## Strain Gauge Theory – Types, formulas and applications

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# Strain Gauge Theory

The strain gauges are based on the resistive properties of the metal. Upon stretching, the metal material loses or increases its electrical resistance. The strain gauge foil is glued on the material, which will be tested using a specially adapted glue for strain gauges. Due to the correct fit to the material being examined, the strain gauge is bent in the same way as the test material, so it reproduces the stresses.

Strain gauge theory tells about resistance elements that are found in a large amount of electronic equipment. They are mainly used to measure different compressive forces, tensile forces and any stresses of materials. They are the main element of automatic systems when it comes to weight measurements e.g. scales, tanks, storage tanks. Properly set strain gauges allow mass measurements, measurements of product dispensing amount, etc.

In order to properly choose the type of strain gauge, it should be determined in what conditions it will operate. You need to know the type of load, the temperature at which it will be operating and the requirements it must have to endure cooperation in that system. Nowadays, resistance strain gauges have taken over most of the market and are most often chosen for retail and laboratory measurements. Their advantages are:

• sensitivity and accuracy of measurement,
• the durability of materials with small dimensions,
• work at high temperatures and with high pressure,
• flexibility.

## Strain gauge applications

The strain gauges are mainly used for following measurements:

• Strain measurement – Whenever the tested material is under high stress or load, they come under strain and that is where strain gauge is doing its job.
• Measurement of other quantities – The change in resistance due to applied load can also be “translated” to measure a variety of other quantities, such as pressure, displacement, force, acceleration etc. They are usually attached to the mechanical transducers (e.g. bellows) for measuring displacement, pressure and other quantities.

Strain gauges are attached to the substrate with a special glue, which bonds them together. The type of glue depends on the required lifetime of the measurement system. Main task of the strain gauge adhesive (bonding material) is to obtain the best possible connection between strain gauge and the surface of the tested sample or object. This allows transferring the object’s deformation with minimal losses to the strain gauge – more precise measurement. Different measurement environments require other types of strain gauge adhesive, because each one influences the measurement process in different way. One of the most important (if not the most important) thing to do is to prepare the surface where the strain gauge’s and tested object’ bonding will be made. Aforementioned surface must be first smoothed (e.g. with special sand paper) and deoiled. Strain gauge must be glued immediately after this process to avoid oxidation or pollution of the prepared area. Without this process, the strain gauge adhesive binding to the surface may generate measurement errors and be unreliable.

The type of strain gauge adhesive and material used for the production of strain gauges are patented secrets of manufacturing companies, thanks to these material parameters the best accuracy and sensitivity are obtained. There are three types of most popular strain gauges:

• tubular strain gauges – made of hose-shaped resistance wires,
• mesh strain gauges – serially connected resistance wires arranged in parallel connected with copper tape,
• foil resistance strain gauges – built on the basis of a tubular resistive mesh, combined with support gasket with metal foil.

## Types of strain gauge adhesive

To obtain the best possible quality of the bonding between strain gauge and the substrate, different requirements must be met. In terms of operating temperature, this led to the availability of various types of strain gauge adhesive materials. These materials can be distinguished in the following way in relation to the application technology:

• Cold-curing strain gauge adhesives – Pretty effortless and easy to apply. We can distinguish single-component adhesives. which start curing with appropriate humidity as well as two-component adhesives that have to be mixed before application. Strain gauge adhesive with very short reaction times are called “superglues”, which are mostly used in experimental measurements.
• Hot-curing strain gauge adhesives – These adhesives require certain curing temperature to make bonding possible, so they can only be used if test object can be brought to such specific environment. In addition, hot-curing adhesive offers a wider temperature application range than cold-curing adhesive, which means that it can be used more precisely

To ensure correct strain gauge calibration process, it is important to know its most important parameters for the corresponding model:

• strain gauge constant,
• resistance in the non-deformed state (without the influence of external strain forces),
• active length of the net (strain gauge base),
• maximum permissible deformation (maximum strain),
• number of measurement cycles (dynamic durability),
• temperature coefficient of resistance (TCR),
• linearity.

## Electrical-resistance strain gauge theory

Electric strain gauges use physical relationships between electrical parameters and material deformation. Depending on the measured units, there is a division of strain gauges:

• magnetostrictive,
• photoelectric,
• piezoelectric,
• capacitive,
• inductive,
• electroresistive.

The electroresistive strain gauges are characterised by good accuracy and sensitivity of strain measurements of the tested sample. The distance of the test sample from the strain gauge can be large in the case of electric strain gauges, which also allows measurement in many measuring places for the material stresses of the test system. Due to the very low value of the inertia parameter of the measurement system, electrofusion strain gauges are well suited for testing fast-changing strains.

## Strain gauge internal construction

The strain gauge system consists of the following components:

• a sensor, which converts mechanical values into electrical values,
• power supply, alternating current generator with measuring bridge,
• DC and AC amplifiers,
• recorder of changes in electrical quantities.

Where: G – voltage generator, R1, R2, R3, R4 – foil strain gauge, R5 – bridge balancing resistor, WS – selector signal amplifier, PF – phaser, rectifier, FD – low pass filter, W – resistance measurement.

Electrical strain gauges are based on the principle of changing the electrical resistance due to the change in the length of the metal used in the strain gauge. The formula for the resistance of strain gauge metal is:

Where: R – strain gauge metal resistance, p – specific resistance of a strain gauge, l – strain gauge length, A – cross-sectional area of the strain gauge.

In resistance theory strain gauge, their length is called the strain gauge base. During the samples work, the length of the strain gauge base changes. At the same time, the length of the metal wire also changes, which directly affects the change of its resistance.
The formula defining changes in deformation in relation to resistance changes is as follows:

Where: k – strain gauge constant, ε – strain, R/R – the variation of the strain gauge resistance.

The two previous formulas describe the basic physical dependencies of the strain gauges electrical resistance. By plunging the Hooke law, it is possible to obtain a tension at a particular point of the sample:

The value of the strain gauge constant depends on the material from which the sensor was produced and constant’s values range from 1.6 to 3.6. Other names for the strain gauge constant are “strain sensitivity” and the “strain gauge sensitivity value”.

A distinguishing feature of electric strain gauges is the method of obtaining deformation values from the material under test. In this process, the strain gauge is used together with a glue specially made for such tasks. To fix the strain gauge correctly, clean the test surface. The whole process of choosing a foil, the shape of a strain gauge and sticking is long. It is necessary to prepare everything carefully and calculate the places of sticking and the shape of the strain gauge.

Electroresistive strain gauge works well as a subset of static, dynamic and moving samples. Their sensitivity is high and their own weight does not affect the discrepancy of measurements. By transmitting the deformations directly to the resistive wire, the transmission and the slippage are avoided. The values of results are dimensionless, so they do not depend on the value of the strain gauge constant. Examination of several places of one sample is not a problem for electric strain gauge. In addition, results can be obtained live through the wireless connection. Thanks to the use of compensation strain gauges, the influence of moisture and temperature on the measurement results is removed.

## Strain Gauge main types

Three types of electric strain gauges are most commonly used: tubular strain gauges, mesh strain gauges, foil strain gauges, with which the foil is the most popular in the industry.

The tubular strain gauge is made of a piece of wire that is covered with foil or paper on both sides. The connectivity of the wire with the electrical circuit provides a copper strip connected to both ends of the wire. Its diameter ranges from 0.02 mm to 0.05 mm.

The mesh strain gauge is made of parallel-laid wires, which are connected by a copper band with a larger width and a lower specific resistance. Due to the use of tape with a larger cross-section, the resistance of the tape deformation is so small, that it is not measurable for the apparatus calibrated under strain gauges. Consequently, the strain gauge has a much lower sensitivity when it comes to measurements in the transverse direction. The execution of wires is constant and their diameter does not exceed 50 micrometers. Connecting copper strips with wires is made using tin soldering. Tin is a problem in measurements at high temperatures above 180 degrees Celsius. The base value of mesh strain gauges is 5 mm to 70 mm.

The foil strain gauge is made of a metal foil, the thickness of which does not exceed 0.025 mm. The method of producing strain gauge foil is similar to printing circuits on flexible films. Thanks to lithographic technique, it is possible to create strain gauges with dimensions and shapes ideally suited to the examination.

## Strain Gauge Rosettes

The strain gauge rosette is a set of strain gauges set in a previously planned way. The number of applied strain gauges depends on the values calculated during the selection of strain gauges to the sample under test.

The strain gauge rosettes can be divided into three types: rectangular rosette (measures deformations in the multi-axis state), T-rosette (measures biaxial stresses), screwed rosette (measures torsion samples).
The strain gauge should be properly prepared by ideally gluing the strain gauge to the material being tested. The glue used should be specially prepared for use in the strain gauge. Its thickness should not exceed 10 micrometers. The operation of the glue should isolate the wires from the material being tested. The glue should withstand high operating temperatures without changing its physical and electrical properties. After proper gluing of the strain gauge, the whole system is covered with a protection layer, it may be a wax.

## Types of Strain Gauges

### Mechanical strain gauge

Mechanical strain gauges are among the most straightforward in the way of making and measuring. The method of measurement consists of measuring the elongation of the blades and reading this value on a scale magnified 1000 times, transmitted by means of mechanical levers. The following figure shows the Huggenberger mechanical strain gauge, its construction consists of a movable, immovable blade, handles, pointer, and scale.

Changing the deformation value activates the lever movement. The Huggenberger strain gauge has a strain gauge base with values ranging from 5 – 100 mm. The strain gauges are a bit heavier because they weigh about 50 g.

### String strain gauge

They are most often used in building constructions. The stress of the surface and the mass of the test material are measured. The main element of a string strain gauge is a tightly tensioned string, adjustable by means of screws. It is placed between two blades.

The measurement is carried out by means of vibrating and resonant vibrations. Strained strings stretched between the blades are subjected to testing.

### Optical strain gauge

Optical strain gauges predominate over mechanical strain gauges in terms of the resolution of strains under investigation. Thanks to the large ratio, they are capable of performing more accurate measurements. The most popular types of strain gauge as optical is the Martens mirror strain gauge.
The operation of the optical strain gauge is based on the study of the length of the measuring base. When changing the length of the base, the mirror rotates and the light falls on the scale, which is read with the telescope. During the operation of this strain gauge, very restrictive conditions must be maintained as to the mechanical shocks of the environment.

### Pneumatic strain gauge

Pneumatic types of strain gauges are distinguished by very high accuracy and measurement resolution. Its operation is based on the dependence of the deformation on the cross-sectional area of the nozzle, which affects the height of the water. Thanks to the height change, we are able to read the change in strain on the scale.

## Strain Gauge formula in Measurements using strain gauges Wheatstone bridge

Commonly used measurement systems are e.g. the Wheatstone bridge system. That bridge consists of four resistors that are connected to the bridge circuit. A voltage source is connected to points A and B. From point C-D, the voltage of the bridge’s imbalance is measured.

The task of the strain gauge bridge is to measure the difference in resistance. To start correct measurements, the bridge should be balanced. It is required that the voltage between C-D points is 0 volts, then the equilibrium condition is met. The Strain Gauge formula resistance is:

The bridge is balanced at the moment of equilibrium, which is achieved by the equilibrium condition recorded above. Non-balanced bridges are also used for strain gauge measurements. An unbalanced bridge occurs only for one combination of resistance values that satisfy the equilibrium condition. When the value of one or several resistors changes, a voltage different from zero will appear between the C-D points.
Voltage ΔU can be calculated from the strain gauge formula:

During measurements using strain gauges, all the resistors used in the strain gauge bridge circuit have the same initial values of resistance. If, however, one strain gauge changes its resistance value then the output voltage is calculated from the strain gauge formula:

The active strain gauge, it is a strain gauge whose resistance is variable under the influence of deformation. After some transformations of the previous pattern, we obtain a formula that shows the dependence of the voltage on ΔU the supply voltage.

Active strain gauges must have an appropriate setting so that the bridge sensitivity is correct:

Deformations with opposite signs in active strain gauges are required for the strain gauges to work properly, this is achieved by the correct connection of strain gauges in the bridge. One of the methods to obtain such a requirement is to glue them on opposite sides of the material being tested. The strain gauges should be connected to the opposite bridge arms if the resistance is the same sign, while if the resistance change is the opposite sign, the strain gauge should be connected to the nearest strain gauge bridge arms. In the case of wrong connection of strain gauges, the voltage at C-D points will not appear, and thus all resistance changes will be compensated and no result will be obtained.

Four load cells connected at the same operating temperature simultaneously obtain the best results in terms of compensating the resistance of strain gauges under the influence of outside temperature.

It is irrelevant whether the same forces act on strain gauges. There are also compensating strain gauges whose task is to compensate the temperature and not to measure deformations as in other strain gauges. The second unexpected use of strain gauges is the combination of two strain gauges together with external resistors, which together form a strain gauge half-bridge also used for temperature compensation. The same applies to the use of only one strain gauge, then you should attach from the outside of the system up to three resistors, which together form the quarter bridge Wheatstone strain gauge. In the case of a bridge system, the strain gauges can work independently in the number of four, two or one. Two or one strain gauge can work for a half bridge. The main principle of the bridge’s balance is that in strain gauge in the judged arms they must change the values of their resistance in opposite directions.

To avoid an undesired thermoelectric effect, the strain gauge bridge is supplied with alternating voltage. It forms at the interface of the wire with the copper leads of the strain gauge and as a result of temperature changes.

## How to use strain gauge

To know how to use strain gauge first step is to measure deformations using a resistive strain gauge one should rely on the phenomenon of resistance change of the conductor, which affects its shape. Examination of the element in terms of vibrations should start from correctly bonding the strain gauge to the tested surface. The tested element influences the strain gauge, whose resistance values change under the influence of the size change. The dependence defining the resistance R, its change ΔR and the strain ε occurs for the equation:

Where constant k is the deformation sensitivity factor of the strain gauge.

The electrical signal contained in the ΔR/R value is converted to voltage signals in measuring devices. The value of this signal is proportional to the strain value ε that affects the strain gauge.

Sources:

1. http://www.imio.polsl.pl/Dopobrania/WM_lab_tensometry.pdf

2. Training materials MEASUREMENTS GROUP Mebtechnik GmbH, Measurements technics in mechanics, Warsaw 1995.

3. http://elektron.pol.lublin.pl/djlj24/pwn/cw1.pdf

4. Hoffmann K., An Introduction to Measurements using Strain Gages, http://www.hbm.com/fileadmin/mediapool/techarticles/hoffmannbook/Hoffmann-book_EN.pdf

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