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:

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:

	A	B	C	D	A’	B’	C’	D’
x	0	0	0	0	1	1	1	1
y	0	0.05	0.05	0	0	0.05	0.05	0
z	0.05	0.05	0	0	0.05	0.05	0	0

Table 1: Unrotated beam dimensions in meters

Similarly, for the rotated geometry, we have:

	A	B	C	D	A’	B’	C’	D’
x	0	0	0	0	1	1	1	1
y	-0.03536	0	0.03536	0	-0.03536	0	0.03536	0
z	0.03536	0.0707	0.03536	0	0.03536	0.0707	0.03536	0

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.

Case	Geometry	Mesh Type	Number of Nodes	Element Type
(A)	Beam – original	Standard – tetrahedral cells	12737	1st order
(B)	Beam – original	Standard – tetrahedral cells	91499	2nd order
(C)	Beam – rotated	Standard – tetrahedral cells	12737	1st order
(D)	Beam – rotated	Standard – tetrahedral cells	91499	2nd 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].

Case	Quantity	[Roark]	SimScale	Error (%)
(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: November 7th, 2023

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