Question

Finite Element Analysis   CVE705

Stiffness Matrix

Problem:

An eight-node element assemblage shown is used in a finite element analysis.

Calculate the diagonal element of the stiffness matrix corresponding to the degree of freedom U₁₀₀ shown.

Use a plane stress case

E = 10,000

v = 0.3

t = 1.0

U100

Answer 1

Under this section, procedure to include the effect of boundarycondition in the stiffness matrix for the finite element analysis will be discussed. The solution cannot be obtained unless support conditions are included in the stiffness matrix. This is because, if all the nodes of the structure are included in displacement vector, the stiffness matrix becomes singular and cannot be solved if the structure is not supported amply, and it cannot resist the applied loads.A solution cannot be achieved until the boundary conditions i.e., the known displacements are introduced. In finite element analysis, the partitioning of the global matrix is carried out in a systematic way for the hand calculation as well as for the development of computer codes. In partitioning, normally the equilibrium equations can be partitioned by rearranging corresponding rows and columns, so that prescribed displacements are grouped together. For example, let considerthe equation of equilibrium is expressed in compact form as: F Kd     (2.4.1) Where, [K] is the global stiffness matrix, {d} is the displacement vector consisting of global degrees of freedom, and {F} is the load vector corresponding to degrees of freedom. By the method of partitioning the above equation can be partitioned in the following manner.

If the overall stiffness matrix is to be formed in half band form then the numbering of nodes should be such that the bandwidth is minimum. For this the labels are put in a systematic manner irrespective of whether the joint displacements are unknowns or restraints. However, if the unknown displacements are labeled first then the matrix operations can be restricted up to unknown displacement labels and beyond that the overall stiffness matrix may be ignored.