J = Polar moment of inertia — a function of the cross-sectional area. For solid circular shafts, J = π(d)4/32. (The modulus of shear strain G is defined in the preceding discussion on shear stress.) Strain gages can be used to determine torsional moments as shown in this equation:

Mt = τ·J (2/d) = γ·G J (2/d)

= γG·πd 3/16

Ø = Mt L/G(J)

This represents the principle behind every torque sensor.

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Wheatstone bridge

Select image to enlarge.

The Wheatstone bridge configuration is capable of measuring small resistance changes; because of its outstanding sensitivity, this circuit is the most frequently used for static strain measurements. Ideally, the strain gage is the only resistor in the circuit that varies, and then only due to a change in strain on the surface. Right, note the signs associated with each gage 1 through 4.

There are two main methods used to indicate the change in resistance caused by strain on a gage in a Wheatstone bridge. Often, an indicator rebalances the bridge, displaying the change in resistance required in micro-strain. A second method includes installing an indicator calibrated in micro-strain to respond to bridge voltage output. This method assumes a linear relationship between voltage out and strain, an initially balanced bridge, and known VIN. In reality, the VOUT strain relationship is nonlinear, but for strains up to a few thousand micro-strain, this error is not significant.

Construction

Strain gages are one of the most important electrical measurement tools for calculating mechanical quantities; as their name indicates, they are used for the measurement of strain. Strain itself can be tensile and compressive strain, designated by a positive or negative value. Strain gages can be used to detect both expansion as well as contraction.

The strain of a design body is always caused by an external influence or an internal effect, in turn caused by forces, pressures, moments, heat, structural material changes, and the like. If certain conditions are fulfilled, the amount or value of the influencing quantity can be derived from the measured strain value. In experimental stress analysis this feature is widely used. Experimental stress analysis uses the strain values measured on the surface of a specimen or structural part to state the stress in the material and also to predict its safety and endurance. Special transducers can be designed for the measurement of forces or other derived quantities — moments, pressures, accelerations, displacements, and vibrations. These transducers generally contain a pressure-sensitive diaphragm with strain gages bonded to it.

Bridge configuration versus output

Bridge configuration affects output, temperature compensation, and compensation of superimposed strains. Here we assume a gage factor of 2.0, Poisson's ratio of 0.3, and no lead wire resistance. This chart is quite useful in determining the meter sensitivity required to read strain values. Temperature compensation as in many of the above configurations is where the gage's thermal expansion coefficient does not have to match the specimen's thermal expansion coefficient. For this reason, many strain gages, regardless of temperature characteristics, can be used with any specimen material. Quarter bridges can have temperature compensation if a dummy gage is used — a strain gage used in place of a fixed resistor. Temperature compensation is achieved when this dummy gage is mounted on a piece of material similar to the specimen which undergoes the same temperature changes as does the specimen, but which is not exposed to the same strain. Strain temperature compensation is not the same as load (stress) temperature compensation, because Young's Modulus of Elasticity varies with temperature. Note: Shear and torsional strain = 2 × ε @ 45° — and the gage position numbers listed here refer to the locations numbered in figure Strain types on the previous page.