A Bi-Metallic Strip Study
Prepared for: NAE496, Ernesto Gutierrez-Miravete.
Prepared by; Tom O'Keeffe
Date: 12/9/99
Outline
General statement
3
Thermal physical model
3
Indirectly heated
4
Directly heated
4
Thermal simulation
4
Indirectly heated
5
Directly heated
7
Structural physical model
8
Simply supported
10
Cantilever supported
11
Directly heated, Cantilever supported
12
I Bibliographies
13
II Macro code
14
III Solution contour plots
19
Appendices
Page 2
General Statement:
In the development of a thermal magnetic circuit breaker, the thermal trip system must be analyzed in order
to determine its performance before prototypes are manufactured. The thermal trip system is comprised of
a bimetallic strip that deflects based on the difference in the coefficients of thermal expansion of the base
materials when it is heated. This deflection engages other parts of the circuit breaker trip system that cause
the circuit breaker to open and to discontinue current flow. In this paper the bimetallic strip’s performance
is analyzed by employing commercial simulation software. The commercial simulation software being used
in this presentation is ANSYS 5.5, multiphysics. The bimetallic strip has two different physical analyses
being considered in the paper such as thermal and structural. Even though the actual situation needs to be
analyzed in the steady state and transient modes, only the steady state analysis will be presented in this
paper. In the thermal analysis the bimetal is modeled as a heat fin. In the structural analysis the bimetal is
modeled in several different supporting modes. In the solution of the combined physical problem, the
thermal problem is first established and solved. The determined temperatures are imposed in the structural
problem to set the displacement field. The solutions to the simplified models are compared to analytical
solutions to determine weather the simulation model can accurately predict the results.
In the comparison section of each analysis there is an error analysis. The error analysis is composed of
several different scenarios. The simplest error scenario uses a relative error formula presented in
percentage. This error gives better insight to the modeling accuracy than the absolute error. A more
complex error analysis is based on the mesh discretization error, which is defined in ANSYS help system
[5]. In this error method the error arises from the difference between the derived value based on the degree
of the freedom solution at the same node but on different elements. Values reported in the comparison
section are sepc (E), sersm (e) and sensm (U). Sepc is the error percentage of the structural model
normalized by the total strain energy over the entire model. Sersm is the energy error over the entire model.
Sensm is the total strain energy over the entire model.
Note that while using the thermal electric brick element, solid69, model discretization errors are not
available, so these errors are only used on the structural model.
 Calculated value 
 100
Errorrel  1 
Exact value 

 e 
Edescr  100 

 E U 
To build the model, a process, which applied the conservation of energy principle, was used to verify the
model thermally. The heat amount applied in the model should equal to the heat came out of the model. In
order to determine the heat amount entering the model, all the elements, which had a heat flux boundary
condition, were summed by finding the total heat flux entering the model and the area, which had the heat
flux. Once the total heat flux and area where determined, then they were multiplied to find the total heat
applied. This was accomplished by accessing the element table through the sequence number and summing
the appropriate item. Please see the output data in the element definition section. Same methodology was
also performed for all the convection elements to determine all the heat leaving the model. The final model
respected the conservation of energy and the heat in equaled the heat out.
Physical Model:
The bimetallic strip used in thermal magnetic circuit breaker trip units are heated by two different modes,
namely directly and indirectly heated. The directly heated bimetal has current, which flows along a
longitudinal path to heat itself. The indirectly heated bimetal has a heater strap, which has current flowing
along it length to heat the strap. The bimetallic strip is attached to the heater strap in one of two ways. In
the preferred way the bimetal is brazed to the heater strap because there is a lower thermal gradient or
thermal resistance across the junction. In the second way the bimetallic strip is riveted to the heater strap.
Page 3
The bimetallic strip used in circuit breakers has a non-uniform cross sectional area. The analytical solution
assumes a uniform cross sectional area so that a uniform cross sectional area simplification is used through
the simulation analysis. Another simplification used in the simulation model applies to the natural
convective heat transfer. In actuality natural convective heat transfer is a based on fluid properties which
are temperature dependent. The simulation model uses a constant natural convective heat transfer
coefficient. Since the relationship between the heat transferred due to natural convection from the surface
temperature to the bulk fluid temperature is weak, the constant convection is assume to have small
implications on the solution.
In real life convective heat transfer is temperature dependent, while the simulation model makes an
assumptions of a constant convection coefficient.
Indirectly heated analytical:
The indirectly heated bimetal is modeled physically as a fined heat radiator. The finned radiator has been
studied and an analytical solution has been determined. The finite length, adiabatic tip finned heat radiator
analytical model is used in this analysis. The derivation of the formulas can be found in any heat transfer
textbook, [1]. For brevity sake only the pertinent formulas are included in this presentation.
T x   Troot  Tambient 
cosh ml  x 
(1)
cosh ml 

Q  hPkATroot  Tambient  tanh ml  (2)

tanh ml 
(3)
ml
m
hP
(4)
kA
Where,
Tr-temperature at root
Tam-ambient temperature
h - convections
P-perimeter
K-conduction
A-conduction cross sectional area
[C]
[C]
[W/m2K]
[m]
[W/mK]
[m2]
Directly heated analytical:
The directly heated bimetallic strip has current flowing through the longitudinal axis for the entire length. A
convective boundary condition is applied to both transverse areas. An analytical solution probably does
exist in the technical boy of knowledge, but it was not found.
Thermal Simulation:
In the simulation model several assumptions where made to make it possibly to directly compare the
analytical model to the simulation model. Due to the fact that the convection area on the sides and the tip of
the bimetal are small compared to the total area, the sides and tip areas are neglected in convective heat
transfer. The simulation model is a three-dimensional model instead of a two-dimensional model because
the impact of non-uniform cross sectional area will be studied later. This also caused some model
implications on symmetric boundary conditions. Since the non-uniform cross sectional areas will be studied
Page 4
later, symmetric boundary conditions will not be imposed. The bimetal name is really a misnomer because
most useful bimetal are composed of three materials or trimetal. In most of the analytical literature the
bimetal studied have only two materials. So the simulation model has the ability to use two or three
materials. The thermal conductivity of the difference base materials has been change to use the same value,
which simplifies comparison between the simulation and the exact solution.
The simulation model uses different element types depending on which physical problem it is solving. The
thermal model uses two different element types. The thermal model uses three-dimensional thermal electric
elements, solid69. These elements have two degrees of freedom, temperature and voltage. These elements
make up the body of the bimetallic strip. Another type of element used in the thermal model is the surface
effect element, surf152. This element has convective heat transfer and/or heat flux boundary condition
surface effect options. By using this element, the heat being transferred through the element can easily be
determined by querying the resultant "forces". This element uses a third node as a repository for heat.
The comparison between the analytical and simulation model for the indirectly heated case use values,
which are based on experimentation. A circuit breaker had thermocouples installed on the bimetallic strip,
heater strap and in the air chamber. After steady state condition was reached the temperatures were taken
and recorded.
Indirectly heated:
Comparison:
The indirectly heated bimetal was modeled in two different modes. In one mode the root temperature was
fixed which allows for a direct comparison between the analytical and the simulation model. In the second
mode the root temperature was not fixed which made it harder for the simulation model to predict the root
and tip temperatures. In both analysis the number of elements are used as a criteria for convergence.
Fixed root temperature:
This comparison is a direct comparison between the analytical and the finite element simulation model.
This comparison varied the size of the elements globally, which indirectly varied the number of elements.
As the table below indicates there are several observations that can be made. One observation is that the
finite element solution is very close the analytical results, small relative errors. The second observation is
that the finite element solution is converged with a small number of elements. This makes it possibly to use
a course grid with essentially no detriment to the accuracy while the execution time is reduced. The model
solution shows the amount of heat being transferred to the surrounding. The solution shows that more heat
is transfer away at the higher temperatures than at the lower temperatures. At the tip of the bimetallic strip
there is not much heat being transferred so the adiabatic tip assumption was reasonable. The model depicts
this phenomenon nicely. Please review table 1 for complete details. Also chart one shows how the tip
temperature converges based on the number of elements.
The following flag configuration and input values were used in the macro code for this analysis.
Root temp 68.7C, ambient temp 50C, convection 10 W/m2K, heat in 0.255 W, conduction 10 W/mK,
struct-0, calib-0, current-0, rfixed-1. Using the above-mentioned exact solution the fin tip temperature is
calculated as 51.31C.
Page 5
Global
Number of Elements
Depth
Element
size
[m]
0.009
36
0.008
42
0.007
72
0.006
81
0.005
90
0.004
156
0.003
225
0.002
600
0.001
2250
0.0005
9000
Tip
Temperature
[C]
1
1
1
1
1
1
1
1
1
1
51.537
51.523
51.513
51.506
51.501
51.493
51.488
51.484
51.482
51.481
Error
[%]
-0.44241
-0.41512
-0.39563
-0.38199
-0.37225
-0.35666
-0.34691
-0.33912
-0.33522
-0.33327
Table 1
Variable root temperature:
This comparison is informative in several ways. One observation is that the finite element model
approximates the exact solution very nicely, small error term. The second observation is the functional
dependency of number of elements in the model to the model's convergence. See the below table for more
details.
The following flag configuration and input values were used in the macro code for this analysis.
Root temp 68.7C, ambient temp 50C, convection 10 W/m2K, heat in 0.255 W, conduction 10 W/mK,
struct-0, calib-0, current-0, rfixed-0. Using the above-mentioned exact solution the fin tip temperature is
calculated as 51.31C.
Chart 1
Page 6
Global
Number of Depth
Element
Elements
size
[m]
0.009
36
0.008
42
0.007
72
0.006
81
0.005
90
0.004
156
0.003
225
0.002
600
0.001
2250
0.0005
9000
Root
temperature
[C]
1
1
1
1
1
1
1
1
1
1
Tip temperature
[C]
68.774
68.943
69.055
69.133
69.189
69.288
69.349
69.395
69.425
69.427
Table 2
51.544
51.542
51.543
51.542
51.541
51.54
51.54
51.539
51.539
51.539
Root
Error
[%]
Tip
Error
[%]
-0.10771
-0.35371
-0.51674
-0.63028
-0.71179
-0.8559
-0.94469
-1.01164
-1.05531
-1.05822
-0.45605
-0.45215
-0.4541
-0.45215
-0.4502
-0.44826
-0.44826
-0.44631
-0.44631
-0.44631
Elements Vs tip Temperature
69.5
69.4
Temperature [C]
69.3
69.2
69.1
Series1
69
68.9
68.8
68.7
0
2000
4000
6000
8000
10000
No of Elements
Directly heated:
This comparison does not have an analytical solution to compare against so the number of elements in the
model is used in determining convergence and errors. Several observations can be made from the
simulation model based on changing the number of elements. One observation is that the solution is
independent of the number of elements. This can have a positive impact on reducing execution time. The
simulation solution is very accurate with respect to calculating the voltage drop across the length of the
bimetal. It is also very accurate at calculating the power dissipated through the electrically passive element.
Even though an exact solution has not been found for this case, it's believed the simulation model is
accurately estimating the temperature distribution in the bimetal. See table below for more details.
The following flag configuration and input values were used in the macro code for this analysis.
Root temp 68.7C, ambient temp 50C, convection 10 W/m2K, heat in 0.255 W, conduction 10 W/mK,
struct-0, calib-0, current-0, rfixed-0. Using the above-mentioned exact solution the fin tip temperature is
calculated as 51.31C.
Page 7
Global Number of
Element Elements
size
[m]
0.009
36
0.008
42
0.007
72
0.006
81
0.005
90
0.004
156
0.003
225
0.002
600
0.001
2250
0.0005
9000
Absolute
Root
temperature
Depth
[m]
Voltage
drop
[C]
92.545
92.545
92.545
92.545
92.545
92.545
92.545
92.545
92.545
92.545
1
1
1
1
1
1
1
1
1
1
Error
Voltage Drop
Loss
[V]
0.075
0.075
0.075
0.075
0.075
0.075
0.075
0.075
0.075
0.075
[W]
2.5312
2.5312
2.5312
2.5312
2.5312
2.5312
2.5312
2.5312
2.5312
2.5312
[%]
3.33E-14
3.33E-14
3.33E-14
3.33E-14
3.33E-14
3.33E-14
3.33E-14
3.33E-14
3.33E-14
3.33E-14
Error
Loss
[%]
0.001975
0.001975
0.001975
0.001975
0.001975
0.001975
0.001975
0.001975
0.001975
0.001975
Table 3
Elements Vs Tip Temperature
100
90
Tip Temperature [C]
80
70
60
50
Series1
40
30
20
10
0
0
2000
4000
6000
8000
10000
No of elem ents
Structural simulation:
The structural simulation is the second section of a two part physical model. In this section we have already
solved the thermal model for the temperature distribution, so we impose the known temperature distribution
into the structural model, set the appropriated boundary conditions and solve for the displacement field. In
the structural analysis we use the same mesh size and element density as in the thermal model as to keep the
same grid. This not essential to the modeling because if the grid is different then the nodal temperatures for
the structural model would be interpolated based on the thermal nodal solution. Before the boundary
conditions are applied to the model the elements are changed from thermal electric elements to structural
elements. For all the details on how the solution is obtained, please review the macro code attached in
appendix II.
The structural analysis is broken down into smaller pieces, which can be analyzed separately to understand
the physics of the situation. Since the actual circuit breaker bimetallic strip problem can not be solved
analytically we use some geometric, material and boundary condition simplifications, which allows for the
finite simulation model to be compared to an analytical formulation. This is not the case for all the
comparisons, so when it is not possible to directly compare solutions we use element size as an indicator of
Page 8
convergence and errors. First case for analysis is a simply support beam with imposed uniform temperature
rise over ambient. This case allows us to compare the simulation solution to the analytical solution. The
next case we analyze is a cantilever beam with an imposed uniform temperature rise over ambient. The
actual support configuration in the circuit breaker is a cantilever type support at the bottom of the bimetal.
The final analysis looks at a cantilever beam with current applied to the bimetal, directly heated.
Note that we are expecting a larger error magnitude in the structural solution than in the thermal solution
when we do not impose fixed temperatures. This is because the thermal solution with its error is the starting
point for the structural solution. When we solve in this way the thermal errors will add to the structural
errors causing the overall error to be sum of the individual errors. In the structural section we will use the
relative errors and the ANSYS energy errors.
Simply supported.
Thermostat bimetals have been used in different industrial applications in for years. They have also been
studied and analyzed for years. Analytical solutions exist, but not for all the different types of support
mechanisms. For the case of a simply supported bimetallic strip an analytical solution exits. For the case of
the cantilever support of the bimetal an analytical solution does not exist or if it exists it was not found.
Texas Instruments, the bimetal manufacturer, publishes formulas for calculating bimetal deflections,
energies and forces as a function of geometry and temperature rise. Their formulas assume a uniform
temperature rise over the bimetal. It's not known how TI has determined these formulas, but it's thought to
be by empirical means. The analytical solution for the simply supported bimetallic strip is presented below.
Again the analytical solution uses the assumption that the bimetal is composed of two different base
materials while the bimetal of interest is composed of three different materials.
1


241   2 t  t0 
h 14  n  1
n


a1
m
a2
E1
n
E2

l2
8
Where,
Rho is the radius of curvature.
E1 is the modulus of elasticity of material one.
E2 is the modulus of elasticity of material two.
N is the ratio of elasticity.
a1 is the thickness of material one.
a2 is the thickness of material two.
m is the ratio of thickness.
h is the overall thickness, a1+a2.
Alpha1 is the coefficient of thermal expansion for material one.
Alpha2 is the coefficient of thermal expansion for material two.
t is the absolute temperature.
to is the ambient temperature.
[m]
[Pa]
[Pa]
[m]
[m]
[m]
[1/C]
[1/C]
[C]
[C]
Comparison
Page 9
The model was broken down into two separate support mechanisms, namely cantilever and simply
supported, the comparisons will be broken down into congruously groups.
Simply supported.
This simulation solution can be directly comparison to the analytical formulation. The simulation model
constrained the nodal translation in the x axis, please see figure ?. All the nodes on both end surfaces, at y
= 0 with 0<x <bimetal thickness and y = l with 0<x <bimetal thickness are constrained against translation.
This includes all the nodes on the exterior end surfaces. Of course no rotational constraints have been
imposed on the simulation model. Since the largest deflection occurs at mid length, these are the
deflections, which are compared between the different formulations in the error analysis. The simulation
material properties of the bimetal have been modified to allow for two base material bimetal and not the
actual three base material bimetal.
The following flag configuration and input values were used in the macro for this analysis.
Ambient 0 C, current 0 amps, heating 0W, struc-1, calib-1, trise=100 C, rfixed-0 and support-0.
Several observations can be made based on running the model with various element sizes. One observation
is that the mid length deflection only varies five percent while the number of elements varies over two
orders of magnitude. Another observation is that the simulation model estimates very nicely the deflections
at mid length. The percent error normalized only varies ten percent when the number of elements varies
over two orders of magnitude. One last point is that the simulation model did not solve for all the runs.
Three runs generated a "maximum negative pivot element error" statement while the matrix was in
assembly. At the time of this writing, the root cause of the error has not been clarified.
Mid point
Esize
Elements Deep Max deflection
sepc
sersm
sensm
SY
SY
Deflection
compress
tension
Error
0.009
36
1
-0.001018
78.563
162.298
100.654
2.34E+11
-2.54E+11
0.104439629
0.008
42
1
-0.000986
78.206
154.008
97.8
2.34E+11
-2.52E+11
0.101156654
0.007
72
1
-0.000966
76.154
124.437
90.13
2.38E+11
-2.46E+11
0.099104795
0.006
81
1
-0.000951
75.78
120.296
89.181
2.39E+12
-2.46E+11
0.097565901
0.005
90
1
-0.00096
75.358
116.541
88.678
2.39E+11
-2.47E+11
0.098489237
0.004
156
1
-0.000955
0.003
225
1-
0.002
600
1
0.001
2250
1-
-
-
-
-
-
-
0.0005
9000
1-
-
-
-
-
-
-
73.055
-
-0.000953
101.377
-
71.81
88.576
-
92.265
2.39E+11
-
86.658
-2.55E+11
-
2.51E+11
0.097976273
-
-2.59E+11
0.097771087
Elements Vs Deflection
Simply supported
0.00E+00
0
50
100
150
200
250
Deflection [m]
-2.00E-04
-4.00E-04
-6.00E-04
Series1
-8.00E-04
-1.00E-03
-1.20E-03
No of elements
Page 10
Table 4
Cantilever supported.
This support type does not allow for a direct comparison between the simulation model and the analytical
solution because the analytical solution does not exist for this support type. So the simulation solution
based on the number of elements will be used to determine the relative error. The percent error normalized
will still be used to determine convergence and relative error. Since the maximum deflection occurs at the
bimetallic tip, these deflections are compared in the error analysis.
The simulation model has a cantilever support at the bottom of the bimetallic strip. The supported end is
considered to be fixed with respect to both translations and rotations in all three-axis directions. This is a
realistic assumption based on the actual clasping mechanism, which is either a brazed/welded or a riveted
connection.
The following flag configuration and input values were used in the macro for this analysis.
Ambient 0 C, current 0 amps, heating 0W, struc-1, calib-1, trise=100 C, rfixed-0 and support-1.
Several observations can be made at this point. One observation is that the tip deflection varies
approximately nine percent while the number of elements varies over two orders of magnitude. The percent
energy error normalized varies 1.2 percent. See the below table for more details.
Element
No of
Tip point
Esize
Elements Depth Max deflection
0.0
36.0
1.0
0.00423
0.0
42.0
1.0
0.00418
0.0
72.0
1.0
0.00410
0.0
81.0
1.0
0.00408
0.0
90.0
1.0
0.00405
0.0
156.0
1.0
0.00400
0.0
225.0
1.0
0.00396
0.0
600.0
1.0
0.00393
0.0
2250.0
1.0
0.00390
0.0
9000.0
1.0
0.00390
sepc
71.2
71.2
70.6
70.7
70.7
69.8
70.0
69.9
70.2
70.3
sersm
133.1
126.6
114.8
112.3
110.2
100.4
97.4
93.4
91.3
90.4
Von Miss Deflection
sensm Max Stress
Error
129.6
6.5E+11
-87.1
123.4
6.6E+11
-84.8
115.8
6.6E+11
-81.2
112.5
6.7E+11
-80.0
110.1
6.7E+11
-79.1
105.8
7.1E+11
-76.6
101.4
7.2E+11
-75.0
97.8
7.3E+11
-73.6
94.1
8.3E+11
-72.3
92.5
8.4E+11
-72.2
Table 5
Elements Vs Deflection
Cantilever supported
4.30E-03
4.25E-03
Deflection [m]
4.20E-03
4.15E-03
4.10E-03
Series1
4.05E-03
4.00E-03
3.95E-03
3.90E-03
3.85E-03
0
2000
4000
6000
8000
10000
No of elements
Page 11
Chart 5
Directly heated cantilever supported
This is the most important case to analyze because this is close to the actual configuration. All the
simplifying assumptions still exist in the model like only two materials in the bimetallic strip, same thermal
conductivity in both materials, uniform cross sectional area, etc. Several new simplification now exist like
the same resistivity for both materials, a current path the full bimetal length. While the thermal solution
shows that the temperature distribution is uniform across the entire length and that there is no variation in
temperature based on the number of elements, the structural simulation should have small errors based only
on structural causes.
The following flag configuration and input values were used in the macro for this analysis.
Ambient 0 C, current 33.75 amps, heat in 0W, struc-1, calib-0, trise =100 C, rfixed-0 and support-1.
The simulation runs show the deflection changes approximately eight percent while the number of elements
change over two orders of magnitude. The simulation also shows convergence of the solution. Another
observation is that the percent energy error normalized changes only one and a quarter percent over the two
orders of magnitude for the number of elements.
tip point
Max deflection
1
0.002859
1
2.83E-03
1
2.77E-03
1
2.75E-03
1
2.74E-03
1
2.70E-03
1
2.68E-03
1
2.65E-03
1
2.63E-03
1
2.63E-03
Esize
Elements Deep
0.009
36
0.008
42
0.007
72
0.006
81
0.005
90
0.004
156
0.003
225
0.002
600
0.001
2250
0.0005
9000
Sepc
71.17
71.16
70.55
70.67
70.72
69.78
69.99
69.90
70.16
70.29
sersm
60.70
57.76
52.36
51.22
50.26
45.82
44.45
42.62
41.65
41.23
sensm
59.13
56.31
52.85
51.35
50.24
48.29
46.28
44.62
42.96
42.23
Elements Vs Deflection
Directly heated, Cantilever support
0.0029
Tip Deflection [m]
0.00285
0.0028
0.00275
Series1
0.0027
0.00265
0.0026
0
2000
4000
6000
8000
10000
No of Elements
Page 12
Appendix I, Bibliography
Mechanical Engineering Reference Manual, Michael Lindeburg, PE, Professional Publications, 1990
Thermostat Metals Designer's Guide, Texas Instruments Incorporated 1991
Heat transfer Textbook, bla bla bla.
Analysis of Bi-Metal thermostats, S. Timoshenko, Journal of the Optical Society of America, 1925, Vol. 11
ANSYS Theory Reference, ANSYS Corp, 9th edition, SAS IP, Inc.
Page 13
Appendix II, Macro code.
! Bimetal analysis for both indirectly heated (current =0) and
! directly heated (current = 25 nominal).
! using bimetal type P250R
! Using mks units
/file,bimetalt
!Start of Pre-Processing
!working plane definition
!wpcsys,-1,0
!wpstyl,,0.001,,,,,0,1,
!wprota,0,0,90
!csys,10
!dsys,10
/prep7
!inputs
bi_l = 0.050
bi_w = 0.015
bi_t = 0.0012
per1 = 1/3.
per2 = 1/3.
per3 = 1/3.
heatfxin = 0.
hc = 25.
ambient = 25.
trise = 0.
current = 25*1.35
struct = 0
calib = 0
support = 1
rfixed = 0
!0.225/(bi_t*bi_w)
!m
!m
!m
!%
!%
!%
!W/m2
!W/m2K
!C
!C
!amps
!0-non struc,1-struc
!calibration, 1-yes(uniform temps), 0-no
!0-simple supported,1-cantilever
!0-root temp not fixed,1-root temp fixed
!key point definition
x1 = 0.
x2 = per1*bi_t
y1 = 0.
y2 = bi_l
z1 = bi_w/2.
k,1,x1,y1,-z1
k,2,x1,y2,-z1
k,3,x1,y2,z1
k,4,x1,y1,z1
!l,1,2
!l,2,3
!l,3,4
!l,4,1
! move working plain
wpoffs,per1*bi_t,0,0
k,5,x2,y1,-z1
k,6,x2,y2,-z1
k,7,x2,y2,z1
k,8,x2,y1,z1
! generating volumns
v,1,2,3,4,5,6,7,8
! generating a component
cm,bi_mat_1,volu
vsel,s,volu,,1
vgen,2,1,1,1,per1*bi_t,0,0,,1,
Page 14
vsel,s,volu,,2
cm,bi_mat_2,volu
!vsel,s,volu,,2
vgen,2,2,2,1,per2*bi_t,0,0,,1,
vsel,s,volu,,3
cm,bi_mat_3,volu
!element type definitions and options
et,1,solid69
!3D thermo-electric
et,2,surf152
!3D surface
et,3,solid45
!3D structural solid
keyopt,1,2,0
keyopt,2,4,1
keyopt,2,5,1
keyopt,2,6,0
keyopt,2,8,5
keyopt,2,9,0
!material definitions
!alloy p
mp,kxx,1,10.
mp,rsvx,1,1.69e-6
mp,dens,1,7.2e3
mp,c,1,0.2
mp,ex,1,124.1e12
mp,nuxy,1,0.27
mp,alpx,1,27.5e-6
vatt,1,,1,0
!
!
!
!
!
!
!
!
!
! alloy b
mp,kxx,2,10.
mp,rsvx,2,0.76e-6
mp,dens,2,8.14e3
mp,c,2,0.385
mp,ex,2,193.0e12
mp,nuxy,2,0.27
mp,alpx,2,18.9e-6
vatt,2,,1,0
! alloy invar
mp,kxx,2,10.
mp,rsvx,2,0.8e-6
mp,dens,2,8.1e3
mp,c,2,0.5
mp,ex,2,147.6e12
mp,nuxy,2,0.33
mp,alpx,2,2.5e-6
vatt,2,,1,0
! alloy invar
mp,kxx,3,10.
mp,rsvx,3,0.8e-6
mp,dens,3,8.1e3
mp,c,3,0.5
mp,ex,3,147.6e12
mp,nuxy,3,0.33
mp,alpx,3,2.5e-6
vatt,3,,1,0
!8.7
!ohm m
!W/mK
!kg/m3
!kg/kjK
!GaPascals
!1/C
!13.4
!ohm m
!kg/m3
!W/mK
!kj/kgK
!GaPascals
!1/C
!10.5
!ohm m
!W/mK
!kg/m3
!kj/kgK
!GaPascals
!1/C
!10.5
!ohm m
!W/mK
!kg/m3
!kj/kgK
!GaPascals
!1/C
allsel
nummrg,kp
!Meshing
esize,0.009
!/eof
mshape,0,3d
mshkey,1
vmesh,all
Page 15
allsel
sftran
!/eof
!Extra node for surface elements
n,100000,5*bi_t,bi_l/10.,0
d,100000,temp,ambient
*if,calib,ne,1,then ! not calibrating, non unifrom temps
!Employing surface elements on element model, convection
allsel
type,2
nsel,s,loc,x,bi_t
nsel,A,loc,x,0.
esln
esurf,100000
esel,r,type,,2
sfe,all,1,conv,0,hc
sfe,all,1,conv,2,ambient
allsel
! heatflux boundary condition
*if,heatfxin,gt,0,then
nsel,s,loc,y,0.
esln
sfe,all,5,hflux,,heatfxin
allsel
*endif
!fixing root temperature, compare against exact
*if,rfixed,eq,1,then !fixed root temp
nsel,s,loc,y,0
d,all,temp,trise
allsel
*endif
*else ! calibrating uniform temps
allsel
d,all,temp,trise
*endif
!Setting the directly heated boundary conditions
*if,current,gt,0,then
!setting the output current
nsel,s,loc,y,bi_l
nsel,r,loc,x,0,bi_t
nsel,r,loc,z,-bi_w/2.,bi_w/2.
cp,1,volt,all
!coupling dof
d,all,volt,0.
!applying current at several nodes
allsel
!setting the input current
nsel,s,loc,y,0.
nsel,r,loc,x,0,bi_t
nsel,r,loc,z,-bi_w/2.,bi_w/2.
cp,2,volt,all
!coupling dof
nsel,s,loc,y,0.
nsel,r,loc,x,0.
nsel,r,loc,z,bi_w/2.
f,all,amps,current !applying current at one node
allsel
*endif
eplot
!Start processing
fini
allsel
save
!Solving thermal model
/solu
antype,static,new
Page 16
solve
!Post processing thermal
/post1
/graphics,full
*get,eerror1,prerr,0,tepc
*get,eerror2,prerr,0,tersm
*get,eerror3,prerr,0,tensm
/output,ztherml,txt
*vwrite,eerror1,eerror2,eerror3
('tepc',3x,f10.3,3x,'tersm',3x,f10.3,3x,'tensm',3x,f10.3)
!prerr
/output
/graphics,power
path,tempdist,2,5,20
ppath,1,,bi_t,0,0
ppath,2,,bi_t,bi_l,0
pdef,tempdist,temp
plpath,tempdist
!set structural loads and boundary conditions.
*if,struct,gt,0,then
/prep7
fini
/filname,bimetals
/prep7
allsel
etchg,ets
etchg,tts
cpdel,1,2,1
allsel
! couple set for displacements
*if,support,eq,1,then
! cantilever
nsel,s,loc,y,0.
cp,3,all,all
d,all,all
*else
!simply supported
nsel,s,loc,y,0 !bi_l/3.
nsel,a,loc,y,bi_l !*2./3.
nsel,r,loc,x,0
d,all,ux,0
*endif
allsel
fini
save
/solu
ldread,temp,,,,,bimetalt,rst
solve
!solving structural model
/post1
/graphics,full
*get,eerror1,prerr,0,sepc
*get,eerror2,prerr,0,sersm
*get,eerror3,prerr,0,sensm
/output,zstruc,txt
*vwrite,eerror1,eerror2,eerror3
('sepc',3x,f10.3,3x,'sersm',3x,f10.3,3x,'sensm',3x,f10.3)
!prerr
Page 17
/output
/graphics,power
plnsol,u,x
*endif
Page 18
Appendix III, Solution Contour plots.
Figure 1, Indirectly heated fin thermal solution.
Page 19
Figure 2.
Directly heated bimetal thermal solution.
Page 20
Figure 3.
Directly heated bimetal voltage solution with boundary conditions
Page 21
Figure 4.
Simply supported bimetal deflection solution with boundary conditions.
Page 22
Figure 5.
Cantilever supported bimetal deflection solution with boundary conditions.
Page 23
Figure 6.
conditions.
Directly heated cantilever supported bimetal deflection solution with boundary
Page 24
Figure 7
Indirectly heated fin thermal distribution.
Page 25