Formulation of Spring Element

Basically you know what is a spring. To model a spring in FEM it is best to use one dimensional spring element with 2 nodes as shown below. The element type is therefore is Linear Spring.

1 and 2 are the node numbers
U1 is translation in x direction at node 1.
U2 is translation in x direction at node 2.
f1 is the force action in x direction at node 1
f2 is the force action in x direction at node 2
K is the stiffness of the spring

Above items are the basic of FEM and the basic items in a idealized spring element. It is assumed that spring obeys hooke’s law and resists forces only in the direction of the spring.

A linear displacement function can be assumed for this element as shown below.

We can input the nodal values to determine the coefficients.

u(0) = u1 = a1
u(L) = u2 = a2L + u1
a2 = (u2 – u1 )/L

Therefore the linear displacement function can be expressed as below in nodal terms

Alternatively, When N1 = 1-x/L , N2 = x/L ( N1,N2 are called shape functions)

Above matrix is the typical format of the displacement function.

Sum of all shape functions should be equal to 1 .

Above figure shows how shape function value varies with the x distance.

Now lets consider the strain-displacement relationship,by assuming constant strain throughout the element

Stiffness of a spring element is calculated by K =AE/L

A= cross section area, E= Young’s Modulus , L = Length of the Element

For the spring element,

Where T is the force in the spring taken as positive for tensile.
Consider the forces in the spring to be positive if tensile and its value being T

This is the system of equations that represent a spring element

The Local Element Stiffness Matrix of Spring is As Below

Above formulation is used to analyse spring elements of a structure. Example problem for springs will be solved in the next post and you will be able to understand how to interpret the spring element in a real situation.