Mechanics of Materials/Printable version


Mechanics of Materials

The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
https://en.wikibooks.org/wiki/Mechanics_of_Materials

Permission is granted to copy, distribute, and/or modify this document under the terms of the Creative Commons Attribution-ShareAlike 3.0 License.

Background

This Wikibook is to provide living content for an undergraduate course in mechanics of materials or strength of materials. The material here will eventually be of sufficient quality and interest for self-learning or prescribed study by instructors/faculty members. One objective is that the material here would facilitate in-class discussions, group projects, or problem-solving that would leverage the instructors' expertise to enhance learning outcomes. Another objective is to facilitate students, instructors, professionals, and interested users adding/curating content to accommodate learning styles that might benefit from a spectrum of insights coming from learners and teachers with diverse backgrounds. The course will draw on material openly available with inspiration from key relevant texts.[1][2][3][4][5][6][7][8][9] There is also a crowd-sourced mind map available (needs updating) on GitHub for this Wikibook on undergraduate mechanics of materials.


Introduction to Mechanics of Materials

Author(s): Aaron D. Mazzeo

1. Introduction to Mechanics of Materials

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Welcome to our undergraduate course on the mechanics of materials. The goal is to crowd-source all the information you would need to learn and understand the concepts taught in a standard university/college level course for mechanical or engineering undergraduates.

1.1 Background

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This Wikibook is to provide living content for an undergraduate course in mechanics of materials or strength of materials. The material here will eventually be of sufficient quality and interest for self-learning or prescribed study by instructors/faculty members. One objective is that the material here would facilitate in-class discussions, group projects, or problem-solving that would leverage the instructors' expertise to enhance learning outcomes. Another objective is to facilitate students, instructors, professionals, and interested users adding/curating content to accommodate learning styles that might benefit from a spectrum of insights coming from learners and teachers with diverse backgrounds. The course will draw on material openly available with inspiration from key relevant texts.[1][2][3][4][5][6][7][8][9] There is also a crowd-sourced mind map available (needs updating) on GitHub for this Wikibook on undergraduate mechanics of materials.

1.2 Why Mechanics of Materials?

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We are "Living in the Material World"

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George Harrison acknowledged we are "living in the material world." And while he was referring to his journey toward enlightenment through his music, materials and structures surround us. Even the earth on which we stand supports us, and its reaction force against the pull of gravity on us is a form of equilibrium.

Raw Materials

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Materials surround us and even in the natural environment, they are often structurally stacked before going from raw material to feedstock to mechanical structures.

Scalability

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What if there were a set of characteristics associated with solids that could help us determine whether a material will break? Well, these material properties exist and have standardized units and defined metrics for their characterization. These parameters also apply across multiples scales of length. In fact, the material properties go beyond only determining the likelihood of failure, yielding, or breaking, but they also help us determine how much deflection/displacement we should expect at a range of scales.

The study of structural mechanics and materials allows us to predict with reasonable accuracy how much a structure will deflect and its safety against catastrophic failure. Indeed, it is a revolutionary concept to test a small sample of material to characterize its elasticity and strength that we can then relate to anticipated levels of stress and strain within an engineered structure. As our ability to characterize materials and predict invisible distributions of stress within structures has improved, our confidence in the predictions and accepted factors have decreased by an order of magnitude [10].

Small Structures

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Microelectromechanical system (MEMS) developed at Sandia National Labs. The largest sprocket is less than 0.5 mm in diameter.

These materials properties are applicable to small structures at the micron scale (1/1000th of a millimeter). Engineering examples include a small micro chain created by engineers at Sandia National Labs. Material properties such as elastic modulus and strength would be relevant to potential yielding and deflection of these micro components. Nonetheless, characteristic material properties, such as those associated with silicone and silicon, can change from their bulk properties at larger sizes in significant fashion as the size of features approaches the micron scale[11][12].

Medium Structures

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Surfer on a surfboard charging a big wave off the coast of California at Mavericks in 2010. The surfboard is what we might consider a medium-sized object with material properties and geometry dictating whether the structure will support anticipated loads.

Examples of medium-sized structures might have the length, width, or thickness of a surfboard. In addition to needing support static loads that might be distributed or concentrated along its surfaces, a surfboard must also be able to withstand repetitive or cyclic loads that could lead to fatigue-based failure.

Large Structures

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The Roman Forum (Forum Romanum) contains examples of fantastically designed structures capable of supporting anticipated loads.

We might think about the loading -- forces and stresses -- acting on the columns of buildings in the Roman Forum. We might envision corrosion and its effects on decreasing the effective cross-sectional areas of structures and how the stresses could build up to lead to mechanical failure.

We can also think about the distribution of stress within the crust of the earth [13]. While we have not introduced a formal definition of stress, you can begin to image that as pieces (e.g., plates) exert forces on each other, there is a buildup of internal loading on the material within these structures, which constitutes stress.

Safety

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Content forthcoming

New Technologies

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Content forthcoming

Sustainability

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Content forthcoming

1.3 Available Formats

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Wikibook

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This Wikibook on Mechanics of Materials is the primary source for content and updates.

Mind Map

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We have started to create MechMatMindMap on GitHub, which has a mind map created in Freeplane. This mind map should mirror the content on this Wikibook. This mind map lags the content here, and we welcome updates from the community.

PDFs

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In the same GitHub project (MechMatMindMap), we will provide a PDF of this Mechanics of Materials Wikibook. The most current printable version/PDF of this Wikibook is available by clicking on the appropriate link on the Mechanics of Materials page.

PowerPoint Slides

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Content forthcoming

YouTube

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Content forthcoming

1.4 How to Use and Contribute Living Content

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We aim to become a living source of content for undergraduate courses in mechanics of materials for independent study and meaningful learning sessions with peers and expert instructors. We invite instructors and students to work together with crowd-sourced content freely available on GitHub and Wikimedia sites using open-source platforms. We hope to create, assemble, and curate high-quality content, examples, problems, and videos to allow university-level mastery in an engaging format.

Students and instructors may use the material curated and created for the teaching and studying of Mechanics of Materials. This material is to provide living content for an undergraduate course in mechanics of materials or strength of materials. The material here will eventually be of sufficient quality and interest for self-learning or prescribed study by instructors/faculty members. One objective is that the material here would facilitate in-class discussions, group projects, or problem-solving that would leverage the instructors' expertise to enhance learning outcomes. Another objective is to facilitate students, instructors, professionals, and interested users adding/curating content to accommodate learning styles that might benefit from a spectrum of insights coming from learners and teachers with diverse backgrounds.

We also hope that students and instructors will contribute original problems or examples in whatever their form to this Wikibook or the GitHub project (MechMatMindMap) to be formatted and incorporated into this body of work.

1.5 Invitation to Contribute

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We welcome contributions, as there is not a set cap on the number of illustrative examples/problems we include, nor do we care if the examples come from learners or instructors. We just ask that the problems be thoughtful and relevant. Contributors should also be open to a shuffling order or categorization of shown examples/problems.

1.6 Learning Objectives and Philosophy

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The hope is that these problems will become part of in-class discussions, small-group learning, projects, or personal study. For classes in which instructors give graded exams, we encourage instructors to create original problems for the students to solve independently. After an exam, the instructors could make the problems available for others to study and add to open-source sites like this Wikibook. This pattern of teaching, learning, and assessment limits ethical concerns/temptations, strengthens the validity of course outcomes, and decreases the anxiety of honest learners who feel they go into exams at a disadvantage to others planning to refer to available databases of solved problems or real-time tutoring services. For engineering students, it also discourages rote memorization and an "arms race" to create an increasing number of "solutions manuals" that inevitably become public. No person or system is flawless or perfect, especially given the stresses and constraints on instructors and learners. Nonetheless, this open-source Wikibook aims to enhance open and transparent learning.

1.7 Illustrative Examples/Problems

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Content Forthcoming

References

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  1. a b Crandall, Stephen H.; Dahl, Norman C.; Lardner, Thomas J. (1999). An Introduction to the Mechanics of Solids (2nd ed.). McGraw-Hill. Invalid <ref> tag; name "CDL_Book" defined multiple times with different content
  2. a b Eshbach, Ovid W., ed. (1936). Handbook of Engineering Fundamentals. John Wiley & Sons, Inc. Invalid <ref> tag; name "Eshbach" defined multiple times with different content
  3. a b Hibbeler, R. C. (2014). Mechanics of Materials. Prentice Hall. Invalid <ref> tag; name "Hibbeler_book" defined multiple times with different content
  4. a b Norton, Robert L. (2014). Machine Design: An Integrated Approach (5th ed.). Prentice Hall. Invalid <ref> tag; name "Norton_book" defined multiple times with different content
  5. a b Ashby, Michael F.; Jones, David R. H. (1980). Engineering Materials 1: An Introduction to their Properties and Applications. International Series on Materials Science and Technology. Pergamon Press, Inc. Invalid <ref> tag; name "AshbyJones_Book" defined multiple times with different content
  6. a b Juvinall, Robert C.; Marshek, Kurt M. (2012). Fundamentals of Machine Component Design (5th ed.). John Wiley & Sons, Inc. Invalid <ref> tag; name "Juvinall_Marshek_Book" defined multiple times with different content
  7. a b Fogiel, Max, ed. (1999). The Handbook of Mechanical Engineering. Research & Education Association. Invalid <ref> tag; name "REA_Handbook" defined multiple times with different content
  8. a b Ucker, Jr., John J.; Pennock, Gordon R.; Shigley, Joseph E. (2017). Theory of Machines and Mechanisms (5th ed.). Oxford University Press. Invalid <ref> tag; name "Ucker_Pennock_Shigley" defined multiple times with different content
  9. a b Shigley, Joseph Edward; Mitchell, Larry D. (1983). Mechanical Engineering Design (4th ed.). McGraw-Hill Book Company. Invalid <ref> tag; name "Shigley_Mitchell"" defined multiple times with different content
  10. Juvinall, Robert C.; Marshek, Kurt M. (2012). Fundamentals of Machine Component Design (5th ed.). John Wiley & Sons, Inc. pp. 273–274.
  11. Xu, Wenwei; Chahine, Nadeen; Sulchek, Todd (2011). "Extreme Hardening of PDMS Thin Films Due to High Compressive Strain and Confined Thickness". Langmuir. 27 (13): 8470–8477.
  12. Chen, Ming; Pethö, Laszlo; Sologubenko, Alla S.; Ma, Huan; Michler, Johann; Spolenak, Ralph; Wheeler, Jeffrey M. (5/2020). "Achieving micron-scale plasticity and theoretical strength in Silicon". Nature Communications. {{cite journal}}: Check date values in: |date= (help); Cite has empty unknown parameter: |1= (help)
  13. Zang, Arno; Stephansson, Ove (2010). Stress Field of the Earth’s Crust. Springer, Dordrecht.


Physical Units and Standards

2. Physical Units and Standards

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Infants know when they want "more" sleep, more milk, more cleaning, and more carrying. They might not attach units to their wants -- hours for sleep, milliliters for milk, diapers per day for cleaning, calories/joules burned by those carrying them -- but they have inherent methods of communicating their needs. As we grow, we become aware of our height (feet, meters) and weight/mass (pounds, kilograms). Indeed, we begin to quantify what we see in our surroundings, form comparisons and attach units to these comparisons. Adults will think about the size of their homes (square feet), their salaries (dollars/year), and time spent in school (years). While comparison may be the thief of joy, we can acknowledge that physical units and standards are necessary when considering the currency of compensation[1].

We involuntarily think and make comparisons when we make purchases or get paid with money, when we specify how much food we will consume and measure quantities when we cook, when we travel/commute over distances, and when we subject ourselves to professional or personal scrutiny regarding our own health. Some of the same metrics/physical units that we use in our everyday life are relevant to mechanical structures and material properties. In fact, when we think of ourselves as mechanical machines, we become aware that our brains are somehow balancing and automatically keeping track of units, loads, mechanics, and materials without a pencil or calculator.

Fun Questions

1 Question: Which of the listed items below is a unit of mass?

slug
newton
metre
joule

2 Question: How much power does a person spend approximately "at rest?"

10 watts
100 watts
1 kW
10 kW

2.1 When Units Get Mixed Up

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In the classic science fiction movie Back to the Future, the DeLorean requires 1.21 Jigowatts of power for its flux capacitor to enable time travel. While the pronunciation of "Jigowatt" might be appropriate[2], the correct spelling would be "gigawatt." The prefix "giga" representing 1 billion or 109. In mechanical engineering, we often use giga when discussing elastic moduli (e.g., the elastic modulus of aluminum is approximately 70 gigapascals (GPa)). We will often use the prefix "mega" representing 1 million or 106 when discussing the strength of materials (e.g., the strength of different metal alloys on the order of 100s of MPa). We can avoid mix-ups by keeping in mind that elastic moduli are often in terms of GPa, while units of strength and stress are often MPa.

 
Artist's rendition of the Mars Climate Orbiter.

A costly mix-up with units occurred in 1998 with the Mars Climate Orbiter. This orbiter was to monitor dust and water vapor, take daily pictures, and help plot an evolutionary map of climate change. Unfortunately, the orbiter did not make it into orbit, as there was a "metric mix-up." The ground software by Lockheed Martin sent calculated trajectories in imperial/US customary units (pound-seconds), but NASA was expecting the results in metric units (Newton-seconds). The thrusters provided an inappropriate amount of force, and the orbiter was lost in the Martian atmosphere. While these types of robotic missions are low cost, the cost for the spacecraft development and launch exceeded $250 million. [3][4][5][6]

There is a reason why many textbooks in physics and engineering dedicate a section to units and physical standards. Units add meaning and help us conceptualize the quantity, size, and magnitude of the quantities of interest. While it is doubtful that one would lose credit for not including units in their scratchwork, keeping track of units prevents embarrassing and costly mistakes. As a courtesy to others, using an appropriate prefix also helps convey understanding of the appropriate measurements, just like the use of an appropriate number of significant figures.

2.2 Units as a Tool for Solving Problems

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We have already emphasized how units help avoid costly mistakes when solving engineering problems. Keeping track of units or manipulating units can also help us solve problems.

Example Problem to Demonstrate How Units Might Help Us Solve a Problem

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At this point, we have not discussed the concept of axial stress, but we might keep track of units to solve a problem involving the calculation of stress. For example, Jumbo the Elephant asks you to calculate the axial stress in MPa for a bar with an axial load of 10 kN axial load to a bar with a cross-section of 1 cm x 5 cm. Go ahead and attempt to solve this question before clicking on the "Answer" below.

Answer

We recall that the units associated with MPa are those of pressure meaning force/area. We can then think in terms of the metric system and the units of relevance to our problem: N and m. We can write N/m2 and then perform the calculations shown in the animated GIF.

 
Animated GIF showing how to use units to calculate stress

We do not need to know that axial stress equals force divided by area. We can guess at the correct answer of 20 MPa by arranging the units appropriately.

Another Example Problem to Demonstrate How Units Might Help Us Solve a Problem

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This question explores how to use units to solve a question relating mechanical, rotational power to electrical power.

Question Relating Rotational Power to Electrical Power and Current

You are a new engineer working at Company XYZ, and your supervisor asks you to calculate how much electrical current you will need to power a motor connected to 120 V outlet, which delivers 1 N m of torque at a speed of 100 rpm.

1.21 GW.
9 A.
900 mA.
90 mA

2.3 Sources of Physical Units

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There are many sources of physical units, which describe material properties or methods for converting between units. Textbooks and websites will often share common material properties. A commercial website such as MatWeb will have both overviews for classes of materials (e.g., an overview of silicone materials) and data for specific grades of materials. The Wikipedia page on Young's modulus also lists approximate values for common materials. Table 2.1 shows some sources for material properties. A mechanical engineer should memorize the following: the elastic/Young's modulus of aluminum (70 GPa), the elastic modulus of steel (~3 X that of aluminum), the density of aluminum (2,700 kg/m3), the density of steel (7,900 kg/m3), and the dimensionless Poisson's ratio of many metal alloys (~0.3).

2.1 Table for Sources of Mechanical Material Properties
Type of Mechanical Property Websites
Density MatWeb, Wikipedia, Wolfram Alpha, Engineering Tool Box
Elastic Modulus MatWeb, Wikipedia, Wolfram Alpha, Engineering Tool Box
Yield Strength MatWeb, Wikipedia, Engineering Tool Box
Ultimate Tensile Strength MatWeb, Wikipedia, Engineering Tool Box
Poisson's Ratio MatWeb, Wikipedia, Wolfram Alpha, Engineering Tool Box

For conversion between units, we can use multiplication and fractions as previously demonstrated. Other online tools that keep track of units automatically when making calculations include Wolfram Alpha , Google Search, and DuckDuckGo. Common conversions in mechanical engineering include those in Table 2.2

2.2 Table for Common Conversions in Mechanics
Base Unit Converted Unit
1 m 3.28 ft
1 ft 0.31 m
1 MPa 145 psi
1 ksi (1,000 psi) 6.89 MPa

2.4 Learning Problems

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References

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  1. O'Toole, Garson (2/6/2021). "Comparison Is the Thief of Joy". Quote Investigator. Retrieved 10/21/2021. {{cite web}}: Check date values in: |access-date= and |date= (help)
  2. "definition and pronunciation of gigawatt". Merriam-Webster. Retrieved 2021-10-21.
  3. "Mars Climate Orbiter". Retrieved 2021-10-22.
  4. "Mars Polar Lander/Deep Space 2". Retrieved 2021-10-22.
  5. "Mars Climate Orbiter". Retrieved 2021-10-22.
  6. "Some Famous Unit Conversion Errors" (PDF). Retrieved 2021-10-22.



Static Equilibrium

3. Static Equilibrium

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3.1 Free Body Diagrams

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3.2 Static Form of Newton's Second Law of Motion

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3.3 Static Equilibrium of Trusses

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3.4 Static Equilibrium of Beams

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