# Matrix Analysis of Framed Structures

## Matrix Analysis of Framed Structures

The advent of the digital computer made it necessary to reorganize the
theory of structures into matrix form, and the first edition of this book was
written for that purpose. It covered the analysis of all types of framed
structures by the flexibility and stiffness methods, with emphasis on the
latter approach. At that time, it was evident that the stiffness method was
superior for digital computation, but for completeness both methods were
extensively discussed. Now the flexibility method should play a less important
role and be characterized as a supplementary approach instead of
a complementary method. The flexibility method cannot be discarded
altogether, however, because it is often necessary to obtain stiffnesses
through flexibility techniques.

This book .was written as a text for college students on the subject of the
analysis of framed structures by matrix methods. The preparation needed
to study the subject is normally gained from the first portion of an undergraduate
engineering program; specifically, the reader should be familiar
with statics and mechanics of materials, as well as algebra and introductory
calculus. A prior course in elementary structural analysis would
naturally be beneficial, although it is not a prerequisite for the subject
matter of the book. Elementary matrix algebra is used throughout the
book, and the reader must be familiar with this subject. Since the topics
needed from matrix algebra are of an elementary nature, the reader can
acquire the necessary knowledge through self-study during a period of
two or three weeks. A separate mathematics course in matrix algebra is
not necessary, although some students will wish to take such a course in
preparation for more advanced work. To assist those who need only an
introduction to matrix algebra, without benefit of a formal course, the

authors have written a supplementary book on the subject.

There are several reasons why matrix analysis of structures is vital to
the structural analyst. One of the most important is that it makes possible
a comprehensive approach to the subject that is valid for structures of all
types. A second reason is that it provides an efficient means of describing
the various steps in the analysis, so that these steps can be more easily
programmed for a digital computer. The use of matrices is natural when
performing calculations with a computer, because they permit large
groups of numbers to be manipulated in a simple and effective manner.
The reader will find that the methods of analysis developed in this book
are highly organized and that the same basic procedures can be followed

in the analysis of all types of framed structures.

This book describes matrix methods for the analysis
of framed structures with the aid of a digital computer. Both the flexibility
and stiffness methods of structural analysis are covered, but emphasis
is placed upon the latter because it is more suitable for computer programming.
While these methods are applicable to discretized structures of all
types, only framed structures will be discussed. After mastering the analysis
of framed structures, the reader will be prepared to study the finite element
method for analyzing discretized continua

All of the structures that are analyzed
in later chapters are calledfi-nmed structures and can be divided into
six categories: beams, plane trusses, space trusses, plane frames, grids,
and space frames. These types of structures are illustrated in Fig. 1-1 and
described later in detail. These categories are selected because each represents
a class of structures having special characteristics. Furthermore,
while the basic principles of the flexibility and stiffness methods are the
same for all types of structures, the analyses for these six categories are
sufficiently different in the details to warrant separate discussionsof them.
Every framed structure consists of members that are long in comparison
to their cross-sectional dimensions. The joints of a framed structure are
points of intersection of the members, as well as points of support and free

ends of members. Examples of joints are points A, B, C, and D in Figs.

When a structure is acted
upon by loads, the members of the structure will undergo deformations (or
small changes in shape) and, as a consequence, points within the structure
will be displaced to new positions. In general, all points of the structure
except immovable points of support will undergo such displacements. The
calculatim.dth~sed..i splacements is.me ssent&l.part of-s tructural anal y-sis ,
as will be seen later in the discussions of the flexibility and stiffness methods.
However, before considering the displacements, it is fist necessarydo

have an understanding of the dehmat ions that purduteh e di.sp1aceme.nts.