4IÔ
DOCK ENGINEERING.
Equating the two values of M, we obtain
dx* 2 1 2
whence, integrating,
EI cfæ= C‘+Miæ - -i 6'- • • <9^
1.0 find what value to attach to the constant (C^) in this expression, we
have the following considération :—Let ß be the slope of the beam at the
origin, B—or, in other words, the inclination of the tangent of the curve to
the horizontal. Then tan 0 = and, in the limit, tan ß= ß. When this
is the case x is so small as to become negligible, and so we can write
C = EI/3,
and, by substitution,
El= El3 +
dx 1 *2 6
Integrating again,
EI^EI^+M^-S^-^.
The constant is omitted in this case because y = 0 when «=0. Again,
since ?/=0 when x = b, we have—
O-Eiø+M^-S^-^. • <W
Now, at a point Q, equally indefinitely near to but on the opposite side
of B, we shall find the bending moment identical in value with that at the
point P. We can therefore write a similar equation in this case, noting
that » has a negative value. Thus—
o=eiø~m2|-s2^+^.
Subtract, and for both Mj and M2 write MB or the bending moment at
the point B, to which they both approximate so closely as to be practically
identical with it and each other. Accordingly,
M'‘ (~^)+ 'V^ ~ - (96)
Again, taking moments about A for the portion A B—
MA = M, + Sä a -
and, similarly, about C for the portion B C—
Mc = MB - S^ - “i2.