A SEMI-GRAPHICAL PROCEDURE CIRCLE IN GEOTECHNICAL ENGINEERING

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A Semi
-
Graphical Procedure Using Mohr’s Circle in Geotechnical Engine
ering


62


A SEMI
-
GRAPHICAL PROCEDURE
USING MOHR’S

CIRCLE IN GEOTECHNIC
AL ENGINEERING

S.K. Jain

Assistant Professor, Department of Civil Engineering, Jaypee University of Information Technology, Waknaghat Distt. Solan
,

H.P.

173 21
5,
India. E
-
mail:
sk.jain@juit.ac.in

K.K. Jain

Professor of Civil Engineering and Dean, Jaypee Institute of Engineering and Technology, Raghogarh, Guna (M.P.)

473 226, India.

E
-
mail:
kk.jain@jiet.ac.in


ABSTRACT:

Traditionally Mohr’s circle has been used via a graphical procedure which requires the use of drawing
instruments (Ranjan & Rao

2
000). A slightly simpler procedure is obtained if the graphical construction is combined with a
set of formulae (Co
duto

1
999).

This paper presents a new
semi
-
graphical approach to Mohr’s circle wherein the right face of
a stress element is chosen as a reference plane, a Mohr circle is
sketched

on a

piece of paper by hand, and a simple calculator
is used, along with add
ed sketch work on Mohr’s circle, in arriving at quantities of interest such as magnitude and orientation
of principal stresses, planes of maximum shear stress, and stresses on arbitrary planes. The procedure is essentially a furth
er
simplification of the p
resentation by Coduto (1999).


1.
INTRODUCTION

The proposed semi
-
graphical procedure stems from a classic
presentation of Mohr’s Circle by Popov (1998). Popov puts
forward two methods of construction and use of Mohr’s
circle.

Method I plots the Mohr’s c
ircle in a straight forward manner
and uses the concept of “pole”

the origin of planes.
Un
-

fortunately, the method does not lend itself to a convenient
use
of a sketch and a calculator

a geometric approach to
Mohr’s
circle proves far more efficient.

In M
ethod II, Popov turns the shear stress axis upside down,
draws the Mohr’s circle, declares the right
-
hand face of the
square element representing stress, as the reference face, and
then labels the state of stress on the reference face on the
Mohr’s circle
as X, the reference point.

Now all motion on the Mohr’s circle must take place relative
to X. It is the movement of the radial line connecting the
center of the circle to X which takes us to a desired plane in
the square element.

Note that turning the sh
ear stress axis upside down by Popov
only means that a positive shear stress is plotted below the x
-
axis
.

In other words, shear stress that tends to rotate the square
element counterclockwise (CCW) is plotted below the x
-
axis
and shear stress that tends t
o rotate the element clockwise
(CW) is plotted above the axis. The sense of shear stress for
any point in the Mohr’s circle is derived the same way.


2
.

MOHR’S CIRCLE IN GEO
TECHNICAL
ENGINEERING

In geotechnical engineering, a compressive stress is taken as

positive. The sign convention for the shear stress is the same
as in the mechanics of solids.

The two methods of Mohr’s circle described by Popov
(1998)
must be carefully adapted for the arbitrarily changed sign
conventions.

Terzaghi (1943), Terzaghi & Pe
ck (1948), and Taylor (1948)
introduced the method of pole to geotechnical engineering.
Since then the method has been universally used in the
pro
-

fession whenever a graphical use of Mohr’s circle was
desired
(Lambe & Whitman

1
969; Murthy

2
003; Powrie

2
00
4).

It may be mentioned that the method of locating a pole in
geotechnical engineering is different than in the mechanics of
solids. In fact, to a geotechnical engineer the method of pole
described by Popov (1998) momentarily appears to be in
error.

The m
ethod II of Popov (1998) is cast into a semi
-
graphical
procedure in Jain
et al
. (2010) for the mechanics of solids.
The procedure can be used in geotechnical engineering
if
the
sign conventions of the mechanics of solids are retained. The
procedure of Jain

et al
. (2010) will yield erroneous answers
if
the sign conventions of geotechnical engineering are
followed.

In this paper, we adapt the procedure of Jain
et al
. (2010),
essentially the method II of Popov (1998), for the sign
conventions of geotechnical e
ngineering.

3.

A SEMI
-
GRAPHICAL PROCEDURE
IN
GEOTECHNICAL ENGINEE
RING

Figure 1 presents the step
-
by
-
step procedure to be used in the
proposed semi
-
graphical approach to Mohr’s circle in
geo
-

technical engineering. The procedure is the same as for the
mech
anics of solids
except

for step 2.

In the procedure for geotechnical engineering, the reference
point X is chosen differently. The shear stress is plotted as is,



IGC 2009
, Guntur, INDIA


A Semi
-
Graphical Procedure Using Mohr’s Circle in Geotechnical Engineering



63


Steps for the Construction and Use of Mohr’s Circle (Geotech)


For the given state of pla
ne stress:


1.

Pick the centre of circle at [(σ
x

+ σ
y
)/2, 0].


2.

Pick the reference point
X


x
, τ
xy
).


3.

Draw the circle passing through
X

on the σ

τ
coordinate system.


4.

Compute
R

from the triangle CXB


5.

Find principal stresses as
,



σ
max

= centre
+
R



σ
min
= centre
-

R


6.

Find
τ
max
=
R


7.

To find σ
x


and τ
x

y


on some plane at angle
θ

with
respect to original square element we proceed as
follows: we rotate on Mohr’s circle from the

reference
point X through an angle


in the same direction as

in
x

y

coordinate system. (Angle
θ

in the square
element is equal to


in the Mohr’s circle). In this
way, we get to the plane having σ
x

, τ
x


y


stresses
.






8.

Determine the orientation of principal planes starting from the reference point
X
. Travel on the Mohr’s circle either
to reach σ
max
or σ
min
, to find

P
.


9.

Determine

S

by traveling from
X

to point having τ
max
.

Fig. 1: The
Procedure

i.e., the positive shear stress is plotted above the axis. This
makes the geotechnical procedure more straightforward and
easier to use than the procedure of the mechanics of solids
given in Jain
et al
. (2010).

As one can see, there is hardly a
nything new here
conceptually.
It’s the way we have set up the scheme of computation and
of working with a sketch which appeals to a novice.

Inasmuch as no formulae need be memorized and no drawing
instruments needed, the proposed procedure becomes
attrac
tive
for field engineers and for students taking examinations.
Once
the procedure is mastered, a student feels that he has a
command over the concept.

Figure 2 presents a problem and its solution. The same
problem
is solved in Das (2002) by transformation
formulas, and in
y



x



x


y


θ


σ
x

σ
y

τ
xy

X

x
, τ
xy
)


-

σ

B

O

τ


x

+ σ
y
)/2

X



x

, τ
x


y

)


-

X

x
, τ
xy
)


-

σ

B

O

τ


x

+ σ
y
)/2



A Semi
-
Graphical Procedure Using Mohr’s Circle in Geotechnical Engine
ering


64

Jain
et al
. (2002) by the procedure of the mechanics of
solids.
All three solutions yield the same answers.



For the state of plane stress shown,
determine (a) the principal planes,
(b) the principal stresses and (c)
stresses on a plane

inclined at 70°
clockwise with the shaded vertical
face. Also sketch the principal
planes and principal stresses.



Fig
.

2: Example Illustration

4.

CONCLUSIONS

The paper states that the graphical procedure for Mohr’s
circle
u
sually employed in the mechanics of solids does not work if
the sign conventions of geotechnical engineering are
followed.

The paper introduces a change in the procedure of the
mechanics of solids so the procedure can be used in
geo
-

technical engineering
. It is found that the resulting procedure
is simpler than the one employed in the mechanics of solids.

The proposed procedure is cast into a step
-
by
-
step format so
the procedure can be used by simply working on a sketch of the
Mohr’s circle and simultaneo
usly using a scientific
calculator.

One does not have to use drawing instruments when using
the
proposed procedure. Experience has shown that students find
it much easier to learn Mohr’s circle if the proposed semi
-
graphical approach is followed in the cl
assrooms.

ACKNOWLEDGEMENTS

S.K. Jain expresses grateful appreciation to Professor E.G.
Henneke for useful discussions on the subject and to
Professor
J.M. Duncan for continuing encouragement toward research
efforts
.

REFERENCES

Beer
,

F. and Johnson
,

G. (198
8)
.

Mechanics of Materials
,
McGraw
-
Hill, New York.

Coduto
,

D.P. (1999)
.

Geotechnical Engineering: Principles
and Practices
,
Prentice
-
Hall of India Pvt. Ltd., New Delhi.

Das
,

B.M. (2002)
.


Principles of Geotechnical Engineering

,

5
th

Ed., Thomson.

Gere
,

J.
M. and Timoshenko
,

S.P. (1984)
.

Mechanics of
Materials
, PWS Publishers, Boston.

Jain
,

S.K. (1991)
.


Lecture Notes, ESM 2030: Mechanics of
Deformable Bodies, Department of Engineering Science
and Mechanics”
,
Virginia Tech,
Blacksburg
, Virginia
USA.

Jain
,

S.
K.
,

Kumar
,

A. and Poonam (2010)
.


A More Practical
Method for Using Mohr’s Circle in the Mechanics of
Solids”,
Proc.

International Conference on Materials,
Mechanics and Management
, College of Engineering
Trivandrum, Kerala, Jan. 2010 (In press)
.

Lambe, T.W. and Whitman, R.V. (1969).
Soil Mechanics
,
John
Wiley, New York.

Murthy
,

V.N.S. (2003)
.

Geotechnical Engineering
,
Marcel
Dekker, New York.

Popov
,

E.P. (1998)
.

Engineering Mechanics of Solids
,
2
nd

edition, Pearson Education, Singapore.

Powrie, W.

(2004).
Soil Mechanics: Concepts and Applications
,

2
nd

Ed., Spon Press, London.

Ranjan
,

G. and Rao
,

A.S.R. (2000)
.

Basic and Applied Soil
Mechanics
,
New Age International Publishers.



120 kN/m
2


300 kN/m
2

40 kN/m
2