Page 148 - Introduction to Naval Architecture

134 STRENGTH

RESPONSE OF THE STRUCTURE

Having determined the shear forces and bending moments it is
necessary to find the stresses in the structure and the overall deflection.
For a beam in which the bending moment at some point x from one
end is M, the stress/at z from the neutral axis of the section is given
by:

where / is the second moment of area about the neutral axis of the
section at x and Z = 7/z is called the section modulus.
The maximum stresses will occur when z is a maximum, that is at the
top and bottom of the section. They will be equal if the neutral axis is
at mid-depth of the section.
This relationship was derived for beams subject to pure bending and
in which plane sections remained plane. Although a ship's structure is
much more complex, applying the simple formula has been found to
give reasonable results.

Calculation of section modulus
The first section modulus to be considered is that for amidships as it
is in that area that the maximum bending moments are likely. Two
cases have to be considered. The first is when all the material is the
same. The second is when different materials are present in the
section.
To contribute effectively to the section modulus, material in the
cross-section must be continuous for a reasonable length fore and aft.
Typically the members concerned are the side and bottom plating,
deck plating, longitudinal deck and shell stiffeners and any longitudi-
nal bulkheads. The structure must not be such that it is likely to
buckle under load and fail to take its fair share of the load. Because
wide panels of thin plating are liable to shirk their load it is usual to
limit the plating contribution to 70 times its thickness if the stiffener
spacing is greater than this. Having decided which structural ele-
ments will contribute, the section modulus is calculated in a tabular
form.
An assumed neutral axis (ANA) is chosen at a convenient height above
the keel. The area of each element above and below the ANA, the first
and second moments about the ANA and the second moments about
each element's own centroid are calculated. The differences of the first
moments divided by the total area gives the distance of the true NA