Strain gauges are used to measure small deflections in objects due to applied loads. Strain measurements are the basis for predicting how long an object will perform without failure due to these loads.
This article explains how strain gauges work, relevant stress and strain background information, and considerations for their proper usage and applications.
Table of Contents
1. What Is Stress? 2. What is Strain? 3. Why Measure Stress and Strain? 4. What is a Strain Gauge? 5. Measurement Principles 6. Wheatstone Bridge 7. Practical Example 8. Basic Strain Gauge Configurations 8.1 Quarter Bridge 8.2 Half Bridge 8.3 Full Bridge 9. Directionality 10. Invention of the Strain Gauge
1. What is Stress?
To understand the goal of strain gauge measurements, it is useful to understand the concepts of stress and strain.
Stress is the internal response of a body to an applied force (Figure 1).
Figure 1: A Load (F) Applied to a Metal Cylinder of Cross Section (A)
Stress is equal to the applied force divided by the cross-sectional area of the object (Equation 1).
Equation 1: Stress equals force divided by area
s is the stress expressed in units of Pa or psi.
F is the applied force.
A is the cross-sectional area of the body.
2. What is Strain?
In response to the applied force the body will deform. It will elongate in the direction of the applied force (Figure 2).
Figure 2: Strain is the change in length divided by the original length
The change in length of the body divided by the original length is called the strain (Equation 2).
Equation 2: Strain equals change in length over original length
e is the strain.
Dl is the change in length of the cylinder.
l is the original length of the cylinder.
3. Why Measure Stress and Strain?
Stress is an important quantity in durability analysis. High stresses can cause fatigue damage and product failure. Parts are typically designed to have a certain “design life”. For example, automotive manufactures may offer a 10 year or 100,000 mile powertrain warranty. They will design the powertrain in such a way that it can withstand the stresses caused by operation for at least the lifetime of the warranty.
If the stress on a part is below the target, then the designer can work on improving the design by removing material to improve other attributes (reduce manufacturing costs, reduce weight, improve fuel economy, etc.).
Stress is a difficult quantity to measure. Some sensors directly measure stress (concrete stress sensors, soil stress sensors), but typically stress is measured indirectly by measuring the strain on an object, and then converting it to stress using Hooke’s law (Equation 3).
Equation 3: Stress equals Young's Modulus times strain when in the elastic region of a material
s is the stress.
E is Young's Modulus (a material property).
ε is the strain.
In the elastic region of a material there is a linear relationship between stress and strain (Figure 3). The slope of the stress-strain curve in the linear region is Young’s modulus.
Figure 3: In the elastic region of a material, the stress and strain have a linear relationship characterized by Young’s Modulus.
Using the resulting stress or strain time histories, the fatigue life of the object of interest can be predicted. See the Knowledge articles “Strain Life Approach” and “What is a SN-Curve” for more information.
4. What is a Strain Gauge?
Strain is measured using a sensor called a strain gauge. A strain gauge is a uniaxial transducer that is typically made of a thin metallic wire bent into a rectangular grid. Strain gauges work by relating the change in resistance of the gauge to the strain in an object.
Important characteristics of the strain gauge are highlighted in Figure 4.
Figure 4: A strain gauge consists of a thin resistive wire which covers a measurement grid area.
Thin Resistive Wire: The “strain sensing” element of a strain gauge is a thin resistive wire that is bent into a rectangular shape.
Active Length: A strain gauge is a uniaxial sensor. It measures the strain in the direction of the active length of the gauge. If the strain field of the test object is known the strain gauge should be laid in the direction of the maximum principal strain. If the strain field is unknown a strain gauge rosette can be used.
Measurement Grid Area: A strain gauge measures the average strain over the measurement grid area. A smaller grid area will result in a more localized strain measurement. However, gauges with smaller grid areas are worse at dissipating heat. This can affect gauge performance.
Transverse Length: Strains acting in a direction other than the active direction of the gauge will introduce errors into the strain measurement. The magnitude of this error is related to the transverse sensitivity of the gauge (i.e. how responsive is the gauge to strains in directions other than the principle direction).
5. Measurement Principles
A strain gauge uses two principles to measure strain. The first is that the resistance of a wire depends on its length and cross-sectional area. If the length or cross-sectional area of the wire changes due to an applied load, then its resistance will change. The resistance of a piece of wire is described by Equation 4.
Equation 4: Change in resistance (R) in a wire based on length, area, and material resistivity.
R is the electrical resistance of the wire expressed in Ohms.
L is the length of the conductor in meters.
r is the resistivity of the material. Resistivity is an intrinsic material property that varies with temperature.
A is the cross-sectional area of the wire.
The second is that the change in resistance of the wire can be related to strain by a property known as the gauge factor (Equation 5).
Equation 5: Gauge factor (GF) equation for strain gauge.
GF is the gauge factor of the strain gauge.
DR/R is the change in resistance of the gauge divided by the nominal resistance.
e is the measured strain.
The nominal resistance of a strain gauge is typically 120 or 350 ohms, although other resistances exist.
When a strain gauge is placed on a test article it is attached in such a way that the strain experienced by the test article is transferred to the gauge.
When a load is applied to the test article, it will deform. The attached strain gauge will also deform. The thin resistive wire will either elongate (tension) or shorten (compression). Because the length and cross-sectional area of the wire has changed, it’s resistance will also change. Equations 4 and 5 can then be used to calculate the measured strain.
6. Wheatstone Bridge
To measure strain, it is required to measure the change in resistance of the strain gauge. In order to measure strain, the change in resistance needs to be converted into a voltage. This is accomplished using a Wheatstone bridge.
A Wheatstone bridge is a voltage divider circuit which can detect small changes in resistance. Two different portrayals of the Wheatstone bridge are shown in Figure 5. The Wheatstone bridge is typically drawn as shown on the left. The schematic on the right may be easier for someone without an electrical background to understand.
Figure 5: Equivalent schematics (right and left) for a Wheatstone bridge consisting for four resistors (R1, R2, R3, R4), a Voltage supply, and a Voltage output.
The bridge is a circuit which consists of a supply voltage and four resistors. The voltage across the bridge, labeled Vout, will be measured.
When a voltage is supplied to the bridge, Vout will be zero if R1 = R2 = R3 = R4. This is called a balanced bridge. Now, imagine R1 is replaced with a strain gauge, G1, that initially has the same resistance as R2, R3, and R4 (Figure 6). If no load is applied to the bridge, the resistance of the strain gauge will not change. Therefore, Vout is still zero.
Figure 6: Wheatstone Bridge with resistor R1 replaced by one Strain Gauge (Yellow, G1)
If the strain gauge experiences a force it will deform. That deformation will cause a small change in resistance of the gauge. Because the resistance has changed, the bridge is no longer balanced, and a voltage will be measured at Vout. This voltage can be calculated according to Equation 6.
Equation 6: Prediction of voltage output using bridge resistors and voltage supply of a wheatstone bridge.
R1, R2, R3, and R4 are the resistances of the Resistors 1, 2, 3, and 4
VOut is the voltage measured across points 1 and 2
VSupply is the voltage supplied to the Wheatstone bridge.
The strain can then be calculated using Equation 7:
Equation 7: Strain output of gauge.
Vout/Vsupply is the ratio of the output voltage to the supply voltage
GF is the gauge factor of the strain gauge.
N is the number of gauges in the Wheatstone bridge.
e is the strain measured by the Wheatstone bridge.
7. Practical Example
A single strain gauge is placed in a Wheatstone bridge (Figure 7). G1, R2, R3, and R4 all have a nominal resistance of 350 Ohms. The supply voltage into the circuit is 5 volts. When a load is applied the resistance of G1 changes by positive 0.07 ohms.
Calculate the output voltage of the Wheatstone bridge and the measured strain.
Figure 7: Strain gauge diagram
V_out can be found by rearranging Equation 6 and solving:
After computing the output voltage, the measured strain can be computed using Equation 7:
Rearranging and solving yields:
Because the measured strain is small and dimensionless, a common way of reporting the value is to multiply it by 10e6 and display the units as mm/m or me:
100 mm/m or 100 me
This example highlights several important aspects of strain gauges:
The output voltage measured from the Wheatstone bridge is very small, usually microvolts. Due to these low levels, gauge measurements can be susceptible to electro-static interference. See the Knowledge article “Simcenter Testlab and Long Strain Cables” for more information.
The output from the Wheatstone bridge can be increased by either increasing the supply voltage or configuring the bridge with more active strain gauges. High supply voltages can cause excessive heating on the gauge and introduce errors due to thermal strain. To choose an appropriate supply voltage consult this community article "Strain Gauges: Selecting an Excitation Voltage".
Strain sensors are ratiometric. Most sensors output a voltage that is proportional to an engineering quantity (e.g. mV/g for an accelerometer). A strain gauge outputs a voltage ratio (Vout/Vsupply) that is proportional to strain (mV/V/e).
8. Basic Strain Gauge Configurations
In order to measure with a strain gauge there must be a complete Wheatstone bridge. If only a single strain gauge is used, as shown below in Figure 8, the rest of the bridge must be completed in the data acquisition system.
Figure 8: Resistors on the measurement system (Simcenter SCADAS) are used to complete the Wheatstone bridge used in strain gauge measurments.
There are three basic strain gauge configurations, all based on the Wheatstone Bridge Circuit. They are the quarter-bridge, half bridge, and whole bridge.
The number and orientation of strain gauges in the Wheatstone bridge will determine the sensitivity of the bridge, the types of strain that can be measured (bending, axial, torsion), and ability to compensate for temperature and electromagnetic interference.
8.1 Quarter Bridge
A quarter bridge is depicted below. It has one (1) active strain gauge and three (3) high precision resistors (Figure 9). A quarter bridge is the most common type of strain gauge setup because it only requires instrumenting one gauge.
Figure 9: Quarter bridge configuration consists of one active gauge and three additional resistors.
However, the quarter bridge has a lower sensitivity than the other configurations and it cannot compensate for errors due to thermal effects on the gauge. A quarter bridge can only measure the superimposed strain on an object. It cannot separate bending and axial strains.
8.2 Half Bridge
A half bridge has two (2) active strain gauges and two (2) high precision resistors (Figure 10). A half bridge requires more effort to instrument than a quarter bridge but has a higher measurement sensitivity.
Figure 10: Half bridge configuration consists of two active gauge and two additional resistors.
Depending on the orientation of the gauges, a half bridge can compensate for thermal strains and measure bending and axial strain.
8.3 Full Bridge
A full bridge has four (4) active strain gauges (Figure 11). A full bridge requires the most effort to instrument but has the highest sensitivity. A full bridge can compensate for thermal effects and employ common mode rejection to reduce the effects of electromagnetic interference.
Figure 11: Full bridge configuration consists of four active gauges.
When properly configured a full bridge can be instrumented to measure bending or axial stresses and can be used to measure the torque on a shaft. Full bridges form the basis of more complex sensors like load cells or pressure transducers.
Strain gauges measure the strain along their active direction. It is very important to consider the orientation of the gauge when instrumenting a test object. An example is shown below in Figure 12.
Figure 12: Left – Strain gauge will measure compressive strain, Right – Compressive strain will not be measured correctly by the gauge.
The cylinder is subjected to a compressive force which will cause a compressive strain in the vertical direction. The gauge on the left-hand side will correctly measure the strain caused by the compressive load because the active length of the gauge is in the same direction as the strain.
The gauge on the right will not measure the correct compressive strain. It will measure the poisson strain due to the applied load and some strain due to the transverse sensitivity of the gauge. It will not measure the total compressive strain due to the applied force. Because of the importance of strain gauge orientation, strain gauge rosettes are commonly used to measure strain fields where the direction of the applied load is unknown.
10. Invention of the Strain Gauge
Modern bonded wire resistant strain gauge (Figure 13) was invented by Edward E. Simmons of the California Institute of Technology (Caltech) and Arthur C. Ruge from the Massachusetts Institute of Technology (MIT) independently in 1938.
Figure 13: Bonded wire resistant stain gauges in rosette formation.
MIT released the rights to Ruge's invention, saying that, while “interesting,” the strain gauge didn’t show much potential.