Page 301 - Introduction to Naval Architecture

286 VIBRATION, NOISE AND SHOCK

In this case the total deflection due to bending and shear
becomes:

Thus the effect of shear is to reduce the frequency. In this simple
example it was seen that r s depended upon the square of the ratio of
the depth of the beam to its length. For shallow beams its effects are
therefore small. Unfortunately a ship is not a shallow beam and the
ship's structure is akin to a box girder which influences greatly the
4 5
value of the shear correction factor. Based on the work of Taylor ' and
6
Johnson the value of r s can be taken as:

where D, B and L are the depth, beam and length of the ship and a =
B/D.
In this formula if D/L is 0.1 and a is 1.7 then r s = 0.256 and the factor
05
affecting frequency is 1/(1.256) = 0.892.
The frequency as calculated by simple bending theory would be
reduced by about 11 per cent for two-node vibration. It can be shown
that for higher modes the effect is much greater. In fact at higher
modes the shear deflection can become dominant.

Structural discontinuities
Full scale experiments have shown that where the structure is
continuous the distribution of stress over depth is reasonably linear apart
from the influence of shear. When there are abrupt changes in section,
in way of superstructures for instance, the picture becomes much more
complex and there is no easy way to determine how the stress varies.
The complication with superstructures arises essentially because
plane sections no longer remain plane and the stress level the
superstructure can take is reduced. It can take no stress at the ends but
towards the centre of a long superstructure it may become reasonably
effective against bending. The effect was touched upon in considering
the efficiency of superstructures in Chapter 7 on strength. A large finite
element analysis would be required to study the problem fully.