MECH3300 Finite Element Methods

1
MECH3300 Finite Element Methods

• Lecture 5 Formulation of approximate stiffness
  matrices
• Planar elasticity problems
• Meshing 2D problems

2
Strain energy in plane stress

• Recall that we can estimate the stiffness of a
  structural finite element by expressing its
  strain energy in terms of its nodal
  displacements.
• For uniaxial stress, strain energy per volume is
  s e /2, the area under an elastic stress-strain
  curve.
• For plane stress in the xy plane, stresses are sx
  ,sy and txy only.
• The strain energy per volume is V 1/2 (sxex
  syey txygxy) or
• Writing stresses and strains as vectors is a
  notation of convenience - they do not behave as
  vectors.

3
Elastic stress-strain relations

• For any linear elastic problem we can write a set
  of linear equations relating stress to strain.
  The equations change with the type of problem (eg
  plane stress, plane strain, solid etc.)
• For plane stress, writing stress in terms of
  strain leads to a elasticity matrix D, where E
  is Youngs modulus and n is Poissons ratio.
• s De or
• D changes with the analysis type
• eg for a solid element D is 6 by 6, as there
  are 3 direct strains and three shear strains at
  some location in a solid element.

4
Strain energy in terms of strain

• Combining the stress-strain relation with the
  expression for strain energy and integrating over
  an element, we get

t is element thickness, and A its area. To
evaluate this integral, we need strains expressed
in terms of nodal displacements - this is done
with the interpolation functions. Now, recall for
the linear interpolation triangle that
x-displacement anywhere in the element is u(x,y)
N1(x,y) u1 N2(x,y) u2 N3(x,y) u3 The
x-strain is There are similar expressions for
y-strain and shear strain. Collectively the
strains, written in a column vector, are e B u,
the terms of the matrix B being either zero or
derivatives of interpolation functions like those
above.
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Strain energy of an element and its stiffness
matrix

• The transpose of e B u is eT uTBT
• Substituting in
• gives
• Hence the element stiffness matrix can be written
  as
• This integral is normally found numerically, as a
  weighted sum of the integrand evaluated at
  integration points (called Gauss points) within
  the element. The terms of B are constant for the
  3 node triangle, but are in general functions of
  position. The number of rows of B varies with
  the number of strain components that are
  relevant. The number of columns of B depends on
  the number of nodal degrees of freedom that the
  element has.

6
Common types of 2D elasticity problems

• Most packages given the user the choice of plane
  stress, plane strain or axisymmetric analysis
  with 2D planar elasticity elements.
• Plane stress occurs on an free surface or in a
  thin plate loaded in-plane.
• A plane stress analysis can also be used to
  analyze a detail like a stress concentration.
• One example is a plate containing a regular grid
  of holes. We do not wish to model all the holes,
  as that would require a very fine mesh. An
  alternative is to mesh a small region of plate
  containing one hole with plane stress elements
  and estimate effective values of E and n, from
  its deformation under load.

Question how to load and restrain this?
7
Plane strain problems

• Plane strain strictly refers to no movement at
  all in the 3rd dimension. This is an ideal which
  is approached in certain cases, where Poissons
  ratio effects tend to be prevented.
• A classic example is the tip of a crack in a
  thick plate. The surrounding material away from
  the crack tip is less stressed and resists the
  large Poisson ratio contraction that would
  otherwise occur in the direction along the crack
  tip, due to the high stresses at the crack tip.
  Hence a situation of plane strain develops at the
  crack tip, except at the ends.

Contraction prevented in this direction
Mesh (portion)
Load
Plane strain is often assumed when modelling a
typical cross-section through something long in
the 3rd dimension.
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Axisymmetric problems

• The most common type of 2D problem - model a
  half-section containing the axis, with each
  element representing a complete ring of material.
• If the loading is radial and axial only, and does
  not vary with angle circumferentially (eg inertia
  of a spinning disk), then the deformation is in
  the plane modeled.

z
Packages usually assume that x radius
x
Region modeled Actual object
9
Stresses and strains in planar elasticity

• For plane stress, there are in-plane stresses sx
  sy txy only. There is out-of-plane direct strain
  ez
• For plane strain there is no out-of-plane strain,
  but there is out-of-plane stress sz
• In an axisymmetric problem, there is out-of-plane
  stress and strain called hoop stress and strain.
• Hoop strain change in circumference/circumferenc
  e Dr/r
• Hence there is no rigid body displacement
  radially, and rigid body motion can be prevented
  by stopping an axial displacement.

10
Combined stresses

• While the state of stress at a point is
  represented by 3 principal stresses, these can
  also be combined to get other measures of stress.
• For isotropic materials, 2 stresses to compare to
  yield are commonly used
• Tresca stress - the diameter of the largest of 3
  Mohrs circles or twice the maximum shear stress
  in a principal plane.
• Von Mises stress - a value proportional to the
  square root of energy of distortion, that part of
  strain energy causing change of shape, as opposed
  to volume change.

t
s
sTr
For plane stress
11
Compatibility in meshing

• Compatibility means that things fit together when
  deformed.
• With finite elements this means displacements
  agree between neighbouring elements, not only at
  the nodes, but all along common edges in 2D or
  all over common surfaces in 3D.
• This means that one element can only join one
  other element, not 2 to 1 or 3 to 1.

This node is not connected to the top element, as
it has no mid-side node. Hence it can move up and
down independently and we have a model of a crack.
This is incompatible.
12
Transition meshes

• To change mesh refinement between two regular
  grids, transition meshes are needed. eg

Packages permit the user to replace an element
with grading subdivisions like these.
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Element distortion and the patch test

• Distorted elements are less accurate,especially
  as 2 sides come close to being parallel. In
  automatic meshing, a tolerance is set on element
  distortion.
• A test of how well elements cope with distortion
  is the patch test. A group of distorted elements
  are loaded in a way that should produce uniform
  stress, and the actual stress in the elements is
  examined to see if it is uniform. eg

P
P
14
Interpolation in natural coordinates

• Interpolation over a quadrilateral or hexahedral
  element can be described most neatly using
  non-Cartesian axes that bisect the sides of the
  element.
• In 2D, coordinates r and s that range from -1 to
  1 across the element are used.
• Interpolation of x-displacement is
• u 1/4 (1r)(1s) u1 1/4(1-r)(1s) u2 1/4
  (1-r)(1-s) u3 1/4 (1r)(1-s) u4
• ie u u1 at r s 1 etc

s
1
2
1
r
1
-1
4
-1
3
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Limits to distortion of quadrilateral elements

• Strains found in r, s coordinates need to be
  transformed to give strains in global x, y
  coordinates when estimating strain energy and
  hence stiffness.
• This transformation can fail if 2 element sides
  are parallel or if a corner angle exceeds 90
  degrees.

Unacceptable quadrilateral elements
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Interpolation on a triangle

• The neatest way to write the interpolation
  functions for a triangular 2D element is to make
  them functions of area coordinates.
• An area coordinate is the fraction of the area
  occupied by a subtriangle, with its apex at the
  point of interest.
• For a 3 node triangle, the interpolation of
  x-displacement is simply
• u S1 u1 S2 u2 S3 u3
• Note at node 1, S1 1. At node 2 or 3, S1 0
  so it works as an interpolation function.

3
S1 A1/A S2 A2/A S3 A3/A
A1
A2
As S1 S2 S3 1 only 2 of these are
independent coordinates.
1
2
A3