# I Beam Under Remote Force¶

## Overview¶

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

• remote force

The simulation results of SimScale were compared to the analytical results derived from [Roark]. The meshes used in case (A) and (C) were created with the fully-automatic-tetrahedralization tool on the SimScale platform. The meshes used in case (B) and (D) were created locally.

Import validation project into workspace

## Geometry¶

Geometry of the I-beam

The I-beam is $$L$$ = 1 m long. The geometry of the cross section is shown below.

Geometry of the cross section

 B 0.06 m H 0.08 m b 0.04 m h 0.06 m

## Analysis type and Domain¶

Tool Type : Code_Aster

Analysis Type : Static

Mesh and Element types :

Case Mesh type Number of nodes Element type
(A) linear tetrahedral 5761 3D isoparametric
(B) linear hexrahedral 5757 3D isoparametric
(C) quadratic tetrahedral 37486 3D isoparametric
(D) quadratic hexrahedral 20749 3D isoparametric

Mesh used for the SimScale case (A)

Mesh used for the SimScale case (B)

## Simulation Setup¶

Material:

• isotropic: E = 205 GPa, $$\nu$$ = 0.28

Constraints:

• Face A is fixed

• A remote force $$F$$ = 1000 N in a distance of 1 m is applied to face B

Remote force

## Reference Solution¶

A remote force can be substituted in a parallel force $$F_0$$ with the same magnitude and a momentum $$M_0$$. This momentum can be calulated with the direction of the force and the distance $$d$$. In this example the connection line between the force and the point of attack is perpendicular to the force and therefore the momentum $$M_0$$ can be calculated as stated in equation $$\eqref{ref1}$$.

$$$\label{ref1} M_0 = F_0 d$$$$$$\label{ref2} w(x) = -\frac{M_0 x^2}{2 E I} -\frac{F_0}{E I}(\frac{1}{2} L x^2 -\frac{1}{6} x^3)$$$$$$\label{ref3} I=\frac{1}{12}(B H^3 - b h ^3)=1.84 \cdot 10^{-6} m^4$$$

The equation $$\eqref{ref2}$$ used to solve the problem is derived in [Roark] with a superposition of the momentum and the force.

## Results¶

Comparison of the deflection along the neutral fiber.

Comparison of the deflection along the neutral fiber

## References¶

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