12 x 12 Stiffness Matrix solved manually

Four hours later I solved manually a 12 x 12 stiffness matrix for a four story and two bays structure with three degrees of freedom in order to understand how the stiffness matrix behaves, before scale it up and code the algorithm to calculate it.

The Finite element method is based in solving the f = K x δ fundamental equation, where f is the forcé, k is the stiffness matrix and δ is the deformation.

Because one of the fundamental characteristics governing the behavior of elastic structures, is the relationship between the applied loads and the displacements with those induce.

Therefore we can use the definition of stiffness, which is the forcé necessary to maintain a unit displacement.

What we do, it is to generate an unit virtual displacement in the structure in order to generate the shape in which the structure bends under the action of the unitary displacement.

In a sense, we tell the structure how to bend under the applied forces in order to find out the coefficients for the stiffness matrix k and solve the fundamental equation.

By choosen three degrees of freedom for the four story structure, we generate the size of the stiffness matrix k, which is three times four.

Once we find out the values for the stiffness matrix k and having the external forces applied to the structure, we proceed to solve the fundamental equation, by calculating the deflections δ.

In order to calculate the deflections it is necessary to invert the stiffness matrix K.

The inversión of a matrix is time consuming and requires real computing power, therefore the electronic computers are very Handy to solve the equation.

So this 12 x 12 matrix is part of the foundation for my MagleV Structures software.