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
(σ
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
2θ
in the same direction as
in
x
–
y
coordinate system. (Angle
θ
in the square
element is equal to
2θ
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
2θ
P
.
9.
Determine
2θ
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
(σ
x
, τ
xy
)

σ
B
O
τ
(σ
x
+ σ
y
)/2
X
′
(σ
x
′
, τ
x
′
y
′
)

X
(σ
x
, τ
xy
)

σ
B
O
τ
(σ
x
+ σ
y
)/2
2θ
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
.
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Mechanics of Materials
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,
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A More Practical
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.
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120 kN/m
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300 kN/m
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40 kN/m
2
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