140 APPLIED MECHANICS 
142. Pure Shear Stress Equivalent to two Normal Stresses.— 
Consider an indefinitely small cube ABCD (Fig. 184) of a strained body. 
Let b be the length of the edges of this cube. Assume that there is no 
stress on the face ABCD or on any interface parallel to it. Suppose that 
the faces AD and BC are subjected to pure shear stress of intensity g, the 
direction of which is parallel to the face ABCD, then by the preceding 
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Fig. 184. Fig, 185. Fia. 186. 
Article there must be shear stresses of intensity g on the faces AB and 
CD, as shown. Imagine the cube divided into two equal parts by a plane 
AC perpendicular to the face ABCD. Consider (Fig. 185) the equili- 
brium of ABC, one of these parts. The resultants QQ of the stresses + 
on AB and BC must balance the resultant R of the stress on AC, and by 
the triangle of forces it is seen that R is perpendicular to AC and equal 
to Q ,/2. The stress on AC is evidently a tensile stress. Let 7 be the 
intensity of the stress on AC, then 
=r? ,/2=Q ,/2= qb? ,/2, therefore r=q. 
Tn like manner, by dividing the cube into two equal parts by a plane 
BD perpendicular to the face ABCD, and considering (Fig. 186) the 
equilibrium of the part ABD, it can be shown that there is a compressive 
stress of intensity g on the face BD. 
Hence a pure shear stress is equivalent to two normal stresses at 45° 
to the shear stress, and each equal in intensity to the shear stress, but one 
is a tensile and the other a compressive stress. 
It is evident that all sections of the cube parallel to the plane AC 
- will be subjected to tensile stress, of intensity g, and all sections parallel 
to BD will be subjected to 
compressive stress of inten- | | | B 
sity g.. Hence if a part of i? 
the interior of the cube be 
mapped out so as to form a 
rectangular solid EF having 
faces parallel to AC and BD, 
as shown in Fig. 187, this 
solid will be subjected 
to tensile and compressive Fic. 187. Fig. 188. 
stresses of intensity equal 
to that of the shear stresses on the faces of the cube ABCD. 
Conversely, it is easy to show that if a cube ABCD (Fig. 188) have 
its faces AD and BC subjected to tensile stress, and also have its faces 
AB and CD subjected to an equal compressive stress, sections parallel to 
AC and BD will be subjected to shear stress of the same intensity. 
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