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This assignment solution was submitted to Amar Sharma for Finite Element Method course at Aligarh Muslim University. It includes: Stress, Distribution, Tapered, Bar, Finite, Elements, Axial, Load, Idealization, Displacement, Models
Typology: Exercises
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Question #1.2:-
Find the stress distribution in the tapered bar shown below using two finite elements under an axial load of P=1N.
Solution:-
Let the bar be considered as the assemblage of two elements as shown below:
Assume the bar to be one-dimensional then we have only axial displacement at any point. As there are three nodes so we have nodal displacements Ø 1 , Ø 2 , Ø 3. These will be unknown.
In each element we assume a linear variation of axial displacement Ø.
where ‘a’ and ‘b’ are constants which can be found out by putting the boundary conditions:
Ø=Ø 1 (e)^ at x = 0 Ø=Ø 2 (e)^ at x = l
So by putting the boundary conditions; we get:
a= Ø 1 (e)^ and b= (Ø 2 (e)^ - Ø 1 (e)) / l(e)
Using principle of minimum potential energy:
where
and
Hence
In Matrix Form
Where
Combining:
Global Load Vector:
Overall Equilibrium Equation:
Inserting Boundary Conditions for nodal displacements:
As Ø 1 =0 So we eliminate the first and first column we have:
And Stresses are:
