Chp 5. Introduction to Theory of plates

Plates : sheet of material whose thickness is small compared with its other dimensions but which is capable of resisting bending, in addition to membrane forces.

• Investigate the effect of a variety of loading and support conditions on the small deflection of rectangular plates.

• Two approaches are presented: an ‘exact’ theory based on the solution of a differential equation and an energy method relying on the principle of the stationary value of the total potential energy of the plate and its applied loading.

Contents

Pure Bending of Thin Plate

Mx , My are bending moment per unit length (uniform along y and x axis)

M > 0 if it’s giving compression on upper surface and tension in lower surface.

Neutral plane ( in the middle of plate) as reference

ρ > 0 M > 0

We have

plane sections remaining plane the direct stresses vary linearly across the thickness of the plate

MxMx

z

ρxNeutral plane

z

σxz

Substituting σx and σy

Let (Flexural rigidity)

If w is the deflection of any point on the plate in the z direction

(Knowing Mx and My deflection of w )

If Mx = My = 0

If My = 0

If Mx = My = M

( Opposite curve direction / antielasticcurve )

(Same curve direction) / Synelastic curve

Plate subjected to Bending and Twisting moment

All M are per unit length

Mxy is a twisting moment intensity in a vertical x plane parallel to the y axis, while Myx is a twisting moment intensity in a vertical y plane parallel to the x axis.

The first suffix gives the direction of the axis of the twisting moment.

All M defined in Figure are all positive.

(two values of α, differing by 90o)

If Mt = 0Mn on two mutually perpendicular planes / principal moments and their corresponding curvatures principal curvatures.

Mxy relates to w ???

On face ABCD

On face ADFE

We know that How to relate v,u with w ???

Similarly

from

from

G = E/ (2*(1+υ))

(due to twisting bending only)

Combined bending and twisting moment

( due to twisting moment)

( due to bending moment)

Plate subjected to a distributed transverse load

q is load per unit area

Qx, Qy are shear force per unit length

Assumptions :

γxy, γyz are neglected

Variation of τxz and τyz are neglected

Resultant shear forces Qxδy and Qyδx are assumed to act through the centroid of faces of the element

Similarly

We have

Equation of equilibrium

Taking moment equilibrium about x

Taking moment equilibrium about y

Find w

Find direct and shear stress

With simply supported edges boundary conditions

Simply supported at x = 0

w = 0 and M = 0

Why ???

Built-in / fixed / Clamp edge boundary conditions

fixed at x = 0

Free edge boundary conditions

free at x = 0 All M and Q = 0

Simply supported at all edges

Plate subjected to uniform vertical loading with simply supported boundary conditions

or

2 BC are sufficient

Solution proposed by

NAVIER

in which m represents the number of half waves in the x direction and n the corresponding number in the y direction.

Find Amn determine w completely determine stresses and strains

We can also proposing

After calculation, we find

Example A thin rectangular plate a x b is simply supported along its edges and carries a uniformly distributed load of intensity q0. Determine the deflected form of the plate and the distribution of bending moment.

amn = 0 for m,n evens ????

Converge rapidly, few first terms give satisfactorily solution

By Taking υ = 0.3

Maximum at the center of plate and for square plate a = b. For five terms it gives

We have these equation in bending subchapter

at z = t/2

for a=b

Similar Procedure to Find Stress ( using Mxy)

Combine bending and in-plane loading of a thin rectangular plate

At the Middle plan due to bending, stresses = 0

In plane direct and shear forces Nx, Ny, Nxy ( per unit of length) are added

If stresses due to Nx , Ny and Nxy are small enough superposition of stress due to bending and twisting

If stress due to Nx, Ny and Nxy are big enough it will affect bending and twisting moment no superposition

Equilibrium on x axis

Small deflection

After calculation, we found that the governing differential equation for a thin plate supporting transverse and in-plane loads

Example Determine the deflected form of the thin rectangular plate of the previous example if in addition to a uniformly distributed transverse load of intensity qo, it supports an in-plane tensile force Nx per unit length.

Expression of transverse load as Fourier’s Series

Boundary conditions

And we have

Etc, etc ….Nx > 0 (tension) w smaller

Nx < 0 (Compression) w higher