1. NUMERICAL METHODS IN APPLIED STRUCTURAL MECHANICS
Lecture notes:
Prof. Maurício V. Donadon

2. Non-linear static problems

3. Introduction

4. Sources of nonlinearities in structural analysis
Geometrical non-linearity
Non-linear material behaviour
Non-linear boundary conditions

5. Geometrical nonlinearities Normal strain-displacement relationships

6. Geometrical nonlinearities Shear strain-displacement relationships

7. Geometrical nonlinearities

8. Nonlinear material behaviour
ELASTIC
ELASTO-PLASTIC
MICROCRACKING
ELASTIC +

9. Nonlinear material behaviour

10. Nonlinear material behaviour

11. Nonlinear boundary conditions
Transient boundary problems:
Boundary conditions change during the analysis!!!

12. Nonlinear boundary conditions

13. Solution methods for non-linear static problems

14. Solution methods
Incremental solutions
Iterative solutions
Combined incremental/iterative solutions
Arc-length method
Quasi-static solutions

15. General form for a static problem
Example: Linear/Nonlinear Spring
K = K0 f(x) Fe

16. Example: Linear/Nonlinear Spring

17. General form for a static problem
Example: Linear/Nonlinear Spring
Trivial solution: Displacement control
Non-trivial solution: Load control which is commonly used in structural analyses!!!

18. Incremental solution

19. INCREMENTAL SOLUTION METHOD BASED ON THE EULER METHOD

20. THE EULER METHOD ALGORITHM

21. Example: Nonlinear Spring

22. Example: Nonlinear Spring

23. Example: Nonlinear Spring

24. Iterative solution

25. ITERATIVE SOLUTION BASED ON THE NEWTON RAPHSON METHOD

26. THE NEWTON RAPHSON ALGORITHM

27. Example: Nonlinear Spring

28. Example: Nonlinear Spring

29. Example: Nonlinear Spring

30. Combined incremental/iterative solutions

31. COMBINED INCREMENTAL/ITERATIVE SOLUTIONS

32. INCREMENTAL/ITERATIVE SOLUTION ALGORITHM

33. Example: Nonlinear Spring

34. Example: Nonlinear Spring

35. Example: Nonlinear Spring

36. Arc-length
Method

37. Highly non-linear structural responses
snap-through
snap-back

39. Arc-length method: Constraint equation
ψ: scale factor, scale forces to the same order of magnitude of the displacements
qef: Fixed external load level vector

40. Arc-length method: Residual force
Equations to be solved simultaneously

41. Arc-length method
Subscript “o” refers to old (previous) iteration
Subscript “n” refers to current iteration

42. Equations to be solved simultaneously Augmented stiffness matrix
Arc-length method
Equations to be solved simultaneously
Augmented stiffness matrix

43. Arc-length method
Error function computation:

44. Spherical Arc-length method
The computational cost associated with the inversion of the augmented stiffness matrix during the iterations is very high because the augmented stiffness matrix is neither symmetric nor banded!
Better solution: Spherical Arc-length!!!!

45. Spherical Arc-length method
Instead of solving both constraint and equilibrium equations simultaneously, one may assume displacement control at single point. Thus, by assuming displacement control the residual forces can be written as follow,

46. Spherical Arc-length method
δpt must be computed for the initial predictor step based on the Forward-Euler tangential predictor and it is fixed because Kt does not change during the iterations

47. Forward-Euler Tangential Predictor
Spherical Arc-length method
Forward-Euler Tangential Predictor
Substituting into the constraint equation we obtain,

48. Spherical Arc-length method

49. Spherical Arc-length method
δλ can be found by substituting the previous equation into the constraint equation,
which leads to the following quadratic equation,

50. Spherical Arc-length method
with,
Which can be solved for δλ.The choice of the root will be based on the cylindrical Arc-length method

51. Spherical Arc-length method
The great advantage of the spherical arc-length over the standard arc-length method is that the former only requires the factorisation of the banded symmetric tangent stiffness matrix. Therefore it avoids the use of the augmented stiffness matrix, which is neither symmetric nor banded!!!

52. Cylindrical Arc-length method
The cylindrical Arc-length consists of setting to zero the scaling factor ψ. In practice, this parameter is not known a priori. Moreover, setting to zero the scaling factor simplifies the choice of the appropriated root of the quadratic equation used to compute δλ, that is,

53. Cylindrical Arc-length method
Choosing the root
Solution 1
Solution 2

54. Cylindrical Arc-length method
Choosing the root

55. Cylindrical Arc-length method
Choosing the root
Having computed δλ, update displacement and load parameter

56. Convergence criterion
Cylindrical Arc-length method
Convergence criterion

57. Quasi-static solutions

58. Example: Nonlinear Spring
DYNAMIC RELAXATION
Example: Nonlinear Spring
K = K0 f(x) M
Fe(t) C

59. DYNAMIC RELAXATION Central difference method
Critical time step computation

60. DYNAMIC RELAXATION Central difference method Displacement field
Velocity field

61. DYNAMIC RELAXATION Damping definition Critical damping
Rayleigh damping

62. EXPLICIT TIME INTEGRATION ALGORITHM
Initial conditions, v0, σ0, n=0, t=0, compute M
Compute acceleration an = M-1Fe,n
Update nodal velocities: vn+1/2 = vn+1/2-α + αΔtan
α = 1/2 if n=0
α = 1 if n>0
Update nodal displacements: un+1 = un+ Δtvn+1/2
Compute strains
Compute stresses
Compute internal forces
Compute residual force vector: Fi - Fe
Update counter and time: n = n+1, t = t+Δt
If simulation not complete go to step 2

63. Example: Nonlinear Spring

64. Example: Nonlinear Spring

65. Example: Nonlinear Spring – 1.0 N/s

66. Example: Nonlinear Spring – 0.1 N/s

67. Example: Nonlinear Spring – 0.01 N/s

68. Example: Nonlinear Spring – Damping effect
Thigh=0.12
Tlow=0.22

69. Example: Nonlinear Spring – Damping effect

70. Example: Nonlinear Spring – Damping effect

71. Example: Nonlinear Spring – Damping effect

72. Example: Nonlinear Spring – Damping effect

73. Over damping effects in dynamic relaxation
Over damping MUST BE AVOIDED in dynamic relaxation methods!
Special care must be taken with over damping
Over damping increases artificially the internal energy of the system!!!!