Copyright © 2000 C. E. Knight
PURPOSE OF THE TUTORIAL
This tutorial is designed to introduce and place strong emphasis on the role of 3-D stress in the process of
mechanical design. Students in engineering are normally introduced to stress in its simplest onecomponent form defined by load divided by area of cross section.. This is a validdefinition of a pure 1-D
state of stress, but in many cases it seems to establish a baseline safe position for which many students
don’t want to venture forth. Carrying this attitude through the mechanical design process is a recipe for
Everything in the mechanical design realm has solid 3-D characteristics. The same is true for the state of
stress in the solid. In many simple cases theeffective state of stress can be reduced to 2-D or 1-D, but only
after careful consideration. In the early stages of mechanical design, the locations of most likely stress
failure and the corresponding stress components acting at those locations must be identified. Once all the
stress components at a given location are determined, they may then be combined to find principal stresses,
maximumshear stress or other measures that are useful for predicting design success or failure. It is very
important to remember that stress components for one location in a machine part should never be
combined with stress components for a different location in the same part.
One of the interesting developments in visualizing the combining of 2 stress components was the
creation of Mohr’scircle. This graphical representation of the 2-D stress transformation equations provides
a quick, accurate and visual protrayal of the 2 state of stress. It finds the principal stresses and a
maximum shear stress (although this maximum shear stress may be quite misleading in 3-D stress).
Review of Two-Dimensional Mohr’s Circle - Graphical Approach
The beginning of a Mohr’s circle representationmust be a stress element sketch of the 2-D state of stress as
shown in the figure.
This shows all potential non-zero 2-D stress components. The graphical Mohr’s circle uses coordinate
pairs of these data to make a plot. They are (σx, τxy ) and (σy , τyx). These two points establish the circle
diameter. By convention normal stresses, σ are positive in tension and negative in compression,however,
the shear stresses, τ, in the Mohr’s circle constructions are taken as positive if they make a cw moment
about the stress element. In the stress element above, τxy is ccw (-) while τyx is cw(+). This convention is
useful for determining the proper orientation of principal stresses and other components relative to the x,y
As an example, assume that σx is positive and τxy ispositive (cw) with σy equal zero. First sketch the
normal stress axis along the horizontal and the shear stress axis along the vertical. Then plot the first
coordinate pair (σx, τxy ) at point A. Then plot the second pair (0, τyx ) at point B. These two points form
the diameter of the circle with its center at point C. Simple geometric triangles can then determine the
circle radius and allprincipal stress and peak shear stress values.
In a second example, assume that σx is smaller than σy , but both are positive, and that τxy is cw. Sketch the
normal stress, σ, and shear stress, τ axes and plot the coordinate pair (σx, τxy ) at point A and then (σy , τyx) at
point B. Connecting these points locates the circle center at point C. Geometrical calculations finish the
The Mohr’s circle gives a complete visual representation of the 2-D state of stress along with accurate
numerical values. However, there is a highly significant factor in mechanical design that has thus far been
neglected. That factor is the influence of the additional 3-D stress components on the design safety.
Three-Dimensional Stress in Mechanical Design
In the real world of...