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# Validation Case: Fixed Beam Under Gravitational Load

This bonded contact gravitational load validation case belongs to solid mechanics. This test case aims to validate the following parameter:

The simulation results of SimScale were compared to the analytical results derived from [Roark]$$^1$$.

## Geometry

Two beam geometries are used for this gravitational load validation. They have a cross-section of 0.05 x 0.05 $$m^2$$ and 1 $$m$$ length (l). The first one consists of unrotated beam geometry, shown below:

The second geometry is rotated 45º around the positive x-axis:

The coordinates for the points in the first geometry are as tabulated below:

Similarly, for the rotated geometry, we have:

## Analysis Type and Mesh

Tool Type: Code Aster

Analysis Type: Linear static

Mesh and Element Types: The meshes for cases A and B were created in SimScale. The standard algorithm was used. The meshes from case A and B were downloaded, rotated by 45º around the positive x-axis, and imported to SimScale. They were used for cases C and D, respectively. With this method, we achieve the same meshes for the rotated and unrotated cases.

Find below the mesh used for case D. It’s a standard mesh with second-order tetrahedral cells.

## Simulation Setup

Material:

• Steel (linear elastic)
• $$E$$ = 205 $$GPa$$
• $$\nu$$ = 0.28
• $$\rho$$ = 7870 $$kg/m³$$

Boundary Conditions:

• Constraints
• Fixed support on face ABCD.
• Gravity (defined under model):
• Cases A and B: 9.81 $$m/s²$$ in the negative z-direction (0, 0, -1);
• Cases C and D: 9.81 $$m/s²$$ rotated 45º around the positive x-direction (0, 1, -1)

## Reference Solution

Converting the gravitational load to a line load $$(w_a)$$:

$$w_{a}l = V.\rho.g \tag {1}$$

Solving $$(1)$$, we have:

$$w_{a}=193.01175\ N/m \tag {2}$$

The moment of inertia $$I$$ is given by:

$$I = \frac {b.h^3}{12} = 5.20833⋅10^{−7}\ m^4 \tag {3}$$

The equation (4) below is derived from [Roark]$$^1$$

$$y(l) = -\frac{w_a l^4}{8 E I} = -2.2597 \cdot 10^{-4}\ m \tag {4}$$

## Result Comparison

The table below shows the SimScale results for the displacement at the free end (face A’B’C’D’) in the gravity direction. Results are compared to the analytical solution by [Roark].

Inspecting the displacements in the z-direction for case B:

References

• W. C. YOUNG, R. G. BUDYNAS. Roark’s Formulas for Stress and Strain. Seventh Edition. McGraw-Hill. 2002. p. 191.

Last updated: November 7th, 2023