Nonlinear Analysis:
Riks Analysis
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Section 3 – Nonlinear Analysis
Objective
Module 5 – Riks Analysis
Page 2
The objective of this module is to show how to determine the
load at the onset of buckling and the displacement controlled
post-buckling response using a Riks analysis in Autodesk
Simulation Multiphysics.
1
0.9
0.8
Load Factor
0.7
Elastic-plastic buckling of
a shallow arch
0.6
0.5
0.4
0.3
0.2
0.1
Load Increment
0
0
50
100
150
Load Factor Curve
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Section 3 – Nonlinear Analysis
Analysis of Unstable Systems
Module 5 – Riks Analysis
Page 3

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Standard Newton-Raphson based
iterative methods use constant load
increments, DF, and iterate to find the
associated displacement increment,
Dd.
These methods run into problems
when instability associated with
buckling is encountered.
As seen in the figure, there is not an
equilibrium point for load increment
DF5 and convergence cannot be
achieved.
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Newton-Raphson
Iterations
F
DF5
DF4
DF3
DF2
DF1
Dd1 Dd3 Dd4
d
Dd2
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Section 3 – Nonlinear Analysis
Riks Methods
Module 5 – Riks Analysis
Page 4
Riks proposed adding an
additional constraint (shown by
the blue line in the figure) that
limits the displacement increment
and makes the magnitude of the
load increment a variable.
Riks/Wempner
Constraint
F
DF5
DF4
DF3
DF2
DF1
Dd1 Dd3 Dd4
E. Riks, An Incremental Approach to the
Solution of Snapping and Buckling Problems,
International J. Solids& Structures, 15, 529
(1979).
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Dd4
d
Dd2
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Section 3 – Nonlinear Analysis
Arc Length Methods
Module 5 – Riks Analysis
Page 5
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Riks’ method led to a class of
solution methods that use a
hypersphere constraint.
These methods, called arc-length
methods, are illustrated in the
figure.
Arc-length methods are easier to
implement into finite element
programs than the original Riks
method.
M.A. Crisfield, A Fast Incremental Iterative
Procedure that Handles Snap-Through,
Computers & Structures, 13, 55 (1981).
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Crisfield
Constraint
F
DF5
DF4
DF3
The constraint radius
or arc-length is held
constant during the
analysis.
DF2
DF1
Dd1 Dd3 Dd4
Dd4
d
Dd2
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Section 3 – Nonlinear Analysis
Arc-length
Module 5 – Riks Analysis
Page 6
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In Simulation, an initial load
increment is used to determine
the magnitude of the
hypersphere constraint.
Typical
Iterations
R
This magnitude is held constant
during the rest of the load
increments.
The objective for each increment
is to determine how much
increase or decrease in the load is
needed to make the
displacements satisfy the
hypersphere constraint equation.
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Constraint
Surface (Typical)
R
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Section 3 – Nonlinear Analysis
Load Factor
Module 5 – Riks Analysis
Page 7
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The load factor is a scalar
multiplied by the external nodal
forces to achieve equilibrium.
The load factor can increase,
decrease and go negative during
an analysis.
The load factor is printed in the
analysis log file during a Riks
Analysis using Autodesk
Simulation Multiphysics
software.
© 2011 Autodesk
Equilibrium Equation
R unb    Fext   Fint 
R unb 

Unbalanced load vector.
Equal to zero when
equilibrium is achieved.
Variable load factor
Fext 
Array of external forces
Fint 
Array of internal forces
caused by displacement
induced stresses
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Reference Section 2: Module 3
PowerPoint Slides
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Section 3 – Nonlinear Analysis
Example Problem
Module 5 – Riks Analysis
Page 8

The elastic-plastic response of an
arch will be determined using
the Riks analysis type in
Autodesk Simulation
Multiphysics.

An elastic-plastic material model
for AISI 1020 cold rolled steel is
used.

The mesh uses two elements
with mid-side nodes through the
thickness to capture the through
thickness stress gradients.
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75 lbs divided
among 15 nodes
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Section 3 – Nonlinear Analysis
Setting Up a Riks Analysis in Autodesk
Simulation Multiphysics

Only a few parameters must be
defined to run a Riks analysis.

The “Total number of steps” controls
how long the analysis will run.

The “Initial load factor increment”
defines the fraction of the load applied
during the first load increment.

The displacements from this first
increment are used to determine the
arc-length used during the rest of the
analysis.
© 2011 Autodesk
Module 5 – Riks Analysis
Page 9
Trial and error is commonly used
to determine values for these
parameters that will achieve
acceptable results.
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Section 3 – Nonlinear Analysis
Load Factor History
Module 5 – Riks Analysis
Page 10

This plot shows the load factor
computed for each load increment.
1
0.9
0.8

The results start to go non-linear at a
load factor of 0.8, but the load factor
continues to increase.
The peak load factor is 0.94 or 94% of
the max load defined in the FEA
Editor.
0.7
0.6
Load Factor

0.5
0.4
0.3
The elastic-plastic buckling
strength of the arch is equal to
0.94 times the applied load.
0.2

The wavering nature of the line is due
to the size of the convergence
tolerance used while seeking a
converged solution.
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0.1
0
0
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20
40
60
Load Increment
80
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120
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Section 3 – Nonlinear Analysis
Deformed Shapes
Module 5 – Riks Analysis
Page 11
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The arch progresses through a
series of shapes during the
event.
The amount of load that it can
support continues to increase
until load increment 20.
After load increment 20 the load
factor decreases indicating that
the arch has buckled.
0% Load
94% Load, Increment 20
77% Load, Increment 50
64% Load, Increment 75
55% Load, Increment 100
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Section 3 – Nonlinear Analysis
Snap Through Problems
Module 5 – Riks Analysis
Page 12

The Riks method is able to handle
snap through problems.
Shallow Arch
Original Shape


Snap through occurs when an elastic
arch is loaded beyond its buckling
load and snaps through to another
shape where it can continue to carry
an increase in load.
Buckled Shape
F
Snap through would have been
encountered in the arch problem
used in this example had the
material not been elastic-plastic.
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d
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Section 3 – Nonlinear Analysis
Module Summary
Module 5 – Riks Analysis
Page 13
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This module has provided an introduction to concepts used in
conjunction with a Riks analysis.
This static analysis method is used to determine the onset of
buckling and the post-buckling response of structural systems.
The method seeks a load factor that will cause the
displacements to lie on a hypersphere constraint surface.
It gives results similar to those obtained from a displacement
controlled experiment.
In general, buckling is not a displacement controlled event and
the true post-buckling response of the system must be
computed using a full nonlinear dynamic analysis.
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