Plate and Shell Theory

8/19/2019 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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Plate and Shell Theory