1. INTRODUCTION INTO FINITE ELEMENT NONLINEAR ANALYSES
Doc. Ing. Vladimír Ivančo, PhD.
Technical University of Košice
Faculty of Mechanical Engineering
Department of Applied Mechanics and Mechatronics
HS Wismar, June 2009

2. CONTENS 1. Introduction 1.1 Types of structural nonlinearities
1.2 Concept of time curves
2. Geometrically nonlinear finite element analysis
3. Incremental – iterative solution
3.1 Incremental method
3.2 Iterative methods
4. Material nonlinearities
5. Examples

3. Introduction 1.1 Sources of nonlinearities
Linear static analysis - the most common and the most simplified analysis of structures is based on assumptions:
static = loading is so slow that dynamic effects can be neglected
linear = a) material obeys Hooke’s law b) external forces are conservative c) supports remain unchanged during loading d) deformations are so small that change of the structure configuration is neglectable

4. Consequences:
displacements and stresses are proportional to loads, principle of superposition holds
in FEM we obtain a set of linear algebraic equations for computation of displacements
where
K – global stiffness matrix
d – vector of unknown nodal displacements
F – vector of external nodal forces

5. Nonlinear analysis – sources of nonlinearities can be classified as
Geometric nonlinearities - changes of the structure shape (or configuration changes) cannot be neglected and its deformed configuration should be considered.
Material nonlinearities - material behaves nonlinearly and linear Hooke’s law cannot be used. More complicated material models should be then used instead e.g.
nonlinear elastic (Mooney-Rivlin’s model for materials like rubber), elastoplastic (Huber-von Mises for metals, Drucker-Prager model to simulate the behaviour of granular soil materials such as sand and gravel) etc.
Boundary nonlinearities - displacement dependent boundary conditions. The most frequent boundary nonlinearities are encountered in contact problems.

6. Consequences of assuming nonlinearities in FEM:
Instead of set of linear algebraic equations
we obtain a set of nonlinear algebraic equations
Consequences of nonlinear structural behaviour that have to be recognized are:
The principle of superposition cannot be applied. For example, the results of several load cases cannot be combined. Results of the nonlinear analysis cannot be scaled.
Bth equations can be considered as conditions of equilibrium of internal and external forces at every node.

7. Only one load case can be handled at a time.
The sequence of application of loads (loading history) may be important. Especially, plastic deformations depend on a manner of loading. This is a reason for dividing loads into small increments in nonlinear FE analysis.
The structural behaviour can be markedly non- proportional to the applied load. The initial state of stress (e.g. residual stresses from heat treatment, welding etc.) may be important.

8. 1.2 Concept of time curves
In order to reflect history of loading, loads are associated with time curves.
Example - values of forces at any time are defined as
where f1 and f2 are nominal (input) values of forces and 1 and 1 are load parameters that are functions of time t.

9. For nonlinear static analysis, the “time” variable represents a pseudo time, which denotes the intensity of the applied loads at certain step.
For nonlinear dynamic analysis and nonlinear static analysis with time-dependent material properties the “time” represents the real time associated with the loads’ application.
The most common case – all loads are proportional to time:

10. 2. Geometrically nonlinear finite element analysis
Example – linearly elastic truss
A0 is initial cross-section
In linear analysis we neglect displacement u in the last equation

11. Condition of equilibrium
where
axial force
cross-section of the truss
engineering strain
A0 is initial cross-section
Initial and current length of the truss are

12. To avoid complications, it is convenient to introduce new measure of strain – Green’s strain defined as
In our example is
hence

13. Example of different strain measures
Logarithmic strain (true strain)

14. The stress-strain relation was measured as
When using Green’s strain the relation should be
This means that constitutive equation should be

15. The new modulus of elasticity is not constant but it depends on strain
If strain is small (e.g. less than 2%) differences are negligible
DL / L0 e eG
E (MPa)
E*(MPa)
E e (MPa)
E*eG(MPa)
0,0000
0,000000
21 000
0,0050
0,005013
20 948
105
0,0100
0,010050
20 896
210
0,0150
0,015113
20 844
315
0,0200
0,020200
20 792
420

16. Assuming that strain is small, we can write
and after substituting into equation
we can derive the condition of equilibrium in the form

17. Consequence of considering configuration changes - relation between load P and displacement u is nonlinear
Generally, using FEM we obtain a set of nonlinear algebraic equations for unknown nodal displacements d

18. 3. Incremental – iterative solution
is tangent stiffness matrix
3. Incremental – iterative solution
Assumption of large displacements leads to nonlinear equation of equilibrium
For infinitesimal increments of internal and external forces we can write
where

19. 3.1 Incremental method
The load is divided into a set of small increments DFi .
Increments of displacements are calculated from the set of linear simultaneous equations
where KT(i-1) is tangent stiffness matrix computed form displacements d(i-1) obtained in previous incremental step.
Nodal displacements after force increment of DFi are

20. Incremental method

21. 3.2 Iterative methods Newton-Raphson method
Consider that di is estimation of nodal displacement. As it is only an estimation, the condition of equilibrium would not be satisfied
This means that conditions of equilibrium of internal and external nodal forces are not satisfied and in nodes are unbalanced forces

22. Correction of nodal displacements can be then obtained from the set of linear algebraic equations
and mew, corrected estimation of nodal displacements is
The procedure is repeated until the sufficiently accurate solution is obtained.
The first estimation is obtained from linear analysis

23. Standard Newton-Raphson (NR) method

24. Modified Newton-Raphson (MNR) method - the same stiffness matrix is used in all iterations

25. Combination of Newton-Raphson and incremental methods

26. 4. Material nonlinearities 4.1 Nonlinear elasticity models
For any nonlinear elastic material model, it is possible to define relation between stress and strain increments as
Matrix DT is function of strains . Consequently, a set of equilibrium equations we receive in FEM is nonlinear and must be solved by use of any method described above

27. 4.1 Elastoplastic material models
The total strains are decomposed into elastic and plastic parts
The yield criterion says whether plastic deformation will occur.
The plastic behaviour of a material after onset of plastic deformations is defined by so-called flow rule in which is the rate and the direction of plastic strains is related to the stress state and the stress rate. This relation can be expressed as

28. Constitutive equation can be formulated as
The tangential material matrix DT is used to form a tangential stiffness matrix KT. When the tangential stiffness matrix is defined, the displacement increment is obtained for a known load increment
As load and displacement increments are final, not infinitesimal, displacements obtained by solution of this set of linear algebraic equation will be approximate only. That means, conditions of equilibrium of internal and external nodal forces will not be satisfied and iterative process is necessary.

29. The problem - not only equilibrium equations but also constitutive equations of material must be satisfied. That means that within the each equilibrium iteration step check of stress state and iterations to find elastic and plastic part of strains at every integration point must be included.
The iteration process continues until both, equilibrium conditions and constitutive equations are satisfied simultaneously.
The converged solution at the end of load increment is then used at the start of new load increment.

31. Example of non-linear static analysis – bending of the beam,
considering elastoplastic material
bilinear material model

32. Detail of finite element mesh – SHELL4T elements are colored according to their thickness

33. Beginning of plastic deformations
Maximal stress sx approaches value of the yield stress
Beginning of plastic deformations
Normal stress distribution in the cross-section at mid of the beam span.

34. Normal stress distribution after increase of the load.

35. COLLAPSE – inability of the beam to resist further load increase

38. Deflection versus load

39. Example of nonlinear dynamic analysis - drop test of container for radioactive waste.
Simulation of a drop from 9 m at an angle to steel target

40. Reduced stresses at time 0,00187 s
Time courses of reduced stresses at selected nodes

41. Drop on side of the lid – check of screwed bolts

42. 0,00165 s.
0,0027 s

43. Maximal displacements

45. Drop along the top on the mandrel

46. Drop along the top on the mandrel - time course of maximal stress in the lid

47. Drop aside on the mandrel

48. Reduced stresses at time 0,00235 s

50. Example:
Study of influence of residual stresses due to arc welding on load-bearing capacity of a thin-walled beam.

51. Coupled thermal and stress analysis in following steps:
Nonlinear transient thermal analysis
temperature dependent thermal material properties c, k and density r
temperature dependent convective heat transfer coefficient
Nonlinear stress analysis
plastic deformations
large displacements
temperature dependent material mechanical properties

52. Temperature field at time t = 5 s, 1st phase of welding

53. Temperature field at time t = 10 s, 1st phase

54. Temperature field at time t = 5 s, 2nd phase

55. Temperature field at time t = 10 s, 2nd phase

56. Temperature field after end of welding

57. fy – yield stress and E20 modulus of elasticity at 20 oC
Reduction coefficients for yield stress and modulus of elasticity
fy – yield stress and E20 modulus of elasticity at 20 oC

58. temperature field
stress field

62. Deflection of the beam during welding

63. Maximum deflection versus load
unannealed
annealed F