Plate Theory slides from WEB.pdf

8/14/2019 Plate Theory slides from WEB.pdf

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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:



Stresses:

Relations Between Forces and Stresses



Stresses:



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:



Governing Equation:

,

Compare the 1 ‐ D equation for straight beam:

Shear forces and bending moments:



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.



Examples: ,

concentrated force P at the center C.

The maximum deflections from analytical solutions are given in the

following table



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,



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:



Plate Elements

Kirchhoff Plate Elements:

4‐Node Quadrilateral Element

DOF at each node:

On each element, the deflection w(x,y) isrepresented 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.



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.



Example:

ANSYS 4‐node quadrilateral plate element.



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;

, , .



Forces in shells:Membrane forces + Bending Moments (plates: bending only)

Exam le: A C lindrical Container.



Shell Elements:

Similarity: bar + simple beam element => general beam element.



EXAMPLE: SIMPLY SUPPORTED PLATES WITH AN ECCENTRIC CIRCULAR HOLE UNDER COMPRESSIVE LOADS

Rectangular plates under compressive loads: with a central circular hole ; with an



Buckling coefficients for simply supported rectangular plates with a central circular hole.



Buckling coefficients for simply supported rectangular plates with an eccentric circular hole

in horizontal direction.

Plate Theory slides from WEB.pdf

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