# 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¶

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)

44494

3D isoparametric

(C)

linear tetrahedral

6598

3D isoparametric

(D)

44494

3D isoparametric

## 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¶

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

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%