July 26-29, 2010
Stress and Strain Measurement
• Safe Load Level monitoring
• Force (indirect measurement by measuring strain of a flexural element)
• Pressure (by measuring strain in a flexible diaphragm)
• Temperature (by measuring thermal expansion of a material)
Strain Gauge (SG)
• Basic Device o
Bonded metal foil – a thin foil of metal (Constantan) o
A grid pattern o
Thin plastic backing material (polymide) o
Large metallic pads for soldering o
Total resistance of the SG is given by
R
= ρ
L
A where
ρ – foil metal resistivity ( Ω -m)
L – total length of the grid line (m)
A – the grid line cross-sectional area (m 2 )
Figure 1. A Typical Strain Gauge

A Cantilever Beam Mounted with a SG
• A SG mounted on a cantilever beam of uniform and symmetric cross section.
• A push-down normal force (stress P is added at the unsupported end), as shown in Figure
2.
P
A
δ x
L
A SG mounted on a cantilever beam of uniform and symmetric cross section
Figure 2
• As the object is deformed, the SG foil is deformed, causing its electric resistance to change
• For the strength materials, the deflection at the unsupported end is δ =
3
P
⋅
⋅
E
L
⋅
3
I
, where P
= normal force, L = length of beam, I = moment of inertia, and E = modulus of elasticity
• The force applied to a solid object resulting the deformation of the object.
• The effect of applied force is called Stress and the resulting deformation is called Strain .
• The Tensile Stress-Strain o
A tensile force applied to an object cause the object to elongate or pull apart o
The stress (unit of pressure) or tensile is defined as
Tensile Stress
=
F
A
(N/m 2 in SI unit or lb/in 2 in the English unit) where F = applied force in N
A = cross-sectional area of the object in m 2 o
The resultant strain is defined as the fractional change in length:
Tensile Strain
ε =
Δ l l

where Δ l l
= change in length in m ( or inch)
= original length in m (or inch)
• The beam develops a tensile stress on the top surface, and total surface stress σ at location A is
σ =
P
⋅ x
⋅ ( t / 2 )
=
E
⋅ ε
I
where t = beam thickness, I = moment of inertia
E = modulus of elasticity or Young’s modulus is given by
=
Stress
=
F
/
A
E
Strain
Δ l / l
For aluminum beam E = 6.89 x 10 10 N/m 2
Therefore we can find P in terms of other terms:
P
=
2 ⋅ x
I
⋅
⋅ t
σ
=
2 I x
⋅
⋅
E t
⋅ ε
Then substituting the applied stress P into deflection δ equation: δ =
3
P
⋅
⋅
E
L
3
⋅
I
δ
ε
=
P
3
Solve ε
=
3
3
⋅
⋅
2 x
⋅
E
⋅
⋅
L
⋅
L t
3
I
⋅ δ
=
2
I
3 ⋅
⋅ x
E
⋅ t
⋅
⋅
ε
E
⋅
⋅
L
3
I
=
2 ε
3 x
⋅
⋅
L
3 t
Gauge Factor (GF)
• Electric Properties o
Attached the SG to an object by a suitable adhesive o
As the object is deformed, the SG foil is deformed, causing its electric resistance to change
• GF is given by
GF
=
Δ
R
ε
/
R where Δ R = change in resistance due to stress ( Ω )
R = original resistance of strain gauge (
ε = strain (in/in, m/m, cm/cm, etc)
Ω )

Example 9.1
If a 120 Ω strain gauge with a GF = 2.0 is used to measure a strain of 100micro ε or 100 με , how much does the resistance of the gauge change from the unloaded state to the loaded state?
Answer:
From the equation:
GF
=
Δ
R
ε
/
R
We can solve it for Δ
R :
GF
Δ
R
⋅
=
ε
R
=
⋅ ε
Δ
R
R
⋅
GF
Δ
R
=(120*100*10 -6 )*2.0 = 120*0.0001*2 = 0.024
Ω
Measuring Δ
R
with a Wheatstone Bridge
• Wheatstone bridge – a circuit for measure small change in resistance
• A four-resistor network with a DC voltage source
• For the static balanced mode, as shown in Figure 3. o
R2 and R3 are precision resistors; R4 – precision potentiometer; R1 – the strain gauge o
Adjust R4 until voltage between A and B is zero volt:
V
AB
= V
A
– V
B
= 0 or
V
A
= V
B
V
I
1
A
R
1
=
I
=
1
I
R
2
1
R
2
V
B
=
I
2
R
2 o
Since we will use a High Impedance Volt Meter to measure the voltage V the meter will draw any current. This condition makes I1 = I4 and I2 = I3.
AB
, so
I
1
=
I
4
=
R
1
V ex
+
R
2
I
2
I
3
= =
R
2
V ex
+
R
3
Substitute I
1
and I
2
into
I
1
R
1
=
R
1
R
4
=
R
R
3
2
I
2
R
2
, we obtain the following expression:

Vex
So we know R2 and R3 accurately, and note the adjusted R4 value for calculating unknown R1 (strain resistance) as
R
1
=
R
2
R
4
R
3
A
R2
R3
I2
I1
I3
I4
R4
B
R1:
Strain gage
Volt
Meter
High
Impedance
Voltmeter
Static Balanced Bridge Circuit

Dynamic Unbalanced Bridge Circuit
• The circuit is shown in Figure 4
Figure
• The output voltage Vout can be expressed as
V out
=
I
1
R
1
−
I
2
R
2
= −
I
1
R
4
+
I
2
R
3
• The input voltage Vex can be express as
V ex
=
I
1
(
R
1
+
R
4
) =
I
2
(
R
2
+
R
3
)
• Solving for Vout in terms of Vex
V out
=
V ex
(
R
1
+
R
4
−
R
2
R
2
+
)
R
1
R
3
• When the bridge is balanced: Vout = 0 and R1 has a known value
• When R1 change value, as the strain gauge is loaded, the Δ Vout is related to Δ R
Δ
V out
V out
=
R
1
R
1
+
+
Δ
R
Δ
1
R
1
+
R
4
−
R
2
R
2
+
R
3
Rearrange the equation to give
Δ
R
1
R
1
=
(
R
4
R
1
1 −
(
Δ
V out
Δ
V ex
V out
V ex
+
−
R
2
R
+
2
R
3
)
R
2
R
+
2
R
3
)
− 1

References
• Strain Gages, Accessories and Instruments, http://www.omega.com/toc_asp/subsectionSC.asp?subsection=E02&book=Pressure
• Strength of Materials, http://en.wikipedia.org/wiki/Strength_of_materials
• Strain Gauge Measurement – A Tutorial, National Instruments Application Note 078, http://www.eidactics.com/Downloads/Refs-Methods/NI_Strain_Gauge_tutorial.pdf
• Strain Rosette for Strain Measurement, efunda Engineering fundamentals, http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/strain_gage_rosette.cf
m
• Electrical Resistance Strain Gage Circuits, http://www.ae.gatech.edu/people/jcraig/classes/ae3145/Lab2/strain-gages.pdf
• Stress, Elasticity and Deformation, http://arch.umd.edu/Tech/Tech_II/Lectures/Stress_Elasticity_and_Deformation.pdf
• Stress and Strain Formulas, Engineering Edge, http://www.engineersedge.com/strength_of_materials.htm
• Beam Diagram and Calculator Input, efunda Engineering fundamentals, http://www.efunda.com/formulae/solid_mechanics/beams/casestudy_display.cfm?case=c antilever_uniformload
• Structure Beam Bending Equation/ Calculation Cantilevered Beam with One Load
Applied at End, http://www.engineersedge.com/beam_bending/beam_bending9.htm