BEAMS AND BENDING 111 
S In designing a reinforced beam, if either /, or /, is assumed, the other 
_ is found from the equation }/,by, = a/;, 
- The value of the ratio m7” is generally taken at 15. 
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Exercises VIIb. 
zz 1. Taking the moment of resistance to bending of section A (Fig. 150) as 
s , find the numbers which will represent the moments of resistance of the 
_. B, C, and D, (In section C the metal is 1 inch thick.) 
2. Denoting the moment of resistance to bending of section A, per square 
inch of section, by 1, 
determine the numbers " 
which will express the pedals e- 5" +4 
moments of resistance of it ! 
the sections B,C, and D io 
square inch.of section. ' 
3. The section E (Fig. 4 
150) has a total depth of A B 
8inches. The flanges are 
5 inches wide and 1} Fia. 150. 
inches thick. The web 
is 1 inch thick. What is the moment of resistance of this section to bending 
when the maximum stréss is 5 tons per square inch? What will the answer be 
if the section is placed with the web horizontal ? 
_ 4 Asolid circular section is 6 inches diameter. A hollow circular section 
is 8 inches diameter outside. Find the internal diameter of the hollow section 
_ gothat it shall have the same area as the solid section ; then, denoting the moment 
of resistance of the solid section by 1 determine the number which will represent 
_ the moment of resistance of the hollow section. 
5. A steel joist has a total depth of 18 inches. The flanges are 7 inches wide 
and 0:94 inch thick. The web is 0°55 inch thick. Determine the section modulus, 
_ @, in inch units (a) by the correct formula, (b) by the formula Z=ah, where a is 
the area of one flange, and / is the total depth. 
; 6. Construct, half full size, the modulus figures for the following sections. 
_ (a) Circle 6 inches diameter. (+) Hollow circle, external diameter 6 inches, internal 
“_ ter 3 inches. (c) Flanged section 6 inches deep, flanges 3} inches wide and 
7 “bce thick, web 1} inches thick. (d) Isosceles triangle, base 5 inches, height 
_ 6 inches. (e) Flanged section 6 inches deep, top flange 2} inches wide and 1} inches 
_ thick, bottom flange 4 inches wide and 14 inches thick, web 1} inches thick. 
_ From these figures determine in cases (a), (6), and (¢) the values of Z, and in 
_ @ases(d) and (e) the values of Z, and Z,. Compare the results 
_ with those obtained by calculation from the correct formule. 
7. AEB and CFD (Fig. 151) are semicircles whose diameters 
AB and CD are parallel and 2} inches apart. AB=2} inches, 
CD=2} inches. AD and BC are straight lines which are per- 
pendicular to one another. The whole figure AEBCFD is the 
modulus figure of a beam section. Construct the beam section, 
8. A cantilever 50 inches long carries a load of 4000 lbs. at 
its free end. The maximum stress due to bending is to be 3000 
per square inch at every cross section. The cross section 
is a rectangle, breadth=a, depth=y. Determine the cross 
section, xx y at 10, 20, 30, 40, and 50 inches from the free end, 
and draw a plan and side elevation of the cantilever (scale, 1 Fig, 151. 
inch to 1 foot) in each of the following cases :— 
(a) y=6 inches, and the lever to be symmetrical about a vertical longitudinal 
section. 
_ (0) x=3 inches, and the top surface of the lever to be horizontal. 
(c) y=3a, the top surface to be horizontal, and the lever to be symmetrical 
about a vertical longitudinal section. ; 
9. Same as Exercise 8, except that the cross section is a circle of diameter y. 
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