Page 49 - Handbook of Civil Engineering Calculations, Second Edition

1.32 STRUCTURAL STEEL ENGINEERING AND DESIGN

STRESSES ON AN OBLIQUE PLANE

A prism ABCD in Fig. 20a has the principal stresses of 6300- and 2400-lb/sq.in.
(43,438.5- and 16,548.0-kPa) tension. Applying both the analytical and graphical meth-
ods, determine the normal and shearing stress on plane AE.

Calculation Procedure:
1. Compute the stresses, using the analytical method
A principal stress is a normal stress not accompanied by a shearing stress. The plane on
which the principal stress exists is termed a principal plane. For a condition of plane
stress, there are two principal planes through every point in a stressed body and these
planes are mutually perpendicular. Moreover, one principal stress is the maximum normal
stress existing at that point; the other is the minimum normal stress.
Let s x and s y the principal stress in the x and y direction, respectively; s n normal
stress on the plane making an angle with the y axis; s s shearing stress on this plane.
All stresses are expressed in pounds per square inch (kilopascals) and all angles in de-
grees. Tensile stresses are positive; compressive stresses are negative.
2
Applying the usual stress equations yields s n s y (s x s y ) cos ; s s /2(s x s y )
1
2
sin 2 . Substituting gives s n 2400 (6300 2400)0.766 4690-lb/sq.in. (32,337.6-
1
kPa) tension, and s s /2(6300 2400)0.985 1920 lb/sq.in. (13,238.4 kPa).
2. Apply the graphical method of solution
Construct, in Fig. 20b, Mohr’s circle of stress thus: Using a suitable scale, draw OA
s y , and OB s x . Draw a circle having AB as its diameter. Draw the radius CD making
an angle of 2 80° with AB. Through D, drop a perpendicular DE to AB. Then OE
s n and ED s s . Scale OE and ED to obtain the normal and shearing stresses on
plane AE.
Related Calculations. The normal stress may also be computed from s n
(s x s y )0.5 (s x s y )0.5 cos 2 .

FIGURE 20