Strain gages for monitoring forces

June 1, 2009
The most universal electrical device for measuring mechanical quantities is the strain gage. Design options and considerations abound.
The engineering measurement called strain is the deformation per unit of length of a material when force is applied to it. In other words, strain is a ratio of a material's change in length from an initial unstressed reference length. Strain gages are elements that sense this change and convert it into an electrical signal. How? These components change resistance as they are stretched or compressed, similar to how wire behaves: Specifically, when wire is stretched, its cross-sectional area decreases, causing its resistance to increase.

To select the proper strain gage type, pattern and size, a designer must understand how the component to be monitored is loaded, and know the principal stress direction. To measure minute strains, components capable of minute resistance changes must be used.

Several strain gage designs depend on the proportional variance of electrical resistance to strain: piezoresistive or semiconductor gages, carbon-resistive gages, bonded metallic wires, and foil resistance gages. However, bonded resistance strain gages are by far the most widely used in stress analysis. These gages consist of a grid of very fine wire or foil bonded to the backing or carrier matrix. The electrical resistance of the grid varies linearly with strain. In use, the carrier matrix is bonded to the surface, force is applied, and the strain is found by measuring the change in resistance. Bonded resistance strain gages are low in cost, can be made with short gage length, are only moderately affected by temperature changes, have small physical size and low mass, and are quite sensitive to strain.

In a strain gage application, the carrier matrix and adhesive must work together to transmit strains from the specimen to the grid. In addition, they serve as an electrical insulator and heat dissipater.

Potential errors and characteristics

In a stress-analysis application, it is important to examine potential error sources prior to taking data. First of all, some gages can be damaged during installation — so strain gage resistance must be checked prior to applying stress. Electrical noise and interference can also alter readings. Shielded leads and adequately insulating coatings may prevent these problems. A value of less than 500 M Ohms (using an ohmmeter) usually indicates surface contamination.

Strain gage selection considerations include: bridge type, quarter, half, or full Wheatstone bridge; gage pattern, grid length and width; resistance; ribbon leads, solder pads (bondable terminal pads are recommended for lead wire connection) or insulated leads; foil measuring grid, constantan or karma; carrier material; availability. Active grid length in the case of foil gages is the net grid length and does not include end loops or tabs. Manufacturers often design foil and carrier dimensions for optimum strain-gage performance.

The resistance of a strain gage is defined as the electrical resistance measured between the two tinned copper leads or two solder pads. Common nominal strain gage resistance values are 120, 350, 500, and 1,000 Ohms. Gage factor or strain sensitivity GF is defined as the ratio of the fractional resistance change to the fractional change in length (strain) along the gage axis. Gage factor is a dimensionless quantity, nominally 2± tolerance, and is published in specifications for strain gages. The exact gage factor of a production lot is determined by sample measurements and is given on the package of strain gages.

Consider temperature

Reference temperature is the ambient temperature for which technical strain gage data are valid, unless temperature ranges are given. Data quoted for strain gages are based on a reference temperature of 23°C. Temperature characteristics vary: Temperature-dependent changes of specific gage grid resistance (in applied gages) depend on the grid's linear thermal expansion coefficient and specimen material. These resistance changes appear as mechanical strain in the specimen.

Representation of apparent strain as a function of temperature is called the temperature characteristic of the strain gage application. To keep these apparent strain changes as small as possible, each strain gage is matched during the production to a certain linear thermal expansion coefficient. Certain manufacturers offer strain gages with temperature characteristics matched to ferritic steel and aluminum.

The service temperature range is the range of ambient temperature in which strain gage use is permitted — and the range in which there is no risk of permanent changes to measurement properties. Service temperature ranges depend on whether static or dynamic values are to be sensed.

For permitted RMS bridge-energizing voltage, the maximum values quoted are only permitted for appropriate application on materials with good heat conduction (for example, steel of sufficient thickness) if room temperature is not exceeded. In other cases, temperature rise in the measuring grid area may lead to measurement errors. Measurements on plastics and other materials with bad heat conduction require the reduction of the energizing voltage or the duty cycle — pulsed operation.

Note: Compensate for temperature effects on gage resistance and factor. This may require temperature measurement at the gage itself with thermocouples, thermistors, or RTDs. Most metallic gage alloys, however, exhibit nearly linear gage factor variation with temperature that is less than ±1% at 0 to 100°C.

Gage output and readings

Total strain is represented by a change in VOUT. If each gage has the same positive strain, the total is zero and VOUT remains unchanged. Bending, axial, and shear strain are the most common types of strain measured; the actual arrangement of strain gages determines output voltage change and which type of strain is measured.

For example, if positive (or tensile) strain is applied to gages 1 and 3 (as labeled in our illustration on the previous page) and a negative (or compressive) strain to gages 2 and 4, then there are four fully active gages, and output will be four times larger. Therefore, greater sensitivity and resolution are possible when more than one strain gage is used. The following equations show the relationships between stress, strain, and force for bending, axial, shear, and torsional strain.

  • Bending or moment strain is equal to bending stress divided by Young's Modulus of Elasticity:

    εB = oB/E where

    oB = Moment stress = MB/Z

    = Fv × l/Z

    and Fv = Vertical load

    Fv × l = Bending moment

    Z = Sectional modulus — a property of the cross-sectional configuration of the specimen. (For rectangles only, Z = bh2/6.) Strain gages used in the bending strain configuration can be used to determine vertical load Fv — more commonly called a bending beam load cell.

    Fv = E ·εB (Z)/l = E ·εB(bh2/6)/l.

  • Axial strain equals axial stress divided by Young's Modulus.

    EA = oA/E where

    oA = FA/A — axial stress oA equals axial load divided by the cross-sectional area. The cross-sectional area for rectangles equals b × d. Therefore, strain gages used in axial configurations can be used to determine axial loads F:

    F (axial) = E ·εAbh

  • Shear strain equals shear stress divided by modulus of shear stress:

    γ = τ/G and

    Shear stress τ = Fv × Q/bI where

    Q = Area moment about neutral axis

    b = Thickness

    I = Moment of inertia

    Both the moment of area Q and the moment of inertia I are functions of the specimen's cross-sectional geometry. For rectangles only Q = bh2/8 and I = bh3/12.

    Shear strain γ is determined by measuring the strain (as illustrated on the previous page) at a 45° angle:

    γ = 2 × ε at 45°

    The modulus of shear strain G = E/2 (1 + μ). Therefore, strain gages used in a shear strain configuration can be used to determine vertical loads Fv — an arrangement more commonly referred to as a shear beam load cell.

    Fv = γ · γ b I/Q

    = G · γ b (bh3/12)/(bh2/8)

    = G · γ bh · 2/3.

  • Torsional strain equals torsional stress τ divided by torsional modulus of elasticity G.

    γ = 2 × ε at 45° = τ/G and

    Torsional stress τ = Mt d/2/J where

    Mt = Torque

    d/2 = Distance from the center of the section to the outer fiber

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.

The information in this feature is courtesy of and copyrighted by Omega Engineering Inc. For the full Omega technical library and additional information, visit or, or call (800) 872-3963.

Wheatstone bridge

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.


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 type Gage position Sensitivity mV/V at 1,000 µε Output per µε at 10-V excitation Temperature compensation Superimposed strain compensation Bending 1/4 1 0.5 5 µV/µε No None 1/2 1, 2 1.0 10 µV/µε Yes Axial Full All 2.0 20 µV/µε Yes Axial Axial strain 1/4 1 0.5 5 µV/µε No None 1/2 1, 2 0.65 6.5 µV/µε Yes None 1/2 1, 3 1.0 10 µV/µε No Bending Full All 1.3 13 µV/µε Yes Bending Shear and torsional 1/2 1, 2 1.0 10 µV/µε at 45°F Yes Axial and bending Full All 2.0 20 µV/µε at 45°F Yes Axial and bending

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.

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