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  • Documentation

    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]\(^1\).


    The geometry used for the case is as follows:

    model of the bar with the bottom corners labeled a to f
    Figure 1: Bar model for validation case

    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.

    sketch of the pinned bar under gravitational load for validation with y as horizontal axis and z as vertical axis
    Figure 2: Validation case sketch

    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.

    CaseMesh TypeNumber of NodesElement Type
    AStandard891st order tetrahedral
    Table 1: Mesh overview
    generated first order standard mesh in simscale for validation which has 89 nodes
    Figure 3: Generated first-order standard mesh with tetrahedral elements

    Simulation Setup


    • 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:

    CaseConstraint CaseSchaum [\(m/s\)]SimScale [\(m/s\)]Error [%]
    ARemote displacement7.088037.08433-0.052
    Table 2: Stress and displacement comparison

    The velocity of the bar can be seen below:

    visualization of velocity  and displacement of a bar under gravitational load by animation from case A
    Animation 1: Velocity and displacement animation from case A


    If you still encounter problems validating your simulation, then please post the issue on our forum or contact us.

    Last updated: July 22nd, 2021