book

Statics For Dummies

Name: Statics For Dummies
Author: PE James H. Allen III, PhD
ISBN: 9780470598948

by PE James H. Allen III, PhD

September 2010

Beginner to intermediate

379 pages

8h 43m

English

For Dummies

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Copyright
About the Author
Author's Acknowledgments
Publisher's Acknowledgments
Introduction
About This BookConventions Used in This BookWhat You're Not to ReadFoolish AssumptionsHow This Book Is OrganizedPart I: Setting the Stage for StaticsPart II: Your Statics Foundation: Vector BasicsPart III: Forces and Moments as VectorsPart IV: A Picture Is Worth a Thousand Words (Or At Least a Few Equations): Free-Body DiagramsPart V: A Question of Balance: EquilibriumPart VI: Statics in ActionPart VII: The Part of TensIcons Used in This BookWhere to Go from Here
I. Setting the Stage for Statics
1. Using Statics to Describe the World around You
1.1. What Mechanics Is All About1.2. Putting Vectors to Work1.2.1. Peeking at a few vector types1.2.2. Understanding some purposes of vectors1.3. Defining Actions in Statics1.4. Sketching the World around You: Free-Body Diagrams1.5. Unveiling the Concept of Equilibrium1.6. Applying Statics to the Real World
2. A Quick Mathematics Refresher
2.1. Keeping Things Accurate and Determining What's Significant2.2. Nomenclature with Little Superscripts: Using Scientific and Exponential Notation2.3. Recalling Some Basic Algebra2.3.1. Hitting the slopes of functions and lines2.3.2. Rearranging equations to solve for unknown variables2.3.3. Sigma notation2.4. Getting into Shapes with Basic Geometry and Trigonometry2.4.1. Getting a handle on important geometry concepts2.4.1.1. Computing angles inside polygons2.4.1.2. Constructing angles created from line segments2.4.1.3. Double-checking angles with degrees and radians2.4.1.4. Recalling the Pythagorean theorem2.4.2. Tackling the three basic identities of trigonometry2.5. Brushing Up on Basic Calculus2.5.1. The power rule: Differentiation and integration of polynomials2.5.1.1. Basic differentiation and tangents to functions2.5.1.2. Basic integration2.5.2. Using calculus to define local maximum and minimum values
3. Working with Unit Systems and Constants
3.1. Measuring Up in Statics3.1.1. The metric system3.1.1.1. Converting to larger and smaller metric units3.1.1.2. Making multiple conversions in one equation3.1.2. U.S. customary units3.1.3. The kip: One crazy exception3.1.4. Never the twain shall meet: Avoiding mixing unit systems3.2. Looking at Units of Measure and Constants Used in Statics3.2.1. Constants worth noting3.2.2. Three common statics units for everyday life3.2.3. All the derived units you'll ever need
II. Your Statics Foundation: Vector Basics

4. Viewing the World through Vectors
4.1. Defining a Vector4.1.1. Understanding the difference between scalars and vectors4.1.2. Taking a closer look at vectors4.1.3. Applying vector basics4.2. Drawing a Vector's Portrait4.2.1. The single-headed arrow approach4.2.2. A two-headed monster: The double-headed arrow approach4.3. Exploring Different Types of Vectors4.3.1. Fixed vector4.3.2. Free vector4.3.3. Sliding vector
5. Using Vectors to Better Define Direction
5.1. Taking Direction from the Cartesian Coordinate System5.2. As a Crow Flies: Using Position Vectors to Determine Direction5.2.1. Describing direction in detail5.2.2. Moving from Point A to Point B and back again5.3. A First Glance at Determining a Vector's Magnitude5.3.1. Recognizing the notation for magnitude5.3.2. Computing the magnitude of a position vector: Pythagoras to the rescue!5.3.2.1. The two-dimensional Pythagorean theorem5.3.2.2. Going vertical: The Pythagorean theorem in three dimensions5.3.2.3. Putting Pythagoras to work5.4. Unit Vectors Tell Direction, Too!5.4.1. Cartesian-vector notation5.4.2. Using unit vectors to create position vectors5.4.2.1. Position vectors can be Cartesian too!5.4.2.2. Relationship between a vector, its magnitude, and its direction5.5. Creating Unit Vectors from Scratch5.5.1. Shrinking down position vectors5.5.2. Using angular data and direction cosines5.5.3. Utilizing proportions and similar triangles5.5.4. Knowing which technique to use
6. Vector Mathematics and Identities
6.1. Performing Basic Vector Operations6.1.1. Adding vectors6.1.2. Subtracting vectors6.1.3. Moving vectors head to tail6.2. What Do You Mean I Can't Multiply Vectors? Creating Products6.2.1. Dot products6.2.2. Cross products6.3. Useful Vector Operation Identities
7. Turning Multiple Vectors into a Single Vector Resultant
7.1. Getting a Handle on Resultant Vectors7.1.1. Depicting a resultant vector7.1.2. Principles of resultants7.1.3. Calculating resultant magnitude and direction7.2. Using Graphical Techniques to Construct Resultants7.3. Using Geometric Methods to Construct Resultants: The Parallelogram Method7.3.1. Useful geometric relationships7.3.1.1. Law of cosines7.3.1.2. Law of sines7.3.2. The parallelogram method7.4. Using Vector Methods to Compute Resultants7.4.1. Using vector addition7.4.2. Calculating the direction of the vector resultant
8. Breaking Down a Vector into Components
8.1. Defining a Vector Component8.2. Resolving a Vector into Cartesian and Non-Cartesian Components8.2.1. Using Cartesian concepts to calculate Cartesian components8.2.1.1. Figuring component magnitudes8.2.1.2. Using scalar magnitudes and directions to create vector components8.2.1.3. Computing vector components in three dimensions8.2.2. Determining components on a non-Cartesian orientation8.2.3. Calculating non-Cartesian components of two-dimensional vectors8.2.3.1. Using the parallelogram method8.2.3.2. Using Cartesian techniques to find non-Cartesian components
III. Forces and Moments as Vectors
9. Applying Concentrated Forces and External Point Loads
9.1. Comparing Internal and External Forces and Rigid and Deformable Bodies9.2. Exploring External Concentrated Forces9.2.1. Normal forces from contact9.2.2. Friction9.2.3. Concentrated loads9.3. Revealing the Unseen with Concentrated Internal Loads9.3.1. Forces in ropes and cables9.3.2. Forces in springs9.3.2.1. Stretch in springs9.3.2.2. Spring constants9.4. Surveying Self Weight as an External Load Value9.4.1. Getting specific on specific gravity and self weight properties9.4.2. Working with lumped mass calculations9.5. Introducing the Principle of Transmissibility
10. Spreading It Out: Understanding Distributed Loads
10.1. Getting a Handle on Some Distributed Load Vocab10.2. Take a (Distributed) Load Off: Types of Distributed Loads10.2.1. Distributed forces10.2.2. Surface loads (pressures)10.2.3. Volumetric loads10.3. Calculating the Resultant of a Distributed Load10.3.1. Uniform and linearly varying forces10.3.1.1. Zero order (uniform) distributions10.3.1.2. First order (linearly varying) distributions10.3.1.3. Other two dimensional distributions10.3.2. Surface loads and pressures in multiple dimensions10.3.3. Avoiding the double integral10.4. Looking at Mass and Self Weight as Distributed Values
11. Finding the Centers of Objects and Regions
11.1. Defining Location for Distributed Loads11.2. Getting to the Center of Centroids11.2.1. Defining a centroid's region type11.2.2. Computing the centroid of a discrete region11.2.2.1. Noting geometric properties of simple shapes11.2.2.2. Building a centroid calculation table11.2.2.3. Including holes in discrete regions11.2.2.4. Handling trapezoidal regions11.2.3. Finding centroids of continuous regions11.2.4. Taking advantage of symmetry11.3. Understanding Centers of Mass and Gravity11.3.1. Center of mass11.3.2. Center of gravity
12. Special Occasions in the Life of a Force Vector: Moments and Couples
12.1. I Need a Moment: Exploring Rotation and Moments of Force12.1.1. Developing rotational behaviors: Meeting couples and concentrated moments12.1.1.1. One force and one distance12.1.1.2. Two parallel forces and a distance: Couples12.1.1.3. No distance? Concentrated moments12.1.2. Taking on torque and bending: Types of concentrated moments12.1.3. Getting a handle on the right-hand rule for moments of force12.2. Calculating a Moment with Scalar Data12.2.1. Planar rotation about a point12.2.2. Determining the magnitude and sense of a two-dimensional couple12.3. Calculating a Moment by Using Vector Information12.3.1. Completing the cross product12.3.2. Using unit vectors to create moment vectors12.4. Using Double-Headed Arrows to Find Moment Resultants and Components12.5. Relocating a Force by Using a Moment: Equivalent Force Couples
IV. A Picture Is Worth a Thousand Words (Or At Least a Few Equations): Free-Body Diagrams
13. Anatomy of a Free-Body Diagram
13.1. Free-Body Diagrams in a Nutshell13.2. Displaying External Forces13.2.1. Portraying concentrated forces13.2.2. Depicting distributed forces13.2.3. Looking at the F.B.D. so far13.2.4. Conveying concentrated moments13.2.4.1. Moments in two dimensions13.2.4.2. Moments in three dimensions13.3. Axial Loads and Beyond: Depicting Internal Forces13.4. Restricting Movements with Support Reactions13.4.1. Three basic planar support reactions13.4.1.1. Rolling along with roller supports13.4.1.2. Freeing up rotation with pinned supports13.4.1.3. Restricting everything with fixed supports13.4.1.4. Moving on up (or down) with inclined supports13.4.2. Three-dimensional support conditions13.4.2.1. Ye olde ball and socket: Pinned supports in three dimensions13.4.2.2. Collar assembly supports13.5. Weighing In with Self Weight
14. The F.B.D.: Knowing What to Draw and How to Draw It
14.1. Getting Your F.B.D. Started14.1.1. Assuming a direction for support reactions14.1.2. Including more than the required info on your F.B.D.14.2. Zooming In with Isolation Boxes14.2.1. Unveiling internal forces14.2.2. Applying rules of application14.2.3. Avoiding problems with incorrect isolation techniques14.3. Using Multiple F.B.D.s
15. Simplifying a Free-Body Diagram
15.1. Presenting the Principle of Superposition15.2. Centering on Centerlines and Lines of Symmetry15.3. Equivalent Systems: Forces on the Move15.3.1. Moving a force: The space potato analogy15.3.2. Moving a moment
V. A Question of Balance: Equilibrium
16. Mr. Newton Has Entered the Building: The Basics of Equilibrium
16.1. Defining Equilibrium for Statics16.1.1. Translational equilibrium16.1.2. Rotational equilibrium16.2. Looking for Equilibrium with Newton's Laws16.2.1. Reviewing Newton's laws of motion16.2.2. The scalar equations that make it happen: The big three16.2.2.1. In two dimensions16.2.2.2. In three dimensions16.3. Identifying Improper Constraints: When Equilibrium Equations Are Insufficient16.3.1. Concurrent force systems16.3.2. Parallel force systems
17. Taking a Closer Look at Two-Dimensional Equilibrium: Scalar Methods
17.1. Tackling Two-Dimensional Statics Problems in Three Basic Steps17.2. Calculating Support Reactions with Two-Dimensional Equilibrium Equations17.2.1. First things first: Creating the F.B.D.17.2.2. Writing the equilibrium equations17.2.2.1. Adding helpful notation17.2.2.2. Summing forces first: Writing two translational equilibrium equations17.2.2.3. Summing moments: Writing the rotational equilibrium equation17.2.2.4. Solving for the unknown reactions17.3. Choosing a Better Place to Sum Moments
18. Getting Better Acquainted with Three-Dimensional Equilibrium: Vector Methods
18.1. Finding a Starting Point18.2. Seeing Equilibrium within Vector Notation18.2.1. Equilibrium in translational behaviors18.2.2. Rotational components18.3. Figuring Support Reactions with Three-Dimensional Equilibrium Equations18.3.1. Establishing the F.B.D.18.3.1.1. Sketching the loads on the F.B.D.18.3.1.2. Writing each load in vector form18.3.2. Writing the equilibrium equations18.3.2.1. Summing forces18.3.2.2. Summing moments18.3.2.3. Setting up to complete the cross product
VI. Statics in Action
19. Working with Trusses
19.1. Identifying Truss Members19.2. The Method of Joints: Zooming In on One Panel Point at a Time19.2.1. Step 1: Drawing isolation boxes19.2.2. Step 2: Applying the equations of equilibrium19.2.3. Step 3: Review and repeat19.3. Drawbacks to the Method of Joints19.4. And Now for My Next Trick: Slicing through the Method of Sections19.4.1. Step 1: Cutting the truss19.4.2. Step 2: Drawing the F.B.D. for the two remaining truss pieces19.4.3. Step 3: Applying the equations of translational equilibrium19.4.4. Step 4: Applying the equation of rotational equilibrium19.4.5. Step 4, continued: Identifying the instantaneous center19.5. Shortcutting the Equation Writing: Zero-Force Members
20. Analyzing Beams and Bending Members
20.1. Defining the Internal Bending Forces20.1.1. And then there were three: Internal forces of two-dimensional objects20.1.2. Strange new three-dimensional effects20.1.2.1. Translation: Another shear force20.1.2.2. Rotation: Torsion and another bending moment20.2. Calculating Internal Loads at a Point20.2.1. Positive moments make you happy!: Yet another two-dimensional sign convention20.2.2. Using the sign convention20.2.3. Computing internal force magnitudes20.3. Writing Generalized Equations for Internal Forces20.3.1. Defining the critical points20.3.2. Establishing the regions of your generalized equations20.3.3. Discovering other useful tricks from generalized equations20.3.3.1. Defining the relationship between shear and moment20.3.3.2. Calculating maximum and minimum shear and moments with calculus20.3.3.3. Plotting a system's internal forces20.4. Creating Shear and Moment Diagrams by Area Calculations20.4.1. Rules to remember when working with area methods20.4.2. Constructing the shear diagram20.4.3. Creating the moment diagram
21. Working with Frames and Machines
21.1. Identifying a Frame and Machine System21.1.1. Defining properties of frames and machines21.1.2. Determining static determinacy21.2. Using the Blow-It-All-Apart Approach to Solve Frame and Machine Problems21.2.1. Breaking it at the hinges21.2.2. Knowing where to start solving frame and machine problems21.2.2.1. Starting on a member with a load21.2.2.2. Solving for the unknown hinge forces as fast as you can21.3. Considering Other Useful Approaches to Common Frame and Machine Problems21.3.1. When more than two members meet at an internal hinge21.3.2. Dealing with pesky pulley problems21.3.2.1. Changing force direction with pulleys21.3.2.2. Creating mechanical advantage with pulleys21.4. Tackling Complex and Unique Assemblies on Machine Problems21.4.1. Pistons and slider assemblies21.4.2. Slotted holes and unidirectional pins
22. A Different Kind of Axial System: Cable Systems
22.1. Defining Nonlinear Structural Behavior22.2. Distinguishing among Types of Flexible Cable Systems22.2.1. Recognizing cables under concentrated loads22.2.2. Picking out parabolic cable systems22.2.3. Identifying catenary cable systems22.3. Solving for Tension in Flexible Cables22.3.1. Concentrated load systems22.3.2. Parabolic cable systems22.3.2.1. Calculating tension when you know sag22.3.2.2. Calculating sag when tension is known22.3.3. Catenary cable systems22.4. Taking a Shortcut: The Beam Analogy for Flexible Cables
23. Those Darn Dam Problems: Submerged Surfaces
23.1. Feeling the Pressure: Understanding Fluid Pressure23.1.1. Dealing with hydrostatic pressure23.1.1.1. Recognizing why zero pressure isn't exactly zero pressure23.1.1.2. Working with a unit width23.1.2. Determining effects from the self weight of water23.2. Making Calculations under (Fluid) Pressure23.2.1. Drawing the fluid F.B.D.23.2.2. Creating the hydrostatic pressure distribution23.2.3. Finding the dead weight of water and dams23.2.3.1. Determining the self weight of water23.2.3.2. Establishing the self weight of a concrete dam23.2.4. Including base reactions for dam structures23.2.5. Applying equilibrium equations23.3. Figuring Partial Pressures on Openings and Gates
24. Incorporating Friction into Your Applications
24.1. Friction: It's More Than Just Heat!24.1.1. Factors affecting friction24.1.2. Types of friction24.2. A Sense of Impending . . . Motion? Calculating Sense24.2.1. Establishing equilibrium when friction is present24.2.2. Finding the friction limit FMAX24.3. Solving Friction Problems by Using Logic and Equations Together24.3.1. Working with friction angles24.3.2. Combining friction and normal forces into a single resultant24.4. Timber! Exploring Tipping24.4.1. Uncovering the tipping point and normal force24.4.2. Moving the normal force to prevent tipping24.4.3. Establishing which friction phenomenon controls, sliding or tipping24.4.3.1. Case 1: Checking sliding before tipping24.4.3.2. Case 2: Checking tipping before sliding24.5. Examining More Common Friction Applications24.5.1. Wedging in on the action24.5.2. Staying flexible with belts and pulleys
VII. The Part of Tens
25. Ten Steps to Solving Any Statics Problem
25.1. Sketches Come First25.2. Determine the Supports25.3. Don't Forget the Applied Loads and Self Weight25.4. Calculate As Many Unknown Support Reactions As You Can25.5. Guess It's a Frame or Machine25.6. Get Out the Dynamite: Separating Pieces from the Problem for Internal Analysis25.7. Assume Directions of Internal Forces25.8. Be Consistent with Your Assumptions25.9. Guess That Three (or Six) Equilibrium Equations Are Necessary25.10. If Friction Is Involved, Guess That the Object Slides
26. Ten Tips for Surviving a Statics Exam
26.1. Find Problems You Know How to Solve26.2. State Your Assumptions26.3. Relax and Remember Your Basic Steps26.4. Identify Your Origin and Coordinate System26.5. Remember Your Vectors26.6. Write Your Equilibrium Equations26.7. Stuck? Draw More Free-Body Diagrams26.8. Draw Your Shear and Moment Diagrams Correctly26.9. Assess Your Answers26.10. Acknowledge Mistakes and Don't Erase

Content preview from Statics For Dummies

Chapter 8. Breaking Down a Vector into Components

In Chapter 7, I show you how to take multiple vectors and combine them into a single resultant behavior, which is a useful skill in helping simplify the number of actions on an object. That works fine if you're interested in examining the combined behaviors of an object, but what happens when you're interested in studying multiple behaviors but only have a single resultant to work with? For this situation, you need to understand how to create multiple behaviors of a resultant, or the components of a resultant vector.

The most useful feature of working with components is that these behaviors let you explore basic behaviors in more detail. For example, when an airplane is coming in for a landing, its approach is actually a vector with a given orientation at a specific speed. However, the pilot must maintain a certain horizontal behavior (which ensures that the plane actually reaches the runway and doesn't overshoot) to land the plane safely while guaranteeing that the vertical descent isn't so fast that it causes the plane to crash into a fiery heap when it hits the ground. The pilot needs to be aware of both the vertical and horizontal behaviors at the same time, for uniquely different reasons.

In this chapter, I show you how to break a single vector back into multiple behaviors that act entirely in Cartesian or non-Cartesian ...

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Statics For Dummies

by PE James H. Allen III, PhD

Chapter 8. Breaking Down a Vector into Components

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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