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• FEM started and developed in the linear analysis of solid and structures

Chapter 4. Formulation of FEM – Linear Analysis in Solid and Structural Mechanics. • FEM started and developed in the linear analysis of solid and structures • Displacement Method (widely used in the analysis of solid) • Mixed formulation (plate & shell, incompressible solid)

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• FEM started and developed in the linear analysis of solid and structures

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  1. Chapter 4. Formulation of FEM – Linear Analysis in Solid and Structural Mechanics • FEM started and developed in the linear analysis of solid and structures •Displacement Method (widely used in the analysis of solid) •Mixed formulation (plate & shell, incompressible solid) • Formulation of displacement-based FEM - Principle of virtual work - Analysis of beam and truss

  2. Symmetric Symmetric

  3. Ex 4.1

  4. Ex 4.1(continued)

  5. General Derivation of Finite Element Equilibrium Equation

  6. Principle of Virtual Work (Displacement) For any small virtual displacement applied on body of equilibrium, the total internal virtual work is equal to the total of external virtual work Virtual displacements are not “real(actual)” displacements but theoretically possible displacements. Assume we know exact displacement field, then we can calculate and . For any given , can be calculated and equation (4.7) hold. If not, is not the correct stress vector.

  7. Ex 4.2 Equilibrium equation for the elastic body is

  8. Ex 4.2 (Continued) Using divergence theorem

  9. Ex 4.3 & Ex 4.4 Stationality of is equivalent to principle of virtual work. Eq 4.7 holds means (equilibrium, compatibility, constitutive law) are satisfied.

  10. Ex 4.5

  11. Ex 4.5(Continued)

  12. Ex 4.5(Continued)

  13. Plane stress (thin plate) Plane strain (long column)

  14. Ex 4.6

  15. Ex 4.6 (Continued) 6(2,4) 3(0,4) (1,1) (-1,1) (-1,-1) (1,-1) 2(0,2) 5(2,2)

  16. Ex 4.6 (Continued) In a general FEA, differential equilibrium is not exactly satisfied at all continuum domain, however, nodal point equilibrium and element equilibrium are always satisfied.

  17. Ex 4.7 (Linearly Varying Load)

  18. Ex 4.7 (continued)

  19. Stress Equilibrium “Exact displacement” means ① DE of equilibrium ② compatibility ③ constitutive law ④ BCs are satisfied • FEA • Structure is discretized as nodes and elements • Externally applied forces are lumped to equivalent global nodal forces • Global nodal forces are equilibrated by element nodal forces that are equivalent to element stress • Compatibility, constitutive law, equilibrium are satisfied at nodes and elements

  20. : element nodal disp in global coord : element nodal disp in local coord : Transformation matrix (sec 2.4)

  21. Ex 4.9

  22. Ex 4.10

  23. Ex 4.10 (Continued) R

  24. Ex 4.11

  25. Ex 4.11 (Continued)

  26. Ex 4.11 (Continued) For element A: LM=[2 3 0 0 0 1 4 5] , LM:connectivity Do i=1,8 Do j=1,8 if (LM(i).eq.0) goto 20 if (LM(j).eq.0) goto 20 K(LM(i),LM(j))=KA(i,j) 10 enddo j 20 enddo i Element B: LM=[6 7 4 5] Element C: LM=[6 7 2 5] Element D: LM=[0 0 0 6 7 8]

  27. Ex 4.11 (Continued)

  28. Imposition of Displacement Boundary Conditions Static condensation

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