2. Question: Find the approximate deflection of a simply supported beam carrying
a symmetrical Triangular load P using Rayleigh Ritz method.
Solution:
Let w(x) denote the deflection of the beam(field variable).The differential
equation formulation leads to following statement of the problem:
i.e it satisfies the governing equation
. /
And the boundary conditions
{
( ) ( )
( ) ( ) ( ) ( )
}
P
3. Total energy i.e. potential energy of the system is given below:
Π= ∫ [ ( ) ( )]
Replace P=8
( )
( )
( )
9
Π= [∫ { ( ) ( )} ∫ { ( ) (
( )
)} ]
Where E=Young’s Modulus
I=Moment of inertia, l=length of beam, P=load on beam
Now we are integrating above equation denoted in different color
separately.
Let here w(x)=
( ) ( ) , where C1 and C2 are constant
After using the value of and w in above equation we have,
∫ 0 ( ( ) ( ) ) .
( )
/1
∫ 6 2 ( ) ( ) ( ) ( ) ( ) ( ) 3
4
( )
57 ( )
6. Therefore, Π= [ ]
i.e. Π= 0 , ( ) ( ) - * ( ) ( ) + , ( )
( ) - , - * , ( ) - , ( ) -+1
where C1 and C2 are independent constant. For the minimum of π we have
{ ( ) } , - =0
{ ( ) }=0
Therefore,
( ) , which is the deflection of beam.
The deflection of beam at the middle point of beam(i.e. at l/2) is given by
( ) ( )
which compare with the exact solution