CHAPTER 1
INTRODUCTION
A column is an upright compression member whose cross-sectional
dimensions are small relative to its overall length. In practice, structural concrete
columns are always subjected to bending moment as wells as axial compression.
Shearing action is produced in columns by horizontal loading.
Furthermore, columns are structural compression members that transmit loads
from the upper floors to the lower levels and then to the soil through the foundations.
Since columns are compression elements, failure of one column in a critical location
can cause the progressive collapse of adjoining floors, and in turn, even the collapse
of the entire structure.
A column is defined in accordance to BS8110 as short when both the ratios
lex/h and ley/b are less than 15 for braced and 10 for unbraced where h is the
dimension of the column section in the plane of bending about the major axis with
the effective height of lex. The dimensions b and ley refer to the minor axis. Short
columns are dominated by the strength limit of the material that is depicted on the
stress/slenderness graph below.
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Figure 1.1: Stress/Slenderness Graph
(www.efunda.com/formulae/ solid_mechanics/columns/index.cfm)
As mentioned earlier, compression members such as columns are mainly
subjected to axial forces. The principal stress in a compression member is therefore
the normal stress,
The failure of a short compression member resulting from the compression
axial force looks like is as explained in Figure 1.2. In summary, the failure of a
compression member has to do with the strength and stiffness of the material and the
geometry (slenderness ratio) of the member.
Figure 1.2: Failure of a Short Compression Member Resulting from the
Compression Axial Force
(www.efunda.com/formulae/ solid_mechanics/columns/index.cfm)
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During the last several decades, the FEM has evolved from a linear structural
analysis procedure to a general technique for solving non-linear, transient, partial
differential equations. An extensive literature on the method exists which describes
the theory necessary to formulate solutions for general classes of problems, as well
as, practical guidelines in its application to problem solution.
R. Courant, who utilized the Ritz method of numerical analysis and
minimization of variational calculus to obtain approximate solutions to vibration
systems, first developed FEA in 1943. Shortly thereafter, a paper published in 1956
by M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp established a broader
definition of numerical analysis. The paper centered on the "stiffness and deflection
of complex structures".
Many problems in engineering are modeled using partial differential
equations (PDE). The set of partial differential equations describing such problems
is often referred to as the strong form of the problem. The differential equations may
be either linear or non-linear. Linear equations are characterized by the appearance
of the dependent variable(s) in linear form only, whereas, non-linear equations
include non-linear terms also.
The FEM Program FEAP may be used to solve a wide variety of problems in
linear and non-linear solid continuum mechanics. The FEM also known as the
Personal Version (FEAPpv) is a computer analysis system designed for: -
i)
Use in an instructional program to illustrate performance of
different types of elements and modeling methods
ii)
In a research, and / or application environment which requires
frequent modifications to address new problem areas or analysis
requirements.”
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The FEA consists of a computer model of a material or design that is stressed
and analyzed for specific results. It is used in new product design, and existing
product refinement. A company is able to verify a proposed design thus enabling
them to perform to the client's specifications prior to manufacturing or construction.
Therefore, Professor Robert L. Taylor from the Department of Civil and
Environmental Engineering, University of California at Berkeley developed the FEA
Program, FEAP. Professor Robert L. Taylor has developed the FEAPpv program as
an open code designed for research and educational use.
For that reason, this thesis is primarily centered on the stress distribution for
short column using the FEA Program FEAP (Theory of Isotropic Linear Elastic
Models), which would also be compared to the classical method of stress
distribution.
The critical paths on how the stress is distributed in a short column using both
the methods would be analysed, compared, tabulated and computed. Furthermore,
the influences of variation in the Young’s Modulus and foundation thickness would
also be studied.
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1.1
Objective of Study
The objective of this study is to analyse the use of the FEA Program FEAPpv
to obtain data on the stress distribution for a short column. This study would also
review the differences between the classical method of stress distribution in a short
column with this FEAPpv program and also the effects and influences of Young’s
Modulus value variation and foundation thickness variation on the column and
foundation by using the FEAPpv program.
1.2
Scope of Study
The main focus of this study is to obtain data on the stress distribution for a
short column using the Finite Element Analysis Program FEAPpv where the column
would be modeled using partial differential equations. Besides this, the study would
also contain: -
i)
the difference between the classical method and finite element
program (FEAPpv)
ii)
the effects and influences of parameters and material variation of the
foundation
iii)
graphical and numerical output of solution results
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1.3
Research Problem
The FEA has become a solution to the task of predicting failure due to
unknown stresses by showing problem areas in a material and allowing designers to
see all of the theoretical stresses within. This method of product design and testing is
far superior to the manufacturing costs, which would increase if each sample was
actually built and tested.
According to BS8110, Clause 3.8.4.6, which explicates shear stress in
columns, has been introduced for the reason being of that there was not a method for
checking shear stress in columns in CP110. Clause 3.8.4.6 in BS8110 explains that
the design shear strength of columns may be checked in accordance with clause
3.4.5.12 (shear and axial compression).
Clause 3.8.4.6 states that for rectangular sections; no check is required where
M/N (moment/axial load) is less than 0.75h (depth) provided that the shear stress
does not exceed 0.8(fcu) 1/2 or 5 N/mm2, whichever is lesser. (British Standard
Institution (1985). “BS8110: Part 1 Structural Use of Concrete.” London: British
Standard Institution)
In majority cases, columns do not have shear problems, but in the case of
high moment and small axial load there could be. In this case, the design shear
strength is checked in accordance to Clause 3.4.5.12 (shear and axial compression,
beam section). (British Standard Institution (1985). “BS8110: Part 1 Structural Use
of Concrete.” London: British Standard Institution)
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Hence, a question of uniformity of stress distribution assumption in a short
column arose leading a further review or clarification by comparing the usage of the
finite element analysis, FEAPpv in acquiring the stress distribution for a short
column to the classical method.
Besides this, this thesis was also ventured into as to dwell into the aspect of
understanding the definition and differences of linear material models and
understanding the concepts and applications of stress concentration factor (near the
support) and stress intensity factor and effect of property materials in analysis and
design.
In summary, the research problems are concentrated on these factors that
were explained above: 
uniformity of stress distribution assumption in a short column

effect of isotropic material properties on stress distribution

stress concentration near support

effect and influences of column foundation thickness increment and Young’s
Modulus value variation
Analysing the finite element program is the main focus in this study. The
information regarding this finite element program was obtained from
www.ce.berkeley.edu/_rlt/feappv/ which is the main website for this program
developed by Professor Robert L. Taylor.
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Therefore, to solve the research problem, several major procedures taken to
develop the required program of stress distribution in a short column are: 
analysing the input data, hence run the FEAPpv program

obtain output data on stress distribution for short column and analyse the
graphical and numerical output of solution results

differentiate the classical method with the FEAPpv program on stress
distribution for short column

analyse the effect of Young’s Modulus value variation and foundation
thickness variation by comparing the graphical output data with the basic
model (of E of 2.5 x 107 and thickness of 0.025m)