Page 140 - Introduction to Naval Architecture

126 STRENGTH

Integrating a second time gives the longitudinal bending moment.
That; is:

2 2
Put the other way, load per unit length = dS/dx = d M/dx .
For any given loading of the ship the draughts at which it floats can
be calculated. Knowing the weight distribution, and finding the
buoyancy distribution from the Bonjean curves, gives the net load per
unit length. Certain approximations are needed to deal with dis-
tributed loads such as shell plating. Also the point at which the net
force acts may not be in the centre of the length increment used and
typically the weight distribution at any point is assumed to have the
same slope as the curve of buoyancy plotted against length. However,
these approximations are not usually of great significance and certain
checks can be placed upon the results. First the shear force and
bending moment must be zero at the ends of the ship. If after
integration there is a residual force or moment this is usually corrected
arbitrarily by assuming the difference can be spread along the ship
length. From the relationships deduced above when the net load is zero
the shear force will have a maximum or minimum value and the
moment curve will show a point of inflexion. Where net load is a
maximum the shear force curve has a point of inflexion. Where shear
force is zero, the bending moment is a maximum or minimum. Besides
causing stresses in the structure the forces acting cause a deflection of
the ship longitudinally. By simple beam theory it can be shown that the
deflection y at any point is given by the equation:

When the ship is distorted so as to be concave up it is said to sag 2nd the
deck is in compression with the keel in tension. When the ship is
convex up it is said to hog. The deck is then in tension and the keel in
compression.
High still water forces and moments, besides being bad in their own
right, are likely to mean high values in waves as the values at sea are the
sum of the still water values and those due to a superimposed wave. The
still water values can be used to determine which are likely to be
stressful ship loading conditions.
The static forces of weight and buoyancy also act upon a transverse
section of the ship as shown in Figure 7.1.
The result is a transverse distortion of the structure which the
structure must be strong enough to resist. In addition these forces can