As promised, this is the continuation of our discussion on HOW LEARNING CAN BE FUN where herein I demonstrate the application of the ‘THEORY’ previously outlined in Part 1, applied to one of the subjects that I teach, THEORY OF STRUCTURES. Here we go.
The lecture on Theory of Structures is about analyzing indeterminate structures. It comprises various techniques of analysis applied onto various types of structures. In the beginning, the students will be introduced to the Chapter of Beam Analysis which comprises three analysis techniques; principle of virtual works, slope deflection and moment distribution methods. These three techniques, despite their differences, would give relatively similar answers. The next chapter to follow is the Truss Analysis which is based on the principle of virtual works. Before I proceed, it is best to show you the pictures of a beam and a truss system to increase your appreciation. The pictures are given below and please, have look at them for a moment.
Figure 1: A beam system
Figure 2: A truss system
Have you looked at them? If you have, it’s great. Indeed, my effort on providing you with the two pictures is the immediate manifestation of the aspects of pattern tracing as well as the Wanabe and Guessing games. Allow me to elaborate. Pattern tracing is about identifying similarities and differences by making comparison. By comparing the two pictures, I believe, although without any training in engineering, a general reader would be able to compare them to some extent. You might say, “Ok, I did see some similarities and differences, so what?” I would say “Yes, you have the right to be impatient, but as you will see, this comparison is just the first of a series of comparisons, so please bear with me”.
On the argument of Wanabe and Guessing games, the effort of showing these pictures is my effort of showing the GRANDUER of the topic. In effect, you might already said “Hmm, it’s quite cool this engineering things” or something like that. If you said this, that means I am triumphant. Is not Wanabe game is about paying respect towards the topics and towards the related ‘heroes’ and ‘heroines’? In this context, I want you to feel the GRANDEUR or GREATNESS of the topic that we are talking about. This is part of Wanabe game. And, to those who agree, comparing the two pictures also brings us down the memory lane of the game we used to play during our childhood, a game which is shown in the figure below. This is an early hint of the Guessing game which we are going to see more as we progress.
Figure 3: Childhood Guessing Game
Enough with the reasoning for the provision of the pictures, lets proceed. In the deliverance of this subject (referring to Theory of Structures), it is usual to finish first the whole chapter on beam. Meaning, the three different techniques are firstly discussed before one proceed to the next chapter of truss analysis. BUT, I DO IT DIFFERENTLY. Instead, once I finished with the beam analysis, I immediately continue my lecture on truss analysis and delay the other techniques of beam for later. My reason is to preserve the continuity of the argument because only this way, I can show the students the concept of LEARNING CAN BE EASY if we hold on to the principle that makes up the argument (etc. 1+2 is 3, 10+20 is 30, remember?). I do this because the beam and the truss analyses are both based on the same principle that is; the Principle of Virtual Works. Allow me to elaborate.
I would not go into the details but you will see some equations. Do not worry about it; just take it as ‘another’ picture because like I said in Part 1, it is the ‘story telling’ that is important, not the equations.
To analyze the beam based on the Principle Virtual Works, there would come a stage where one is required to calculate the reaction force of a so called redundant. This calculation is given as:
Current or the orthodox (I would say) method of teaching is to teach the student how to get all the variables, m, M, E, I and to carry out the integration and that’s it. At the end of the semester, the student will repeat what has been taught and SCORE hence the prevalence of the typical exam orientation concept. But again, I DO IT DIFFERENTLY.
In my teaching, I would show the derivation of Equation 1. By now, you should say “What is the need to derive it? Even if the students know how to derive the equation, it’s still the application of the equation is what important, isn’t?” I would reply “Yes, you are right and the derivation is indeed a waste, if and only if, you fail to repeat it for other cases”. You will see that, my derivation is not a waste; in fact A MUST after I show you the same derivation is applied to the truss system. If I don’t show this derivation, how can I ever show that 1+2 is 3 and 10+20 is 30, right? This is what I meant earlier in Part 1 when I said “math can either kill the fun or make things fun”. Be patient and let’s proceed.
But there is one dilemma, how low or deep should I go with the derivation? This is a 3rd year subject you see, and a complete derivation would require me to dig into the previous year’s topics. And since I am constraint by both space and time, I got to decide on what is appropriate. And also, you must know one thing; this dilemma is also caused by the fact that our student is so poor in fundamentals.
THIS IS HOW I DO IT. Since the topic based on Principle of Virtual Works, I just got to start with the mathematical statement of the principle:
INTERNAL WORK = EXTERNAL WORK
Equation 2: Principle of Virtual Work
Do I elaborate on this? The answer is no. What I would do is just to ask the student, “Have you ever heard of this statement during your 2nd or 1st year?”. They would say yes and I would reply “Ok, good enough”.
Next I would write down the statement of strain energy of beam bending as follows:
Do I elaborate more on this? No, I would just ask the student, “Have you ever heard of Strain Energy?” They would say yes and that is enough for me. And then I continue by saying, “Please know that the INTERNAL WORK as shown in Equation 2 can also be considered as strain energy if either one of the derivative terms in Equation 3 is assumed to be caused by the virtual displacement. What I mean is, INTERNAL WORK can be given as below”.
If you compare Equations 3 and 4, you would notice how similar they are except for the tilde sign, which refers to the virtual displacement. I would ask the students to spend some time comparing Equations 3 and 4 as I just did with you, with the intention that they then can realize how two mathematical ‘entities’ which seem and sound so different at first; strain energy and internal work, are actually very related like ‘cousins’. Not only I want them to realize this, but I also like them to appreciate the generality of mathematics although at this point I even doubt their understanding on the word ‘generality’ itself, but never mind. I believe they are starting to.
And you readers, do you realize that when you compare Equations 3 and 4, what you did is actually repeating what we did earlier with the pictures of the beam and the truss. To make it obvious to you, let me put both equations side by side as below. It’s exactly like Figure 3 isn’t?
Figure 4: Principle of Virtual Works Guessing game
Ok listen. Equation 4 is where I will begin my derivation in getting Equation 1. This is how deep I have decided to go after considering what is appropriate. Why I keep saying this? This is because I can always go deeper into the fundamentals. For example, the strain energy can actually be discussed further from the Principle of Minimum Potential Energy point of view, which derivation can be shown as another Euler-Lagrange equations, or I can even go to the lowest of Brachistocrone’s curve of Johann Bernoulli; the original argument on calculus of variation. But, like I said, due to the constraint of space and time, a teacher must decide on the limit of the discussion. However, deciding what is appropriate, as far as the limit is concerned, is something that I cannot teach. It is totally depend on the teachers to decide and this is what differentiates a good teacher from the mediocre. The only advice that I can give is, the more the teachers know about his or her topic as well as the state of the students are in, the better the decisions would be. But if you ask, ‘how a teacher can know more?”, I would then reply, a teacher must keep reading, reading, reading, thinking, thinking, thinking, researching, researching, researching and living, living, living scholarly . It’s a one way ticket!
Let’s do the derivation in Part 3. (To be continued..)