An Interactive Learning Environment for Engineering
Physics
Project Report submitted in support of the degree of
Master of Engineering
Katie Breakspear
April 5,2007
Abstract
When an engineer is selecting a beam for any given purpose,it is imperative that they
can calculate certain properties of the beam,such as its shear force,bending moment
and deﬂection proﬁles.A useful tool would therefore be a computer program that would
output diagrams of these three values for the entirety of the beams length.This would
allow an engineer to spend time on more important calculations than the complex algebra
involved in creating these diagrams.
Educationally,it is important for an inexperienced engineer to build up intuition about
how these diagrams will look for a beam with any given loading or supporting.Although
many commercial structural analysis programs exist to calculate shear force,bending mo
ment and beam deﬂection,these programs are not build in a way to optimise learning for
student engineers.Therefore,a computer program that calculated these three diagrams
and was build with the objective of helping inexperienced engineers would be useful in
order for these student engineers to familiarise themselves with diﬀerent beam situations.
This report details the making of such a computer program.
Contents
1 Introduction 4
1.1 Background..................................
4
1.2 Project plan..................................
5
1.3 Project objectives...............................
5
1.4 Problems encountered............................
6
1.5 Structure of the report............................
8
2 Analysis and Design 9
2.1 Deﬁning project requirements........................
9
2.2 Deﬁning program requirements.......................
10
2.3 Designing the program............................
11
2.4 Choice of programming language......................
11
2.5 Visual design of the program.........................
13
2.6 Designing the actual code and creating an implementation plan.....
14
3 Mathematical Reasoning 16
3.1 Deﬁning shear,bending moment and beam deﬂection...........
17
3.2 Details to consider..............................
18
1
3.3 Calculating shear,bending moment and beam deﬂection.........
19
4 Implementation 26
4.1 A simple scenario...............................
26
4.2 Building the program’s complexity.....................
31
4.3 Creating a usable program..........................
31
4.4 The help ﬁle..................................
34
5 Testing 35
5.1 Testing for errors...............................
35
5.2 Random testing................................
36
5.3 Testing by sample users...........................
37
6 Conclusions 38
6.1 Suggestions for future development.....................
38
6.2 Conclusion...................................
39
Bibliography and References 40
Appendices 42
A Help File 42
B Questionnaire on Beam Program 43
2
List of Figures
1.1 Original and redesigned versions of the program..............
7
2.1 The design of the point load form......................
13
2.2 Design of the Main Menu...........................
13
3.1 Types of statically determinate beams...................
19
3.2 Example of a freebody diagram.......................
20
3.3 Example of a shear force illustration....................
21
3.4 An example of a bending moment illustration...............
23
3.5 Another example of a bending moment illustration............
24
4.1 A simple scenario...............................
27
4.2 The completed library form.........................
27
4.3 Examples of forms needing user inputs...................
28
4.4 Code deﬁning a data structure........................
29
4.5 Example of the ‘onscreen’ freebody diagram...............
30
4.6 An example of an error message.......................
32
4.7 Figure to show a ﬁxed support form and its editable version.......
33
3
Chapter 1
Introduction
A beam is a structural element which carries a transverse but not an axial load.It can
be diﬃcult to tell without calculation what the graphs of shear force,bending moment
and beam deﬂection for a given beam might look like.In engineering,the maximum and
minimum magnitudes of both the bending moment and shear force are used as primary
considerations when selecting a beam.
This report will detail the creation of a computer programthat will calculate these values
pictorially.
1.1 Background
Within the course Engineering Physics EMAT1005,taught at the University of Bristol,
shear force,bending moment and deﬂection of beams are taught to ﬁrst year Engineering
Mathematics students.This type of teaching is reﬂected country wide in similar degree
courses.At present,students are expected to understand these concepts quickly and
develop intuition about how diﬀerent beams deform after a relatively small amount of
teaching and practice.Often,students will not complete the suﬃcient number of examples
necessary to gain a full understanding of these concepts.Additionally miscalculation can
result in false intuition being developed.This can be due to the detailed algebra necessary
for each problem,and therefore the time it takes to solve every situation.Consequently a
computational tool has been suggested to help students studying this degree course.This
4
tool is intended to allow students to spend time improving their understanding rather
than completing detailed,time consuming algebra.
Whilst programs already exist on the internet to perform these types of calculation,
they have not been tailored to suit this degree course.Of the programs found,many
concentrate only on the results and when requesting inputs from the user can become
confusing.In a teaching situation where young,engineers are assimilating many new
ideas it is important to provide a program that is as clear as possible since these concepts
are being learnt for the ﬁrst time and confusion at this point could be very detrimental.
The aim of this project is therefore to produce a comprehensive userfriendly program
which will compute the diagrams of the bending moment,shear force and beamdeﬂection
for a beam with any given loading and supporting.
1.2 Project plan
This project was intended to run a total of 20 weeks with the ﬁnal deadline in March
2007.Originally,it had been planned that the implementation of the project would be
ﬁnished by week 14 leaving ample time for testing and making changes to the program,as
well as being able to complete the written report by week 20.However,with the change
of programming language (as detailed in section 1.4) and the change in implementation
strategy (as detailed in section in section 1.4) the project plan was rewritten many times.
The ﬁnal program implemented was started in week 13 and testing was not completed
until the ﬁnal week.
1.3 Project objectives
The main objectives of the project are:
•
To write a program that is as userfriendly as possible,with no room for user error.
•
To use the same mathematical methods for calculating these graphs as engineers
already use but in a more structured manner.
5
•
To display three graphs,one showing beamdeﬂection,one showing bending moment
and one showing shear force.
•
To be able to load and save data.
•
To be able to print the results.
•
To be able to show a free body diagram as an engineer would draw.
1.4 Problems encountered
During the course of this project,we encountered a number of problems.In October,
when the original choice of programming language was considered,we decided to write
the programusing the computing package ‘Bordland JBuilder 9 Developer’.The decision
to use this language was made after careful consideration as to which the most useful
and eﬀective language for this type of program (see section 2.4).
However,after struggling with JBuilder for many weeks,limitations of the language were
apparent.More and more obstacles were posed until ﬁnally progress seemed to reach a
complete stand still.The diﬃculty of ‘dragginganddropping’ from one panel to another
proved too hard a problem to solve in the given time (if a tangible output was to be
created).The choice of programming language was reconsidered and ‘Delphi 7’ was then
chosen to continue the project.Delphi 7 was chosen due to its renowned IDE (Integrated
Development Environment) and graphics package and the fact that it was a language
known to the project supervisor.This is the language that the ﬁnal project is written in.
Whilst the program may have been started in an altogether diﬀerent language to the one
that the ﬁnal project was written in,we have not found that using this ﬁrst language was
a complete waste of time.Using a computing package similar to Delphi helped in the
simplest stages of programming,even though the syntax for both languages is entirely
diﬀerent.This was simply due to having created a very similar program already.It also
helped organisation of the code within the simpler parts of programming.
Not long after changing programming language,it became clear that the diﬃculty found
in using JBuilder was partly due to the design of the program not being ideal for the
language in which it was being written.Again,many programming issues were manifest,
6
the most apparent still being the diﬃculty of ‘dragginganddropping’ from one panel to
another.This proved to be a problem for the design originally envisaged and planned
for.Whilst this was the case for both programming languages used,the decision was
made to carry on using the language ‘Delphi 7’ and adapt the design of the program so
that it could be accommodated by Delphi.This amendment needed to occur without any
detrimental eﬀect to the userfriendly interface which had been designed in the earlier
plan.The two designs can be seen in ﬁgure 1.1.Whilst the original program had the
advantage of being self contained in one window,an MDI(Multiple Document Interface)
application,the use of many windows in the redesigned version was carefully planned in
order to avoid confusion.
(a) Original design
(b) Redesigned view of the program
Figure 1.1:Original and redesigned versions of the program
This report will detail only the ﬁnal program,which has been written using ‘Delphi 7’.
7
1.5 Structure of the report
In order to create a program that would fulﬁl the speciﬁcations of the project,it was
important to follow a full software development life cycle.For this reason,the report has
been broken down into the main parts of this life cycle:
•
Analysis  this section explains what the program intends to achieve,both in com
puting and noncomputing terms.
•
Design  The choice of programming language,design and organisation of the pro
gram as well as the project is addressed here.The organisation of data structures
within the programas well as the overall userfriendliness are particularly important
points to be discussed.
•
Implementation  The actual implementation is discussed including problems found
and solved,or avoided.This chapter includes screen shots of the program as it was
implemented illustrating the written points.
•
Testing  In order to produce a working program,it is necessary to test all compo
nents in a program for boundary conditions and misuse as well as using a random
testing process.Feedback is needed in order for improvements to be made.
•
Deployment  this is the part of the life cycle where the user has the software.A
developer must provide for the maintenance and updating of this software as well
as producing documentation about the program including help ﬁles (which will be
described in section 4.4).
Although this report is part of the documentation to accompany the program cre
ated,it does not give details about the program that the user would ultimately
deploy,instead it merely describes the process of its development.Due to time
constraints,there is no documentation to go alongside the program,and for this
reason we will not include a section on deployment within this report.
Although the above ﬁve areas can often be fuzzy and blend into one another,it is neces
sary to follow such a life cycle in order to produce an optimal ﬁnal program.
8
Chapter 2
Analysis and Design
Within the software development life cycle,the analysis (or speciﬁcation) of the problem
and the design of the solution are considered paramount to the success of the program.
Whilst an eﬃcient program can be written without either a speciﬁcation or a written
design,the result will rarely be what was originally intended,even if the end user is also
the developer who will implement the code.Before the implementation of the program
can occur,it is import for a developer to know exactly what is expected from their
program.This process must be undertaken when creating a program in order for a
developer to learn exactly what the objectives and speciﬁcations of the project will be.If
the analysis and design components of the software life cycle are not completed,the end
programwill often performthe main task intended but do so ineﬃciently.Also secondary
functions intended for the program may be sacriﬁced due to poor planning.The design
and speciﬁcation of this project are outlined in this chapter.
2.1 Deﬁning project requirements
This project is intended to show a user certain properties of a loaded beam and help
them to develop engineering intuition about how beams react under diﬀerent loading
conditions.For this reason it is important to be as clear as possible and to be careful to
avoid any possible confusion within both the design of the programand the user interface.
As an educational tool,this is key to the success of the product.Many design features
9
will come from this speciﬁcation.Thus the need to have simple windows,easy data input
and an accessible help ﬁle.The project’s main requirement is to act as a useful tool for
the course Engineering Physics EMAT10005 undertaken as part of the degree Engineering
Mathematics,studied at the University of Bristol.However,it is also intended to be of
use as a professional aid.This could,for example,be in industry,as a tool for an engineer
who wants to ﬁnd out the properties of a beam without the need for long calculations.
2.2 Deﬁning program requirements
This program is designed to be used by engineers with many diﬀerent levels of experi
ence,both in terms of the mathematical content as well as computing experience.It
is therefore important to keep the program as simple as possible whilst provide a user
friendly interface accessible to the most inexperienced computer users.We must also be
careful not to frustrate more experienced users.The programming language chosen will
be a RAD (Rapid Application Development) application,allowing the program a fast
runtime,with an IDE (Integrated Development Environment) and GUI (Graphical User
Interface),(see section 2.4).
Within this program,many inputs are required fromthe user.Auser must specify exactly
how the beam is supported and loaded.This will take place within a number of separate
windows.In order to make this as clear and accessible as possible,we intend to have
a number of diﬀerent ways to input data.Within each window,the position of a load
or support on the beam is determined either by percentage,actual distance or through
using a visual aid.Exactly what this aid will be is dependent on what programming
package will be used.Since this program is intended to be entirely interactive,we have
decided that there should be draganddrop facilities within the freebody diagram part
of the program.This means that a user can move a load or support with no need to
open a separate window which makes the program as concise as possible.Another way
of doing this is for a user to be able to double click on a load or support which calls a
window of all the values to do with that entity.This will be useful if the user wishes to
edit or delete a support.It is also important for users to be steered in the right direction
as far as input is required,therefore error messages will appear if a wrong type of input
is entered.
10
2.3 Designing the program
In the initial discussion about the project,it was decided that in order for the program
to meet the speciﬁcations outlined in section 2.2 and we must:
•
Create a simple nonMDI program with a main window where the ‘drawing of the
problem will take place.A separate window will exist alongside the main window,
acting as a library for all the loads and supports oﬀered
•
Create an interactive design with drag and drop facilities.Ideally an engineer will
drag a load or support along the beam within the ‘drawing’ which will refresh the
graphs automatically
•
Develop a menu bar with many options,including File,Create and Help options.
The ‘create’ option will take the user to a submenu where they can choose which
graphs they want to create.
•
Allow printing,outputting the beam diagram,as well as the three graphs of shear
force,bending moment and beam deﬂection
•
Display the three ﬁnal graphs in separate windows with labels showing the positions
of the supports and the extremal values of shear,bending and beam displacement
2.4 Choice of programming language
During the initial discussion regarding this project,it was important to determine which
programming language would be most suitable for the task in hand.It was paramount
to look for a program with a good graphics package,ideally with an easy to use IDE
(interactive development platform) in order to aid programming as much as possible.
Only RAD (Rapid Application Development) packages were considered since these types
of programs allow programmers to quickly build working programs,ideal for this type
of application to be input,where a developer is inexperienced.RAD programs are also
known to have a large emphasis on simplicity and usability and oﬀer a number of tools
to help build graphical user interface quickly and easily often with the help of a GUI
11
(Graphical user interface) themselves.With these objectives in mind there were three
obvious choices:
•
Borland Delphi 7
This program uses a base language of Pascal.Delphi is the most well established
of the programs considered.Its’ most popular use is for development of desktop
applications.The main disadvantage is that there are few books published on this
language as it is not as popular as C++ or Java,however,it is a language known
to the project supervisor.
•
Borland C++ Builder
This programhas a base language of C++.This is an environment also produced by
the CodeGear subsidiary of Borland and is very similar to Delphi.Most components
developed in Delphi can be used in C++ Builder with no modiﬁcation,although
the reverse is not true.C++ Builder was developed simply as a modiﬁed version
of Delphi with the diﬀerent base language.
•
Borland JBuilder 9 Developer
This program has a base language of java.JBuilder is a java IDE from Borland.It
has won several awards as the most powerful IDE for professional java programming.
The ﬁrst version of the system was mostly written in Delphi,excluding the form
designer and the component library.JBuilder is a constantly developing software
which can work with multiple application servers and can be used for developing
web services.
The advantages and disadvantages of the three languages were compared and we decided
to proceed with Delphi.We had already attempted to use JBuilder for this program
and encountered major problems (as detailed in section 1.4).Because C++ and Delphi
are so similar,we considered it better to opt for the program which the developers had
originally built,rather than one which had been modiﬁed to use another base language.
12
2.5 Visual design of the program
We designed the program on paper before implementation.The most important aspect
of the design is to ensure that all the correct information is requested in the correct form
from a user.Figure 2.1 shows an example of one of the designs drawn.
Figure 2.1:The design of the point load form
The aim when designing forms is to make the windows as simple as possible,whilst al
lowing the user freedom of input so that the program will be able to do exactly what
the user wants,within the given constraints.The main window is not designed on paper
before implementation since this window would appear blank until data was input,and
even then would be a simple as possible.The main window,however,is going to be the
one from which all procedures are called,once the initial loading of the beam has been
determined,therefore this is the only window with a menu.The items that this menu
will include can be seen in ﬁgure 2.2.
Figure 2.2:Design of the Main Menu
13
2.6 Designing the actual code and creating an im
plementation plan
It is very important to design how the program will actually work in terms of where
procedures are called from and how and where information is saved.All forms are
designed to be called from either the main form or the library form.This is because
these are the only forms from which a user can determine what the program should do
next.The mathematical procedures are to take place within the relevant forms,even
though sometimes the same code is used in more than one form,for example when
calculating reaction forces.Thus,each form will be as self contained as possible.The
data to be input by the user is intended to be saved in a data structure.Data structures
are a very important part of any program and allow data to be saved in order to be
used again within the program.Data structures can also be set up in order to only save
certain parts of the data.Data structures are also very useful for loading and saving
data.Which data structure will be used will depend on the type of input.Initially,we
plan three main data structures:
•
The ﬁrst data structure holds all the necessary information about each support.
•
The second data structure holds all the necessary information about each load.
•
The third data structure contains all the information encompassed in the previous
two structures,as well as any other information entered about the beam.
It is possible that other data structures will be needed when programming but these have
not been planned for as they are not an integral part of the code.
Since Delphi has been chosen to programin,only this relatively small amount of planning
has occurred prior to the implementation of the code.This is partly due to the RAD
nature of Delphi.We intend to design a simple working program and build up from this.
Further code will be designed at each obvious point when making the program more
complex and thus we decide,that our aimis to develop a programthat can become bigger
and more complex.Another reason for this approach is because neither the program nor
the programming language being used are known,therefore it will not be clear exactly
how the program works until programming has started.The implementation plan is
14
therefore just to start coding for the simplest case and build up the program from there.
First,we aim to input a complete range of loads and supports and compute the results.
Then,we will reﬁne the program by adding other features such as a printing option and
help ﬁle.
15
Chapter 3
Mathematical Reasoning
At present,many engineers use a standard ‘intuitive’ way of calculating bending moments,
shear forces and deﬂections of beams.This is done through studying free body drawings of
the beam with all loads and supports detailed.After this,sets of simultaneous equations
are determined by taking moments around a support or load.
However,there are many problems with this method of calculation.Not only is it time
consuming but the algebra involved can get very complex and errors can creep in.This
could put an entire project in jeopardy.Errors may also occur with this method of cal
culation within the freebody diagram.With a complexly loaded beam,the free body
diagram can become very confusing and therefore all loads or supports may not be ac
counted for when making calculations.
Within this project,only statically determinate beams have been considered.Statically
indeterminate beams involve more complex algebra and have not be described below.
In this chapter,we will show the mathematics behind how to create shear force,bending
moment and beam deﬂection diagrams.
16
3.1 Deﬁning shear,bending moment and beam de
ﬂection
In order to draw diagrams of shear,bending and deﬂection we must ﬁrst discuss exactly
what these words mean.
Shear force,V,can be described as “The force that tends to cause atoms of a structure
to slide past each other” [1].
Put simply,shear force is an internal force which acts in a direction perpendicular to
the beams longitude in order to keep the beam remaining in equilibrium for all values of
displacement along the beam.
The bending moment,M (or internal torque) of a beam,is intrinsically linked with the
shear force (as is the beam deﬂection which will be discussed later).Mathematically,it
can be described as the “algebraic sum of moments about the neutral axis of any cross
section of a beam” [2].and can be denoted by:
M =
V (x)dx
(3.1)
where x = distance from the origin of coordinates
Beam deﬂection,ν,is the distance that a beam moves from it’s equilibrium value in the plane
perpendicular to its longitude at any given point and can be found by combining expressions
for curvature,κ,as shown in equations 3.2 and 3.3.A beam deﬂection diagram gives an
image of exactly how the beam will bend and therefore look under diﬀerent loading/supporting
conditions.
κ ≈
d
2
ν
dx
2
(3.2)
κ =
M(x)
EI
(3.3)
where EI is known as the structural rigidity
Therefore,by combining equations 3.2 and 3.3 and integrating,we get that:
17
ν =
M(x)
EI
dxdx
(3.4)
where υ =
EIν
L
2
and L = length of the beam
For the purpose of this computer program,as well as the mathematical reasoning,equation 3.4
is scaled in order for ν not to be dependent on EI.Therefore equation 3.4 becomes:
υ =
M(x)dxdx
(3.5)
The value of deﬂection,υ,we are now calculating is just a scaled version of the actual deﬂection.
Each diagram that is to be drawn,plots the relevant value (i.e.shear,bending or deﬂection)
against x,the distance along the beam from the origin of coordinates.
3.2 Details to consider
Since we are only considering statically determinate beams,the loads allowed will be:
•
a concentrated (point) load
•
a linearly distributed load
•
a nonlinearly distributed load  this has not be input and therefore will not be described
in this chapter
The supports allowed will be:
•
a ﬁxed support (cantilever)  this will not allow translation or rotation in any direction
•
a pin support  allows rotation but not translation
•
a roller support  allows translation in one direction but not rotation
this has not be input and therefore will not be described in this chapter
As has been mentioned before,the only types of beam being considered in this approach are
statically determinate ones which means that there are only relatively few ways of supporting
the beam.These can be seen in ﬁgure 3.1.
18
Figure 3.1:Types of statically determinate beams
When considering a distributed load,it is necessary to ask more information of a user than
for a point load.The start and end point of the load must be known as must the equation
of linearity.This can either be deduced by asking for the load at the start and end points,or
as a function of intensity in equation format.In our program the start and end load will be
requested.
3.3 Calculating shear,bending moment and beam
deﬂection
The method of calculation detailed is considered as if looking at a prismatic bar with constant
cross section throughout its length.To investigate internal stresses such as shear and bending
moment,an imaginary cut is made perpendicular to the to the longitudinal axis of the bar.
19
Shear Force
Section 3.1 showed that in order to calculate bending moment and beam deﬂection,we ﬁrst
need to ﬁnd the shear force.In order to solve this problem,an engineer often draws a freebody
diagram.This is where a simple,twodimensional diagram is drawn showing exactly how the
beam will be supported and loaded (for example,see ﬁgure 3.2).
Figure 3.2:Example of a freebody diagram
As ﬁgure 3.2 shows,every support has a reaction force.These reaction forces can be determined
through using two simple equations,see equation 3.7 and 3.6.
M = 0
(3.6)
L = 0
(3.7)
where L = load value
Once all the forces and reactions on the beamare known,then it is simple to calculate the shear
force.
An engineer will often calculate shear force in sections where in each section the properties of
the beam change.For the program,a loop will run,calculating the shear force for every unit
along the beams length.The description below explains how shear force.
The value for shear force changes throughout the beam,since as the beam’s properties change,
greater or smaller internal forces are needed in order to keep the beam from shearing.
Firstly an imaginary cut takes place along the beam’s length and an illustration is drawn of
what remains between this point and the origin of coordinates (for example,see ﬁgure 3.3).
20
Figure 3.3:Example of a shear force illustration
From this diagram,we can deduce that for this case:
V = R−L
or in general
V =
R−
L
(3.8)
When the illustration become more complicated,for example,including distributed loads,the
calculations become more diﬃcult as equations 3.9 and 3.10 show.
Equation 3.9 shows how to calculate shear force if the value of displacement,x,falls between
the start and end points of a shear load.
V =
R−
L
P
−
L
DL
(x −x
DL
) −
2(L
DR
−L
DL
)(x −x
DR
)
2
x
DL
−x
DR
(3.9)
where L
P
= point load,L
DL
= the load value at the left hand side of a distributed load,L
DR
= the load value at the right hand side of a distributed load,x
DL
= the position of the left
hand side of a distributed load,x
DR
= the position of the right hand side of a distributed load,
x = displacement along the beam
21
Equation 3.10 shows how to calculate shear if the value of displacement falls outside the bounds
of a distributed load.
V =
R−
L
P
−
(L
DL
(x −x
DL
) −
2(L
DR
−L
DL
)(x −x
DR
)
2
x
DL
−x
DR
)
(3.10)
+
(L
DR
(x −x
DR
) +
2(L
DR
−L
DL
)(x −x
DR
)
2
x
DL
−x
DR
)
In order to plot the shear force diagram,the shear force (calculated by drawing a simple shear
force diagram and using equations 3.9 and 3.10 accordingly with the forces seen on the illustra
tion) is plotted against displacement.
Bending Moment
Once shear force has been calculated,the bending moment can be determined.This is done
using equation 3.1.
In order to calculate bending moment,we integrate equation 3.1 to give equation 3.11.
M = V x +C
(3.11)
where C = a constant
This constant,C,is diﬀerent for each value of displacement,x.
An engineer will often ﬁnd it easier to take moments about the neutral axis,using equation 3.6
in order to ﬁnd the bending moment.This is done in the following way.
Firstly,a bending moment illustration is drawn in much the same way as the shear force illus
tration was drawn,see ﬁgure 3.4.
22
Figure 3.4:An example of a bending moment illustration
Then,using equation 3.6,taking moments about the neutral axis,we can deduce that for this
example:
M +V x −Lx
L
= 0
or
M = Lx
L
−V x
where x
L
= the position of the load on the beam
For a more general case:
M =
Lx
L
−
Rx
R
−V x
(3.12)
Again for a distributed load,the algebra becomes slightly more complex,the easiest way of
calculation now involves using integration of shear,since C=0 since one support must be at
displacement,x=0 for a statically determinate load.No equation has been shown here,since
equation3.1 is used which has been seen previously.
As we have shown,if a beam is loaded complexly then it is necessary to user 3.1,therefore we
shall always use this equation in future calculations.
If a ﬁxed support is used for the beam then the calculation includes one more variable,this is
because another moment,M
R
,associates itself with a ﬁxed support.This is easiest to show
23
pictorially and an example of a bending moment can be seen in ﬁgure 3.5
Figure 3.5:Another example of a bending moment illustration
Mathematically,this does not make the bending moment any harder to ﬁnd but before any
calculations can occur,M
R
must be calculated by taking moments over the entirety of the
beam.The bending moment for any displacement value can be determined after this,for
example (with no distributed loads,using the ﬁrst method discussed),using equation 3.13.
M = M
R
+
Lx
L
−
Rx
R
−V x
(3.13)
or for a more general case:
M = M
R
+
V dx
(3.14)
where M
R
= 0 for a pin support
Using equations 3.14 as shown,we ﬁnd bending moment for each value of displacement,x,and
plot on a diagram.This is how we create the bending moment diagram.
24
Beam Deﬂection
Beam deﬂection is the most diﬃcult of the three values to calculate.In for this value to be
calculated,equation 3.5 must be evaluated.This involves integrating twice and gives equation
3.15.
υ =
Mx
2
2
+C
1
x +C
2
(3.15)
where C
1
and C
2
are constants
We know that at a ﬁxed support,both curvature of the beam as well as deﬂection are equal to
0 therefore,C
1
and C
2
are equal to 0 for the beams length.The constants do not change since
there is only one support acting on the beam for such statically determinate examples.
However,constants C
1
and C
2
can be diﬃcult to ﬁnd for situations when the beam is supported
by pin supports.Sometimes it is necessary for an engineer to solve dozens of simultaneous
calculations in order to ﬁnd values for these constants.
It was discovered that for the case of two pin supports,equation 3.16 gives the beam deﬂection
for all cases along the beam.
υ = θ
0
x +
M(x)dxdx
(3.16)
where θ
0
is a constant which can be calculated by using equation 3.17
θ
0
= −
M(x)dxdx
(3.17)
where the x value used is the displacement of the second support from the origin of coordinates
and there are no constants
We have shown howto calculate beamdeﬂection and this value is plotted again the displacement,
x in order to create the beam deﬂection diagram.
25
Chapter 4
Implementation
This section was the most timeconsuming part of the project.Firstly,it was important to
become familiar with the IDE used and Delphi as a program,whilst also learning the language
‘Pascal’,the base language of Delphi.
The quickest way to learn how to use Delphi was to build very simple applications to see how
the program worked.This involved creating ‘Hello World’ type applications and building them
up in order to use more and more diﬀerent objects,or example buttons,as well as using more
than one form within an application and seeing how procedures could interact over a number
of forms.
Following an initial period spent learning how to use Delphi,the program was slowly built up
into its ﬁnal version.
4.1 A simple scenario
Within this problem,a simple scenario of a loaded beam is a beam supported by pin supports
at either end with one point load in the middle (as seen in ﬁgure 4.1).
26
Figure 4.1:A simple scenario
In order to start building this scenario,the main window including the menu and the library
window is set up.Next,the library form is set up as a canvas,so pictures of the loads and
supports can be drawn,in order to direct the user into inputting the correct data.After ‘drawing
these pictures and placing buttons next to them,the library form is complete,see ﬁgure 4.2.
Figure 4.2:The completed library form
Both of these forms are conﬁgured with a ‘bssingle’ border style,meaning they cannot be resized
(this is something that can be addressed at a later date if project requirements dictate.This
is line with our aim of a program that can develop and grow to meet future needs).All other
forms are created with this type of border style in order to give the program continuity.
The next step is to allow the buttons shown on the library form to link to other forms where
the appropriate inputs will be requested.In order to do this,template forms are implemented,
as similar as possible to the designs previously produced (for example,those shown in ﬁgure 2.1
in section 2.5).Code for each button is then written creating a new form each time a button
is activated.This means that every time a point load is added,it will be added on a diﬀerent
form to the previous load input.The creating of a new form for every new load or support is
completed for ease of making changes to the values of the load or the distance along the beam,
27
which can be implemented at a later date.
In this simple case scenario,the point load is the ﬁrst load (and button) to have a formassociated
with it.The ﬁnished form can be seen in ﬁgure 4.3(a) alongside the ﬁnished pin support form
(see ﬁgure 4.3(b)) which has been implemented in much the same way.
(a) Form to input point loads
(b) Form to input pin supports
Figure 4.3:Examples of forms needing user inputs
As seen in the screenshot of this form,the user is given a number of diﬀerent ways to enter the
data.The load can be input normally,or using the ‘updown option,which means that a user
can input the value using these buttons to choose an integer value for the load.This can be
either negative or positive.The position along the beam can be entered in one of four ways;
either by position along the beam,approximate percentage along the beam (which can be done
manually or by use of another ‘updown’ option) or by placing the load using the track bar.The
track bar is seen in terms of the position along the beam,as shown by the value at the end of
the track bar which will change for diﬀerent beam lengths.Operating the cancel button means
that none of the inputs are saved.However,the ‘OK’ button has been set up in order to save
the inputs speciﬁed.
28
The next step in order to create this simple scenario is to set up a number of data structures to
save the inputs provided by the user.These data structures are conﬁgured within the library
form since this is where the load and support forms are called from.These data structures are
known in Delphi as records.One such record can be seen in ﬁgure 4.4.
Figure 4.4:Code deﬁning a data structure
Within this snippet of code,TLoad has been declared as a record (or a data structure) in
which certain parameters can be saved.These are the load type,load values,positions along
the beam and the values that would deﬁne polynomial loading conditions (for use with non
uniformly distributed loads).This data structure is used within the program every time data
about a load is saved.An array of type TLoad and of length max
loads exists in which all the
values for each load are stored.This data structure is then ordered after every input so that
the ﬁrst load will be the one closest to the left hand side of the beam.The loads after this will
be in order of their displacement across the beam.Similar data structures were implemented
for the supports as well as for the beam in general.Thus the data can be easily saved.
Every time the main form is in view,the ‘paint’ procedure is called.This procedure draws
the saved loads and supports on the beam,accessing the data from arrays of type TLoad and
TSupport.These arrays are called my
supports and my
loads.An example of a beam with
these supports and loads ‘painted’ on can be seen in ﬁg 4.5.
29
Figure 4.5:Example of the ‘onscreen’ freebody diagram
Once the free body diagram has been drawn,the next step is to create a form showing the shear
force diagram.In order to create this,the mathematical calculations shown in Chapter 3 must
be input.
Firstly the reactions at the two ends are calculated using equations 4.1 and 4.2:
L = 0
(4.1)
where L = load value
M = 0
(4.2)
where M = Moment
A set of axes is drawn on a new form and then shear value is plotted for each value of the
displacement,x.The bending moment diagram is drawn in a similar way.Finally,the beam
deﬂection diagram is drawn.This proves to be more diﬃcult than either of the previous dia
grams due to calculations of constants.Whilst the simple case that we are looking at is easy
to calculate,we need to generate generic code that can work for many loads of this type.
Our simple case scenario for the program is now running and ready to use.
30
4.2 Building the program’s complexity
Now the simple program is working correctly,it is important to create a program that is of
use for a more general case.However,it is important to do this in an ordered way.Since only
statically determinate beams are being considered,there are relatively few inputs to consider.
The ﬁrst new support to be input was the ﬁxed support.This can be ﬁxed on either the right
end of the beam or the left,dependant on user input.The calculations of shear force for this
type of beam are very simple since there is only one reaction force to be calculated in the
simple statically determined set.However,the moment around this ﬁxed support needs to be
calculated when considering bending moment.Since the beam is ﬁxed at one end there are
no constants to consider in the beam deﬂection diagram and these three diagrams are created
fairly quickly without error.
The case of using two simple pin supports is expanded in order that one support can move from
the end point of the beam.Computing the shear and bending moment are no harder than in
the simple case shown in section 4.1.
After this,we decide that a linearly distributed load will be input.This has already proved to be
a problem whilst designing the form on paper and is just as diﬃcult to create computationally.
We discover that it is not possible to implement a double track bar within Delphi.Therefore the
formis created as designed but without the track bar.This is in order to preserve simplicity and
continuity of design within the program.Drawing this type of load on the freebody diagram
proves to be more taxing than any of the previous drawings.This is due to the diﬃcult visual
nature of the load which requires a large amount of space.It is diﬃcult to preserve clarity in the
freebody diagram whilst this type of load is being shown.In the end,we decide to only place
the beginning and end points of the load on the beam with a line joining them to show a user
that this is a distributed load.This type of load provides a greater challenge when calculating
shear,bending moment and beam deﬂection.However,the code is input in a similar way to
that of the simple point load.
4.3 Creating a usable program
Whilst this programruns fairly well,it is noted that there are many small problems which makes
the program diﬃcult to run and does not allow the user an enjoyable computing experience.It
is important for a program to be as user friendly as possible,whilst giving no room for error,
31
although it is widely known that creating a totally error free program is near impossible.
A system for catching incorrect inputs such as punctuation or letters in the numeric boxes is
created,showing the user an error box with an easy to understand message (for example,see
ﬁgure 4.6).
Figure 4.6:An example of an error message
Since,the system is also designed so that only statically determined beams are be allowed,if a
user tries to enter a statically indeterminate beam,the system outputs an error message and
does not allow them to do so.An error message is also output if a user enters more than one
support or load in the same position (including more than one load with exactly the same start
and end points for a distributed load).
To make the program more user friendly,we make the freebody diagram in the main window
editable,so that when a support or load is doubleclicked it brings up the values input in the
original form so that they can be changed or deleted.This results in some slight changes in the
form,with the cancel box becoming a delete box,for an example see ﬁg 4.7.
32
(a) An input form for a ﬁxed support
(b) An input form to edit a ﬁxed support
Figure 4.7:Figure to show a ﬁxed support form and its editable version
Another change is to the shear,bending and beamdeﬂection diagrams.The outputs are scaled in
order for themto ﬁt within their windows and on the axes drawn and labels are created showing
the displacement along the beam of the supports as well as the maximum and minimum values
along the yaxis (indicating shear,bending moment or beam displacement).Whilst working on
these forms,an additional label is input in the top left hand corner of the screen.This label
tracks the mouse position and displays the coordinates at the point on the graph where the
mouse is.At the point along the axis where the mouse is placed,the shear,bending moment
or displacement of the beam is be calculated and displayed.
Within the design plan,we stated that only once the programwas running,would we implement
other features such as printing,loading data or saving data.These features are then added in
order to make the program as helpful to a user as possible.The added features are mainly to
do with the menu on the main screen and consist of:
•
Using the open/save and save as facility
33
•
Creating all three force diagrams at once
•
Allowing the program to restart if a user decides to change the loading and supporting
of a beam entirely
•
Exiting the program via a menu option
•
Printing/which uses a ‘print screen’ method where the program creates a form with all
the possible data i.e.freebody diagram,shear stress diagram,bending moment diagram
and beam deﬂection diagram.This form is then be seen by the computer,saved as a
bitmap and printed with the default printer.
After this ﬁnal bit of implementation,the only obvious part of the program missing is the help
ﬁle.
4.4 The help ﬁle
After implementing the program,we decide that a help ﬁle is very beneﬁcial to a user.This
will explain how to use the program as well as exactly what it can do.The beneﬁts of writing
either a manual or including an ‘online help ﬁle to attach to the program are considered and
we decide that writing an ‘online ﬁle is better.In an online ﬁle,a user will be able to access
the entire help ﬁle without leaving their desk and it will be more ﬂexible than a written aid.
However,since creating such a help ﬁle will involve learning yet another computing package,a
simple written overview for the program is completed instead with a view to creating an online
help ﬁle at a later date (see Appendix A).
34
Chapter 5
Testing
Once the program is complete,it is tested to make sure it gives correct answers.Parts of the
testing happened continuously throughout the implementation process in order to make sure the
program was progressing as planned and the given functions were yielding the correct results.
However,within the software development cycle,outlined in section 1.5,it is necessary to allow
for a certain amount of time on testing the program when the developer ﬁrst believes it to be
complete.The program is tested mainly to detect what it can do and to make sure it will not
stop working due to errors.It is also necessary to test how userfriendly the interface is and see
how useful this program is deemed to be.This is discussed in section 5.3.
5.1 Testing for errors
It is necessary for this program to be tested with boundary conditions as well as by inputting
incorrectly formatted data.We also try to make the program fail in other possible ways such
as miskeying.Testing is done in this way as extensively as possible,so that it is as error free
as possible.We aim to identify errors,amend the program and produce a truly viable output.
35
Table 5.1 shows the tests completed when testing for errors for each formrequiring a user input.
Type of test
Error Box shown as expected?
Inputting a letter into a load edit box
Yes
Inputting punctuation into a position edit box
Yes
Inputting a negative value into a position edit box
Yes
Inputting two loads/supports in the same position
Yes
Inputting two ‘minus’ signs in a load edit box
Yes
Inputting a distributed load with the start
position further along the beam than the end position
Yes
Allowing a beam with three supports
Yes
Allowing a diﬀerent type of statically indeterminate
beam,e.g.a ﬁxed support and a pin support
Yes
Entering a percentage greater than 100% in the
No error box appears,
percentage box
however the program
resets this value to 100%
Deleting the value in the load box
No error box needed
Deleting the value in the position or percentage box
Error box provided
by Delphi
Table 5.1:Table to show the results of error testing
An error box also appears if a user tries to create a diagramwhen the beamis not in equilibrium.
5.2 Random testing
Sample questions from previous Engineering Physics examination papers are input into the
program and the output is compared against the sample answer sheets for the relevant ex
aminations.We also test the program extensively using many other diﬀerent combinations of
loads and supports to make sure the diagrams output are correct.In order to make sure these
are correct,manual calculations are made and the diagrams drawn from these calculations are
compared to the diagrams output by the program.
36
Table 5.2 shows the results of the tests carried out
Loads Included
Supports Included
Results as expected
Point
Linearly Distributed
Pin
Fixed
no
no
yes,2
no
yes
yes
no
yes,2
no
yes
no
yes
yes,2
no
yes
yes
yes
yes,2
no
yes
no
no
no
left
yes
yes
no
no
left
yes
no
yes
no
left
yes
yes
yes
no
left
yes
no
no
no
right
yes
yes
no
no
right
yes
no
yes
no
right
yes
yes
yes
no
right
yes
Table 5.2:Table to show the results of random testing
5.3 Testing by sample users
Once the error testing and random testing had taken place it was intended that we would ask
ﬁrst year students from the degree Engineering Mathematics who were completing the course
Engineering Physics EMAT10005 and who are studying this type of bending moment,shear
force although and beam deﬂection to participate in a testing and evaluation session for the
program.They would be asked to complete a number of problems on the program and then
complete a short questionnaire about their experiences using the program.This questionnaire
would ask how user friendly the student found the program as well as asking for any comments
or problems that they may have found in using the program and ways in which it might be
improved or be made more user friendly.The research found by this would be collated in order
to make improvements.This questionnaire can be seen in Appendix B.
37
Chapter 6
Conclusions
We have developed a computer program that outputs diagrams of shear force,bending moment
and beamdeﬂection for statically determinate beams.Whilst certain modiﬁcations to the design
were made after learning the possibilities and limitations of Delphi (for example,not being able
to drag and drop on the freebody diagramleading to the loads and supports only being editable
through a doubleclick method),the program produced works and outputs correct results.
6.1 Suggestions for future development
Using results collated through testing as well as experience gained through completing this
project,it is possible to suggest ways in which the program could be improved.
Initially,we intended to create a program that would run for a beam with any possible type of
support or loading.At present,the program works for statically.determinate beams.Ideally
we would also program situations where the beam becomes statically indeterminate.
This program does not currently allow nonlinearly distributed loads to be input.This would
involved signiﬁcantly more programming time and we have therefore not progressed this appli
cation.
We can develop this program further,for example,being allowed to determine how many loads
can be input (the default is set to be 10),or the length of the beam.A more comprehensive help
ﬁle would improve the program as would being able to toggle whether to see the load values
and load and support positions on the freebody diagram,or to leave it purely as a diagram.
38
Improvement on all of the above points could be made in time,but a great deal more testing
on a wide range of users would need to be completed in order to maximise its eﬀectiveness.
6.2 Conclusion
A successful program is one that has been shown to meet the needs of the speciﬁcation as well
as the objectives (outlined in section 1.3).This program:
•
Is userfriendly,with no user error
•
Uses the same structured mathematical methods as engineers currently use (see Chapter
3).
•
Displays three graphs showing beam deﬂection,bending moment and shear force.
•
Is not yet able to load and save data.However,since all the data to be saved to ﬁle is
ordered and saved within the program in three data structures,inputting load and save
options would be a relatively easy addition.
•
Can print results.
•
Shows a freebody diagram
This list shows that all except one of the objectives (detailed in section 1.3) have been met.
Whilst all testing has proved the program to give the correct outputs,only extensive use of the
program will really show if it is entirely errorfree.Further testing,as detailed in section 5.3 is
needed in order to do this as well as to further reﬁne the program.
Overall,the program created,has met expectations of the program envisaged and would be a
useful tool for an engineer.
39
Bibliography
References
[1]
Foundations of Engineering Foundations of Engineering:Glossary
Retrieved from:http://highered.mcgrawhill.com/sites/0072480823/student
view0/glossary.html
[2]
UniStates UniStates RMT Glossary
Retrieved from:http://www.unistates.com/rmt/explained/glossary/
rmtglossaryb.html
[3]
Author:James M.Gere,Stephen P.Timoshenko
Title:Mechanics of materials
Publisher:Brooks Cole
Year:2000
Bibliography
[4]
Delphi Help Files
[5]
Author:Kent Reisdorph
Title:Sams teach yourself Borland Delphi 4 in 21 days
Publisher:Sams
Year:1998
[6]
Author:Marco Cant
Title:Mastering Delphi 7
Publisher:Sybex Inc.
Year:2003
40
[7]
Author:Rozlog,Mke;Goetz,Geoﬀrey;Sung,Nguyen
Title:Mastering JBuilder
Publisher:John Wiley & Sons Inc
Year:2003
[8]
About:Delphi Programming Getting Started with Delphi
Retrieved from:http://delphi.about.com/od/beginners/Getting
Started
with
Delphi.htm
[9]
EMAT10005 Engineering Physics EMAT10005 Engineering Physics
http://www.enm.bris.ac.uk/admin/courses/EMAT10005.htm
[10]
Soft4Structures General information
Retrieved from:http://www.soft4structures.com/Beam/sm
general
info.html
41
Appendix A
Help File
Beam Program Overview
In order to begin using the program please insert a load or a support from the library seen to
the left of the main window.
Input the position of this support or load and if a load has been input then it’s value also.
Click OK to proceed or Cancel to go back to the main form without saving.
Insert more loads/supports in this manner.
If you need to edit or delete a load or support,just double click,and an editable box will appear.
When the beam is drawn as intended,choose a menu option to perform another task.
To exit the program,choose File then Exit,or click the ‘x’ in the top right corner of the main
window.
42
Appendix B
Questionnaire on Beam Program
I am currently developing a computer program to solve algebraic problems for shear,bending
moment and beam deﬂection and would very much appreciate your views on the program I
have created.Please use the program provided to complete a number of example problems
from Chapter 4 (Shear Forces and Bending Moments) and Chapter 9 (Deﬂection of Beams) of
“Mechanics of Materials” (Gere,Timoshenko) [3].
Did the program give the correct result?
Was the program easy to use?
Does this program save time compared with your normal method of calculation?
Are you conﬁdent with the result it produces?
Would you use this program in future if it was made available?
Please use the space below to make any comments on improvements you would like to see.
43
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