STRAIN GAGE

Force, Stress, and Strain

If an object receives an external force from the top, it internally generates a repelling force to maintain the original shape. The repelling force is called internal force and the internal force divided by the cross-sectional area of the object (a column in this example) is called stress, which is expressed as a unit of Pa (Pascal) or N/m². Stress is the force an object generates inside by responding to an applied external force. Suppose that the crosssectional area of the column is A (m²) and the external force is P (N, Newton). Since external force = internal force, stress, σ (sigma), is:

       σ = P/A   (Pa or N/m²)     (1)

 When a bar is pulled, it elongates by ΔL, and thus it lengthens to L (original length) + ΔL (change in length). The ratio of this elongation (or contraction), ΔL, to the original length, L, is called strain, which is expressed in ε (epsilon):

        ε₁ = ΔL (change in length) / L (original length)

Strain in the same tensile (or compressive) direction as the external force is called longitudinal strain. Since strain is an elongation (or contraction) ratio, it is an absolute number having no unit. Usually, the ratio is an extremely small value, and thus a strain value is expressed by suffixing “x10^-6 (parts per million) strain,” “μm/m” or “με.”

The pulled bar becomes thinner while lengthening. Suppose that the original diameter, d₀, is made thinner by Δd. Then, the strain in the diametrical direction is:

       ε₂ = –Δd / d₀

Strain in the orthogonal direction to the external force is called lateral strain. Each material has a certain ratio of lateral strain to longitudinal strain, with most materials showing a value around 0.3. This ratio is called Poisson’s ratio, which is expressed in ν (nu):

       ν = |ε₂ / ε₁| = 0.3

With various materials, the relation between strain and stress has already been obtained experimentally. Figure below graphs a typical relation between stress and strain on common steel (mild steel). The region where stress and strain have a linear relation is called the proportional limit, which satisfies the Hooke’s law.

       σ = E.ε       or       σ/ε = E

The proportional constant, E, between stress and strain in the equation above is called the modulus of longitudinal elasticity or Young’s modulus, the value of which depends on the materials. As described above, stress can be known through measurement of the strain initiated by external force, even though it cannot be measured directly.

Polarity of Strain

There exist tensile strain (elongation) and compressive strain (contraction). To distinguish between them, a sign is prefixed as follows:

• Plus (+) to tensile strain (elongation)
• Minus (–) to compressive strain (contraction)

Strain Gage

A strain gage is a sensor whose resistance varies with applied force; It converts force, pressure, tension, weight, etc., into a change in electrical resistance which can then be measured.

Structure of Strain Gages

There are many types of strain gages. Among them, a universal strain gage has a structure such that a grid-shaped sensing element of thin metallic resistive foil (3 to 6μm thick) is put on a base of thin plastic film (15 to 16μm thick) and is laminated with a thin film.

Principle of Strain Gages

Metal wires can be used as strain gages. Stretching of the wire changes its geometry in a way that acts to increase the resistance. For a metal wire, we can calculate the gage factor  as follow:

        
        R = ρL/A = ρL/πr² = 4ρL/πD²
       dR = (4L/πD²) dρ + (ρ/πD²) dL− (8ρL/πD³) dD
       dR/R = dρ/ρ + dL/ L − 2dD/D 

Then

         K = (dR/R)/(dL/L) = (dρ/ρ)/(dL/L) + 1 − (2dD/D)/(dL/L)

Since

       −(dD/D)/(dL/L)

is defined as Poisson’s ratio, v, we have the Gage Factor:

       K = 1 + 2υ + (dρ/ρ)/(dL/L)

For different metals, this quantity depends on the material properties, and on the details of the conduction mechanism. In general, metals have gage factors between 2 and 4.

Now, since the stress times the area is equal to the force, and the fractional change in resistance is equal to the gage factor times the fractional change in length (the strain), and stress is Young’s modulus times the strain, we have 

       F = σA = EA(dL/L) = (EA/K)(dR/R)

or

       dR/R=FK/EA

So the fractional change in resistance of a strain gage is proportional to the applied force and is proportional to the gage factor divided by Young’s modulus for the material. Clearly, we would prefer to have a large change in resistance to simplify the design of the rest of a sensing instrument, so we generally try to choose small diameters, small Young’s modulus, and large gage factors when possible. The elastic limits of most materials are below 1%, so we are generally talking about resistance changes in the 1%–0.001% range. Clearly, the measurement of such resistances is not trivial, and we often see resistance bridges designed to produce voltages that can be fed into amplification circuits.

Examples of strain gage applications

There is an uncountable number of different applications for strain gages. Only a few are listed here:

Experimental stress analysis. Diagnosis on machines and failure analysis. 

• multi axial stress fatigue testing, proof testing
• residual stress
• vibration measurement
• torque measurement
• bending and deflection measurement
• compression and tension measurement
• strain measurement

Sensors for machines, automotive, research etc.

• force measurement in machine tools
• aerospace
• impact sensors
• dental sensors
• medical sensors
• automotive, motor sport
• Biometrics
• tension sensors
• web tension
• force on hydraulic or pneumatic press

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Pressure Sensor

Ashcroft Indust Digital 4-1/2" Pressure Gauge

Pressure is force per unit area. Sensing it follows the same principle as the sensing of force that is measuring the displacement of an appropriate member of the sensor in response to pressure. The range of methods is quite large and includes thermal, optical as well as magnetic and electrical principles. Earliest sensors were purely mechanical.

Pressure Units

The basic SI unit is pascal, where 1 pascal (Pa) = 1 (N/m2). Other often units are:

• bar: 1 bar = 0.1 Mpa
• torr: 1 torr = 133Pa
• millibar: 1 millibar = 100Pa = 1.333 torr
• microbar: 1 microbar = 0.1 Pa

Types of Pressure Sensor

Pressure sensors come in four basic types :

• Absolute pressure sensors: pressure sensed relative to absolute vacuum.
• Differential pressure sensors: the difference between two pressures on two ports of the sensor is sensed
• Gage pressure sensors: the pressure relative to ambient pressure is sensed. (Most common)
• Sealed gage pressure sensor: the pressure relative to a sealed pressure chamber (usually 1atm at sea level or 14.7 psi) is sensed.

How Does Pressure Sensors Work

Pressure metrology is the technology of transducing pressure into an electrical quantity. Normally, a diaphragm construction is used with strain gauges either bonded to , or diffused into it, acting as resistive elements. Under the pressure-induced strain,the resistive values change.

In capacitive technology, the pressure diaphragm is one plate of a capacitor that changes its value under pressure-induced displacement.

Figure 1. Pressure sensing using diaphragm technology

Pressure sensing using diaphragm technology measures the difference in pressure of the two sides of the diaphragm. Depending upon the relevant pressure, we use the terms ABSOLUTE, where the reference is vacuum (1st picture), GAUGE, where the reference is atmospheric pressure (2nd picture), or DIFFERENTIAL, where the sensor has two ports for the measure of two different pressure.

The piezoresistive pressure sensor, or silicon cell

Figure 2. Pressure sensor consists of a micro-machined silicon diaphragm with piezoresistive strain gauges

This type of pressure sensor consists of a micro-machined silicon diaphragm with piezoresistive strain gauges diffused into it, fused to a silicon or glass backplate. The resistors have a value of approx. 3.5 kOhm. Pressure induced strain increases the value of the radial resistors (r), and decreases the value of the resistors (t) transverse to the radius. This resistance change can be high as 30%. The resistors are connected as a Wheatstone Bridge, the output of which is directly proportional to the pressure.

Figure 3. The resistors are connected as a Wheatstone Bridge

Leadouts from the bridge

Figure 4. Leadout from the bridge

1. Gold or aluminium wires are welded to the aluminium contacts on the chip and to the glass feed-through, pins of the header.
2. TAB (Tape Automated Bonding). The contacts on the chip have a gold dot.

A pretinned felxible printed circuit is directly soldered to these gold dots and the other end to a PC-board, or the header. In the first method, the sensor must be fixed on the header. The TAB printed circuit, however, holds the sensor in place itself.

Low cost sensors

Figure 5. Low cost sensors

Low cost sensors are devices where the they are exposed to the media without protection. The glass feed-through and the silicon cell is mounted in a plastic housing with pressure ports for positive and negative pressure (1st picture). The silicon sensor with the TAB print is fixed between two plastic mouldings with pressure ports (2nd picture). The silicon sensor is bonded to a brass pressure port. The contacts are made either by gold wires to soldering pins, or by TAB flexible printed circuit (3rd picture).

Piezoresistive OEM Pressure Transducer

Figure 6. Piezoresistive OEM Pressure Transducer

The silicon sensor on the glass feed-through is mounted in a stainless steel housing, isolated by a thin stainless steel diaphragm and filled with silcone oil. The pressure acts on the diaphragm and is transfered through the oil onto the sensor. These transducers are fully tested for temperature and linearity and the compensation resistor values given on the individual test sheets.

  Pressure Transducers

Figure 7. Pressure transducer

Pressure transducers are pressure measuring instruments, ready to use. It is an OEM transducer with pressure port, integrated compensation resistors and a cable or connector. Transducers give an unamplified signal into a separate instrumentation amplifier or indicator. They can be considered as passive bridges, being interchangeable between different manufacturers.

Pressure Transmitters.

Figure 8. Pressure transmitter

In pressure transmitters, the full signal conditioning circuitry is integrated in the housing. The sensor signal is conditioned into standard output signals of 0...100mV, 0...10V, 0.5...4.5V,and 4-20mA. Normally, the signal is independent from the excitation (i.e. 8...28V), but in ratiometric transmitters, the signal is proportional to the excitation. The accuracy of a transmitter is best described by an error band. This band covers all errors over the full pressure and temperature range. Typical errors are also given. The typical error describes the accuracy which can normally be expected in a measurement.

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