Chap12 Strain Measurements

7/28/2019 Chap12 Strain Measurements

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Strain Measurements

Engineering calculations are often based on stress. If we want to do

experiments to confirm our theory, we need to measure the result ofstress rather than stress directly. Stress results in the deformation of

material, which is called strain. For most engineering materials, there

is a rather simple relationship between stress and strain.

a Ea

a dL

LL

2L

1

L1

LL

1


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Lateral Strain, Poissons Ratio

If we stress a rod by pulling on it,and is stretches axially as a result, it

will also get thinner. This behavior

is quantified by Poissons ratio:

lateral strain

axial strain

L

a

This is a property of the material.


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General Stress States

y y

E

x

E

x x

E

y

E

x E x y

1 2

y

E y x

1 2

These equations relate the 2-D stress field to the

2-D strain field. I will assume that you alreadyknow this.


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We measure strain in one or more directions and infer the stress state

from that. In general, in order to know the 3-D stress state, wewould need 3 components of strain. In some cases (like pure axial

stress) we may be able to reduce the number of required

components. I will teach you more about the instrumentation side of

this topic, and it will be left to you to figure out how to get the stress

state from the measurements.


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12.3 Electrical Resistance Strain Gage

Ruge, 1940s


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Rosettes


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Installation


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The gauge length limits the spatial

resolution of the sensor.

Connection to the bridge is made

at the solder tabs.

The backing material needs to be

made of something that can:

Withstand the temperatures

encountered

Transmit strain but electrically

insulate

Accept the bonding adhesive


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12.4 Metallic Gauges

R LA

LCD2

If you have a conductor of resistivity , the resistance across that

conductor is

If you strain this conductor axially, its length will increase while its

cross sectional area will decrease. Taking the total differential ofR,

dR R

d

R

LdL

R

CD2 d CD2

1CD2

Ld dL 2L dDD

dR

R

dL

L 2

dD

D

d


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Metallic Gauges

dR /R

dL /L1 2

dD/D

dL /Ld/

dL /L

dR

RdL

L 2

dD

Dd

a dL

L

L dD

D

L

a

FdR /R

dL /LdR /R

a1 2v

d/

dL /L

For most strain gauges, = 0.3. If the resistivity is not a function of

strain, thenFonly depends on poissons ratio, andF~ 1.6.

Gage factor


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dRR dLL 1

d

1

1

E

d/

dL1/L

dR /R

dL /L1 2

1E

Piezoresistance

Coefficient


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Gage Factor

1

F

R

R

F dR /RdL /L

dR /Ra

1 2v d/dL /L

FandR are supplied by the

manufacturer, and we measureR.


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Example

A typical strain gauge uses constantan (55% copper, 45% nickel)

which has a resistivity of 49 X 10-8

W

m. The strain gauge must be120W nominally (why?). If the diameter is 0.025 mm, how long

does it need to be?

R eL

AcL = 12 cm


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12.5 Selection and Installation

Read on your own


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12.6 Circuitry for Metallic Strain Gage

Most commercial strain gages are 120 W, have a gage factor near 2, and

can measure 1 microstrain (1 part in a million).

1

F

RR

R 120 2 1E 6 0.00024W

Clearly, our work is cut out for us in terms of the measurement.

h i id i i


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12.8 The Strain Gage Bridge Circuit

eo

ei

R

1/R

4

2

R1/R

1

F

RR

eo eiF

4 2F


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eo eiF

4 2F

If we assume some typical values for the excitation voltage (8V) and the

gage factor (2), then we can see that the second term in the denominator

is not significant:

eo 16

4 4

eo eiF

4

so

id i h d i


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12.8.1 Bridges with 2 and 4 strain gages

The bending strain on the top gage is equal

and opposite of the one on the bottom.


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eo eiR

2

R1

R2

R

4

R3

R4

makeR2 =R4 =R

eo eiR

R1

R

R

R3

R

eo eiR

R1

1R

R3

1

eo

ei R

1

R1

1

R3

M l i l G B id


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Multiple Gauge BridgeMost strain gauge measurement systems allow us to make 1, 2, 3 or all 4

legs of the bridge strain gauges. There are many reasons to do this that

we will talk about now.

Going back to our fundamental bridge equations from chapter 6,

Eo EiR

1

R1

R2

R

3

R3

R4

Say that unstrained, all of these have the

same value. If they are then strained,

the resultant change isEo

is

dEo Eo

Rii1

4

dRi

Eo

M l i l G


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Multiple GaugesMake the following assumptions:

All gauges have the same nominal resistance (generally true)All gauges have matched gauge factors (must be purchased as set)

Then:

EoEi

F

4

1

2

4

3 Eo

A S i d C i


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Apparent Strain and Compensation

Things like temperature can change the resistance of a gauge and our

system may interpret this as strain. Sometimes our gauge may be

subject to strains other than the one we are interested in.

Compensation is removing these effects by using multiple gauges. As

an example, say you have a beam under axial stress and a bending

moment, and you are interested in the axial stress only:

x 12My /bh3 FN /bhThe two gauges see the

same axial strain but

opposite bending

strains eo

ei F

41 4

1

a1 b14

a4 b4e

oei

F

2a

T t C ti


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Temperature CompensationThe resistance of a strain gauge changes with temperature. In addition,

changing its temperature may cause strain in the gauge making it even

more sensitive to temperature.

C ti


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Compensation

12 8 2 B id C t t


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12.8.2 Bridge Constant

kA

B

k= the bridge constant

A = the actual bridge output

B = the output you would get with a single gage.

12 8 3 L d i E


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12.8.3 Lead-wire Error

Since we are looking at very small changes in resistance, the lead wires

can create significant errors. We handle this the same way we discussedfor RTDs.

We have wire especially made for strain gagemeasurements which has three conductors

12 10 T t C ti


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12.10 Temperature Compensation

R1

R2

R

3

R4

R1

RtR

2 Rt

R

3

R4

If the temperature of the

specimen changes, then

both gages will change their

resistance similarly

12 11 C lib ti


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12.11 Calibration

R RgRs

Rg Rs Rg

Rg2

Rg Rs

1

F

Rg

Rg

e 1

F

Rg

Rg Rs

12 16 1


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12.16.1

Multiple gages in series

12 17 1 C S iti it


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12.17.1 Cross Sensitivity

eL

Kt L /a po

1poKt100

Semicond ctor Strain Ga ges


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Semiconductor Strain Gauges

The gauges we have been talking about are made of metal. We can

also make them out of semiconductors, which is how the strain

gauges in our pressure sensors are made. These are dominated by the

piezoresistive component of the change in resistance and have

several advantages and disadvantages:

Pros:

Very high gauge factors (up to 200)

Higher resistance

Longer fatigue life

Lower Hysteresis

Smaller

High frequency response

Cons:

Temperature sensitivity

Nonlinear output

More limited on maximum strain

Mostly used for construction of

transducers

Hysteresis of Strain Gauges


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Hysteresis of Strain Gauges

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Chap12 Strain Measurements

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