4I8
DOCK ENGINEERING.
If A B (fig. 401) be a portion of a weightless beam between any two
supports, P Q, with bending moments, y} and y.2, at A and B respectively,
due to some external system of loading, it is
clear that the line of moments between A and
B will be right, and by a simple application of
geometrical principles
3/(^1 + «2) = yi<«2+ y^v - ■ (99)
If, however, the beam be not weightless,
or be loaded with a weight of w Ibs. per foot
run, the curve of moments is parabolic and the equation becomes______
y (æi + «2) = yi*2 + 3/2 ah + 2 xiX^ (æi + *2)- • • (100)
The foregoing relationship is, of course, conditional upon there being no
point of support between A and B. When such is not the case, and there
is an upward reaction, B, at the point, X, we must expand the expression
still further into
y^ + x2) = yix2 + y.^ + -x^^ + x2) - R^^. (101)
Re-arrange, and divide throughout by aq æ2,
R = 2/1 + ^
æl æ2
^ 9 (^l ^ ^2)’
(102)
an equation which is identical with the value of RB given above, when the
notation has been adapted thereto.
The second equation (100) in the preceding group yields us an expres-
sion for the current bending moment at any point, X, intermediate between
the points of support.
Mxb = MBa; + MC(6 - æ) + ^-^(6 _ xy
(103)
If we revert to the case in which there are no moments at the end
supports, we may derive the amounts of reaction at these points very
readily, as follows :—
From equation (97) we have
2MB(a + 6) = - ™(a3 + b3),
or.
- Mb = g (a2 - ab + 62). . . . (104)
Also, fiom a considération of the conditions of equilibrium to the left of B,
w Cl
iyib - KÅa - —.