Mohr's Circle for 2-D Stress Analysis

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, Boston, 1990.

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 s₁ and s₂, 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 s₁ and s₂ . 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_xq , s_yq , and t_xyq 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_xq = s_x cos ² q + s_y sin² q + 2 t_xy sin q cos q

(2)

and

t_xyq = - (s_{x -} s_y ) cos² q + t_xy (cos² q - sin² q)

(3)

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

s_xq = (s_x + s_y)/2 + (s_x - s_y)/2 cos 2q + t_xy sin 2q

(4)

and

t_xyq = - (s_x - s_y)/2 sin 2q + t_xy cos 2q

(5)

The expression for the normal stress s_yq may be obtained by replacing q in the relation for s_xq in Eqn. 3 by q + 90^o ,  it turns out to be

s_yq = (s_x + s_y)/2 - (s_x - s_y)/2 cos 2q - t_xy sin 2q

(6)

From the relations for s_xq and s_yq , one obtains the circle equation:

(s_xq - s_ave)² + t²_{xyq =} R²_m

(7)

where

s_ave = (s_x + s_y)/2 = (s_xq + s_yq)/2 ;    R_m = [(s_x - s_y)² / 4 + t²_xy]^1/2

(8)

This circle is with radius R_m and centered at C = (s_{ave  ,} 0) if  let s = s_xq and  t = -t_xyq 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 = (s₁ , 0 )

(10)

which corresponds to the point at which q = q_p that gives the principal stress s₁ ! 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 = (s₂ , 0 )

(13)

which corresponds to the point at which q = q_p + 90^o that gives the principal stress s₂ ! To this end, one can pick the maximum normal stresses as

s_{max =} max (s₁ , s₂), s_{min =} min (s₁ , s₂)

(14)

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

t_max = R_m = [(s_x - s_y)² / 4 + t²_xy]^1/2

(15)

which corresponds to the apex of the Mohr's circle at which q = q_p + 45^o !

(The end.)

Mohr's Circles for 3-D Stress Analysis

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 s₁ , s₂ , and s₃ , 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 s₁ , s₂ , and s₃ .

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_xv²_x + s_yv²_y + s_zv²_z + 2 t_xyv_xv_y + 2 t_yzv_yv_z + 2 t_xzv_xv_z

(18)

There exist three sets of direction cosines, n₁, n₂, and n₃ - the three principal axes, which make s_n achieve extreme values s₁ , s₂ , and s₃ - 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:

(sI₃ - T₃)n = 0

(19)

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

det(sI₃ - T₃) = s³ - As² + Bs - C = 0

(20)

where

A = s_x + s_y + s_z

(21)

B = s_xs_y + s_ys_z + s_xs_z - t²_xy - t²_yz - t²_xz

(22)

C = s_xs_ys_z + 2 t_xyt_yzt_xz - s_xt²_yz- s_yt²_xz- s_zt²_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 s₁, s₂ , and s₃ , one has the following equations:

s₁ + s₂ + s₃ = A

(24)

s₁s₂ + s₂s₃ + s₁s₃ = B

(25)

s₁s₂s_{3 =} C

(26)

Numerically, one can always find one of the three roots of Eqn (20), e.g. s₁ , 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. s₂ and s_{3 .} To this end, one can re-order the three roots and obtains the three principal stresses, e.g.

s₁ = max (s_{1 ,} s_{2 ,} s₃)

(27)

s₃ = min (s_{1 ,} s_{2 ,} s₃)

(28)

s₂ = (A - s₁ - s₂ )

(29)

Now, substituting s₁ , s₂ , or s₃ into Eqn (19), one can obtains the unit vectors n₁, n₂, or n_3, respectively, of the corresponding principal axes.

Similar to Fig. 3, one can imagine a cube with their faces normal to n₁, n₂, or n_3. For example, one can do so in Fig. 3 by replacing the axes X,Y, and Z with n₁, n₂, and n₃ , respectively,  replacing  the normal stresses s_x , s_y , and s_z with the principal stresses s₁ , s₂ , and s₃ , respectively, and removing the shear stresses t_xy , t_yz , and t_zx .

Now, pay attention to the new cube with axes n₁, n₂, 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 n₃ axis remain equal to zero, and the normal stress s₃ is perpendicular to the plane spanned by n₁ and n₂ 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 n₃ 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 n₁ and n₂ 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 =} s₁

(30)

s_{min =} s₃

(31)

t_{max =} (s₁  - s₃)/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   n₁, n₂, and n₃  , respectively.

(The end.)

 Mohr's Circles for Strain and for Moments and Products of Inertia

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