Bending Beam Strain Gage

The bending beam strain gage is an excellent method for measuring the force exerted by an object. It is quite scalable, it can measure from grams to tons with little effort. The approach shown here is self compensating for temperature and has an output that is large enough to be easily measured with readily available components. The implementation shown here was originally conceived to measure a liquid fill process by weight. 

Aluminum and steel are popular choices for beam material. They are commonly available in many useful preformed sizes and strain sensors are available with built in compensation for thermal expansion. Other materials are possible, including titanium, plastic, brass or just about anything else, but special consideration must be given when using these materials as they may not have linear strain characteristics or may require additional compensation for temperature. The example shown here is for an aluminum bar. Additional tables for brass, stainless and steel and titanium are provided.

Figure 1
Typical Bending Beam Load Cell

The equations describing the stress on the bar when a force (F) is applied is fairly simple and direct. The stress is dependent upon the thickness or height of the bar (H), width or beam of the bar (B), distance or length away from the force (L) and the Force or weight (F) .

Stress = F*L*6/(B*H^2)

The Strain on the bar is the above equation with Young Modulus in the denominator: Youngs Modulus is a constant for a given material. Table 1 has example Youngs modulus for some common materials.

Strain = F*L*6/(Y*B*H^2)

Table 1 Youngs Modulus and Recommended Maximum Stress for Common Materials

The maximum recommended stress in Table 1 has a safety factor of approximately 2 based upon the yield strength of the materials. The maximum force sensitivity is achieved by having the maximum measured force create these stresses. In other words: the more you bend it, the more there is to measure. However, all materials have a limit of elasticity; the point of no return, if it is bent any further, it stays bent and becomes something other than the original. Our assumption is half way to the maximum elastic limit is safe and gives sufficient bend to allow measurement. Be aware that your design should never exceed these numbers or you will have a broken gage. For example if your maximum force is 20 pounds and the bending beam is designed to to measure 20 pounds; a problem would exist if someone dropped the 20 pound weight on to the beam. The resulting force from even a foot drop would most likely deform and damage the beam. Some type of mechanical limit to restrict the beams' motion and/or a larger, less sensitive beam are appropriate solutions to this problem.

From the stress equation, we could now make a bending bar that would be appropriate for our force measurement. So where does the strain fit in? The strain needs to be known to select the strain gage. A strain gage is nothing more than a piece of resistive wire that is attached to the top and bottom of the bending bar. When the bar bends, the top piece of resistive wire is stretched and the bottom piece is compressed. The resistance varies in both due to these physical changes. This is what we measure electrically. Commercially available strain gages are actually multiple pieces of resistive wire that are all stretched or compressed. This magnifies the strain effect.

Two gages are used to cancel thermal effects and multiply the strain effect. When used as the top and bottom legs of a Wein bridge Figure 2, the increase in resistance in one leg, while decreasing the resistance in the other leg, makes the circuit twice as sensitive as a circuit with one gage. Additionally, if only one gage were used, the resistance would increase and decrease as the temperature were raised and lowered within the metal bar and appear as weight being added or removed from the load cell. When two gages are used as top and bottom legs of the bridge circuit, their effects cancel, because their resistive ratio remains the same. When the temperature go up, both gages stretch, both resistances go up, when the temperature goes down, both resistances go down. The ratio between the two, however remains the same and there is no effect at the output.

Figure 2
Strain Bridge Example Using 2 Gages

The electrical output signal from a strain gage is very small, on the order of milivolts. This is not easy to measure. Most strain gages have an instrument amplifier attached to the gage output to provide voltages that are more useful. An instrumentation amplifier is just a fancy term for an amplifier that is immune to the noise and unwanted external effects while accurately reproducing the input signal. The external effects and noise that are common to this application are temperature drift and stray ac noise, like 60 Hertz noise from the power system. Figure 3 is a schematic for an instrumentation amplifier that is effective at minimizing both. The first stage of amplification is a chopper stabilized amplifier manufactured by Texas Instruments and now multiply sourced. The part has been around for many years and is one of my personal favorites, because it is simple to use and has excellent drift and stability characteristics.  The amplifier gain has been set to 100. This stage also has a single pole of filtering with a cutoff at about 6 Hertz to attenuate the unwanted noise. The second amplification stage is a less expensive and more common part manufactured by National Semiconductor and again, multiply sourced. It has a gain of  10 and a single pole filter with a cutoff of 2 Hertz. This does not need to be chopper stabilized because the gain is much lower than the first amplifier.

Figure 3
Typical Instrumentation Amplifier

There are several other unique features to this configuration. The design operates from one supply voltage rather than the traditional plus minus supply configuration. This approach minimizes the costs involved in powering the unit, however it also means that the amplifier output is not likely to be zero volts when the weight on the load cell is zero. This detail was not important in this design. If it is important to your design; add the minus supply. Be careful not to exceed the Choppers' maximum voltage of 15 volts.

The instrumentation amplifier traditionally has adjustments for zero and for the gain. One control allows the user to correct for variances in the construction of the strain gages while the other control allows the user to adjust for the inaccuracies of the amplifier circuits. I do not like the fact that these two controls are not independent of one another. The configuration shown here has a control to set the zero point of the gage (zero is actually 3 volts) The second adjustment is the trip point for the voltage comparitor. This approach provides a go no-go signal as the output; exactly what was required for my application. The output can be used to operate a relay or other electronic switch.

The attached Excel spread sheet is an aid for calculating the size of the bar required to measure the maximum force and translate it into the actual output voltages seen at the output of the instrumentation amplifier. Figure 4 is an example using a maximum weight of 20 pounds and and amplifier gain of 1000. The concept here is to minimize the amount of work to make the load cell while maximizing the electrical output. Please note that the maximum strain is shown to be about 1/2 the allowable maximum strain. This is part of the safety feature put into the design of this particular device. After the units are filled, there is a good chance that they will accidentally dropped on the load cell creating a force much higher that the maximum anticipated measured force. In addition to the reduced allowable strain, this design also has a mechanical limit built into the load cell to limit the travel of the bending bar.

The three columns solve for different values within the column. Use these as an iterative process to create a bar that requires minimal machining to produce an acceptable load cell that will fit within the dimensions of your application. The example is for an aluminum bar. Aluminum was selected because the environment it is going into is wet and it is easy to obtain in many sizes and machine. The first element in column 1 is: Maximum load pounds, this is the maximum force placed on the bar. Distance from Force to Support for Bar and Distance from Force to center of gage require two entries, because the bar will be longer than where the measurement is made. The maximum force sustained by the bar will be at its attachment point while the value measured at the gage will be measured from the center of the strain gage. The distance from the center of the gage to the attachment point should be minimized if it is at all possible. Thickness of the bar can be any value; it is often set to some standard stock size. Thickness of the bar can be any value; it is often set to some standard stock size.

To use the Excel spread sheet for calculating beam size, there are several things specific to the design that must be known before hand:

1. Maximum weight to be measured
2. Two of the three dimensions for the beam
3. Material to be used as a beam
4. Target strain value for the strain gage (Typically 1000 uStrain)

Our example:
1. Maximum weight : 20 Lbs.
2. Two of the three dimensions for the beam: 1x .125 inches
3. Material to be used as a beam: Aluminum
4. Target strain value for the strain gage (Typically 1000 uStrain): 1x10-3 strain

These values were inserted into column 1, Calculate Bar Width. The result was .768 inches. Using this information and knowing that Aluminum stock can be purchased with a nominal .125 x .750 inches, these values were inserted into column 2. The target location for the center of the gage was very close to the target of 1 inch (.977 calculated). These values 1x.125x.750 were substituted into column 3 and the actual strain was calculated to be 1.02x10-3 strain, again, very close to our desired strain. The maximum stress was checked against the maximum for aluminum 20,000psi and found to be acceptable for the application. The amplifier information was then inserted into the Circuit Variables column and also found to be acceptable for this design

Bending Bar Load Cell Calculator
	Calculate Bar Width		Calculate Gage Location		Calculate Strain Value	Circuit Variables
Maximum Load Lbs.	20	Pounds	20		20	Reference voltage	6.00
Distance from force to Support for Bar	1.162	Inches	1.137		1.137	Amp Gain	1000.00
Distance from force to center of gage	1	Inches	0.977		1	Offset Voltage	3.00
Thickness of Bar	0.125	Inches	0.125		0.125	Output voltage	9.14
Target Strain Value	1.00E-03		1.00E-03		1.02E-03
Youngs Modulus	1.00E+07	Aluminum	1.00E+07		1.00E+07
Width of BAR	0.768	Inches	0.750		0.750
Maximum Stress	11620	Pounds	11638		11643
Youngs Modulus for Aluminum	1.00E+07
Youngs Modulus for 316 Stainless	2.80E+07
Youngs Modulus for Hot Rolled Steel	3.00E+07
Youngs Modulus for Brass	1.60E+07
Youngs Modulus for Titanium	1.57E+07
Material	Maximum Recommended Stress
T6061-T0 Aluminum	8000	lb/in2	55	N/mm2	(mPa)
T6061-T6  Aluminum	20000	lb/in2	140	N/mm2	(mPa)
Hot Rolled Steel	13000	lb/in2	90	N/mm2	(mPa)
Cold Rolled Steel	20000	lb/in2	140	N/mm2	(mPa)
Brass, yellow Hard	20000	lb/in2	140	N/mm2	(mPa)
316 Stainless	30000	lb/in2	210	N/mm2	(mPa)
Titanium	35000	lb/in2	245	N/mm2	(mPa)
Copyright © 2002 Electro Technical Products, Inc.

Figure 4
Example Load Cell Design
Download a copy of the Calculator

There are several resources on the web to aid in the selection and design of your strain gage and load cell. I list a few that I thought were helpful below. Some are commercial and some are individuals that have also wrestled with this technology:

Richard Nakka's Experimental Rocketry Web Site This site has a different load cell design approach, covers the fundamentals
Omega Engineering Strain Gage Technical Data Fundamentals of Strain Gage selection
Omega Engineering Strain Gage  Strain gage and Load Cell design information

Copyright (C) 2002 Electro Technical Products, Inc.

Please feel free to contact the author with comments or suggestions: tpeterick