WORLD FOR CIVIL

Topic 8.2: Special Topics I - Stresses on Incline Planes

In Diagram 1a we have shown a rectangular section in simple tension with an axial force F apply as shown to each end. The normal stress on the end on the rectangular section is simply given by Normal Stress = F/A. However if we cut the section at an angle theta as shown in Diagram 1b, the force F is no longer perpendicular to the inclined plane area. To determine the stresses on the inclined plane area we break the force into components perpendicular and parallel to the incline plane. (Again as shown in Diagram 1b.) The area of incline plane can be written as the cross sectional area of the rectangular section divided by the cosine of angle theta, or A' = A/cos (theta).

We can then write the axial and shear stress on the inclined area as follows:
Axial Stress = F cos (theta) / A /cos (theta) = (F/A) cos²(theta)
Shear Stress = F sin (theta) / A /cos (theta) = (F/A) sin(theta) * cos(theta)
or, using a trigonometric identity, we can rewrite the shear stress as
Shear Stress = (F/2A)Sin (2*theta), and finally writing symbolically, and in terms of the normal stress on the rectangular area, we have:
; and
where these relationships allow us to calculate the axial and shear stress on an incline plane section at an angle theta.

From our relationships we can determine the maximum stresses. The maximum axial stress is just the initial normal stress on the rectangular cross section = F/A. The maximum shear stress occurs at theta = 45^o, and is equal to F/(2A), which is half of the maximum axial stress. We now look at a simple example of this application.

Example
In Diagram 2, we have shown a square, 2" by 2", section in tension with a normal force of 2000 lb. acting on each end. We would like to know the axial and shear stress on a 30^o incline plane cut through the section.

As shown in Diagram 2, we first calculate the normal stress on the square cross section, and find Normal Stress (sigma) = 500 lb/in². We next apply the our formula for axial and shear stress on an incline plane (again, as shown in Diagram 2), and find that the normal axial stress on the 30^o incline plane is 375 lb/in², and the shear stress on the 30^o incline plane is 217 lb/in².

Up to this point we have considered a section with an axial stress only in one direction. We will shortly look at the general case where we have several axial and shear stresses acting. We will wish to determine what are called the principal stresses and the principal planes.