A graphical method of constructing the shear and normal stress tractions on any plane given two principal stresses. This only works in 2-D. Equations exist to compute the shear and normal stress tractions in 3-D. Luckily there are geologic situations where the maximum and minimum principal stress traction are the crucial ones to consider, and a 2-D perspective is adequate. The assumption is that the plane continues in and out of the plane perpendicularly (meaning it would contain the intermediate principal stress). plane max principal stress Given: 2 principal stress tractions and a plane. Find: the shear and normal traction on that plane. min principal stress min principal stress max principal stress

Given: 2 principal stress tractions and a plane. Find: the shear and normal traction on that plane. plane 1) Construct a circle enclosing the strain ellipse. Radius = max principal stress. 2) Construct a perpendicular line to the plane through the center of the ellipse. This can be worked the other way, from a stress traction to find the plane it acts on. 3) From where the perpendicular intersects the enclosing circle draw a line parallel to the direction of the minor ellipse axis (the least principal stress direction) until in intersects the stress ellipse boundary. 4) Construct a vector from where the line constructed in step 3 intersects the ellipse to the center of the ellipse – this is your answer. Draw the corresponding vector on the other side of the plane. The yellow vectors represent the stress traction on this plane given the 2-D stress state.

How to graphically construct the shear and normal traction on the plane of interest? 1) From the tail of the yellow vector (the traction on the given plane, draw a vector that is perpendicular to the plane. This is the normal traction component, drawn in green here. The next natural question might be as to whether these stresses will cause faulting along that plane. We will get to that. 2) For the shear traction component, draw a vector from the head of the normal traction component to the head of the total traction on the plane (the yellow vector) parallel to the plane. These are depicted in brown here, and one can see that in this perspective the shear should be dextral. If the diagram was scaled you could estimate the magnitude.

Here is a practice example for you Here is a practice example for you. You can either print the page of and draw on that or use the shape tools in PowerPoint to complete it within this file. max principal stress min principal stress plane min principal stress max principal stress

Another example to ‘play’ with Another example to ‘play’ with. Note here that the difference between the max and min principle stress is less than in the previous examples. plane min principal stress max principal stress max principal stress min principal stress