1

COMBINED AXIAL AND SHEARING STRESS

In general when a body is subject to combined loadings , every element will be subject to combined normal stresses fx and fy together with the shearing stress sxy and syx as shown.

fy

syx

syx

sxy

sxy

fxfx x

y

Note: normal stresses fx and fy are considered positive for tensile stress negative for compressive; sxy and syx are positive if it creates clockwise rotation about the center hence sxy =-syx

fy

2

Stresses on an inclined plane

fx

sxy

fy

syx

s

f

x

y

If stresses fx,fy,sxy are known the normal stress and the shearing stress On a plane inclined at an angle θ to the x -axis can be obtained by the followingformulas or by Mohr’s Circle

2222

SinSCosffff

f xyyxyx

3

222

CosSSinff

S xyyx

Principal Stresses – there are certain values of the angle θ that will lead to maximum and minimum values of f for given set of stress fx,fy and sxy. These maximum and minimum values that f may assume are termed principal stresses and can be computed by:

2

2

max )(22 xy

yxyx Sffff

f

2

2

min )(22 xy

yxyx Sffff

f

4

Principal Planes : The angle θp between the x -axis and the planes on which the principal stress occur. The are always two values of θp that will the equation below . The maximum stress occur on one of these planes and the minimum onthe other. The planes defined by angle θp are known as principal planes. The shearing stress on these plane are always zero.

yx

xyP ff

S

22tan

Maximum and Minimum shearing stress. The maximum and minimum shearing stress is given by:

2

2

max )(2 xy

yx sff

S

5

The angle between the x-axis on which the maximum and minimum shearing stresses occur are determined by the equation

xy

yx

S

ff

22tan

MOHR’S CIRCLE:

The visual interpretation of the formulas given above can be interpreted by Moh’r circle, without the necessity of memorizing them

fy

syx

fy

fx

syx

sxysxy

fx

y

x

6

(fx,sxy)

fy,-syx

fmax

fmin

(fx + fy)/2

fy

fx

(fx-fy)/2

R

2θ f

s

sxy

-syx

0,0

7

Rules for applying Mohr’s Circle

1. On the rectangular f - s axes , plot points having the coordinates ( fx,sxy) and ( fy, syx). In plotting these points assume tension as plus and compression as minus for fx and fy and shearing stress as plus when its moment about the center of the element is clockwise. Note that sxy=-syx

.2. Join the points plotted by a straight line. This line is the diameter whose center is on the f-axis.

3. the radius of the circle to any point on its circumference represents the axis directed normal to the plane whose stress components are given by the coordinates of that point.

4. The angle between the radii to selected points on the Mohr’s circle is twice the angle between the normals to the planes represented by these points. The rotational sense of this angle corresponds to the rotational sense of the actual angle between the normals to the planes. ( If the N –axis is actually at a counterclockwise θ from the x-axis then on the Mohr’s circle the radius is laid off at a counterclockwise angle 2θ from the x -radius

8

Problem:The state of plane stress at a point with respect to the xy axes is as shown In the figure. Using Mohr’s Circle, determine a) the principal stress and principal planes. b) the maximum in plane shear stress; c) the equivalent state of stress with respect to the x’y’ axes. Show all results on sketches ofproperly oriented elements.

MPa 40xf MPa 20yf MPa 16yxS MPa 16xyS

)16,40( x )16,20(y

x

y20 MPa

40 MPa

16 MPa

Solution Given

-16 MPa500

x’

y’

9

f

S

40,-16

20,16

x

y

40

20

302

2040

10

2

2040

16

1630,0

Radius of Mohr Circle

22 )16()10( R

MPaR 87.18

R

R

1010

f

S

x

y

40

20

30 10

16

16

18.87

18.87

Principal stresses: Located on the Mohr Circle wherethe shearing stress is zero

maxf

minf

Rf 30max

87.1830max fMPa 87.48max f

Rf 30min

MPaf 13.11min

Principal planes 2

10

162tan

0582 029

12

f

S

x

y

40

20

30 10

16

16

18.87

18.87

2

a

b

1

211.13 MPa

48.87 MPa

290

00 148902

Sketch for principal stresses and planes

Maximum in plane shear stress: equalTo the radius of the circle

MPaS 87.18max

Angle 01 74148

2

1

12

a30 MPa

30 MPa

740

18 .87MPa

18.87 MPa

Sketch for Maximum in plane shear stress

f

S

x

y

40

20

30 10

16

16

18.87

18.87

2

State of stress with respect to x’y’ axes

0100

X’

y’

13

f

S

x

y

40

20

30 10

16

16

18.87

18.87

20100

X’

y’

042581002100

042

042RCos042RCos

'xf

'yf

''yxS

''xyS

04230' RCosf x 04287.1830' Cosf x

MPaf x 44'04230' RCosf y

MPaf y 98.15'

0'' 42RSinS yx

0'' 4287.18 SinS yx

MPaS yx 63.12''

MPaS xy 63.12''

14

X’

Y’

15.98 MPa

44 MPa

500

12.63 MPa 12.63 MPa

Sketch for state of stress in the x’y’ axes

15

Problem

A section of a beam is subject to combined tensile stress of 120 MPa and shearing stress of 80 MPa. Determine the maximum, minimum normal and shearing stress in the section.

15

MPa 120xf 0yf MPa 80xyS MPa 80yxS

)80,120(x )80,0( y

Solution Given

16

f

S

y

120,80

0,-80

x

602

0120

60,0

maxfminf

maxS80

80

Radius of Mohr Circle

22 )80()60( R

MPaR 100

R

R

Maximum normal stress

Rf 60max

10060max f

MPaf 160max

Minimum normal stress

Rf 60min

10060min f

MPaf 40min

Maximum shearing stressMPaRS 100max

17

ProblemFor the soil element shown in figure, calculate the normal and shear stresson plane AB.

x

y

300 kPa

120 kPa

40 kPa

A

B

kPa 120xf kPa 030yf kPaSxy 40 kPa 40yxS

)40,120(x )40,300( y

Given

040

f

Ssolution

(-120,40)x

(-300,-40)y

2102

)300120(

(-210,0)

40

40

2

0802

A

B

ABf

ABS

Radius of Mohr Circle

22 )40()120210( R

kPaR 49.98

R

R

120210

402tan

0242

22180

802 AB plane of angle o

0000 768024180

Normal stress in plane AB

)210( RCosfAB

)7649.98210( 0CosfAB kPafAB 17.186

Shear stress in plane AB

sinRSAB 076sin49.98ABS kPaSAB 56.95

19

ProblemFor the soil element shown in figure, calculate the maximum and minimumPrincipal stress and the normal and shear stress on plane CD. 035

x

y

120 kPa

80 kPa

40 kPa

C

D

kPaSxy 40 kPa 40yxSGiven

kPaf x 80 kPaf y 120

)40,80( x )40,120(y

f

S

(-80,-40)

(-120,40)

(100,0)

y

x

1002

)80120(

202

)80120(

240

40

Radius of Mohr Circle

22 )40()20( R

kPaR 72.44

20

402tan

05.632

R

RMaximum Principal stress

maxf

)100(max Rf

)72.44100(max f

kPaf 72.144max

Minimum Principal stress

minf

)100(min Rf )72.44100(min f

kPaf 28.55min

21

21

f

S

(-80,-40)

(-120,40)y

x

100

20

05.6340

40

R

R

clockwise 70)35(2 00

Angle of plane CD

070

C

D

Normal stress in plane CD

CDf

)100( RCosfCD

0000 5.46705.63180

)5.4672.44100( 0CosfCD kPafCD 25.69

Shear stress in plane CD

CDS

RSinSCD 05.4672.44 SinSCD

kPaSCD 47.32

22

Problems

1. A section of a beam is subject to combined tensile stress of 120 MPa and shearing stress of 80 MPa. Determine the maximum, minimum normal and shearing stress in the section.

2. At a certain section of a stressed material , the stresses are fx = 50 MPa

fy = -10 MPa,sxy =-20 MPa. Determine the maximum and minimum normal stresses.

3. At a certain point on a stressed body, the principal stresses are fx =80 MPa and fy =-40 MPa. Determine the normal and shearing stress on the plane whose normal is at +300 with the x –axis.

4. At a certain point on a stressed body, the stresses are fx = 40 MPa,

fy =-5 MPa, and sxy = 30 MPa. What are the principal planes ?