L9-Stress & Strain

7/30/2019 L9-Stress & Strain

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STRAIN & STRESSMEASUREMENT and ANALYSIS

Ch-12; Beckwith

All structural members deform tosome extent when subject to

loadingresult in displacement orstrains.

For simple axial loading

dimensionlinearstrainedfinal

lengthgageordimensionlinearstrainaxial

2

1

11

12

L

L

L

L

L

LL

L

dL

a

a

2

Since the strain quantity

is very smallcommonlymultiplied with onemillionresulting numberis called microstrain (-strain or ppm).

The stress-strain relationfor a uniaxial condition(such as simple tensiontest or at the outer fiber ofa beam in bending

In simple uniaxial loadingin elastic range, thelateral stress results asper following relationship

limitalproportionthebelowkeptisstress

theaslongsomaterialsmostforconstantaisE

stresstheofdirectionin thestrainthe

stressuniaxial

modulussYoung'

a

a

a

a

E

E

strainlateral

ratiosPoisson'

L

a

L


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In case of biaxial loading When x is applied

Strain in the x direction = x

/E

Strain in y-direction because of

Poissons ratio = - x/E

When y is applied

Strain in the y-direction = y/E

Strain in y-direction because of

Poissons ratio = - y/E

Net strains are expressed as

Solving equations simultaneously for x and y

When a stress z exists acting in the third direction,

the more general 3-D relations are

EE

xy

y

yx

x

and

22 1

)(

and1

)(

yy

y

xx

x

EE

)]([1

)]([1

)]([1

yxzz

xzyy

zyxx

E

E

E

4

Strain Measurement

Electrical type strain gages consist of: Simple resistive

Capacitive

Inductive

Photoelectric

Resistive type are most common

Inductive and capacitive gages are more rugged, and maintain calibration over a long period of time

Inductive gages are used for permanent installations suchas on rolling-mill frames for monitoring roll loads.

Torque meters often use strain gages including inductiveand capacitive

Other strain measuring techniques include: Optical methods such as photo elasticity, Moir

techniques and holographic interferometry.


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The Electrical Resistance Strain Gage 1856Lord Kevin demonstrated that resistance of Cu and Fe wire

changes when these are subjected to mechanical strains. First wire resistance strain gage was made by Carlson by 1931. In 1938used of bonded wire gage 1950sfoil-type gage was introduced

6

The Metallic Resistance

Strain Gage Assume an initial conductor length

L having x-section area CD2 (D issectional dimension and C isproportionality constt.)

If the conductor is strained axially intension, the lateral dimensionshould reduce as a function ofPoissons ratio.

Assuming that resistivity shouldremain constt., with strain, then thegage factor should be a function ofPoissons ratio alone, and in the

elastic range should not vary muchfrom 1+2(0.3)=1.6.

F for metallic gages is essentially aconstt. in the usual range ofrequired strains, and its value(determined experimentally) isreasonably consistent for a givenmaterial.

)4(/

/21

/

/

/factorGage

/

/ratiosPoisson'

strainlateral

strainaxialNow

/

/

/

/21

/

/

writtenbemaywhich

2

Eq.1byEq.3Dividing

)3(2)(1

)2()(

2)(

atingDifferentiC.exceptchangequantitiesallthat

assumemaywestrainedisconductortheif

)1(

2

22

2

2

LdL

dRdR

LdL

RdRF

LdL

DdD

D

dD

L

dL

LdL

d

LdL

DdD

LdL

RdR

d

D

dD

L

dL

R

dR

D

dDLdLLd

CD

CD

LDdDCdLLdCDdR

CD

L

A

LR

a

L

a


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8

Selection and Installation Factors for Bonded Metallic Strain Gages

Five gage parameters to govern the performance:

1) Grid material and configuration2) Backing material3) Bonding material and method4) Gage protection5) Associated electrical circuitry

Desirable properties of grid include:1) High gage factor, F2) High electrical stability3) High yield strength4) High resist ivity,

5) Low temperature sensitivity Most important worrisome factor Two factors involved even with compensation circuits

Differential expansion between grid and grid paper Change in resistivity with temperature

6) High endurance limit

7) Good workabi lity8) Good solderability or weldability9) Low hys teres is10) Low thermal emf when joined to other material

Must be avoided if dc circuitry used

11) Good corrosion resistance Corrosion can lead to development of miniature rectifier More serious in ac than in dc


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Circuitry for the Metallic Strain Gage

The relationship between gage factor, resistivity and straincan be expressed as

Three circuit arrangements are used for this purpose1. The simple voltage-divider of potentiometer or ballast circuit2. The Wheatstone bridge

3. The constt. current circuit

0.0002%ofchangeresistanceatoamountswhich

00024.0)000001.0)(120)(2(

gageinmeasuredbemustchangeresistance

ngcorrspondithehence,eqpt.,commercial

withdetectableareppm1ofStrain

120,0.2

areconstt.gagetypical

1

gg

g

g

g

FRR

RF

R

R

FThe Bridge Constant

k= A/B (bridge output / gage output)

k= the bridge constant

A = the actual bridge output

B = the output from the bridge ifonly a single gage, sensing maxstrain, where effective.

10


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Temperature Compensation

The Adjacent-Arm Compensating Gage

Fig.12.6 & 12.7 Initial electrical balance is obtained when

If the gages in arms 1 and 2 are alike and mounted on similarmaterials and if both the gages experience the same resistance shift,Rt, caused by temperature change, then

The bridge remains in balance and output is unaffected by thechange in temperature. In this case, the compensation gage is calleda dummy gage.

Self-Temperature Compensation

If the temp compensation is not obtained by dummy gage, e.g. the

temp gradient between the two parts is sufficiently great, or aballast circuit is used rather than bridge circuitIn thesesituations, self-compensation is highly desired.

4

3

2

1

R

R

R

R

4

3

2

1

R

R

RR

RR

t

t

12

Two types: Melt gage

Some control over the temp sensitivity of grid material is possiblethrough proper manipulation of alloy and processing particularlycold working.

E.g. Fig.12.9; Practical compensation obtained in temp range50250 F

Dual-element gage Makes use of two grid elements connected in series in one gage

assembly. The two elements have different temp characteristics and are

selected so that the net temp induced strain is minimized whenthe gage is mounted on the specified test material.

Performance is similar to melt-type gage.


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Stress-Strain Relationships Strain gages are generally used for one of two reasons:

1. To determine stress conditions through strain measurements, or2. To act as secondary transducers calibrated in terms of such quantities as force,

pressure, displacement, etc. Require good grasp on stress-strain relationship

The Simple Uniaxial Stress Situation

limitalproportionthebelowkeptisstress

theaslongsomaterialsmostforconstantaisE

stresstheofdirectionin thestrainthe

stressuniaxial

modulussYoung'

a

a

a

a

E

E

strainlateralratiosPoisson'

L

a

L

14

The Biaxial Stress Situation

If the test is on a free surface, the condition is termed asbiaxial. E.g. outer surface or shell of a pressure vessel.

Hoop stresses acting circumferentially

Longitudinal stresses

22 1

)(and

1

)(

yy

yxx

x

EE

Eqs. 12.4

These eqs. are useful in twodirections and gives complete

stress-strain picture only whentwo directions coincide withthe principal directions.

If the principal directions arenot known, at least 3 strainmeasurement must be, madeusing a 3-element rosette.


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Gages connected in series Fig. 12.19 Percentage change in resistance, dR/R is counted not dR alone. Resistance change in one arm will be three times what it would be for

single gage. Total resistance will also be three times as great. The only advantage is that of averaging to eliminate incorrect readings

resulting from eccentric loading.

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Special Problems

Cross-Sensitivity Strain gages are arranged with most of the strain-sensitive

filament aligned with the sensitive axis of the gage. However, unavoidably, a part of grid may be aligned transversely. The transverse portion senses the strain in that direction and its

effect is superimposed on the longitudinal output. This is known as cross-sensitivity. The error is small, seldom exceeding 2 or 3%.

Plastic Strains and the Post Yield Gage These gages have been developed extending the usable range to

approx. 10% to 20%. Grid material in very ductile condition is used, which is literally

caused to flow with the strain in the test material.

The primary problem in developing an elastic-plastic grid is toobtain a gage factor that is the same under both conditions.

Fatigue Applications of Resistance Strain Gages In general, the vulnerable point is the discontinuity formed at the

juncture of grid and the lead wire. Strain level is also the most important factor in determining the life. Isoelastic grid material performs better than does constantan.


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Cryogenic Temperature Applications Performance of resistive gages can change unpredictably at sub-

zero temp. Adhesives and backings become glass-hard and quite brittle.

Mechanical properties of certain grid materials are drasticallyreduced.

Large changes in the resistivities may be encountered. And theeffective values are dependent to a great degree on

Trace elements, and Previous mechanical working of the material

High Temperature Applications Max temp for short period use of gages:

Paper 180oF Epoxy 250oF Glass-filled phenolic base 600oF

Primary limiting factors are decomposition of cement and

carrier material For higher temp applicationsceramic base insulation must The grid may be of

The strippable support Free-element type, or Weldable type

18

Free element type gage involves constructing the gage on thespot.

Either brushable or flame-sprayed ceramic bonding material is used.

The process demands considerable skill and careful baking

Flame spray involves use of plasma-type oxyacetylene gun

Leads must be attached by spot welding

Lead-wire temperature-resistance variation may also presentproblems

A weldable strain gage consist of a resistance element surroundedby a ceramic-type insulation and encapsulation within a metalsheath.

Creep Creep in the bond between gage and test surface

It is of importance only in static strain testingprimarily in long

duration The loading cycle is not repeated

Under these conditions, the gage creep will result in direct errorsequal to the magnitude of the creep.

If the load can be slowly cycled, the creep will appear as ahysteresis loop in the result.

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L9-Stress & Strain