When I finished writing this Part 3, I realized that the early part of the writing might be a bit confusing and boring, especially for those who do not like math but believe me, towards the end, it will be very interesting, and this I promise you. And frankly speaking, I need to show you these derivations because I need them in Part 4 (by this time you might say, arrrrrrggghhh, there will be Part 4, bila nak habis daaa?) and I am really sorry about this. But, I really hope you can endure. Thanks for reading and let’s continue.
To derive Equation 1 from Equation 4, I would start by writing down the relationship between the curvature and the moment force, M, which can be given as:
Do I go deeper? I do, just a bit. I would draw the following diagram:
Figure 5: Beam curvature
“Do you remember seeing the above figure? “ I would say. Then they say “Yes”. I will continue to say “You can also remember that derivation of Equation 5 involved the use of Figure 5, right”. They would say “Right”. I would then proceed.
Now, you may wonder why I keep asking the students this way, and why I am satisfied just by the simple answer ‘yes’ from the students on these kinds of question? Simply because, THE DAMAGED IS ALREADY DONE, ok? The students are so poor in the fundamentals, to teach them all over again would take years. But, having said this, we should also realize that they still have some sort of ‘GRAPHICAL MEMORIES’ of those things they have learned previously. They can still somehow ‘see’ or ‘visualize’ in their head, Equation 5, or Equation 1 or 2 or whatever although they most probably no longer understand the meaning of it. Realizing this, we teachers, can MINIMIZE THE DAMAGE by revisiting these equations rather quickly. It’s all about deciding on the limit of the discussion, remember? (See Part2). But, what can we expect from this ‘quick revisit’ approach?
1. We hope that by invoking the ‘GRAPHICAL MEMORY’ somehow able to show and strengthen the continuity and the pattern of the discussion
2. We also hope that it can increase the degree of appreciation towards what have been learned and may be in their free time, they will take their own initiative to revisit the subject by themselves
3. If they can appreciate what they have missed, they will try not to miss anything like that in the future
The above points are what I meant by MINIMIZING THE DAMAGE. Ok, let’s proceed. For easy reading (not to bother you to go back and forth between Part 2 and Part 3), I rewrite herein Equation 1 as follows:
And to make our discussion (thus our derivation) simpler, let’s derive only the numerator of Equation 1 above (it’s the ‘story telling’ that is important, not the equations, remember?), that is:
Let’s call the above as Equation 6. As I mentioned in Part 2, we are going to derive Equation 6 from Equation 4. Again, for easy reading, I rewrite Equation 4 herein as:
Looking at both Equations 5 and 4, even ‘secondary school’ math will say that Equation 5 can be inserted into Equation 4 to give:
Now, Equation 5 is the curvature caused by the ‘actual’ loading (don’t worry about this technicality). Similarly, for the curvature caused by the ‘virtual’ loading, the corresponding moment, m can be given as:
One more time, even ‘secondary school’ math will say that Equation 8 can be inserted into Equation 7 to give:
Walla! Didn’t I promise you the derivation of Equation 6 from Equation 4, did I? I just did exactly that. Ok, we have finished with anything that is relevant as far as beam is concerned. Next, we are going to discuss the analysis of truss and this is actually when the true fun started. Let me put herein the picture of a truss, in case you forget how it looks.
Figure 2: A truss system
In the upcoming discussion you will witness the true Guessing game in action, how the pattern ultimately becomes obvious, how math is the sole enabler and finally the true appreciation towards those who have invented all these; Euler and Lagrange hence the prevalence of the Wanabe concept. The whole idea that underlies this writing will be jammed, nailed and finalized in the next few paragraphs, so you just got to stay awake for a bit longer. You have gone this far, you have survived the torment of Part 1 and Part 2, let’s finish with what we have started, let us be enlightened once and for all!!! Aja! Aja! Fighting!
To analyze a truss system, despite its ‘formidable’ look (refer to Figure 2 above), basically one just needs to fill-up the following form:
The orthodox way of teaching this is simply to tell the students how to get A, E, L, F, u, H and F’ and then fill up this form. You might wonder “That’s it? This easy?” I would reply “Yes, that’s it, this easy”. And, at the end of the semester, the student will repeat what has been told and SCORE! That’s it, this easy! BUT I DO IT DIFFERENTLY.
Ok follow me carefully. When it comes to this stage, I would tell the students, “Let’s play a game of guessing”. “Do not write and do not even look at the book, let’s use our mental strength” I would say. “Identify the similarities and differences between Table 1 and Equation 1”. Let me give you readers both of them, side by side to ease your reading.
Excited, the students will say, “Fu is somehow corresponds to mM, uu corresponds to mm and AE corresponds to EI”. Even you can be surprised by how easy it is to the extent of asking “This is so obvious, why are you telling me all this?” It is obvious because it is easy. This is what I meant by 1+2 is 3, 10 +20 is 30 and so on (see Part 1).
You may not believe it, but every time the students get to this stage, they would be in vibrant mode (I would say), where they sit up straight, their eyes shine and they exhibit all the positive body gestures. They start to love this guessing game. Believe me you, if a student gets an answer right all by himself or herself, he or she would feel like ‘on top of the world”; these are the ‘highs’ I mentioned in Part 1 and we teachers should keep feeding these ‘highs’ until they become ‘knowledge addicts’.
And I usually would picking up on the momentum and ask them another interesting question, “Can anybody tell me, why in Table 1, there exists the variable L which is absent in Equation 1?” Many of them usually get it right to suspect it got to do with the integration in Equation 1. “It got to do with the integration” they would say. I then reply “You are right, but how, why?” By this time, they would all go crazy thinking about it, just like a group of children being given a ‘teka-teki’, they think and think and think, and many of them think out loud. And for me, I give myself a minute or two to watch and enjoy the atmosphere. Oh, how I love being a teacher.
And you readers, are you dying to know as well? If you are, I just like to say “Oh, how I love being a teacher”.
“The reason being”, the students would all wait in silence, waiting for me to continue, “The variable L exists because Fu/AE is constant”. Some of them would sigh “Of course” and they continue “if Fu/AE is constant, it can be taken out from the integration sign”. What they mean is:
If Fu/AE is constant, then:
It is simple math, it is SPM math and it is indeed easy. And what usually surprises them more is what I am about to tell you next. “There is more? Hmmm, I likeeeeee”, I wish you say this to yourself.
The constant nature of Fu/AE can also describe the difference in the physical condition between a truss and a beam. For example, “Why is it constant in truss and not in beam?” would be one way to question it. What I am about to tell you is the proof of how mathematics is needed to understand ‘things’. Let refer to Figure 6 below which are the common arrangement for beam and truss systems:
Figure 6: Domain of integration
You see, the integrations as in Equation 1 and Equation 9 should be carried out over the domains of the systems as graphically shown in Figure 6. As can be seen, the difference between the two arrangements is that, whilst there are forces acting on the domain of the beam, there are no forces acting on the domain of the truss. Realizing this, one can conclude in words that, “since there is nothing disturbing whatever in the domain of the truss, therefore whatever inside this domain will not change and remain constant, hence the reason for Fu/AE being constant”. Now, are you amazed? You should be amazed because of the following:
1. How we keep doing exactly the same thing as we did with our Childhood guessing game (Figure 3 of Part 2) where we compare side by side two ‘pictures’ which supposed to be similar but not quite
2. How the abstract nature of mathematics can actually describe a physical phenomenon. This is also the materialization of I what meant by “math can either kill the fun or make things fun”
3. How things can easily be explained if we just hold on to the principle that makes up the argument in the first place (i.e. 1+2 is 3, 10+20 is 30, 100+200 is 300 and so on).
4. How, whatever I have theorized in Part 1 are so true
I think it is better for me to adjourn our discussion for tonight. In Part 4, I will reveal to you how EA would correspond to EI and with that, I promise, will wrap up the whole discussion. I will also give some insights on what actually went wrong with our present educational system and how can we overcome this. Until then goodnight and please do not give up reading this….
(to be continued..)