42° DOCK ENGINEERING.
Take any point, P, between B and W, at a distance, x, from B, the
origin of co-ordinates.
Then, as already established, for the equilibrium of the portion B P,
= Mb - S^.
aa:2 1
Integrate
EiÉ-c + M-«-s4
and as ,/æ = t“ @ when x is indefinitely small, so in the limit,
C = EI/3,
and
Elg=EI^+MB£C-Sl^. . . (109)
Integrating again,
2 3
Ely = El/Ja: + Mb^ - Sæ . (110)
There is no constant, since x and y vanish together.
These equations hold good for all values of x between x= 0 and x = dv
Next, let the point, P, lie between Wj and C, and remove the origin of
co-ordinates to C. Then
K1<fæ2 -
Integrating and determining the constant as before,
EIE-EI“-S4 - . ••(Hl)
and again,
Ely= Elaæ - So *3...............(112)
Now, these two pairs of equations, though possessing different co-
ordinates, have two conditions in common, viz.:_
(1) At the point, Wp the value for y must be the same in each case.
(2) At the same point the slope or gradient is the same, but measured in
opposite directions —-i.e.,
+ (^) - 0.
Hence, substituting dx for x in equation (110) and (b - dj for x in equation
(112), we deduce from the first condition,
EI/^+Mb^ - Sj^- = Ela(6 - ^ - S0 ^-1^13. . (113)
Also, substituting likewise in equations (109) and (111), and using the
second relation,
EI^M^-S^fEIa-So .(L-^ . (114)