- BEAMS AND BENDING 107 
esol J-HK. Hence the strip HK subjected to the stress / will have 
_ the same resistance as the strip mn subjected to the stress /,,. 
____ If the above construction be repeated for a sufficient number of strips 
(the construction 
fora strip above, M 
XX is shown by “ 
_ dotted lines), and 
the points joined esi) iad 
RP 
° 
z 
_ Thesectionmodu- _ 145. 
Tus figure has the ea 
property that the sum of the moments of the areas of all the strips 
parallel to XX about XX multiplied by /, will be the moment of re- 
_ sistance of the section. Hence if d, is the distance between the centres 
of gravity of the parts of the section modulus figure on opposite sides of 
_ XX measured perpendicular to XX, and if a, is the area of the figure, 
then /,Z,=/,a,d,, and Z, =a,d,. 
If ins of projecting the various strips on to the base line Y,Y, 
__ they be projected on to Y,Y,, which is parallel to XX, and passes through 
the highest point of the section, the construction will give another section 
_ modulus figure whose area is a,, and the distance corresponding to d, will 
_ bed,, and then Z,=a,d,. But since Z,=I/y,, and Z,=I/y,, it follows 
that Z,=Z,y,/y.. Hence if Z, is found from a modulus figure, Z, can 
readily be deduced from it without drawing another figure. 
_ The positions of the centres of gravity of the parts of the modulus 
figure on opposite sides of the neutral axis may be determined by one of 
_ the methods described in Chapter V. . 
4 A section modulus would only be determined in practice from a section 
modulus figure when the section was such that, not having definite, or 
sufficiently simple, mathematical properties, its moment of inertia could 
not be determined by the usual method. Even then it may be quicker 
to find moment of inertia by one of the methods described in 
rv. 
114. Beams of Uniform Strength. — Considering resistance to 
bending, a beam is said to be of uniform strength when the maximum 
_ stress is the same at every cross section. If / is the maximum stress, Z 
_ the modulus of the section, and M the bending moment, then for every 
section M=/Z, and for uniform strength M/Z is constant. 
In practice it is seldom possible to make M/Z constant for the whole 
length of a beam. For example, in a beam supported at the ends and 
loaded on the top, M vanishes at the ends where the shearing force is a 
maximum, and sufficient area of cross section must be provided to resist 
the shearing force. 
115. Strength of Wheel Teeth.—A tooth of a wheel is a cantilever 
on which the load acts at different points of the length during the time 
of its contact with another tooth on another wheel. The direction of the 
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