x 
148 APPLIED MECHANICS . 
g? 
Also, sin? d= S 5 sin? 6, andeos?p=% cos? 0, 
therefore, 7?=p? sin? 0 +q? cos? 0, and tan d= ; P tan 0. 
Draw OX parallel to P, OY parallel to Q, and Os parallel to R. 
Make Os=7. Draw sm and sn parallel to OX and OY respectively. 
Let On=x=r sin ¢, and On=y=r cos ¢. Then, substituting # for 
2 caf 
y sin ¢, and y for 7 cos ¢ in the equation at the foot of p. 147, tan 1, 
which is the equation to an ellipse whose semi-axes Oa and Ob. are equal 
to p and q respectively. 
The point s may be found graphically as follows. Draw OT perpen- 
dicular to LN, to meet a circle with centre O and radius Oa at #, and 
a circle with centre O and radius Ob at T. Through T and ¢ draw 
parallels to OX and OY respectively to meet at s. 
OX and OY are principal axes of stress, and the ellipse, whose semi- 
axes are Oa and OJ, is called the ellipse of stress. 
If the stresses y and g have opposite signs, that is, if one is tensile 
and the other compressive, then Of’ =p must be measured in the opposite 
direction from O. The construction being completed as before, Os’ will 
be the direction and intensity of the resultant stress on the interface LN. 
148. Shear Stresses in Beams.—The existence of a transverse shear 
stress in beams has been discussed in Art. 99, p. 87, and in Art. 141, 
p. 139, it has been shown that a shear stress in one plane i is always accom- 
panied by a shear stress of equal intensity in planes at right angles to — 
> ee ego eet 
Fig. 200. Fig. 201. 
that plane ; hence there is shear stress in horizontal longitudinal sections 
of a horizontal beam. The object of this Article is to determine the 
intensity of the longitudinal shear stress at any point in a beam, and also 
to show how the intensity of the transverse shear stress varies at different 
points in the depth of the beam. 
Before discussing the general case of a beam of any section, it will be | 
advantageous to first consider the simple case of a beam of rectangular — 
section. Fig. 200 shows a portion of a rectangular beam of depth d and 
breadth b. YY and Y’Y’ are two transverse sections very near to one — 
another, and W, at a distance x from YY, is the resultant of all the external — 
forces acting on the beam to the right of YY or Y’Y’. The bending moment 
‘at YY is Wa, and if /, is the maximum stress at this section due to the 
bending moment Wz, then Wx = 4bd?f,. The bending moment at YY’ 
iq 
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