plate and shell theory
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Plate and Shell Elements
Plate Theory
∙ Flat plate
∙ Lateral loading
Applications:
∙ Shear walls
∙ Floor panels
∙ Bending behavior dominates ∙Shelves
∙ …
Forces an Moments Act ng on t e P ate:
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Stresses:
Relations
Between
Forces
and
Stresses
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Stresses:
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Thin Plate Theory ( Kirchhoff Plate Theory)
Assumptions (similar
to
those
in
the
beam
theory):
A straight line along the normal to the mid surface remains straight and normal to the
deflected mid surface after loading, that is, these is no transverse shear deformation:
Displacement:
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Strains:
Note that there is no stretch of the mid surface
due to the deflection (bending) of the plate.
Stresses (plane stress state):
or Main variable: deflection w=w(x,y)
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Governing Equation:
,
Compare the 1‐D equation for straight beam:
Shear forces and bending moments:
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Boundary
Conditions:
where n is the normal direction of the boundary. Note that the given values in the
boundary conditions shown above can be non‐zero values as well.
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Examples:
,
concentrated force P at the center C.
The maximum deflections from analytical solutions are given in the
following table
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Thick Plate Theory (Mindlin Plate Theory)
e c ness o a p a e s no n , e.g.,
(L = a characteristic dimension of the plate), then the thick plate theory by Mindlin should
be applied. This theory accounts for the angle changes within a cross section, that is,
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New
relations:
Note that if we imposed the conditions (or
assumptions) that
then we can recover the relations applied in
the thin plate theory.
Main variables:
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Plate Elements
Kirchhoff Plate Elements:
4‐Node Quadrilateral Element
DOF at each node:
On each element, the deflection w(x,y) is
represented by
where Ni, Nxi and Nyi are shape functions.
The
stiffness
matrix
is
still
of
the
form
where
B
is
the
strain‐
displacement
matrix,
and
E
the
stress
strain
matrix.
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Mindlin Plate
Elements:
‐ ‐
DOF
at
each
node:
On each element:
∙ Three independent fields.
∙ Deflection w(x,y) is linear for Q4, and quadratic for Q8.
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Example:
ANSYS 4‐node quadrilateral plate element.
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Shell
Elements
Shells
– Thin
structures
witch
span
over
curved
surfaces.
Examples:
∙ Sea shell, egg shell (the wonder of the nature);
∙ Containers, pipes, tanks;
∙ Car bodies;
, , .
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Forces in shells:
Membrane forces + Bending Moments (plates: bending only)
Exam le: A C lindrical Container.
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Shell Elements:
Similarity: bar + simple beam element => general beam element.
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Buckling coefficients for simply supported rectangular plates with an eccentric circular hole
in horizontal direction.
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Buckling coefficients for simply supported rectangular plates with an eccentric circular hole
in horizontal direction.
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