Practical Strain Measurements
Introduction
With today’s emphasis on product liability and energy efficiency, designs must not only be lighter and stronger, but also more thoroughly tested than ever before. This places new importance on the subject of experimental stress analysis and the techniques for measuring strain. The main theme of this application note is aimed at strain measurements using bonded resistance strain gages. We will introduce considerations that affect the accuracy of this measurement and suggest procedures for improving it. We will also emphasize the practical considerations of the strain gage measurement, with an emphasis on computer controlled instrumentation.
Stress and Strain
The relationship between stress and strain is one of the most fundamental concepts from the study of mechanics of materials and is of paramount importance to the stress analyst. In experimental stress analysis we apply a given load and then measure the strain on individual members of a structure or machine. Then we use the stress strain relationships to compute the stresses in those members to verify that these stresses remain within the allowable limits for the particular materials used.
Strain
When a force is applied to a body, the body deforms. In the general case this deformation is called strain. In this application note we will be more specific and define the term STRAIN to mean deformation per unit length or fractional change in length and give it the symbol (ε).
This is the strain that we typically measure with a bonded resistance strain gage. Strain may be either tensile (positive) or compressive (negative). See Figure 2. When written in equation form ε = ΔL/L , we see that strain is a ratio and, therefore, dimensionless. To maintain the physical significance of strain, it is often written with units of inch/inch. For most metals the strains measured in experimental work are typically less than 0.005000 inch/inch.
Since practical strain values are so small, they are often expressed in microstrain which is ε x 10-6 (note this is equivalent to parts per million or ppm) and is expressed
by the symbol με. Still another way to express strain is as percent strain, which is ε x 100. For example: 0.005 inch/inch = 5000 με = 0.5%.
As described to this point, strain is fractional change in length and is directly measurable. Strain of this type is also often referred to as normal strain.
Shearing Strain
Another type of strain, called SHEARING STRAIN is a measure of angular distortion. Shearing strain is also directly measurable, but not as easily as normal strain.
If we had a thick book sitting on a table top and we applied a force parallel to the covers, we could see the shear strain by observing the edges of the pages. See
Figure 3. Shearing strain, γ, is defined as the angular change in radians between two line segments that were orthogonal in the undeformed state. Since this angle is very small for most metals, shearing strain is approximated by the tangent of the angle.
Poisson Strain
A bar is shown with a uniaxial tensile force applied, like the bar in Figure 1. The dashed lines show the shape of the bar after deformation, pointing out another phenomenon, that of Poisson strain. The dashed lines indicate that the bar not only elongates, but that its girth contracts. This contraction is a strain in the transverse direction due to a property of the material known as Poisson’s ratio. Poisson’s ratio, υ, is defined as the negative ratio of the strain in the transverse direction to the strain in the longitudinal direction. It is interesting to note that no stress is associated with the Poisson strain. Referring to Figure 4, the equation for Poisson’s Ratio is υ = - εt / ε1. Note that υ is dimensionless
Normal Stress
While forces and strains are measurable quantities used by the designer and stress analyst, stress is the term used to compare the loading applied to a material with
its ability to carry the load. Since it is usually desirable to keep machines and structures as small and light as possible, the parts should be stressed, in service, to
the highest permissible level. STRESS refers to force per unit area on a given plane within a body. The bar in Figure 5 has a uniaxial tensile force, F, applied along the x-axis. If we assume the force to be uniformly distributed over the cross-sectional area, A, the average stress on the plane of the section is F/A. This stress is perpendicular to the plane and is called NORMAL STRESS, σ expressed in equation form, σ = F/A. and has units of force per unit area. Since the normal stress is in the x-direction and there is no component of force in the y-direction, there is no normal Stress in that direction. The normal stress is in the positive x-direction and is tensile.
Shear Stress
Just as there are two types of strains, there is also a second type of stress called SHEAR STRESS. Where normal stress is normal to the designated plane, shear
stress is parallel to the plane and is expressed by the symbol, τ. In the example shown in Figure 5, there is no y-component of force, therefore, no force parallel to
the plane of the section, so there is no shear stress on that plane. Since the orientation of the plane is arbitrary, what happens if the plane is oriented other than normal to the line of action of the applied force? Figure 6 demonstrates this concept with a section taken on the n-t coordinate system at some arbitrary angle, φ, to the
line of action of the force.
We see that the force vector, F, can be broken into two components, Fn and Ft, that are normal and parallel to the plane of the section. This plane has a cross- sectional area of A’ and has both normal and shear stresses applied. The average normal stress, σ, is in the n-direction and the average shear stress, τ is in the t-direction. Their equations are: σ = Fn/A’ and τ = Ft/A’. Note that it was the force vector that was broken into components, not the stresses, and that the resulting stresses are a function of the orientation of the section. This means that stresses (and strains), while having both magnitude and direction, are not vectors and do not follow the laws of vector addition, except in certain special cases, and they should not be treated as such. We should also note that stresses are derived quantities, computed from other measurable quantities, and are not directly measurable.
Principal Axes
In the preceding examples the x-y axes are also the PRINCIPAL AXES for the uniaxially loaded bar. By definition, the principal axes are the axes of maximum
and minimum normal stress. They have the additional characteristic of zero shear stress on the planes that lie along these axes. In Figure 5 the stress in the x-direction is the maximum normal stress, and we noted that there was no force component in the y-direction and therefore zero shear stress on the plane. Since there is no force in the y-direction, there is zero normal stress in the y-direction, and in this case zero is the minimum normal stress. So the requirements for the principal axes are met by the x-y axes. In Figure 6 the x-y axes are the principal axes since that bar is also loaded uniaxially. The n-t axes in Figure 6 do not meet the zero shear stress requirement of the principal axes. The corresponding STRAINS on the principal axes are also maximum and minimum and the shear strain is zero.
The principal axes are very important in stress analysis because the magnitudes of the maximum and minimum normal stresses are usually the quantity of interest.Once the principal stresses are known, the normal and shear stresses in any orientation may be computed. If the orientation of the principal axes is known, through knowledge of the loading conditions or experimental techniques, the task of measuring the strains and computing the stresses is greatly simplified.
In some cases we are interested in the average value of stress or load on a member, but often we want to determine the magnitude of the stresses at a point. The material will fail at the point where the stress exceeds the load-carrying capacity of the material. This failure may occur because of excessive tensile or compressive
normal stress or excessive shearing stress. In actual structures, the area of this excessive stress level may be quite small. The usual method of diagramming the
stress at a point is to use an infinitesimal element that surrounds the point of interest. The stresses are then a function of the orientation of this element and, in one particular orientation, the element will have its sides parallel to the principal axes. This is the orientation that gives the maximum and minimum normal stresses on the point of interest.
Stress-Strain Relationships
Now that we have defined stress and strain, we need to explore the stress-strain relationship. It is this relationship that allows us to calculate stresses from the
measured strains. If we have a bar made of mild steel and incrementally load it in uniaxial tension and plot the strain versus the normal stress in the direction of
the applied load, the plot will look like the stress-strain.
We can see that, up to a point called the proportional limit, there is a linear relationship between stress and strain. Hooke’s Law describes this relationship. The slope of this straight line portion of the stress-strain diagram is the MODULUS OF ELASTICITY or YOUNG’S MODULUS for the material. The modulus of elasticity, E, has the same units as stress (force per unit area) and is determined experimentally for
materials. Written in equation form this stress-strain relationship is σ = E * ε. Some materials, for example, cast iron and concrete, do not have a linear portion
to their stress-strain diagrams. To do accurate stress analysis studies for these materials it is necessary to determine the stress-strain properties, including Poisson’s ratio, for the particular material on a testing machine. Also, the modulus of elasticity may vary with temperature. This variation may need to be experimentally determined and considered when performing stress analysis at temperature extremes. There are two other points of interest on the stress- strain diagram in Figure 7, the yield point and the ultimate strength value of stress. The yield point is the stress level at which strain will begin to increase rapidly with little or no increase in stress. If the material is stressed beyond the yield point, and then the stress is removed, the material will not return to its original size but will retain a residual offset or strain.
The ultimate strength is the maximum stress developed in the material before rupture.
The examples we have examined to this point have been examples of uniaxial forces and stresses. In experimental stress analysis the biaxial stress state is the most common. Figure 8 shows an example of a shaft with both tension and torsion applied. The point of interest is surrounded by an infinitesimal element, Figure 8 with its sides oriented parallel to the x-y axes. The point has a biaxial stress state and a triaxial strain state (remember Poisson’s ratio). The element, rotated to be aligned with the principal (p-q) axes, is also shown in Figure 8. Figure 9 shows the element removed with arrows added to depict the stresses at the point for both orientations of the element. We see that the element oriented along the x-y axes has a normal stress in the x-direction, zero normal stress in the y-direction, and shear stresses on its surfaces. The element rotated to the p-q axes orientation has normal stress in both directions but zero shear stress as it should, by definition, if the p-q axes are the principal axes. The normal stresses, σp and σq, are the maximum and minimum normal stresses for the point. The strains in the p-q direction are also the maximum and minimum, and there is zero shear strain along these axes. Appendix C gives the equations relating stress to strain for the biaxial stress state. If we know the orientation of the principal axes, we can then measure the strain in those directions and compute the maximum and minimum normal stresses and the maximum shear stress for a given loading condition. We don’t always know the orientation of the principal axes, but if we measure the strain in three separate directions, we compute the strain in any direction including the principal axes directions. Three and four element rosette strain gages are used to measure the strain when the principal axes orientation is unknown. The equations for computing the orientation and magnitudes of the principal strains from 3-element rosette strain data are found in Appendix C.
Measuring Strain
Stress in a material can’t be measured directly. It must be computed from other measurable parameters. Therefore, the stress analyst uses measured strains
in conjunction with other properties of the material to calculate the stresses for a given loading condition. There are methods of measuring strain or deformation
based on various mechanical, optical, acoustical, pneumatic, and electrical phenomena. This section briefly describes several of the more common methods
and their relative merits.
Gage Length
The measurement of strain is the measurement of the displacement between two points some distance apart. This distance is the GAGE LENGTH and is an important
comparison between various strain measurement techniques. Gage length could also be described as the distance over which the strain is averaged. For example, we could (on some simple structure such as the part in Figure 10) measure the part length with a micrometer both before and during loading. Then we would subtract the two readings to get the total deformation of the part. Dividing this total deformation by the original length would yield an average value of strain for the entire part.
The gage length would be the original length of the part. If we used this technique on the part in Figure 10, the strain in the reduced width region of the part would be locally higher than the measured value because of the reduced cross-sectional area carrying the load. The stresses will also be highest in the narrow region and the part will yield there before the measured average strain value indicates a magnitude of stress greater than the yield point of the material.
Ideally, we want the strain measuring device to have an infinitesimal gage length so we can measure strain at a point. If we had this ideal strain gage we would place
it in the narrow portion of the specimen in Figure 10 to measure the high local strain in that region. Other desirable characteristics for this ideal strain measuring device are small size and mass, easy attachment, high sensitivity to strain, low cost, and low sensitivity to temperature and other ambient conditions.
Mechanical Devices
The earliest strain measurement devices were mechanical in nature. We have already considered an example using a micrometer to measure strain and observed a problem with that approach. Extensometers are a class of mechanical devices used for measuring strain that employ a system of levers to amplify the minute strains to a level that can be read. A minimum gage length of 1/2 inch and a resolution of about 10 με is the best that can be achieved with purely mechanical devices. The addition of light beam and mirror arrangements to extensometers improves resolution and shortens gage length, allowing 2 με resolution and gage lengths down to 1/4 inch.
Still another type of device, the photoelectric gage, uses a combination of mechanical, optical, and electrical amplification to measure strain. This is done by using a light beam, two fine gratings, and a photocell detector to generate an electrical current that is proportional to strain. This device comes in gage lengths as short as 1/16 inch but it is costly and delicate. All of these mechanical devices tend to be bulky and cumbersome to use, and most are only suitable for static strain measurements.
Optical Methods
Several optical methods are used for strain measurement. One of these techniques uses the interference fringes produced by optical flats to measure strain. This device is sensitive and accurate but the technique is so delicate that laboratory conditions are required for its use.
Brittle Coatings
Brittle coating techniques are another way to indicate static strain. They are often used in conjunction with strain gages. The test object is coated with a brittle lacquer and the load is applied in increments, when possible. The lacquer will crack first in the region of highest surface strain and the cracks will be perpendicular to the highest tensile strain. A representation of the full strain field is obtained indicating where to locate the strain gages and in what orientation. Under favorable conditions a reasonable estimate of the magnitude of the strain can be obtained. Getting goad data from brittle lacquer coatings is as much art as science. For high-temperature applications, a brittle ceramic coating may be used instead of the lacquer.
Electrical Devices
Another class of strain measuring devices depends on electrical characteristics which vary in proportion to the strain in the body to which the device is attached.
Capacitance and inductance strain gages have been constructed but sensitivity to vibration, mounting difficulties, and complex circuit requirements keep them from being very practical for stress analysis work. These devices are, however, often employed in transducers. The piezoelectric effect of certain crystals has also been
used to measure strain. When a crystal strain gage is deformed or strained, a voltage difference is developed across the face of the crystal. This voltage difference is proportional to the strain and is of a relatively high magnitude. Crystal strain gages are fairly bulky, very fragile, and not suitable for measuring static strains.
Probably the most important electrical characteristic, which varies in proportion to strain, is that of electrical resistance. Devices whose output depend on this
characteristic are the piezoresistive or semiconductor gage, the carbon-resistor gage, and the bonded metallic wire and foil resistance gages. The carbon resistor gage is the forerunner of the bonded resistance wire strain gage. It is low in cost, can have a short gage length and is very sensitive to strain. A high sensitivity to temperature and humidity are the disadvantages of the carbon-resistor strain gage.
The semiconductor strain gage is based on the piezoresistive effect in certain semiconductor materials such as silicon and germanium. Semiconductor gages have elastic behavior and can be produced to have either positive or negative resistance changes when strained. They can be made physically small while still maintaining a high nominal resistance. The strain limit for these gages is in the 1000 to 10000 με range with most tested to 3000 με in tension. Semiconductor gages exhibit a high sensitivity to strain but the change in resistance with strain is nonlinear. Their
resistance and output are temperature sensitive and the high output, resulting from changes in resistance as large as 10-20%, can cause measurement problems when using the devices in a bridge circuit. However, mathematical corrections for the temperature sensitivity, the nonlinearity of output, and the nonlinear characteristics of the bridge circuit (if used), can be made automatically when using computer controlled instrumentation to measure strain with semiconductor gages. They can be used to measure both static and dynamic strains. When measuring dynamic strains, temperature effects are usually less important than for static strain measurements, and the high output of the semiconductor gage is an asset.
The bonded resistance strain gage is by far the most widely used strain measurement tool for today’s experimental stress analyst. It consists of a grid of very fine wire, or more recently, of thin metallic foil bonded to a thin insulating backing called a carrier matrix. The electrical resistance of this grid material varies linearly with strain. In use, the carrier matrix is attached to the test specimen with an adhesive. When the specimen is loaded, the strain on its surface is transmitted to the grid material by the adhesive and carrier system. The strain in the specimen is found by measuring the change in the electrical resistance of the grid material. The bonded resistance strain gage is low in cost, can be made with a short gage length, is only moderately affected by temperature changes, has small physical size and low mass, and has fairly high sensitivity to strain. It is suitable for measuring both static and dynamic strains. The remainder of this application note deals with the instrumentation considerations for making accurate, practical strain measurements using the bonded resistance strain gage.
The Bonded Resistance Strain Gage
The term “bonded resistance strain gage” can apply to the nonmetallic (semiconductor) gage or to the metallic (wire or foil) gage. Wire and foil gages operate on the same basic principles and both can be treated in the same fashion from the measurement standpoint. The semiconductor gage, having a much higher sensitivity to strain than metallic gages, can have other considerations introduced into its measurement. We will use the term STRAIN GAGE or GAGE to refer to the BONDED METALLIC FOIL GRID RESISTANCE STRAIN GAGE throughout the rest of this application note. These foil gages are sometimes referred to as metal film gages.
Strain gages are made with a printed circuit process using conductive alloys rolled to a thin foil. The alloys are processed, including controlled atmosphere heat treating, to optimize their mechanical properties and temperature coefficient of resistance. A grid configuration for the strain sensitive element is used to allow higher values of gage resistance while maintaining short gage lengths. Gage resistance values range from (30 to 3000) Ω, with 120 Ω and 350 Ω being the most commonly used values for stress analysis. Gage lengths from 0.008 inch to 4 inches are commercially available. The conductor in a foil grid gage has a large surface area for a given cross-sectional area. This keeps the shear stress low in the adhesive and carrier matrix as the strain is transmitted by them. This larger surface area also allows good heat transfer between grid and specimen. Strain gages are small and light, will operate over a wide temperature range, and can respond to both static and dynamic strains. They have wide application and acceptance in transducers as well as stress analysis.
In a strain gage application, the carrier matrix and the adhesive must work together to faithfully transmit the strains from the specimen to the grid. They also serve as an electrical insulator between the grid and the specimen and must transfer heat away from the grid. Three primary factors influencing gage selection are operating temperature, state of strain (including gradients, magnitude and time dependence), and stability requirements for the gage installation. The importance of selecting the proper combination of carrier material, grid alloy, adhesive, and protective coating for the given application cannot be over- emphasized. Strain gage manufacturers are the best source of information on this topic and have many excellent publications to assist the customer in selecting the proper strain gages, adhesives, and protective coatings.
Gage Factor
When a metallic conductor is strained, it undergoes a change in electrical resistance, and it is this change that makes the strain gage a useful device. The measure of this resistance change with strain is GAGE FACTOR,GF. Gage factor is defined as the ratio of the fractional change in resistance to the fractional change in length (strain) along the axis of the gage. Gage factor is a dimensionless quantity and the larger the value, the more sensitive the strain gage.
It should be noted that the change in resistance with strain is not just due to the dimensional changes in the conductor, but that the resistivity of the conductor material also changes with strain. The term gage factor applies to the strain gage as a whole, complete with carrier matrix, not just to the strain sensitive conductor. The gage factor for constantan and nickel-chromium alloy strain gages is nominally 2 and various gage and instrumentation specifications are usually based on this nominal value.
Transverse Sensitivity
If the strain gage were a single straight length of conductor, and of small diameter with respect to its length, it would respond to strain along its longitudinal axis and be essentially insensitive to strain perpendicular or transverse to this axis. For any reasonable value of gage resistance, it would also have a very long gage length. When the conductor is in the form of a grid to reduce the effective gage length, there are small amounts of strain sensitive material in the end loops or turnarounds that lie transverse to the gage axis. This end loop material gives the gage a non-zero sensitivity to strain in the transverse direction. TRANSVERSE SENSITIVITY FACTOR, Kt is defined by and is usually expressed in percent. Values of Kt range from 0 to 10%. To minimize this effect, extra material is added to the conductor in the end loops and the grid lines are kept close together. This serves to minimize the resistance in the transverse direction. Correction for transverse sensitivity may be necessary for short, wide grid gages, or where there is considerable misalignment between the gage axis and the principal axis, or in rosette analysis where high transverse strain fields may exist. Data supplied by the manufacturer with the gage can be entered into the computer controlling the instrumentation and corrections for transverse sensitivity made to the strain data as it is collected.
Temperature Effects
Ideally we would prefer the strain gage to change resistance only in response to the stress induced strain in the test specimen, but the resistivity and the strain sensitivity of all known strain sensitive materials vary with temperature. Of course that means the gage resistance and the gage factor will change when the temperature changes. This change in resistance with temperature for a mounted strain gage is a function of the difference in the thermal expansion coefficients between the gage and the specimen and of the thermal coefficient of resistance of the gage alloy. Self- temperature compensating gages may be produced for specific materials by processing the strain sensitive alloy such that it has thermal resistance characteristics that compensate for the effects of the mismatch in thermal expansion coefficients between the gage and the specific material. A temperature compensated gage produced in this manner is accurately compensated only when mounted on a material that has a specific coefficient of thermal expansion. Table 2 is a list of common materials for which self temperature compensated gages are available.
Thermal expansion coefficients of some common materials for which temperature compensated strain gages are available
The compensation is only effective over a limited temperature range because of the nonlinear character of both the thermal coefficient of expansion and the thermal coefficient of resistance. The gage manufacturer supplies information specifying the accuracy of the temperature compensation in the form of an APPARENT STRAIN curve. This is a plot of temperature-induced apparent strain versus temperature, for the gage, mounted on a specific material with a specified coefficient of thermal expansion. The equation for this curve can be obtained from the gage manufacturer by applying curve fitting techniques to the graph supplied with the gage, or by generating an apparent strain curve with the actual gage after installation.
If we monitor the temperature at the gage during the strain measurement we can solve this equation to compensate for temperature-induced apparent strain. A later section of this application note discusses temperature compensation using this technique and also a correction method for using a temperature compensated gage on a material with a thermal coefficient of expansion different from that used by the manufacturer for the apparent strain curve. The manufacturer also supplies data, usually in the form of a graph, that shows how gage factor varies with temperature, so that the strain data can also be corrected for this temperature effect.
The Measurement
From the gage factor equation we see that it is the FRACTIONAL CHANGE in resistance that is the important quantity rather than the absolute resistance value of the gage. Let’s see just how large this resistance change will be for a strain of 1 με. If we use a 120 Ω strain gage with a gage factor of +2, the gage factor equation tells us that 1 με applied to a 120 Ω gage produces a change in resistance of
ΔR = 120 x 0.000001 x 2 = 0.000240 Ω,
or 240 μΩ. That means we need to have microohm sensitivity in the measuring instrumentation. Since it is the fractional change in resistance that is of interest and since this change will likely be only tens of milliohms, some reference point is needed from which to begin the measurement. The nominal value of gage resistance has a tolerance equivalent to several hundred microstrain and will usually change when the gage is bonded to the specimen, so this nominal value can’t be used as a reference.
An initial, unstrained, gage resistance is used as the reference from which strain is measured. Typically, the gage is mounted on the test specimen and wired to the instrumentation while the specimen is maintained in an unstrained state. A reading taken under these conditions is the unstrained reference value and applying a strain to the specimen will result in a resistance change from this value. If we had an ohmmeter that was accurate and sensitive enough to make the measurement, we would measure the unstrained gage resistance and then subtract this unstrained value from the subsequent strained values. Dividing the result by the unstrained value would give us the fractional resistance change caused by strain in the specimen. In some cases it is practical to use just this method, and these cases will be discussed in a later section of this application note. A more sensitive way of measuring small changes in resistance is with the use of the Wheatstone bridge circuit, and in fact, most instrumentation for measuring static strain uses this circuit.
With today’s emphasis on product liability and energy efficiency, designs must not only be lighter and stronger, but also more thoroughly tested than ever before. This places new importance on the subject of experimental stress analysis and the techniques for measuring strain. The main theme of this application note is aimed at strain measurements using bonded resistance strain gages. We will introduce considerations that affect the accuracy of this measurement and suggest procedures for improving it. We will also emphasize the practical considerations of the strain gage measurement, with an emphasis on computer controlled instrumentation.
Stress and Strain
The relationship between stress and strain is one of the most fundamental concepts from the study of mechanics of materials and is of paramount importance to the stress analyst. In experimental stress analysis we apply a given load and then measure the strain on individual members of a structure or machine. Then we use the stress strain relationships to compute the stresses in those members to verify that these stresses remain within the allowable limits for the particular materials used.
Strain
When a force is applied to a body, the body deforms. In the general case this deformation is called strain. In this application note we will be more specific and define the term STRAIN to mean deformation per unit length or fractional change in length and give it the symbol (ε).
This is the strain that we typically measure with a bonded resistance strain gage. Strain may be either tensile (positive) or compressive (negative). See Figure 2. When written in equation form ε = ΔL/L , we see that strain is a ratio and, therefore, dimensionless. To maintain the physical significance of strain, it is often written with units of inch/inch. For most metals the strains measured in experimental work are typically less than 0.005000 inch/inch.
Since practical strain values are so small, they are often expressed in microstrain which is ε x 10-6 (note this is equivalent to parts per million or ppm) and is expressed
by the symbol με. Still another way to express strain is as percent strain, which is ε x 100. For example: 0.005 inch/inch = 5000 με = 0.5%.
As described to this point, strain is fractional change in length and is directly measurable. Strain of this type is also often referred to as normal strain.
Shearing Strain
Another type of strain, called SHEARING STRAIN is a measure of angular distortion. Shearing strain is also directly measurable, but not as easily as normal strain.
If we had a thick book sitting on a table top and we applied a force parallel to the covers, we could see the shear strain by observing the edges of the pages. See
Figure 3. Shearing strain, γ, is defined as the angular change in radians between two line segments that were orthogonal in the undeformed state. Since this angle is very small for most metals, shearing strain is approximated by the tangent of the angle.
Poisson Strain
A bar is shown with a uniaxial tensile force applied, like the bar in Figure 1. The dashed lines show the shape of the bar after deformation, pointing out another phenomenon, that of Poisson strain. The dashed lines indicate that the bar not only elongates, but that its girth contracts. This contraction is a strain in the transverse direction due to a property of the material known as Poisson’s ratio. Poisson’s ratio, υ, is defined as the negative ratio of the strain in the transverse direction to the strain in the longitudinal direction. It is interesting to note that no stress is associated with the Poisson strain. Referring to Figure 4, the equation for Poisson’s Ratio is υ = - εt / ε1. Note that υ is dimensionless
Normal Stress
While forces and strains are measurable quantities used by the designer and stress analyst, stress is the term used to compare the loading applied to a material with
its ability to carry the load. Since it is usually desirable to keep machines and structures as small and light as possible, the parts should be stressed, in service, to
the highest permissible level. STRESS refers to force per unit area on a given plane within a body. The bar in Figure 5 has a uniaxial tensile force, F, applied along the x-axis. If we assume the force to be uniformly distributed over the cross-sectional area, A, the average stress on the plane of the section is F/A. This stress is perpendicular to the plane and is called NORMAL STRESS, σ expressed in equation form, σ = F/A. and has units of force per unit area. Since the normal stress is in the x-direction and there is no component of force in the y-direction, there is no normal Stress in that direction. The normal stress is in the positive x-direction and is tensile.
Shear Stress
Just as there are two types of strains, there is also a second type of stress called SHEAR STRESS. Where normal stress is normal to the designated plane, shear
stress is parallel to the plane and is expressed by the symbol, τ. In the example shown in Figure 5, there is no y-component of force, therefore, no force parallel to
the plane of the section, so there is no shear stress on that plane. Since the orientation of the plane is arbitrary, what happens if the plane is oriented other than normal to the line of action of the applied force? Figure 6 demonstrates this concept with a section taken on the n-t coordinate system at some arbitrary angle, φ, to the
line of action of the force.
We see that the force vector, F, can be broken into two components, Fn and Ft, that are normal and parallel to the plane of the section. This plane has a cross- sectional area of A’ and has both normal and shear stresses applied. The average normal stress, σ, is in the n-direction and the average shear stress, τ is in the t-direction. Their equations are: σ = Fn/A’ and τ = Ft/A’. Note that it was the force vector that was broken into components, not the stresses, and that the resulting stresses are a function of the orientation of the section. This means that stresses (and strains), while having both magnitude and direction, are not vectors and do not follow the laws of vector addition, except in certain special cases, and they should not be treated as such. We should also note that stresses are derived quantities, computed from other measurable quantities, and are not directly measurable.
Principal Axes
In the preceding examples the x-y axes are also the PRINCIPAL AXES for the uniaxially loaded bar. By definition, the principal axes are the axes of maximum
and minimum normal stress. They have the additional characteristic of zero shear stress on the planes that lie along these axes. In Figure 5 the stress in the x-direction is the maximum normal stress, and we noted that there was no force component in the y-direction and therefore zero shear stress on the plane. Since there is no force in the y-direction, there is zero normal stress in the y-direction, and in this case zero is the minimum normal stress. So the requirements for the principal axes are met by the x-y axes. In Figure 6 the x-y axes are the principal axes since that bar is also loaded uniaxially. The n-t axes in Figure 6 do not meet the zero shear stress requirement of the principal axes. The corresponding STRAINS on the principal axes are also maximum and minimum and the shear strain is zero.
The principal axes are very important in stress analysis because the magnitudes of the maximum and minimum normal stresses are usually the quantity of interest.Once the principal stresses are known, the normal and shear stresses in any orientation may be computed. If the orientation of the principal axes is known, through knowledge of the loading conditions or experimental techniques, the task of measuring the strains and computing the stresses is greatly simplified.
In some cases we are interested in the average value of stress or load on a member, but often we want to determine the magnitude of the stresses at a point. The material will fail at the point where the stress exceeds the load-carrying capacity of the material. This failure may occur because of excessive tensile or compressive
normal stress or excessive shearing stress. In actual structures, the area of this excessive stress level may be quite small. The usual method of diagramming the
stress at a point is to use an infinitesimal element that surrounds the point of interest. The stresses are then a function of the orientation of this element and, in one particular orientation, the element will have its sides parallel to the principal axes. This is the orientation that gives the maximum and minimum normal stresses on the point of interest.
Stress-Strain Relationships
Now that we have defined stress and strain, we need to explore the stress-strain relationship. It is this relationship that allows us to calculate stresses from the
measured strains. If we have a bar made of mild steel and incrementally load it in uniaxial tension and plot the strain versus the normal stress in the direction of
the applied load, the plot will look like the stress-strain.
We can see that, up to a point called the proportional limit, there is a linear relationship between stress and strain. Hooke’s Law describes this relationship. The slope of this straight line portion of the stress-strain diagram is the MODULUS OF ELASTICITY or YOUNG’S MODULUS for the material. The modulus of elasticity, E, has the same units as stress (force per unit area) and is determined experimentally for
materials. Written in equation form this stress-strain relationship is σ = E * ε. Some materials, for example, cast iron and concrete, do not have a linear portion
to their stress-strain diagrams. To do accurate stress analysis studies for these materials it is necessary to determine the stress-strain properties, including Poisson’s ratio, for the particular material on a testing machine. Also, the modulus of elasticity may vary with temperature. This variation may need to be experimentally determined and considered when performing stress analysis at temperature extremes. There are two other points of interest on the stress- strain diagram in Figure 7, the yield point and the ultimate strength value of stress. The yield point is the stress level at which strain will begin to increase rapidly with little or no increase in stress. If the material is stressed beyond the yield point, and then the stress is removed, the material will not return to its original size but will retain a residual offset or strain.
The ultimate strength is the maximum stress developed in the material before rupture.
The examples we have examined to this point have been examples of uniaxial forces and stresses. In experimental stress analysis the biaxial stress state is the most common. Figure 8 shows an example of a shaft with both tension and torsion applied. The point of interest is surrounded by an infinitesimal element, Figure 8 with its sides oriented parallel to the x-y axes. The point has a biaxial stress state and a triaxial strain state (remember Poisson’s ratio). The element, rotated to be aligned with the principal (p-q) axes, is also shown in Figure 8. Figure 9 shows the element removed with arrows added to depict the stresses at the point for both orientations of the element. We see that the element oriented along the x-y axes has a normal stress in the x-direction, zero normal stress in the y-direction, and shear stresses on its surfaces. The element rotated to the p-q axes orientation has normal stress in both directions but zero shear stress as it should, by definition, if the p-q axes are the principal axes. The normal stresses, σp and σq, are the maximum and minimum normal stresses for the point. The strains in the p-q direction are also the maximum and minimum, and there is zero shear strain along these axes. Appendix C gives the equations relating stress to strain for the biaxial stress state. If we know the orientation of the principal axes, we can then measure the strain in those directions and compute the maximum and minimum normal stresses and the maximum shear stress for a given loading condition. We don’t always know the orientation of the principal axes, but if we measure the strain in three separate directions, we compute the strain in any direction including the principal axes directions. Three and four element rosette strain gages are used to measure the strain when the principal axes orientation is unknown. The equations for computing the orientation and magnitudes of the principal strains from 3-element rosette strain data are found in Appendix C.
Measuring Strain
Stress in a material can’t be measured directly. It must be computed from other measurable parameters. Therefore, the stress analyst uses measured strains
in conjunction with other properties of the material to calculate the stresses for a given loading condition. There are methods of measuring strain or deformation
based on various mechanical, optical, acoustical, pneumatic, and electrical phenomena. This section briefly describes several of the more common methods
and their relative merits.
Gage Length
The measurement of strain is the measurement of the displacement between two points some distance apart. This distance is the GAGE LENGTH and is an important
comparison between various strain measurement techniques. Gage length could also be described as the distance over which the strain is averaged. For example, we could (on some simple structure such as the part in Figure 10) measure the part length with a micrometer both before and during loading. Then we would subtract the two readings to get the total deformation of the part. Dividing this total deformation by the original length would yield an average value of strain for the entire part.
The gage length would be the original length of the part. If we used this technique on the part in Figure 10, the strain in the reduced width region of the part would be locally higher than the measured value because of the reduced cross-sectional area carrying the load. The stresses will also be highest in the narrow region and the part will yield there before the measured average strain value indicates a magnitude of stress greater than the yield point of the material.
Ideally, we want the strain measuring device to have an infinitesimal gage length so we can measure strain at a point. If we had this ideal strain gage we would place
it in the narrow portion of the specimen in Figure 10 to measure the high local strain in that region. Other desirable characteristics for this ideal strain measuring device are small size and mass, easy attachment, high sensitivity to strain, low cost, and low sensitivity to temperature and other ambient conditions.
Mechanical Devices
The earliest strain measurement devices were mechanical in nature. We have already considered an example using a micrometer to measure strain and observed a problem with that approach. Extensometers are a class of mechanical devices used for measuring strain that employ a system of levers to amplify the minute strains to a level that can be read. A minimum gage length of 1/2 inch and a resolution of about 10 με is the best that can be achieved with purely mechanical devices. The addition of light beam and mirror arrangements to extensometers improves resolution and shortens gage length, allowing 2 με resolution and gage lengths down to 1/4 inch.
Still another type of device, the photoelectric gage, uses a combination of mechanical, optical, and electrical amplification to measure strain. This is done by using a light beam, two fine gratings, and a photocell detector to generate an electrical current that is proportional to strain. This device comes in gage lengths as short as 1/16 inch but it is costly and delicate. All of these mechanical devices tend to be bulky and cumbersome to use, and most are only suitable for static strain measurements.
Optical Methods
Several optical methods are used for strain measurement. One of these techniques uses the interference fringes produced by optical flats to measure strain. This device is sensitive and accurate but the technique is so delicate that laboratory conditions are required for its use.
Brittle Coatings
Brittle coating techniques are another way to indicate static strain. They are often used in conjunction with strain gages. The test object is coated with a brittle lacquer and the load is applied in increments, when possible. The lacquer will crack first in the region of highest surface strain and the cracks will be perpendicular to the highest tensile strain. A representation of the full strain field is obtained indicating where to locate the strain gages and in what orientation. Under favorable conditions a reasonable estimate of the magnitude of the strain can be obtained. Getting goad data from brittle lacquer coatings is as much art as science. For high-temperature applications, a brittle ceramic coating may be used instead of the lacquer.
Electrical Devices
Another class of strain measuring devices depends on electrical characteristics which vary in proportion to the strain in the body to which the device is attached.
Capacitance and inductance strain gages have been constructed but sensitivity to vibration, mounting difficulties, and complex circuit requirements keep them from being very practical for stress analysis work. These devices are, however, often employed in transducers. The piezoelectric effect of certain crystals has also been
used to measure strain. When a crystal strain gage is deformed or strained, a voltage difference is developed across the face of the crystal. This voltage difference is proportional to the strain and is of a relatively high magnitude. Crystal strain gages are fairly bulky, very fragile, and not suitable for measuring static strains.
Probably the most important electrical characteristic, which varies in proportion to strain, is that of electrical resistance. Devices whose output depend on this
characteristic are the piezoresistive or semiconductor gage, the carbon-resistor gage, and the bonded metallic wire and foil resistance gages. The carbon resistor gage is the forerunner of the bonded resistance wire strain gage. It is low in cost, can have a short gage length and is very sensitive to strain. A high sensitivity to temperature and humidity are the disadvantages of the carbon-resistor strain gage.
The semiconductor strain gage is based on the piezoresistive effect in certain semiconductor materials such as silicon and germanium. Semiconductor gages have elastic behavior and can be produced to have either positive or negative resistance changes when strained. They can be made physically small while still maintaining a high nominal resistance. The strain limit for these gages is in the 1000 to 10000 με range with most tested to 3000 με in tension. Semiconductor gages exhibit a high sensitivity to strain but the change in resistance with strain is nonlinear. Their
resistance and output are temperature sensitive and the high output, resulting from changes in resistance as large as 10-20%, can cause measurement problems when using the devices in a bridge circuit. However, mathematical corrections for the temperature sensitivity, the nonlinearity of output, and the nonlinear characteristics of the bridge circuit (if used), can be made automatically when using computer controlled instrumentation to measure strain with semiconductor gages. They can be used to measure both static and dynamic strains. When measuring dynamic strains, temperature effects are usually less important than for static strain measurements, and the high output of the semiconductor gage is an asset.
The bonded resistance strain gage is by far the most widely used strain measurement tool for today’s experimental stress analyst. It consists of a grid of very fine wire, or more recently, of thin metallic foil bonded to a thin insulating backing called a carrier matrix. The electrical resistance of this grid material varies linearly with strain. In use, the carrier matrix is attached to the test specimen with an adhesive. When the specimen is loaded, the strain on its surface is transmitted to the grid material by the adhesive and carrier system. The strain in the specimen is found by measuring the change in the electrical resistance of the grid material. The bonded resistance strain gage is low in cost, can be made with a short gage length, is only moderately affected by temperature changes, has small physical size and low mass, and has fairly high sensitivity to strain. It is suitable for measuring both static and dynamic strains. The remainder of this application note deals with the instrumentation considerations for making accurate, practical strain measurements using the bonded resistance strain gage.
The Bonded Resistance Strain Gage
The term “bonded resistance strain gage” can apply to the nonmetallic (semiconductor) gage or to the metallic (wire or foil) gage. Wire and foil gages operate on the same basic principles and both can be treated in the same fashion from the measurement standpoint. The semiconductor gage, having a much higher sensitivity to strain than metallic gages, can have other considerations introduced into its measurement. We will use the term STRAIN GAGE or GAGE to refer to the BONDED METALLIC FOIL GRID RESISTANCE STRAIN GAGE throughout the rest of this application note. These foil gages are sometimes referred to as metal film gages.
Strain gages are made with a printed circuit process using conductive alloys rolled to a thin foil. The alloys are processed, including controlled atmosphere heat treating, to optimize their mechanical properties and temperature coefficient of resistance. A grid configuration for the strain sensitive element is used to allow higher values of gage resistance while maintaining short gage lengths. Gage resistance values range from (30 to 3000) Ω, with 120 Ω and 350 Ω being the most commonly used values for stress analysis. Gage lengths from 0.008 inch to 4 inches are commercially available. The conductor in a foil grid gage has a large surface area for a given cross-sectional area. This keeps the shear stress low in the adhesive and carrier matrix as the strain is transmitted by them. This larger surface area also allows good heat transfer between grid and specimen. Strain gages are small and light, will operate over a wide temperature range, and can respond to both static and dynamic strains. They have wide application and acceptance in transducers as well as stress analysis.
In a strain gage application, the carrier matrix and the adhesive must work together to faithfully transmit the strains from the specimen to the grid. They also serve as an electrical insulator between the grid and the specimen and must transfer heat away from the grid. Three primary factors influencing gage selection are operating temperature, state of strain (including gradients, magnitude and time dependence), and stability requirements for the gage installation. The importance of selecting the proper combination of carrier material, grid alloy, adhesive, and protective coating for the given application cannot be over- emphasized. Strain gage manufacturers are the best source of information on this topic and have many excellent publications to assist the customer in selecting the proper strain gages, adhesives, and protective coatings.
Gage Factor
When a metallic conductor is strained, it undergoes a change in electrical resistance, and it is this change that makes the strain gage a useful device. The measure of this resistance change with strain is GAGE FACTOR,GF. Gage factor is defined as the ratio of the fractional change in resistance to the fractional change in length (strain) along the axis of the gage. Gage factor is a dimensionless quantity and the larger the value, the more sensitive the strain gage.
It should be noted that the change in resistance with strain is not just due to the dimensional changes in the conductor, but that the resistivity of the conductor material also changes with strain. The term gage factor applies to the strain gage as a whole, complete with carrier matrix, not just to the strain sensitive conductor. The gage factor for constantan and nickel-chromium alloy strain gages is nominally 2 and various gage and instrumentation specifications are usually based on this nominal value.
Transverse Sensitivity
If the strain gage were a single straight length of conductor, and of small diameter with respect to its length, it would respond to strain along its longitudinal axis and be essentially insensitive to strain perpendicular or transverse to this axis. For any reasonable value of gage resistance, it would also have a very long gage length. When the conductor is in the form of a grid to reduce the effective gage length, there are small amounts of strain sensitive material in the end loops or turnarounds that lie transverse to the gage axis. This end loop material gives the gage a non-zero sensitivity to strain in the transverse direction. TRANSVERSE SENSITIVITY FACTOR, Kt is defined by and is usually expressed in percent. Values of Kt range from 0 to 10%. To minimize this effect, extra material is added to the conductor in the end loops and the grid lines are kept close together. This serves to minimize the resistance in the transverse direction. Correction for transverse sensitivity may be necessary for short, wide grid gages, or where there is considerable misalignment between the gage axis and the principal axis, or in rosette analysis where high transverse strain fields may exist. Data supplied by the manufacturer with the gage can be entered into the computer controlling the instrumentation and corrections for transverse sensitivity made to the strain data as it is collected.
Temperature Effects
Ideally we would prefer the strain gage to change resistance only in response to the stress induced strain in the test specimen, but the resistivity and the strain sensitivity of all known strain sensitive materials vary with temperature. Of course that means the gage resistance and the gage factor will change when the temperature changes. This change in resistance with temperature for a mounted strain gage is a function of the difference in the thermal expansion coefficients between the gage and the specimen and of the thermal coefficient of resistance of the gage alloy. Self- temperature compensating gages may be produced for specific materials by processing the strain sensitive alloy such that it has thermal resistance characteristics that compensate for the effects of the mismatch in thermal expansion coefficients between the gage and the specific material. A temperature compensated gage produced in this manner is accurately compensated only when mounted on a material that has a specific coefficient of thermal expansion. Table 2 is a list of common materials for which self temperature compensated gages are available.
Thermal expansion coefficients of some common materials for which temperature compensated strain gages are available
The compensation is only effective over a limited temperature range because of the nonlinear character of both the thermal coefficient of expansion and the thermal coefficient of resistance. The gage manufacturer supplies information specifying the accuracy of the temperature compensation in the form of an APPARENT STRAIN curve. This is a plot of temperature-induced apparent strain versus temperature, for the gage, mounted on a specific material with a specified coefficient of thermal expansion. The equation for this curve can be obtained from the gage manufacturer by applying curve fitting techniques to the graph supplied with the gage, or by generating an apparent strain curve with the actual gage after installation.
If we monitor the temperature at the gage during the strain measurement we can solve this equation to compensate for temperature-induced apparent strain. A later section of this application note discusses temperature compensation using this technique and also a correction method for using a temperature compensated gage on a material with a thermal coefficient of expansion different from that used by the manufacturer for the apparent strain curve. The manufacturer also supplies data, usually in the form of a graph, that shows how gage factor varies with temperature, so that the strain data can also be corrected for this temperature effect.
The Measurement
From the gage factor equation we see that it is the FRACTIONAL CHANGE in resistance that is the important quantity rather than the absolute resistance value of the gage. Let’s see just how large this resistance change will be for a strain of 1 με. If we use a 120 Ω strain gage with a gage factor of +2, the gage factor equation tells us that 1 με applied to a 120 Ω gage produces a change in resistance of
ΔR = 120 x 0.000001 x 2 = 0.000240 Ω,
or 240 μΩ. That means we need to have microohm sensitivity in the measuring instrumentation. Since it is the fractional change in resistance that is of interest and since this change will likely be only tens of milliohms, some reference point is needed from which to begin the measurement. The nominal value of gage resistance has a tolerance equivalent to several hundred microstrain and will usually change when the gage is bonded to the specimen, so this nominal value can’t be used as a reference.
An initial, unstrained, gage resistance is used as the reference from which strain is measured. Typically, the gage is mounted on the test specimen and wired to the instrumentation while the specimen is maintained in an unstrained state. A reading taken under these conditions is the unstrained reference value and applying a strain to the specimen will result in a resistance change from this value. If we had an ohmmeter that was accurate and sensitive enough to make the measurement, we would measure the unstrained gage resistance and then subtract this unstrained value from the subsequent strained values. Dividing the result by the unstrained value would give us the fractional resistance change caused by strain in the specimen. In some cases it is practical to use just this method, and these cases will be discussed in a later section of this application note. A more sensitive way of measuring small changes in resistance is with the use of the Wheatstone bridge circuit, and in fact, most instrumentation for measuring static strain uses this circuit.