# Fixed Beam Under Gravitational Load¶

## Overview¶

The aim of this test case is to validate the following functions:

The simulation results of SimScale were compared to the analytical results derived from [Roark]. The meshes used in (A) and (B) were created with the parametrized-tetrahedralization-tool on the SimScale platform and then downloaded. Both meshes were then rotated and uploaded again. These were used in case (C) and (D). With this method we achieve the exact same mash for the rotated and non-rotated case.

Import validation project into workspace

## Geometry¶

Geometry of the beam

The beam has a cross section $$A$$ of 0.05 x 0.05 m² and a length $$l$$ = 1.0 m.

## Analysis type and Domain¶

Tool Type : CalculiX/Code_Aster

Analysis Type : Static

Mesh and Element types :

Case Mesh type Number of nodes Element type
(A) linear tetrahedral 6598 3D isoparametric
(B) quadratic tetrahedral 44494 3D isoparametric
(C) linear tetrahedral 6598 3D isoparametric
(D) quadratic tetrahedral 44494 3D isoparametric

Mesh used for the SimScale case (A)

Mesh used for the SimScale case (C)

## Simulation Setup¶

Material:

• isotropic: E = 205 GPa, $$\nu$$ = 0.3, $$\rho$$ = 7870 kg/m³

Constraints:

• Face ABCD is fixed: $$d_x=d_y=d_z=0.0$$

• Case (A) and (B): gravitational load $$g$$ = 9.81 m/s² pointing in the negative z-direction: $$(0,0,-1)$$
• Case (C) and (D): gravitational load $$g$$ = 9.81 m/s² now rotated +45° around the x-axis: $$(0,1,-1)$$

## Reference Solution¶

$$$\label{ref1} w_a l= V \rho g$$$$$$\label{ref2} w_a = 193.01175 \frac {N} {m}$$$$$$\label{ref3} I=\frac{bh^3}{12}=5.20833 \cdot 10^{-7} m^4$$$$$$\label{ref4} y(l) = -\frac{w_a l^4}{8 E I} = -2.2597 \cdot 10^{-4} m$$$

The equation $$\eqref{ref4}$$ used to solve the problem is derived in [Roark]. In equation $$\eqref{ref1}$$ the gravitational load is converted in a line load $$w_a$$.

## Results¶

Comparison of the displacement at the free end in the direction of the gravitation vector.

Comparison of the displacements at the free end in [m]
Case Tool Type [Roark] SimScale Error
(A) CalculiX -2.2597E-4 -2.1340E-4 5.56%
(B) CalculiX -2.2597E-4 -2.2559E-4 0.17%
(C) CalculiX -2.2597E-4 -2.1340E-4 5.56%
(D) CalculiX -2.2597E-4 -2.2560E-4 0.16%
(A) Code_Aster -2.2597E-4 -2.1340E-4 5.56%
(B) Code_Aster -2.2597E-4 -2.2559E-4 0.17%
(C) Code_Aster -2.2597E-4 -2.1340E-4 5.56%
(D) Code_Aster -2.2597E-4 -2.2560E-4 0.16%

## References¶

 [Roark] (1, 2, 3) (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh