# Bimetallic Strip Under Thermal Load

## Overview

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

• thermomechanical solver with different materials

The simulation results of SimScale were compared to the analytical results derived from [Roark]. The mesh used was created locally consisting of quadratic hexahedral elements and uploaded to the SimScale platform.

Import validation project into workspace

## Geometry

The bimetallic strip has a length of l=10m

$l=10m$

, width of w=1m

$w=1m$

and total height of h=0.1m

$h=0.1m$

with each strip thickness of ta=tb=0.05m

${t}_{a}={t}_{b}=0.05m$

.

## Analysis type and Domain

Tool Type : CalculiX

Analysis Type : Thermomechanical

Mesh and Element types :

Mesh type Number of nodes Number of 3D elements Element type
quadratic hexahedral 3652 600 3D isoparametric

## Simulation Setup

Material:

Upper strip:

• isotropic: Ea
${E}_{a}$

= 200 GPa, ν

$\nu$

= 0, ρ

$\rho$

= 7870 kg/m³, κ

$\kappa$

= 60 W/(mK), γa

${\gamma }_{a}$

= 1e-5 1/K, Reference temperature = 300 K

Lower strip:

• isotropic: Eb
${E}_{b}$

= 200 GPa, ν

$\nu$

= 0, ρ

$\rho$

= 7870 kg/m³, κ

$\kappa$

= 60 W/(mK), γb

${\gamma }_{b}$

= 2e-5 1/K, Reference temperature = 300 K

Initial Conditions:

• uniform: To
${T}_{o}$

= 300 K

Constraints:

• Node N1 fixed in all directions
• Node N2 fixed in x and y direction
• Node N3 fixed in y direction

Temperature:

• T
$T$

= 400 K on face ABCD and EFGH

Contact:

• Bonded contact with automatic ‘Position tolerance’ between two strips.

## Reference Solution

(1)

K1=4+6tatb+4(tatb)2+EaEb(tatb)3+EaEbtatb=16

${K}_{1}=4+6\frac{{t}_{a}}{{t}_{b}}+4{\left(\frac{{t}_{a}}{{t}_{b}}\right)}^{2}+\frac{{E}_{a}}{{E}_{b}}{\left(\frac{{t}_{a}}{{t}_{b}}\right)}^{3}+\frac{{E}_{a}}{{E}_{b}}\frac{{t}_{a}}{{t}_{b}}=16$

(2)

dx=6l(γbγa)(TTo)(ta+tb)(tb)2K1=0.0015 m

(3)

dz=3(l)2(γbγa)(TTo)(ta+tb)(tb)2K1=0.075 m

(4)

σ=(γbγa)(TTo)EaK1[3tatb+2EaEb(tatb)3]=50 MPa

The equation (1)(2)(3) and (4) used to solve the problem is derived in [Roark]. Equations (2) and (3) are the displacements of the bimetallic strip in x and z direction respectively. Whereas, equation (4) is the normal stress in x direction at the bottom surface.

## Results

Comparison of the x and z displacements computed on node N3 and σxx

${\sigma }_{xx}$

computed on node N2 with [Roark] formulations.

Comparison of the displacements and stress
Quantity [Roark] SimScale Error
dx (m) 0.0015 0.0015 0%
dz (m) 0.075 0.074975 0.03%
σ (Mpa) 50 48.79 2.42%

## References

 [Roark] (1, 2, 3, 4) (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh