Applied Mechanics of Solids (A.F. Bower) Chapter 7: Intro to FEA - 7.1
Guide to using FEA software
Introduction to Finite Element Analysis
in Solid Mechanics
Most practical design calculations involve components
with a complicated three-dimensional geometry, and may also need to account for
inherently nonlinear phenomena such as contact, large shape changes, or
nonlinear material behavior.
problems can only be solved using computer simulations.
The finite element method is by far the most
widely used and versatile technique for simulating deformable solids.
This chapter gives a brief overview of the
finite element method, with a view to providing the background needed to run
simple simulations using a commercial finite element program.
More advanced analysis requires a deeper
understanding of the theory and implementation of finite element codes, which
will be addressed in the next chapter.
WARNING: It is deceptively easy to
use commercial finite element software: most programs come with a nice
user-interface that allows you to define the geometry of the solid, choose a material
model, generate a finite element mesh and apply loads to the solid with a few
mouse clicks.
If all goes well, the
program will magically turn out animations sho contours
and much more besides.
It is all too easy, however, to produce
meaningless results, by attempting to solve a problem that does not have a well
by using an inappropri or simply using
incorrect settings for internal tolerances in the code.
In addition, even high quality software can
contain bugs.
Always treat the results
of a finite element computations with skepticism!
7.1 A guide to using finite element software
The finite element method (FEM) is a computer
technique for solving partial differential equations.
One application is to predict the deformation
and stress fields within solid bodies subjected to external forces.
However, FEM can also be used to solve
problems involving fluid flow, heat transfer, electromagnetic fields,
diffusion, and many other phenomena.
The principle objective of the displacement based
finite element method is to compute the displacement
field within a solid subjected to external forces.
To make this precise, visualize a solid deforming
under external loads.
Every point in the
solid moves as the load is applied.
displacement vector u(x) specifies the motion of
the point at position x in the undeformed solid.
Our objective is to determine u(x).
is known, the strain and stress fields in the solid can be deduced.
There are two general types of finite element analysis
in solid mechancis.
In most cases, we
are interested in determining the behavior of a solid body that is in static equilibrium.
This means that both external and internal
forces acting on the solid sum to zero.
In some cases, we may be interested in the dynamic behavior of a solid body.
Examples include modeling vibrations in structures, problems involving
wave propagation, explosive loading and crash analysis.
For Dynamic
Problems the finite element method solves the equations of motion for a
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essentially a more complicated version of
∑ F
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Naturally, in this case it must
calculate the motion of the solid as a
function of time.
For Static
Problems the finite element method solves the equilibrium equations
∑ F
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In this case, it may not be necessary to
calculate the time variation of motion.
However, some materials are history dependent (e.g metals deformed in
the plastic regime).
In addition, a
static equilibrium problem may have more than one solution, depending on the
load history.
In this case the time
variation of the solution must be computed.
For some applications, you may also need to solve
additional field equations.
For example,
you may be interested in calculating the temperature distribution in the solid,
or calculating electric or magnetic fields.
In addition, special finite element procedures are available to calculate
buckling loads and their modes, as well as natural frequencies of vibration and
the corresponding mode shapes for a deformable solid.
finite element calculation, you
will need to specify
1.The geometry of the solid.
This is done by generating a finite element mesh for the solid.
The mesh can usually be generated
automatically from a CAD representation of the solid.
2.The properties of the material.
This is done by specifying a constitutive law for the solid.
3.The nature of the loading applied to the solid.
This is done by specifying the boundary conditions for the problem.
4. If your
analysis involves contact between two more more solids, you will need to
specify the surfaces that are likely to come into contact, and the properties
(e.g. friction coefficient) of the contact.
5.For a dynamic analysis, it is necessary to specify initial conditions for the
This is not necessary for a
static analysis.
6.For problems involving additional fields, you may need
to specify initial values for these field variables (e.g. you would need to
specify the initial temperature distribution in a thermal analysis).
You will also need to specify some additional aspects
of the problem you are solving and the solution procedure to be used:
1. You will need to specify whether the computation should take into
account finite changes in the geometry of the solid.
2. For a dynamic analysis,
you will need to specify the time period
of the analysis (or the number of time increments)
For a static analysis you will need to decide whether the problem is
linear, or nonlinear.
Linear problems
are very easy to solve.
problems may need special procedures.
4. For a static analysis
with history dependent materials you will need to specify the time period of
the analysis, and the time step size (or number of steps)
5. If you are
interested in calculating natural frequencies and mode shapes for the system,
you must specify how many modes to extract.
6. Finally, you will need to specify what the
finite element method must compute.
steps in running a finite element computation are discussed in more detail in
the following sections.
Finite Element Mesh for a 2D or 3D component
The finite element mesh is used to specify the
geometry of the solid, and is also used to describe the displacement field
within the solid.
A typical mesh (generated
in the commercial FEA code ABAQUS) is shown in the picture to the right.
A finite element mesh may be three dimensional, like
the example shown.
Two dimensional
finite element meshes are also used to model simpler modes of deformation.
There are three main types of two dimensional
finite element mesh:
1.Plane stress
2.Plane strain
3.Axisymmetric
In addition, special types of finite element can be
used to model the overall behavior of a 3D solid, without needing to solve for
the full 3D fields inside the solid.
Example beam elements and truss
These will be discussed in a
separate section below.
Stress Finite Element Mesh : A plane stress finite element mesh is used to
model a plate - like solid which is loaded in its own plane.
The solid must have uniform thickness, and
the thickness must be much less than any representative cross sectional
dimension.
A plane stress finite element
mesh for a thin plate containing a hole is shown in the figure to the right.
Only on quadrant of the specimen is modeled,
since symmetry boundary conditions will be enforced during the analysis.
Strain finite element mesh : A plane strain finite element mesh is used to
model a long cylindrical solid that is prevented from stretching parallel to
For example, a plane strain
finite element mesh for a cylinder which is in contact with a rigid floor is
shown in the figure.
Away from the ends
of the cylinder, we expect it to deform so that the out of plane component of
displacement
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There is no need to solve for
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therefore, so a two dimensional mesh is sufficient to calculate
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As before, only one quadrant of the specimen is
meshed: symmetry boundary conditions will be enforced during the analysis.
Axisymmetric
finite element mesh: An axisymmetric mesh is used to model a solids that
has rotational symmetry, which is subjected to axisymmetric loading.
An example is shown on the right.
The picture compares a three dimensional mesh of an
axisymmetric bushing to an axisymmetric mesh.
Note that the half the bushing has been cut away in the 3D view, to show
the geometry more clearly.
In an axisymmetric
analysis, the origin for the (x,y)
coordinate system is always on the axis of
rotational symmetry.
that to run an axisymmetric finite element analysis, both the geometry of the
solid, and also the loading applied to the solid, must have rotational symmetry
about the y axis.
7.1.2 Nodes
and Elements in a Mesh
A finite element mesh is defined by a set of nodes together with a set of finite elements, as shown in the sketch on
the right.
Nodes: The
nodes are a set of discrete points within the solid body.
Nodes have the following properties:
Every node is assigned an integer number,
which is used to identify the node.
convenient numbering scheme may be selected
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the nodes do not need to be numbered in order,
and numbers may be omitted.
For example,
one could number a set of n nodes as
100, 200, 300… 100n, instead of
coordinates.
For a three dimensional finite element
analysis, each node is assigned a set of
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coordinates, which specifies the position of
the node in the undeformed solid.
two dimensional analysis, each node is assigned a pair of
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coordinates.
For an axisymmetric analysis, the
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axis must coincide with the axis of rotational
displacements.
When the solid deforms, each node moves to a
new position.
For a three dimensional
finite element analysis, the nodal displacements specify the three components
of the displacement field u(x) at each node:
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For a two dimensional analysis, each node has
two displacement components
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The nodal displacements are unknown at the
start of the analysis, and are computed by the finite element program.
4.Other nodal
degrees of freedom. For more complex
analyses, we may wish to calculate a temperature distribution in the solid, or
a voltage distribution, for example.
this case, each node is also assigned a temperature, voltage, or similar
quantity of interest.
There are also
some finite element procedures which use more than just displacements to
describe shape changes in a solid.
example, when analyzing two dimensional beams, we use the displacements and
rotations of the beam at each nodal point to describe the deformation.
In this case, each node has a rotation, as
well as two displacement components.
collection of all unknown quantities (including displacements) at each node are
known as degrees of freedom.
A finite element program will compute values
for these unknown degrees of freedom.
are used to partition the solid into discrete regions.
Elements have the following properties.
1.An element
Every element is assigned an integer number,
which is used to identify the element.
Just as when numbering nodes, any convenient scheme may be selected to
number elements.
2.A geometry.
There are many
possible shapes for an element.
the more common element types are shown in the picture.
Nodes attached to the element are shown in
In two dimensions, elements are
generally either triangular or rectangular. In three dimensions, the elements
are generally tetrahedra, hexahedra or bricks.
There are other types of element that are used for special purposes:
examples include truss elements (which are simply axial members), beam
elements, and shell elements.
3.A set of
These are simply the sides of the element.
4.A set of
nodes attached to the element.
The picture on the right shows a typical
finite element mesh.
Element numbers are
shown in blue, while node numbers are shown in red (some element and node
numbers have been omitted for clarity).
All the elements are 8 noded quadrilaterals.
Note that each element is connected to a set
of nodes: element 1 has nodes (41, 45, 5, 1, 43, 25, 3, 21), element 2 has
nodes (45, 49, 9, 5, 47, 29, 7, 25), and so on.
It is conventional to list the nodes the nodes in the order given, with
corner nodes first in order going counterclockwise around the element, followed
by the midside nodes.
The set of nodes
attached to the element is known as the element
connectivity.
5.An element
interpolation scheme.
The purpose of a finite element is to
interpolate the displacement field u(x)
between values defined at the nodes.
This is best illustrated using an example.
Consider the two dimensional, rectangular 4
noded element shown in the figure. Let
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denote the components of displacement at nodes
a, b, c, d.
The displacement at an arbitrary point within
the element can be interpolated between values at the corners, as follows
=(1−ξ)(1−η)
+ξ(1−η)
  +ξη
+ (1−ξ)η
=(1−ξ)(1−η)
+ξ(1−η)
 +ξη
+ (1−ξ)η
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aaaakiaaykW7cqGHRaWkcqaH+oaEcqaH3oaAcaWG1bWaaSbaaSqaai
aaikdaaeqaaOWaaWbaaSqabeaacaGGOaGaam4yaiaacMcaaaGccqGH
RaWkcaaMc8UaaiikaiaaigdacqGHsislcqaH+oaEcaGGPaGaeq4TdG
MaamyDamaaBaaaleaacaaIYaaabeaakmaaCaaaleqabaGaaiikaiaa
dsgacaGGPaaaaaaaaa@9DCB@
/B,            η=
/H 
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hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
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kiaac+cacaWGibGaaGPaVdaa@53D6@
You can verify for yourself that the displacements
have the correct values at the corners of the element, and the displacements
evidently vary linearly with position within the element.
Different types of element interpolation scheme
The simple example described
above is known as a linear
Six noded triangles and 8 noded
triangles are examples of quadratic
elements: the displacement field varies quadratically with position within the
In three dimensions, the 4
noded tetrahedron and the 8 noded brick are linear elements, while the 10 noded
tet and 20 noded brick are quadratic.
Other special elements, such as beam elements or shell elements, use a
more complex procedure to interpolate the displacement field.
Some special types of element interpolate both the
displacement field and some or all components of the stress field within an
element separately. (Usually, the displacement interpolation is sufficient to
determine the stress, since one can compute the strains at any point in the
element from the displacement, and then use the stress
MathType@MTEF@5@5@+=
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
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strain relation
for the material to find the stress). This type of element is known as a hybrid element.
Hybrid elements are usually used to model
incompressible, or nearly incompressible, materials.
6.Integration
One objective of a finite element analysis is
to determine the distribution of stress within a solid.
This is done as follows.
First, the displacements at each node are
computed (the technique used to do this will be discussed in Section 7.2 and
Chapter 8.)
Then, the element
interpolation functions are used to determine the displacement at arbitrary
points within each element.
displacement field can be differentiated to determine the strains.
Once the strains are known, the stress
MathType@MTEF@5@5@+=
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4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
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strain relations
for the element are used to compute the stresses.
In principle, this procedure could be used to
determine the stress at any point within an element.
However, it turns out to work better at some
points than others.
The special points
within an element where stresses are computed most accurately are known as integration points.
(Stresses are sampled at these points in the
finite element program to evaluate certain volume and area integrals, hence
they are known as integration points).
For a detailed description of the locations of
integration points within an element, you should consult an appropriate user
The approximate locations of
integration points for a few two dimensional elements are shown in the figure.
There are some special types of element that use fewer
integration points than those shown in the picture.
These are known as reduced integration elements. This type of element is usually less
accurate, but must be used to analyze deformation of incompressible materials
(e.g. rubbers or rigid plastic metals).
7.A stress
MathType@MTEF@5@5@+=
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa
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strain relation
and material properties.
Each element is occupied by solid
The type of material within
each element (steel, concrete, soil, rubber, etc) must be specified, together
with values for the appropriate material properties (mass density, Young’s
modulus, Poisson’s ratio, etc).
Special Elements
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E7@
Beams, Plates, Shells and Truss elements
If you need to analyze a solid with a special geometry
(e.g. a simple truss, a structure made of one or more slender beams, or plates)
it is not efficient to try to generate a full 3D finite element mesh for each
member in the structure.
Instead, you
can take advantage of the geometry to simplify the analysis.
The idea is quite simple.
Instead of trying to calculate the full 3D
displacement field in each member, the deformation is characterized by a
reduced set of degrees of freedom.
Specifically:
1.For a pin jointed truss, we simply calculate the displacement of each joint in the
structure. The members are assumed to be in a state of uniaxial tension or
compression, so the full displacement field within each member can be
calculated in terms of joint displacements.
2.For a beam, we calculate the displacement and rotation of the cross section along the beam.
These can be used to determine the internal
shear forces bending moments, and therefore the stresses in the beam. A three
dimensional beam has 3 displacement and 3 rotational degrees of freedom at each
3.For a plate, or shell, we again calculate the displacement and rotation of the plate
section. A three dimensional plate or shell has 3 displacement and two rotational degrees of freedom at
each node (these rotations characterize the rotation of a unit vector normal to
the plate).
In some finite element
codes, nodes on plates and shells have a fictitious third rotational degree of
freedom which is added for convenience
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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but you will find that attempting to impose
boundary conditions on this fictitious degree of freedom has no effect on the
deformation of the structure.
an analysis using truss, beam or plate elements, some additional information
must be specified to set up the problem:
1.For a truss analysis, each member in the truss is a
single element.
The area of the member’s
cross section must be specified.
2.For a beam analysis, the shape and orientation of the
cross section must be specified (or, for an elastic analysis, you could specify
the area moments of inertia directly).
There are also several versions of beam theory, which account
differently for shape changes within the beam.
Euler-Bernoulli beam theory is the simple version covered in
introductory courses.
Timoshenko beam
theory is a more complex version, which is better for thicker beams.
3.For plates and shells, the thickness of the plate must
In addition, the deformation
of the plate can be approximated in various ways
MathType@MTEF@5@5@+=
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for example, some versions only account for
transverse deflections, and neglect in-plane shearing and stretching of the
more complex theories account for this behavior.
Calculations using beam and plate theory also differ
from full 3D or 2D calculations in that both the deflection and rotation of the beam or plate must
be calculated.
This means that:
1.Nodes on beam elements have 6 degrees of freedom
MathType@MTEF@5@5@+=
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aaaaaaa8qacaWFtacaaa@37E6@
the three displacement components, together
with three angles representing rotation of the cross-section about three axes.
Nodes on plate or shell elements have 5 (or
in some FEA codes 6) degrees of freedom.
The 6 degrees of freedom represent 3 displacement components, and two
angles that characterize rotation of the normal to the plate about two axes (if
the nodes have 6 degrees of freedom a third, fictitious rotation component has
been introduced
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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aaaaaaa8qacaWFtacaaa@37E6@
you will have to read the manual for the code
to see what this rotation represents).
2.Boundary conditions may constrain both displacement
and rotational degrees of freedom.
example, to model a fully clamped boundary condition at the end of a beam (or
the edge of a plate), you must set all displacements and all rotations to zero.
3.You can apply
both forces and moments to nodes in an analysis.
in an analysis involving several beams connected together, you can connect the
beams in two ways:
1.You can connect them with a pin joint, which forces the beams to move together at the
connection, but allows relative rotation
2.You can connect them with a clamped joint, which forces the beams to rotate together at the
connection.
In most FEA codes, you can create the joints by adding
constraints, as discussed in Section 1.2.6 below. Occasionally, you may also
wish to connect beam elements to solid, continuum elements in a model: this can
also be done with constraints.
Material Behavior
A good finite element code contains a huge library of
different types of material behavior that may be assigned to elements.
A few examples are described below.
Elasticity.
You should alreadly be familiar
with the idea of a linear elastic material.
It has a uniaxial stress
MathType@MTEF@5@5@+=
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4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFuacaaa@37E7@
strain response
(valid only for small strains) as shown in the picture
stress--strain law for the material may be expressed in matrix form as
−ν
−ν
−ν
−ν
−ν
−ν
1+ν
1+ν
1+ν
]+αΔT[
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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aGimaaqaaiabgkHiTiabe27aUbqaaiabgkHiTiabe27aUbqaaiaaig
daaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGim
aaqaaiaaicdaaeaacaaIXaGaey4kaSIaeqyVd4gabaGaaGimaaqaai
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aGymaiabgUcaRiabe27aUbqaaiaaicdaaeaacaaIWaaabaGaaGimaa
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9oGBaaaacaGLBbGaayzxaaWaamWaaeaafaqabeGbbaaaaeaacqaHdp
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4maaqabaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaaGc
baGaeq4Wdm3aaSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiabeo8aZn
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aSIaeqySdeMaeuiLdqKaamivamaadmaabaqbaeqabyqaaaaabaGaaG
ymaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaI
WaaaaaGaay5waiaaw2faaaaa@9CE0@
Here, E and v are Young’s modulus and Poisson’s
ratio for the material, while
MathType@MTEF@5@5@+=
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denotes the thermal expansion
coefficient.
Typical values (for steel)
are E=210 GN/
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKaaGiaab2gakm
aaCaaajeaibeqaaiaabkdaaaaaaa@385D@
α=5×
  
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hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
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aaaaa@4228@
More extensive tables of values are given in
Section 3.1.5.
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa
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plastic material
behavior. Recall that the uniaxial stress
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
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strain curve for
an elastic
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4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
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0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
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plastic solid
looks something like the one shown on the right:
The material behaves elastically until a critical
stress (known as the yield stress) is reached.
If yield is exceeded, the material deforms permanently.
The yield stress of the matierial generally
increases with plastic strain: this behavior is known as strain hardening.
The conditions necessary to initiate yielding under
multiaxial loading are specified by a yield
criterion, such as the Von-Mises or Tresca criteria.
These yield criteria are built into the
finite element code.
The strain hardening behavior of a material is approximated
by allowing the yield stress to increase with plastic strain.
The variation of yield stress with plastic
strain for a material is usually specified by representing it as a series of
straight lines, as shown in the picture.
Boundary conditions
Boundary conditions are used to specify the loading
applied to a solid.
There are several
ways to apply loads to a finite element mesh:
Displacement
boundary conditions.
displacements at any node on the boundary or within the solid can be
specified.
One may prescribe
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or all three.
For a two dimensional
analysis, it is only necessary to prescribe
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caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaa
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaa
Various symbols are used to denote displacement
boundary conditions applied to a finite element mesh: a few of these are
illustrated in the figure on the right.
Some user-interfaces use small conical arrowheads to indicate
constrained displacement components.
For example, to stretch a 2D block of material
vertically, while allowing it to expand or contract freely horizontally, we
would apply boundary constraints to the top and bottom surface as shown in the
Observe that one of the nodes on the bottom of the
block has been prevented from moving horizontally, as well as vertically.
It is important to do this: the finite
element program will be unable to find a unique solution for the displacement
fields if the solid is free to slide horizontally.
During the analysis, the finite element program will
apply forces to the nodes with prescribed displacements, so as to cause them to
move to their required positions.
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
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caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaa
component of displacement is prescribed, then
the force will act in the
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa
direction.
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaa
is prescribed, the force acts in direction
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa
and so on. Notice that you cannot directly apply a rotation to a node attached to a 2D or 3D solid.
Rotations can, however, be applied to the
nodes attached to certain special types of element, such as beams, plates and
shells, as well as rigid surfaces.
conditions: Most finite element codes can automatically enforce symmetry
and anti-symmetry boundary conditions.
Prescribed
forces. Any node in a finite element mesh may be subjected to a prescribed
The nodal force is a vector, and
is specified by its three (or two for 2D) components,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaGGOaGaamOramaaBaaaleaacaaIXa
aabeaakiaacYcacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa
dAeadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3AB5@
Notice there is no direct way to apply a moment to a 3D solid
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E6@
you would need to do this by applying two
point forces a small distance apart, or by applying contact loading, as
outlined below.
Moments can be applied
to some special types of element, such as shells, plates or beams.
Distributed
A solid may be subjected to
distributed pressure or traction acting on ints boundary.
Examples include aerodynamic loading, or
hydrostatic fluid pressure. Distributed traction is a
vector quantity, with physical dimensions of
force per unit area in 3D, and force per unit length in 2D.
To model this type of loading in a finite
element program, distributed loads may be applied to the the face of any
boundary condition at boundary nodes.
If no displacements or forces are prescribed at a boundary node, and no
distributed loads act on any element faces connected to that node, then the
node is assumed to be free of external force.
External body forces may act
on the interior of a solid.
Examples of
body forces include gravitational loading, or electromagnetic forces.
Body force is a vector quantity, with
physical dimensions of force per unit volume.
To model this type of loading in a finite element program, body forces
may be applied to the interior of any element.
Probably the most common way to load a solid
is through contact with another solid.
Special procedures are available for modeling contact between solids
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E6@
these will be discussed in a separate section
In some cases, one may wish
to apply a cycle of load to a solid.
this case, the prescribed loads and displacements must be specified as a
function of time.
guidelines concerning boundary conditions.
When performing a static analysis, it is very important to make sure
that boundary conditions are applied properly.
A finite element program can only solve a problem if a unique static
equilibrium solution to the problem exists.
Difficulties arise if the user does not specify
sufficient boundary constraints to prevent rigid body motion of a solid.
This is best illustrated by example.
Suppose we wish to model stretching a 2D
solid, as described earlier.
examples to the right show two correct ways to do this.
The examples below show various incorrect ways to
apply boundary conditions.
In each case,
one or more rigid body mode is unconstrained.
Constraints
You may sometimes need to use more complicated
boundary conditions than simply constraining the motion or loads applied to a
Some examples might be
1.Connecting
different element types, e.g. beam elemen
2.Enforcing periodic
boundary conditions
3.Constraining a
boundary to remain flat
4.Approximating the
behavior of mechanical components such as welds, bushings, bolted joints, etc.
You can do this by defining constraints in an analysis.
At the most basic level, constraints can simply be used to enforce
prescribed relationships between the displacements or velocities of individual
nodes in the mesh.
You can also specify
relationships between motion of groups
Contacting Surfaces and Interfaces
In addition to being subject to forces or prescribed
displacements, solid objects are often loaded by coming into contact with
another solid.
Modern finite element codes contain sophisticated
capabilities for modeling contact.
Unfortunately,
contact can make a computation much more difficult, because the region where
the two solids come into contact is generally not known a priori, and must be determined as part of the solution.
This almost always makes the problem nonlinear
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E6@
even if both contacting solids are linear
elastic materials.
In addition, if there
is friction between the contacting solids, the solution is history dependent.
this reason, many options are available in finite element packages to control
the way contacting surfaces behave.
are three general cases of contact that you may need to deal with:
1.A deformable solid contacts a stiff, hard solid whose
deformation may be neglected.
case the hard solid is modeled as a rigid
surface, as outlined below.
2.You may need to
model contact between two deformable solids
3.The solid comes into contact with itself during the
course of deformation (this is common in components made from rubber, for
example, and also occurs during some metal forming operations).
you model contact, you will need to
1.Specify pairs of surfaces that might come into
One of these must be designated
as the master surface and the other
must be designated as the slave surface. (If
a surface contacts itself, it is both a master and a slave.
2.Define the way the two surfaces interact, e.g. by
specifying the coefficient of friction between them.
a stiff solid as a rigid surface:
many cases of practical interest one of the two contacting solids is much more
compliant than the other.
include a rubber in contact with metal, or a metal with low yield stress in
contact with a hard material like a ceramic.
As long as the stresses in the stiff or hard solid are not important,
its deformation can be neglected.
this case the stiffer of the two solids may be idealized as a rigid surface.
Both 2D and 3D rigid surfaces can be created,
as shown in the figure.
A rigid surface (obviously) can’t change its shape,
but it can move about and rotate.
motion is defined using a reference point
on the solid, which behaves like a node.
To move the solid around during an analyisis, you can define
displacement and rotational degrees of freedom at this node.
Alternatively, you could apply forces and
moments to the reference point.
Finally, in a dynamic analysis, you can give the rigid solid appropriate
inertial properties (so as to create a rigid projectile, for example).
a contact pair
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E7@
master/slave surfaces: Whenever you set up
a finite element calculation that involves contact, you need to specify pairs
of surfaces that may come into contact during the analysis.
One of each pair must be designated the master surface, the other must be
designated the slave surface.
This rather obscure finite element terminology refers
to the way that contact constraints are actually applied during a
computation.
The geometry of the master
surface will be interpolated as a smooth curve in some way (usually by
interpolating between nodes).
surface is not interpolated.
each individual node on the slave surface is constrained so as not to penetrate
into the master surface.
For example,
the red nodes on the slave surface shown in the figure would be forced to
remain outside the red boundary of the master surface.
For a sufficiently fine mesh, the results should not
be affected by your choice of master and slave surface.
However, it improves convergence (see below to learn what this means) if you choose the
more rigid of the two surfaces to be the master surface.
If you don’t know which surface is more
rigid, just make a random choice.
run into convergence problems later, try switching them over.
parameters You can define several parameters that control the behavior of
two contacting surfaces:
1.The contact formulation - `finite sliding’ or `small
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E6@
specifies the expected relative tangential
displacement of the two surfaces.
`Finite sliding’ is the most general, but is computationally more
demanding.
`Small sliding’ should be
used if the relative tangential displacement is likely to be less than the
distance between two adjacent nodes.
2.You can specify the relationship between the contact
pressure and separation between the contacting surfaces. Alternatively, you can
assume the contact is `hard’
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E6@
this means the interface can’t withstand any
tension, and the two contacting surfaces cannot inter-penetrate.
3.You can specify
the tangential behavior of the interface
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E6@
for example by giving the friction
coefficient.
Initial Conditions and external fields
For a dynamic analysis, it is necessary to specify the
initial velocity and displacement of each node in the solid.
The default value is zero velocity and
displacement.
In addition, if you are solving a coupled problem
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf
ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa
aaaaaaa8qacaWFtacaaa@37E6@
one involving coupled deformation and heat
flow, for example - you may need to specify initial values for the additional
field variables (e.g. the temperature distribution)
7.1.9 Solution
procedures and time increments
The finite element method calculates the displacement
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiaahIhadaahaaWcbe
qaaiaadggaaaGccaGGPaaaaa@373B@
at each node in the finite element mesh by
solving the equations of static equilibrium or the equations of motion.
In this section, we briefly outline some of
the solution procedures, and the options available to control them.
or Nonlinear Geometry As you know, you can simplify the calculation of
internal forces in a structures by neglecting shape changes when solving the
equations of equilibrium.
For example,
when you solve a truss problem, you usually calculate forces in each member
based on the undeformed shape of the
structure.
You can use the same idea to simplify calculations
involving deformable solids.
In general,
you should do so whenever possible.
However, if either
1.You anticipate that material might stretch by more
than 10% or so or
2.You expect that some part of the solid might rotate by
more than about 10 degrees
3.You wish to calculate buckling loads for your
you should account for finite geometry changes in the
computation.
This will automatically
make your calculation nonlinear (and so more difficult), even if all the
materials have linear stress-strain relations.
stepping for dynamic problems:
For a dynamic problem, the nodal
displacements
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiaahIhadaahaaWcbe
qaaiaadggaaaGccaGGSaGaamiDaiaacMcaaaa@38E4@
must be calculated as a function of
The displacements are calculated
by solving the equations of motion for the system, which look something like
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWHnbWaaSaaaeaacaWGKbWaaWbaaS
qabeaacaaIYaaaaOGaaCyDaaqaaiaadsgacaWG0bWaaWbaaSqabeaa
caaIYaaaaaaakiabgUcaRiaahUeacaWH1bGaeyypa0JaaCOraiaacI
cacaWG0bGaaiykaaaa@4036@
where M and
K are called mass and stiffness
Both M and K can be functions
There are 2.5 ways to integrate this
1.The most direct method is called explicit time integration, or explicit
dynamics and works something like this.
Remember that for a dynamic calculation, the values of u and
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0Jaamizaiaahwhaca
GGVaGaamizaiaadshaaaa@3947@
are known at t=0.
We can therefore
compute M and K at time t=0, and then
use them to calculate the acceleration
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWHHbGaeyypa0JaamizamaaCaaale
qabaGaaGOmaaaakiaahwhacaGGVaGaamizaiaadshadaahaaWcbeqa
aiaaikdaaaaaaa@3B0E@
F(t)−Ku
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWHHbGaeyypa0JaaCytamaaCaaale
qabaGaeyOeI0IaaGymaaaakmaabmaabaGaaCOraiaacIcacaWG0bGa
aiykaiabgkHiTiaahUeacaWH1baacaGLOaGaayzkaaaaaa@3ED4@
The acceleration can
then be used to find the velocity
v(Δt)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWH2bGaaiikaiabfs5aejaadshaca
GGPaaaaa@377D@
and displacement
u(Δt)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiabfs5aejaadshaca
GGPaaaaa@377C@
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R
Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
caGaaeqabaGadeaadaaakeaacqqHuoarcaWG0baaaa@3525@
v(Δt)=v(0)+aΔt          u(Δt)=u(0)+v(0)Δt+
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9
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This procedure can
then be applied repeatedly to march the solution through time.
2.The second procedure is called implicit time integration or implicit
The procedure is very
similar to explicit time integration, except that instead of calculating the
mass and stiffness matrices at time t=0,
and using them to calculate acceleration at t=0,
these quantities are calculated at time
MathType@MTEF@5@5@+=
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This is a bit more time consuming to do, however, because it involves
more equation solving.
3.The 2.5th method is called Modal Dynamics and only works if M and K are constant. In this
case one can take the Fourier transform of the governing equation and integrate
it exactly.
This method is used to solve
linear vibration problems.
following guidelines will help you to choose the most appropriate method for
your application:
1.For explicit
dynamics each time step can be calculated very fast.
However, the method is stable only if
MathType@MTEF@5@5@+=
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is very small
MathType@MTEF@5@5@+=
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specifically, the time interval must be
smaller than the time taken for an elastic wave to propagate from one side of
an element to the other.
This gives
Δt<L
ρ(1−2ν)/2μ(1−ν)
MathType@MTEF@5@5@+=
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GBcaGGPaaaleqaaaaa@4628@
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is the mass density of the solid,
MathType@MTEF@5@5@+=
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is its shear modulus and
MathType@MTEF@5@5@+=
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its Poisson’s ratio.
Explicit dynamics works best for rapid,
transient problems like crash dynamics or impact.
It is not good for modeling processes that
take place over a long time.
If elastic
wave propagation is not the main focus of your computation, you can sometime
speed up the calculations by increasing the density
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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(but you have to be careful to make sure this
does not affect the results).
called mass scaling.
2.For implicit
dynamics the cost of computing each time step is much greater.
The algorithm is unconditionally stable,
however, and will always converge even for very large
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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This is the method of choice for problems
where inertial loading is important, but rapid transients are not the focus of
the analysis.
Dynamics only works for linear
elastic problems.
It is used for
vibration analysis.
Solution Procedures for Static Problems: If a problem involves contact,
plastically deforming materials, or large geometry changes it is nonlinear.
This means that the equations of static
equilibrium for the finite element mesh have the general form
MathType@MTEF@5@5@+=
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where F() denotes
a set of b=1,2…N vector functions of
the nodal displacements
MathType@MTEF@5@5@+=
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a=1,2…N, and N is the number of nodes in the mesh.
The nonlinear equations are solved using the
Newton-Raphson method, which works like this.
You first guess the solution to the equations
MathType@MTEF@5@5@+=
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aaaaaaa8qacaWFtacaaa@37E6@
MathType@MTEF@5@5@+=
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Of course (unless you are a genius) w won’t satisfy the equations, so you
try to improve the solution by adding a small correction
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