SoM_test0

Stress and Strain

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Learning Objectives

After completing this chapter you should be able to:

Define normal and shear stress and strain
Discuss the relationship between design stress, yield stress and ultimate stress
Design members under tension, compression and shear loads
Determine members deformation under tension and compression

Mechanical Stress

This section discusses the effects of mechanical loads (forces) acting on members. Chapter 3 will cover the effects of thermal loads (thermal expansion).

Normal, tensile and compressive stresses

Tension or compression in a member generate normal stresses; they are called “normal” because the cross-section that resists the load is perpendicular (normal) to the direction of the applied forces. Both tensile and compressive stresses are calculated with: If a member has a variable cross-section, the area that must be used in calculations is the minimum cross-sectional area; this will give you the maximum stress in the member, which ultimately will govern the design.

Shear stresses

In shear the cross-section area that resists the load is parallel with the direction of applied forces. In addition to that, when estimating the shear area you must factor in how many cross-sections contribute to the overall strength of the assembly. For instance, if you consider the pin of a door hinge as subjected to a shear load, you have to count how many cross-sections resist the load. The formula for calculating the shear stress is the same: In a punching operation the area that resists the shear is in the shape of a cylinder for a round hole (think of a cookie cutter). Therefore the area in shear will be found from multiplying the circumference of the shape by the thickness of the plate. Hint: when looking at textbook figures you will observe that two forces are indicated. This does not mean that the force you use in the formula is (2 × Force P), but simply indicates that one is the Action force and the second one is the Reaction.

Strain and Modulus of Elasticity

Normal Strain

A member in tension or compression will elastically deform proportional with, among other parameters, the original length. Strain, also called unit deformation, is a non-dimensional parameter expressed as: If you choose to use a negative value for compression strain (reduction in length) then you must also express the equivalent compression stress as a negative value.

Modulus of Elasticity

The stress – strain curve is generated from the tensile test. Over the elastic region of the graph the deformation is direct proportional with the load. Dividing the load by the cross-section area (constant) and the deformation by the original length (constant) leads to a graphical representation of Strain vs. Stress. The constant ratio of stress and strain is Young's Modulus or Elastic Modulus, a property of each material.

Reasonable Answers

When solving normal stress - strain problems, especially in the SI system, you may be able to judge if your answers are reasonable or not. Consider the following example:

A 1 m long, 20 mm diameter, A 36 Carbon Steel bar (Yield strength 248 MPa, Ultimate strength 400 MPa, Elastic Modulus =200 GPa) suspends a 6 metric tons load. Evaluate the stress and the strain in the bar.

Bibliography

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This is where you can add appendices or other back matter.

Introduction and Units

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Learning Objectives

Become familiar with study procedure and expectations
Perform units conversions within SI and US Customary systems

Study Procedure

Delivery of this course is based on the Applied Strength of Materials for Engineering Technology textbook, by Dr. Barry Dupen. You have access to the textbook in electronic format and/or hard copy. For best results I suggest following this sequence:

Before commencing a new chapter, read the corresponding theory from the textbook and review the example problems. Some topics were already covered in Applied Physics but you will benefit from a brief review. Moving to a new topic will be announced in class.
Classroom lectures:
- Instructor will review the theoretical concepts and answer questions
- Instructor will demonstrate solving selected problems. Hand notes of these solutions will be available through D2L.
- Students will solve assigned problems in small groups, with guidance from instructor
Individual work
- Students will solve on their own assigned problems to self-evaluate their understanding.
- Instructor will provide guidance and feedback during posted office hours or Tutorial Sessions
Evaluation – selected topics will be assessed through quizzes, announced in class

Strength of Materials is a “methodical” discipline. This means that it deals in general with standard/classical questions that usually have an established method of solving them. When solving problems students often follow steps and procedures that were previously demonstrated in class or in the textbook. These approaches are logical and never students would be expected to memorize them. However, it is important for students to practice solving questions on their own since this will help them see patterns in questions, provide them with problems solving experience and help them complete the exercise in the allotted time. It is recommended to work at home between 2 and 3 hours for each hour of class lecture. This effort will be different for each student. For best efficiency consider attending the scheduled tutorials where you can reach to your instructor for help.

Units and Conversions

Like in many other engineering disciplines calculations may be performed in both systems of units, US Customary and SI. While Canada has officially adopted the SI (metric) system in 1970, economic cooperation with US companies requires engineering graduates to be fluent in both systems. Some computational software that you will use may be available only in US Customary units, being developed in US, and mostly for American users. It is therefore imperative to be able to complete calculations in both systems of units and to be able to convert between systems. When solving problems, if the data is given in SI units, complete the solution in SI units. Similarly for US Customary units; there is not need to switch the system of units. When performing conversions please observe the following.

Conversions within the SI system of units

In the metric system prefixes are added to base and derived units to form names and symbols that are multiples of SI units. A list of commonly used SI prefixes is given in Appendix A. For a complete list see Figure 1.

Fig. 1 - SI System Prefixes

SI Examples:

1. Convert 0.2 km to cm

When performing SI conversions it is easy to see if your answer is reasonable or not. For instance if you move from a large unit (kilo) to a smaller one (centi), the resulting value should be greater.
Looking at Fig. 1, you may also consider moving the decimal point to the right, three steps from Kilo to base and two more steps from base to your final answer. This is an alternative approach to performing SI conversions.

2. Convert 50 000 cW to kW

Note that some units may be presented with a less commonly used prefixes. For instance, while “centimeter” is frequently used, “centiwatts” not so much. However, you should be able to identify the prefix and the unit it applies to.

3. Convert 300 000 cm³ to dam³

You may look at this conversion as follows:

Pay extra attention when using powers, as in volume or area conversions.

Conversions within the US Customary system of units

For the purpose of this course most of the US Customary conversions will deal with linear dimensions. The conversion factors that we use are presented in Appendix A. At some point you are expected to remember the most used factors such as 1 ft = 12 in or 1 yd = 3 ft.

US Customary Examples:

4. Convert 1.2 yards to inches 5. Convert 2 square feet to square inches

Exercises

The hoop stress in a pressure vessel is calculated with the formula where p is the design pressure, d_i is the inside diameter and t is the wall thickness.
1. If p = 4450 kPa, d_i = 1.8 m and t = 20 mm, determine the hoop stress in the wall, in MPa.
2. If p = 645 psi, d_i = 6 feet and t = ¾ in, determine the hoop stress in the wall, in ksi.
To determine the dead load on a foundation you are required to estimate the weigh of a spherical tank (V=4/3 πr³), full with a liquid of given density. Tank mass is negligible compared to the mass of the product. Determine its weight based on the following:
1. Diameter = 200 cm, density = 1.12 g/cm³. Answer in N.
2. Diameter = 80 in., density = 70 lb/ft³. Answer in lb.

Poisson's Ratio and Thermal Expansion

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Cover Image

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"HÜTTE" Engineering Handbook and pocket slide rule mechanical calculator

Pressure Vessels

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Bolted and Welded Joints

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Ch1Fig1

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SI Prefixes from Technical Physics

Ch1Fig2

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Ch1Fig4

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Ch1Fig6

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Ch1Fig7

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Ch1Fig8

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Ch1Fig3

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Ch1Fig5

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Ch2Fig1

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Ch2Fig2

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Ch2Fig3

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Ch2Mott1

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Ch2Fig4

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Main Body

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Authors

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Cover

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About

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Buy

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Access Denied

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Book Information

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Acknowledgements

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The adoption of this textbook and the further development of study materials for our students was made possible with support from the Open Education Working Group at BCIT. I would like to use this opportunity to thank Dr. Barry Dupen for sharing his engineering knowledge and work with the rest of us and for making this book freely available to all students. Special appreciation goes to my colleagues:

Mr. Serhat Beyenir – for encouraging me to develop this work, for providing feedback each step of the way and for help when I needed it
Dr. Sanja Boskovic – for inspiring us to do more and to keep learning but also for reviewing the work

To all students who provided me with feedback, thank you!

Editors

The following students of the Power and Process Engineering 2^nd year of studies brought suggestions and contributions to this book: 2017 - 2018 P&P 2^nd Year

John Smith	Mark Twain	Brad Pitt	Isaac Asimov
Jane Doe	Dwayne Johnson	George Clooney	Marie Curie

Preface

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Objectives

Strength of Materials is a topic taught in all Mechanical Engineering related programs and as such is part of the Power and Process Engineering curriculum in British Columbia Institute of Technology. This course is a continuation to the principles of Statics and Applied Mechanics taught to second year Power and Process students, in Level 1. Successful completion of this course will also help Power Engineering (PE) graduates in advancing their careers as the topic is part of their 2^nd and 1^st class examinations curriculum. The focus in this subject will be mostly on the “Applied” part of Strength of Materials while still covering theoretical aspects. The problem sets are developed or selected in such way that they are relevant to Power Engineers.

Textbook Format

In British Columbia Institute of Technology we have delivered this course following the “Applied Strength of Materials” textbook written by Robert L. Mott. In my years of teaching the subject, alongside with the students, we have used the 4th and 5th edition of the textbook. I often noticed that students did not have the textbook or they had “offshore” printed editions amended with footnotes like “not for use in North America”. When the cost of such a textbook was about 15% of the original one, I found it hard to reason with them. For a few years the textbook was discontinued by the publisher and our institute managed to secure internal publishing rights on a cost-recovery basis. Things changed again when the textbook was upgraded to the 6^th Edition, undoubtedly improved but also more expensive. And this brings us to the current textbook… A fact well know to students and acknowledged by educators is that college and university textbooks are expensive and not having access to one may affect student’s success. Typically, these books establish the knowledge foundation of lucrative careers and are also part of the technical library of many working professionals. They are undoubtedly valuable but also expensive. It came as a surprise for me to find a first-rate Strength of Materials textbook that covers all the topics that our students need to know (and then some) and... free. The textbook that we follow is written by Dr. Barry Dupen, Associate Professor in the Mechanical Engineering Technology Department of Indiana University – Purdue University Fort Wayne. The book is licensed under Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) and is available for download here: http://opus.ipfw.edu/mcetid_facpubs/55/ It was Dr. Dupen’s intention to not include end of the chapter questions as he provides new sets with every course delivery. This gave us the opportunity and challenge to create our own problem sets. When creating our questions we had Power Engineering in mind and tried to make them relevant to their profession. At the same time, when we felt that further clarifications or procedures would bring value to the textbook and help to our students, we added them.

Properties of Areas

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Torsion in Round Shafts

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Beam Reactions, Shear Diagrams, and Moment Diagrams

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Stresses in Beams

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Beam Design

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SoM_test0

Stress and Strain

Learning Objectives

Mechanical Stress

Strain and Modulus of Elasticity

Reasonable Answers

Bibliography

Introduction and Units

Learning Objectives

Study Procedure

Units and Conversions

Exercises

Poisson's Ratio and Thermal Expansion

Cover Image

Pressure Vessels

Bolted and Welded Joints

Ch1Fig1

Ch1Fig2

Ch1Fig4

Ch1Fig6

Ch1Fig7

Ch1Fig8

Ch1Fig3

Ch1Fig5

Ch2Fig1

Ch2Fig2

Ch2Fig3

Ch2Mott1

Ch2Fig4

Main Body

Authors

Cover

Table of Contents

About

Buy

Access Denied

Book Information

Acknowledgements

Editors

Preface

Objectives

Textbook Format

Properties of Areas

Torsion in Round Shafts

Beam Reactions, Shear Diagrams, and Moment Diagrams

Stresses in Beams

Beam Design