16_Direct Methods

Three main ways to derive finite element equations.

 

1. Direct method or Direct equilibrium method for structural analysis problems

2. Energy Methods and principle of virtual work

3. Weighted Residual Methods

 

Of the three methods, the direct method is the simplest and can clarify the physical understanding of the FEM. However, it is limited to finding the stiffness matrix of a one-dimensional element.

In this post, the direct method will be explained.

 

Before that, it is essential to undertand the stiffness matrix.

Define the stiffness matrix as follows. The element stiffness matrix $[k]$ can be representedd by the following matrix.

 

 

Through $[k]$, the nodal displacement ${d}$ can be related to the nodal force ${f}$.

In the case of a structure composed of multiple elements, the stiffness matrix $[k]$ connects the global node displacement ${d}$ of the entire structure to the global force ${F}$.

 

 

In the above equation, $[K]$ represents the global stiffness matrix, not the element stiffness matrix.

 

 

In the case of a 1-dimensional linear spring, the axial forces acting on the nodes at the ends of the spring are ${f_{1x}}$ and ${f_{2x}}$, and the displacements are ${u_{1}}$ and ${u_{2}}$.

 

The deformation of the spring caused by the tensile force is denoted as $\delta $, where ${u_{1}}$ has a negative value as it is in the opposite direction to the x-axis, and ${u_{2}}$ has a positive value.

 

Through $T = k\delta $, which represents the relationship between force and strain, the following equation can be derived.

 

 

Since the spring isin a state of force equilibrium, it satisfies the following equation.

 

 

It can be represented in matrix form as follows.

 

 

The stiffness matrix is a symmetric, square, and singular matrix.

 

 

The force/strain relationship of a spring with two connected elements can be represented by a matrix as follows, where the susperscript denotes the element number and the subscript denotes the node number.

 

 

The force equilibrium equation, equating external and internal forces, can be represented as follows.

 

 

Therefore, representing the external forces acting on each node in terms of stiffness and displacement results in the following expression.

 

 

In matrix form.

 

 

Thus, using the direct method, the global stiffness matrix for a spring composed of two elements can be determined.