Validation Case: Reaction Force analysis of a statically indeterminate system

Overview

This article is part of the series of FEA validation cases performed using OnScale Solve.

A bar assembly with three parts is fixed at two ends and force loads are applied at the part interfaces. This type of system is statically indeterminate. The reaction forces at both ends from the Solve simulation are compared against analytical results.

The following features of a linear mechanical static analysis are included in this validation test.

• Force Load
• Restraints
• Reaction Forces (outputs)

Geometry

The geometry is avalable as an Onshape document.

The problem setup is shown in fig. 1. The dimensions are:

• Cross Sectional Area = 0.1~\text{m}~\text{by}~0.1~\text{m}
• L_{1} = 0.4~\text{m}
• L_{2} = 0.3~\text{m}
• L_{3} = 0.3~\text{m}

Material

• Structural Steel:
  • Young’s Modulus (E) = 200~\text{GPa}
  • Poisson’s Ratio (\nu) = 0

Physics

• Restraints: Fixture at the Top and Bottom faces.
• Force Load:
  • F_{1} \rightarrow 500~\text{N}
  • F_{2} \rightarrow 1,000~\text{N}

Meshing

A second-order tetrahedral mesh with 1642 vertices is used for this study.

Reference Solution

The solution can be derived from [1]. The force equilibrium equation is:

R_{C} + R_{B} - F_{1} - F_{2} = 0

The compatibility equation is derived from the change in length of each section and from the condition of zero change in overall length. Uniform material properties and cross-sectional area means that the modulus and area do not affect the solution.

\frac{R_{B} \cdot L_1}{A \cdot E} = \frac{R_{C} \cdot (L_2 + L_3)}{A \cdot E}

By solving both equations simultaniously, the result is:

R_{B}= \frac{F_1 + F_2}{1 + \frac{L_1}{L_3+L_2}} = \frac{500 + 1000}{1 + \frac{0.4}{0.3+0.3}} = 900~\text{N}

R_{C}= \frac{F_1 + F_2}{1 + \frac{L_2+L_3}{L_1}} = \frac{500 + 1000}{1 + \frac{0.3+0.3}{0.4}} = 600~\text{N}

Results comparison

The table below compares the values obtained using the analytical methods described above and the OnScale Solve analysis.

References