How to find the stiffness matrix for a beam element?

[chiraganand]

I need to find the stiffness matrix of a beam element from the basic priciples. I am not allowed to use Lagrange's polynomial or other methods. Need to find the stiffness matrix from basic principles ie to find at each node k11 and so on..
I have attached a screenshot of the problem.

When i try to find the deflection at each node by using deflection formulae for beams i am not able to find out the answers. the formula i use id EI (d^2 y/ d^2 x)= M , i am not able to find the answers. Please help me out in this

 

[Valkarie]

problem.The stiffness matrix of a beam element is related to the deflection of the beam at each node. The equation you are using (EI(d^2y/d^2x) = M) is an equation for the bending moment of the beam, which is related to the force in the beam. It is not an equation for the deflection of the beam.To find the stiffness matrix from basic principles, you need to solve for the displacement of the beam at each node. To do this, you can use the equations of static equilibrium to solve for the forces in the beam and then use the equations of static deformations (strain-displacement relations) to solve for the displacement of the beam at each node. Once you have the displacement of the beam at each node, you can use the equations of kinematic compatibility to solve for the stiffness matrix. Hope this helps.