Download Introduction – What do load cells do

Technical Contribution Report
Design and Selection of Load Cells
ME405 – Senior Design Practicum
Spring 2000
Dr. Radford
Dr. Sakurai
Dr. Smith
March 1, 2000
Scott W. Gallagher
Contents
INTRODUCTION: WHAT DO LOAD CELLS DO? ............................................................................... 1
SECTION 1: THE STRAIN GAGE AND THE WHEATSTONE BRIDGE .......................................... 2
THE STRAIN GAGE...................................................................................................................................... 2
THE WHEATSTONE BRIDGE ........................................................................................................................ 4
SECTION 2: DESIGN AND MANUFACTURE OF LOAD CELLS ...................................................... 9
THE SPRING ................................................................................................................................................ 9
LOAD MECHANISMS ..................................................................................................................................10
The Bending Beam Configuration ........................................................................................................10
Direct (Axial) Loading .........................................................................................................................14
Shear Loading ......................................................................................................................................15
SELECTION OF THE SPRING MATERIAL ......................................................................................................16
SELECTION OF THE STRAIN GAGE ..............................................................................................................17
SELECTION OF THE ADHESIVE ...................................................................................................................19
ASSEMBLING THE LOAD CELL ...................................................................................................................20
SECTION 3: SELECTION OF COMMERCIAL LOAD CELLS ..........................................................21
LOAD CELL PROPERTIES............................................................................................................................21
FAMILIES OF LOAD CELLS .........................................................................................................................24
INTERPRETING LOAD CELL SPECIFICATION SHEETS ..................................................................................25
The Futek L2330 ..................................................................................................................................25
The Omega LCDB Series .....................................................................................................................29
SECTION 4 – DESIGN OF A LOAD CELL FOR HPV AERODYNAMICS .......................................32
BACKGROUND ...........................................................................................................................................32
PROPOSED LOAD CELL CONFIGURATIONS .................................................................................................32
ANALYSIS OF LOAD CELL CONFIGURATIONS ............................................................................................34
SELECTION OF A CONFIGURATION .............................................................................................................38
SELECTION OF A STRAIN GAGE .................................................................................................................38
CONCLUSIONS ...........................................................................................................................................39
APPENDIX A – CALCULATION OF EFFECT OF BEAM WEIGHT ON STRAIN .........................40
BEAM 1 – FIXED AT BOTH ENDS .................................................................................................................40
BEAM 2 – CANTILEVER BEAM ....................................................................................................................41
REFERENCES ............................................................................................................................................42
Introduction: What do load cells do?
In this document, the design and selection of load cells will be examined in depth,
but first it is important to know what purpose these devices serve. As the name implies,
load cells are used to measure applied loads or forces.
The basic function of a load cell is that of a transducer – “a device which
transforms one type of energy into another.”1 The load cell transforms an input energy
(mechanical energy in the form of strain resulting from an applied force) into an energy
output (electrical energy in the form of an induced voltage difference). The induced
voltage can be amplified, converted to a digital signal, and read into a computer. With
the appropriate software and knowledge of the workings of a load cell, the resulting
signal can be converted into a digital readout of the initial load. The block diagram for
this system can be seen in Figure (I-1).
Figure I-1. Schematic of load measurement system
The importance of the load cell in this system is readily apparent. By
transforming the applied force into a voltage signal, it plays a crucial role in creating an
easily readable, digital display of the force to be measured. The applications of load cells
are thus ubiquitous, as they may be used whenever a force needs to be measured.
“Strain Gage Based Transducers – Their Design and Construction,” Interactive Guide to Strain Gage
Technology, Release 111099, July 1999. Measurements Group, November 16, 1999,
<http://www.measurementsgroup.com/guide/ta/sgbt/sgbtndex.htm>, page 2.
1
1
Section 1: The Strain Gage and the Wheatstone Bridge
The fundamental building blocks of a load cell are the strain gage and the
Wheatstone bridge circuit. The strain gage is the physical mechanism in which
mechanical energy is converted into electrical energy and the Wheatstone bridge circuit is
the configuration in which the change in the strain gage may be observed. In this section,
the basic workings of these two elements shall be examined.
The Strain Gage
A strain gage is basically a thin wire folded several times so that a great length of
wire can occupy a small space, as in figure (1-1). At each end of the wire, there are wide
solder pads, so the gage may be wired into an electronic circuit. The gage is mounted
onto the surface where the strain is to be measured, so that when the surface experiences
strain, the strain gage likewise experiences strain.
Figure 1-1. A basic strain gage
Basic circuit theory shows that the resistance in a linear electronic component is
related to its length and cross-sectional area as follows.
R
 *
A
Where: R is the resistance of the component []
 is the material’s resistivity [*m]
 is the length of the component [m]
A is the cross-sectional area of the component [m2]
From this relationship, it follows that when the gage experiences strain, its
resistance will vary accordingly.
R
 *
dR 
A

* d
A
In this analysis, it is assumed that the change in cross-sectional area is negligible
compared to the change in the length of the gage wire. The validity of this assumption
can be quickly verified through an examination of the Poisson’s ratio relationship of
these quantities. Here,  is the axial length of the wire, d is the diameter of the wire, d
is the change in axial length of the wire, dd is the change in the diameter of the wire, and
 is Poisson’s ratio.
2
d d
d
  *
d

d d

  * d
d

Since the strain gage wire is long and thin, the ratio of the diameter to the axial
length is nearly zero. Thus, the ratio of the change in diameter (and hence, area) to the
change in axial length is nearly zero. Thus, the change in resistance of the strain gage is
directly proportional to the change in length of the gage. For convenience, this
relationship is typically made non-dimensional and restated in terms of a constant of
proportionality, the gage factor k.
dR
d
k*
R

In this equation, and henceforth, R refers to the unloaded resistance and  refers to
the unloaded length. The gage factor is best determined experimentally, by applying
known loads to the strain gage and measuring the resulting change in resistance.
Typically, its value is approximately 2.
It is now apparent how strain causes a change in resistance in a single strain gage.
However, only the resistance across the gage can be measured. Since the change in
resistance is very small compared to the total resistance, its effects on the total resistance
cannot be easily discerned. Thus, it is necessary to isolate this resistance change in a
circuit so it can be accurately measured.
3
The Wheatstone Bridge
The circuit that makes this measurement possible is the Wheatstone bridge, shown
in figure 1-2. This circuit, which was invented in 1843 by Sir Charles Wheatstone,2
allows unknown resistances to be measured with respect to known resistances.
Figure 1-2. Basic Wheatstone bridge circuit
The fundamental relationship between the applied (excitation) voltage Vi and the
output voltage Vo can be derived in terms of the four bridge (strain gage) resistances R1,
R2, R3, and R4 by using the intermediate voltages Va and Vb.
R2
Va  Vi *
R1  R2
Vb  Vi *
R3
R3  R4
Vo  Va  Vb
Vo
R3
R2


Vi R1  R2 R3  R4
Vo
R2 * R4  R1 * R3

Vi ( R1  R2 ) * ( R3  R4 )
Typically, the bridge is initially balanced, with all four resistances being equal. It
is readily apparent that this situation leads to an output voltage of zero. When a load is
applied, it may act on one, two, three, or all four resistors, creating the following voltage
ratio.
Vo
( R  R2 )( R4  R4 )  ( R1  R1 )( R3  R3 )
 2
Vi ( R1  R2  R1  R2 ) * ( R3  R4  R3  R4 )
“Applying the Wheatstone Bridge,” HBM Application Notes, Karl Hoffmann, 1996. HBM Weighing
Technology, November 16, 1999, <http://www.hbmwt.com/appnotes/wheatstn/wheatstn.htm>.
2
4
As mentioned above, the change in resistance is much smaller than the initial
resistance, so in the reduction of this equation, second order terms are neglected. The
initial resistances are also assumed to be equal, as per standard practice. However, this
assumption does not mean that all R terms are equal. These assumptions lead to the
following simplified relationship.
Vo R * R2  R * R4  R * R1  R * R3

Vi
4R
Vo 1  R2 R4 R1 R3 
 *




Vi 4  R
R
R
R 
Vo 1  R1 R2 R3 R4 

 *



Vi 4  R1
R2
R3
R4 
This equation states the general relationship between the output voltage and the
change in resistance of the four load cells, and it can also be stated in terms of the strain
in each resistor.
Vo k  d1 d 2 d 3 d 4 

 *



Vi 4  1
2
3
 4 
Vo k
 *  1   2   3   4 
Vi 4
Now that a general relationship between the output voltage and the strain in the
four legs of the bridge has been established, the specific outputs from different loading
situations can be examined.
The first possible loading scenario is when only one of the legs is subjected to
strain, as seen in figure 1-3. This configuration is known as a quarter bridge, as one
quarter of the bridge is under strain. Obviously, the strain in members 2, 3, and 4 is zero,
so the output voltage is directly proportional to the strain in member 1.
Vo k
   1 
Vi 4
Figure 1-3. Quarter bridge circuit
5
Another possible configuration is the half bridge, shown in figure 1-4. In this
circuit, two of the legs are under strain. It is important that one cell be in tension and one
in compression, or else their respective strains may cancel each other out. The mounting
arrangement to realize this constraint depends upon the physical make-up of the load cell,
which will be discussed in detail in Part 2. The maximum output signal for a half bridge
will occur if the strains in members 1 and 2 are equal in magnitude with opposite signs.
Vo k
   1   2 
Vi 4
Vo
k
 ( )
Vi max 2
Figure 1-4. Half bridge circuit
A third possible configuration, albeit an uncommon one, is when three legs of the
bridge are loaded, as in figure 1-5. This three-quarters bridge arrangement produces a
greater output when R1 and R3 are subjected to the same kind of force (tension or
compression) and R2 is subjected to the opposite kind. Again, the maximum possible
output results when all three forces are equal in magnitude, with members one and three
having one sign and member 2 having the opposite sign.
6
Vo k
   1   2   3 
Vi 4
Vo
3k

( )
Vi max
4
Figure 1-5. Three-quarters bridge circuit
The final, and most widely used, configuration is the full bridge (figure 1-6). In
this arrangement, the load on gages 1 and 3 should have the opposite sign as that on
gages 2 and 4. As in the other arrangements, the maximum output voltage occurs when
all four legs of the bridge have the same magnitude of strain.
Vo k
   1   2   3   4 
Vi 4
Vo
 k
Vi max
Figure 1-6. Full bridge circuit
7
The advantages of the full bridge circuit are fairly obvious. Primarily, it can
produce the greatest output voltage for a given applied load. Also, in this circuit, it is
possible to have two strain gages in compression and two in tension, resulting in a
balanced load reading.
Another important advantage of the full bridge is that it compensates for external
disturbances, such as temperature variations. The strain in the strain gages is directly
effected by fluctuations in temperature, so the strain measured across each strain gage is
caused in part by the mechanical load and in part by deformation due to temperature
variation. If all of the gages in the bridge have the same properties, the temperatureinduced strain will have the same magnitude and sign in all gages.
 n   nmechanical   ntemperature
 1temp   2temp   3temp   4temp
Therefore, in a full bridge circuit, the complete voltage-strain relationship is:
Vo k
   1mech   1temp   2 mech   2temp   3mech   3temp   4 mech   4temp 
Vi 4
Vo k
   1mech   2 mech   3mech   4 mech 
Vi 4
By inspection, it is obvious that the temperature components of strain do not
cancel out in the quarter bridge and three-quarters bridge circuits. Thus, it is necessary to
use a compensation gage in these configurations. The compensation gage is another
strain gage with the same properties as the gage(s) being subjected to a load, and it must
be exposed to the same temperature variation as the active gage(s), but it is not loaded
mechanically. The resulting strain in this gage is due exclusively to temperature effects,
and it can be used to compensate for the temperature-induced strain in the active gage(s).
Fundamentally, the strain gage and Wheatstone bridge are not complicated or
overwhelming measurement tools. At the heart of the load cell, the strain gage is the real
transducer that produces an electrical output from a mechanical input, while the
Wheatstone bridge configuration allows the electrical output to be measured accurately,
making up the electronic component of the load cell.
8
Section 2: Design and Manufacture of Load Cells
Now that the characteristics of a strain gage and a Wheatstone bridge circuit are
known, they must be applied to the design of an entire load cell. In addition to the
electronic circuitry of the Wheatstone bridge, the design of a load cell requires careful
consideration of several factors, including the physical properties of the strain gages, the
properties of the load bearing member, the positioning of the strain gages on the spring,
the mounting and bonding of the gages, and protection from the elements.
The Spring
The load-bearing member is a crucial component in the load cell. This part must
be strong enough to withstand the applied load, yet at the same time it must react with a
uniform strain in the area where the strain gages are mounted. There are certain design
constraints that must be considered for all load-bearing members.
The natural frequency of the system should be high so that oscillations do not
disturb the load cell. To achieve this end, the load-bearing member should have a high
rigidity to mass ratio. Material selection and careful design of the shape/orientation of
the load-bearing piece can achieve this goal. Manufacturing techniques come into play as
well. If the spring is machined from one initial block of material rather than being
assembled, its rigidity will be higher.
Another consideration is the location of the strain gage(s). First, the structure
must be designed such that the strain gages can easily and accurately be mounted.
Second, the strain in this region must be considered. Obviously, it is desirable to have a
strain concentration at these points so the strain gages can create an output, but the strain
must be kept within a certain range to avoid causing permanent deformation in the strain
gages. A strain of ~1500in/in works well, as it produces a discernable output without
deforming the gages. This strain must also be uniform in the strain gage area to ensure
that the output from the gage is indeed linear. The strain level in the rest of the assembly
is ideally minimized to prevent wear on the load cell and increase the cell’s stiffness.
9
Load Mechanisms
Now some different loading situations and their corresponding strain gage
mounting arrangements will be examined. Generally, load cells transmit strain to the
strain gages via one of three stress mechanisms: bending, axial stress, and shear (figure 21).3 The mechanism of choice is a function of the force to be measured, as well as the
desired mechanical response of the load cell. In this analysis, each mechanism will be
analyzed in turn to determine the advantages, disadvantages, and best applications of
each type of loading.
Figure 2-1. The three types of loading on a load cell:
(a) bending, (b) axial stress, (c) shear
The Bending Beam Configuration
One of the simplest and most common load cell configurations is that of a
bending beam. The most ubiquitous example of a bending beam is the classic cantilever
beam, mounted rigidly at one end while a force is applied to the other end (figure 2-2).4
Strain gages are simply affixed to the upper and lower surfaces of the beam, so that two
are in compression and two are in tension. The positive strain on the upper surface and
The load cell drawings in this section are taken (some directly, some with artistic license) from “Strain
Gage Based Transducers – Their Design and Construction,” Interactive Guide to Strain Gage Technology,
Release 111099, July 1999. Measurements Group, November 16, 1999,
<http://www.measurementsgroup.com/guide/ta/sgbt/sgbtndex.htm>.
3
4
In this section, R1 and R3 shall be in tension and R2 and R4 shall be in compression.
10
negative strain on the lower surface induce strain in the gages, thus creating a voltage
across the bridge.
Figure 2-2. Basic cantilever beam loading
The obvious attraction about this design is its simplicity and ease of
manufacturing. There aren’t any complicated parts to machine, and the gages are all
mounted on flat, open surfaces. However, this arrangement has many undesirable
characteristics. First, its deflection is generally large, giving the whole system a low and
hence obtainable natural frequency, so vibrations can cause problems here.
The cantilever beam can be modified by making it thinner at the area of strain
measurement than in the rest of its mass (figure 2-3). This adaptation creates a greater
strain concentration at the location of the strain gages, and the extra thickness on the rest
of the beam reduces deflection by making it more rigid.
Figure 2-3. Modified cantilever beam loading
Unfortunately, it still tends to make the natural frequency high because of the
added mass. So, another modification can be made (figure 2-4). In this situation, the end
is hollowed out, decreasing the mass of the spring. However, even with all of these
modifications, the cell still fails to comply with one of the primary constraints of load cell
manufacturing, that the strain be uniform over the strain gage. In this case, the strain
decreases with distance from the fixed end of the beam. Another source of trouble for the
cantilever beam is that as it deflects, the force is no longer applied at the same location on
the beam, causing more non-linearity in the output. So it is obvious that if a linear output
is desired, the cantilever beam is not the best choice for a load cell.
11
Figure 2-4. Hollow modified cantilever beam
One way to eliminate some of the difficulties associated with the cantilever beam
while still maintaining a simple design is to fix the beam at both ends and load it in the
middle (figure 2-5). This configuration assures that the force will always be applied at
the same location on the beam, and it greatly reduces the deflection of the beam,
increasing the natural frequency. Additionally, the beam experiences double bending,
allowing both tension and compression strain gages to be installed on the top of the beam.
However, this design fails to eliminate the strain gradients inherent in the beam, so a
certain amount of non-linearity can be expected in the output.
Figure 2-5. Central beam loading
Figure 2-6. Other double bending configurations
There are other possible configurations for double bending (figure 2-6).
Arrangement (a) is basically the central double bending device cut in half, and
arrangement (b) is a double beam version of arrangement (a). The advantages of
arrangement (b) are that the applied loads are coaxial, reducing the amount of off-axis
loads. However, the thin beams on which the strain gages are mounted give this design a
very low stiffness, and it can potentially be deformed beyond the linear range of the strain
gages. Again, the solution to this problem is to increase the load-bearing area of these
beams in the regions where strain is not being measured. An example of this principle is
12
the classic “binocular” design (figure 2-7). This thickening greatly increases the overall
stiffness of the load cell without really losing any precision, as there is still a strain
concentration in the region of the strain gages.
Figure 2-7. Binocular load cell
The principles of double bending can also be easily extended to three-dimensional
load cell configurations, such as a wheel (figure 2-8).
Figure 2-8. Wheel configuration
This system has the advantage of a low profile and extra stiffness. It should also
be noted that this particular arrangement has eight strain gages, or simply two on each leg
of the Wheatstone bridge. Obviously, more spokes may be employed to achieve greater
stiffness.
Most other bending beam load cells are variations on the principles presented thus
far: the thickness is minimized locally at the strain gage mounting sites to achieve a strain
concentration and is maximized elsewhere to increase the stiffness of the cell as a whole.
Through these innovations, a poor load cell configuration (the cantilever beam) can
actually become a decent configuration, provided the applied load is not too high. These
cells are limited by the fact that by being thinner at the strain gage sites, they are
inherently weakened at those points.
13
Direct (Axial) Loading
The next loading mechanism is axial loading. The analysis of this kind of loading
is perhaps the most straightforward of any of the three loading mechanisms. Quite
simply, the load-bearing element in axial loading is a column in tension or compression.
The strain gages can be mounted parallel to the axial load or perpendicular to it (figure 29). From the Poisson’s ratio relationship between longitudinal (axial) strain and crosssectional strain, it is evident that if the column is in tension, strain gages mounted
perpendicular to the load will be in compression, and vice versa.
Figure 2-9. A column load cell
The column load cell generally offers greater stiffness than the bending beam
configuration, particularly in compression. However, this load mechanism is not ideal.
First, there is no localized strain concentration where the gages are mounted. Secondly,
the stiffness of the column varies slightly when it is in tension or compression, giving the
output a nonlinear characteristic. Third, Poisson’s ratio dictates that the change in
resistance is not the same in all four gages, which further contributes to the non-linearity.
The direct stress loading presents some constraints for the column as well. Specifically,
the column must be long enough to have a uniform strain field, a necessary ingredient for
the desired linear output characteristic. Further, the resulting length to area ratio makes
the column vulnerable to off-axis loads. Hence, compensation for these effects becomes
necessary. One solution to this problem is to secure the column with diaphragms, as in
figure 2-10.
14
Figure 2-10. Column load cell with diaphragms
There are various ways in which column load cells may be combined for more
strength. They can be added in parallel or hollowed out to provide increased resistance
for non-axial forces, while still maintaining their axial load characteristics. These
configurations create load cells with all the stiffness of an axial load cell, but without the
characteristic non-linearity associated with direct-force loading.
Shear Loading
The third load mechanism is shear loading. With this type of loading, an applied
force may be measured with minimal deflection. The basic shear-web load cell (figure 211) is a cantilever beam with a section machined out, creating a local I-beam. The
cantilever beam as a whole has a large enough cross section that deflections are small. In
fact, they are too small for a surface-mounted strain gage to generate an intelligible
output signal. Strain gages are mounted at 45-degree angles to the neutral axis, where
they are under strain from shear forces. In this manner, the deflection is reduced, as are
vibrations.
Figure 2-11. Schematic of a shear load cell
15
Selection of the Spring Material
In addition to the configuration and load mechanism on the spring, the
composition of the spring itself also influences the characteristics of the load cell. The
basic mechanical requirement of the spring is that it behave like a very stiff, high
precision spring while having a surface area large enough for the mounting of strain
gages. When a material must be selected to be the spring, its mechanical and thermal
properties are both considered, as well as the practicality of manufacturing a spring from
this material.
Ideally, a spring will exhibit linear elastic deformation within the range of applied
forces, both in tension and compression. Obviously, if the spring (on which the strain
gages are mounted) does not exhibit linear characteristics, the output from the strain
gages will not be linear either. Generally, the ultimate strength of the spring material is
not a primary concern, as the stresses to which it is subjected in a load cell are
considerably less than its ultimate strength. Likewise, fatigue strength is not a major
criterion, as the strain gage is more likely to fail in fatigue than the spring. However,
creep is a factor in the selection of spring materials, as they are often subjected to steady
–state loads. If a spring becomes deformed in its typical applications, it is basically
worthless.
Thermal factors are important because the spring has the possibility of
transferring heat with the strain gages. A high thermal conductivity is ideal because
temperature gradients are less likely to develop in a material with high thermal
conductivity. Temperature gradients are undesirable because they cause inconsistent
mechanical properties within the spring. Further, if the strain gages do start heating up, a
spring with high thermal conductivity enables the strain gages to cool down more
quickly, reducing the possibility of thermal effects in them.
Lastly, manufacturing issues must be considered in the selection of a spring
material. The material selected should be easily machinable, as the spring will be
stronger if it is machined from a single piece of stock, rather than put together. Also, the
spring is typically hardened after being machined, so materials that distort during the
hardening process are undesirable. Obviously, these factors must be weighed against one
another, as no material completely satisfies all of these criteria. However, some
commonly used spring materials are 4140 and 4340 alloy steels (high modulus for high
capacity cells) and 2024-T4 and 2024-T351 aluminum (low modulus).
16
Selection of the Strain Gage
The strain gage itself must also be selected and installed with care. As mentioned
above, an area of high strain concentration is planned for the strain gages, and it is of
utmost importance that they be installed in this zone. It is critical that the active gage
length (typically 1.5 to 3.2 mm) not exceed the area of maximum strain, or else there will
be a strain gradient within the strain gage, and linearity will be lost. A uniform strain
under the gage also reduces the risk of fatigue failures in the gage.
The geometry of the strain gage is also considered. Generally, the selection of a
gage geometry amounts to a choice between a wide grid and a thin grid (figure 2-12). As
far as performance is concerned, a wide grid is usually the better choice, as it can
dissipate more power. However, thin grids are usually less expensive.
Figure 2-12. Different grid widths
If the load cell geometry calls for two strain gages to be installed in series close to
each other, a multiple grid pattern, which is effectively a strain gage with two grids on it,
may be used (figure 2-13).5
Figure 2-13. A dual-grid strain gage
This sort of package saves time and uncertainty on installation and alignment.
Wiring is quicker as well, since the two grids on this gage share a common solder tab.
The likelihood of property variation between different gages is also lessened, as the two
grids tend to have almost identical properties.
In addition to its geometry, the gage’s composition must also be taken into
account. First, the gage backing, which is responsible for protecting the fragile gage
material, isolating it electrically from the spring material, and transferring the spring’s
strain to the gage, must be chosen well. The three principle varieties of strain gage
backing are polymides, epoxies, and reinforced epoxies. Polymides are virtually
unbreakable, but they tend to creep, making them a poor choice for high-accuracy
5 “Strain Gage Based Transducers – Their Design and Construction,” Interactive Guide to Strain Gage Technology, Release 111099,
July 1999. Measurements Group, November 16, 1999, <http://www.measurementsgroup.com/guide/ta/sgbt/sgbtndex.htm>, page 51.
17
applications. However, new polymides reportedly do not exhibit the same creep as their
predecessors.
Next, the grid alloy must be selected.6 The main alloy types used today are
constantan, Karma, isoelastic, and platinum-tungsten. Constantan, a copper-nickel alloy,
offers solid linearity, is widely available in many configurations, and can be easily
soldered. However, one major drawback of constantan is that it experiences a drift in
resistance when exposed to high temperatures (>150 F). Karma, a nickel-chromium
alloy, also exhibits great linearity, and it offers a higher resistivity and resistive stability
at high temperatures than constantan. It also has a high fatigue life. Of course, karma
has its share of shortcomings; namely it is difficult to solder and expensive to
manufacture. Isoelastic alloys are characterized by their good fatigue life and high gage
factor. Unfortunately, they are also difficult to solder and they cannot be selftemperature compensated. Finally, there are platinum-tungsten alloys, which also have a
good fatigue life and high gage factor, but cannot easily be temperature compensated.
For most standard strain gage applications, constantan is the most practical of these
alloys, though the others may be useful in the areas of constantan’s shortcomings, namely
at high temperatures.
The next grid issue to be considered is the grid resistance. In the previous section
it was shown that the output of a strain gage circuit is a function of strain, gage factor,
and grid resistance. From that relationship, it should be obvious that the grid resistance is
an important characteristic of a load cell. Furthermore, the amount of power dissipated in
a strain gage is directly related to the grid resistance.
V2
P V *I 
R
In most load cell applications, an increase in the excitation voltage can produce a
greater output signal. However, the power dissipated as heat in the strain gages increases
with the square of the voltage, so an increase in excitation voltage can produce a lot of
heat in the strain gage. Any extra heat being dissipated in the strain gage is usually not
desirable, as heat can produce thermal effects, causing non-linearity in the output. So if a
higher excitation voltage were really needed, a high grid resistance would be desirable, as
it reduces the amount of energy dissipated as heat.
The gages must also be compensated for thermal effects in the anticipated
operating range, as well as for creep. The thermal effects are compensated by selecting
an S-T-C (self-temperature-compensation) number to match the thermal expansion
behavior of the spring material. To compensate for creep, it is necessary to design the
strain gage creep to offset the spring creep. Such design is best accomplished through
trial and error.
6
In many cases, the strain gages are sold from catalogues, with the alloy/backing combination already
determined, so the selection process described here may not necessarily be realistic. However, it is still
important to know what function each component serves and to understand the properties of the available
materials.
18
Selection of the Adhesive
Now that the proper strain gage has been chosen, it is time to bond it to the spring.
This step in load cell assembly is extremely important, as the choice of adhesive must
serve as a second backing, transmitting strain from the spring to the strain gage and heat
from the strain gage to the spring without transmitting any electrical charges or strains
from the strain gage to the spring. The bond between the strain gage and the spring must
be rigid and strong enough to completely transmit strain to the gage over the operating
temperature range. The adhesive should also be easy to prepare and apply.
Some of the adhesives available today are cyanoacrylate adhesive, epoxy
adhesives, and high-performance adhesives. Cyanoacrylate is easy to prepare and apply,
making it suitable for routine, low-accuracy applications. The weakness of this adhesive
is its sensitivity to moisture, which gives it a life of only a few months. Epoxy adhesives,
particularly unfilled epoxy adhesives, can bond strain gages with tiny gluelines. Their
cure time tends to be longer than that of cyanoacrylate, but the resulting bonds last
longer. High-performance adhesives give even thinner gluelines than epoxies, and their
bonds can withstand higher temperatures. If high accuracy is needed, then highperformance adhesives are the best choice.
19
Assembling the Load Cell
Once all of the necessary components have been carefully chosen, the load cell
can be built. At this stage, the primary concern it protecting the strain gages, so they will
be useful for a long time. As the circuit is wired together, there are several potential
dangers to the strain gages.
First, the soldering iron can potentially heat up the strain gage alloy, causing
unwanted thermal effects. For this reason, the use of a temperature-regulated soldering
iron is strongly recommended so the temperature of the solder never gets any higher than
it absolutely needs to be. Care must also be taken to touch the surfaces to be soldered
with the soldering iron for only as long as is absolutely necessary.
Another major concern at this point is the possibility of corrosion of the strain
gages. Corrosion may result from impurities left in touch with the gage, such as flux
residue. For this reason, great care should be taken to make sure that there is not any flux
left on the strain gages after soldering.
The wires themselves raise some other concerns, such as wire symmetry,
temperature gradients, and effective insulation of charge carrying wires. The idea of wire
symmetry is to make the wires between the gages have a uniform length, to insure that
any measured resistance changes are a result of applied strain. Also, no stray wire should
be present in the bridge circuit. Essentially, any unnecessary wires in the circuit are
potential sources of inaccuracies.
Once the circuit has been carefully wired, there is still one more step that must be
taken to ensure that the load cell will have a long service life. It needs to be protected
from the environment. Strain gages tend to be sensitive to moisture and other
contaminants, so the strain gage must be tightly sealed against these corrosive agents.
The gage can be sealed with a hermetic seal, wax, or rubber. The hermetic seal is the
best absolute protection from environmental contaminants, sealing it within metal
enclosures. However, this method is often cost prohibitive, in which case alternative
seals should be used. Microcrystalline wax is a great organic sealant, within a certain
temperature range, as it tends to crack at low temperatures and at high temperatures it
tends to melt. Betyl rubber provides a good barrier against moisture, and it is easy to
apply. On the other hand, silicon rubber provides a good strong reinforcement, but it
doesn’t do much to protect the gage from moisture.
Sealing the strain gage is the final step in the design and construction of a load
cell. Next, the cell must be tested, and finally, put to work.
20
Section 3: Selection of Commercial Load Cells
In order to be a discriminating purchaser of commercial load cells, one must be
familiar with the various load cell properties and how they relate to design constraints.
Further, it is important to have a basic knowledge of the general classes of load cells on
the market. Finally, being able to read and understand load cell specification sheets is a
must.
Load Cell Properties
The number of commercially available load cells is astonishing. Thus, when
purchasing a load cell, it is necessary to be highly selective and to choose only the cell
that best satisfies the given design constraints. There are several factors to be considered
when purchasing a load cell, including performance parameters and other constraints
(i.e., cost, space considerations).
Performance parameters are likely to be the primary considerations in the
purchase of a load cell. After all, if the cell does not perform as the design demands, it is
worthless to the design. Depending on the application, any of the following factors may
be critical to the selection process: accuracy, compensated temperature, creep, deflection,
hysteresis, natural frequency, non-linearity, non-repeatability, operating temperature,
rated load, rated output, resolution, safe overload, temperature shift, and zero return.
The accuracy, or tolerance of average deviation of the actual output from
theoretical output, will be an important factor in applications where there is little margin
for error in measurements.
The compensated temperature is the range of temperatures in which the load cell
is effectively temperature compensated. This parameter isn’t generally a very big issue in
the selection of a cell, as long as the anticipated operating temperatures are within the
compensated range. However, in high-temperature applications, temperature
compensation will become a major concern, so the compensation range should not be
overlooked.
Creep refers to the phenomenon of load cell output changing with time under a
constant load. If the load cell is to be subjected to a steady-state loading situation, creep
should be minimal; or else it will be impossible to collect reliable, consistent data for the
output.
A load cell’s deflection between no-load and its rated load plays a major role in
the its performance if vibrations are a concern. The stiffness, or spring constant of the
cell is defined as its rated load divided by its deflection. If vibrations are a potential
hazard, a stiff load cell is desirable, so the deflection should be minimal.
Hysteresis is measured by increasing the load until it reaches some arbitrary value
(usually half of the rated output), then starting again with a larger load (usually the rated
output) and decreasing it until it reaches the same arbitrary value used before. The
hysteresis is then the difference between the measured output when the load is increased
to that value and that when the load is decreased to that value. Essentially, this
measurement quantifies the difference in output when the load is being reduced from that
when it is being increased. As such, if the cell will need to measure variable loads,
hysteresis should be taken into account.
21
Natural frequency is the frequency at which a system will oscillate out of control.
The natural frequency of the load cell itself may not necessarily be as important as the
natural frequency of the whole system (the load cell plus the object from which the force
is being measured). Ideally, the system’s natural frequency should be sufficiently high as
to be unreachable in its normal use. This frequency is a function of the cell’s deflection
and rated load, so if frequency is a concern, particular attention should be paid to those
two properties.
Non-linearity is defined as “the maximum deviation of the calibration curve from
a straight line drawn between the no-load and rated load outputs.”7 The ideal calibration
curve is linear, meaning that the output is directly proportional to the load. It is thus
obvious that the non-linearity of a load cell should be very small for all applications.
Non-repeatability is another property that should always be small. In any data
acquisition application, the results have to be repeatable in order to be valid. Therefore,
if a load cell has a high non-repeatability factor, it cannot be used for serious data
acquisition.
The operating temperature, like the compensated temperature, is usually
important as long as the operating conditions of the load cell will be within this range.
The range of operating temperature is generally wider than that of compensated
temperature, meaning that the cell can still operate at some temperatures outside the
compensated range, but the output is subject to temperature effects. Again, for hightemperature applications, special attention should be paid to the operating temperature
range.
The rated load is usually one of the first considerations in load cell selection. This
load is the maximum load that the cell is designed to measure, so it obviously cannot be
ignored. Obviously, a load cell whose rated load is greater than the maximum load to be
measured is essential. As mentioned above, the rated load also effects the stiffness of the
cell and consequently the natural frequency of the system as well. So if a high natural
frequency is desired, a high rated load is recommended.
The rated output is the output voltage at the rated load. This parameter generally
is not of primary concern, as a small rated output load cell can easily be connected to an
amplifier to increase the magnitude of the output.
In precision measurement, the resolution of a load cell is important. Defined as
“the smallest change in mechanical input which produces a detectable change in the
output signal,”7 the resolution of a load cell governs the degree of precision to which
mechanical inputs can be measured. Unfortunately, there is an obvious conflict between
maximizing resolution and maximizing stiffness, as a stiffer load cell (one with a higher
rated load) will tend to have poorer resolution than load cell with a lower rated load. So,
when the optimization of two parameters is mutually exclusive, the individual application
must be investigated to determine which parameter is more important.
Safe overload is maximum load that can be applied without causing permanent
deformation in the load cell. Obviously, the safe overload of the selected load cell should
be greater than the maximum anticipated load, but this shouldn’t even be a factor, as the
7
Term Glossary, November 1999. Futek Advanced Sensor Technology, November 15, 1999,
<http://www.futek.com/glossary.asp>.
22
rated load, which is less than the safe overload, should always be greater than the
maximum anticipated load.
Temperature shift, or the shift in output due to temperature changes in the load
cell, is basically a measure of temperature compensation. If the load cell is properly
compensated, the temperature shift should be very small.
Zero return measures the difference in the output measured at zero load before a
load is applied and after the load is removed. Effectively, zero return is a measure of the
variation in output from repeated loading. If the zero return value is a significant portion
of the desired precision, that load cell is unsatisfactory.
In most engineering applications, performance constraints are not the only
requirements that can dictate the selection of parts. Among other constraints, two major
considerations are cost and space. Load cells cost anywhere from $40 to $2500, and in
many situations, available funds dictate how much money can be spent on a load cell. In
most cases, the cost of a load cell is another of many factors that must be balanced.
Space is important because in some situations, the load cell must fit in a small, or oddly
shaped space, so the dimensions of the mounting space should be known before load cells
are seriously researched.
23
Families of Load Cells
While the sheer number of commercial load cells on the market can seem
overwhelming at first, it may be some comfort that they are sorted into different classes,
making the selection of a cell for a particular application easier. Some families of load
cells are bending beam, load button, s beam/z beam, donut, pancake, column, miniature,
and load/force washer. Each of these groups has its own general advantages,
disadvantages, and specific applications.
Bending beam load cells are basic cells designed to measure small forces (0-500
lb.). A small rated output, coupled with the bending beam mechanics of these cells, gives
them a relatively low rigidity. However, if rigidity is not a major concern, these cells are
worthwhile because of their low cost. They can be used to measure force, displacement,
or pressure, and are primarily for small measurements in which vibrations are not an
issue.
Load button cells are designed to measure compressive forces over a wide range
of loads (5-100000 lb.). They are compact, making them ideal for applications in which
space is a factor. The one real disadvantage of these cells is that they cannot be used to
measure tensile forces.
S beam/z beam cells on the other hand are designed for tension measurement (2525000 lb.). These cells are in-line, rather than simply being mounted at one end and
subjected to a force at the other. For this reason, they are excellent for measuring tensile
forces, but they would not be a prime choice for measuring compression.
Donut cells will work in either tension or compression (50-50000 lb.). Their
primary use is in space limited applications measuring clamping forces.
Pancake cells can also measure both tension and compression (300-250000 lb.),
and like the donut cells, they are compact. They are also characterized by high precision
and ability to make eccentric measurements.
Column load cells utilize a column mounting for the strain gages, making them
best suited for compression measurements (875-225000 lb.). These cells tend to be very
rigid, virtually eliminating system vibrations. The drawback to column cells is their high
load range and consequent low resolution, making them impractical for measuring small
forces.
As their name implies, miniature load cells are compact. There are many
configurations of miniature load cells, capable of measuring from 5 to 100000 lb. Their
common use is space limited applications, and within that framework, there are miniature
gages that can measure tensile and compressive forces.
The last family of load cells is that of load/force washers. The primary use of
these hollow cells is to measure fastener-clamping forces (2000-200000 lb.), and they
also feature a compact design, making them suitable for space limited applications.
24
Interpreting Load Cell Specification Sheets
Once the appropriate class of load cell has been selected, the right cell must be
determined from the specifications of each load cell. In this report, two sample
specification sheets will be examined to familiarize the reader with the data contained
therein. The cells examined shall be Futek model FP10124-00060-C and the Omega
LCDB series.
The Futek L2330
(a)
25
(b)
Figure 3-1. (a) Specifications of the L2330, (b) dimensions of the L2330
26
This sheet (figure 3-1) is a concise summary of many of the salient properties
described earlier in this section, and it is thus beneficial to quickly run through the data
presented here to familiarize the reader with the nomenclature and unit conventions used
by load cell manufacturers.
Capacity:
the Rated Load of the cell, or the maximum load that it is designed to
measure. The Capacity or Rated Load is given in pounds or Newtons.
Thread size:
the size of the hole threads at the point of loading in the cell. This
information is important for connecting the load cell to the load.
Material:
the material from which the load cell is made. Knowing the material, one
can estimate the relative stiffness of the load cell.
Resistance:
the no load resistance of each strain gage in the circuit, given in ohms.
Rated Output: the voltage output at the Rated Load. The units here are typically
mV of output per V of excitation.
Safe Overload: the maximum load that the load cell can withstand without suffering
permanent deformation. This quantity is given as a percentage of the
Rated Load. In this case, the Safe Overload is 1000% of the Rated Load,
or 100 pounds.
Excitation:
the input voltage across the Wheatstone bridge circuit, in volts. In this
case, “rec.” means recommended, which implies that this cell may work at
other excitation voltages, but 10 volts is the manufacturer’s
recommendation.
Bridge Resistance: the net no load resistance across the Wheatstone bridge, in ohms. In
the case of a full bridge where all four resistances are equal, the equivalent
resistance is equal to the resistance in each leg.
Non-Linearity: the maximum deviation of the calibration curve from an ideal, linear
curve. The Non-Linearity is expressed in volts, as a percentage of the
Rated Output.
Hysteresis:
the difference between the output at ½ of the Rated Load for an increasing
load and that at ½ of the Rated Load for a decreasing load, expressed as a
percentage of the Rated Output.
Non-repeatability: the difference between outputs for an identical load under identical
conditions taken at different times, expressed as a percentage of the Rated
Output.
Creep:
the change in output with time for a constant load without external
disturbances, expressed as a percentage of the load.
Zero Balance: the no load output at the recommended excitation voltage, expressed as a
percentage of the Rated Output.
Temp. Shift Zero: the change in no-load output with respect to temperature changes,
expressed as a percentage of the Rated Output per degree Fahrenheit.
27
Compensated Temp: the temperature range within which the load cell may safely be
assumed to be temperature compensated, in degrees Fahrenheit or degrees
Centigrade.
Operating Temp: the temperature range within which the load cell can function, provided
it is properly compensated. Operating Temperature is expressed in
degrees Fahrenheit or degrees Centigrade.
Price:
the cost, in dollars, of the load cell.
Dimensions: the physical dimensions of the load cell.
Although this chart can answer many questions about the load cell, it does not
cover everything. The deflection and stiffness of the cell are conspicuously absent, as is
the resolution. Unfortunately, most spec sheets do not contain all of the salient data, but
that information does exist. Usually a call to the manufacturer will yield any pertinent
properties not listed in the sheets. Furthermore, different manufacturers list different
properties, sometimes with different nomenclature, in their specifications. To illustrate
this point, the Omega LCDB series will also be examined.
28
The Omega LCDB Series
Figure 3-2. Specification sheet for the Omega LCDB series
29
Although this chart (figure 3-2) conveys much of the same information as the
Futek chart, much of the nomenclature is different, and it covers a whole series of load
cells rather than one sole cell within the series, so it merits explanation.
Rated Output: the same characteristic as the Rated Output in the other chart, in mV/V of
excitation. Here, the tolerances are also listed, as are the Rated Outputs
for the 100 and 200 pound capacity load cells in this series.
Excitation:
again, this property is the same as in the Futek chart, only here a
maximum allowable value is listed. It also specifies that only dc voltage
may be applied.
Calibration:
the type of load for which this cell is calibrated. In this case, it is only
calibrated for compression.
Accuracy:
a measurement of how close the real output value is to the theoretical
output. Accuracy is expressed as a percentage of the Full Scale Output.8
Linearity:
the same quantity referred to as Non-Linearity in the Futek chart, given
here as a percentage of the Full Scale Output.
Hysteresis:
the same quantity as in the Futek chart, given here as a percentage of the
Full Scale Output.
Repeatability: the same quantity referred to as Non-repeatability in the Futek chart, given
here as a percentage of the Full Scale Output.
Zero Balance: the same quantity as in the Futek chart, given here as a percentage of the
Full Scale Output.
Creep in 20 min: the change in output over a 20 minute loading period for a constant load
without external disturbances, expressed as a percentage of the load.
Operating Temp. Range: the same as in the Futek chart.
Compensated Temp. Range: the same as in the Futek chart.
Thermal Effects: the change in output with a change in temperature, given for both no
load and for a loaded output. The Thermal Effects are expressed as a
percentage of the Full Scale Load per degree Fahrenheit for no load and as
a percentage of the load per degree Fahrenheit for a loaded cell.
Max Safe Load: the same quantity referred to as Safe Overload in the Futek chart, here
given for axial loads as well as for side loads, as a percentage of the Full
Scale Load.
Bridge Resistance: the same quantity as in the Futek chart, with tolerances given as well.
Full Scale Deflection: the deflection of the load-bearing member of the load cell at Full
Scale Load. In this series, the higher capacity load cells have a lower
deflection.
8
The Full Scale Output is defined as the difference between the maximum output and the minimum output.
In this case, the cell is only calibrated for forces in one direction, so the Full Scale Output is equal to the
Rated Output.
30
Construction: a description of the materials used in the load cell, including the nature of
stress in the different cells in the series.
Electrical:
the wires connecting to the load cell.
Weight:
the weight of the load cell itself, given in pounds or kilograms.
Although they do not necessarily use the same reference points or nomenclature,
both of these specification sheets provide a wealth of pertinent information about their
respective load cells, and many of the key considerations in load cell selection are
covered here. Of course, the on-line glossaries and technical support are invaluable for
answering any other questions that one may have about these cells. Essentially, these
charts have shown that it is not hard to extract useful data from spec sheets.
In this section, various load cell characteristics and their potential applications
were discussed. So it is now possible to form an idea of which characteristics are crucial
to a given application and what approximate values of those characteristics should be
sought. Next, the general classes of load cells were examined, and the strengths and
weaknesses of these groups were found. Hence the search for load cells may be quickly
narrowed to one or two groups depending on their desired use. Finally, the comparison
of the actual characteristics of two real load cells from different manufacturers illustrated
the process of critically examining different cells and selecting those which fulfill the
requirements of the problem, completing a structured, efficient methodology for finding
the right load cell.
31
Section 4 – Design of a Load Cell for HPV Aerodynamics
Background
In the second iteration of the HPV aerodynamics instrumentation system (Fall
1999), a Futek load cell was used to measure horizontal (drag) forces on a 1/12 scale
model HPV fairing. For the third iteration (Spring 2000), two additional load cells (one
per supporting rod) were needed to measure vertical (lift) forces. It was thus decided to
construct these load cells rather than buying them, as the cost of constructing them is
significantly less than that of purchasing them ($200 vs. $800). It was also proposed that
the structural integrity of the force balance would be enhanced if the load cells were to be
integrated into the force balance.
Proposed Load Cell Configurations
With the above considerations in mind, two designs were proposed. Of the three
possible loading mechanisms, only the bending beam was practical for this project. An
axial loading cell would be far too stiff for an application where the maximum possible
force is ~5 pounds. Likewise, a shear load cell would be too stiff for this design, and it
would be complicated to manufacture, which could cause problems given the limited
time available to complete the project. The material to be used is 6061 Aluminum alloy.
Of the bending beam family, the first proposed option is a beam load cell, fixed at both
ends for added rigidity (figure 4-1). Both beams are integrated into one member, which
is mounted on the horizontal linear bearing. The force is applied in the middle of each
beam, approximated here as a point load.9 The beam has four strain gages mounted on
top of it, with their centers at 1/8, 3/8, 5/8, and 7/8 of the total length, the points of
maximum strain. The gages mounted at 1/8 and 7/8 of the length are in tension, while
those mounted at 3/8 and 5/8 of the length are in compression. Because the magnitude of
strain in each gage is equal and two are in tension while two are in compression, this
arrangement satisfies the requirements of the full bridge load cell, as described in section
1 of this document. Additionally, the gages would all be the same make and model,
satisfying the temperature compensation requirements.
9
In reality, the rods which support the scale model fairing are mounted on pin joints which transfer the
force to the beam as a point force.
32
Figure 4-1. Proposed load cell number 1.
The second proposed load cell arrangement is simply two cantilever beams, again
integrated into a single member (figure 4-2). Since this design application is a low
magnitude loading situation, the non-linearity inherent in cantilever beam cells should not
be a problem. The sole member is affixed to the top of the horizontal linear bearing, as in
the first proposed design. Here, the maximum strain occurs where the beam meets its
support, but the gages cannot be placed there due to mounting considerations. In order to
function as a full bridge circuit, the cell must have four gages: two in tension and two in
compression with equal and opposite strain. The equal and opposite strains are attained
by mounting the gages on the top and on the bottom of the beam. The gages on the
bottom of the beam cannot be mounted too close to where the beam meets the support,
for installation purposes. Thus, the location of L/4 is proposed to balance the need to
maximize strain and the need to facilitate mounting. Also, it should be noted that two
gages are mounted side by side on top and two gages are mounted on the bottom of each
beam, resulting in two full bridge load cells.
Figure 4-2. Proposed load cell number 2.
33
Analysis of Load Cell Configurations10
Each of these proposed designs is analyzed here, starting with the required strain
at the middle of the strain gage. This strain is found from the required output voltage
ratio, which is 2mV/V, to ensure compatibility with the amplifier.11
Vo
 GF * 
Vi

Vo
1
*
Vi GF
0.002V 1
*
V
2
  0.001
This means that the strain at max deflection should be approximately 0.001 in/in to
ensure that the amplifier does not overload. Thus, the max strain is assumed to be
constant at 0.001 in the following analysis. The task is then to find the dimensions
(length, width, height) that satisfy this requirement. The length is fixed at 2 inches
because of the limited available space in the instrumentation box and the relative position
of the two rods. The width can be varied from .25 in to .75 in for the fixed end beam, but
it cannot be smaller than 0.35 in for the cantilever beam, as this beam must have two
gages mounted side-by-side. Other quantities held constant are the modulus of the beam-assumed to be 107 lbf/in2 (Aluminum will be used) and the applied force—assumed to
be 5 lbf (the maximum force given by fluid dynamic analysis of the problem).

The first analysis presented is that of the fixed-end beam, as performed in
Engineering Equation Solver (EES). Here, three iterations are performed, with beam
widths of 0.25, 0.5, and 0.75 inches.
First iteration – Input
E = 107
L
x1 =
8
x1
L
M1 = F ·
– F·
2
8
L = 2
F = 5
x2 = 3 ·
b = 0.25
 = 0.001
x3 = 5 ·
L
8
L
8
L
x4 = 7 ·
8
M2 = F ·
x2
L
– F·
2
8
M3 = M2
M4 = M1
Moment = M2
s =  ·E
h =
d =
6·
Moment
s·b
F · L3
1
192 · E ·
· b · h3
12
Vout = 2 · 
k =
F
d
10
Here, the weight of the beams is neglected. For justification of this assumption, see appendix A.
Note that the gage factor is assumed to be 2 in this analysis. In commercially available metal foil load
cells, this is indeed the case. There are semiconductor load cells with a gage factor of 150-180 on the
market, but their diminutive size renders them impractical to install.
11
34
First Iteration – Output
b=0.25 [in]
δ=0.01721 [in]
E=1.000E+07 [lbf/in^2]
ε=0.001 [in/in]
F=5 [lbf]
h=0.03873 [in]
k=290.5 [lbf/in]
L=2 [in]
Moment=0.625 [lbf*in]
σ=10000 [lbf/in^2]
V_out=0.002 [V/V]
Second Iteration – Input
E = 107
L = 2
x2 = 3 ·
F = 5
b = 0.5
s =  ·E
L
x1 =
8
x3 = 5 ·
 = 0.001
x1
L
M1 = F ·
– F·
2
8
L
8
L
8
L
x4 = 7 ·
8
M2 = F ·
x2
L
– F·
2
8
h =
d =
M3 = M2
M4 = M1
6·
Moment
s·b
F · L3
1
192 · E ·
· b · h3
12
Vout = 2 · 
Moment = M2
Second Iteration – Output
k =
F
d
b=0.5 [in]
δ=0.02434 [in]
E=1.000E+07 [lbf/in^2]
ε=0.001 [in/in]
F=5 [lbf]
h=0.02739 [in]
k=205.4 [lbf/in]
L=2 [in]
Moment=0.625 [lbf*in]
σ=10000 [lbf/in^2]
V_out=0.002 [V/V]
Third Iteration – Input
L
8
x1
L
– F·
2
8
s =  ·E
E = 107
x1 =
L = 2
L
x2 = 3 ·
8
x2
L
M2 = F ·
– F·
2
8
h =
L
x3 = 5 ·
8
M3 = M2
d =
F = 5
b = 0.75
 = 0.001
L
x4 = 7 ·
8
M1 = F ·
M4 = M1
Moment = M2
6·
Moment
s·b
F · L3
1
192 · E ·
· b · h3
12
Vout = 2 · 
k =
F
d
35
Third Iteration – Output
b=0.75 [in]
δ=0.02981 [in]
E=1.000E+07 [lbf/in^2]
ε=0.001 [in/in]
F=5 [lbf]
h=0.02236 [in]
k=167.7 [lbf/in]
L=2 [in]
Moment=0.625 [lbf*in]
σ=10000 [lbf/in^2]
V_out=0.002 [V/V]
The second analysis is that of the cantilever beam, also performed in EES, for
widths of 0.35 inches, 0.5 inches, and 0.75 inches.
First Iteration – Input
7
E = 10
Moment = F · (L – x)
s =  ·E
L = 2
F = 5
b = 0.35
 = 0.001
x =
L
4
h =
d =
6·
Moment
s·b
F · L3
1
3 ·E·
· b · h3
12
Vout = 2 · 
k =
F
d
First Iteration – Output
b=0.35 [in]
δ=0.03136 [in]
E=1.000E+07 [lbf/in^2]
ε=0.001 [in/in]
F=5 [lbf]
h=0.1134 [in]
k=159.5 [lbf/in]
L=2 [in]
Moment=7.5 [lbf*in]
σ=10000 [lbf/in^2]
V_out=0.002 [V/V]
x=0.5 [in]
36
Second Iteration – Input
E = 107
Moment = F · (L – x)
s =  ·E
L = 2
F = 5
h =
b = 0.5
 = 0.001
x =
L
4
d =
6·
Moment
s·b
F · L3
1
3 ·E·
· b · h3
12
Vout = 2 · 
k =
Second Iteration – Output
F
d
b=0.5 [in]
δ=0.03748 [in]
E=1.000E+07 [lbf/in^2]
ε=0.001 [in/in]
F=5 [lbf]
h=0.09487 [in]
k=133.4 [lbf/in]
L=2 [in]
Moment=7.5 [lbf*in]
σ=10000 [lbf/in^2]
V_out=0.002 [V/V]
x=0.5 [in]
Third Iteration – Input
E = 107
Moment = F · (L – x)
s =  ·E
L = 2
F = 5
h =
b = 0.75
 = 0.001
x =
L
4
d =
6·
Moment
s·b
F · L3
1
3 ·E·
· b · h3
12
Vout = 2 · 
k =
F
d
37
Third Iteration – Output
b=0.75 [in]
δ=0.0459 [in]
E=1.000E+07 [lbf/in^2]
ε=0.001 [in/in]
F=5 [lbf]
h=0.07746 [in]
k=108.9 [lbf/in]
L=2 [in]
Moment=7.5 [lbf*in]
σ=10000 [lbf/in^2]
V_out=0.002 [V/V]
x=0.5 [in]
Selection of a Configuration
The fixed-end beam has a required thickness (height) of 0.02981 inches to
0.03873 inches and the cantilever beam has a required thickness of 0.07746 inches to
0.1134 inches. These results indicate that the cantilever beam is the better choice for this
application, because a thickness of 0.039 inches would render the fixed beam load cell
fragile, greatly increasing the possibility of failure of the member. In theory, the fixed
beam cell does provide greater stiffness for a given force, so it would most likely be a
better choice for higher force applications. However, the maximum force here is 5
pounds, so all that extra stiffness is unnecessary. A quick examination of the results for
the cantilever beam cell reveals that the stiffness increases as the width decreases. It is
thus advantageous to minimize the width.
Selection of a Strain Gage




The strain gage to be used in this system must meet the following criteria:
Size
- The width of two gages must be minimal (0.35 in max for 2 gages), so as to
minimize the width of the beam. The gages can be either two single grid gages or
one double grid gage.
- The length and width of the gages must be large enough to facilitate installation
by inexperienced amateurs (1/4 in x 1/8 in minimum).
Cost
- The total cost for all 8 gages should not exceed $100.
Voltage
- Maximum allowable voltage must be at least 10 volts to be compatible with
current amplifiers and power source.
Temperature Compensation
- Gages must be temperature compensated for mounting on Aluminum.
38
Given these constraints, the following gages were found from HBM inc.
Model double grid –
mm(inch)/Ohm
DK 13.G – 3 (0.125)/350
DK 13.C – 6 (0.25)/350
DK 23.K – 6 (0.25)/350
length
mm/(in)
9.5(.37)
12.5(.49)
12.5(.49)
width
mm/(in)
8.5(.33)
8.5(.33)
8.5(.33)
Max Supply
Voltage (V)
10
14
13
Cost ($)
57 (for 5)
62 (for 5)
54 (for 5)
Of these gages, the second choice seems to be the best, as its max supply voltage
is four volts higher than our requirement, giving it a built-in safety factor. The size of all
three gages is approximately the same, and all three meet the required size specifications.
So, since it best fits all of the design requirements, the DK 13.C – 6 (0.25)/350 is the gage
of choice for this project
Conclusions
Now that a load cell configuration has been determined and a strain gage has been
selected to fit this configuration, there still remains the task of integrating these parts into
two functional load cells within an instrumentation system. Once the design is approved
and parts are purchased, the cantilever beams need to be machined from aluminum stock
and the strain gages need to be mounted, soldered and sealed. Finally, the LABVIEW
interface must be reprogrammed to take data from three gages and record the data to a
text or spreadsheet file.
39
Appendix A – Calculation of Effect of Beam Weight on Strain
To determine whether or not the weight of the beam itself would have a
significant effect on the strain for the two beam load cells, the resulting strain in each
beam, subjected to a uniform load, was calculated. In these calculations, the linear
weight density λ was assumed to be the only force acting upon the beams (aside from
reactive forces).
Beam 1 – fixed at both ends
l= g ·
g = 170
7
E = 10
b = 0.35
c =
h
2
3
I =
1
· b · h3
12
d =
l· L4
384 · E · I
h = 0.1134
L = 2
1
12
Mmax =
·b ·h
l· L2
12
s =
Mmax · c
I
 =
s
E
Results
b=0.35 [in]
c=0.0567 [in]
δ=3.825E-07 [in]
E=1.000E+07 [lbf/in^2]
ε=1.735E-07 [in/in]
γ=170 [lbf/ft^3]
h=0.1134 [in]
I=0.00004253 [in^4]
L=2 [in]
λ=0.003905 [lbf/in]
M_max=0.001302 [lbf*in]
σ=1.735 [lbf/in^2]
40
Beam 2 – cantilever beam
g = 170
l= g ·
E = 107
b = 0.35
h = 0.1134
I =
d =
L = 2
c =
h
2
1
12
3
·b ·h
1
· b · h3
12
l· L4
8 ·E·I
Mmax =
l· L2
2
s =
Mmax · c
I
 =
s
E
Results
b=0.35 [in]
c=0.0567 [in]
δ=0.00001836 [in]
E=1.000E+07 [lbf/in^2]
ε=0.000001041 [in/in]
γ=170 [lbf/ft^3]
h=0.1134 [in]
I=0.00004253 [in^4]
L=2 [in]
λ=0.003905 [lbf/in]
M_max=0.007809 [lbf*in]
σ=10.41 [lbf/in^2]
For the fixed-end beam, the strain resulting from the weight of the beam is 4
orders of magnitude less than that resulting from the applied force, so it can be neglected.
Likewise, the strain resulting from the weight of the cantilever beam is 3 orders of
magnitude less than that resulting from the applied force, so it too can be neglected.
41
References
Alciatore, David G. and Michael B. Histand. Introduction to Mechatronics and
Measurement Systems. New York: McGraw-Hill Publishing, 1999.
Load Categories. November 1999. Futek Advanced Sensor Technology. November 15,
1999. <http://www.futek.com/category.asp?loadtype=load>.
“FP10124-00060-C.” Load Cells. November 1999. Futek Advanced Sensor
Technology. November 17, 1999.
<http://www.futek.com/productpf.asp?product=fp10124%2D00060%2Dc>.
Term Glossary. November 1999. Futek Advanced Sensor Technology. November 15,
1999. <http://www.futek.com/glossary.asp>.
“Applying the Wheatstone Bridge.” HBM Application Notes. Karl Hoffmann. 1996.
HBM Weighing Technology. November 16, 1999.
<http://www.hbmwt.com/appnotes/wheatstn/wheatstn.htm>,
<http://www.hbmwt.com/appnotes/wheatstn/wheatst2.htm>.
“Beam Load Cell Performance.” HBM Application Notes. Paul Becquart. 1996. HBM
Weighing Technology. November 16, 1999.
<http://www.hbmwt.com/appnotes/beamperf.htm>.
42
“Load Cell Terminology.” HBM Application Notes. 1999. HBM Weighing
Technology. November 16, 1999.
<http://www.hbmwt.com/appnotes/define.htm>.
“Strain Gage Product Index.” HBM Product Index. 1999. HBM Weighing Technology.
January-February 2000. <http://www.hbmwt.com/gageland/index.htm>.
“Load Cell Overview.” Load Cell Technical Tip Sheets. November 1999.
Measurement Specialists. November 16, 1999.
<http://www.measurespec.com/Tips/principles.htm>.
“Load Cell Troubleshooting Tips.” Load Cell Technical Tip Sheets. November 1999.
Measurement Specialists. November 16, 1999.
<http://www.measurespec.com/Tips/loadcell.htm>.
“Strain Gage Based Transducers – Their Design and Construction.” Interactive Guide to
Strain Gage Technology. Release 111099. July 1999. Measurements Group.
November 16, 1999.
<http://www.measurementsgroup.com/guide/ta/sgbt/sgbtndex.htm>.
New Digital Smart Load Cells. August 07, 1998. Nordisk Transducer Teknik Denmark.
November 16, 1999. <http://www.ntt.dk/smartLC.htm>.
43
“LCDB Series: 0-50 to 0-1000 lb.” Load Cells, Force Sensors and Torque Transducers.
November 1999. Omega Engineering. November 17, 1999.
<http://www.omega.com/Pressure/pdf/LCDB.pdf>.
“Glossary.” Technical Reference Section. November 1999. Omega Engineering.
November 17, 1999. <http://www.omega.com/techref/glossary.html>.
“Strain Gage Technical Data.” Technical Reference Section. November 1999. Omega
Engineering. November 15, 1999.
<http://www.omega.com/techref/glossary.html>.
44