GEOENG2 Mohr's Circle

8/13/2019 GEOENG2 Mohr's Circle

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Mohrs Circle Analyses

Geotechnical Engineering 2

(GEOENG2)

Jonathan Rivera Dungca


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Stress Acting on Other Planes

Some problems may require computation of the

stresses on other planes

We can obtain these stresses using a graphical

representation called a Mohrs circle, which

was developed by the German engineer Otto

Mohr (1835-1918)


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Mohrs Circle

It describes the two

dimensional stresses

at a point in a material

It considers the

stresses acting on

each side of a twodimensional element

and plots them on a s

vs. tdiagram


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Mohrs Circle

Each point on the

circle represents

the normal and

shear stresses

acting on one

side of anelement oriented

at a certain angle


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Principal Stresses

If the soil element is

rotated to a certain

angle, the shear

stresses will be zero

The planes on each

side of this element

are known aspr incipal planes


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Principal Stresses

The stresses acting

on them are known

as pr incipal

stresses

The major p r inc ipal

stress, 1, is also

the greatest normalstress that acts on

any plane


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Principal Stresses

The minor pr inc ipalstress, 3, is the smallest

normal stress that acts on

any plane

These two stresses act at

right angles to each other

In a 3D analysis, there

would also be an

intermediate princ ipal

stress, 2which are at right

angles to both s1 ands3


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Magnitudes ofs1and s3

2

2

3 22 zxzxzx

t

ssss

s

2

2

122 zx

zxzxt

sssss


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Angle between szand s1

31

311 2cos21

sssss zz

where:s1= major principal stress

s3= minor principal stress

sx= horizontal stress

sz= vertical stress

tzx= shear stress acting on ahorizontal plane

z= angle between

szand

s1


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Principal Stresses Beneath a

Circular Loaded Area


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Mohrs Circle that represents the

combined effects of both the

Geostatic and the Induced Stress

To develop such a circ le, compu te the

geostat ic xand zand the induced x, z,

and zx, then add them by superposi t ion

and use the combined values to develop

the Mohrs Circle. Do not attempt tocombine 1and 3values by

superposi t ion


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Stresses on Other Planes

After constructing the Mohrs circle, we can obtain

the normal and shear stresses that act on any

plane through that point

ssss

s 2cos22

3131

ss

t 2sin2

31


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Greatest Shear Stress, tmax

It occurs on the plane Gand H in the figure. These

planes are oriented at 450

angles from the principal

planes, and the shearstress acting on them is

equal to the radius of the

Mohrs Circle

2

31max

sst


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Procedure to compute the stresses

on a given plane1.Draw a soil element that is

aligned with the x and z

axes.

2.Compute sx, sz, txzand tzxand mark these stresses on

the soil element.

3.Plot the points sx

, txz

and

sz, tzxon an s, t diagram

then use these points to

draw the Mohrs circle. The

center of the circle is at2zx ss

s


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on a given plane4. Compute s1and s3

5. Compute the angle zbetween sz and s1

6. Compare the positions ofthe points on the Mohrs

circle that represents szand s

1

to determine if z

extends clockwise or

counter-clockwise from sz.


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on a given plane7. Draw another soil element

that is rotated at an angle

z from the first soil

element. The sides of thiselement are the principal

planes. Mark the stresses

s1 and s3 on this soil

element. Since the sidesof this element are the

principal planes, the shear

stresses are zero.


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on a given plane8. Compute for the value of

tmax(if required). It acts on

the planes oriented 450

from the principal planes.9. Draw a third soil element

with one of its sides

oriented in the direction of

the plane on which thestresses are to be

computed. Mark the

stresses as s

and t on this

element


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on a given plane10.Determine the angle

betweensand s1, then

locate the point on the

Mohrs circle thatrepresents the plane. This

point is located at an angle

2from the point that

represents s1. Be sure tofollow the sign convention:

angles measured

clockwise from s1are

positive.


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on a given plane

11.Compute for sand t.


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Example

The horizontal and vertical stresses at a certain

point in a soil are as follows:sx= 2100 lb/ft

2, sz= 3000 lb/ft2, tzx= -300lb/ft

2

a. Determine the magnitudes and directions of themajor and minor principal stresses.

b. Determine the magnitude and directions of themaximum shear stress.

c. Determine the normal and shear stresses actingon a plane inclined at 350clockwise from the x-axis


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Example


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Example

Solution (a):

2

2

2

2

2

1

/3091

)300(2

30002100

2

30002100

22

f tlb

xz

zxzx

t

sssss

2

2

2

2

2

3

/2009

)300(2

30002100

2

30002100

22

f tlb

xz

zxzx

tssss

s


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Example

Solution (a) Continued :

0

1

31

311

17

20093091

20093091)3000(2cos

2

1

2cos

2

1

ss

sss

zz


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Example


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Example

Solution (b):

231max /541

2

20093091

2f tlb

sst


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Example

Solution (c):

2

3131

/2419

)3517(2cos

2

20093091

2

20093091

2cos22

f tlb

sssss

2

31

/525

)3517(2sin2

20093091

2sin

2

f tlb

sst


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Example

GEOENG2 Mohr's Circle

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