stress, strain, and strain gages primer

Stress, Strain, and Strain GagesIntroduction

Stress:When a material is loaded with a force, stress at some location in the material is defined asforce per unit area.

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For example, consider a wire or cylinder, anchored at the top, and hanging down. Suddenly,some force F (e.g. a hanging weight) starts pulling at the bottom, as sketched:

A is the original cross-sectional area of the wire, and L is the original wire length.

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In this situation, the material will experience a stress, called an axial stress, denoted by thesubscript a, and given by

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Notice that the dimensions of stress are the same as those of pressure, i.e. force per unitarea.

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Strain:In the above simple example, the wire will stretch vertically as a result of the force. Strain isdefined as the ratio of increase in length to original length.

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Specifically, when force is applied to the wire, its length L increases, while itscross-sectional area, A decreases, as sketched:

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The axial strain is defined asm

The dimensions of strain are unity, i.e. strain is nondimensional.m

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Hooke's law:It turns out that stress is linearly proportional to strain for elastic materials.m

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Stress, Strain, and Strain Gages

http://spike.me.psu.edu/~ME082/Learning/Strain/strain.html (1 di 11) [04/05/2001 13.29.16]

Mathematically, this is expressed by Hooke's law, which states

where E = Young's modulus, also called the modulus of elasticity.

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Young's modulus is assumed to be constant for a given material.m Note that Hooke's law breaks down when the strain gets too high. On a typical stress-straindiagram, Hooke's law applies only in the elastic stress region, in which the loading isreversible. Beyond the elastic limit (or proportional limit), the material starts to behaveirreversibly in the plastic deformation region, in which the stress vs. strain curve deviatesfrom linear, and Hooke's law no longer holds.

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In this learning module, only the elastic stress region is considered.m

Wire ResistanceThe electrical resistance R of a wire of length L and cross-sectional area A is given by

where is the resistivity of the wire material. (Do not confuse with density, for which the samesymbol is used.)

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The electrical resistance of a wire changes with strain:As strain increases, the wire length L increases, which increases R.m As strain increases, the wire cross-sectional area A decreases, which increases R.m For most materials, as strain increases, the wire resistivity also increases, which furtherincreases R.

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The bottom line is that wire resistance increases with strain.l In fact, it turns out that at constant temperature, wire resistance increases linearly with strain.l Mathematically,

where S is the strain gage factor, defined from the above equation, i.e.

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S is typically around 2.0 for commercially available strain gages S is also dimensionless.l

Strain Gage



The principle discussed above, namely that a wire's resistance increases with strain, is key tounderstanding how a strain gage works.

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A strain gage consists of a small diameter wire (actually an etched metal foil), which is attachedto a backing material (usually plastic). The wire is looped back and forth several times to createan effectively longer wire. The longer the wire, the larger the resistance, and the larger the changein resistance with strain. A sketch is shown below:

On this sketch, four loops of metal foil are shown, providing an effective total foil length L eighttimes greater than if a single wire, rather than a looping pattern, was used. Actual commerciallyavailable strain gages have even more loops than this. The ones used in the ME 82 lab have sevenloops.

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The direction of the applied strain is indicated on the sketch. The connecting wires or leads go toan electronic circuit (discussed below) which measures the change in resistance.

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How are strain gages used? A practical laboratory application is discussed here.l Consider a beam undergoing axial strain; the strain is to be measured.l A strain gage is glued to the surface of the beam, with the long sections of the etched metal foilaligned with the applied axial strain as sketched:

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As the surface stretches (strains), the strain gage stretches along with it. The resistance of the straingage therefore increases with applied strain. Assuming the change in resistance can be measured,the strain gage provides a method for measuring strain.

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Typical Strain Gage ValuesHere are some typical values for resistance, strain gage factor, and strain, along with the predictedvalues of change in resistance:

The electrical resistance, R, of a strain gage (with no strain applied) is typically either or . The most widely used commercially available strain gages are .

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The strain gage factor, S, of the metal foil used in strain gages is typically around 2.0.m In typical engineering applications with metal beams, axial strain, , is in the range of 10-6

to 10-3.m

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Using these values and the above equation for resistance as a function of strain and strain gagefactor,

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Notice how small dR is!l For a typical strain gage, the change in resistance expressed as a ratio with R is dR/R = 2 x10-6 to 2 x 10-3.

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This is the main problem when working with strain gages: One cannot simply stick in an ohmmeter to measure the change in resistance, because dR/R is so small. Most ohm meters do not havesufficient resolution to measure changes in resistance that are three to 6 orders of magnitudesmaller than the resistance itself.

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Strain Gage Electronics

As discussed above, since dR/R is very small and difficult to measure directly, electronic circuits must be



designed to measure the change in resistance rather than the resistance itself. Fortunately, there arecircuits available to do just that.The Wheatstone Bridge

A clever circuit to measure very small changes in resistance is called a Wheatstone bridge.l Here is a schematic diagram of a simple Wheatstone bridge circuit:l

As seen in the sketch, a supply voltage is supplied across the bridge, which contains four resistors(two parallel legs of two resistors each in series). The output voltage is measured across the legs inthe middle of the bridge.

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In the analysis here, it is assumed that the measuring device (voltmeter, oscilloscope,computerized digital data acquisition system, etc.) used to measure output voltage Vo has aninfinite input impedance, and therefore has no effect on the circuit.

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The output voltage Vo can be calculated using Ohm's law. Namely,l

How does the Wheatstone bridge work? Well, if all four resistors are identical, i.e. R1 = R2 = R3 =R4, the bridge is balanced since the same current flows through the left leg and the right leg of thebridge. For a balanced bridge, Vo = 0.

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More generally (as can be seen from the above equation), a Wheatstone bridge can be balancedeven if the resistors do not all have the same value, so long as the numerator in the above equationis zero, i.e. if

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In practice, the bridge will not be balanced automatically, since "identical" resistors are notactually identical, with resistance varying by up to several percent. Thus, a potentiometer (variableresistor) is sometimes applied in place of one of the resistors in the bridge so that minoradjustments can be made in order to balance the bridge. An arrow through the resistor indicatesthat its resistance can vary, as sketched below:

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In the above sketch, resistor R2 was arbitrarily chosen to be replaced by a potentiometer, but anyof the four could have been used instead.

Quarter Bridge CircuitTo measure strain, one of the resistors, in this case R3, is replaced by the strain gage. (Note thatone of the other resistors may still be a potentiometer rather than a fixed resistor, but that will notbe indicated on the circuit diagrams to follow.)

Again, an arrow through the resistor indicates that its resistance can vary - this time because R3 isan active strain gage, not a potentiometer.

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With only one out of the four available resistors substituted by a strain gage, as in the aboveschematic, the circuit is called a quarter bridge circuit.

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The output voltage Vo can be calculated from Ohm's law, as given above. Namely,l

Let the initial resistance of the strain gage (with no load) be and let.

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The bridge is therefore initially balanced when R3 = R3i, since R3iR1 - R4R2 = 0, and Vo is thuszero.

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Unbalanced Quarter Bridge Circuit - to measure strainIn normal operation, the Wheatstone bridge is initially balanced as above. Now suppose strain isapplied to the strain gage, such that its resistance changes by some small amount dR3, i.e. R3changes from R3i to R3i + dR3.

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Under these conditions the bridge is unbalanced, and the resulting output voltage Vo will not bezero, but can be calculated as follows:

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Note that the numerator was simplified by invoking the initial conditions, i.e. R3iR1 - R4R2 = 0.The denominator was simplified by recognizing, as pointed out above, that the resistance of astrain gage changes very little as strain is added, i.e. dR3

Finally, keeping to two significant digits (since S and the supply voltage are only given totwo digits),

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Half Bridge CircuitSuppose one mounts two active strain gages on the beam, one at the front and one at the back assketched:

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Also suppose that both strain gages are put into the Wheatstone bridge circuit, as shown:

Note that resistors R1 and R3 have been replaced by the two strain gages.

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Since half of the four available resistors in the bridge are strain gages, this is called a half bridgecircuit.

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After some algebra, assuming that both strain gage resistances change identically as the strain isapplied, it can be shown that

Or, if all the resistors are initially the same (e.g. ), this reduces to

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Compared to the quarter bridge circuit, the half bridge circuit yields twice the output voltage for agiven strain. One can say, alternately, that the sensitivity of the circuit has improved by a factor oftwo.

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One might ask why R1 (rather than R2 or R4) was chosen as the resistor to replace with the secondstrain gage. It turns out that R1 is used for the second strain gage if its strain is of the same signas that of R3.

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To prove the above statement, suppose all four resistors are strain gages with initial values R1i,R2i, etc. The corresponding changes in resistance due to applied strain are dR1, dR2, etc. It can beshown (via application of Ohm's law) that the output voltage varies as follows:

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As can be seen, the terms with dR1 and dR3 are of positive sign, and therefore contribute to apositive output voltage as strain is increased. However, the terms with dR2 and dR4 are of negativesign, and therefore contribute to a negative output voltage as strain is increased.

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In the above beam example, in which both strain gages measure the same strain, it is appropriate tochoose only R1 for the second strain gage. If R2 or R4 had been chosen instead, the output voltagewould not change at all as strain was increased, because of the signs in the above equation! (Thechange in resistance of the two strain gages would cancel each other out.)

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Example - The ME 82 Lab ExperimentAs an example, consider the lab experiment to be conducted in this class. A cantilevered beam isclamped to the lab bench, and a weight is applied at the end of the beam as sketched:

A strain gage is attached on the top surface of the beam, and another is attached at the bottomsurface, as shown.

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As the beam is strained due to the applied force, the top strain gage is stretched (positive axialstrain), but the bottom strain gage is compressed (negative axial strain).

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If the beam is symmetric in cross section, and if the two strain gages are identical, it turns out thatdRbottom = -dRtop.

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In this case, if R1 and R3 were chosen for the two strain gages in the bridge circuit, the Wheatstonel



bridge would remain balanced for any applied load, since dR1 and dR3 would cancel each otherout.Obviously, the half bridge circuit should be constructed in this case with either R3 and R2 or R3and R4 as the two resistors to be substituted by the strain gages.

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The circuit to be built in the lab for this experiment will use R3 for the top strain gage and R4 forthe bottom strain gage, with the Wheatstone bridge circuit wired as shown below:

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Circuit analysis for this case yields

i.e. the two strain gages effectively double the sensitivity of the Wheatstone bridge circuit, whencompared to a quarter bridge circuit.

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Full Bridge CircuitOne can actually substitute strain gages for all four resistors in a Wheatstone bridge. The result iscalled a full bridge circuit, as sketched below:

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Warning: One must be very careful with the signs when wiring a full bridge circuit!l If the wiring is done properly (e.g. R1 and R3 have positive strain, while R2 and R4 have negativestrain), the sensitivity of the full bridge circuit is four times that of a quarter bridge circuit, i.e.

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In general, if n is defined as the number of active gages in the Wheatstone bridge, i.e.n = 1 for a quarter bridge,n = 2 for a half bridge, andn = 4 for a full bridge,then the strain can be generalized to

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On a final note, it is not always necessary to initially balance the bridge. In other words, supposethere is some initial non-zero value of bridge output voltage, namely

This voltage represents the reference output voltage at some initial conditions of the experiment,which may not necessarily even be zero strain.

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One can still calculate the strain by using the output voltage difference rather than the outputvoltage itself, as follows:

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Take a quiz on the material in this learning module.



psu.eduStress, Strain, and Strain Gages

stress, strain, and strain gages primer

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