1. FORCE-STRAIN-STRESS
measurements

2. 9.1 - INTRODUCTION
Strain is the amount of deformation of a body due to an applied force. More specifically, strain is defined as the fractional change in length.
Strain can be positive (tensile),, or negative (compressive). Although dimensionless, strain is sometimes expressed in units such as mm/mm.
In practice, the magnitude of measured strain is very small. Therefore, strain is often expressed as micro strain (µЄ), which is Є x 10-6.

3. Eq. 9.1
Where ε is the strain, and δL is the extension or contraction, L original length.
When a bar is strained with a uniaxial force, a phenomenon known as Poisson Strain causes the cross section of the bar to contract in the transverse, or perpendicular direction. The magnitude of this transverse contraction is a material property indicated by its Poisson's Ratio.

4. The Poisson's Ratio (y) of a material is defined as the negative ratio of the strain in the transverse direction (perpendicular to the force) to the strain in the axial direction (parallel to the force). Poisson's Ratio for steel, for example, ranges from 0.25 to 0.3.
Eq. 9.2
Eq. 9.3
Eq. 9.4

5. Where εl, and εt are the longitudinal , and transverse strain 
Stress is the intensity of the applied force per unit area
Eq. 9.5
The constant connecting stress and the strain in elastic
materials is the modulus of elasticity, E.
Eq. 9.6

6. 9.2 SPRING BALANCE
Fig Spring Balance

7. Spring balance is frequently used to indicates the magnitude of the applied force
F=kx Eq. 9.7
Where F is the applied force, k spring stiffness, x is the spring displacement from its equilibrium position.

8. Fig. 9.3 hydraulic load cell
Applied load increases the oil pressure, which is indicated by the pressure gauge.

9. 9.4 PROVING RING
Fig. 9.4 Proving ring

10. Proving ring is an elastic ring which is employed for force measurements. The force deflection relation of such ring is;
F=kx Eq. 9.8 
Where F is the applied force, k Ring stiffness constant, x is the ring deflection. Ring deflection is measured by a micrometer attached to the ring. Deflection measurements may be made within ±0.5 µm. Proving ring is used as a calibration standard for large testing machines.

11. 9.5 STRAIN GAUGE TRANSDUCER
Electric resistance strain gauges are used to measure force, stress, and strain. Strain gauge transducer is a device whose electrical resistance varies in proportion to the amount of strain.

12. Fig. 9.5 Electric resistance Strain gauge
9.5.1 STRAIN GAUGE CONFIGURATION
Fig. 9.5 Electric resistance Strain gauge

13. The metallic strain gauge consists of a very fine wire or, more commonly, metallic foil arranged in a grid pattern, Fig. 9.5.
The grid pattern maximizes the amount of metallic wire or foil subject to strain in the parallel direction. The grid is bonded to a thin backing, called the carrier, which is attached directly to the test specimen.
Unstrained resistance ranges from 120 to several hundred ohms. Because a significant length of wire or foil is necessary to provide high unstrained resistance, metal strain gauges cannot be made extremely small. The Gauge provides an output proportional to the average strain in their active area.

14. 9.5.2 STRAIN GAUGE INSTALLATION
Metal strain gauges are usually bonded (glued) to the surface where strain is to be measured
It is very important that the strain gauge be properly mounted onto the test specimen so that the strain is accurately transferred from the test specimen, though the adhesive and strain gauge backing, to the foil itself.

15. 9.5.3 STRAIN GAUGE PRINCIPLE
The resistance of a metal wire is given by
Eq. 9.10
Where R is the resistance of the wire, ρ is the wire material resistivity, L wire length, and d is the wire diameter.

16. Suppose this wire is now stressed by the application of a force F
Suppose this wire is now stressed by the application of a force F .Then we know that the material elongates by some amount. It is also true that in such a stress-strain condition, although the wire lengthens, its volume will remain nearly constant. It follows that if the volume remains constant and the length increases, then the area must decrease by some amount Because both length and area have changed, we find that the resistance of the wire will have also changed.

17. Eq. 9.9
Where δR, δL, δρ, and δd are the changes in wire resistance, length, resistivity, and diameter. R, L, ρ , and d are the resistance, length, resistivity, and diameter of the wire.
Eq. 9.12 Eq
Eq. 9.14

18. Where G is a constant depend on the wire material, and called the gauge factor.
The change in the resistance is given by Equation 9.14, which is the basic equation that underlies the use of metal strain gauges because it shows that the strain converts directly into a resistance change.

19. 9.5.4 STRAIN GAUGE MEASURING PRINCIPLE
The basic technique of strain gauge measurement involves attaching (gluing) a metal wire or foil to the element whose strain is to be measured. As stress is applied and the element deforms, the strain gauge material experiences the same deformation, if it is securely attached. Because strain is a fractional change in length, the change in strain gauge resistance reflects the strain of both the gauge and the element to which it is secured.

20. 9.5.5 STRAIN GAUGE MATERIAL
Strain gauges are made of materials that exhibit significant resistance change when strained. This change is the sum of three effects. First, when the length of a conductor is changed, it undergoes a resistance change approximately proportional to change in length. Second, in accordance with the Poisson effect a change in the length of a conductor causes a change in its cross-sectional area and a resistance change that is reverse proportional to change in area.  

21. Third, the piezoresistive effect, a characteristic of the material, is a change in the bulk resistivity of a material when it is strained. All strain gauge materials exhibit these three properties, but the piezoresistive effect varies widely for different materials.
Strain gauges are made of specially formulated alloys with relatively large piezoresistive effects. Silicon alloys are commonly used to manufacture of strain gauges.

22. 9.5.6 SPECIFICATION OF STRAIN GAUGES
In Ω Resistance
In watt Power
Gauge factor;
In mm*mm Dimensions
Are the main specification of the electric strain gauge
Strain gauges are available commercially with nominal resistance values from 30 to 3000 Ω, with 120, 350, and 1000 Ω . being the most common values.

23. A fundamental parameter of the strain gauge is its sensitivity to strain, expressed quantitatively as the gauge factor (G). Gauge factor is defined as the ratio of fractional change in electrical resistance to the change in length strain
Eq. 9.15
The Gauge Factor for metallic strain gauges is typically around 2.

24. 9.5.7 STRAIN GAUGE CIRCUIT WHEASTONE RIDGE CIRCUIT
In practice, the strain measurements rarely involve quantities larger than a few millistrain . Therefore, to measure the strain requires accurate measurement of very small changes in resistance. For example, suppose a test specimen undergoes a strain of A strain gauge with a gauge factor of 2 will exhibit a change in electrical resistance of only 2.(500 x 10-6) = 0.1 %. For a 120 Ω gauge, this is a change of only 0.12 Ω.

25. To measure such small changes in resistance, strain gauges are almost always used in a bridge configuration with a voltage excitation source. The general Wheatstone bridge consists of four resistive arms with an excitation voltage, VEX, that is applied across the bridge.
The bridge is said to be balanced (output voltage, Vo, is zero)if
R1R3=R2R

26. Any change in resistance in any arm of the bridge will result in a nonzero output voltage.
The output voltage Vo could be determined by the following equation
Eq. 9.17

27. SINGLE ACTIVE (QUARTER)BRIDGE
If we replace the resistance in one arm of the bridge with an active strain gauge, any changes in the strain gauge resistance will unbalance the bridge and produce a nonzero output voltage.

28. If the nominal resistance of the strain gauge is designated as RG, then the strain-induced change in resistance can be expressed as
Eq. 9.18
The bridge equation 9.16, can be rewritten to express Vo as a function of strain
Eq. 9.19

29. HALF BRIDGE
The sensitivity of the bridge to strain can be doubled by making both gauges active in a half-bridge configuration. This half-bridge configuration, yields an output voltage that is linear and approximately doubles the output of the quarter-bridge circuit, Eq.9.20.
Eq. 9.20

30. FULL BRIDGE
Finally, you can further increase the sensitivity of the circuit by making all four of the arms of the bridge active strain gauges in a full-bridge configuration. the output voltage is given in eq
Eq.9.21

31. 9.5.8 SIGNAL CONDITIONING FOR STRAIN GAUGE
Bridge Completion
Strain gauges are offered in several different configurations: quarter-bridge, half-­bridge, and full bridge. For quarter and half-bridge strain gauges. The used instrumentation should provide bridge completion, adding the necessary resistors to complete a Wheatstone bridge:
Excitation
Strain gauges require voltage excitation to generate a voltage representing strain. This voltage source should be constant and at a level recommended by the strain gauge manufacturer.

32. Amplification
Strain gauges typically provide small signal levels. It is therefore important to have accurate instrumentation to amplify the signal before it is displayed.