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Module
4
Analysis of Statically
Indeterminate
Structures by the Direct
Stiffness
Method
Version 2 CE IIT, Kharagpur
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23
The Direct Stiffness
Method: An
Introduction
Version 2 CE IIT, Kharagpur
Instructional Objectives:
After reading this chapter the student will be able to
1. Differentiate between the direct stiffness method and the displacement
method.
2. Formulate flexibility matrix of member.
3. Define stiffness matrix.
4. Construct stiffness matrix of a member.
5. Analyse simple structures by the direct stiffness matrix.
23.1 Introduction
All known methods of structural analysis are classified into two distinct groups:
(i)
force method of analysis and
(ii)
displacement method of analysis.
In module 2, the force method of analysis or the method of consistent
deformation is discussed. An introduction to the displacement method of analysis
is given in module 3, where in slopedeflection method and moment distribution
method are discussed. In this module the direct stiffness method is discussed. In
the displacement method of analysis the equilibrium equations are written by
expressing the unknown joint displacements in terms of loads by using load
displacement relations. The unknown joint displacements (the degrees of
freedom of the structure) are calculated by solving equilibrium equations. The
slopedeflection and momentdistribution methods were extensively used before
the high speed computing era. After the revolution in computer industry, only
direct stiffness method is used.
The displacement method follows essentially the same steps for both statically
determinate and indeterminate structures. In displacement /stiffness method of
analysis, once the structural model is defined, the unknowns (joint rotations and
translations) are automatically chosen unlike the force method of analysis.
Hence, displacement method of analysis is preferred to computer
implementation. The method follows a rather a set procedure. The direct stiffness
method is closely related to slopedeflection equations.
The general method of analyzing indeterminate structures by displacement
method may be traced to Navier (17851836). For example consider a four
member truss as shown in Fig.23.1.The given truss is statically indeterminate to
second degree as there are four bar forces but we have only two equations of
equilibrium. Denote each member by a number, for example (1), (2), (3) and (4).
Let
i
α
be the angle, the
i
th member makes with the horizontal. Under the
Version 2 CE IIT, Kharagpur
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and
, the joint
E
displaces to
E’
. Let
u
and
v
be its
vertical and horizontal displacements. Navier solved this problem as follows.
x
P
y
P
In the displacement method of analysis
u
and
v
are the only two unknowns for
this structure. The elongation of individual truss members can be expressed in
terms of these two unknown joint displacements. Next, calculate bar forces in the
members by using force–displacement relation. Now at
E
, two equilibrium
equations can be written viz.,
0
=
∑
x
F
and
0
=
∑
y
F
by summing all forces
in
x
and
y
directions. The unknown displacements may be calculated by solving
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This note was uploaded on 06/30/2009 for the course CE 358 taught by Professor Trifunac during the Fall '07 term at USC.
 Fall '07
 Trifunac

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