1 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Lecture notes: Prof. Maurício V. Donadon NUMERICAL METHODS IN APPLIED STRUCTURAL MECHANICS

2 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Non-linear static problems

3 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Introduction

4 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Sources of nonlinearities in structural analysis Geometrical non-linearity Non-linear material behaviour Non-linear boundary conditions

5 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Geometrical nonlinearities Normal strain-displacement relationships

6 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Geometrical nonlinearities Shear strain-displacement relationships

7 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Geometrical nonlinearities

8 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Nonlinear material behaviour ELASTIC ELASTO-PLASTIC ELASTIC + MICROCRACKING

9 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Nonlinear material behaviour

10 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Nonlinear material behaviour

11 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Nonlinear boundary conditions Transient boundary problems: Boundary conditions change during the analysis!!!

12 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Nonlinear boundary conditions

13 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Solution methods for non-linear static problems

14 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Incremental solutions Iterative solutions Combined incremental/iterative solutions Arc-length method Quasi-static solutions Solution methods

15 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 General form for a static problem K = K 0 f(x) FeFe Example: Linear/Nonlinear Spring

16 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Linear/Nonlinear Spring

17 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 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 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Incremental solution

19 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 INCREMENTAL SOLUTION METHOD BASED ON THE EULER METHOD

20 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 THE EULER METHOD ALGORITHM

21 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

22 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

23 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

24 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Iterative solution

25 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 ITERATIVE SOLUTION BASED ON THE NEWTON RAPHSON METHOD

26 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 THE NEWTON RAPHSON ALGORITHM

27 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

28 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

29 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

30 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Combined incremental/iterative solutions

31 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 COMBINED INCREMENTAL/ITERATIVE SOLUTIONS

32 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 INCREMENTAL/ITERATIVE SOLUTION ALGORITHM

33 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

34 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

35 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

36 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length Method

37 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Highly non-linear structural responses snap-through snap-back

38 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method: Single DOF

39 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method: Constraint equation b: scale factor, scale forces to the same order of magnitude of the displacements

40 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method: Residual force Equations to be solved simultaneously

41 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method

42 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method

43 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method Equations to be solved simultaneously Augmented stiffness matrix

44 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method Error function computation:

45 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method Constant arc-length algorithm:

46 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Arc-length method Variable arc-length algorithm:

47 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method: b=0 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! The scale factor is unknown “a priori” Better solution: Spherical Arc-length!!!!

48 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method

49 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method

50 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method

51 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method

52 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method

53 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method Choosing the root Solution 1Solution 2

54 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Spherical Arc-length method The predictor solution

55 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256

56 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Quasi-static solutions

57 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 DYNAMIC RELAXATION Example: Nonlinear Spring K = K 0 f(x) M F e (t) C

58 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Central difference method DYNAMIC RELAXATION Critical time step computation

59 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Central difference method DYNAMIC RELAXATION Displacement field Velocity field

60 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Damping definition DYNAMIC RELAXATION Critical damping Rayleigh damping

61 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 EXPLICIT TIME INTEGRATION ALGORITHM 1. Initial conditions, v 0, σ 0, n=0, t=0, compute M 2. Compute acceleration a n = M -1 F e,n 3. Update nodal velocities: v n+1/2 = v n+1/2-α + αΔta n 4.α = 1/2 if n=0 5.α = 1 if n>0 6.Update nodal displacements: u n+1 = u n + Δtv n+1/2 7.Compute strains 8.Compute stresses 9.Compute internal forces 10. Compute residual force vector: F i - F e 11.Update counter and time: n = n+1, t = t+Δt 12.If simulation not complete go to step 2

62 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

63 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring

64 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – 1.0 N/s

65 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – 0.1 N/s

66 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – 0.01 N/s

67 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – Damping effect T low =0.22 T high =0.12

68 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – Damping effect

69 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – Damping effect

70 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – Damping effect

71 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Example: Nonlinear Spring – Damping effect

72 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 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!!!!