Page:Elementary Text-book of Physics (Anthony, 1897).djvu/128

114 The combination of tangential stresses which maintain equilibrium may be considered from another point of view. For, if we examine the triangular prism of which the cross-section is $$ABD$$ (Pig. 35), and to which the tangential stresses $$S$$ and $$S'$$ are applied, it appears at once that equilibrium will obtain when a force equal to the resultant of $$aS$$ and $$aS'$$, where $$a$$ is the area of each of the square faces of the prism, is applied to the face of which $$AB$$ is the trace. The area of this face is $$aX\sqrt{2}$$, and if $$X$$ represent the pressure on this face, the force applied to it is $$aX\sqrt{2}$$. But $$S$$ equals $$S'$$, and the resultant of $$aS$$ and $$aS'$$ is $$aS\sqrt{2}$$; whence $$X = S$$. A similar pressure acts in the opposite direction upon the face of the similar prism $$ACB$$. These pressures are positive, that is, they are tensions which tend to separate the parts of the body to which they are applied. If we compound the tangential stresses in another manner by taking as the element of the combination the stresses applied to the faces $$AD$$ and $$AC$$, it is at once evident that they are equivalent to a negative pressure $$S$$ upon the diagonal face $$CD$$. A similar pressure acts across the same face toward the other prism $$CBD$$. We may therefore consider the set of stresses constituting the couples in the plane $$ACBD$$ as equivalent to a positive pressure or tension in the direction of one diagonal and a negative pressure in the direction of the other diagonal. This combination of couples, or its equivalent tension and pressure, is called a shearing stress.

101. Superposition of Stresses.—Stresses, whether pressures or tensions, being vector quantities, are compounded like other vector quantities, and, in particular, when they are in the same line, are added algebraically.

Suppose a cube so subjected to stress that equal and opposite pressures, which we will assume to be directed outward from the cube, act on two opposite faces, and that the other faces experience no stress. Such a stress is called a longitudinal traction. We will show that this form of stress may be obtained by the combination of a stress made up of equal tensions acting on each face of the cube, and of two shearing stresses.

In Fig. 36 let $$P$$ represent the value of the longitudinal