Deriving stiffness of a node in finite element theory

[chandran]

I want to derive the theory of finite element using my own understanding. With this derivation i want to understand the fea theory of rod element

I call the nodes as "points" in a structure.

This scenario is for a simple one dimensional displacement of points(that is the single degree of freedom nodes)

i have 4 points names 1,2,3,4 consecutively along a line and the displacement of each point is named as u1,u2,u3,u4. and force at each point as f1,f2,f3,f4. The displacements will occur only along a line.

There are elements between the nodes signifying the stiffness. The elements have stiffness named,K1,K2,K3.

here K1 is the element between point 1 and 2 K2 between point 2 and 3 and K3 between point 3 and 4.

Now i am writing an equation for displacement of point 1

u1=f1/stiffness. How can i derive the stiffness.

I can say the stiffness is equal to K1 and say u=f1/K1 if and only if u2=0. But u2 may have a definite displacement in this scenario.

so what is the equation of u1

 

[lance]

unless you are actually trying to model some sort of physical material (ie a rod made out of steel), you can choose any numerical value you want for your stiffnesses.

The point of finite element analysis, is to take the governing differential equation of a system and solve for its weak form (integral form). I don't think you're approach will teach you all that much about FEA.

I would suggest, looking into the Galerkin method for solving a 1-dimensional rod problem. you can find a solution by hand and it should give you a better idea of what FEA is all about.

if you have trouble finding any info on that, I would be happy to write out an example.

 

[Mech_Engineer]

You have the basic equation:

F=k*u

For a single link, say link 1. It has two nodes, node 1 and node 2; two displacements, u1 and u2; two forces, F1 and F2. Link 1 has one stiffness, k1.

It ends up being two equations with two unknowns, where we consider the displacement of each node unknown, and the force at each node along with the link's stiffness known.

F1=k1*u1
F2=k1*u2

You can then think of the system of equations in terms of a linear algebra equation with matrices.

[F]=[[k]]*

Once the two equations are described as a linear algebraic equation, you can add in two more equations, and see how they would be added into one stiffness matrix...

Does that point you in the right direction?

 

[chandran]

lance,
can you write out an example