System of Two Springs and its Calculations


In the process of finite element stress analysis, elastic body is first divided into individual connected parts, that are the finite elements of the structure. Elements which are connected at points are the nodes. The process of dividing the domain into elements is called discretization and the pattern of elements is the mesh. Later on, using a stiffness matrix, and the loads applied to the nodes of a single element and the resultant nodal displacements of the structure. The final step is to combining together all the stiffness matrices of each and individual elements into a single large matrix which is the global stiffness matrix. Now we shall illustrate this process using linear elastic springs.

The below figure shows a shows a system of springs, each of stiffness k but with a distribution of forces. In the particular spring which is identified by the nodes 1 and 2 in that system of springs, and there are the forces F1 and F2 respectively. At equilibrium F1 + F2 = 0 or F2 = −F1 . System at equilibrium condition, which consists of a pair of springs, with different stiffnesses k1 and k2 . As it follows that


Free Body Diagram of Pair of Springs


From above diagram, We can write the equation for each spring in matrix form:

equation in matrix form

As we start assembling the equilibrium equations, which describing the actual behavior of the system of two springs, and the displacement compatibility conditions, which will relate element displacements to the system displacements,  and it is written as:


And therefore, substituting these  displacements in the above equation:

substitute equation

In these above two equations, we use the notations for “f” to represent the amount of force exerted on the element “j” and at node “i“.

By expanding the each equation in matrix form:

expanding equation

Now summing member by member:

summing equation

Let we refer to the free-body diagrams of each of the three nodes:

three nodes

Final Form of the Equation:

From the above equations, we can write the final form of the equation

final form

Where the stiffness matrix [K] is:

s.matrix for two springs

Conditions and brief record of points has written down for system stiffness matrix is:

  • MATRIX IS SYMMETRIC, It is the expression with all the linear systems are referred to the right-angled coordinate systems;
  • MATRIX IS SINGULAR, As no boundary conditions are applied to arrest rigid body motion of the system;
  • Matrix system is simply a enfold of the discrete element stiffness matrices, with proper assignment of the element nodal displacements and connected stiffness coefficients to system nodal displacements.

Hence, these is the calculations and conditions for system of two springs in Finite Element Theory.

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Hello, I am an aircraft structural analyst with industrial experience and a master degree on aerospace structures. Currently working for an aerospace company as a stress
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