Ex 4'12

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Ex 4.12
Transformation Matrix
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Ex 4.13
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4.2.3 Generalized Coordinate Models for Specific
Problems
Choice of mathematical models i.e, each element
as bar, beam, plate, solid
In Ex. 4.6
Polynomials are used to approximate disp since
they are easy to differentiate to get strain.
(The higher the degree the better the approx )
More effective FE are isoparametric formulation.
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(No Transcript)
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Elementary Beam (Technical)
Kirchhoff Plate
Shear correction factor k
Reissner-Mindlin Plate
Timoshenko Beam
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Kirchhoff theory - Shear deformations are
neglected. - The straight line remains normal
to the midsurface during deformation. Reissner
Mindlin theory - Shear deformation are
included. - The straight line remains straight
and in general not normal to the midsurface
during deformation. Flat rectangular shell
element
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For more whose adjacent element is almost
coplanar such as folded plate or curved shell ,
the stiffness matrix is singular or
ill-conditioned. Therefore it is difficult to
solve for equilibrium. To avoid this problem, add
a small stiffness is in direction, i.e
,
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k Factors for various Cross Sections ( Cowper
,G.R., J of applied mechanics , June, 1966,p335)
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4.2.4 Lumping of Structure Properties and Loads
Ex 4. 21
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Analyses with and without Consistent Loading
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4.3 Convergence of Analysis Results
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A FE solution should converge as the number of
elements is increased to the analytical solution
of the DE of mathematical model.? monotonic
convergence Errors Round-off, time integration,
iteractive, mode selection, linearization,
discretization
PVW
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Monotonic convergence Elements must be
complete? and compatible?? complete
Displacements of FE should represent the rigid
body displacement and the constant
strain state ? compatible Displacements within
and across the elements must be
continuous Number of element rigid body mode
Element DOF Number of element straining mode
(natural mode)
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Calculation of Stresses
Compatible (conforming) ?displacement and their
derivatives are continuous.Displacement
continuity does not mean stress
continuity.Coarse FE model ?more
difference.?Reason compatibility and
constitutive equation are exactly satisfied
while stress equilibrium are approximated.Thus
Stresses in
general more accurate at integration points than
at the modal point. Hence, for a least square fit
, use higher order functions than that of
stresses from the assumed displacement function (
)
Average nodal stress bilinearly extrapolate
from Gauss point stress and then average
Least square procedure requires more computation
than average scheme
NASTRAN
ABAQUS
s
s
m,b
m,b
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4.4 Incompatible(nonconforming) and mixed FE
In displacementbased FEM, the assumed
displacement function are complete and
compatible. Those solution converges in the SE
monotonically to the exact solution.For shell
problem, compatibility is hard to
maintain. Incompatible displacement-based models
Relax compatibility condition ? No guarantee of
monotonic convergence.Note that the size of FE
gets smaller, each element should approach a
constant strain condition. Patch test ( BM Irons
A Razzague ) Boundary nodal forces (or
displacement BC ) ? constant stress For
displacementbased incompatible elements, if the
patch test is passed, convergence is insured