Beam Element Stiffness Matrix Deriviation

[hkanj1993]

Dear Analysts,

I have been trying to understand the process of derivation of the stiffness matrix of the classic Euler–Bernoulli beam element.
I started with deriving the shape functions using a third order polynomial.

Then I could write the approximation for the displacement function:

where N1,N2,N3, and N4 are:

Trying to derive the first element of the stiffness matrix goes something like this:

applying integration by parts we get:

for N­1*N1'''':

My problem is that the integration of the error will give a trivial zero. I would appreciate if someone would point out where my calculation is going wrong.

Sorry for the long post, many thanks and best regards.

 

[FEA way]

Did you compare this with derivations that can be found in literature ? Here's a short list of books where you may find such information:
- "Introduction to Finite Element in Engineering" Chandrupatla and Belegundu
- "A First Course in Finite Elements" Fish, Belytschko
- "Introduction to the Finite Element Method" Ottosen, Petersson
- "A First Course in the Finite Element Method" Logan
- "Fundamentals of Finite Element Analysis" Hutton

 

[hkanj1993]

- "Introduction to Finite Element in Engineering" Chandrupatla and Belegundu

Thanks for the suggestions, I'll see if I can find the answer in this book.