CHAPTER IX 
COMPOUND STRAINS AND STRESSES 
139. Directions of Stresses Generally Parallel to one Plane.—If 
a portion of a strained body be taken which is a right prism it will be 
found that in the majority of practical cases this prism may be selected 
so that on its parallel ends there is no stress whatever, and if this is so it 
is easily shown that the directions of the stresses on the remaining faces 
must be parallel to the planes of the ends of the prism. In the web of a 
girder, for instance, there is usually no stress on the sides or on plane 
sections parallel to the sides, and the tensile, compressive, and shearing 
stresses are all in directions parallel to the sides of the web. 
In considering the equilibrium of a right prismatic element, on the 
ends of which there is no stress, it is most convenient to represent this 
element with its ends parallel to the plane of the paper upon which it is 
projected ; the directions of the stresses considered are then all parallel to 
that plane. 
In the articles and exercises of this chapter it will be assumed, unless 
otherwise stated, that the directions of the stresses considered are parallel 
to the plane of the paper, and that the plane sections upon which the 
stresses act are perpendicular to that plane. 
In proving the propositions connected with stresses in a strained body 
it is convenient to consider an element of it which is a right prism, selected 
as described above, and in many cases it is necessary to assume that the 
element is indefinitely small to allow of the stresses being of varying inten- 
sities, because if the stress on a surface is not of uniform intensity, the 
stress on an indefinitely small area of that surface may be considered 
as of uniform intensity. 
140. Stresses on an Oblique Section of a Bar subjected to Direct 
Tension or Compression.—Let AB (Fig. 182) be a bar subjected to a 
direct pull or push by a load P 
which is uniformly distributed attitit p~dciit 
over its ends. If a@ is the area D 
of the cross section of the bar, 
and p the intensity of the 
stress on it, then P=pa. Con- 
sider an oblique section CD 
inclined at an angle @ to the gs - ¥ 
cross section. The area of Tuttyt? FTTtTTs 
this oblique section is a/cos 0. Fra. 182 
Considering the equilibrium of ae 
the part ACD, the force P is balanced by a force N perpendicular to 
CD, and a force Q in the plane of CD. The force N is the resultant 
138 
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