Mohrs Circle – Principal Stress and Stress Transformation

Mohr's circle is a graphical representation of the transformation of normal and shear stress as the orientation of a plane through a stressed point is changed. It provides a convenient method of determining principal stresses, maximum shear stress and the stresses acting on an inclined plane.

Although the same results can be obtained directly from the stress transformation equations, Mohr's circle remains useful for visualising the relationship between stress components and plane orientation.

Contents

• Plane Stress
• Stress and Sign Convention
• Constructing Mohr's Circle
• Principal Stresses
• Maximum Shear Stress
• Plane Stress Transformation
• Principal Plane Orientation
• Three-Dimensional Mohr's Circles
• Combined Loading of a Shaft

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Plane Stress

For a plane-stress condition in the x-y plane, the non-zero stress components may be:

• σ_x – normal stress acting on the x face;
• σ_y – normal stress acting on the y face;
• τ_xy and τ_yx – complementary shear stresses.

The out-of-plane stress components are zero:

$${\sigma }_z={\tau }_{xz}={\tau }_{yz}=0$$

For an ordinary stress element in static equilibrium, moment equilibrium requires the complementary shear stresses to have equal magnitude:

$${\tau }_{xy}={\tau }_{yx}$$

As the orientation of the element is changed, the normal and shear stress components acting on its faces change. At particular orientations the shear stress becomes zero. The normal stresses acting on those planes are the principal stresses.

Principal Stress Ordering

For a general three-dimensional stress state the principal stresses are conventionally ordered:

$${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$$

For plane stress, one principal stress is the zero out-of-plane stress. The two in-plane principal stresses must therefore be considered together with zero before assigning the ordered labels σ₁, σ₂ and σ₃.

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Stress and Sign Convention

Normal tensile stress is taken as positive and compressive stress as negative on this page.

Shear-stress sign conventions used with Mohr's circle vary between engineering texts. In particular, some conventions plot positive shear upward on the Mohr diagram while others plot it downward.

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Constructing Mohr's Circle

For a plane stress state defined by σ_x, σ_y and τ_xy, the centre of Mohr's circle lies on the normal-stress axis at:

$${\sigma }_{\text{avg}}=\frac{{\sigma }_x+{\sigma }_y}{2}$$

The radius of the circle is:

$$R=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$

The equation of the circle is therefore:

$${\left(\sigma -{\sigma }_{\text{avg}}\right)}^2+{\tau }^2=R^2$$

The intersections of the circle with the normal-stress axis correspond to the two in-plane principal stresses. The top and bottom of the circle represent the maximum magnitude of in-plane shear stress.

Examples of Mohr's circles for different stress states are shown below.

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Principal Stresses

The two in-plane principal stresses are:

$${\sigma }_{1,2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$

Equivalently:

The shear stress on the principal planes is zero.

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Maximum Shear Stress

For the plane stress transformation, the maximum in-plane shear-stress magnitude is equal to the radius of Mohr's circle:

$${\tau }_{\text{max,in-plane}}=R$$

Therefore:

$${\tau }_{\text{max,in-plane}}=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$

The normal stress acting on these maximum in-plane shear planes is:

$$\sigma ={\sigma }_{\text{avg}}=\frac{{\sigma }_x+{\sigma }_y}{2}$$

After ordering all three principal stresses:

$${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$$

the absolute maximum shear stress is:

$${\tau }_{\text{max,absolute}}=\frac{{\sigma }_1-{\sigma }_3}{2}$$

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Plane Stress Transformation

Consider axes x'-y' rotated counterclockwise through an angle θ from the original x-y axes.

The original derivation considers the forces acting on an inclined elemental plane. If its area is A_θ, its projected areas are:

The following trigonometric identities are useful:

Applying force equilibrium and simplifying gives the plane-stress transformation equations:

$${\sigma }_{x\prime }=\frac{{\sigma }_x+{\sigma }_y}{2}+\frac{{\sigma }_x-{\sigma }_y}{2}\cos 2\theta +{\tau }_{xy}\sin 2\theta$$ $${\sigma }_{y\prime }=\frac{{\sigma }_x+{\sigma }_y}{2}-\frac{{\sigma }_x-{\sigma }_y}{2}\cos 2\theta -{\tau }_{xy}\sin 2\theta$$ $${\tau }_{x\prime y\prime }=-\frac{{\sigma }_x-{\sigma }_y}{2}\sin 2\theta +{\tau }_{xy}\cos 2\theta$$

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Principal Plane Orientation

The principal planes occur where the transformed shear stress is zero:

$${\tau }_{x\prime y\prime }=0$$

Therefore:

$$\tan \left(2{\theta }_p\right)=\frac{2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}$$

The two principal planes are perpendicular to one another in the physical element. On Mohr's circle their corresponding points are separated by 180°.

$$2{\theta }_p=\mathrm{atan2}\left(2{\tau }_{xy},{\sigma }_x-{\sigma }_y\right)$$

The planes of maximum in-plane shear are oriented 45° from the principal planes:

$${\theta }_s={\theta }_p\pm 45°$$

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Three-Dimensional Mohr's Circles

A general three-dimensional stress state has three principal stresses:

$${\sigma }_1\ge {\sigma }_2\ge {\sigma }_3$$

Three Mohr circles can be drawn, corresponding to the principal-stress pairs:

• σ₁ and σ₂;
• σ₂ and σ₃;
• σ₁ and σ₃.

The centres of the three circles are:

Their radii are:

The largest circle is the σ₁-σ₃ circle, and its radius gives the absolute maximum shear stress:

$${\tau }_{\text{max}}=\frac{{\sigma }_1-{\sigma }_3}{2}$$

A plane-stress condition is therefore a special three-dimensional case in which one of the three principal stresses is zero.

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Combined Loading of a Shaft

A common engineering application of Mohr's circle is a shaft subjected to a combination of axial load, bending moment and torque.

At a point on the shaft surface, the axial and bending stresses combine to give a longitudinal normal stress:

$${\sigma }_x={\sigma }_{\text{axial}}+{\sigma }_{\text{bending}}$$

For a circular shaft:

$${\sigma }_{\text{axial}}=\frac{F}{A}$$ $${\sigma }_{\text{bending}}=\frac{My}{I}$$

The torque produces torsional shear stress:

$$\tau =\frac{Tr}{J}$$

At a free surface where there is no radial normal stress, this produces the common plane-stress state:

$${\sigma }_x={\sigma }_{\text{axial}}+{\sigma }_{\text{bending}},{\sigma }_y=0,{\tau }_{xy}=\frac{Tr}{J}$$

The principal stresses and maximum shear stress can then be obtained directly from Mohr's circle or from the equations given above.

For ductile-material design, the resulting multiaxial stress state can subsequently be assessed using the Tresca or von Mises yield criteria.

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Related RoyMech Mechanics References

• Component Loading – direct, shear, bending and torsional loading.
• Stress and Strain – basic elastic stress and strain relationships.
• Yield and Failure Criteria – Tresca, von Mises and principal-stress criteria.
• Column and Strut Buckling – stability of compression members.
• Thick Cylinders – radial, circumferential and longitudinal stresses in thick-walled cylinders.
• Rotating Discs and Cylinders – radial and circumferential stresses caused by rotation.
• Torsion – torsional shear stress and angular deformation.
• Stress-Strain Curve – material yielding and tensile behaviour.
• Loaded Flat Plates – stress and deflection of loaded plates.
• Beam Design and Analysis – beam bending and shear stresses.

Other Mechanics References

• Vectors
• Statics
• Kinematics
• Kinetics and Dynamics
• Mechanical Linkages
• Simple Machines

External Mohr's Circle References

1. Tech.Plym – Mohr's Circle – lecture notes including three-dimensional Mohr's circle.
2. eFunda – Mohr's Circle – engineering reference on Mohr's circle and stress transformation.
3. Solid Mechanics – Wikibooks – general solid-mechanics reference.
4. Solid Mechanics: Stress – Mohr's Circle for Plane Stress – detailed reference on plane-stress Mohr's circle.

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