WHY STIFFNESS MATRIX IS IMPORTANT IN FEA

 Hey guys, 

Let's talk about some important thing which is the stiffness matrix. The stiffness matrix is used in every type of analysis either it is static or dynamic. So what is the stiffness matrix as its word suggest the stiffness of an object or stiffness of anything 

The stiffness matrix is a matrix which is the stiffness of any elements or the structure which is expressed in the matrix form for solving the governing equation of the problem to get the solution.

The stiffness matrix is the key to the treasure to the FEA. Bcoz if there is no stiffness matrix associated with your problem then we cant solve the FEA problem. The stiffness matrix is denoted by this [k].

The matrix mainly depends on the properties which are the material and the geometric structure.

For deriving the matrix for any geometry or shape of elements there are three types of methods which is...

1. Direct method: The  direct method is easy to understand but difficult to program so the commercial software doesn't use this method

2.Variational method: This method is easy to understand but moderate in the program point of view

3.Weighted residual method: This method is difficult to understand but easy from the coding point of view this is the most common method which is used in the software.

So let's dive into it what is the stiffness matrix?

Stiffness [K] is given as = Force(F)/Displacement(dx)

For better understanding, let's consider the two rods having the same dimensions but different material say, steel and aluminum.

The rod is fixed at the one end and the force is applied in the other end in a downward direction.

So what is stiffness?

The steel rod is stiffer because it has the high elastic modulus

Now consider the same material but the one rod is having a more cross-sectional area than another rod. In this case, the rod having more c/s has stiffer 

If talking about the elements we have general the two elements are in use that is quad and tria in this case if we compare the elements, the tria elements have stiffer then the quad elements. That's why we are generally going with the quad element mesh. We will be discussing more on the meshing in the upcoming session.

The stiffness matrix is also categorized by the two types which are 

1.Local stiffness matrix

2.Global stiffness matrix 

The local is for only the one problem to solve, but if there are various constrained in the problem or we can say,  for the whole system then we use the global matrix which is larger then the local one.

The governing equation for the static is given as 

[K]{u}={F}

where,   [K]=stiffness matrix, {u}=Nodal displacement, {F}=Force vector

In which force is usually known and the displacement is unknown and the [K] is characterized property of elements.  

For the example of the stiffness matrix method we have to know about the degrees of freedom so, for now, we will stop here and we will continue with the example in the upcoming blog the example for this matrix is like the mathematical formulation that needs to understand. That will take some time. In the next post, we will talk about DOF's and meshing for better understanding. 

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