Chap 5. Isoparametric Finite Element Matrices

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Chap 5. Isoparametric Finite Element Matrices
Generalized coord FE models polynomials x, y,
z generalized coords Isoparametric
FE Relationship between the element
displacements at any point and the element nodal
point displacements using shape function. It
means we dont compute
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Isoparametric Derivation of Bar(Truss) Element
Stiffness Matrix
X, Y global coord r natural coord
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• Advantages of Isoparametric FE over General Coord
  FE
• Easily handle curved boundaries
• Easily construct element disp func since

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Implicit Function Theorem
Let denotes a set of coord and
as Cartesian coord. Let
where are
differentiable. It can be solved for as a
differentiable func of if Jacobian matrix
of is non-singular, i.e.,
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, in general, exits except for the elements
such as distorted or folded.
where is the matrix F evaluated at the
points weighting factors The more integration
points the more accurate. Once we know H, then
are the same as in
chap.4.
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Example 5.5
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For 2-D element, we need to compute
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Triangular Elements Degenerated triangle from
quadrilateral
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Example 5.15
Using the conditions
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Example 5.15 (continued)
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Example 5.15 (continued)
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Triangular Elements by Area Coord.
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Basic convergence requirements for Isoparameteric
elements
Monotonic convergence compatible disp
continuous with neighbor complete rigid
body disp constant strain Compatibility is
satisfied if the elements have the same nodes on
the common face and coords and disps on the
common face are defined by the same interpolation
functions Completeness requires rigid body
displacements and constant strain.
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For general geometric shape, isoparametric
elements always have the capability to represent
the rigid body modes and constant strain stress.
Therefore convergence is guaranteed. If dimension
of and are the same in then J is the
square matrix and can be inverted. in this
case, the element matrix correspond directly to
the global disp. If the order of global
coordination system is higher than the other of
natural coord system such as truss or plane
element, transformation to the global coordinate
should be included in the formulation.
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Ex 5.22
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Ex 5.23 Shear Stress Energy
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Ex 5.25
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Reissner Mindlin Plate
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General Cross-Section Isoparametric Beam with
Offset
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General Cross-Section Isoparametric Beam with
Offset
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General Cross-Section Isoparametric Beam with
Offset
T1 the global to local transformation T2 the
nonprincipal to principal transformation T3 the
translational transformation due to offset
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General Cross-Section Isoparametric Beam with
Offset
The translational matrix for displacement in
calculating kinetic energy bilinear form
The translational matrix for displacement in
calculating strain energy bilinear form
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Bilinear Isoparametric Plate with Warping
fa general forces (i.e. forces and moments)
applied in the actual plane fe general forces
applied in the mean plane ua general
displacements occurred in the actual plane ue
general displacements occurred in the mean plane
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Plate Checking
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Numerical Integration
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Gauss Quadrature
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Gauss Quadrature
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Gauss Quadrature
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Gauss Quadrature
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Polynomial
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Newton-Cotes