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Stiffness Coefficients 2-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Aligarh Muslim UniversityMathematical Methods for Numerical Analysis and Optimization

This assignment solution was submitted to Amar Sharma for Finite Element Method course at Aligarh Muslim University. It includes: Beam, Element, Generate, Stiffness, Coefficient, Direct, Approach, Equilibrium, Deflection

Typology: Exercises

2011/2012

Uploaded on 07/08/2012

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ASSIGNMENT # 06
Title: Problem # 5.19
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  • ASSIGNMENT #
    • Title: Problem # 5.

PROBLEM # 5.

For the beam element considered in problem 5.18 generate expressions for the stiffness coefficients K 12 , K 22 , K 32 and K 42

Solution:

Direct approach is used to solve the problem i.e. basic deflection relations and static equilibrium equations. So first we do the solution comprising the deflection relations. For our convenience we replace the symbol used in the figure by the most familiar symbols i.e replacing , , , by , , , respectively.

So for the second column of the stiffness matrix i.e. K 12 , K 22 , K 32 and K 42 .we have all the other constraints equal to zero, while the θa =1. The θ is the rotation produced by the force V.

At any section along the length of the beam we have relation for bending moment which is given as

---------------------------- (ii)

Now from statics we get

------------------------------ (iii)

Now by taking sum of moments at a

----------------------------- (iv)

Thus from (i), (ii), (iii) ,(iv) we get,