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SimScale DocumentationValidation CasesFixed Beam Under Gravitational Load

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:

  • Gravitational load

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

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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:

fixed beam gravitational load validation simscale
Figure 1: Unrotated beam geometry.

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

rotated geometry fixed beam validation simscale
Figure 2: Rotated beam geometry.

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

ABCDA’B’C’D’
x00001111
y00.050.05000.050.050
z0.050.05000.050.0500
Table 1: Unrotated beam dimensions in meters.

Similarly, for the rotated geometry, we have:

ABCDA’B’C’D’
x00001111
y-0.0353600.035360-0.0353600.035360
z0.035360.07070.0353600.035360.07070.035360
Table 2: Rotated beam dimensions in meters.

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.

CaseGeometryMesh TypeNumber of NodesElement Type
(A)Beam – originalStandard – tetrahedral cells127371st order
(B)Beam – originalStandard – tetrahedral cells914991st order
(C)Beam – rotatedStandard – tetrahedral cells127372nd order
(D)Beam – rotatedStandard – tetrahedral cells914992nd order
Table 3: Mesh characteristics.

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

tetrahedral second order mesh rotated
Figure 3: Second order standard mesh used for Case D.

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].

CaseQuantity[Roark]SimScaleError (%)
(A)Displacement at the free end \([m]\)-2.2597e-4-2.1480e-4-5.20
(B)Displacement at the free end \([m]\)-2.2597e-4-2.2560e-4-0.16
(C)Displacement at the free end \([m]\)-2.2597e-4-2.1480e-4-5.20
(D)Displacement at the free end \([m]\)-2.2597e-4-2.2560e-4-0.16
Table 4: Comparison of SimScale’s results against an analytical solution.

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

fixed beam under gravitational load results
Figure 4: Case B, showing the displacements in the z-direction.

References

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

Last updated: October 8th, 2020

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