Chapter 11
Matrix Stiffness Method
Matrix stiffness method (MSM) is a modern powerful method of analysis of
engineering structures. Its effective andwidespread application is associated with
availability of modern computers and effective computer programs. The MSM
allows performing detail analysis of any sophisticated 2D and 3D engineering
structure and takes into account different features of a structure and loading.
The method demands a set of new concepts. They are finite element, global and
local coordinate systems, possible displacements of the ends, ancillary diagrams,
initial matrices, stiffness matrix of separate element and structure in whole, etc.
This method uses the idea of the displacement method and contains its further
development: arbitrary structure should be presented as a set of finite elements and
three aspects of any problem – a static, geometrical, and physical – should be pre-
sented in matrix form. The MSM does notdemand of constructing bending moment
diagram caused by unit primary unknowns in the primary system. Instead it is nec-
essary to prepare few initial matrices according to strong algorithms and perform
matrix procedures by computer using the standard programs.
This chapter contains the detailed discussion of MSM, all ancillary diagrams and
initial matrices are constructed by hand. Mainly such presentation of material allows
reader to see the internal logic and the features of the method, to understand physical
meaning of each step, and to find the corresponding result in analysis of a structure
by displacement method in canonical form.
At the present time, MSM is developed with great detail. The reader can find out
in literature the different presentations of MSM. In our book, we will consider this
method in simplest form and apply it for analysis of the planar bar structures only.
Among them are the statically indeterminate beams, frames, trusses, and combined
structures subjected to different external exposures (loads, settlements of supports,
change of temperature, influence lines).
11.1 Basic Idea and Concepts...............................................
The fundamental concepts of MSM are the following: the finite elements, the pos-
sible displacements of the ends and degree of kinematical indeterminacy (degree of
freedom), the global and local coordinate systems.
I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-911,cSpringer Science+Business Media, LLC 2010
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