Mohr’s Circle Rules and Comments
July 1, 2012
2D Mohr’s Circle
1. Mohr’s circle is plotted on a cartesian coordinate system as shown in Figure 1, where positive
normal stress coordinate, σ, is to the right and positive shear stress coordinate, τ , is downwards.
y
σy
F
(σy , −τxy ) τxy
G
τxy
σmin ( σx +σ
2 , 0)
y
σmax +σ σx σx
C 2θ
(σx , τxy ) τxy θ x
A
E τxy σy
+τ
Figure 1: Positive coordinate system for Mohr’s circle.
Shear stress is measured downwards so that rotations on Mohr’s circle occur in the same
direction as rotations on the stress block currently being analyzed.
2. The center of the circle, C, is located at coordinate ( σx +σ
2
y
, 0).
3. A reference point, A, is created for the given stress block at coordinate (σx , τxy ).
4. Points C and A define the radius of the Mohr’s circle for the given stress block.
5. From the center and having the radius, Mohr’s circle may be drawn.
6. The principle stresses will lie on the σ axis. The principle stresses are often referred to as
σmax = σ1 and σmin = σ2 .
7. The maximum in plane shear stress is located at the top, point F (or bottom, point E) of the
circle and is located above (or below) the σavg coordinate.
8. For the given stress block, after Mohr’s circle has been constructed, the state of stress at any
angle, θ, from our given stress block may be found. This may be accomplished by moving 2θ
away from radius, CA, on Mohr’s circle and reading off the resulting (σx , τxy ) values. The stress
σy is 90 degrees away on the stress block, so it is 180 degrees away on Mohr’s circle and can be
read from the circle also (see point G). Positive angles are always measured counterclockwise
usually from the positive x-axis of the given stress block, or from radius CA on Mohr’s circle.
3D Mohr’s Circle
Using the equalities τxy = τyx , τxz = τzx and τyz = τzy the state of stress at any point may be written
in matrix form as follows:
τxx τxy τxz
τ = τxy τyy τyz (1)
τxz τyz τzz
The eigenvalues of the stress matrix are the principle stresses. That is, the principle stresses are the
roots of the characteristic equation found by taking the determinant of τ − σI, where σ can be any
one of the three eigenvalues (or principle stresses). After some algebra the characteristic equation is
σ 3 − (τxx + τyy + τzz )σ 2 + (τxx τyy + τyy τzz + τzz τxx − τxz
2 2
− τxy 2
− τyz )σ
2 2 2
− (τxx τyy τzz − τxx τyz − τyy τxz − τzz τxy + 2τxy τyz τxz ) = 0 (2)
The above equation may be rewritten as
σ 3 − I1 σ 2 + I2 σ − I3 = 0, (3)
where I1 , I2 and I3 are the stress invariants defined as
I1 = τxx + τyy + τzz = trτ (4a)
2 2 2 1
(trτ )2 − tr(τ 2 )
I2 = τxx τyy + τyy τzz + τzz τxx − τxz − τxy − τyz = (4b)
2
2 2 2
I3 = τxx τyy τzz − τxx τyz − τyy τxz − τzz τxy + 2τxy τyz τxz = detτ . (4c)
In two dimensions the above characteristic equation (2) simplifies to
σ 2 − (τxx + τyy )σ + (τxx τyy − τxy
2
)=0 (5)
since τzz = τxz = τyz = 0.
So for a given stress block with given stresses τij the principle stresses σ1 , σ2 and σ3 may be found.
From the largest and smallest of these three stresses the maximum shear stress is found as
σmax − σmin
τmax = (6)
2
Suppose for example the principle stresses were (σ1 = 50 ksi, σ2 = 10 ksi, σ3 = −15 ksi ). Then
3D Mohr’s circle can be plotted as shown below in Figure 2.
All possible stress
states in the dark
grey region
+σ
σ3 σ2 σ1
+τ
Figure 2: 3D Mohr’s Circle Example
