OBJECTIVES & ASSUMPTIONS

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OBJECTIVES & ASSUMPTIONS Z Project Objectives: Construct a template for the 5-node pyramid element Conduct higher-order patch tests Verify element formulation Determine the optimal element in bending

2 OBJECTIVES & ASSUMPTIONS Z Project Objectives: Construct a template for the 5-node pyramid element Conduct higher-order patch tests Verify element formulation Determine the optimal element in bending X Y h Project Assumptions: Rectangular base in X-Y plane with dimensions a and b, orthogonal to and symmetric about X and Y axes Height, h, and apex oriented along Z axis Poisson s ratio is neglected (ν = 0) 15 total DOFs (3 per node) a PYRA5 b

3 PREVIOUS WORK: TEMPLATES & RANK Template Theory: The master stiffness matrix, K, can be decomposed into a basic stiffness matrix, K b, and a parameterized higher-order stiffness matrix, K h Rank and Parameters: The rank of the master stiffness matrix is 9 (15 DOFs 6 RBM) The rank of the basic stiffness matrix is 6 (derived from the 6 RBM) The correct rank of the higher order stiffness matrix must be 3 Accordingly, 3 parameters are used to optimize the higher order stiffness matrix

4 PREVIOUS WORK: FINDING K B The basic stiffness matrix, K b, is a function of the element volume, V, the element force lumping matrix, L, and the element material matrix, E. It is defined by the following equation: The force lumping matrix, L, was found by: Determining the direction cosines of pyramid face (dependent upon element geometry) Resolving facial tractions with Cauchy s Formula and an assumed stress field Converting tractions to forces and lumping at nodes

5 PREVIOUS WORK: NUMERICAL K B & EIGENVALUES Mathematica Output of Numerical K b : Mathematica Output of Numerical K b Eigenvalues:

6 FINDING K H : ASSUMED DISPLACEMENTS Apply 6 constant curvature bending modes to a prism consisting of 6, 5-node pyramids Compute displacement fields from curvatures: 9

7 FINDING K H : ASSUMED DISPLACEMENTS Create the basic-mode matrix, G r, which spans the element s 6 rigid body modes, and the constant-strain matrix, G c, which spans the element s 6 constant strain states:

8 FINDING K H : ASSUMED DISPLACEMENTS K h must be orthogonal to rigid body modes and constant strain states To ensure orthogonality, use a geometric projector, H h, derived from the assumed displacement fields: Ideally, the following relationships should hold true: In our case, we were unable to obtain orthogonality and forced to pursue a different formulation method

orthogonality Average value of h-strain over element is 0 Enables

9 FINDING K H : ENERGY ORTHOGONALITY Develop G r and G C in an identical manner Use a weak orthogonality called energy orthogonality Average value of h-strain over element is 0 Enables orthogonality between K h and rigid body modes and constant strain states Egyptian Pyramid

10 FINDING K H : STRESS HYBRID METHOD & FILTERING K h determined for each face as if it were an independent element using the stress hybrid method Produces an excessive K h rank of 12 compared to the expected rank of 3 Filter each K h using P rc to arrive at a rank of 3

11 FINDING K H : PARAMETERIZATION Parameterized grouping of symmetric faces 125 combined with combined with kept independent Apply β i (i=1 3) to three combinations to match rank of filtered K h matrices Combine three filtered & parameterized 15 x 15 K h matrices into single K h

12 FINDING K Combine K b found through force lumping formulation with parameterized K h found through energy orthogonality and the stress hybrid method Limited to previously described geometry in Mathematica code: Rectangular base is centered in the X-Y plane and the apex is along the Z-axis

13 CHECKING THE RANK OF K, K B, & K H K must have rank of 9: K b has rank of 6 Filtered K h has rank of 3 The ranks combine linearly Verify numerically:

14 NUMERICAL KH

15 NUMERICAL K & EIGENVALUES

specified geometry Conduct patch tests Verify element

16 FUTURE WORK Improve modules for geometry based on nodal coordinates and arbitrary geometry rather than specified geometry Conduct patch tests Verify element formulation Find optimal parameters for bending Food Pyramid

17 LIMITATIONS AND LESSONS LEARNED K b and K h applicable only to pyramid geometry previously described Cannot easily re-orient pyramid to perform patch tests More in depth knowledge of templates needed No clear procedure for template formulation Race against time no guarantee a given approach will work

18 Disclaimer: Not the actual value of our pyramid Thanks for your time! Questions? Special thanks to Professor Felippa for all his question answering and module help!

. D CR Nomenclature D 1

. D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the