INTRODUCTION INTO FINITE ELEMENT NONLINEAR ANALYSES

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INTRODUCTION INTO FINITE ELEMENT NONLINEAR
ANALYSES

• Doc. Ing. Vladimír Ivanco, PhD.
• Technical University of Košice
• Faculty of mechanical Engineering
• Department of Applied Mechanics and Mechatronics

HS Wismar, June 2010
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• CONTENS
• Introduction
• Types of structural nonlinearities
• Concept of time curves
• Geometrically nonlinear finite element analysis
• Incremental iterative solution
• Incremental method
• Iterative methods
• Material nonlinearities
• Examples

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• INTRODUCTION
• 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 Hookes 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

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• 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
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Nonlinear analysis sources of nonlinearities
can be classified as

1. Geometric nonlinearities - changes of the
   structure shape (or configuration changes) cannot
   be neglected and its deformed configuration
   should be considered.
2. Material nonlinearities - material behaves
   nonlinearly and linear Hookes law cannot be
   used. More complicated material models should be
   then used instead e.g.

nonlinear elastic (Mooney-Rivlins 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.

1. Boundary nonlinearities - displacement dependent
   boundary conditions. The most frequent boundary
   nonlinearities are encountered in contact
   problems.

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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

1. 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.

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1. Only one load case can be handled at a time.
2. 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.
3. 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.

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b) 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.
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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
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2. Geometrically nonlinear finite element analysis
Example linearly elastic truss
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Condition of equilibrium
where
axial force
cross-section of the truss
engineering strain
Initial and current length of the truss are
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To avoid complications, it is convenient to
introduce new measure of strain Greens strain
defined as
In our example is
hence
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Example of different strain measures
Logarithmic strain (true strain)
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The stress-strain relation was measured as
When using Greens strain the relation should be
This means that constitutive equation should be
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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) EeG(MPa)
0,0000 0,0000 0,000000 21 000 21 000 0 0
0,0050 0,0050 0,005013 21 000 20 948 105 105
0,0100 0,0100 0,010050 21 000 20 896 210 210
0,0150 0,0150 0,015113 21 000 20 844 315 315
0,0200 0,0200 0,020200 21 000 20 792 420 420
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Assuming that strain is small, we can write
and after substituting into equation
we can derive the condition of equilibrium in the
form
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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
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4. Incremental iterative solution
Assumption of large displacements leads to
nonlinear equation of equilibrium
For infinitesimal increments of internal and
external forces we can write
- tangent stiffness matrix
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a) Incremental method
The load is divided into a set of small
increments . Increments of displacements
are calculated from the set of linear
simultaneous equations
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a) Iterative methods
Newton-Raphson method
i-th iteration of displacements is computed from
the set of linear algebraic equations
then computed displacements di are checked by
substitution into the condition of equilibrium.
If the displacements are not accurate, the
condition is not satisfied
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Example of non-linear static analysis bending
of the beam, considering elastoplastic material
bilinear material model
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Detail of finite element mesh SHELL4T elements
are colored according to their thickness
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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.
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Normal stress distribution after increase of the
load.
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COLLAPSE inability of the beam to resist
further load increase
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Deflection versus load
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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
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Reduced stresses at time 0,00187 s
Time courses of reduced stresses at selected nodes
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Drop on side of the lid check of screwed bolts
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0,00165 s.
0,0027 s
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Maximal displacements
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Drop along the top on the mandrel
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Drop along the top on the mandrel - time course
of maximal stress in the lid
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Drop aside on the mandrel
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Reduced stresses at time 0,00235 s
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Example
Study of influence of residual stresses due to
arc welding on load-bearing capacity of a
thin-walled beam.
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• 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

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Temperature field at time t 5 s, 1st phase of
welding
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Temperature field at time t 10 s, 1st phase
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Temperature field at time t 5 s, 2nd phase
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Temperature field at time t 10 s, 2nd phase
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Temperature field after end of welding
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Reduction coefficients for yield stress and
modulus of elasticity
fy yield stress and E20 modulus of elasticity
at 20 oC
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temperature field
stress field
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Deflection of the beam during welding
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Maximum deflection versus load
F