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# 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:

• 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.

## 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.
• 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:

The velocity of the bar can be seen below: