Problem 2-1

An air-filled rubber ball has a diameter of 150 mm. If the air pressure within it is increased until theball's diameter becomes 175 mm, determine the average normal strain in the rubber.

Given: d0 150mm d 175mm

Solution:

d d0d0

0.1667mmmm

Ans

Problem 2-2

A thin strip of rubber has an unstretched length of 375 mm. If it is stretched around a pipe having anouter diameter of 125 mm, determine the average normal strain in the strip.

Given: L0 375mm

Solution:

L 125( ) mm

L L0L0

0.0472mmmm

Ans

Problem 2-3

The rigid beam is supported by a pin at A and wires BD and CE. If the load P on the beam causes theend C to be displaced 10 mm downward, determine the normal strain developed in wires CE and BD.

Given: a 3m LCE 4m

b 4m LBD 4m

LCE 10mm

Solution:

LBDa

a bLCE LBD 4.2857 mm

CELCE

LCECE 0.00250

mmmm

Ans

BDLBD

LBDBD 0.00107

mmmm

Ans

Problem 2-4

The center portion of the rubber balloon has a diameter of d = 100 mm. If the air pressure within itcauses the balloon's diameter to become d = 125 mm, determine the average normal strain in therubber.

Given: d0 100mm d 125mm

Solution:

d d0d0

0.2500mmmm

Ans

Problem 2-5

The rigid beam is supported by a pin at A and wires BD and CE. If the load P on the beam is displace10 mm downward, determine the normal strain developed in wires CE and BD.

Given: a 3m b 2m c 2m

LCE 4m LBD 3m

tip 10mm

Solution:

LBDa

LCEa b

=LCE

a btip

a b c=

LCEa b

a b c tip LCE 7.1429 mm

LBDa

a b c tip LBD 4.2857 mm

Average Normal Strain:

CELCE

LCECE 0.00179

mmmm

Ans

BDLBD

LBDBD 0.00143

mmmm

Ans

Problem 2-6

The rigid beam is supported by a pin at A and wires BD and CE. If the maximum allowable normalstrain in each wire is max = 0.002 mm/mm, determine the maximum vertical displacement of the loadP.

Given: a 3m b 2m c 2mLCE 4m LBD 3m

allow 0.002mmmm

Solution:

LBDa

LCEa b

=LCE

a btip

a b c=

Average Elongation/Vertical Displacement:

LBD LBD allow LBD 6.00 mm

tipa b c

aLBD

tip 14.00 mm

LCE LCE allow LCE 8.00 mm

tipa b c

a bLCE

tip 11.20 mm (Controls !) Ans

Problem 2-7

The two wires are connected together at A. If the force P causes point A to be displaced horizontally 2mm, determine the normal strain developed in each wire.

Given:

a 300mm 30deg

A 2mm

Solution:

Consider the triangle CAA':

A 180deg A 150 deg

LCA' a2A

2 2 a A cos A

LCA' 301.734 mm

CALCA' a

a

CA 0.00578mmmm

Ans

Problem 2-8

Part of a control linkage for an airplane consists of a rigid member CBD and a flexible cable AB. If aforce is applied to the end D of the member and causes it to rotate by = 0.3°, determine the normalstrain in the cable. Originally the cable is unstretched.

Given:

a 400mm b 300mm c 300mm

0.3deg

Solution:

LAB a2 b2 LAB 500 mm

Consider the triangle ACB':

C 90deg C 90.3 deg

LAB' a2 b2 2 a b cos C

LAB' 501.255 mm

ABLAB' LAB

LAB

AB 0.00251mmmm

Ans

Problem 2-9

Part of a control linkage for an airplane consists of a rigid member CBD and a flexible cable AB. If aforce is applied to the end D of the member and causes a normal strain in the cable of 0.0035 mm/mmdetermine the displacement of point D. Originally the cable is unstretched.

Given: a 400mm b 300mm c 300mm

AB 0.0035mmmm

Solution:

LAB a2 b2 LAB 500 mm

LAB' LAB 1 AB LAB' 501.750 mm

Consider the triangle ACB':

C 90deg=

LAB' a2 b2 2 a b cos C=

C acosa2 b2 LAB'

2

2 a b

C 90.419 deg

C 90deg 0.41852 deg

0.00730 rad

D b c( ) D 4.383 mm Ans

Problem 2-10

The wire AB is unstretched when = 45°. If a vertical load is applied to bar AC, which causes =47°, determine the normal strain in the wire.Given:

45deg 2deg

Solution:

LAB L2 L2= LAB 2 L=

LCB 2L( )2 L2= LCB 5 L=

From the triangle CAB:

A 180deg A 135.00 deg

sin BL

sin ALCB

=

B asinL sin A

5 L B 18.435 deg

From the triangle CA'B:

'B B 'B 20.435 deg

sin 'BL

sin 180deg A'LCB

=

A' 180deg asin5 L sin 'B

L

A' 128.674 deg

'C 180deg 'B A'

'C 30.891 deg

sin 'BL

sin 'CLA'B

= LA'Bsin 'Csin 'B

L LAB 2 L

ABLA'B LAB

LABAB 0.03977 Ans

Problem 2-11

If a load applied to bar AC causes point A to be displaced to the left by an amount L, determine thenormal strain in wire AB. Originally, = 45°.Given: 45deg

Solution: ACL

L=

LAB L2 L2= LAB 2 L=

LCB 2L( )2 L2= LCB 5 L=

From the triangle A'AB:

A 180deg A 135.00 deg

LA'B L2 LAB2 2 L LAB cos A=

LA'B L2 2 L2 2 L L=

ABLA'B LAB

LAB=

ABL2 2 L2 2 L L 2 L

2 L=

AB12

LL

21

LL

1=

Neglecting the higher-order terms,

AB 1L

L

0.51=

AB 112

LL

..... 1= (Binomial expansion)

AB12

LL

= Ans

Alternatively,

ABLA'B LAB

LAB= AB

L sin2 L

=

AB12

LL

= Ans

Problem 2-12

The piece of plastic is originally rectangular. Determine the shear strain xy at corners A and B if theplastic distorts as shown by the dashed lines.

Given:

a 400mm b 300mm

Ax 3mm Ay 2mm

Cx 2mm Cy 2mm

Bx 5mm By 4mm

Solution:

Geometry : For small angles,

Cxb Cy

0.00662252 rad

By Cya Bx Cx

0.00496278 rad

Bx Axb By Ay

0.00662252 rad

Aya Ax

0.00496278 rad

Shear Strain :

xy_B xy_B 11.585 10 3 rad Ans

xy_A xy_A 11.585 10 3 rad Ans

Problem 2-13

The piece of plastic is originally rectangular. Determine the shear strain xy at corners D and C if theplastic distorts as shown by the dashed lines.

Given:

a 400mm b 300mm

Ax 3mm Ay 2mm

Cx 2mm Cy 2mm

Bx 5mm By 4mm

Solution:

Geometry : For small angles,

Cxb Cy

0.00662252 rad

By Cya Bx Cx

0.00496278 rad

Bx Axb By Ay

0.00662252 rad

Aya Ax

0.00496278 rad

Shear Strain :

xy_D xy_D 11.585 10 3 rad Ans

xy_C xy_C 11.585 10 3 rad Ans

Problem 2-14

The piece of plastic is originally rectangular. Determine the average normal strain that occurs along thdiagonals AC and DB.

Given:

a 400mm b 300mm

Ax 3mm Ay 2mm

Cx 2mm Cy 2mm

Bx 5mm By 4mm

Solution:

Geometry :

LAC a2 b2 LAC 500 mm

LDB a2 b2 LDB 500 mm

LA'C' a Ax Cx2 b Cy Ay

2

LA'C' 500.8 mm

LDB' a Bx2 b By

2

LDB' 506.4 mm

Average Normal Strain :

ACLA'C' LAC

LACAC 1.601 10 3 mm

mmAns

BDLDB' LDB

LDBBD 12.800 10 3 mm

mmAns

Problem 2-15

The guy wire AB of a building frame is originally unstretched. Due to an earthquake, the two columnsof the frame tilt = 2°. Determine the approximate normal strain in the wire when the frame is in thiposition. Assume the columns are rigid and rotate about their lower supports.Given:

a 4m b 3m c 1m 2deg

Solution:

180= 0.03490659 rad

Geometry : The vertical dosplacement is negligible.

Ax c Ax 34.907 mm

Bx b c( ) Bx 139.626 mm

LAB a2 b2 LAB 5000 mm

LA'B' a Bx Ax2 b2

LA'B' 5084.16 mm

Average Normal Strain :

ABLA'B' LAB

LAB

AB 16.833 10 3 mmmm

Ans

Problem 2-16

The corners of the square plate are given the displacements indicated. Determine the shear strain alongthe edges of the plate at A and B.

Given: ax 250mm ay 250mm

v 5mm h 7.5mmSolution:

At A :

tan'A2

ax h

ay v=

'A 2 atanax h

ay v'A 1.52056 rad

nt_A 2'A nt_A 0.05024 rad Ans

At B :

tan'B2

ay v

ax h=

'B 2 atanay v

ax h'B 1.62104 rad

nt_B 'B 2 nt_B 0.05024 rad Ans

Problem 2-17

The corners of the square plate are given the displacements indicated. Determine the average normalstrains along side AB and diagonals AC and DB.

Given: ax 250mm ay 250mm

v 5mm h 7.5mm

Solution:

For AB :

LAB ax2 ay

2

LA'B' ax h 2 ay v 2

ABLA'B' LAB

LAB

LAB 353.55339 mm

LA'B' 351.89665 mm

AB 4.686 10 3 mmmm

Ans

For AC :

LAC 2 ay LAC 500 mm

LA'C' 2 ay v LA'C' 510 mm

ACLA'C' LAC

LACAC 20.000 10 3 mm

mmAns

For DB :

LDB 2 ax LDB 500 mm

LD'B' 2 ax h LD'B' 485 mm

DBLD'B' LDB

LDBDB 30.000 10 3 mm

mmAns

Problem 2-18

The square deforms into the position shown by the dashed lines. Determine the average normal strainalong each diagonal, AB and CD. Side D'B' remains horizontal.

Given:

a 50mm b 50mm

Bx 3mm

Cx 8mm Cy 0mm

'A 91.5deg LAD' 53mm

Solution:

For AB :

By LAD' cos 'A 90deg b By 2.9818 mm

LAB a2 b2 LAB 70.7107 mm

LAB' a Bx2 b By

2 LAB' 70.8243 mm

ABLAB' LAB

LAB

AB 1.606 10 3 mmmm

Ans

For CD :

Dy By Dy 2.9818 mm

LCD a2 b2 LCD 70.7107 mm

LC'D' a Cx2 b Dy

2 2 a Cx b Dy cos 'A

LC'D' 79.5736 mm

CDLC'D' LCD

LCD

CD 125.340 10 3 mmmm Ans

Problem 2-19

The square deforms into the position shown by the dashed lines. Determine the shear strain at each ofits corners, A, B, C, and D. Side D'B' remains horizontal.

Given: a 50mm b 50mm

Bx 3mm

Cx 8mm Cy 0mm

'A 91.5deg LAD' 53mm

Solution: 'A 1.597 radGeometry :

By LAD' cos 'A 90deg b By 2.9818 mm

Dy By Dy 2.9818 mm

Dx LAD' sin 'A 90deg Dx 1.3874 mm

In triangle C'B'D' :

LC'B' Cx Bx2 b By

2

LC'B' 54.1117 mm

LD'B' a Bx Dx LD'B' 48.3874 mm

LC'D' a Cx Dx2 b Dy

2

LC'D' 79.586 mm

cos B'LC'B'

2 LD'B'2 LC'D'

2

2 LC'B' LD'B'=

B' acosLC'B'

2 LD'B'2 LC'D'

2

2 LC'B' LD'B' B' 101.729 deg B' 1.7755 rad

D' 180deg 'A D' 88.500 deg D' 1.5446 rad

C' 180deg B' C' 78.271 deg C' 1.3661 radShear Strain :

xy_A 0.5 'A xy_A 26.180 10 3 rad Ans

xy_B 0.5 B' xy_B 204.710 10 3 rad Ans

xy_C 0.5 C' xy_C 204.710 10 3 rad Ans

xy_D 0.5 D' xy_D 26.180 10 3 rad Ans

Problem 2-20

The block is deformed into the position shown by the dashed lines. Determine the average normalstrain along line AB.

Given: xBA 70 30( )mm yBA 100mm

xB'A 55 30( )mm

yB'A 1102 152 mm

Solution:

For AB :

LAB xBA2 yBA

2 LAB 107.7033 mm

LAB' xB'A2 yB'A

2 LAB' 111.8034 mm

ABLAB' LAB

LAB

AB 38.068 10 3 mmmm

Ans

Problem 2-21

A thin wire, lying along the x axis, is strained such that each point on the wire is displaced x = k x2

along the x axis. If k is constant, what is the normal strain at any point P along the wire?

Given: x k x2=

Solution:

xxd

d=

2 k x= Ans

Problem 2-22

The rectangular plate is subjected to the deformation shown by the dashed line. Determine the averagshear strain xy of the plate.Given: a 150mm b 200mm

a 0mm b 3mm

Solution:

atanb

a

1.146 deg

19.9973 10 3 rad

Shear Strain :

xy

xy 19.997 10 3 rad Ans

Problem 2-23

The rectangular plate is subjected to the deformation shown by the dashed lines. Determine the averagshear strain xy of the plate.Given: a 200mm a 3mm

b 150mm b 0mm

Solution:

atana

b

1.146 deg

19.9973 10 3 rad

Shear Strain :

xy

xy 19.997 10 3 rad Ans

Problem 2-24

The rectangular plate is subjected to the deformation shown by the dashed lines. Determine theaverage normal strains along the diagonal AC and side AB.

Given: a 200mm b 150mm

Ax 3mm Ay 0mm

Bx 0mm By 0mm

Cx 0mm Cy 0mm

Dx 3mm Dy 0mm

Solution:

Geometry :

LAC a2 b2 LAC 250 mm

LAB b LAB 150 mm

LA'C a Cx Ax2 b Cy Ay

2

LA'C 252.41 mm

LA'B Ax2 b2

LA'B 150.03 mm

Average Normal Strain :

ACLA'C LAC

LACAC 9.626 10 3 mm

mmAns

ABLA'B LAB

LABAB 199.980 10 6 mm

mmAns

Problem 2-25

The piece of rubber is originally rectangular. Determine the average shear strain xy if the corners Band D are subjected to the displacements that cause the rubber to distort as shown by the dashed lines

Given: a 300mm b 400mm

Ax 0mm Ay 0mm

Bx 0mm By 2mm

Dx 3mm Dy 0mm

Solution:

AB atanBya

AB 0.38197 deg

AB 6.6666 10 3 rad

AD atanDxb

AD 0.42971 deg

AD 7.4999 10 3 rad

Shear Strain :

xy_A AB AD

xy_A 14.166 10 3 rad Ans

Problem 2-26

The piece of rubber is originally rectangular and subjected to the deformation shown by the dashedlines. Determine the average normal strain along the diagonal DB and side AD.

Given: a 300mm b 400mm

Ax 0mm Ay 0mm

Bx 0mm By 2mm

Dx 3mm Dy 0mm

Solution:

Geometry :

LDB a2 b2 LDB 500 mm

LAD b LAD 400 mm

LD'B' a Bx Dx2 b Dy By

2 LD'B' 496.6 mm

LAD' Dx2 b Dy

2 LAD' 400.01 mm

Average Normal Strain :

BDLD'B' LDB

LDBBD 6.797 10 3 mm

mmAns

ADLAD' LAD

LADAD 28.125 10 6 mm

mmAns

Problem 2-27

The material distorts into the dashed position shown. Determine (a) the average normal strains x and y , the shear strain xy at A, and (b) the average normal strain along line BE.

Given: a 80mm Bx 0mm Ex 80mm

b 125mm By 100mm Ey 50mm

Ax 0mm Cx 10mm Dx 15mm

Ay 0mm Cy 0mm Dy 0mm

Solution:

xAC Cx Ax xAC 10.00 mm

yAC Cy Ay yAC 0.00 mm

AC atanCxb AC 79.8300 10 3 rad

Since there is no deformation occuring along the y- and x-axis,

x_A yAC x_A 0 Ans

y_AxAC

2 b2 b

b

y_A 0.00319 Ans

xy_A AC

xy_A 79.830 10 3 rad Ans

Geometry :

BxCx

Byb

= BxByb

Cx Bx 8 mm

By 0mm

ExDx

Eyb

= ExEyb

Dx Ex 6 mm

Ey 0mm

LBE Ex Bx2 Ey By

2 LBE 94.34 mm

LB'E' a Ex Bx2 Ey Ey By

2 LB'E' 92.65 mm

BELB'E' LBE

LBEBE 17.913 10 3 mm

mmAns

Note: Negative sign indicates shortening of BE.

Problem 2-28

The material distorts into the dashed position shown. Determine the average normal strain that occursalong the diagonals AD and CF.

Given: a 80mm Bx 0mm Ex 80mm

b 125mm By 100mm Ey 50mm

Ax 0mm Cx 10mm Dx 15mm

Ay 0mm Cy 0mm Dy 0mm

Solution:

xAC Cx Ax xAC 10.00 mm

yAC Cy Ay yAC 0.00 mm

AC atanCxb AC 79.8300 10 3 rad

Geometry :

LAD a2 b2 LAD 148.41 mm

LCF a2 b2 LCF 148.41 mm

LA'D' a Dx Ax2 b Dy Ay

2

LA'D' 157.00 mm

LC'F a Cx2 b Cy

2

LC'F 143.27 mm

Average Normal Strain :

ADLA'D' LAD

LADAD 57.914 10 3 mm

mmAns

CFLC'F LCF

LCFCF 34.653 10 3 mm

mmAns

Problem 2-29

The block is deformed into the position shown by the dashed lines. Determine the shear strain atcorners C and D.

Given: a 100mm Ax 15mm

b 100mm Bx 15mm

LCA' 110mmSolution:

Geometry :

C asinAx

LCA'C 7.84 deg

C 0.1368 rad

D asinBx

LCA'D 7.84 deg

D 0.1368 rad

Shear Strain :

xy_C C

xy_C 136.790 10 3 rad Ans

xy_D D

xy_D 136.790 10 3 rad Ans

Problem 2-30

The bar is originally 30 mm long when it is flat. If it is subjected to a shear strain defined by xy = 0.0x, where x is in millimeters, determine the displacement y at the end of its bottom edge. It is distorteinto the shape shown, where no elongation of the bar occurs in the x direction.

Given: L 300mm xy 0.02 x=

unit 1mmSolution:

dydx

tan xy=

dydx

tan 0.02 x( )=

0

yy1 d

0

Lxtan 0.02 x( ) unit( ) d=

y0

30xtan 0.02x( ) unit( ) d

y 9.60 mm Ans

Problem 2-31

The curved pipe has an original radius of 0.6 m. If it is heated nonuniformly, so that the normal strainalong its length is = 0.05 cos , determine the increase in length of the pipe.Given: r 0.6m 0.05 cos=Solution:

L Ld=

L0

90degr0.05 cos d=

L0

90deg0.05 r cos d

L 30.00 mm Ans

Problem 2-32

Solve Prob. 2-31 if = 0.08 sin .

Given: r 0.6m 0.08 sin=

Solution:

L Ld=

L0

90degr0.08 sin d=

L0

90deg0.08 r sin d

L 0.0480 m Ans

Problem 2-33

A thin wire is wrapped along a surface having the form y = 0.02 x2, where x and y are in mm. Originathe end B is at x = 250 mm. If the wire undergoes a normal strain along its length of = 0.0002x,

determine the change in length of the wire. Hint: For the curve, y = f (x), ds = 1dydx

2dx .

Given: Bx 250mm 0.0002 x= unit 1mm

Solution:

y 0.02x2=dydx

0.04x=

ds 1dydx

2dx=

ds 1 0.04x( )2 dx=

L sd= L0

Bxx0.0002x( ) 1 0.04x( )2 d=

L unit( )0

250

x0.0002x( ) 1 0.04x( )2 d

L 42.252 mm Ans

Problem 2-34

The fiber AB has a length L and orientation If its ends A and B undergo very small displacements uA and vB, respectively, determine the normal strain in the fiber when it is in position A'B'.

Solution:

Geometry:

LA'B' L cos uA2 L sin vB

2=

LA'B' L2 uA2 vB

2 2L vB sin uA cos=

Average Normal Strain:

ABLA'B' L

L=

AB 1uA

2 vB2

L2

2 vB sin uA cos

L1=

Neglecting higher-order terms uA2 and vB

2 ,

AB 12 vB sin uA cos

L1=

Using the binomial theorem:

AB 112

2 vB sin uA cos

L.. 1=

ABvB sin

L

uA cos

L= Ans

Problem 2-35

If the normal strain is defined in reference to the final length, that is,

'n

= p'p

s' ss'

lim

instead of in reference to the original length, Eq.2-2, show that the difference in these strains isrepresented as a second-order term, namely, n - 'n = n 'n.

Solution:

nS' S

S=

n 'nS' S

SS' S

S'=

n 'nS'2 2 S S' S2

S S'=

n 'nS' S 2

S S'=

n 'nS' S

SS' S

S'=

n 'n n 'n= (Q.E.D.)