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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

IBeamUnderRemoteForce-geometry

Geometry of the I-beam

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

IBeamUnderRemoteForce-crosssection

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
IBeamUnderRemoteForce-mesh-a

Mesh used for the SimScale case (A)

IBeamUnderRemoteForce-mesh-b

Mesh used for the SimScale case (B)

Simulation Setup

Material:

  • isotropic: E = 205 GPa, \(\nu\) = 0.28

Constraints:

  • Face A is fixed

Loads:

  • A remote force \(F\) = 1000 N in a distance of 1 m is applied to face B
IBeamUnderRemoteForce-load

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 \(\eqref{ref1}\).

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

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.

IBeamUnderRemoteForce-result

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