experimental stress analysis - strain gauges

8/3/2019 Experimental Stress Analysis - Strain Gauges

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COVENTRY UNIV ER SITY

Faculty of Engineering & Compu tin g

EXPERlMENT AL STRESS ANALYSIS

including

BRITTLE I~ACQUERS

STRAIN GAUeiE ROSETTES

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We have faith in m odem structures and m achines. Subconsciously w e assum e that engineers

have made calculations to verify that they w ill not fail in use and operation. This is rarelythe case. .

The civil engineer w ill rely on a mass of accumulated know ledge to select steel girders to

support a wall in a building at the design stage, but some engineering components are socomplex in their shape and loading that no em pirical data exists to ease the designers task.

In the past he has resorted to large safety factors to m inim ise the chances of catastrophic

failure. However. todays designer is subject to com mercial pressures and perform ance

criteria w hich dem and an efficient use of resources.

C onsequently experim ental stress analysis and strain m easurem ent techniques have gained.

prominence in rec en t ye ars.

These meth od s are p articularly u sefu l w hen d esig nin g ag ain st fatig ue failu re. A fatig ue crack

in a dynam ically loaded com ponent m ay not occur until thousands of load cycles have been

applied. and crack propagation to failure may take many thousands more cycles. An

experim ental analysis to predict regions of fatigue failure can obviate expensive failures and

justify the initial high cost of the equipm ent used.

The p rin cip al tech niqu es u sed are -

i) Brittle lacquers.

ii) Strain gauges.

i) BRITTLE LACQUERS

The component to be tested is sprayed with a co atin g w hich cu res in air or under the

ap plicatio n o f heat and becomes brittle. When loaded (in th e same manner as inservice, though to a far lesser intensity) the coating cracks w hen its 'threshold strain '

(strain sensitivity) is reached. S ince the crack w ill occur w here strain is g rea te st, an

instant indication of stress concentration is given. The cracks w ill be at right anglesto m axim um principal strains so are invaluable for determ ining the positioning of

strain g au ges fo r accu rate strain m easu rem en t.

Spraying lacquers on to com ponents is preferred to painting since a uniform thickness

is achieved. O ther lacquers are in wax or powder form and require preheating to

melt and even out the coating. Thicknesses vary but 1rnm is typical. Strain

sensitivity of the coatings is found from applying know n loads to rectangular section

'c oa t c alib ra tio n bars'. M easurem ent of component strain cannot be considered

accurate to more than 10-20% .

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APPUCATIONS FOR BRITILE COATINGS 1N EXPERIMENTAL STRESS

ANALYSIS

Itmaybe advantageous to use brittle lacquers for experimental stress analysis where:-

1) Determining an area of high stress concentration is required, rather

than a specific stress level.

2) Untrained personnel are used to conduct the analysis.

3) An inexpensive analysis is required.

4) Creep effects are to be observed.

5) Investigations in adverse environmental conditions are required.

6) Plastic yielding is investigated.

and 7) Where miniature components are to be analysed.

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Crack pattern in a

ceramic tile when

struck with ahammer.

Brittle lacq uer

crack pattern on arifle stock after

firing.

Brittle lacquer testto in vestigate

stresses ill the

lower jaw bone

when biting.

Crack pattern in a

fir tree coupling.



ii) RESISTANCE STRAIN" GAUGES.

PERSPECTIVE:-

In 1856 Lord Kelvin observed. that if a metal wire was strained its electrical resistancechanged. However. the s urfa ce s tra in of a component was not measured . u sing this

principle until 1938 in the U.S.A.

It is now the commonest method of performing experimental stress analysis. Though

it can be used on small components, its greatest application is on large structures like

aircraft fuselages where a great number of gauges can be employed and many

readings are automatically recorded and analysed.

CONSTRUCTION & PRINCIPLE:-

There are two main types of strain gauges.

a) Fine metal wire approximately O.025mm in dia. is formed into a grid and

bonded to a thin paper backing.

b) Metal foil gauges where the grid of wire filaments is obtained by "printing"

on to metal foil.

Metal Foil Stram Gauge

Strain in a component causes the gauge wire cemented to it to be strained and the

resulting change in resistance is measured by means of a Wheatstone bridge. The

four arms of the bridge exhibit resistance. One of these arms is the active strain

gauge. A battery supplies th e electromotive force and a galvanometer detects

potential difference across two apex points. To allow detennination of the directionsand magnitudes of principal stresses, gauges are normally bonded to components in

groups of three. Such a group is termed a rosette.

Wheatstone Bridge

The commonest gauge materials are copper nickel alloys (55%/45%) and nickel,chromium iron alloys. Resistances from 600 to 10,0000 ohms. Safe working current

is approximately 25mA.

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AXIAL & CROSS SENSITIVITYw

Strains are calculated using the formula -

e =1.dR

Ka.Rwhere e = strain

dR = change in resistance

Ka = gauge factor - a measure of the

axial sensitivity of the gauge.

& R = resistance.

Cross sensitivity (Kc) needs consideration in wire gauges. Itoccurs because some of

the grid wire is unavoidably normal to the gauge direction. Kc/Ka is generally about

0.02 and a correction factor is applied. Foil gauges have almost zero cross

sensi tivity .

TEMPERATURE & HUMIDITY.

Temperature can change the resistance of strain gauges. This can be solved by fixing

a dummy gauge with the same resistance, as one arm in the Wheatstone Bridge, so

that strain changes due to temperature affect each gauge equally and balance is

unaltered. Humidity causing corrosion can change wire resistance. Waterproofing

of gauges can be achieved by coating them in various sealants.

ATIACHING STRAIN GAUGES.

Component surfaces must be clear of rust, scale and grease and be slightly

roughened. A thin coat of cement is applied to the gauge which, when fixed to the

component must be bonded evenly and securely all over its surface.

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RSsUME OF R ELEV AN T FORMU LAE FOR A II-AXIAL STRESS SYSTEM

a1

:;: Principal Stresses" 281

= Principal Strains81

1 1 =Pois son's Ra t ioE =Modulu s o f E l a stic ity

8~ = strain along ox

y

8y = Strain along oy

' Y ; :; y : ; : Shearing strain

e, = Strain onplane at L O to ox

x 0' = angle (5) of principalstrain axes to plane ox

e = 8 cos2(1 + e sin28 + 'VsinOcosO~ ~ y I

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s T R A i N GAU G E AN AL Y SIS

WORKED EXAMPLES

1) A delta strain gauge rosette on an alum inium alloy component gave micros tr ai n

readings of 100, -400 and 250. Find the principal stress values and directions

( E = 70 GNlm 2, 1 1 = 0.29)

x

100

y e x lcr)

Using 8, = 8:&cos28 + 8,sin28 + ')'sin8cos8

Then s , % = 100

SIlO" = ( 100 x !)( E, X !)"Y if = -400 ..(1)

8240• = ( 1.00 x !)( 8, X ~ ) +"Y i f = 25a .. (2)

By adding (1) and (2) we find e , =-133.3(x 1<r)

'Y =750

Usinge 1 J { e ~ - e , , ) 1 + "/....1 = _ ( S + 8 ) ± ~ ~. -J

82 2 x Y 2

Then 81 = 1 (100 - 133.3) ± 21i(100 + 133.3)2 + 750282 2'

= +376

-409(x 1cr)

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Using

Using

a n d a =2

01 = 70 X 109

(376 - 0.29 x 409)Xl.0-61-0.292

01 = 19.67 X 106 N/m2 = 19.67 MPa

02 = 70 X 109

( - 409 + 0~29 (376» x 10-6

1-0.292

= - 22.8 X 106 N/m2 = -22.8 MPa

( J" I n >

tan 2 = - . . .. .. .. . . .. ;- ~ - - : -( e ; c - e,)

750ta n 2 (J = -=-:~--:--"""""!'"=----;:, .....

100 - ( -133.3)

= 3.2143

= 72.72

(J = 36.360

OR 126.360

Clockwise relative to strain e,

-8-



2) A rectangular rosette of strain gauges (shown) gave the following strain reading

Gauge A + 450 X 10-4

B + 200 x 1O~

c -200 x IQ-4

Determine the magnitude and directions of the principal stresses if E for the material

is 209 GN/m2 and U is 0.3

cB If 8I & 82 are the principal strains

and e~ the strain in a direction at

L8 to the direction of B I then we

can use the formulae -

* The normal form of this equation is

But 'Y sy = a on the principal axes.

Thus (1)

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Adding (1) and (3) 250 x 10' = ~l + e2 (4)

Substituting (4) in (2)

200 X 10-6 = 125 X 10-6 + { ( 8t - < = 2 ) Cos (900 + 28)

Substituting (4) in (1)

Dividing (5) by (6 )

75 X 10-6=

- { (81 - 82) sin 26

~ (61 - 82) Cos 2825 X 10-6

- n , 231 = tan 28

28 = - 13° OR 180 - 13 = 167°

6=83°30'

(So gauge A is 83 e 3 0' anticlockwise from the direction of 81 )

From (6)

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So

E, + £2 = 250 X 10-6

£1 - £2 = -667 X 10-6

Adding

2 e 1 = -417 x 10-6

... el = -208.5 x 10-6

Subtracting

2 E2 = 917 X 10-6

:. £2 = 458.5 X 10-6

209 X 109

-6lIi. - = - - - " " " ' : : " ' " ( -208.5 + 0..3 x 458.5) x 101 - 0.32

= -16 ..30 MPa

ur = 209 x 109 (458.5 - 208.5 x 0..3) x 10-6

1 - 0..32

= 90.94 MPa

NOTE: al is normally the largest principal tensile stress

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CONSTRUCIlNG A STRAIN CffiCLEFROM THREE KNOWN $TRAINS

SY S TE 1'v.IO F TIIR E E K NOWN STRA IN S

d b a

(~-aJ

o

x

1. On the line da mark off the known strains fa I Cb , Ec to an

appropriate scale.

2. Draw perpendiculars to da from a. b & c.

3. From a convenient point X~ an the perpendicular thro' B, draw linescorresponding to the known strain directions of ea & ea to intersect the

p erp en dic ula rs from a & c at A & C respectively .

4. Construct perpendicular bisectors from XA & XC to meet at 'Y' which

becomes the centre of Mohr's strain circle.

5. With centre Y. radius YC or YA, draw the circle, fixing B on Xb.

6. Draw in the vertical shear strain axis, perpendicular to da through d and the

horizontal linear strain axis through Y to fix o.7. Join points A, B & C to Y. These radii are in the same anguw order as the

original strains. but the angles between them will be 2x those on the strainsystem.

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The principal strains are then e1

and c1

a s dep ic te d.

The principal stresses can be determ ined from :-

,

The construction overleaf works irrespective of the measured values of strain andwhatever the angles between the gauges the process is simplified, however, if the

sequence of the gauges is in ascending or descending order and the included anglebetween max and min is less than 1800 :-

Emax Eintennediate

€ m i n

Analytical detennination of principal strains from rosette readings.

Applied to the three values of 8 from the rosette gauges, gives three simultaneous

equations from which e x I e , & " Y . r : y can be found. The principal strains can be

determined from -

81 or Bl = ~ ( e x + e , . ) ± ~ { { B z - e , Y + " ' ( x , . 2

The direction of the principal strain axes are then calculated from

tan 28 = " 1 - " ,( e J : - B y )

Ro sette g au ges in common use

a re rec tangula r ( t J : : : 0 t 45, 901;1)

and delta (8 = 0 I 60 I 12 a (I) •

For each the X axis is chosen

to coincide with (J. 0

so eo III: ~ (e~ + S'J') + ; (e~ - e,) - B ;.

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SOLUTIONS TO WORK ED EXAM PLES USINGTHE M cCLINTOCK CONSTRUCTION

1)

(lcr).

@ 36.36° clockwise to eS

£:1 = -409 X 10-6

F rom con stru ctio n

e1= 376 x 10-6

@ 126.360 clockwise to BB

(Stresses calculatedas delineated on

PageS).

- 1 4 -



2)

(1~

From cons truc tion

~--------_'~450

l:•.. 458.5 x 10"

@ 6- 301 antlclocJadse relative to e4

(Stresses calculated on page I, )

£1 • -208.5 X 10-6

@ 83· 30' clockwise relative to 84

d b a

c

£450

x~

2 e =130

OQ'eJ from A

or 16'1 Q from A



TUTORIAL SHEET

ROSETTE ANALYSIS

Fig I

IF ig 2

1. Figure 1 shows a 45° rosette strain gauge mounted on the surface of a component.

When the component is loaded the strains recorded by gauges A, Band Care

+ 0.00108, -0.00054 and + 0.00050 respectively. Evaluate the principal stresses

and the inclination of the greater principal plane relative to gauge A if Poisson's

ratio =0.3 and Young's Modulus = 206 kN/mm2 for the component.

2. Fig.1 shows a 45° rosette strain gauge mounted on the surface of a component.

When the component is loaded the strains recorded by the gauges A, B and Care,

- .00040, - .00085, - .00109 respectively. Evaluate the maximum shear stressand planes relative to gauge A, if Poisson's ratio =0.3 and Young's

Modulus = 206kN/mm2 for the component.

3.Figure 2 shows a 1200 rosette strain gauge mounted on the surface of a component.

When the component is loaded the strains recorded by gauges A, B and C are

0.00036, - 0.00090 and 0 respectively. Evaluate the principal strains, maximum

shear strains and directions of these relative to gauge A.

Indicate how one would wire these gauges into bridge circuits and how one would

compensate far changes in resistance due to temperature variations.

4 .

A state of plane stress exists at the free surface of a body. Strain measurements are

taken at a point P on such a free surface by the 120° rosette shown in figure 2. The

measurements are

EA = 100J..L mm/rom, E8 = -500 ~ ro m /m m , ec = 200 J .! m m /m m . If the

material obeys Hooke's law, cornpu te the principal stresses and the direction 0f theprincipal planes relative to the gauge A. Assume Young's Modulus =200 kN/mm2

and Poisson's Ratio = 0.3.

experimental stress analysis - strain gauges

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