UNIT 5
Introduction to Theory of Plates and Shells
Assumptions
● Thickness is smaller than the other dimension.
● Displacements are much smaller than plate thickness.
● Governing equations are based on undeformed geometry.
● Materials follow Hook’s Law.
● The middle surface is always unstrained during bending.
● The Plane normal to the middle surface remains normal.
● Rotatory inertia is negligible. Transverse shear strains also negligible.
Let us consider a thin plate in xy plane as given in the figure
Figure 5.1 A thin plate
Moments
Moment wrt to yaxis let us consider a cubical small element as shown in figure we will get
Figure 5.2 Small differential element
dFxx =σxx dz Moment wrt to xaxis (similarly) Twisting moment of shear stress Using the section module formula of the beam – 
Shear forces
Vertical equilibrium of plate
Substituting the above equation we get Shear force according to thickness

Governing Equation
For Qx –
w) =

Biharmonic governing equation
There are three possibilities of boundary condition
 Simply supported
Figure 5.3 Middle plane
For x constant –
w( x , y ) = 0
For edge y constant
W=0
2. Clamped
For x constant –
W=0
For y constant
W=0
3. Free edge
For x constant
Bending moment
Moment xy and shear –
For y constant
Bending moment
=0
Shear force
5.3 Navier’s method for a simply supported rectangular plate
Navier solution for lateral deflection of a simply supported rectangular plate having dimension a and b with distributed q as shown in the figure
Figure 5.4 Simply supported plate
Where m, n=1,3,5……
Multiply both side by sin(jπx/a) sin(kπy/b) and integrate wrt x and y from 0 to a
and 0 to b Integrating wrt x
Integrating wrt y
At m = j
= Similarly at n=k –
Now the equation(1) will be –
Substituting w and q(x,y) in the fourthorder governing differential equation
Now the W
Integrating equation (2) wrt x and y And Substituting the above two values in the equation number (2) we will get Now the equation (1) will be –

This is the equation of deflection for UDL using the Navier method for method.
5.4 Levy’s Method of Solution for Rectangular Plates
Let us consider a thin plate in xy plane in which at y=0 and y=b are simply supported and at x=0 and x=a can have any support like(clapped, free or simply supported) as shown in the figure –
Figure 5.5 Neutral plane of thin plate
A suitable governing equation for this case will be
For an isotropic plate Let us consider this is a square plate a=b=1m, n=1 made of steel and n=1 then Because it is a differential equation of fourthorder. Hence it will give two solution Where λ is the roots of the equation. If For isotropic plate r12 = π2, now the roots will be and The solution And The final solution will be

The above equation is the solution for the plate using the Levy theorem.
Key Takeaway:
 KCPT
2. 3. 4. 5.

6. Navier Method For plate (deflection)

7. Levy theorem For plate

REFERENCES
 Book Mechanics by Fridtjov Irignes Chapter7
 Sadd 9.3, Timoshenko Chapter11
 Module 9 version 2 ME, IIT Kharagpur
 Book solid mechanics 2nd by Kelly