RF Alter RF24D24 (see the MSC.NASTRAN Reference Manual, Chapter 15). 8. Rigid elements are ignored in heat transfer problems. 9. See the MSC.NASTRAN Reference Manual, Section 5.5, for a discussion of rigid elements. 10. The m-set coordinates specified on this entry may not be specified on other entries that define mutually exclusive sets. See the MSC.NASTRAN Quick Reference Guide, Appendix B for a list of these entries. 11. The constraint coefficient matrix is affected by the order of the Gi Ci pairs on the RSPLINE entry. The order of the pairs should be specified in the same order that they appear along the line that joins the two regions. If this order is not followed then the RSPLINE will have folds in it that may yield some unexpected interpolation results. 12. The independent degrees of freedom which are the rotation components most nearly parallel to the line joining the regions should not normally be constrained.

S2-82NAS105, Section 2, May 2005 RSSCON – BULK DATA ENTRY Defines multipoint constraints to model clamped connections of shell-to-solid elements.

S2-83NAS105, Section 2, May 2005 RSSCON – BULK DATA ENTRY (Cont.)

S2-84NAS105, Section 2, May 2005 RSSCON – BULK DATA ENTRY (Cont.) Remarks: 1. RSSCON generates a multipoint constraint that models a clamped connection between a shell and a solid element. The shell degrees of freedom are put in the dependent set (m-set). The translational degrees of freedom of the shell edge are connected to the translational degrees of freedom of the upper and lower solid edge. The rotational degrees of freedom of the shell are connected to the translational degrees of freedom of the lower and upper edges of the solid element face. Poissons ratio effect are considered in the translational degrees of freedom. 2. The shell grid point must lie on the line connecting the two solid grid points. It can have an offset from this line, which can be not be more than 5% of the distance between the two solid grid points. The shell grid points that are out of the tolerance will not be constrained, and a fatal message will be issued. This tolerance is adjustable. Please see PARAM, TOLRSC, and PARAM, SEPIXOVR. 3. When using the TYPE = ELAM option. a. The elements may be p-elements. The solid elements are CHEXA, CPENTA, and CTETRA with and without midside nodes. The shell elements are CQUAD4, CTRIA3, CQUADR, CTRIAR, CQUAD8, or CTRIA6. b. In case of p-elements, the p-value of the shell element edge is adjusted to the higher of the p-value of the upper of lower solid p-element edge. If one of the elements is an h-elements, then the p-value of the adjacent edge is lowered to 1.

S2-85NAS105, Section 2, May 2005 RSSCON – BULK DATA ENTRY (Cont.) Remarks: (Cont.) c. Both the shell and solid elements have to belong to the same superelement. This restriction can be bypassed using SEELT entry to reassign the downstream boundary element to an upstream superelement. d. When a straight shell p-element edge and a solid p-element are connected, the geometry of the shell edge is not changed to fit the solid face. When a curved shell p-element edge and a solid p-element are connected, the two solid edges and solid face are not changed to match the shell edge. e. It is not recommended to connect more than one shell element to the same solid using the ELEM option. If attempted, conflicts in the multipoint constraint relations may leads to UFM When using TYPE_GRID option a. The GRID option does not verify that the grids used are valid shell and/or solid grids. b. The hierarchical degrees of freedom of p-element edges are not constrained. The GRID option is therefore not recommended for p-elements. c. The grids in the GRID option can be different superelements. The shell grid must be in the upstream superelement. 5. It is recommended that the height of the solid elements face is approximately equal to the shell elements thickness of the shell. The shell edge should then be placed in the middle of the solid face.

S2-86NAS105, Section 2, May 2005 RSSCON – BULK DATA ENTRY (Cont.) Remarks: (Cont.) 6. The shell edge may coincide with the upper or lower edge of the solid face. 7. The RSSCON entry, unlike MPCs, cannot be selected through the Case Control Section. 8. Forces of multipoint constraints may be recovered in the linear structured sequences (SOLs 101 through 200) with MPCFORCE Case Control command. 9. The RSSCON is ignored in heat-transfer problems. 10. The shell edge may coincide with the upper or lower edge of the solid face. 11. The RSSCON entry, unlike MPCs, cannot be selected through the Case Control Section. 12. Forces of multiple constraints may be recovered in the linear structure solution sequences (SOLs 101 through 200) with the MPCFORCE Case Control command. 13. The RSSCON is ignored in heat-transfer problems. 14. The m-set coordinates (shell degrees of freedom) may not be specified on other entries that define mutually exclusive sets. See the MSC.NASTRAN Quick Reference Guide, Appendix B for a list of these entries.

S2-87NAS105, Section 2, May 2005 RSSCON – BULK DATA ENTRY (Cont.) Figure 1. Shell Elements Connected to the Faces of Solid Elements

S2-88NAS105, Section 2, May 2005