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

2. Examples of Geometric Nonlinearity

3. Shallow truss problem

4. Shallow truss problem
Initial configuration z w W L
E, A

5. Shallow truss problem formulation
Equilibrium equation
Strain-displacement relationship for the bars

6. Shallow truss problem formulation
Internal force in the bar
Resultant equilibrium equation

7. Shallow truss problem simulation
Truss dimensions and properties:
EA = 50 MN
z = 25 mm
L = 2500 mm
Ks = 1.35 N/mm
Ks = 1.0 N/mm
Ks = 0.0 N/mm

8. Shallow truss problem simulation
Solution methods to be tested under load control:
Incremental solutions (EULER)
Iterative solutions (N-R)
Combined incremental/iterative solutions (EULER + N-R)
Quasi-static solutions (DYNAMIC RELAXATION)

9. Shallow truss problem simulation
ANALYTICAL SOLUTION

10. Shallow truss problem
EULER METHOD

11. Shallow truss problem
EULER METHOD

12. Shallow truss problem
EULER METHOD

13. Shallow truss problem
EULER METHOD

14. Shallow truss problem
EULER METHOD

15. Shallow truss problem
N-R METHOD

16. Shallow truss problem
N-R METHOD

17. Shallow truss problem
N-R METHOD

18. Shallow truss problem
N-R METHOD

19. Shallow truss problem
N-R METHOD

20. Shallow truss problem
N-R METHOD

21. Shallow truss problem
N-R METHOD

22. DYNAMIC RELAXATION – Ks=1.35
Shallow truss problem
DYNAMIC RELAXATION – Ks=1.35

23. Shallow truss problem
DYNAMIC RELAXATION

24. DYNAMIC RELAXATION – 500 N/s – Ks=1.35
Shallow truss problem
DYNAMIC RELAXATION – 500 N/s – Ks=1.35

25. DYNAMIC RELAXATION – 50 N/s – Ks=1.35
Shallow truss problem
DYNAMIC RELAXATION – 50 N/s – Ks=1.35

26. DYNAMIC RELAXATION – 5 N/s – Ks=1.35
Shallow truss problem
DYNAMIC RELAXATION – 5 N/s – Ks=1.35

27. DYNAMIC RELAXATION – Ks=1.0 – Ks=1.35
Shallow truss problem
DYNAMIC RELAXATION – Ks=1.0 – Ks=1.35

28. DYNAMIC RELAXATION – Ks=1.0
Shallow truss problem
DYNAMIC RELAXATION – Ks=1.0

29. DYNAMIC RELAXATION – Ks=1.0
Shallow truss problem
DYNAMIC RELAXATION – Ks=1.0

30. DYNAMIC RELAXATION – Ks=1.0
Shallow truss problem
DYNAMIC RELAXATION – Ks=1.0

31. DYNAMIC RELAXATION – Ks=0.0
Shallow truss problem
DYNAMIC RELAXATION – Ks=0.0

32. DYNAMIC RELAXATION – Ks=0.0 – Disp. control
Shallow truss problem
DYNAMIC RELAXATION – Ks=0.0 – Disp. control

33. DYNAMIC RELAXATION – Ks=0.0 – Disp. control
Shallow truss problem
DYNAMIC RELAXATION – Ks=0.0 – Disp. control

34. ARC-LENGTH METHOD – Ks=1.35
Shallow truss problem
ARC-LENGTH METHOD – Ks=1.35

35. ARC-LENGTH METHOD – Ks=1.0
Shallow truss problem
ARC-LENGTH METHOD – Ks=1.0

36. ARC-LENGTH METHOD – Ks=0.0
Shallow truss problem
ARC-LENGTH METHOD – Ks=0.0