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Fasteners, Connections, and Power Screws
Springs
Brakes and Clutches
Flexible Machine Elements
Appendix A: Physical and Mechanical Properties of Materials 573 Appendix B: Stress-Strain Relationships 583
Appendix C: Stress Intensity Factors for Some Common Crack Geometries 591 Appendix D: Shear, Moment, and Deflection of Selected Beams 597
Appendix E: Dimensions of Threaded Fasteners 601 Index 605
Preface
Learning Tools
Case studies are printed with a light color background and are placed just before the end-of-chapter summary. The styles of the end-of-chapter problems are designed to cover each stage in modern learning taxonomies.
Web Site
Case studies aim to reference the topic of the chapter and place it in the right design framework, so that students have no doubt that the chapter is relevant and important. After the summary, the chapter has a list of key words that the student can use to study or help with jargon when needed.
Contents
Case studies are usually qualitative descriptions of important modern applications of the chapter's machine elements, but at a depth that requires an understanding of the chapter material. The chapter also focuses on squeeze film and hydrostatic bearings, which use different pressure generating mechanisms.
What’s New in This Edition
Additional material data are included for the fracture mechanics approaches to fatigue design. In Chapter 8, a streamlined discussion of typical surface finishes in machine elements, and manufacturing processes used to produce them, has been prepared.
Acknowledgments
About the Authors
Fundamentals
Introduction
Load, Stress, and Strain
Deformation
Lubrication, Friction, and Wear
Introduction
Symbols
- What is Design?
- Design of Mechanical Systems
- Design as a Multidisciplinary Endeavor
- Design of Machine Elements
- Safety in Mechanical Design
The role of the design engineer is to predict the conditions under which failure is likely to occur. The design of the elevator can be 50% overloaded before the safety switch shuts down the motor.
Design Procedure 1.1: The Safety Hierarchy
Government Codes and Industry Standards
In many cases, engineers must rely on government codes and industry-issued standards for design criteria. In addition, maintenance is simplified because standard bolts can be purchased anywhere, making replacement parts readily available.
Manufacturing
Life Cycle Engineering
Case Study 1.2: Sustainable Manu- facturing in the Production of Nike
Computers in Design
Computers allow virtual integration of all phases of the design process, whether technical or. Currently, a wide variety of polymers can be used, as well as metals and ceramics.
Catalogs and Vendors
Rapid prototyping has been particularly useful in design visualization and rapid detection of design errors. Finite element analysis (FEA) is the most widely used computational method for solid and fluid mechanics analysis and heat transfer.
Units
The stiffness of each element is known and is expressed in the form of a stiffness matrix for that element. By combining all the stiffness matrices, applying kinematic and stress boundary conditions, and solving for unknown stresses or displacements, complicated geometries and load conditions can be easily analyzed.
Case Study 1.3: Loss of the Mars Climate Orbiter
Unit Checks
Design Procedure 1.2: Procedure for Unit Checks
Significant Figures
However, when numbers begin or end with a zero, it is difficult to tell how many significant figures there are. Since force is accurate to three digits, acceleration can only be calculated to the accuracy of .
Case Study 1.4: Design and Manufacture of the Invisalign
The laser only cures a small depth of the polymer, so a part can be built on a tray that is gradually lowered into a vat of photopolymer while layers or slices of the desired geometry are traced and rasterized by the laser. Instead, the stereolithography machine produces patterns of the desired incremental positions of the teeth (Fig. 1.4c).
Summary
Once a treatment plan has been drawn up, the computer-based information must be used to manufacture the Aligners. These are sent to the orthodontist and new aligners are given to the patient as needed, usually every two weeks or so.
Key Words
Recommended Readings
Questions
Qualitative Problems
Quantitative Problems
Calculate the force required by the firefighter to hold the nozzle if the mass flow of water is 3 tons/h and the speed of the water is 100 km/h. It is raining and the extra mass due to water on the car is 1.349 kg.
Design and Projects
Assume that the frictional force is the viscosity times the surface area times the speed of the moving surface and divided by the lubricant film thickness. The nozzle diameter is small compared to the hose diameter, so the force on the nozzle from the water is.
Load, Stress, and Strain
V transverse shear force, N W applied normal load, N wo load per unit length, N/m x,y,z Cartesian coordinate system, m x0,y0,z0 rotated Cartesian coordinate system, stress m γ shear.
Subscripts
Introduction
Critical Section
Design Procedure 2.1: Critical Section and Loading
- Load Classification and Sign Convention
- Support Reactions
- Static Equilibrium
- Free-Body Diagram
- Supported Beams
- Shear and Moment Diagrams
- Method of Sections
Solution: Figure 2.5 shows a diagram of the forces acting on the ladder due to the weight of the painter as well as the weight of the ladder. A spring attached to the bottom of the sphere exerts a downward force of 150 N. Find: The forces acting on the two wires.
Design Procedure 2.2: Drawing Shear and Moment Diagrams by the
Direct Integration
Frax= 2m tox= 6m the displacement diagram is linear with respect to tox so that the moment diagram becomes quadratic. Atx = 6m, it is known that the moment will have a value of 8 kN-m by summing the areas of the displacement diagram segments.
Singularity Functions
Since the shear diagram is positive, the moment the results are negative they must be according to Eq. The slope of the moment curve is equal to the value of the shear curve, as can be seen from the derivation of Eq.
Design Procedure 2.3: Singularity Functions
This can be seen by summing the areas on the intercept diagram, remembering that areas below the abscissa are considered negative. It can be seen that the largest magnitude of the shear stress is at x = 2m and has a value of |V|max = 8 kN.
Design Procedure 2.4: Shear and Moment Diagrams by Singularity
The diagram at the top shows individual displacements, and the diagram below shows the composition of these displacement components. The diagram above shows individual moments; the diagram at the bottom is the composite moment diagram.
Functions
- Stress
- Stress Element
- Stress Tensor
- Plane Stress
- Mohr’s Circle
Of the three stresses acting on a given surface, the normal stress is denoted byσ and the shear stress byτ. Any other combinations of the normal and the shear direction will produce a negative shear stress.
Design Procedure 2.5: Mohr’s Circle
- Three-Dimensional Stresses
- Octahedral Stresses
- Strain
- Strain Tensor
- Plane Strain
- Summary
The principal normal stresses (i.e., the extreme values of normal stress) are located at the locations where the circle intersects the x-axis. Find: Draw Mohr's circle and find the most important normal and shear stresses in the x-y plane.
Summary of Equations
Determine the reactions and maximum shear force and bending moment for each section of the beam. Ans. Use the singularity functions to determine the shear force and bending moment. Ans. Mmax= 25kN-m.
Introduction to Materials and Manufacturing
Introduction
This is followed by a discussion of the major classes of solids, namely metals, polymers, ceramics and composites. Finally, the chapter concludes with a discussion of the major classes of manufacturing processes and their effects on material properties.
Ductile and Brittle Materials
- Ductile Materials
- Brittle Material
The cost of the design phase of the product life cycle is usually low, usually less than 5% of the total cost. Assume that the midplane of the plate experiences neither tension nor compression and thus will experience no extension.
Classification of Solid Materials
- Metals
- Ceramics and Glasses
- Polymers and Elastomers
- Composites
A certain critical fiber length is necessary for effective reinforcement and stiffening of the composite material. Find: Determine the required fiber-matrix bond strength to fully develop the strength of the fiber reinforcement.
Stress-Strain Diagrams
- Metals
- Ceramics
- Polymers
The stress-strain behavior of ceramics is not usually determined by the tensile test used for metals. After an initial linear zone, the stress-strain diagram becomes markedly nonlinear, with typically a strain of 0.01, as in Fig.
Properties of Solid Materials
- Density
- Modulus of Elasticity, Poisson’s Ratio, and Shear Modulus
- Strength
- Resilience and Toughness
- Hardness
- Thermal Conductivity
- Linear Thermal Expansion Coefficient
- Specific Heat Capacity
- Archard Wear Constant
Solution: The key parameter to use in this evaluation is the resistance of the two materials. Hardness is related to the reflection of the indentation: the higher the reflection, the harder the material.
Stress-Strain Relationships
Since the beam can only rotate about the hinge pin, the deflection δ1atP1 can be described by the rotation angle α, and this angle is the same when describing the deflection δ2 atP2.
Two-Parameter Materials Charts
- Stiffness versus Density
- Strength versus Density
- Stiffness versus Strength
- Wear Rate versus Limiting Pressure
- Young’s Modulus versus Relative Cost
CFRP: Carbon Fiber Reinforced Polymer KFRP: Kevlar Fiber Reinforced Polymer GFRP: Graphite Fiber Reinforced Polymer. Likewise, for steel and carbon fiber reinforced plastic S. Rubber has a maximum elastic strain of 60%, while carbon fiber reinforced plastic springs and steel have a maximum elastic strain of steel springs will be five times heavier than carbon fiber reinforced plastic springs.
Effects of Manufacturing
- Manufacture of Metals
- Manufacture of Polymers
- Manufacture of Ceramics
- Selection of Manufacturing Processes
Cores will be used to produce the void area of the part shown in (a). f) The mold cover half is assembled by securing the face pattern plate to the flask with alignment pins and adhesive inserts to form the sprue and riser. g). The balloons are then pressurized to counteract the buoyant forces in the fluid, which can lift the head. l) After the metal solidifies, the casting is removed from the mold. m) Spray and risers are cut and recycled, and the casting is cleaned, inspected and heat treated (where necessary). Source: Courtesy of Steel Founders.
Case Study: The Maker Movement
Summary
Determine whether the strength of the composite will be determined by the strength of the fiber or the bond between the fiber and the matrix. The answer. The fiber determines the strength of the composite. What is the ratio of strains experienced by the most compliant material mentioned in this chapter compared to the stiffest? Ans. 1.125×105.
Stresses and Strains
Introduction
Readers familiar with these topics may wish to proceed to Section 4.3, which introduces the concepts of normal stress and strain, or Section 4.4, which deals with torsion. Bending is discussed in Section 4.5 and the effect of shear in bending is presented in Section 4.6.
Properties of Beam Cross Sections
- Centroid of Area
- Area Moment of Inertia
- Parallel-Axis Theorem
The unit for moment of inertia and polar moment of inertia is length raised to the fourth power. Find: the moment of inertia of the plane of the circle about the x and y axes and the polar moment of inertia about the center of gravity.
Theorem
- Radius of Gyration
- Section Modulus
- Mass Moment of Inertia
- Normal Stress and Strain
- Torsion
- Stress and Strain
- Power Transfer
- Bending Stress and Strain
- Straight Member
- Bending Stress
- Curved Member
- Transverse Shear Stress and Strain
Find: (a) the maximum moment that can be applied to the beam. b) Stresses at points A, B and C when the maximum moment is applied. Thus, the shear force and moment of inertia of the area are the same for both parts of this case. a) The width of the point where τ is calculated is 50 mm. b).
Case Study: Design of a Shaft for a Coil Slitter
Summary
The stresses and strains associated with bending were explained, as well as the importance of the neutral axis. Deformations associated with combined bending and transverse loading were not covered in this chapter, but are considered more fully in Chapter 5. Surface moment of inertia, also called second surface moment, a property of a cross section that relates the bending stress , the applied moment and the distance from the neutral axis, m4 the average normal load of the normal stress divided by. center of the area geometric center of an area, m. mass moment of inertia product of the mass of the element and the square of the distance of the element from the axis, kg-m2 elastic deformation with normal stress divided by the axis parallel to the original length theorem a theorem that allows the calculation of. moment of inertia about each axis. rate of power doing work, or product of force and velocity or torque and angular velocity, Nm/s. the radius of gyration which when squared and multiplied by the area gives the surface moment of inertia, m. The section modulus of the modulus of inertia divided by the farthest distance from the centroidal axis to the outer fiber of the solid, m3. spring rate normal load divided by elastic strain, N/m. Torsional loading resulting in twisting of the shaft.
Calculate the deformation of the column, the spring rate and the stress in the column. Calculate the deformation of the pipe, the spring rate, and the maximum and minimum compressive stresses in the pipe.
Deformation
Introduction
This chapter attempts to quantify the deformation that can occur in a wide variety of machine elements. This chapter focuses on describing the deformation under distributed loading, such as that occurs in a beam.
Moment-Curvature Relation
Assume that the cross-section is constant along the beam and that the material is the same throughout, meaning that the moment of area, I, and the modulus of elasticity, E, are constant. Find: The slope and deformation at anyx, and the location and value of the maximum slope.
Singularity Functions
Design Procedure 5.1: Deflection by Singularity Functions
Solution: Figure 5.2a shows a free-body diagram of a simply supported beam, while Fig. 5.2b has a beam section. From Table 2.2, for concentrated forces, the load intensity equation for the forces shown in Fig. a) Integrating twice gives the moment as.
Singularity Functions
Method of Superposition
The solution to the original problem then takes the form of a superposition of these solutions. The solution assumes that the deflection of the beam is linearly proportional to the applied load. 5.12).
Cantilever Through Superposition
Strain Energy
- Normal Stress
- Shear Stress
- Transverse Shear Stress
- General State of Stress
Integrating over the entire volume gives the strain energy stored in the element due to the shear stress as . This is the overall strain energy due to transverse shear stress for a rectangular section.
Castigliano’s Theorem
The ratio of the energy stored in the two parts can be expressed as: For torsion, Table 4.1 should be used for J, the moment of inertia in the polar region for a circular cross-section.
Design Procedure 5.2: Procedure for Using Castigliano’s Theorem
Summary
If a load of 50 kN is applied at the location shown in the Sketchh, what will be the deflection of the beam. Derive an expression for the deflection of the free end in the z direction using Castigliano's theorem.
Failure Prediction for Static Loading
- Introduction
- Stress Concentration
- Stress Concentration Factor Charts
- Flow Analogy
- Introduction to Fracture Mechanics
- Modes of Crack Growth
- Fracture Toughness
Figures 6.5 and 6.6 show the stress concentration factor for a round bar with a fillet and a groove respectively. Note that as the radius of the discontinuity decreases, the stress concentration increases.
Design Procedure 6.1: Fracture Mechanics Applied to Design
Failure Prediction for Uniaxial Stress State
The dimensions of the leaf spring are such that the width is 10 times the thickness. The design stress of Eq. a) From Table 4.1 for a rectangular cross-section. 4.45) gives the magnitude of the bending design stress axis.
Failure Prediction for Multiax- ial Stress State
- Ductile Materials
- Brittle Materials
- Selecting Failure Criteria
Solution: Figure 6.16b shows the tension element at the top of the bar against the wall. (c) Mohr's circle representation of stresses. 2.16), the principal normal stresses in biaxial tension can be written as. 2.19), the maximum and principal shear stresses can be written as. 6.8), the MSST predicts that failures will be avoided if.
Design Procedure 6.2: Selection of a Failure Criterion
Designers are often reluctant to follow this procedure, as IFT is the most mathematically complex criterion. However, those presented in this chapter are by far the most widely used and a suitable yield criterion can usually be selected from those presented.
Case Study: Stress Concentration Factors for Complicated Geometries
Summary
Determine the applied load that causes yielding at the edge of the hole. Answer.92.43 kN. Find the factor of safety by a) Maximum Shear Stress Theory (MSST) Ans. b) Distortion-Energy Theory (DET) Ans.
Fatigue and Impact
Introduction
Thus, the voltage at any point around the paper clip will vary as a function of time. Repeated bending of the paper clip will eventually exhaust the material, resulting in failure.
Fatigue
Bending causes compressive and tensile stresses on opposite sides of the paper clip, and these stresses change with the direction of bending. Most of the engineering design experience in the field of fatigue is based on an experimental understanding of the behavior of carbon steel.
Design Procedure 7.1: Methods to Maximize Design Life
Cyclic Stresses
Acyclic stress is time dependent, but the variation is such that the stress sequence repeats itself. Find: the average stress, stress range, stress amplitude, and stress ratio for the shear bolt.
Strain Life Theory of Fatigue
A stress cycle (N = 1) consists of applying and removing a load once and then applying and removing the load again in the opposite direction. The advantage of the Manson-Coffin equation is that it provides insight into important properties when determining fatigue strength.
Fatigue Strength
- Rotating-Beam Experiments
- Regimes of Fatigue Crack Growth
- Microstructure of Fatigue Failures
- S-N Diagrams
Note that the crack growth rate can be even less than one atomic distance of the material per cycle. Near the origin of the fatigue crack (point B in Fig. 7.5) the surface is polished or very smooth.
Design Procedure 7.2: Staircase Approach
Fatigue Regimes
- Low-Cycle Fatigue
- High-Cycle, Finite-Life Fatigue
- High-Cycle, Infinite-Life Fatigue
The S-N diagram (Fig. 7.7a) shows different types of behavior as the number of cycles to failure increases. If the fatigue strength is given and the number of cycles to failure is desired.
Stress Concentration Effects
Assume from strain energy theory that the equivalent stress is σe=√. Find: Determine the fatigue stress concentration factors for bending and torsion of the drive shaft if the shaft material has a tensile strength of 965 MPa. Remember that the shaft is dimensioned such that the shear and bending stresses are equal (τ =σ).
The Modified Endurance Limit
- Surface Finish Factor
- Size Factor
- Reliability Factor
- Temperature Factor
- Miscellaneous Effects
Relieving residual tensile stresses on a surface is generally beneficial, but equal or larger grains can be detrimental from a fatigue standpoint. This is an important tool for fatigue design because it represents one of the only strategies that increases the fatigue strength of materials, and this increase can be very large.
Design Procedure 7.3: Estimation of Endurance Limit
- Cumulative Damage
- Influence of Nonzero Mean Stress
- Ductile Materials
- Brittle Materials
- Influence of Multi-Axial Stress States
- Simple Multiaxial Stress
- Complex Multiaxial Stresses
- Fracture Mechanics Approach to Fatigue
- Linear Impact Stresses and Deformations
A straight circular rotating beam with a diameter of 30 mm and a length of 1 m has an axial load of 30 000 N applied at the end and a stationary radial load of 400 N. The beam is cold drawn and the material is AISI 1040 steel. Given: For the beam given in Example 7.5, the bending moment varies between 50 and 200 Nm.