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INTRODUCTION INTO FINITE ELEMENT NONLINEAR ANALYSESDoc. Ing.
Vladimr Ivano, PhD.Technical University of Koice Faculty of
Mechanical EngineeringDepartment of Applied Mechanics and
Mechatronics
HS Wismar, June 2009
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CONTENS1. Introduction1.1 Types of structural nonlinearities 1.2
Concept of time curves2. Geometrically nonlinear finite element
analysis 3. Incremental iterative solution3.1 Incremental method3.2
Iterative methods4. Material nonlinearities5. Examples
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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 loadingd) deformations are so small that
change of the structure configuration is neglectable Introduction
1.1 Sources of nonlinearities
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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 displacementswhere K
global stiffness matrixd vector of unknown nodal displacementsF
vector of external nodal forces
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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 Hookes law cannot be used.
More complicated material models should be then used instead
e.g.Nonlinear analysis sources of nonlinearities can be classified
asnonlinear 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.Boundary nonlinearities - displacement
dependent boundary conditions. The most frequent boundary
nonlinearities are encountered in contact problems.
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we obtain a set of nonlinear algebraic equationsConsequences of
assuming nonlinearities in FEM:Instead of set of linear algebraic
equationsConsequences 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.
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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.
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1.2 Concept of time curvesIn 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 analysisExample
linearly elastic truss
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Condition of equilibriumwhereengineering strainaxial
forcecross-section of the trussInitial 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 asIn our example
ishence
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Example of different strain measuresLogarithmic strain (true
strain)
Graf3
000
0.010050.00995033090.01
0.02020.01980262730.02
0.030450.02955880220.03
0.04080.03922071320.04
0.051250.04879016420.05
0.06180.05826890810.06
0.072450.06765864850.07
0.08320.07696104110.08
0.094050.08617769620.09
0.1050.09531017980.1
0.116050.10436001530.11
0.12720.11332868530.12
0.138450.12221763270.13
0.14980.13102826240.14
0.161250.13976194240.15
0.17280.14842000510.16
0.184450.15700374880.17
0.19620.16551443850.18
0.208050.17395330710.19
0.220.18232155680.2
0.232050.19062035960.21
0.24420.19885085870.22
0.256450.20701416940.23
0.26880.21511137960.24
0.281250.22314355130.25
0.29380.2311117210.26
0.306450.23901690050.27
0.31920.24686007790.28
0.332050.25464221840.29
0.3450.26236426450.3
0.358050.27002713720.31
0.37120.27763173660.32
0.384450.28517894220.33
0.39780.2926696140.34
0.411250.30010459250.35
0.42480.30748469970.36
0.438450.31481073980.37
0.45220.32208349920.38
0.466050.32930374710.39
0.480.33647223660.4
0.494050.34358970440.41
0.50820.35065687160.42
0.522450.35767444430.43
0.53680.36464311360.44
0.551250.37156355640.45
0.56580.37843643570.46
0.580450.38526240080.47
0.59520.39204208780.48
0.610050.398776120.49
0.6250.40546510810.5
0.640050.41210965080.51
0.65520.41871033490.52
0.670450.42526773540.53
0.68580.43178241640.54
0.701250.43825493090.55
0.71680.44468582130.56
0.732450.45107561940.57
0.74820.4574248470.58
0.764050.46373401620.59
0.780.47000362920.6
Green's strain
Logarithmic strain
Engineering strain
Dl / l0
strain
List1
0.01
Engineering strainGreen's strainLogaritmic starin
Dl/l0ee*elog
0000
0.010.010.010050.0099503309
0.020.020.02020.0198026273
0.030.030.030450.0295588022
0.040.040.04080.0392207132
0.050.050.051250.0487901642
0.060.060.06180.0582689081
0.070.070.072450.0676586485
0.080.080.08320.0769610411
0.090.090.094050.0861776962
0.10.10.1050.0953101798
0.110.110.116050.1043600153
0.120.120.12720.1133286853
0.130.130.138450.1222176327
0.140.140.14980.1310282624
0.150.150.161250.1397619424
0.160.160.17280.1484200051
0.170.170.184450.1570037488
0.180.180.19620.1655144385
0.190.190.208050.1739533071
0.20.20.220.1823215568
0.210.210.232050.1906203596
0.220.220.24420.1988508587
0.230.230.256450.2070141694
0.240.240.26880.2151113796
0.250.250.281250.2231435513
0.260.260.29380.231111721
0.270.270.306450.2390169005
0.280.280.31920.2468600779
0.290.290.332050.2546422184
0.30.30.3450.2623642645
0.310.310.358050.2700271372
0.320.320.37120.2776317366
0.330.330.384450.2851789422
0.340.340.39780.292669614
0.350.350.411250.3001045925
0.360.360.42480.3074846997
0.370.370.438450.3148107398
0.380.380.45220.3220834992
0.390.390.466050.3293037471
0.40.40.480.3364722366
0.410.410.494050.3435897044
0.420.420.50820.3506568716
0.430.430.522450.3576744443
0.440.440.53680.3646431136
0.450.450.551250.3715635564
0.460.460.56580.3784364357
0.470.470.580450.3852624008
0.480.480.59520.3920420878
0.490.490.610050.39877612
0.50.50.6250.4054651081
0.510.510.640050.4121096508
0.520.520.65520.4187103349
0.530.530.670450.4252677354
0.540.540.68580.4317824164
0.550.550.701250.4382549309
0.560.560.71680.4446858213
0.570.570.732450.4510756194
0.580.580.74820.457424847
0.590.590.764050.4637340162
0.60.60.780.4700036292
List1
Green's strain
Logaritmic starin
Engineering strain
Dl / l0
strain
List2
List3
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The stress-strain relation was measured asWhen using Greens
strain the relation should beThis means that constitutive equation
should be
-
If strain is small (e.g. less than 2%) differences are
negligibleThe new modulus of elasticity is not constant but it
depends on strain
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3.1 Incremental methodThe load is divided into a set of small
increments DFi . Increments of displacements are calculated from
the set of linear simultaneous equations Nodal displacements after
force increment of DFi arewhere KT(i-1) is tangent stiffness matrix
computed form displacements d(i-1) obtained in previous incremental
step.
-
Incremental method
-
3.2 Iterative methodsNewton-Raphson methodConsider that di is
estimation of nodal displacement. As it is only an estimation, the
condition of equilibrium would not be satisfiedThis means that
conditions of equilibrium of internal and external nodal forces are
not satisfied and in nodes are unbalanced forces
-
Standard Newton-Raphson (NR) method
-
Modified Newton-Raphson (MNR) method - the same stiffness matrix
is used in all iterations
-
Combination of Newton-Raphson and incremental methods
-
4. Material nonlinearities 4.1 Nonlinear elasticity modelsFor
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
-
4.1 Elastoplastic material modelsThe total strains are
decomposed into elastic and plastic partsThe 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
-
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.
-
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.
-
Example of non-linear static analysis bending of the
beam,considering elastoplastic material
-
Detail of finite element mesh SHELL4T elements are colored
according to their thickness
-
Beginning of plastic deformationsNormal stress distribution in
the cross-section at mid of the beam span. Maximal stress sx
approaches value of the yield stress
-
Normal stress distribution after increase of the load.
-
COLLAPSE inability of the beam to resist further load
increase
-
Deflection versus load
-
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
-
Reduced stresses at time 0,00187s Time courses of reduced
stresses at selected nodes
Graf2
0
0
0
000
0.000000000100
0.00001566400
0.2187800
6.997500
10.1140.00000028520.0000004291
18.8640.0127850.019006
20.011.25891.7724
23.5363.59844.5916
31.6434.48046.4578
38.60914.18919.565
49.08616.7623.9
89.79613.42721.591
109.6320.01429.192
139.4225.86643.271
167.5453.08683.318
191.1863.293.781
212.9777.855122.48
243.6889.502142.73
264.68107.43164.27
299.61121.83181.14
339.16132.01220.01
356.9140.78238.35
364.11156.35260.85
371.47175.72309.43
378.59190.65338.02
386.56212.23354.5
395.03235.19360.2
402.69250.79365.36
409.93264.9370.36
415.94279.42374.5
421.91294.28380.75
428.14303.84385.27
433.95312.51389.14
438.38320.61393.21
442.44328.27397.34
446.15336.63400.93
451.56344.65403.93
456.76352.55407.15
454.41358.89409.71
445.74364.97413.82
463.77381.35420.39
466.27359.58380.27
473.57353.05389.33
477.32382.21380.06
480.87367.25398.42
480.15391.88423.85
464.74364.22424.5
454.94406.58426.2
402.91361.12422.21
424.28356.72402.53
464.44322.06323.93
300.01328.12
318.59306.76
337.85339.7
uzol 727
uzol 3437
uzol 3440
t [ms]
s [MPa]
A3_2_strmax
XY POINT LIST
ND=727
Pt#TIMEVON-1
12.50E-064.56E-152.50E-034.56E-159.90E-151.46E-14
23.98E-065.00E-153.98E-035.00E-151.15E-141.39E-14
34.60E-068.33E-154.60E-038.33E-151.24E-141.61E-14
45.29E-068.94E-155.29E-038.94E-151.08E-141.82E-14
51.02E-059.71E-111.02E-029.71E-115.66E-139.64E-13
61.63E-051.57E-051.63E-021.57E-052.85E-074.29E-07
72.60E-052.19E-012.60E-022.19E-011.28E-021.90E-02
83.54E-057.00E+003.54E-027.00E+001.26E+001.77E+00
94.49E-051.01E+014.49E-021.01E+013.60E+004.59E+00
105.39E-051.89E+015.39E-021.89E+014.48E+006.46E+00
116.35E-052.00E+016.35E-022.00E+011.42E+011.96E+01
127.25E-052.35E+017.25E-022.35E+011.68E+012.39E+01
138.20E-053.16E+018.20E-023.16E+011.34E+012.16E+01
141.24E-043.86E+011.24E-013.86E+012.00E+012.92E+01
151.74E-044.91E+011.74E-014.91E+012.59E+014.33E+01
162.24E-048.98E+012.24E-018.98E+015.31E+018.33E+01
172.74E-041.10E+022.74E-011.10E+026.32E+019.38E+01
183.24E-041.39E+023.24E-011.39E+027.79E+011.22E+02
193.74E-041.68E+023.74E-011.68E+028.95E+011.43E+02
204.24E-041.91E+024.24E-011.91E+021.07E+021.64E+02
214.74E-042.13E+024.74E-012.13E+021.22E+021.81E+02
225.24E-042.44E+025.24E-012.44E+021.32E+022.20E+02
235.74E-042.65E+025.74E-012.65E+021.41E+022.38E+02
246.24E-043.00E+026.24E-013.00E+021.56E+022.61E+02
256.74E-043.39E+026.74E-013.39E+021.76E+023.09E+02
267.24E-043.57E+027.24E-013.57E+021.91E+023.38E+02
277.74E-043.64E+027.74E-013.64E+022.12E+023.55E+02
288.24E-043.71E+028.24E-013.71E+022.35E+023.60E+02
298.74E-043.79E+028.74E-013.79E+022.51E+023.65E+02
309.24E-043.87E+029.24E-013.87E+022.65E+023.70E+02
319.74E-043.95E+029.74E-013.95E+022.79E+023.75E+02
321.02E-034.03E+021.02E+004.03E+022.94E+023.81E+02
331.07E-034.10E+021.07E+004.10E+023.04E+023.85E+02
341.12E-034.16E+021.12E+004.16E+023.13E+023.89E+02
351.17E-034.22E+021.17E+004.22E+023.21E+023.93E+02
361.22E-034.28E+021.22E+004.28E+023.28E+023.97E+02
371.27E-034.34E+021.27E+004.34E+023.37E+024.01E+02
381.32E-034.38E+021.32E+004.38E+023.45E+024.04E+02
391.37E-034.42E+021.37E+004.42E+023.53E+024.07E+02
401.42E-034.46E+021.42E+004.46E+023.59E+024.10E+02
411.47E-034.52E+021.47E+004.52E+023.65E+024.14E+02
421.52E-034.57E+021.52E+004.57E+023.81E+024.20E+02
431.57E-034.54E+021.57E+004.54E+023.60E+023.80E+02
441.62E-034.46E+021.62E+004.46E+023.53E+023.89E+02
451.67E-034.64E+021.67E+004.64E+023.82E+023.80E+02
461.72E-034.66E+021.72E+004.66E+023.67E+023.98E+02
471.77E-034.74E+021.77E+004.74E+023.92E+024.24E+02
481.82E-034.77E+021.82E+004.77E+023.64E+024.25E+02
491.87E-034.81E+021.87E+004.81E+024.07E+024.26E+02
501.92E-034.80E+021.92E+004.80E+023.61E+024.22E+02
511.97E-034.65E+021.97E+004.65E+023.57E+024.03E+02
522.02E-034.55E+022.02E+004.55E+023.22E+023.24E+02
532.07E-034.03E+022.07E+004.03E+023.00E+023.28E+02
542.12E-034.24E+022.12E+004.24E+023.19E+023.07E+02
552.17E-034.64E+022.17E+004.64E+023.38E+023.40E+02
A3_2_strmax
000
000
000
000
000
000
000
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000
000
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