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2 Analysis of Stress
2.18 Mohr’s Stress Circle
A graphical means of representing the stress relationships was discovered by
Culmann (1866) and developed in detail by Mohr (1882), after whom the graphical
method is now named.
2.19 Mohr Circles for Two-Dimensional Stress Systems
Biaxial Compression (Fig. 2.14a)
The biaxial stresses are represented by a circle that plots in positive σ space, passing
through stress points σ 1 , σ 2 on the τ = 0 axis. The centre of the circle is located on
the τ = 0 axis at stress point
1
2 (σ 1 + σ 2 ). The radius of the circle has the magnitude
1
2 (σ 1 − σ 2 ), which is equal to τ max .
Biaxial Compression/Tension (Fig. 2.14b)
Here, the stress circle extends into both positive and negative σ space. The centre
of the circle is located on the τ = 0 axis at stress point
1
2 (σ 1 + σ 2 ) and has radius
1
2 (σ 1 − σ 2 ). This is also the maximum value of shear stress, which occurs in a direction at 45° to the σ 1 direction. The normal stress is zero in directions ± θ to the
direction of σ 1 , where
cos 2θ = −
σ 1 + σ 2
σ 1 − σ 2
Biaxial Pure Shear (Fig. 2.14c)
Here, the circle has a radius equal to τ zy , which is equal in magnitude to τ yz , but
opposite in sign. The centre of circle is at σ = 0, τ = 0. The principal stresses σ 1,
σ 2 are equal in magnitude, but opposite in sign, and are equal in magnitude to τ zy .
The directions of σ 1 , σ 2 are at 45° to the directions of τ zy , τ yz .
2.20 Construction of Mohr’s Circle for Two- Dimensional
Stress System
Sign Convention
For the purposes of constructing and reading values of stress from Mohr’s circle, the
sign convention for shear stress is as follows.