Chapter 4

CHAPTER 4
4. Introduction to Matrix Method of Structural Analysis

4.1 Introduction
Although a study of classical methods is essential for developing an understanding of structural
behavior and the principles of structural analysis, the analysis of large structures by using these
hand calculation methods can be quite time consuming. With the availability of inexpensive, yet
powerful, microcomputers, the analysis of structures in most design offices is routinely
performed today on computers using software based on matrix methods of structural analysis.
In contrast to classical methods, matrix methods were specifically developed for computer
implementation; they are systematic (so that they can be conveniently programmed), and general
(in the sense that the same overall format of the analytical procedure can be applied to the
various types of framed structures). It will become clear as we study matrix methods that,
because of the latter characteristic, a computer program developed to analyze one type of
structure (e.g., plane trusses) can be modified with relative ease to analyze another type of
structure (e.g., space trusses or frames). As the analysis of large and highly redundant structures
by classical methods can be quite time consuming, matrix methods are commonly used. A set of
simultaneous equations that describe the force–displacement characteristics of the structure are
developed and can be programmed easily on a computer. The force-displacement relations are
based on exact solutions of the underlying differential equation. However, classical methods are
still preferred by many engineers for analyzing smaller structures, because they provide a better
insight into the behavior of structures. Classical methods may also be used for preliminary
designs, for checking the results of computerized analyses, and for deriving the member force–
displacement relations needed in the matrix analysis. Furthermore, a study of classical methods
is considered to be essential for developing an understanding of structural behavior.
Matrix methods do not involve any new fundamental principles; but the fundamental
relationships of equilibrium, compatibility, and member force-displacement relations are now
expressed in the form of matrix equations, so that the numerical computations can be efficiently
performed on a computer. Therefore, familiarity with the basic operations of matrix algebra is a
prerequisite to understanding matrix structural analysis.
Although both the flexibility (force) and the stiffness (displacement) methods can be expressed in
matrix form, the stiffness method is more systematic and can be more easily implemented on
computers. Thus, most of the commercially available computer programs for structural analysis
are based on the stiffness method. This method can be used to analyze statically determinate as
well as indeterminate structures.
In the matrix stiffness method of analysis, the structure is considered to be an assemblage of
straight members connected at their ends to joints. A member is defined as parts of the structure
for which the member force displacement relations to be used in the analysis are valid. In other
words, given the displacements of the ends of a member, one should be able to determine the
forces and moments at its ends by using the force displacement relations. A joint is defined as a
structural part of infinitesimal size to which the member ends are connected. The members and
joints of structures are also referred to as elements and nodes, respectively.
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Degrees of Freedom
The degrees of freedom of a structure are the independent joint displacements (translations and
rotations) that are necessary to specify the deformed shape of the structure when subjected to an
arbitrary loading.
Matrix structural analysis methods can also be grouped into stiffness (displacement) or flexibility
(force) methods.
4.2 Stiffness Methods
In the stiffness method, the primary unknowns are displacements (translation/rotation). The
unknown displacements are solved first by equilibrium equations. Then the unknown
forces/moments are evaluated using compatibility equations and member force-displacement
relations. Most commercially available structural analysis software is based on the stiffness
method. This is because the method is more systematic and requires the same procedure for both
statically determinate and indeterminate structures.
When using the stiffness method to solve continuous beam problems, the nodes of each element
can be located:
•At the supports
•At points where the members are connected together
•At points where external loads/moments are applied
•At points where cross-sectional properties change suddenly
Each node has two degrees of freedom: a vertical displacement and a rotation
•Axial deformations are usually neglected
•Transformation matrices are not needed because global and local coordinates are parallel to
each other
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The process for using the stiffness method to analyze beams is:
 Establish the stiffness matrix for each element
 Assemble the element matrices to form the structure stiffness matrix
 Use the force-displacement relations to solve unknown vertical displacements and
rotations
 Finalize external reactions and internal shear/moment
Local and global stiffness matrix for a beam

Derivation of the stiffness matrix for a slender beam element:
q is the member end force/end moment matrix

k=k′ is the member stiffness matrix of the beam
d is the member end displacement/rotation matrix
Assembling the structure stiffness matrix for the entire beam (K) is
 dependent on identifying the location of each element in the member stiffness matrix (k)
 determined by adding the respective member stiffness matrices in the proper entries
Global force-displacement relationship

 The global force matrix (Q) and global displacement matrix (D) are related by the
structure stiffness matrix (K)
Q=KD
 The Q matrix contains known applied external loads/moments and unknown reactions
forces/moments at each DOF
 The D matrix contains unknown displacements or rotations for unconstrained DOFs, and
zero displacements or rotations for constrained DOFs
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Partition the Q = KD matrix into two parts
For now, we will focus on beams with no support displacement (also known as unyielding
support). For this case, Dknown is a zero vector.
First solve D unknown
Second solve Qunknown
4.3 Flexibility Methods

In the flexibility method, the primary unknowns are forces/moments. The unknown
forces/moments are solved first by compatibility equations. Then the unknown displacements are
evaluated using equilibrium equations and member force-displacement relations.
Compatibility equation (e.g. deflection at B is zero).
fBB is the deflection per unit force (flexibility coefficient)

Choice of redundant is arbitrary.
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The total number of compatibility equations is equal to the number of redundant or DOTI.
Summary of Steps:
 Determine the degree of total indeterminacy
 Select redundant reaction force(s)/moment(s) and the primary structure
 Calculate deflection(s)/rotation(s) of the primary structure at the same direction as
selected redundant force(s)/moment(s) (e.g. using deflection tables)
 Calculate the flexibility coefficient(s)
 Apply compatibility equation(s) to determine the redundant force(s)/moment(s)
 Apply equilibrium equations to the real structure to determine the rest of the reaction
forces/moments
 If required, draw the AFD, SFD and BMD
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CHAPTER 4

4. Introduction to Matrix Method of Structural Analysis

Local and global stiffness matrix for a beam

q is the member end force/end moment matrix

Global force-displacement relationship

Second solve Qunknown

4.3 Flexibility Methods

Compatibility equation (e.g. deflection at B is zero).

fBB is the deflection per unit force (flexibility coefficient)

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