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STRENGTH OF MATERIALS. 245
Beams fastened at one end and loaded at the other may be
reduced in size toward the loaded end and still have the same
strength. Suppose the beam to be fastened in the wall at X
(Fig. 23) and loaded at the other end with a given load, this load
will then have the greatest breaking effect upon
the beam at X; at half way between X and d fig. 23.
the load has only half the breaking effect, at c
only one-quarter the effect. Therefore, the beam
may be tapered off toward b in such proportion
that the square of the height a is equal to
three-quarters the square of the height at X.
The square of the thickness at b is one-half the
square of the thickness at X, and the square of the thickness at
c is one-quarter the square of the height at X.
Example.
An iron bracket is four feet long, projecting from a wall
(as Fig. 23). It is strong enough when 24 inches high at X.
How high will it have to be at a, b and c?
Solution
:
X = 242 = 576 Height at X = 24"
a = V % X 576 = V432 Height at a — 20.78"
b — V y
2 X 576 == V288 Height at b = 16.97"
c — V X X 576 = \/l44 Height at c — 12" •
The curved boundary line of such a beam is a parabolic
curve, because the property of a parabola is that the square of
the length of any one of the vertical lines (ordinates) is in pro-
portion as their distance from the extreme point d. By this
construction one-third of the material may be
saved and the same strength be maintained.
If the load is distributed along the
whole length of the bracket instead of at its
extreme end, it should "have the form shown in
Fig. 24.
Square and Rectangular Wooden Beams.
The strength increases directly as the width and as the
square of the thickness. The strength decreases in the same
proportion as the length of the span increases.
Example 1.
Find the ultimate breaking load in pounds of a spruce
beam 6 inches square and 8 feet long, when supported under
both ends and loaded at the center.
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