This document provides an overview of strain and stress measurement techniques. It discusses tensile testing and how axial stress results in both axial and lateral strains. It also covers biaxial stress states, 3D stress-strain relationships, and the use of strain gages to measure stress. Strain gages operate on the principle of variable resistance, where an applied strain causes a change in electrical resistance. Signal conditioning circuits like the Wheatstone bridge are used to detect small resistance changes indicating strain. Multiple strain gages can be arranged in rosettes and oriented to measure stresses in different directions. Proper calibration is important for accurate strain measurement.

1. Chapter 11
Strain and Stress:
Measurement and Analysis
ME 310 – Engineering Experimentation
I
Spring, 2004
University of Kentucky

2. s/Strain Curve (1-D stress state
Tensile test (axial stress/strain)

3. Axial and Lateral Strain
Positive strain in the axial direction
results in negative strain in the
lateral or
transverse direction. The ratio
is known as Poisson’s ratio, a material
property which is constant over the
elastic
range

4. Biaxial Stress (on a surface):
Consider stress in the x and y
direction applied separately:
εx results in a strain σx/E and a
lateral strain – εx = -νσy/E; a
similar effect occurs when σy is
applied. The net strains are
therefore:
Solving for the stresses:

5. 3-D Stress State & Stress/Strain Relations

6. The strain Gage
Bonded wire type: Foil strain
gages:

7. Strain Gage Principle: Variable Resistance
Electrical Resistance: a material’s resistance to the flow of
electricity (Unit: ohm)
e.g., a length of wire L with cross-sectional area A:
ρ = electrical “resistivity” of the material (ohm•m)
Variable Resistance Sensor Examples:
Strain Gage
Thermistor
Thermocouple

8. Strain Gage Principle: Variable Resistance

9. Example:
A certain strain gage has a gage factor of 2.0 and a
nominal resistance of
120 ohm. What resistance change should one expect
for a strain of 1 x
10-6 (or 1 μstrain or 1 ppm)

10. Stage 2: Signal Conditioning
• 2 Most Common Circuits Used for
Resistance
Type Strain Gages
– Voltage dividing potentiometer
(Ballast circuit)
– Wheatstone Bridge
• Must be very sensitive circuits with
high
resolution
– Strain is commonly very small (1
microstrain)

11. Example (cont.):
If we use a ballast circuit to detect this
change in resistance, what will the
change in output voltage be if the
ballast resistor is 120 ohm and the
input voltage is 8 V? Ballast circuit schematic
Thus, our voltage indicator would have to detect one part in 106, which is
not feasible

12. Dynamic Strain Measurement:
Here the ballast circuit can be used if we put a capacitor in
series with it
to remove the dc component.

13. Wheatstone Bridge Circuit (Review)

14. Wheatstone Bridge Circuit
Gage 2 is located close to Gage 1 to
compensate for temperature changes
(the bridge is rebalanced before the
load is applied
This is, of course, the same as the
voltage change for the ballast
circuit, but here we do not
have the 4 V offset to contend with.
The output is small, but can be
amplified (and filtered
to remove noise), as required.

15. Multiple arms sensitive to strain
Using more than one gage can multiply
output and compensate for temperature
and other unwanted strains

16. Multiple arms sensitive to strain
The bridge constant k:
k=2 for the situation shown
output of bridge
output of primary gage
k =

17. Multiple arms sensitive to strain

18. Gage Orientation and Interpretation of Results
Gage Orientation and Interpretation of Results

19. Example
Gage 1 is sensitive to axial strain as
well as temperature change AND any
unintentional bending.

20. Strain Gage Rosette
Gage located on shell of cylindrical pressure vessel.
In many situations, one is interested in multi-axial stress, such as
the biaxial stress found in pressure vessels.
To completely measure the stress state it takes at least three
strain gages. This is called a strain gage rosette. Use Mohr’s
circle to determine principal stresses.

21. Calibration of Strain Gages
What is the normal calibration procedure? Why won’t this work for strain gages?
• introduce resistance change via shunt resistor
• determine output voltage
• equate to “equivalent” strain

22. Example
If the output voltage due to this resistance change is –
100 mV, then the
sensitivity is

23. Basic Strain Measuring System
To scope or
meter readout or
DAQ device

24. Note to students:
See lab notes for further
details.

25. Chapter 12
Measurement of Vibration,
Acceleration, Force and
Torque

26. Goals of the Chapter
• To understand the principles of operation of
typical
vibrometers and accelerometers
• Understand the measurement of force using
– mechanical weighing systems
– transducers based on elastic deformation
– strain-gage load cells
– piezoelectric load cells
– hydraulic and pneumatic systems
• Understand the measurement of torque using
mechanical,
hydraulic, electric and passive dynamometers
• Other topics: distance (length), mass, velocity,
etc.

27. Vibration and Acceleration

28. Mechanical Motion Measurands:
• Displacement (linear or angular motion)
• Velocity
• Acceleration
• “Jerk”
Usually, these quantities vary with time:
• Vibration: a continuous signal that varies
with time
• Shock: a transient and/or interrupted
time signal
Examples of each?

29. Information Usually of Interest:
• Amplitude
• Frequency
• Harmonic components (Fourier
spectrum)
• Statistical information (e.g.,
probability that
displacement exceeds a certain value).

30. Classification of Motion Transducers:
• Vibrometers: sensitive to either displacement or
velocity (called “vibration pickups” by industry)
• Accelerometers: sensitive to acceleration
We often are forced to measure a certain quantity
simply because it is proportional to our transducer
output (e.g., the accelerometer) which must then be
transformed to the quantity we are most interested in
(such as displacement).

31. The Seismic Instrument (Vibrometer)
Natural frequency of primary
transducer is (K/M)1/2
• When Ωt >> (K/M)1/2, Sm~0, due
to M’s inertia
• Secondary transducer
measures either displacement
or velocity of housing (which is
the same as the support).
• Voltage dividing pot
• Variable capacitance
• Strain gage(s)
• Variable reluctance (magnet
with a coil of wire) – most
common – velocity
transducer

32. Frequency Response of the Vibrometer:

33. Vibrometers (cont.)
• Must be used at frequencies well
above
resonance of the primary transducer
• For use at low frequency,
vibrometers need to be
large (large mass)
• This makes them delicate
• Calibration is cumbersome

34. Piezoelectric Accelerometers
• By far the most common method to measure
either vibration or shock
• High sensitivity
• Small (compared to vibrometers)
• Rugged
• High output impedance and low charge
output
means that a suitable charge amplifier must
be
located close to the transducer
• ICP accelerometers have built in electronics

35. The Piezoelectric Principle

36. ession Mode Piezoelectric Accelerometer:

37. Shear Mode Accelerometer:

38. Note ICP circuitry – charge amplifier – built into these accelerometers
Compression and Shear Mode Accelerometers:

39. Signal Conditioning the Accelerometer Output:
Piezoelectric devices have a high output impedance
and therefore require
signal conditioning to transform them to a low output
impedance.

40. Useful Frequency Range:
Useful frequency range of a piezoelectric accelerometer:
• Low-frequency limit determined by capacitive nature of
the device (no DC output)
• High-frequency limit determined by the natural
frequency
Practical ranges vary from a few hertz to 50 kHz

41. The Laser Vibrometer (Michelson-type Interferometer):
Reflector tape is attached to vibrating surface
• Number of fringes per unit time (counted by
photodetector) is
proportional to velocity
• Noncontacting measurement
• Measurement of hot or rotating surfaces
• Standoff distance is about 1 m or less
Figure 12.7

42. The Laser Vibrometer:
Measuring the modes of
vibration of a brake disc
assembly

43. Force, Torque & Power

44. The Analytical Balance
• Operates on the principle of moment comparison
• Both equal- and unequal-arm balances are commonly used
• When “null” reading is achieved, the arm is in balance, and
the moments are equal

45. Multiple-Lever Systems
• Used to measure large
weights
• Large weights W are
measured in terms of pan
weight Ws and poise weight
Wp (zeroed with
counterpoise)
• Assuming Wp is at 0, a
weight W placed on the
platform will be balanced by
a pan weight Ws such that
(equilibrium)
• Moving the poise weight Wp
to the right along the beam
allows fine adjustment to
achieve equilibrium with

46. The Pendulum Scale
• Downward motion of load
rod causes sectors to rotate
about points A, causing
counterweights to swing out
• Motion stops when load and
counterweight moments are
in equilibrium
• Downward motion of
equalizer bar is transformed
to rotary motion via a rack
and pinion
• Dial of weight scale is
calibrated to read true
platform weight

47. Elastic Transducers
• In elastic transducers, application of a load to an
elastic
member typically results in a linear deflection;
deflection is
calibrated to applied load
• Factors to consider:
– sensitivity (small forces to cause large
displacements)
– settling time (depends on damping and natural
frequency)
– stress (don’t want to exceed elastic limit of member –
may
cause failure, or nonlinearity in the form of hysteresis)
– calibration adjustment (e.g., to remove deflection due
to
weighing pan)

48. Elastic Transducers (see Figure 12.27)
What are some other ways to determine the force F ?

49. Elastic Transducers: The Proving Ring
• Considered to be the standard for
calibrating materials testing
machines
• Range in capacity from 300 to
300,000 lbf [1300 N to 1.3 MN]
• Repeatable measurements are
obtained with the aid of a
vibrating
reed (A) – contact with the
micrometer (B) is indicated by a
decrease in the vibration
amplitude
• Assumption: thin ring (radial
thickness of ring is small
compared
to radius)

50. Elastic Transducers: The Proving Ring

51. Strain-Gage Load Cells
• Measures load in terms of
strain
(instead of using total
deflection
as a measure of load)
• Resistance strain gages are
typically mounted on one of
the
elastic members shown
previously
• Use of more than one gage
provides temperature
compensation
Which gage or gages are for
temperature
compensation? What is the bridge
constant here?

52. Proving-Ring Strain-Gage Load Cells
• Left: bridge output is a function of bending
strains only
• Right: axial strains are not canceled out

53. Strain-Gage Load Cells: Temperature
Sensitivity
• Temperature causes dimensional variation (e.g. cross-sectional
area of steel tension member changes 0.15% per 100 oF)
• Temperature causes variation in Young’s modulus: ~ 2.5%
decrease in E per 100 oF increase
• Young’s modulus change is the
more important factor: introduce
temperature compensation resistors
Rs to decrease sensitivity of strain
gage bridge as E decreases

54. Piezoelectric Load Cells
• Useful for measuring impact or dynamic loads
• Essentially a capacitive sensor – cannot measure
static loads
• Typical output: 10 – 20 pC/lbf (need charge
amplifier)
• Multi-axis cells can be obtained by combining
quartz elements
with different properties (obtained by varying
slicing plane)
• Desirable qualities:
– Wide ranges of working load
– Excellent frequency response
– high stiffness
– High resolution
– Small size

55. Hydraulic Load Cell
• Applied load is balanced by a pressure acting
over a resisting area;
pressure becomes a measure of the applied
load
• Floating piston is used to eliminate accuracy
problems caused by
friction
• Hydraulic load cell: bridge ring provides seal
and allows small
piston movement
• Capacities to 5,000,000 lbf
and accuracies of +/- 0.5%
of reading or +/- 0.1% of
capacity have been attained

56. Pneumatic Load Cells
• Principle similar to hydraulic load cell –
pressure measurement
• Designed to regulate the balancing
pressure through the use of a
position-controlling bleed valve –
diaphragm seeks position that
will result in proper airflow to support
the load
• “Roll” provided to allow diaphragm
area to remain constant
• Capacities of up to 80,000 lbf;
error as small as 0.1% of full
scale

57. Torque Measurement: It’s about power
• Every engine is designed to develop power; by definition
power = knowledge
power = work ÷ time
= force ﾗ distance ÷ time
=force ﾗ velocity = torqueﾗ
rotation rate
• Power output required to move a vehicle at a given
velocity; here,
force can be equal to drag or the desired thrust for
example
• Since the vast majority of engines & motors develop
power
through rotating elements, measuring power through
torque is
important
– Pumps, fans, gas turbines, steam turbines
– Automobile engines -- Otto, Diesel, Stirling (rotary)

58. Torque Measurement Dynos
• Absorption dynamometers: dissipate mechanical
energy; used to measure torque of power sources such
as engines or electric motors;
power = torque x angular velocity
P = Tω
[watts (W) = N·m/s = J/s = hp · 746]
• Driving dynamometers: both measure torque and
power and supply energy to the tested device; used to
measure performance of pumps and compressors
• Transmission dynamometers: passive devices that
simply sense torque. Also known as torque meters

59. Prony Brake
Most common absorption
dynamometer; dry friction
converts mechanical energy to
heat

60. Example
A Prony brake is used to measure the torque of a small
engine. The force
on the platform scale is F = 120 N (or 26.98 lbf), the moment
arm r = 75
cm (or 2.46 ft) and the angular speed is 20 rev/s. How much
torque and
power is the engine producing at these conditions?
power = torque x angular velocity
[watts (W) = N·m/s = J/s = hp · 746]

61. • Uses fluid friction rather than dry
friction to provide the dissipation;
housing is free to rotate on trunnion
bearings
Water Brake
Capacity is a function of speed and
water level
• Power absorption is approximately a
function of the cube of the speed
• Water brakes have larger capacity
than prony brakes because heat is
dissipated more easily through the
water

62. Eddy-current dynamometers
• Absorption type only – no drive capability
• When a conducting material moves through a magnetic flux
field,
current flows. If the conducting material is an isolated
conductor,
only circular (eddy) currents occur; these produce a magnetic
field that opposes the magnetic field of the field coils
• This causes the cradled external housing to turn, allowing the
torque to be measured
• Mechanical energy is
converted to heat which
must be dissipated,
usually by water cooling

63. Other types of electric
dynamometers
• Cradled DC Dynamometer (DC motor/generator)
– Both absorption (DC generator) and driving (DC motor)
modes
– Cradling in trunnion bearings permits determination of
reaction torque
DC or AC motors – these are cradled to permit
torque calculation
• In all cases, bearing friction must be minimized

64. Note to Students:
• We have skipped sections 12.1-12.4 and
12.6 and
have chosen to focus on vibration,
acceleration,
force, torque and power in Chapter 12.
Please
read through the sections on linear distance,
displacement, mass, and velocity
measurements.