Validation Case: Static Analysis of an I-Beam Under Remote Force

This validation case belongs to solid mechanics. This test case aims to validate the following parameters:

• Remote force boundary condition

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

Geometry

The geometry used for the case is as follows:

The beam has a length \(L\) of 1 \(m\), with the cross-section dimensions as shown:

Parameter	Value [\(m\)]
B	0.06
H	0.08
b	0.04
h	0.06

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Static Linear

Mesh and Element Types:

Tetrahedral meshes were computed using SimScale Standard mesh algorithm and manual sizing. The table below shows an overview of the mesh characteristics.

Case	Mesh Type	Number of Nodes	Element Type
A	Standard	3345	1st order tetrahedral
B	Standard	20742	2nd order tetrahedral

Simulation Setup

Material:

• Linear Elastic Isotropic:
  • \( E = \) 205 \(GPa \)
  • \( \nu = \) 0.28

Boundary Conditions:

• Constraints:
  • Face A is fixed
• Loads:
  • Remote force of 1000 \(N\) applied at a distance \(d = \) 1 \(m\) on face B

Reference Solution

The analytical solutions for the deflection \(w\) at the free end of the beam are given by the following equations. The remote force \(F\) is substituted with a force and moment pair \(M\):

$${M = Fd} \tag{1}$$

$$ w = \frac{F L^3}{3 E I} + \frac{M L^2}{2 E I} \tag{2} $$

$$ I = \frac{1}{12} ( B H^3 – b h^3) \tag{3} $$

The computed reference solution is:

$$ M = 1000\ Nm $$

$$ I = 1.84×10^{-6}\ m^4 $$

$$ w = 2.209×10^{-3}\ m $$

Result Comparison

Comparison of displacement DZ of the center point of face B with the computed reference solution \(w\):

CASE	DZ [\(10^{-3}\ m\)]	\(w\) [\(10^{-3}\ m\)]	Error
A	2.14109	2.209	3.07 %
B	2.2121	2.209	-0.14 %

Comparison of the neutral fiber deflection shapes:

And finally, a plot showing the deformed shape and the magnitude of DZ displacement:

Related Tutorials

• Linear Static Analysis of a Crane

Last updated: September 9th, 2022