Topic: Cantilever Beam and Wheatstone Bridge
Background on Strain Gages and Wheatstone Bridges
A strain gage is a specific (and the most common) type of resistive sensor. A resistive sensor is a based
on a principle that a change in some physical variable will produce a change in the resistance of a wire.
By measuring this change in resistance, the student can determine the corresponding change in the
physical variable.
In this lab, the physical variable of interest is strain, which is defined as
DL
(1)
e= .
L
The resistance of a wire is given by
L
(2)
R=r ,
A
where  is the resistivity (-m) of the material, L is the nominal length of the wire, and A is the nominal
cross section of the wire. Note that the resistivity of the wire is a constant property of the wire material,
but the resistance of the wire is a function of the wire geometry. Therefore it should be fairly obvious
that a change in the length of the wire (ΔL) will produce a change in the resistance. For more details
about the theory of strain gages, consult Section 3.2.2 of the course book.
Strain gages can be used in a number of various applications. For instance, the stress on a material is not
directly measureable but essential to predicting material failure. However, if strain can be measured
then the stress can be determined by using Hooke’s law (for example). In practical usage, a load is
applied to a strain gage that causes a deflection ΔL, and a measurement system provides the user with a
ΔR reading. However, in order to determine the actual ΔL, the user must know the relationship between
ΔR and ΔL. The strain gage must be calibrated by applying known deflections ΔL and measures the
corresponding values of ΔR. A curve fit of this calibration data generates a calibration curve, which
provides the equation that relates ΔR and ΔL. If the two properties are linearly related, the calibration
curve will take the generic form of
æ DR ö
DL
(3)
= a0 + a1ç ÷,
è Rø
L
where a0 and a1 are some constants. This calibration curve can then be used to determine some
unknown deflection. The local gage factor is the instantaneous slope of this curve, whereas the
engineering gage factor is the slope over the entire region of strain investigated (essentially the
DR R
sensitivity) and is defined as Ge =
.
DL L
In many applications strain gages are used in a Wheatstone bridge, which consists of four resistors in an
electrical circuit. Any one (or some or all) of these resistors may be replaced with a strain gage(s) and
the resulting circuit can be used to measure deflections. If one is replaced it is known as a quarter
bridge, if two are replace a half bridge, and if all four resistors are strain gages it is called a full bridge.
A common application is using strain gages on a cantilever beam in a Wheatstone bridge configuration
to measure an unknown force, such as weight, based on the deflection of the beam. Figure 1 shows a
schematic of a four-arm bridge, and a similar configuration is used here.
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Figure 1 – Cantilever beam with four strain gages.
As can be seen, strain gages 1 and 4 are on top of the beam and strain gages 2 and 3 are on the bottom of
the beam. Therefore strain gages 1 and 4 experience a tensile strain (are stretched) and strain gages 2
and 3 experience a compressive strain. If the relationship between strain and resistance is linear, then
under some load F the changes in resistance will be
R1¢ = R1 + dR1 ü
(4)
ýtensile
R4¢ = R4 + dR4 þ
and
R2¢ = R2 - dR2 ü
(5)
ýcompressive.
R3¢ = R3 - dR3 þ
As discussed in Section 2.5 of the course book, when the four strain gages have an equal nominal
resistance (i.e., R1 = R2 = R3 = R4 = R) then the deflection method Wheatstone bridge equation reduces to
the linear equation
æ dR ö
(3)
E o = E iç ÷ µ F .
èRø
Using known weights, a calibration curve can be established that relates the weight W to the output
voltage on a digital meter Eo,
(4)
E o = a0 + a1W ,
where a0 and a1 are some constants. Once an unknown weight is known, an unknown mass or density
easily follows.
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