Introduction

Essential Mechanics: Statics and Strength of Materials with MATLAB and Octave

Formal Description:

Essential Mechanics - Statics and Strength of Materials with MATLAB and Octave combines two core engineering science courses - “Statics” and “Strength of Materials” - in mechanical, civil, and aerospace engineering. It weaves together various essential topics from Statics and Strength of Materials to allow discussing structural design from the very beginning. The traditional content of these courses are reordered to make it convenient to cover rigid body equilibrium and extend it to deformable body mechanics.The e-book covers the most useful topics from both courses with computational support through MATLAB/Octave. The traditional approach for engineering content is emphasized and is rigorously supported through graphics and analysis. Prior knowledge of MATLAB is not necessary. Instructions for its use in context is provided and explained. It takes advantage of the numerical, symbolic, and graphical capability of MATLAB for effective problem solving. This computational ability provides a natural procedure for What if? exploration that is important for design. The book also emphasizes graphics to understand, learn, and explore design. The idea for this book, the organization, and the flow of content is original and new. The integration of computation, and the marriage of analytical and computational skills is a new valuable experience provided by this e-book.

Essential Mechanics - Statics and Strength of Materials with MATLAB and Octave, can be used in the earliest core engineering science course in mechanical, civil, and aerospace engineering. It is probably a required service course for most other engineering disciplines. In this book it represents an integration of topics from Statics and Strength of Materials to make it useful for design. If you have already graduated these courses you will have been introduced to the topics through through adjoining courses probably titled “Statics” and “Strength of Mechanics”. Many would have had a single course with Strength of Materials content following Statics in sequence.

Statics is about calculating loads (forces and couples) assuming a rigid structure. Strength of materials assumes a deformable structure and uses the loads determined in Statics to study the deformation of structure as well as the stress capacity of the structure. A significant portion of the effort in the Strength of Materials course is applying Statics initially to the problem.

For engineering design, which attempts to design structures that do not fail structurally during their life time, the critical knowledge is acquired through the Strength of Materials course which depends significantly on Statics. The course on Statics merely lays the preliminary calculation of the loads that the individual parts of the structure must carry during the operation. With just “Statics” the students cannot design as they cannot determine if the product or device will fail. “Statics” is just an special applied “Physics” course dealing with equilibrium. Students are forced to recollect their “Statics” when they continue with “Mechanics of Materials” in a two-course sequence. This is a challenge in today's learning environment. It is also a waste of opportunity in design instruction. Design is complete when analyzed for failure. The failure is related to the stresses that the structure must endure. This book attempts to combine and thread the calculations of the loads with the stresses to ensure that the structure will not fail and the approach should make intuitive sense.

Another advantage of proceeding in this direction is to make available the extra course to continue with advanced applications of mechanics that could involve numerical approaches to structural design. These are essential for real world structural design - using software that deliver solutions through finite and other discrete element methods.

Let us consider a simple example to illustrate this connection between Statics and Strength of Materials in design:

Figure. A simple design problem

In this problem you seek the answer to the question : Will he be rudely awakened or the rope will hold the weight?

You can reformulate the question : Will the rope fail?

“Statics” will solve for the load carried by the rope.

“Statics” cannot answer the question by itself.

“Strength of Materials” will use this load to calculate if the stress in the rope will cause failure.

The question can only be answered by the combination of “Statics” and “Strength of Materials”

Calculating the stress is such a simple extension that it is a shame to postpone it to another course and make the student wait to understand design.

In summary - Statics ignores the deformation of the structure. In Strength of Materials we allow the structure to deform. A design goal is to ensure the deformation is elastic, that it the structure returns to the undeformed state once the loads are removed. The actual material properties will determine this behavior. In this book we will combine this analysis.

How to calculate the answer?

The solution in most of structural engineering problems is to use the Laws of Physics – in this case the simplified Newton’s Law for stationary objects – called an equilibrium equation.

We solve this problem by:

Simplifying the figure by making assumptions

- this is a physical model representing the actual problem

Identifying the information that is known

- like the angle the rope makes with the horizontal line

Solving for the unknowns (or missing information) using the natural law or Newton’s Law– in this case the force in the rope - using algebra and/or calculus and/or geometry - this is the mathematical model

These are standard procedure followed by all text books in Statics

Calculating the stress in the rope and checking that it will not rupture, and sometimes calculate the sag in the hammock

This is the the application of Strength of Materials

This book solves this problem immediately instead of waiting for another course

Checking that the number makes sense

- this comes must become a part of your engineering instinct - through practice and experience

Additionally in this book we use the computational tool MATLAB/Octave to solve the problem.

The required MATLAB code is integrated into the book. It is not a pre-requisite.

To be effective, you should do all of the above in a

Simple

Consistent

Effective, and very importantly through an

Easily remembered process

Why MATLAB?

Many times engineering calculations can be extensive. Many times the graphical solution help understand the solution and develop an instinct. Often there are several unknown that must be simultaneously solved. Designing is about answering What If? questions. Programming the calculations is a smart way to deal with design. If things go well then programming reinforces engineering knowledge. Programming can be learned by using MATLAB as a highly efficient calculator that can effortlessly combine symbolic, numeric, graphical, and textual computation in an integrated manner.

The MATLAB code in this book will also run on Octave except in specific cases because of syntax and parsing issues. Sometimes it is just a matter of a single line that must be commented out or included or changed. The code uses symbolic variables in most exercises and you will need to include the corresponding toolbox in Octave. The symbolic analysis in Octave is parsed to the Python module sympy. Knowledge of Python is not necessary. The Octave code is verified to run for at least one example in each section where MATLAB code appears. The Octave used in the book is GNU Octave, version 5.1.0. Remember it is the same file with the MATLAB code with suggested changes if any. For this book programming in Octave is same as programming in MATLAB. You will type in the MATLAB code in the Octave editor. The MATLAB code runs on Version 2015a. It should run on later versions without problems. This was tested with some random examples from the book. The MATLAB version was 2019a.

Important Note about Materials: Till recently the material meant metals. Today, design is all about new engineered materials. Most new aircraft designs use composite materials, a mixture of metal and plastic that is light and can handle all the stresses that the aluminum alloys can. As a benefit - it is lighter than the metal it will replace. Same performance but at lower weight and hence more efficient. Recently Ford announced that its new trucks will be made of Aluminum instead of steel and thereby saving over 600 lb in weight. This also improves engine performance because of reduced weight. Another extreme example is synthetic biology where you can design organisms through software. Engineering new materials is an exciting new field of rewarding pursuit.Computational multi-physics is developing new models for additive manufacturing. Nano-materials are being embedded in new concrete structures. All of these require new areas today rely of computational skills.

To the Users:

Please consider this as a WARNING. The best way to view this e-book is through a laptop or a desktop screen with at least 90 columns of text data. This will prevent inadvertent code wrapping. Both MATLAB and Octave are not kind when this event occurs. They will flag errors that may take time and effort to fix, apart from being very frustrating. To avoid parsing errors both MATLAB and Octave require to be informed about continuation of code on the next line. They only allow it certain circumstances.

To the Student:

Thank you for for picking up this e-book. I really hope you find this useful. I am sure you might come across some errors and if you find them kudos - pat yourself on the back since you understand the material you are trying to learn. Initially you might find it easy to copy and run these codes and maybe think “ this author dude certainly knows his MATLAB”. If you always did it you will continue to just confirm my expertise. However, if you played with the code, improved it, made it do additional interesting things in the same context, used it to explore more design issues, then Thank You. I hope you will share you work with others. I wrote this book specially for you. Please drop me a line about what can make it more useful. I will certainly use it in my next book.

To the Instructor:

Thank you for picking up the book, recommending it, or even adopting it. I am hopeful it works for you. You will find that the analysis in this book does not lack substance compared to the other texts out there. It does not discount or shorten the development of the technical topic or the idea behind it. You will see that the development is not weak in graphics either. It might appear that there is a lot going on all the time with the code. Students may be able to run with it if you start it for them. Here, I encourage you to consider coding is just copy, paste, edit, and extend. The words Essential Mechanics at the beginning of the tile is to keep the text as small as possible by focusing on important and necessary topics. I will look forward to your comments. Please also consider assigning students to come up with their problems and share them with all of us. This e-book is released under Creative Commons licensing. You can add, subtract, and change topics and make it your own.

P. Venkataraman

January 2020.

Rochester, NY, USA.