Analyzing a Linear Elastic Spring System Using FEM and ANSYS

1. Introduction to Linear Elastic Springs

A linear elastic spring is a mechanical device that supports axial loading. Its deformation (elongation or compression) is directly proportional to the applied load, as described by Hooke’s Law:

where:

  • (F) = applied force,
  • (k) = spring stiffness (N/m),
  • (\Delta L) = deformation.

In this problem, we have three springs with stiffness values (3k), (2k), and (k), supporting equal weights (W = 76.982 , N).

 

 

2. Finite Element Method (FEM) Procedure

The FEM general procedure involves the following steps:

  • Formulate Individual Element Stiffness Matrices

Each spring is treated as a finite element, and its stiffness matrix is derived. For example, the stiffness matrix for Element (1) (with stiffness (3k)) is:

  • Assemble the Global Stiffness Matrix

The global stiffness matrix ([K]) is assembled by superposing the individual element matrices. The resulting system matrix is:

  • Apply Equilibrium Conditions

The governing equation for the system is:

[K] * {U} = {F}

where ({U}) is the displacement vector and ({F}) is the force vector.

  • Solve for Nodal Displacements

The system of equations is solved to find the displacements (U_1, U_2, U_3, U_4).

 

 

 

 

3. Hand Calculations

Using the given data ((k = 1000 , N/m), (W = 76.982 , N)), the system of equations is solved to determine the displacements. The results are:

  • (U_1 = -76.982 , mm) (Top Spring),
  • (U_2 = -153.96 , mm) (Middle Spring),
  • (U_3 = -230.95 , mm) (Bottom Spring).

4. ANSYS Modeling and Results

The system was modeled in ANSYS using the following steps:

Material and Geometry

    • Material: Structural steel (density = 7850 kg/m³).
    • Geometry: Defined springs and weights.

Simulation

    • Applied boundary conditions and loads.
    • Solved for nodal displacements.

Results

The ANSYS results matched the hand calculations perfectly:

Spring Position Deformation (mm)
Top Spring -76.982
Middle Spring -153.96
Bottom Spring -230.95

 

5. Verification and Validation

The results were verified by:

  1. Ensuring consistency with the mathematical model.
  2. Comparing ANSYS results with hand calculations.
  3. Confirming that numerical errors were within acceptable limits.

6. Conclusion

This problem demonstrates the power of FEM and ANSYS in solving structural analysis problems. By combining theoretical calculations with simulation tools, we can achieve accurate and reliable results.

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