chapter 3 - Bentham Science

Use of Spreadsheets for Analyses Applications of Spreadsheets in Education The Amazing Power of a Simple Tool 19
The use of Excel in educating engineers with elementary structural mechanics is first illustrated.
The conventional approach to the solution of problems in engineering statics (i.e., calculation of
shear forces and bending moments) would always require solving simultaneous equations for determining
the values of the support reactions. It is demonstrated herein that with the use of Excel
the solution could be obtained very differently by an intuitive approach. For example, the value of
a vertical support reaction of a simply supported beam can be determined by trial-and-error until
equilibrium is satisfied (i.e., end-moments at free ends approaching zero values). Embodied in this
trial-and-error process is the educational objective of having the student visualize the conditions
of equilibrium through graphical display of the forces and moments surrounding the beam while
different reaction values are being keyed in. Essentially, the focus of attention is on the physical
conditions of the beam as displayed graphically, which is in contrast to the conventional treatment
of the problem by linear algebra. This alternative approach to engineering statics is particularly
useful to students who are introduced to the concepts for the first time. The teaching of the conventional
solution method could be introduced subsequently when the underlying concepts have been
well understood.
The conventional method of finding deflections of a beam would require finding the values
of constants in polynomial expressions based on pre-defined boundary conditions. When Excel
is employed, the solution can be obtained intuitively in two steps: (i) deforming the beam with a
constant or variable curvature along its length and (ii) rotating the beam (as a rigid body) in the
vertical plane about a support until the beam is leveled with all the supports. The trial-and-error
procedure of deforming and rotating the beam about one of its supports serves to engage the students
into finding the deflection profile by intuition while alleviating the need to become heavily involved
with algebra. The educational attribute can be further enhanced by the use of interactive graphics
display in Excel. A similar intuitive approach could be employed for determining the value of the
reaction at a redundant (say interior) support through visualizing the beam gradually leveling with
the support point as the reaction value is gradually varied. A full description of the programming is
presented in Section 2.2.
Section 2.3 explores the use of standard matrix operations (involving addition, multiplication,
and inversion of matrices and vectors) for analyzing lateral resisting elements including building
frames, structural walls, or wall-frame elements. The flexibility matrix of a lateral resisting element
can be constructed readily once the deflection of the element at every floor level in the building
has been identified. The flexibility matrix can be inverted in Excel to give the stiffness matrix.
Importantly, stiffness matrices representing contributions by individual lateral resisting elements
can be summed to represent joint actions that are facilitated by diaphragm actions of the building
floors.
Section 2.4 illustrates the use of Excel for undertaking the more advanced (dynamic) analysis
of single-story and multi-story buildings. The simulation of the response displacement time-history
of a single-degree-of-freedom (SDOF) system can be accomplished by what is known as the central
difference method of step-by-step time integration as is illustrated in Section 2.4.1. The forward
marching algorithm featured in this solution method can be implemented easily by a column array
in Excel. Such a column array can be constructed from top to bottom using the fill down command.
Interestingly, the use of a column array for finding the solution for a SDOF system can be expanded
into a rectangular array for solving multiple systems, the results of which can be used for plotting
the response spectra of the applied excitations. The rectangular array can be constructed from