Page 179 - Introduction to Naval Architecture

STRENGTH 165

comparison with similar successful designs, rather than to obtain
absolute values of stress or deflection.
It is not appropriate in this book to deal with the analysis of
frameworks in detail. There are many textbooks available to which
reference should be made for detailed explanations and for an
understanding of all the underlying assumptions. Very briefly, however,
the methods of analysis available are:

{1) Energy methods. These are based on the theorem of Castigliano
which postulates that the partial derivative of the total strain
energy with respect to each applied load is equal to the
displacement of the structure at the point of application in the
direction of the load.
(2) Moment distribution methods. This is an iterative process. All
members of the framework are initially considered fixed rigidly
and the bending moments at the joints calculated. Then one
joint is relaxed by allowing it to rotate. The bending moment
acting is distributed between the members forming the joint
according to their inertias and lengths. Half the distributed
moment is transmitted to a member's far end which is still held
rigid. Joints are relaxed in turn and the process repeated until
the moments are in balance.
(3) Slope-deflection methods. If M is the bending moment at some
point along a beam the area under the curve of M/EI between
two points on the beam gives the change in slope between those
points. Further, if the moment of the curve between the points
is taken about the first point, the moment gives the perpendicu-
lar distance of the first point from the tangent at the second
point. By expressing the changes in deflection at the ends of
portal members in terms of the applied loads and the
(unknown) moments at their ends, a series of equations are
produced which can be solved to give the unknown moments.

FINITE ELEMENT ANALYSIS

Mention has been made several times of finite element analysis
techniques which are the basis of modern computer based analysis
methods in structures and hydrodynamics. These are very powerful
techniques using the mathematics of matrix algebra. In this book it is
only possible to give the reader a simplified explanation of the
principles involved in the method. The structure is imagined to be split
up into a series of elements, usually rectangular or triangular. The
corners where the elements meet are called nodes. For each element an