Validation Case: Pinned Bar Under Gravitational Load

This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters:

• Gravitational load
• Remote displacement
• Nodal velocities

The simulation results of SimScale were compared to the results presented in [Schaum’s]\(^1\).

Geometry

The geometry used for the case is as follows:

The bar is a square bottom with a cross-section area of 0.01 \(m^2\) and the length of the bar is 1.0 \(m\). Point B is located at the middle of the edge AC and point E is at the middle of the edge DF.

The bar is released from a 45° angle (\(\theta_{start}\)) and the velocity at the free end of the bar at 180° (\(\theta_{end}\)) is compared with the analytical solution from the document above.

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Dynamic

Mesh and Element Types:

The mesh for case A consists of 1\(^{st}\) order elements and it was created with the Standard meshing algorithm in SimScale with a region refinement applied.

Case	Mesh Type	Number of Cells	Number of Nodes	Element Type
A	Standard	216	104	1st order tetrahedral

Simulation Setup

Material/Fluid:

• Steel (linear elastic)
  • Isotropic: \(E\) = 205 \(GPa\), \(\nu\) = 0.3, \(\rho\) = 7870 \(kg/m^3\)

Initial and/or Boundary Conditions:

• Constraints:
  • Remote displacement:
    • Face ACDF with origin as an external point.
    • All DOF are fixed except rotation around the x-axis.
  • Fixed edge:
    • Edge BE with all displacements fixed.
• Load:
  • Gravitational load (\(g\)) = 9.81 \(m/s^2\) in the -z direction

Reference Solution

The analytical solutions for stress and displacement are calculated with the equations below:

$$C = \frac{3g}{2l}{cos(\theta_{start})}\tag{1}$$

$$ \omega = \sqrt{\frac{3g}{l}{\left[cos\left(\theta_{start}\right)-cos\left(\theta_{end}\right)\right]}}\tag{2} $$

$$ v = l\omega \tag{3}$$

The equations used to solve the problem are derived in [Schaum’s].

The constant of integration C is given by equation (1). It is then used to calculate the angular velocity at the free end of the bar in equation (2) with \(\theta_{start}\) = 45° and \(\theta_{end}\) = 180°. With the angular velocity, the magnitude of the velocity can be calculated with equation (3).

The magnitude of velocity (\(v\)) obtained then is 7.08803 \(m/s\).

Result Comparison

Comparison of the velocity obtained from SimScale against the reference results obtained from [Schaum’s]\(^1\) is given below:

Case	Constraint Case	Schaum [\(m/s\)]	SimScale [\(m/s\)]	Error [%]
A	Remote displacement	7.08803	7.08433	-0.052

The velocity of the bar can be seen below:

Last updated: June 16th, 2025