# Pinned Bar Under Gravitational Load

## Overview

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

• dynamic analysis
• nodal velocities
• remote displacement

The simulation results of SimScale were compared to the analytical results in [Schaums]. The mesh used in (A) was created with the parametrized-tetrahedralization-tool on the SimScale platform and the mesh used in (B) and (C) was meshed with Salome. Two different simulation setups were made to compare the remote displacement boundary condition with the fixed boundary condition.

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## Geometry

The bar has a cross section A

$A$

of 0.1 x 0.1 m² and a length l

$l$

= 1.0 m. The point B is located in the middle of edge AC. The same applies to point E and edge DF.

The bar is released from an angle θstart

${\theta }_{start}$

= 45° and the velocity v

$v$

of the free end at θend

${\theta }_{end}$

= 180° is compared with the analytical solution from [Schaums].

## Analysis type and Domain

Tool Type : Code_Aster

Analysis Type : Dynamic

Mesh and Element types :

Case Mesh type Number of nodes Element type
(A) linear tetrahedral 51 3D isoparametric
(B) linear hexahedral 189 3D isoparametric
(C) linear hexahedral 189 3D isoparametric

## Simulation Setup

Material:

• isotropic: E = 205 GPa, ν
$\nu$

= 0.3, ρ

$\rho$

= 7870 kg/m³

Constraint Case ‘remote displacement’:

• Face ACDF is subject to a remote displacement with the origin as external point, all DOF are fixed except the rotation around the x-axis

Constraint Case ‘fixed edge’:

• Edge BE all displacements fixed

$g$

= 9.81 m/s²

## Reference Solution

C=3g2lcos(θstart)(1)

$\begin{array}{}\text{(1)}& C=\frac{3g}{2l}cos\left({\theta }_{start}\right)\end{array}$

ω=3gl[cos(θstart)cos(θend)]−−−−−−−−−−−−−−−−−−−−−√(2)

$\begin{array}{}\text{(2)}& \omega =\sqrt{\frac{3g}{l}\left[cos\left({\theta }_{start}\right)-cos\left({\theta }_{end}\right)\right]}\end{array}$

ω=vl(3)

$\begin{array}{}\text{(3)}& \omega =\frac{v}{l}\end{array}$

The equations used to solve the problem are derived in [Schaums]. The constant of integration C is given by equation (1)

$\text{(1)}$

. It is then used to calculate the angular velocity of the free end of the bar in equation (2)

$\text{(2)}$

. With the angular velocity the magnitude of the velocity can be calculated with equation (3)

$\text{(3)}$

. With θstart

${\theta }_{start}$

= 45° and θend

${\theta }_{end}$

= 180° the resulting velocity is v

$v$

= 7.08803 m/s.

## Results

Comparison of the velocity at the free end of the bar obtained with SimScale with the results derived from the equations presented in [Schaums].

Comparison of the velocity at the free end of the bar
Case Constraint Case [Schaums] SimScale Error (%)
(A) remote displacement 7.08803 7.07417 -0.20%
(B) remote displacement 7.08803 7.07418 -0.20%
(C) fixed edge 7.08803 7.0854 -0.04%

## References

 [Schaums] (1, 2, 3, 4, 5) (2011)”Schaum’s Outline of Engineering Mechanics Dynamics”, E. Nelson, Charles Best, W.G. McLean and Merle Potter