17-11-2012, 05:12 PM
Plastic Analysis and Design of Steel Structures
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Structural Analysis— Stiffness Method
Introduction
Computer programs for plastic analysis of framed structures have
been in existence for some time. Some programs, such as those developed
earlier by, among others, Wang,1 Jennings and Majid,2 and
Davies,3 and later by Chen and Sohal,4 perform plastic analysis for
frames of considerable size. However, most of these computer programs
were written as specialist programs specifically for linear or
nonlinear plastic analysis. Unlike linear elastic analysis computer
programs, which are commonly available commercially, computer
programs for plastic analysis are not as accessible. Indeed, very few,
if any, are being used for daily routine design in engineering offices.
This may be because of the perception by many engineers that the
plastic design method is used only for certain types of usually simple
structures, such as beams and portal frames. This perception discourages
commercial software developers from developing computer
programs for plastic analysis because of their limited applications.
Contrary to the traditional thinking that plastic analysis is performed
either by simple manual methods for simple structures or by
sophisticated computer programs written for more general applications,
this book intends to introduce general plastic analysis methods,
which take advantage of the availability of modern computational
tools, such as linear elastic analysis programs and spreadsheet applications.
These computational tools are in routine use in most engineering
design offices nowadays. The powerful number-crunching
capability of these tools enables plastic analysis and design to be performed
for structures of virtually any size.
Degrees of Freedom and Indeterminacy
Plastic analysis is used to obtain the behavior of a structure at collapse.
As the structure approaches its collapse state when the loads are increasing,
the structure becomes increasingly flexible in its stiffness. Its
flexibility at any stage of loading is related to the degree of statical indeterminacy,
which keeps decreasing as plastic hinges occur with the
increasing loads. This section aims to describe a method to distinguish
between determinate and indeterminate structures by examining the
degrees of freedom of structural frames. The number of degrees of freedom
of a structure denotes the independent movements of the structural
members at the joints, including the supports. Hence, it is an indication
of the size of the structural problem. The degrees of freedom of a structure
are counted in relation to a reference coordinate system.
External loads are applied to a structure causing movements at
various locations. For frames, these locations are usually defined
at the joints for calculation purposes. Thus, the maximum number
of independent displacements, including both rotational and translational
movements at the joints, is equal to the number of degrees of
freedom of the structure. To identify the number of degrees of freedom
of a structure, each independent displacement is assigned a number,
called the freedom code, in ascending order in the global coordinate
system of the structure.
Treatment of Internal Loads
So far, the discussion has concerned externally applied loads acting
only at joints of the structure. However, in many instances, externally
applied loads are also applied at locations other than the joints, such
as on part or whole of a member. Loads being applied in this manner
are termed internal loads. Internal loads may include distributed
loads, point loads, and loads due to temperature effects. In such cases,
the loads are calculated by treating the member as fixed-end, and
fixed-end forces, including axial forces, shear forces, and bending
moments, are calculated at its ends. The fictitiously fixed ends of
the member are then removed and the effects of the fixed-end forces,
now being treated as applied loads at the joints, are assessed using
the stiffness method of analysis.