2. ECIV 720 A Advanced Structural Mechanics and Analysis Non-Linear Problems in Solid and Structural Mechanics Special Topics

3. Introduction Nonlinear Behavior: Response is not directly proportional to the action that produces it. P 

4. Introduction Recall Assumptions Small Deformations Linear Elastic Behavior

5. Introduction Linear Behavior

6. Introduction A. Small Displacements Inegrations over undeformed volume Strain-displacement matrix does not depend on

7. Introduction B. Linear Elastic Material Matrix [E] does not depend on

8. Introduction C. Boundary Conditions do not change (Implied Assumption) Constraints do not depend on

9. Introduction If any of the assumptions is NOT satisfied NONLINEARITIES Material Assumption B not satisfied Geometric Assumption A & or C not satisfied

10. Classification of Nonlinear Analysis Small Displacements, small rotations Nonlinear stress-strain relation

11. Classification of Nonlinear Analysis Large Displacements, large rotations and small strains – Linear or nonlinear material behavior

12. Classification of Nonlinear Analysis Large Displacements, large rotations and large strains – Linear or nonlinear material behavior

13. Classification of Nonlinear Analysis Change in Boundary Condition

14. Classification of Nonlinear Analysis

15. Nonlinear Analysis Cannot immediately solve for {d} Iterative Process Required to obtain {d} so that equilibrium is satisfied

16. Solution Methods

17. Newton-Raphson

18. Newton Raphson With initial conditions

19. Modified Newton-Raphson

20. SPECIAL TOPICS Boundary Conditions Elimination Approach Penalty Approach Special Type Elements

21. Boundary Conditions – Elimination Approach Consider B C u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn u 1 =a Singular, No BC Applied

22. Boundary Conditions – Elimination Approach Boundary Conditions u 1 =a

23. Boundary Conditions – Elimination Approach Consequently, Equilibrium requires that Since u 1 =a known, DOF 1 is eliminated from

24. Boundary Conditions – Elimination Approach ……… K ff u f =P f + K fs u s

25. Boundary Conditions – Elimination Approach k ii k ij k ik k il k im uiui ujuj ukuk ulul k ji k jj k jk k jl k jm k ki k kj k kk k kl k km k li k lj k lk k ll k lm k li k lj k lk k ll k lm umum = PiPi PjPj PkPk PlPl PmPm K ff K fs K sf K ss uf uf Pf Pf usus PsPs

26. Boundary Conditions – Elimination Approach k ii k ij k ik k il k im uiui ujuj ukuk ulul k ji k jj k jk k jl k jm k ki k kj k kk k kl k km k li k lj k lk k ll k lm k li k lj k lk k ll k lm umum = PiPi PjPj PkPk PlPl PmPm K ff K fs uf uf Pf Pf usus K ff u f + K fs u s =P f K sf K ss PsPs K sf u f + K ss u s =P s u f = K ff (P f + K fs u s )

27. Boundary Conditions Penalty Approach u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn k=C large stiffness Boundary Conditions u 1 =a

28. Boundary Conditions Penalty Approach u1u1 u2u2 u3u3 u4u4 uiui u i+1 u n-1 unun P4P4 PiPi PnPn k=C large stiffness Consequently, for Equilibrium Contributes to 

29. Boundary Conditions Penalty Approach The only modifications Support Reaction is the force in the spring

30. Choice of C Rule of Thumb Error is always introduced and it depends on C Penalty approach is easy to implement

31. Changing Directions of Restraints 1 2 3 4 x,u y,v 

32. Changing Directions of Restraints 1 2 3 4 e.g. for truss

33. Changing Directions of Restraints

34. 1 2 3 4 Introduce Transformation In stiffness matrix…

35. Connecting Dissimilar Elements Simple Cases 1 2 3 4 5 6

36. 1 2 3 4 5 6 a b L 

37. Hinge Beam

38. Connecting Dissimilar Elements Simple Cases Beam Stresses are not accurately computed

39. Connecting Dissimilar Elements Eccentric Stiffeners

40. Use Eccentric Stiffeners Connecting Dissimilar Elements Eccentric Stiffeners 1 2 3 4 Slave Master

41. Connecting Dissimilar Elements Eccentric Stiffeners bb

42. bb 3,4 Slave 1,2 Master

43. Connecting Dissimilar Elements Eccentric Stiffeners bb The assembly displays the correct stiffness in states of pure stretching and pure bending The assembly is too flexible when curvature varies – Use finer mesh

44. Connecting Dissimilar Elements Rigid Elements Rigid element is of any shape and size Generalization of Eccentric Stiffeners – Multipoint Constraints Use it to enforce a relation among two or more dof

45. Connecting Dissimilar Elements Rigid Elements e.g. 1 2 3 a b 1-2-3 Perfectly Rigid Rigid Body Motion described by u 1, v 1, u 2

46. Connecting Dissimilar Elements Rigid Elements

47. Elastic Foundations Strain Energy RECALL

48. Elastic Foundations RECALL

49. Elastic Foundations Additional stiffness Due to Elastic Support RECALL

50. Elastic Foundations + RECALL

51. Elastic Foundations – General Cases Soil x y z Foundation Plate/Shell/Solid of any size/shape/order Winkler Foundation

52. Elastic Foundations – General Cases Winkler Foundation Stiffness Matrix s is the foundation modulus H are the Shape functions of the “attached element”

53. Winkler Foundations Resists displacements normal to surface only Deflects only where load is applied Adequate for many problems

54. Other Foundations Resists displacements normal to surface only They entire foundation surface deflects More complicated by far than Winkler Yields full matrices

55. Elastic Foundations – General Cases Soil x y z Infinite

56. Infinite Elements

58. Use Shape Functions that force the field variable to approach the far-field value at infinity but retain finite size of element Use conventional Shape Functions for field variable Use shape functions for geometry that place one boundary at infinity or

59. Shape functions for infinite geometry Element in Physical Space Mapped Element Reasonable approximations

60. Shape functions for infinite geometry Node 3 need not be explicitly present