If you want to know the principal stresses and maximum shear stresses, you can simply make it through 2-D or 3-D Mohr's circles!

You can know about the theory of Mohr’s circles from any text books of Mechanics of Materials. The following two are good references, for examples.

1.
Ferdinand P. Beer and E. Russell Johnson, Jr,
"Mechanics of Materials", Second Edition, McGraw-Hill, Inc, 1992.

2 .
James M. Gere and Stephen P. Timoshenko, "Mechanics of Materials", Third
Edition, PWS-KENT Publishing Company,

The 2-D stresses, so called plane stress
problem, are usually given by the three stress components
s_{x },
s_{y }, and t_{xy}
, which consist in a two-by-two
symmetric matrix (stress tensor):

(1)

What people usually are
interested in more are the two principal stresses s_{1} and s_{2}, which
are the two eigenvalues of the two-by-two symmetric
matrix of Eqn (1), and the
maximum shear stress t_{max} ,
which can be calculated from s_{1} and s_{2} . Now, see the Fig. 1 below, which represents that a state of plane
stress exists at point **O** and that it is defined by the stress components
s_{x }, s_{y }, and t_{xy}
associated with the left element in the Fig. 1.
We propose to determine the stress components s_{x}_{q}_{ }, s_{y}_{q}_{ }, and t_{xy}_{q}_{ }_{ }associated
with the right element after it has been rotated through an angle q about the z axis.

Fig. 1 Plane stresses in different orientations

Then, we have the following relationship:

s_{x}_{q}_{ }_{ }= s_{x }cos ^{2}^{ }q + s_{y }sin^{2}^{ }q + 2 t_{xy}
sin q cos q

(2)

and

t_{xy}_{q}_{ }_{ }= - (s_{x -} s_{y }) cos^{2}^{ }q + t_{xy} (cos^{2}^{ }q - sin^{2}^{ }q)

(3)

Equivalently, the above two equations can be rewritten as follows:

s_{x}_{q}_{ }_{ }= (s_{x} +_{ }s_{y})/2 + (s_{x} -_{ }s_{y})/2 cos 2q + t_{xy} sin 2q

(4)

and

t_{xy}_{q}_{ }_{ }= - (s_{x} -_{ }s_{y})/2 sin 2q + t_{xy} cos 2q

(5)

The expression for the normal
stress s_{y}_{q}_{ }may be
obtained by replacing q in the
relation for s_{x}_{q}_{ }in Eqn. 3 by q + 90^{o}^{ }^{ },
it turns out to be

s_{y}_{q}_{ }_{ }= (s_{x} +_{ }s_{y})/2 - (s_{x} - _{ }s_{y})/2 cos 2q - t_{xy} sin 2q

(6)

From the relations for s_{x}_{q}_{ }and_{ }s_{y}_{q}^{ }, one obtains
the circle equation:

(s_{x}_{q }- s_{ave})^{2}^{ }_{ }+ t^{2}_{xy}_{q}_{ }_{ = }R^{2}_{m}

(7)

where

s_{ave}_{ }= (s_{x} +_{ }s_{y})/2 = (s_{x}_{q} + _{ }s_{y}_{q})/2 ; *R*_{m} = [(s_{x} -_{ }s_{y})^{2 }/ 4 + t^{2}_{xy}]^{1/2}

(8)

This circle is with radius^{
}*R*_{m} and centered at
**C**_{ }=_{ }(s_{ave}_{ ,}_{ }0) if let s = s_{x}_{q}_{ }_{ }and t = -t_{xy}_{q}_{ }_{ }as shown in Fig. 2 below - that is right the
Mohr's Circle for plane stress
problem or 2-D stress problem!

Fig. 2 Mohr’s circle for plane (2-D) stresses

In fact, Eqns.
4 and 5 are the parametric equations for the Mohr's circle! In Fig.
2, one reads that the point

**X = **(s_{x} , -t_{xy}
)

(9)

which corresponds to the point at which q = 0 and the point

**A = **(s_{1} , 0 )

(10)

which corresponds to the point at which q = q_{p }that gives the principal stress s_{1 }! Note that

tan 2q_{p = }2t_{xy} /(s_{x} - _{ }s_{y})

(11)

and the point

**Y = **(s_{y} , t_{xy}
)

(12)

which corresponds to the point at which q = 90^{o} and the point

**B = **(s_{2} , 0 )

(13)

which corresponds to the point at which q = q_{p }+_{ }90^{o} that gives the principal stress s_{2 }!
To this end, one can pick the maximum normal
stresses as

s_{max }_{=
}max (s_{1 }, s_{2}), s_{min }_{= }min (s_{1 }, s_{2})

(14)

Besides, finally one can also read the maximum shear stress as

t_{max }=_{
}*R*_{m} = [(s_{x} -_{ }s_{y})^{2 }/ 4 + t^{2}_{xy}]^{1}^{/2}

(15)

which corresponds to the apex of the Mohr's circle at which
q = q_{p }+_{ }45^{o} !

(The end.)

The 3-D stresses, so called
spatial stress problem, are usually given by the six stress components s_{x} , s_{y} , s_{z} , t_{xy} , t_{yz} , and t_{zx} , (see Fig.
3) which consist in a three-by-three
symmetric matrix (stress tensor):

(16)

What people usually are
interested in more are the three principal stresses s_{1} , s_{2} , and s_{3} ,
which are eigenvalues
of the three-by-three symmetric matrix of Eqn
(16) , and the three maximum shear
stresses t_{max1} , t_{max2} , and
t_{max3}
, which can be calculated from s_{1} , s_{2} , and s_{3} .

Fig. 3 3-D stress state represented by axes parallel to X-Y-Z

Imagine that there is a plane cut through the
cube in Fig. 3, and the unit normal vector **n** of the cut plane has the direction
cosines v_{x}_{
}, v_{y}_{ }, and v_{z}_{ },
that is

**n = (**v_{x} ,
v_{y} ,
v_{z})

(17)

Then the normal stress on this plane can be represented by

s_{n}_{ }_{ }= s_{x}v^{2}_{x}^{ }+ s_{y}v^{2}_{y}^{ }+ s_{z}v^{2}_{z}^{ }+ 2 t_{xy}v_{x}v_{y}_{
}+ 2 t_{yz}v_{y}v_{z}_{ }+ 2 t_{xz}v_{x}v_{z}

(18)

There exist three sets of
direction cosines, **n**_{1}, **n**_{2}, and **n**_{3} - the three principal axes, which make s_{n }achieve
extreme values s_{1 }, s_{2 }, and s_{3 }- the
three principal stresses, and on the corresponding cut planes, the shear
stresses vanish! The problem of finding the principal stresses and their
associated axes is equivalent to finding the eigenvalues
and eigenvectors of the following problem:

(s**I**_{3} - **T**_{3})**n = 0**

(19)

The three eigenvalues of Eqn (19) are the roots of the following characteristic polynomial equation:

det(s**I**_{3} - **T**_{3}) = s^{3} - As^{2} + Bs - C = 0

(20)

where

A = s_{x }+ s_{y }+ s_{z}

(21)

B = s_{x}s_{y} + s_{y}s_{z} + s_{x}s_{z}_{ } - t^{2}_{xy} - t^{2}_{yz} - t^{2}_{xz}

(22)

C = s_{x}s_{y}s_{z}_{ }+ 2 t_{xy}t_{yz}t_{xz} - s_{x}t^{2}_{yz}- s_{y}t^{2}_{xz}- s_{z}t^{2}_{xy}

(23)

In fact, the
coefficients A, B, and C in Eqn (20) are invariants
as long as the stress state is prescribed (see e.g. Ref. 2) . Therefore,
if the three roots of Eqn (20) are __s___{1}, __s___{2 }, and __s___{3 },_{ }one has the following equations:

__s___{1 }+ __s___{2 }+ __s___{3 }= A

(24)

__s___{1}__s___{2} + __s___{2}__s___{3} + __s___{1}__s___{3 }= B

(25)

__s___{1}__s___{2}__s___{3 =} C

(26)

Numerically, one can always
find one of the three roots of Eqn (20), e.g.
__s___{1 }, using
line search algorithm, e.g. bisection algorithm.
Then combining Eqns (24)and (25), one obtains a
simple quadratic equations and therefore obtains two other roots of Eqn (20), e.g. __s___{2} and __s___{3 .}_{ }_{ }To this
end, one can re-order the three roots and obtains the three principal stresses,
e.g.

s_{1 }= max (__s___{1 ,} __s___{2 ,} __s___{3})

(27)

s_{3}_{ }= min (__s___{1 ,} __s___{2 ,} __s___{3})

(28)

s_{2}_{ }= (A - s_{1 }- s_{2 })

(29)

Now, substituting s_{1 }, s_{2 }, or s_{3}_{ }_{
}into Eqn
(19), one can obtains the ** unit**
vectors

Similar to Fig. 3, one can imagine a
cube with their faces normal to **n**_{1}, **n**_{2}, or **n**_{3. }For
example, one can do so in Fig. 3 by replacing the axes **X**,**Y**, and **Z**
with **n**_{1}, **n**_{2}, and **n**_{3 }, respectively, replacing the normal
stresses s_{x} , s_{y} , and s_{z} with
the principal stresses s_{1 }, s_{2 }, and s_{3 },
respectively, and removing the shear stresses t_{xy} ,
t_{yz} , and t_{zx}
.

Now, pay attention to the new cube with
axes **n**_{1}, **n**_{2}, and **n**_{3. }Let the cube be rotated about the axis **n**_{3,} and then the corresponding transformation of stress
may be analyzed by means of Mohr's circle as if it were a transformation of
plane stress. Indeed, the shear stresses exerted on the faces normal to the **n**_{3 }axis remain equal to zero, and the normal stress s_{3}_{ }_{ }is perpendicular to the plane spanned by **n**_{1 }and **n**_{2 }in
which the transformation takes place and thus, does not affect this
transformation. One may therefore
use the circle of diameter *AB* to determine the normal and shear stresses
exerted on the faces of the cube as it is rotated about the **n**_{3 }axis (see Fig. 4). Similarly, the circles of diameter *BC*
and *CA *may be used to determine the stresses on the cube as it is
rotated about the **n**_{1
}and **n**_{2 }axes,
respectively.

Fig. 4 Mohr’s circles for spatial (3-D) stresses

What if the rotations are about the axes rather
than principal axes? It can be shown that any other transformation of axes
would lead to stresses represented in Fig. 4 by a point located within the area
which is bounded by the biggest circle with the other two circles removed!

Therefore, one can obtain the maximum/minimum normal and shear stresses from Mohr's circles for 3-D stress as shown in Fig. 4!

Note the notations above (which may be different from other references), one obtains that

s_{max =} s_{1}

(30)

s_{min =} s_{3}

(31)

t_{max =} (s_{1 }- s_{3})/2 = t_{max2}

(32)

Note that in Fig. 4,_{ }t_{max1 }, t_{max2 }, and t_{max3 }are the maximum shear stresses obtained while the
rotation is about _{ }**n**_{1}, **n**_{2}, and **n**_{3 }_{ }, respectively.

(The end.)

Mohr's circle(s) can be used for strain analysis, the moments and products of inertia, and other quantities as long as they can be represented by two-by-two or three-by-three symmetric matrices (tensors).

(The end.)

By *Jing Li* (Email: jili@vt.edu) and *Rakesh** K. Kapania* (Email: rkapania@vt.edu)

Copyright © 1997-2007