Solutions Manualto accompany

MACHINE ELEMENTS INMECHANICAL DESIGN

Robert L. Mott

Fourth Edition

PEARSON---1'11'111 jl'PHall

10 9 8 7 6 5 4 3 2 1

Copyright © 2004 hy Pearson Education, Inc., Upper Saddle River, New Jersey 07458.Pearson Prentice Hall. All rights reserved. Printed in the United States of America. This publication is protected byCopyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage m aretrieval system, or transmissiou in any form or by any means, electronic, mechanical, photocopying, recordmg, orlikewise. For information regarding permission(s), write to: Rights and Permissions Department.

Pearson Prentice HalJTM is a trademark of Pearson Education, Inc.Pearson® is a registered trademark of Pearson picPrentice Hall® is a registered trademark of Pearson Education, Inc.

Instructors of classes using Mott, Machine Elements in Mechanical Design, Fourth Edition, may reproduce materialfrom the solutions manual for classroom use.

ISBN 0-13-140873-9

CONTENTS

1

iii

Solutions Manual for

Machine Elements in Mechanical Design, 4th ed.By: Robert L. Mott

Table of Contents...... ii

MDESIGN software Included on the CD in the book , iii

Spreadsheets included on the CD in the Solutions Manual , x

Solutions to end-of-chapter problems

Chapter 1: The Nature of Mechanical Design 1

Chapter 2: Materials in Mechanical Design...... 3

Chapter 3: Stress and Deformation Analysis... ... ... ... . .. ... ... ... ... ... ... ... .. 9

Chapter 4: Combined Stresses and Mohr's Circle 31

Chapter 5: Design for Different Types of Loading , 42

Chapter 6: Columns , 64

Chapter 7: Belt Drives and Chain Drives , 77

Chapter 8: Kinematics of Gears .,. ... ... ... .. . ... . .. .. . .. . .. . 87

Chapter 9: Spur Gear Design _. 108

Chapter 10: Helical Gears, Bevel Gears, and Wormgearing 151

Chapter 11: Keys, Couplings, and Seals 180

Chapter 12: Shaft Design .. . .. .. .. .. ... .. .. .. .. 184

Chapter 13: Tolerances and Fits , 206

Chapter 14: Rolling Contact Bearings 211

Chapter 16: Plain Surface Bearings 217

Chapter 17: Linear Motion Elements 224

Chapter 18: Fasteners, 226

Chapter 19: Springs 228

Chapter 20: Machine Frames, Bolted Connections, and Welded Joints 243

Chapter 21: Electric Motors and Controls 254

Chapter 22: Motion Control: Clutches and Brakes , 257

cted bye m ang, or

naterial

MACHINE ELEMENTS IN MECHANICAL DESIGN, 4TH EDITION

By: Robert L. Mott

Published by Prentice-Hall, Inc.

MDES/GN - Mechanical Design Software

Comments On Its Use For Problem Solutions

IntroductionTh.e.designof machine elements jnherently .involves extensive procedures, complex calculations,

and many design decisions. Data must be found from numerous charts and tables. Furthermore, design

is typically iterative, requiring the designer to try several options for any given element, leading to the

repetition ·ofdesign calculations with new data or new design decisions. This is especially true·for

complete mechanical devices containing several components as the interfaces between components are

considered. Changes to one component often require changes to mating elements. Use of computer

aided mechanical design software can facilitate the design process by perfonning many of the tasks while

leaving the major design decisions to the creativity and judgment of the designer or engineer.

We emphasize that users of computer software must have a solid

understanding of the principles of design and stress analysis toensure that design decisions are based on reliable foundations. We

advlsetbat·thesoftware be used only after mastering a given designmethodology by careful study and practicing manual techniques.

Then, data with known results can be applied to the software as a

check on the understanding of the program's Input and output.

Included in this book is the MDES/GN mechanical design software created by the TEDATA

Company. Derived from the very successful MDESIGN mec software produced for the European market,

the U.S. version of MDESIGN employs standards and design methods that are in typical use in North

America. Many of the textual aids and design procedures come directly from this book, Machine Elements

in Mechanical Design.

Topics for which the MDESIGN software can be used as a supplement to this book include:

Beam stress analysis Beam deflections Mohr's circle Columns

Belt drives Chain drives Spur gears Helical gears

Shafts Keys Power screws Springs

Rolling contact bearings Plain surface bearings Bolted connections Fasteners

Clutches BrakesSpecial icons are placed in the book at places where use of one of the 28 modules in the

software is pertinent.

v

Basic Features of MOESIGNMOESIGN soflware is a useful tool for problem solving and design. Users must first ensure that

the software is appropriatetothe nature of the problemlo be solved. While the 28 modules cover a wide

range of applications within the field of machine design, not every problem can be solved. The soflware is

very user friendly and each module contains several textual and graphic aids that explain the technical

"bases on which the module is constructed .and.the data that must be entered by the user. Input screens

prompt the user to define the problem and to make basic design decisions. Several modules contain

extensive data bases that allow the user to consider multiple optional designs and compare them to

select the more optimum choice. Some modules also provide a parametric analysis feature.

After all necessary data are entered, the user applies the Calculate feature to cause the module

to perform the required analyses and produce the output in an easy to read format. The Calculate feature

is represented on the tooioaras an image of a blue calculator with a yellow gear. Altematively, the user

can use the function key, F10. Input and output can be reported in several different units selected by the

user. Conversions, if needed, are automatically accomplished by the software.

The output always contains lists of pertinent input values and computed results. Many modules

augment the basic output with graphical displays of the results and pictorial views of the element being

designed. Some modules include cautionary comments when results do not meet design requirements,

advising the user to redesign the element.

DeKfiptions of the Modules

The following sections list the 28 modules organized into 13 groups and provide brief discussions

about each module. Included are the chapter where each module can be used, a basic description of the

type of problems that can be solved, the type of input data required, and the type of output results that are

produced. NOTE: The computed results from MOESIGN may not match exactly those reported in the

book for example problems or end of chapter problems. Values for some parameters may be selected by

a user from charts and graphs In the book, whereas the program may compute those giving slightly

different results. For some modules, particularly Ball and Roller Bearings, databases in the program aredifferent from those in the book.

Columns Group: Pertinent to Chapter 6 in the book.

Column Analysis Module: Sections 6-6, 6.7, 6·11 and 6.12

Centrally loaded straight or crooked columns and eccentrically loaded straight columns are

analyzed.lnput data required are: column length, end fIXity, material, cross section shape and dimensions,amount of crookedness if any and amo t f . . .. ,un 0 eccentncity of the hne of action of the load if any. Output forstraight, ce~tratty loaded columns includes critical buckling load and allowable load. Output for crooked

and eccentncally loaded columns includes maximum stresses. For eccentrically loaded columns, thernaxmum lateral deflection of the column is also computed.

-u_-------------------Column Design Module: Section 6-10

Given the required load on a column of a given length with specified end fixity, the program

computes the minimum acceptable diameter of'acolumn with a solid circular cross section.

Seams Group:Pertinent to Chapters 3 and 5 in the book.

Statically Detenninate Beams Module:This is a very extensive beam analysis program that allows the application of loads in virtually any

direction including concentrated forces, distributed loads, concentrated moments, and torques. It offers

visualization of the beam, its supports, and its loading pattem. Careful data entry is advised.

Statically Indeterminate Beams Module:This program analyzes beams on more than two supports with a large variety of loading types.

Combined Stress and Mohr's Circle: Pertinent to Chapters 4 and 5Given the stress condition on an element, the program completes Mohr's circle, presents

maximum principal stresses, maximum shear stress, and the orientation of those elements. The Mohr's

circle is drawn automatically.

V-Selt Drives: Pertinent to Chapter 7, Section 7-4This tool for designing V-belt drives is based on an extensive data base of ratings for three sizes

of V.belts. The input data screen allows the specification of power to be transmitted, speeds, approximate

center distance, and service factor. The program presents a list of suitable belt types, sizes, and standard

lengths, and sheave sizes from which the user selects one combination. The performance of that design

is then displayed. Numerous subsequent altemative designs can be easily created.

Chain Drives: Pertinent to Chapter 7, Section 7-6An aid for designing roller chain drives, this program includes an extensive data base of the latest

power ratings for roller chain drives from the American Chain Association. The input data screen allows

the specification of power to be transmitted, speeds, approximate center distance, number of chains, and

service factor. The program presents a list of suitable chain sizes and sprockets from which the user can

select one combination. Single or multiple strands of chain can be used.

Gearing Group: Pertinent to Chapters 8, 9, and 10Separate modules are provided for Spur Gearing, Helical Gearing, Bevel Gearing, and

wormgearing. Each module basically follows the procedures described in the book. Input data include

the diametral pitch, number of pinion teeth, input speed, output speed, certain material factors, overload

factor, reliability factor, and others depending on the module. The program provides guidance for the

quality number and the specification of the number of teeth in the gear and its face width. The geometry

factors for bending and pitting resistance are computed by the program for some modules, while others

require the user to input those using tables and charts from the book or from charts reproduced in the

vii

t d sizes of pertinent geometric features, forces on gearhel screens. Outputs include actual outpu spee , '.

teeth, and tooth stresses. The acceptability of stresses is evaluated by the ~rogram with suggestions for

the types of materials to be specified. Multiple designs can be tried very quickly to work toward anoptimum final result.

Pertinent to Chapter 11Keys Group:

separate modules are provided for Parallel Keys and Woodruff Keys, using design pro~dures

similar to those presented in the book. Input can be either the torque to be transmitted or the .combination

of power and shaft speed. The shaft diameter isspecified,likely based on prior stress analysts, Matenals

for the key, the shaft, and the hUbof the power transmitting element are selected from a list taken from

Appendix 3 in the book.The yield strengths are then automatically inserted. The usespeclnss a design

factor and the length of the hub. The program determines the appropriate size forthe 'key according to the

shaft size and the stress analyses using procedures similar to those outlined in the book.

Bearings Group: Pertinent to Chapters 14 (Rolling contact) and 16 (Plain surface)

Ball and Roller Bearings: Chapter 14

This program aids in the specification of it commercially available bali or roller bearing fromextensive databases for t6 different types of bearings from two widely known manufacturers, FAG and

SKF. The user selects the preferred type, and provides detafor redial and thrust loads, speeds, desired

hours of 'life, and size limitations. New factorsihat are not discussed in the book are also specified:

Viscosity Grade for the lubricant to be applied (typical choice is 66); Operating Temperature (typical

choice is 40"C); and Contamination Factor (tYPical choice is 1.0). The program then produces a list of

possible'bearings, organized in oroer of the bore size. The User selects oneandtl1e program completes

the calculation of performance, giVing the projected life of the bearing in hours that is compared with the

desired life. SUbsequent trials may be done qulte quickly to work toward an optimum design.

NOTE: The basicdynarnic /()~ data .repodedfr1r any [Jivenbearing wiN not match those given in

the book because the bearings come from different Sources and because an additional tector, a23. calledthe Material and Lubrication Factor, is applied. The value of this factor is prOprietary to a givenmanufacturer andfypically yields a higher value for basic dynamic load, C, than is reported in the book.

Plain Surface Bearings: .Chapter 16, Section 16-6

This program assists in the deSign of boundary lubricated plain Surface bearings using the

methOd presented in Section 16-5 of the book. The user enters data for radial/oad, speed, minimum shaft

diameter, and the desired ratio of bearing length to diameter. The program then computes the required

pVvalue and recommends a material from Table 16-1 in the book that has an acceptable rating for pv,

NOTE: Data for the wear factor, K, and the coefficient of dynamic friction are not available in thebook version of the software and are repOrted in the output W!'th I

-va ues of 0.000.

Vl1l

Power Screws: Pertinent to Chapter 17 Linear Motion Elements; Section 17-2

The procedures described in 'Section 17-2 of the book are implemented by this program to design

a power screw with Acme threads having a 14 1/2' thread angle. The user enters data for the load to be

moved, the distance traveled, and the time to move the load. The material for the screw isseleeled from

lists of steels, aluminums, cast irons, copper alloys, bronzes, and zinc alloys. The strength data are

automatically entered by the program. The design value for the coefficient of frielion is entered. The

output includes the dimensions of the standard Acme screw threads from Table 17-1 in the book. The

program checks the tensile stress in the screw and the shear stress 'in the threads. The minimum length

of MQagement'of the threads with a nut is reported. The program also computes the torque required \0

raise and lower the load, the efficiency, the linear speed of the nut, the rotational speed of the screw, and

the power required to drive the screw. !fthe screw is loaded in compression,it should be analyzed for

column buckling using the Column Analysis module.

Joints Group: Pertinent to Chapters 18

Fasteners: Chapter 18, section 18-4

This module completes the procedure outlined in Section 18-4 in the book for designing and

analyzing bolted joints that provide clamping loads. Input data include the total load applied to the joint

and the number of bolts. A Demand Factor, k, is specified that gives the allowable percent of the proof

load of the screw material, often taken to be 75%. The factor, k1, is based on the lubrication present and

is setteo.tsfor typical conditions urness the threads are thoroughly cleaned. After starting the Calculate

process.awmoowasks fortheSAEmaterial grade forthe bolt as listed ln Table 18-1 in the book. The

output includes the required size of bolt, taken from the list in Table 16-4 in the book, and the tightening

torque.

Springs Group: Chapter 19

This group contains three modules for Helical Compression Springs, Helicat Extension

Springs, and Helical Torsion Springs. Each module implements the design procedures developed in

the book and uses data from the book for spring wire sizes and material properties.

Helical Compression Springs: Section 19..0

Method 1 for the design of helical compression springs illustrated in Example Problem 19-2 is

used in this module. The user supplies values for forces and lengths, end type, wire type, type of service

{Iight,average,severe),and an inilialestimateofthe mean diameter of the spring and the design shear

stress (typicaltyinthe range from 60 to 140ksi). The program determines an appropriate wire diameter,

computes the actual stresses, and outputs the geometry of the spring.

Helical Extension Springs: Section 19-7

The method for designing helical extension springs illustrated in Example Problem 19-4 is used in

this module. The user supplies values for forces and lengths, end type, wire type, type of service (light,

average, severe), and an initial estimate of the mean diameter of the spring and the design shear stress

.. f 80 t 140 kSI·) The program determines an appropriate wire diameter,(tYPically In the range rom a .

computes the actual stresses, and outputs the geometry of the spring.

Helical Torsion Springs: Sect jon 19-8

The method for designing helical torsion springs illustrated in Example Problem 19-5 is used in

this module. The user supplies values for moments and angles of rotation, end type, wire type, type of

service (light, average, severe), and an initial estimate of the mean diameter of the spring and the design

bending stress (typically in the range from 11010 190ksi). The program determines an appropriate wire

diameter, computes the actual stresses, and outputs the geometry of the spring.

Joints Group:

Bolted Connections: Chapter 20, Section 20-3

This module determines the minimum required diameter of the bolts in a connection subjected to

direct shear and shear due to a moment applied to the member similar to that shown in Figure 20-5 in the

book. The bolted connection may be comprised of any array of bolls for which the distance from the

centroid of the array 10any.individual bolt is th.e same. Examples area rectangular alTay .of four.bolts or a

circular array of any number of bolts. The analysis considers only a single force to be applied to the

connection. If more than one force is applied, the resultant of all applied forces must be determined by the

user for input to the program. Both the magnitude and the Orientation of the line or action .of the force

must be known. The program assumes that the bolls are subjected to single shear. If they are in doubleshear, the applied force should be divided by 2.0.

Inputs required are the shear lead, the ·number of bolts, the perpendicular distance from the line

of action oflhe load to the centroid of the boll pattem, the radial distance from the centroid to any boll, the

x and y distances from the centroid to the bolt, and the angle of inclination of the applied load. The angle,Cl, is zero for a vertically downward load. Graphic and textual helps are provided.

The output consists of the forces on Ihe bott, the required diameter and the nearest standard bolldiameter as Shown in Table 18-4 in the book.

Welded Joints: Chapter 20, Section 20-4

The method Of Section 20-4 of the :book is used in this module to compute the required size ofweld to carry .a specified force using a weld geometry selected from those shown in Figure 20-8 in thebook. This figure is included on help screens in the program.

Inputs required are the type of jOint, key dimensions of the joint, the location of the load, and theallowable force per inch of weld length taken from Table 2Q,3 that 'ISshe hi. th

wn ona e p screen In eprogram. The term, bending console length, ab, is the distance, a, shown under the Bending column ofFigure 20-8. Similarly, Ihe term, twistirrg console length at, is the d'istan·ce a f th T. I n

., . , ,rom e J orslon co um .OUtputs inclUde geometry factors ·of the weld pattern, bending momenl, twisting moment, forces

on the weld, and Ihe required weld leg size.

x

Clutches and Brakes Group: Chapter 22This group contains five modules for Cone or Plate-Type Clutches or Brakes and Short Shoe,

Long Shoe, and Band Brakes. Each module implements the design procedures developed in the book.

Plate-Type Clutch or Brake: Section 22-11The procedure illustrated in Example Problem 22-6 is used in this module for the analysis of a

plate-type clutch or brake. The user inputs the desired friction torque, the nonmalactuation force

available, the coefficient of friction, the rotational speed, and a reasonable value for the ratio of the

outside radius to the inside radius (say 1.50). The program computes the required mean radius, the

outside radius, the inside radius, the frictional power absorbed and the wear ratio (hplin\

Cone Clutch or Brake Section 22-13

The procedure illustrated in Example Problem 22-7 is used in this module for the analysis of a

cone clutch or brake. The user inputs the desired friction torque, the mean radius of the drum, the cone

angle, and the coefficient offriction. The program computes the required axial actuation force.

Short Shoe Drum Brake: Section 22-14

The procedure illustrated in Example Problem 22-8 is used in this module for the analysis of short

shoe drum brakes. The user inputs the friction torque desired and proposes the drum diameter,

coefficient of friction, and basic dimensions of the actuation system. The program computes the required

actuation force.

Long Shoe Drum Brake: Section 22-14

The procedure illustrated in Example Problem 22-9 is used in this module for the analysis of long

shoe drum brakes. The user inputs the drum speed, friction torque desired, drum radius, coefficient of

friction, design value for maximum pressure, and the basic dimensions of the actuation system. The

program computes the required width of the brake pad, the actuation force, the frictional power, and the

wear ratio (hp/in\

Band Brake Section 22-15

The procedure illustrated in Example Problem 22-10 is used in this module for the analysis of

band brakes. The user inputs the drum speed, braking torque desired, design value for maximum

pressure, drum radius and width, coefficient of friction, the angle of wrap of the band on the drum, and the

basic dimensions of the actuation system. The program computes the friction torque, the actuation force,

the friction power, and the average wear ratio (hp/in2). Subsequent iterations are quickly done to work

toward an optimum design.

MACHINE ELEMENTS IN MECHANICAL DESIGNFourth Edition

Robert L. Mott

Prentice-Hall Publishing Company

Description of Spreadsheets Included on CD in Solutions Manual

Introduction

The Solutions Manual for this book inciudes a computer disk that contains 26 computational aids that arekeyed to the book. The files are written as Microsoft Excei spreadsheets uSing Version 2002 onWindowsXP.

Many of the spreadsheets appear in the text. Others were prepared to Produ~ solutions for the.Solutions Manual. The given spreadsheets include data and results from certain figures In the text. fromcertain example problems, or for certain problems from the end of chapters containing the analysts anddesign procedures featured in the programs.

The following sections give brief descriptions of each spreadsheet. Many are discussed in the text inmore extensive detail. It is expected that you will verify all of the elements of each spreadsheet beforeusing them for solutions to specific problems.

Using the Spreadsheets: It is recommended that you maintain the givenspreadsheets as they initially appear on the disk, considering them to be master copies. To use aprogram for SOlving other problems, call it up in Excel and use the ·Save as. command to nameit something different. For instance, the Original program cal/ed Column Analysis should beconsidered the master. Use "Save as" and call it, for example, Column Analysis _ Working. Thenuse that version for general problem Solving.

You should study the concepts and the solution techniques for each type of problem before using thespreadsheets. You should work sample problems by hand first. Then enter the appropriate data into thespreadsheet to verify the solution. In most spreadsheets, the data that need to be entered are identifiedby gray-shaded areas and by italic type.

Descriptions of Spreadsheets

The descriptions are given here in the order that the Subjects for the spreadsheets are covered in thetext. The files are listed on the disk in alphabetical order.

Co~umnAnalysis: Chapter 6. Analyzes straight columns of uniform cross section to detemnine thecntlca~ buckling load and the allowable load. The spreadsheet shows results for Example Problem 6-1 asgiven In Figure 6-9 on page 242. U.S. Customary units are used. A description is given in Section 6-8.The process IS esse~tlally the same as that shown in the flow chart of Figure 6-4. Note that a shortmacro program In Visual BaSICISused to decide w~ether the column is iong (EUler) or short (J. B.Johnson) and to complete the calculation of the cntlcal bUCkling load Be th t Excel programenables macros. . sure a your

Column Analysis SI: .Chapter 6. Same as COlumn AnalYSis: exce t SI't d The solution toExample Problem 6-21S shown as given in Figure 6-10 on page 243. p UOis are use .

CirCUlar Column Analysis: Chapter 6. Special version of Col A I '. t .cproperties of a COlumnwith a solid cirCUlar cross Section are c umn na ysls In which th~ g.eome nspreadsheet can be used as an iterative des' t I omputed When the diameter IS input. Thea circular cross section to carry a given load Ig~ oOFto determine the required diameter of a column with

. ee Igure 6-14 on page 249.

xii

;

Crooked Column Analysis: Chapter 6. Section 6-11. Analyzes the allowable load on a column ofconstant cross section with a given amount of crookedness. Data from Example Problem 6-4 are used

as shown in Figure 6-16 on page 252.Eccentric Column Analysis: Chapter 6. Section 6-12. Computes the required yield strength of thematerial and the resulting maximum deflection of the middle of a column that is loaded eccentrically.Data from Example Problem 6-6 are used as shown in Figure 6-18 on page 256.

Chain Drive Design: Chapter 7. Design of roller chain drives as described in Section 7-6. User mustobtain rated power data from Tables 7-5, 7-6, or 7-7 to specify a suitable chain number and number of

teeth in the smaller sprocket.Contact Ratio-Spur GeatS: Chapter 8. Computes the contact ratio for spur gears using the procedure

shown on page 317 in Section 8-4.Bevel Gear Geometly: Chapter 8. Computes the geometric features of straight bevel gears using theformulas listed in Table 8-7 in Section 8-8 and illustrated in Example Problem 8-6 on page 337. Twoidentical programs are shown side-by-side. One shows the results of Example Problem 8-6 and the other

can be used to solve any given problem.Wormgearing Geometry, CD, VR: Chapter 8. Computes essential geometric features of a worm andwormgear, the center distance between their shafts, and the velocity ratio. Uses procedure from Section8-10 as illustrated in Example Problem 8-7. The spreadsheet was used to complete Problems 52-57 at

the .end of the chapter.Gear Geometry: Chapter 8. Computes the geometric features of spur and helical gears using therelationships in Sections 8-4 and 8-7. Can be used for Problems 1-9 and 41-44.

GeatS VR Design: Chapter 8. Aids in the specification of the number of teeth in a pinion and gear toproduce a specified velocity ratio. Uses a procedure similar to that shown in Section 8-13 on pages 350-357 and illustrated in Table 8-9. An integer is entered for the number of teeth in the pinion. Theprogram computes the required approximate number of teeth in the gear to produce the given velocityratio. The user then enters an integer for the actual number of gear teeth. The program identifies thecombination of numbers of teeth that produces the minimum differential between the desired ratio andthe actual ratio. The spreadsheet was used to complete Problems 62-65 at the end of the chapter.

Spur Gear Forces: Chapter 9. Computes the tangential, radial, and normal forces on spur gear teethof a given design transmitting a given power at a given pinion speed. It uses the method of Section 9-3.The spreadsheet was used to complete Problems 1-6 at the end of Chapter 9. The results for Problems 1

and 2 are shown in the master.Spur GeatS-Design-U.S.: Chapter 9. Performs a complete design analysis for a pair of spur gears,including the essential geometry, tangential force, required bending stress number, and required contactstress number. All modifying factors for stress calculations as described in Sections 9-8 to 9-12 areincluded. The data from Example Problem 9-5 are shown in the given spread sheet as illustrated inFigure 9-28 on pages 416-417. An extensive discussion of the spreadsheet is given in Section 9-14 onpages 415-419. An added feature of the spreadsheet on the CD is the computation of the requiredhardness (HB) for through-hardened Grade 1 steel using the equations in Figures 9-10 and 9-11. Theuser can then specify suitable materials and list them at the bottom of the spreadsheet.

Geometry Factor-I-Pitting: Chapter 9. Computes the value of the geometry factor, I, used in thecalculation of contact stress for spur gears in Equation 9-25 on page 401 of the text. Program uses the

algorithm from Appendix A20.Spur Gears-Design-U.S .•With I: Chapter 9. Same as Spur Gears-Design except the geometry factor, I,is computed within the program instead of being input by the user. The program Geometry Factor·l-Pitting is integrated within $pur GeatS-Design. One additional input value is needed for the pressure

angle.Spur GeatS-Design-SI: Chapter 9. Similar to Spur GeatS-Deslgn: except SI metric data are used asdescribed in Section 9-13 and illustrated in Example Problem 9-6. Data from Example Problem 9-6 are

used in the given spreadsheet.

XIII

Spur Gears-Capacity-U.S.: Chapter 9. Section 9-16. Determi.nes the ~ower transmitting capaclty of agiven set of spur gears considering both bending strength and pitting resistance. The user must Input theallowable bending stress and allowable contact stress based on the material specified for the pinion andthe gear using Figures 9-10 to 9-15 and Tables 9-3 and 9-4. The spreadsheet includes the computationof the required bending stress number, Sat, and contact stress number, sac, based on user-enteredhardness (HB) for through-hardened Grade 1 steel using the equations in Figures 9-10 and 9-11. Theuser must transcribe these values into the spreadsheet if, in fact, this kind of material is specified.

Plastic Gears· Design: Chapter 9. Completes the design of plastic gears using the procedure frompages 440-1. Data are shown for Example Problem 9-6.

Helical Gears-Design: Chapter 10. Computes the forces on helical gear teeth as described in Section10-2 and illustrated in Example Problem 10-1. Completes the design analysis for a pair of helical gearsas described in Sections 10-3 to 10-5 and illustrated in Example Problem 10-2. Used for the solutions toProblems 1-11 at the end of Chapter 10.

Helical Gears-capacity: Chapter 10. Similar to Spur Gears-Capacity: with modifications for thespecial geometry of helical gear teeth. Used for the solutions to Problems 12 and 13 at the end ofChapter 10. The user must input the allowable bending stress and allowable contact stress based on thematerial specified for the pinion and the gear using Figures 9-10 to 9-15 and Tables 9-3 and 9-4. Thespreadsheet includes the computation of the required bending stress number, sa" and contact stressnum~r, s~,based on user-entered hardness (HB) for through-hardened Grade 1 steel using theequations In Figures 9-10 and 9-11. The user must transcribe these values into the spreadsheet if infact, this kind of material is specified. '

Bevel Gears - Design: Chapter 10. Computes forces and stresses on bevel gears.

Wormgearing - Design: Computes some geometry values, forces, and stresses for wormgearing.

Shaft Design: Chapter 12. Computes the minimum acceptable diameter for shafts using Equation 12-2~~n ':th bending and torsion are present and Equation 12-16 when only vertical shearing stress is~odifyi~9 f:~~so~~~~~~r~~s :or torques, forces, bending moments, pertinent material strengths,at several selected sections of t~en~~'ftand .~tressconcentration factor. The program is typically appliedgiven spreadsheet uses data from one I as ;. ustrated in Design Ex~mple 12-1 on pages 548-552. TheFigure 12-19 on page 561. A discussio ocafth

lonon the shaft .In.Deslgn Example 12-1 as illustrated inno e spreadsheet IS In Section 12.10.

Foree Fits: Chapter 13, Section 13-8 Str f .between mating members assembled ~ith e~es or Force Fits. Computes the pressure at the interfaceresulting stresses and deformations for the ~ I~terference fit (See Section 13-6.) Also computes thefrom Example Problem 13-2 are Shown in th a Ing members usmq the procedure on pages 587-8. Datae example.Spring Design-Method 1: Chapter 19 Section .method from Example Problem 19-2 to d' 19-6. The given spreadsheet uses data and theto fit given geometrical limitations. See ~~J~rn~~f~ helical compression spring for a given loading andSpring Design-Method 2' Ch . e on page 753 and the accompanying discussion.

. . . apter 19. Similar to S rI DdeSigning to a set of geometricallimitalions S Ex P ng eslgn·Method 1 without the restriction ofand the aCCOmpanying diSCUSSion. . ee ample Problem 19-3, Figure 19-17 on page 753,

xiv

CHAPTER 1THE NATURE OF MECHANICAL DESIGN

Problems 1 - 14 require the specification of functions and designrequirements for design projects and have no unique solution.

l/s.1 o = 1·7S" I"'· )( ZS.Y/1n~/M" 'I¥-S ",.,.,h1'T

\;6.1 L. #6;=r X O•.3~Y4 ~/;=r= /Y-O /111

~ r = / Z $'$0 L..6·/N x C)./130 N' -""/1.4 'IN 2 1'118 ;V'nov

V8.1 A:I '1.12. IN ":» 611.5".2. /tnAlf Y/N 1.". Z6S8 /1Ifmt z.

I22J z == /~ • .s IN 3 )( /.« 39 NOt' /I'II""!;0/1 ':1 => Z. ¥3NO G'"",~

IU.I .r'" 88.01# I' x '1./62. )(/".s",...",.".,VHI';' 3.'" X/f/ -",,/nl"

IZ/,j /11 !4PP<:1V4IX A-/b/ /f.eeA.s A-~ t;/VlFN 1/./ /~cHes z.

It.. '700 __ ~x / 1,11 ~ 6yS". 2. />"'I"'" 2. /./6 .IN ~

L z. . I< Z X~ : II -= /. 36 .W '2.

\22) P = 7.0'A.I? X 7'¥.r.? w/~P = 5.09 NO' W = .s:S9.J IN

IZ3.1 S.... - /Z? ~ )( ~. 89S" 1'11'a./AM. 2 87~/11?0-

~ UrT D3 (}.o3S';ft'f;L~O.{'7S'/m lrO'-II"'G"=:i/=AJ<I.=t;r()~)"LV = 71' (O,03S "",,)"l.)( 0.6,S,..., _ 6.'19 '1./0-«"",3

yo / -'I J_/'IIlSS = DEAlSIT)! '" V. 7680 ~/"..1""o.'I?1tIO Att

3:o ~'l8"7

J

� r- lao 1.8-111 J( eJ,1130/ll'/Ift/LO'/J/ = 2tJ ..3 N'_

e: 35")( ffRAII/180' =CJ.b/12.4t:1.

Sellar. '19 & 180LNN/.ss" = S'/'1LI-IN/PF62EI!'

scsu:> r/s ~ Zo.") #,,,,,/0,6/11*0 .... ~].3JtI,,,.,/R-40.

£1'11:1/(;'/ : f'OWE~' nMFE= J2.~J.p ~ Ii-}, l 5"DIIYJ

D041 1/'Iel!J(

E = /.011</0 " Fr.Ltl/VFAI!.

x 52 W'A::S" • ssru::r:LlJ )(.2600 ~'ItrlHl or •A P ~

€ .. /,()j XJo" Fl"'LI! l I,j.s~ :t t I.o/Y·". X loW ~.LLY~M. IT'LI1 .t N'M/.! J600.5

E = ~.B l/O' Wi/~..,It - 38.8 MIN-ill/FAX!

127·1 VISCOS'II·Y"'.)t .. 3.7$ I/;yN. I.au,s J( IWI6L:: 5'101.11.JIN 1.,.f4YfV FrO. FT ..

ke.1 LIFE" : l7.so~{;'1J< ~ X f:DI'1IN, .. .2CSOAYS SY.&ks!'lIN j) A '/ fo YEJ"tIl,t

UfT := f.6olC/O' Kt=1I0LtJTlaNS

CHAPTER 2MATERIALS IN MECHANICAL DESIGN

1. Ultimate tensile strength is the apparent stress at thepeak of the stress-strain curve.

2. Yield point is the value of the apparent stress from thestress-strain curve at which there is a large increase instrain with no increase in stress. It is the point wherethe stress-strain curve exhibits a horizontal slope.

3. Yield strength is the apparent stress from the stress-strain curve at which there is a large increase in strainwith little increase in stress for materials that do notexhibit a yield point. The offset method is used bydrawing a line parallel to the straight part of the stress-strain curve through a value of 0.2\ on the strain axis.

4. Many low alloy steels exhibit a yield point.5. The proportional limit is the apparent stress on the stress-

strain curve at which the curve deviates from a straightline. At this value, the material is usually stillelastic. The elastic limit is the apparent stress at whichthe material is deformed plastically and will not return toits original size and shape.

6. Hooke's law applies to that portion of the stress-straincurve that is a straight line for which stress isproportional to strain.

7. The modulus of elasticity is a measure of the stiffness ofa material.

8.. The percent elongation is a measure of the ductility of amaterial.

9. The material is not ductile. Materials having a percentelongation greater than 5\ are considered to be ductile.

10. poisson's ratio is the ratio of the lateral strain in amaterial to the axial strain when subjected to a tensileload.

11. From ~Q. 2-5 G = E/[2(1+Vl] = (114 GPal/[2(l+O.33l]G = 42.9 GPa

12. Hardness - 52.8 HRC (Approximate; Appendix J~)

13. Tensile strength ~ 235 ksi (Approximate; Appendix 19)

14.-17. Errors in given stat_ents:

14. A hardness of HB 750 is extremely hard, characteristic ofthe hardest steels in the as-quenched or surface hardenedcondition. Appendix 3 shows annealed steels to havehardness values in the approximate range of HB 120 to 230.

15. Hardness on the HRB scale is normally limited to HRB 100.16. Hardness on the HRC scale is normally no lower than HRC 20.17. The relationship between hardness and tensile strength is

only valid for steels.16. Charpy and Izod tests measure impact strength.19. Iron and carbon. Other elements are often present.20. In addition to iron and carbon, AISI 4340 steel contains

nickel, chromium, and molybdenum. (Table 2-3)21. Approximately 0.40% carbon in AISI 4340 steel.22. Low-carbon:

Medium-carbon:High-carbon:

Less than 0.3%0.30% to 0.50%0.50% to 0.95%

23. Typically a bearing steel contains 1.0% carbon.Lead is added to AISI 12L13 steel to improve machinability.Shafts are often made from AISI 1045 4140, 4640, 5150,6150, and 8650 steels. (Table 2-1/),Gears are often made f5150, 6150, and 8650 rom AISI 1045, 4140, 4340, 4640,steels. (Table 2-lf)The blades of a post hole diresistance, high strength gger should have good wearsteel is a reasonable ch ~ and good ductility. AISI 1060

Ol.ce.

AISI 5160 OQT 1000 is hicontaining approXimate~ gh-carbon, chromium steel,It was heat treated by ~ O'fO% carbon and 0.60% chromium.temperature, quenched ine~i ng above its upper criticaldegrees Fahrenheit It h 1, and then tempered at 1000ksi or 1040 MPa) a~d as fairly high strength (Sy ~ 151

good ductility (14% elongation) .

24.25.

26.

27.

26.

4

31. Induction hardening is a heat treating process in which thearea to be hardened is subjected to a high-frequencyelectric current created by a coil, inducing current flownear the surface of the part and causing local heating.After sufficient time to bring the surface to a temperatureabove the upper critical temperature of the material, thepart is quenched to harden the surface.

32. Some carburizing grades of steels are AISI 1015, 1020,1022, 1117, l11S, 411S, 4320, 4620, 4820, 8620 and 9310.The carbon content ranges from 0.10% to 0.20%.

29. In general, a high hardness with good ductility aredesirable for machine parts and tools subjected to impactloads as seen by a shovel. A hardness of HRC 40corresponds to approximately HE 375 and is consideredmoderately hard. While this is a good level, even a highervalue up to HRC 50 (HB 475) would be better, providedductility is fairly high, say about 15% elongation.Appendix 3 shows some forms of oil-quenched AISI 1040 andnone listed have sufficiently high hardness. Appendix 4-1shows the same material quenched in water and tempered.AISI 1040 WQT 700 has a hardness of HB 401 (HRC 43) withapproximately 20% elongation and a yield point of 92 ksi.

30. Through hardening involves heating the entire part followedby quenching to achieve the hardened condition. Except forsome variation in thick sections, the part is hardenedthroughout. But no chemical composition changes occur. Incarburizing, the chemical composition of the surface ischanged by the infusion of carbon. Thus, carburizingresults in a hard surface while the core is softer.

33. The AISI 200 and 300,series of stainless steels arenonmagnetic.

34. Chromium gives stainless steels good corrosion resistance.35. ASTM f\qq:tstructural steel is used for most wide-flange

beams.36. HSLA structural steels are high-strength, low-alloy steels

having yield strengths in the range of 42 - 100 ksi (290 -700 MPa.

37. Three types of cast iron are gray iron, ductile iron, andmalleable iron.

3S. ASTM A4S-S3, Grade 30 is a gray iron with a tensilestrength of 30 ksi (207 MPa); no yield strength; less than1% elongation (brittle); modulus of elasticity (stiffness)of 15x106 psi (103 GPa).

Problem 38. (continued)ASTM A536-84, Grade 100-70-03 is a ductile,iron with atensile strength of 100 ksi (689 MFa); a y~eld strength of70 ksi (483 MFa); 3% elongation (brittle); modulus o~elasticity (stiffness)of 22xl06 psi (152 GPa) .ASTM A47-84, Grade 35018 is a malleable iron with a tensilestrength of 53 ksi (365 MFa); a yield strength of 35 ksi(241 MFa); 18% elongation (ductile); modulus of elasticity(stiffness) of 25xl06 psi (172 GPa) .ASTM A220, Grade 70003 is a malleable iron with a tensilestrength of 85 ksi (586 MFa); a yield strength of 70 ksi(483 MFa); 3% elongation (brittle); modulus of elasticity(stiffness) of 26xl06 psi (179 GPa) .

39. Powdered metals are preformed in a die under high pressureand sintered at a high temperature to fuse the particles.Re-pressing after sintering is sometimes used.Parts made from Zamak 3 zinc casting alloy typically havegood dimensional accuracy and smooth surfaces, a tensilestrength of approximately 41 Ksi (283 MPa), a yieldstrength of 32 Ksi (221 MFa), 10% elongation, and a modulusof elasticity of 12.4xlO' psi (85 GPa). (Appendix 10)

40.

41. Type D tool steels are typically used for stamping dies,punches, and gages. (Table 2-5)

42. The suffix 0 on aluminum 6061-0 indicatescondition. the annealed43. The suffix H on aluminum 3003-H14 indicates h 'strain hardened. t at ~t was44. The suffix T on aluminum 6061-T6 'treated. ~ndicates that it was heat45. Aluminum 7001-T6 has the h' h~ 98 ksi (676 MFa)' y' ld ~g est strength; tensile strength, ~e strength = 91 ksi (627 MFa) .

Aluminum all 606oy 1 is one of the most versatile.Three typical uses of t't 'structures, chemical pr~can~~m alloys are aerospacehardware, ess~ng equipment, and marineBronze is an alloy ofphosphorus, nickel z,Copper with tin, aluminum lead,, ~nc, mangan 'ese, or silicon.

46.47.

48.

49. Bronze C86200 is a manganese bronze casting alloy with atensile strength of 95 ksi (655MPa); yield strength of 48ksi (331 MFa); 20% elongation (ductile); modulus ofelasticity of 15x106 psi (103 GPa) .

50. Bronze is used for gears and bearings.51. Thermosetting plastics undergo a chemical change during

forming resulting in a structure of cross-linked molecules.The process cannot be reversed or repeated. Thermoplasticmaterials can be formed repeatedly by reheating because themolecular structure is essentially unchanged duringprocessing.

PEr;52. a) Gears: Nylon, polycarbonate, acetal,~polyurethane

elastomer, phelolic. b) Helmets: ABS and polycarbonates.c) Transparent shield: Acrylic. d) Structural housing: per,ABS, polycarbonate, acrylic, PVC, phenolic, polyester/glasscomposite. e) Pipe: ABS, PVC. f) Wheels: Polyurethaneelastomer. g) Switch parts: polyimide, phenolic, PIT.

53. Designers of parts to be made from composite materials cancontrol 1) base resin, 2) reinforcing fibers, 3) amount offibers, 4) orientation of fibers, 5) number of layers, 6)overall thickness, 7) orientation of layers, 8)combinations of types of materials.

54. Composite materials are comprised of two or more differentmaterials, typically a resin reinforced by fibers.

55. Resins used for composites include polyesters, epoxies,polyimides.

56. Reinforcing fibers used for composites are glass, boron,aramid, and carbon/graphite.

57. Sporting equipment is made from glass/epoxy, boron/epoxy,and graphite/epoxy composites.

58. Aerospace structures are made from glass/epoxy,boron/epoxy, graphite/epoxy, and aramid/epoxy composites.

59. Sheet molding compound is typically a glass/polyestercomposite.

60. SMC's are used for auto and truck body panels and largehousings.

61. Reinforcing fibers are produced as continuous filaments,chopped fibers, roving, fabric, yarn, and mats.

1

.,...-------------------62.

63.

64.

65.

66.

67.

68.

69.

Wet processing of composites involves,the layup of fabricreinforcing sheets on a form, saturat~on of the sheetswith the resin, and curing under heat and pressure.preimpregnated composite mat~rials are ~roduced with theresin already on the fibers ~n a conven~ent form, calleda prepreg. The prepreg is layered onto the form andcured.SMC's are preimpregnated fabric sheets formed in a moldand cured simultaneously under heat and pressure.Pultrusion is a process of coating the fiberreinforcement as it is pulled through a heated die toproduce a continuous form such as tubing, structuralshapes, rod, and hat sections used to stiffen aircraftstructures.In the filament winding process, continuous filaments areplaced around a mandrel in a controlled pattern and thencured. The process is used for pipe, pressure vessels,rocket motor cases, containers and enclosures.Specific strength is the ratio of the strength of amaterial to its specific weight.speci~ic stiffness is the ratio of the modulus ofelast~city of a material to its specific weight.ManY,c~osites have significantly higher values ofspec~f~c strength and specific stiffness than metals.

70 • 73 refer to P1gure 2-lZ and Table 2-~.General conclusions from Questions 70 - 73: The specificstrengths,of the metals listed range from 0.l94xl06 to1.00xI06 ~n approximat 1 f 'stiffnesses'are very e y a actor of 5.0. The specif~capproximat 1 nearly equal for all metals listed,e y 1.0xI0S in The 'f ' hcomposites I' d . spec~ ~c strengths of t e~sterange 1 87 t ' ,than any of the metal . 0 4.86xI06 ~n, much h~gherstiffness about 2/3 t~t G~ass/epoxy has a specificcomposites listed ran 0 the metals. The otherthe metals. ge from 2.2 to 8. .3 times as stiff as

See Section 2-18 for answers to Questions 74 to 78.

CHAPTER 3STRESS AND DEFORMATION ANALYSIS

Direct Tension and Compressiono .Lr= F/A; A= 1T1/8~-/z>-)/'1=/lI.J1_ 1.-

(J""" = LiS-ooN'/ /'//-I-' 2 =- 'J/. 8AI'/..........l. = 3/, .B 1'11"0...3 "" Pi = (lfSOtJH)( 7..5V",,_) J( /IJ"hI-'-tJIZ

eA LZ(}7JC/o'fh'/-mZ){NI-~_""z..) Af'7~ -. "..,,,.,

0-= P/A= J50°o/{;rLlo)/~.0h1""': 'If.6/7 1"0...

0-; '/1'1 = ;;'ON03/V/fo·.]o)"..".;z.: 66.1Ml'o-

t:r= F/A= 8i.OL/S/tfJ.'I"/N)~= 53'SPsi

a:« %c /"DOLo/Hro.37r"AI)'f::/fJ,20"~5J

&. .. S'O«J#/o:= PIA; A = (;Z_,..')" /'1'/". ....,;()~ 1''IY'''''/f'f~·3'/.1ItfPA.

PI.. (S(J(J(JN'(./&SCAOfAOl) S7 z,9Z. ~ bu.),; • -:I J = "".,,....

4A (c II/",.,,..,,)(/Y'I--~) Ga.) ,4/.1/ /a zo ; ~ = 24'lt;I' ...= 2IJ?Xlo'3/11/--. ...; fa tJ.Z?)A" .....

• ) IIISI.86~; c=2,/)7~"-.,; t- (J.Z.17".,~

c) /)f/CT'//,e- /l!otJ~ E·,<n,~AJ..; "-P. 3'~/Wf,,",

J) At..ul'I/I/tI"'f l06/-r,; Ge. 69~PA,; [. tI,83tJ.,..,w

~) PTlloI'rJ"" {i-6AL-W; E":>'//~$P __./ $'= ().5tJ3~

~)he; c= 25//DI'1A. =2J//D""'/-._· ;1"=ZJ.8AOf-

g.) PHe/ll4/,/(. .; C = ?saoI1PG.. ~ 5'~ ?S6,.".,_)V07tO: S"'r.as~ IS CLD~lF 1"lI ~ l.Il..n;'?ltT5'Fo~ f AND ,.,.

.~-------------------

-=

P rEAf = '/.,4; p:::: s:

A. (j..LS1._ 2. 01t.)lN 1.= /'02./,N"

(p. 00'11" )(/o x/(j6 I.'PAI")(I.IJZ /N1.)

!'= ".0'''·

<T = Ph = ~6'-%"2./N~ = 2.5tJi pst'

o E/I'/s r:::.().. ZSdD(7S) -F" (6tJ)

Fe ~ ZSb/J(7%0) ::::j/ZS LiS """!1rNtlld po~ ,II Ac

a: ~...e... .. 3/zs L8 : .s"9s P3iA (i.S»J(J •.s,)/.l/l.

Z Fp' S"~" = ISIJO LS

Fit = lS'oo,!Coz,..t"'f(J'~) =/0 t.1 t.S

a:> ~; .eEQO.A" ~...... ,.~ASiJlL4

A" /161U3 = O.as89/N~=.fN),1.IS fJDfJ l.S""&

,:..,..,:--~~o =('41,; r:::.{'1to;s8Cf) = ".2.7¥/A.

$ -AS'" F. = /soo t.lS = /S"DOI..8. ::: Z £9981.&.) 'If. Z. SA'I 9 .z SAN uS")

..q = Zc"~8DDO ... a.IV /# '1. ;. L)zVY;;:U) - Co '1SS/~

II

ISAI _--6;;""'- ~c.IO.s~p s.lS

FI~. '3'/;;'"'UhJ.tOINT })

e:o

" Jil'

S.LSAN 6

t9D~~' - S;U~N~o .. ID.~.J.AI • CDIt, : Ao (j411Ju1.4'1 A", • ~(

$r.te1US:114,,8' ~

SD :

/2

ttllA ·0= 6axJ(&)-rIla:t>Vl.) -1., (JI) rFe,." IODOO '-,

fltF ·0 a/l-COO{,) +6oe.O&.1~.t(;') 9I!,,- 8aoou I

"'C .tP-:.....l3:.--~..A..~~...&.-~,..L

~,~----18 ----._t,«.t • 80110c.A K" .. All ~&.. Ail (0",)

Ail .. f{./U" 11tJoo/1'" -10 ofll L4 ttl'll'.lip: Aa c.eoae"" 100.etO.&)= 60'OU (&IS.

Bo"OItili."'liD 60C10L&

.:JLt.ooou /jE ~ 0 ,..60.0 -I'tCS~ & ...0

Sir t= A4.Mo II ;000 a ,P"D{C.t) -'- : ZSl»u Tit..".tr._-6( .....:..~ ~.,

BC: Ifl c.a & ~ 8£ Co.s~ .. 100#'/''-') +ZRD/tJ,tJ=

IJ~ $(.::to 1SD01.4 CD"'~•••••

l'l#lO I.C Be" c» (#CL.

C,. • s« /~S 9 - ~o Iii. 6 • 11.SH LIJ ell"t'.C~ = I~CDO- CF ~. ,. 12Dco-IZs»o/IJ,r);: ZOIlDU ~

c.f!

--......1Q'£Srtrnsl'.p

fo, •~o c D;. :::6000/0.911::: ../9,I'Mcrp • 7soo/0.9" ."771(1,*diD -000,. =ZSfJO/D.'It'l"'"SI's"":I1"Cl!' ... ·Zooo/o.VIIl ""-'1132."'-~4 e -lOcco/~.f1= - "/'J1.";1&. .. -7stIA/z.n" -30.' ,.M11"C/' = -/l~l.l'l. - -#/,S-pA

£,:

AR.~AS OF ,#11',<i46tS :{n-PI. AS'} It,)!I'D, D~, 1:" - Z'tl/fllfl)'" (J.9"8 ,IN 'I.

BO) IIF J C. (" - D.mIN'"

A-6) BCI eF - J.CM./)~ Z.Va./Hl.

N. rr I eo""t'ltt ~/"" tftC",'aJ ""itT 't''HI oa» Fil (Dl_AI 4"aLI/I("

Itc (z.u·) (/.Yo) + zLU.IfO){D.S)lt>] ': '/.1/1 ,N 1.-

11"- F/A" (S"l.OOO Lo/"',"'IIN~) • /I 'I'II,..w

/3

�----------

Direct Shear Stress F7S

lIb. I PIN VIII'" 0.50,,;. i o au C3:1! SItEItIl. z 1/ sa ---j cA-.s" l.(1TD/·d = l. tr ((}.SOJ7! = 0.3'12.-711>1A;r-- FA<

r =- F·/A.s "II'1/:H/JEil. Be: eos:» /J:o =(z.soD)(7.J)-~,4-c L60)rAc. = ZSoo (7f-:J "'3/l..n..1S

t<t1JO By= fA(.:- JIUL(J

U'I::O ZSQ) L C.

J<eftlt.-n'to/r /lrtJ~ B= Jllz.JL~2.S00?--: '1bO]..U. 8y

P 1N.s A "'''0 Co. '. T= "'Y,4,- = 311.5Lo/d.J"1 ZJ IN ~:= 7'JSd '$/

U"'L6.

rlfr. P3-as

17 8 ...- ~~ = '1"0'- Lfj/ ~ _ /Oldo rs:rIN_: I := 7~ /O.3'1bJN _ 17•-"-----

Fo1/. fJs ~o·;FIl:= 1"7 /..t. s: SHMfl. FDt.l~ ()JI VPPE:R....f'JN.J.

IfSS<.IHt= POl/l!lE SHt:JIfL: As = 2 no/(, " l ""(O.l5'~ .:O.BB.U IN ....

r:- Fs//f.J ~ //67l%.8IJ.J'INL =- 1.12./ ('[; "

Lowee. PIN: Fs = /SIJIJLi1r -e, F~A-s= ISCDL%'88J'IJO/'-:::' /698 I's/

[20,1 PtlN c» - FIG P:J-ZQ As = (fep/I'!,) t:-.: !l.$I1+ 2.00 1-I,S#+,fd, ~ I- zrJfo.p,o)

AJ =(8,5>'IN){o,Ol.Ol/v) = tJ.!>!J "v ..."/-:; FS/

A_ 5Z DOOLt5/ 1 = /1J/t.'tJol's/

/ IT.!> - / p,Si:J IN _......:' _

~ PIINCfi--!=/Cr. P3-Z/. PEIf!JI'I ~ 60 ~ z.(~o) r-Z(7.S) +-3r-;:5Y ==2t>S.7_""

A s ~ {eEI.HI.J t: ' {z.os; 7)(:z.oJ = "11/.'I....."" 2-

r'= PYA _ 2Z-S000H = S'f?H./_~1=5Y7/1Po..s- 91/. lI __ .. ~.~Torsion

,- -~p

TLe- t;.:i'

TrD 'I rr-(S-O)" II .r.:r= 3"i: = S~ Aft.- • 6.///)(AT .-._41

e- 6JOOlt/o3;it.--)f6lS'l)--) = O.0/38R.+i'x/90fJl!4: o:YlCi!iIf.

(80 1VO~ '%._~) {6./f/ )(loS'4"-") 17'RAo

1"" ...I:. = .-L.: 88.0 U·IN. = 'lDo.J n;~p 1TD~6 IT(O.YDIH)'/;6

T= 6.3 000 (.p)/,., = 6.$000 Old Af)/se,ollPn =

/<L...I:... /i.. 3'S"1.4-IN = 31. Z-?(J p~;r, TTU.LS"/NYp6

r- P/~ .. 28X'03N,A"t/0SRADA' ~ 6ZZ.,N •.-

& _ ". C.Dv-dV) _ IT[ -f'D v_ 3D"] 1n 1'1'1 = 8$"~O/Hf"" 3P - /6 D - /1. (4IOAff-)

3r=- L _ 6~2. 1'1.- )( /0 __ - ?2.~.M -7Z.YI"IP ..e,. I~o _ ....] _ - ~ ...

-r t: • _ reiD" ,4'1) _ ,,[+'0"-30"1__ v=/.7J~AID~_V{J -;:-F • .r - sz, - 04-

{9. (6~z. ,1/'''''')(~O()/fOf'-) x /O~AOfe: ,./oJ AlI-_.a){l.718mr_y) /1'01

•(UJl8R~ ~ =/. AU-

ff/JAII ~. ';.;..."-

/~

Noncircular Members in Torsion

Ill.\ S~vAAE '. a." !S-- ;Q =t ,.Z.18a.,,1 -31.:° Itn/1OfJ 2Fllr.3-I.

j:.= 0"1/'0.-"" S;SIXIO ~- JII 12.10,.y.",.,/' ~) 10'_ "1".B~ - 7tJ.8;+J1'...

-;. /~ = (- /ZlSD_- .....

8_ r L_ =(Z.30X/t'JN''''''''''){~-)= ().()339/l4P,J( ~ -/.fjl'. - t!r K LBlJUO'~_"')(5'.sIX/~";..,-") ,Mil -

IZ'1·1 'V/t..- fh6Oj/.SD"'O.'10 (fIr;. P3-~)C,:: 0.?8 ~ Cz.:: 0.10J(. ,. C', 11': ;::d.78OrS4/= 3, 9s- III V

Q'" Oz.1l.3 .. o,» (UD)3 .. Z. 3" ,,v 3

r ID 6CD LI·/JI· UJI. 87 •r: = : ,. TT PS,~ Z.3' IN:s

8,...r L. :: (Id 6C0 Lt.·IM) ('/'1.1111)~ I<. 1//.S"lC/ll "%.1.) J.9S'/"'v ·O.O/OJ i41J t.~ • 0, STJo

C trVtD

a. ..z-" Ilf ; b" 'l.D,,,; t· ",/t1fll/~ ~.t}= /.8'111'" 'Or;;}:).l"/ ,"';I.=~.sFroJ oJ l ..78 IN.

'" z~ {a.-c)"(,,·t:J"l- UI)'/"'IJ{/9'1,)40.ef,)'" v,,= ().+b -Zt: = . = ~,O"I IN(z,o +''1.0 -UOo/dfJ]

Q:: z» (a.-c)(b -t) .. 2(O,IOft)(;,891)C3.B"lI) = /.60¥ ",,3ra T'Q:: (6000 L.~N"X/,6()'IINS):: 9(z.y LI..,,,,

f!} ::::t. r::..= (JHY 1.4'/),)(78111)~k ~ = O,O3l. £4o~· = /.83"IISUO'L4/;J/l.){J.,D'lI;./I'I) 1TAAO

'ft,-=

Beams

I~

fr-+

,.) '!-I/J SCHePlIL~ Yo ""'IEs:::3.Z/S 11'1 s

CL) VOLlJ""e" AJtL = (z..s #/)'-,. lUI". - Iso/II'; ,Mr=tl.l>:J4XNJ)f!so'J). ,lAr .. 212 L~.

b) v= ,<t..L,= (j.ZS){3.SfI){IU)'" szs/A/J ~,Mr= &2.81)(5"20)= J'I'lL.6

c) V • fl·7~)(.s:,00)(llo)r /0$0 /N)'; ~:: (;Z.8~)(/.so)" 2 91L.B'1. \ ~

rJ) v.: .!!.£... ><L = .,,(3.tollil ~ /1.4/11.= 8lfBl1J'" )..t.T~(g.Z83){81{6): 2¥OL8'f 'f .)

e) 'l.7U/Fr{IOFr)= 17.0 t.Co (.t;) '10 Ll/FrCtoFr)' YOOL.A

~) 10.7' La/,.rC /0 Pr) e 107.,1.1J 1/:,4'11. ~/7YIN·){n.o;;~~.~8l:o;'lV~AAi- = O.ozti3'lJ8.o1} e ".7~L8/Fr

FRoGI'! CfltSEC;APPS"DDf. /IJ.: a."'3',tA~ La/UJ,tN; P=/ZMI..4~~"'3o..,,·P"·

"f"'A1t - p(J.. (3L "'-¥<t. ...)= IU»{3,)C3(/zc)"'-l{(34) ...] _ l..UI IIWz/IE"I ZY(Jof/o').r - I_ pa,.1. (3L-'14..) Il.oo(Jd"[3{/U)-'f(~OJ _ 1.8"

t1•.. -tc. - b€I = 600"'0'5 .r -.r lit(~£E Nstr PFU;fF)

/7

.~-----------------133.1 (cOAl r) Y jI , 2.z..91

a) r = a';'l~ .. (z.so)'jn.=3,ZI./,Ij '1'",.. - 3.%' =()""liI

/. S""'" = ~c." ~ (}.S?2 IN.. ,., ".Z6

11) :z - Hh~·O.'ZS)(3".so~;:. 'I. '111#: 'VNt=(J·SIIINj n'"0,'/18 /~]

c) Ia H~~:::~I(J(l.U%l.::: Z.Z3/N 'I : 7,.,,,,,or /.421 ,IAI )1" ;. ;,83' ,,~.

sr o" rr(3.~O)'fd) r- -;;;r- ... 6Y :::3. '181/ : 1'IW· 4.0$1'1 ,-.1<1~ 1'&::. (}."6'///I.

e) r- I..DS IN' ; "t/'lQ"" tJ.31S""N; '6= (}.30'?N/.

of) I'I:: '1,23 /1./ ; Z,.,llr ""O.2.Y7 ,IAI ; 1& = (J.Z-D2nJ

'}) x» 7,2J#/"; ~ :::4.31 S"/N j 7-:: (J,ZS8IN.

IJ( N)

o

8tJtJI t!""N tPoN Z, 11'"

.., .8.... ,J .1 I .11, •:'" I' It'~ ·all/J -./ ,slll"l- ~~..IIt'lN

II 8/1

l40·11

1-+-.5"'1 _ -38'.,"" ,n -q9'l3l'l <, ;

V I ,ZJ? :l\:' i~z.f-

,A " I

~ ~

IS

a)S/nl'u CANTIl-Eli'" - C~$r 6. .-MPtfNln.,r I¥-l: I" 7.ZJ,/N"

.AI\, a -f? JC .... {JCt-.:9 =-(.gpo)(7'BJ2. [X"7Z)- 'Ie]~-o.2.18 /.y. :::"t"(11 ,~~.~~~w A

:r,- +'-6* .. '(8/.y; 0...= 6;t. "'7V",; -;xp8 p =9'-Ali."'r~= ~p(L1.(~;(;...0.J= -8CO(?Z)z [UlI) -74 =0.'881.y --?'a

6E:r sa • .t.Il'.x7. zs:

6) $rPPrut:rn3 ChNTlI.f1Yflll. - eMG h -!tPP • .lJDIX /'{-3

""'II .. -I' -;r.J.b:J (3C,-.C, ~-'t· rz erc:(,0'1f/'''''I8'''' ;a..."'- ')1.lh;'" 'lit": tld'/"'''; L=/iI;t./U/A; ....=~.ZY.INCia a, L( Lt-II);: 'JZ (IZIJ)(IU)= /.'I~2 )(,1"/,,3

cs =l!.+.v(L+b) +a.L.=(J9z)(ll.8)i-72.Clu)= '!.090x.t" /NZ.

I'IiA. ·(afJtJ) ('tS)t('18) \i(U.fL)(II') -f#/X/I'I(f/,U =1J.O'l7/J(=/{A·U IZ(J4Il"'Jr7.1.3)(llD)'~ L-

,,~. -p Q"ZN'"'] [3L1." -",,"'{3L-4.B/2.£.rL

;(800)(7zi(' Z'I) . '3(;ZlCY8) -(z~y,(IU)-7z)1:?J,o$'Z /N-"'S1'4 Iz.(J'~"'X7.U)(l2i;Jt ~

13.7.1 <r= 1'1/., • 5" MIlt':./ .... ,) ,,""'....8000 Le-," ... "1 JS - a ".Il1O III12. iHlO U/",,"

5~l/utF&r Bell", oe .3.I II 1.631'; Sa/.'!"!1'J3;:t=Z.Z'! 1"'1

'fill .. + P£,o.. 0 c. (L -I- a;;6 IL

= l!-SOO)l30)CVO)(Ul) C?o"Iiol ~O./"O"I'(IU",'){Z.ZI')( 70) vI'

!fA -'tAl +7Al. = ... No -. %.1'1=-0.07YINDo .. AJ

/9

~

~0Il1 5"00.'

ZQ.3o i 'to

«.= 100:

1/ I 'I..

(loS) D ,...1 -+---+----1

$'ilO

It <t.~ 30 'I'

e.L:7- .

•• .10 L-I{QQL"o

=-il./$3M

:: faL z 10,06'1/5') = ('!D.J(U)(1o)7. t o.Ob'!l> )1"C2. EX l.{/fl1,/~'){ ~ -;'I'J - o./IZ IN ~p

s/ore : o./L 0: Yo/7ao 0,5?1. melV PaJ"rclrCUJS~ ro1)''''M,I<C'<>S41''

1'"(,." "'F' f?"C.l =-().IS3~.I/2. =-0.0'1/"" /)owN

Beams with ConcentratedBending Moments

A. = 9'f$<4 R~=."s /Z'lSL8

V I(...) Dt- ~I--....:.-~I

-8fIS

~~ I . lIS

~

~ . I("-~~

. , .

l. • '''''

F60

y z.o r----..J(LI.) Q~---t---.J.

I---_.J-yo

o_~_~20

Yo.

I·'----~-I---...,·t,a/O'"

V(u) O~ __ ~ __ - •.....01 J.,.

/10

"7(u.".) 0 ~ .....,~----;

II

EJ )'" =Ztb us·.§] I $40 ........ I Z.40 I

jf

X f ~1l'I' I~'N;< )(

Irs ZU-J<I

4= Itt"" .. 3'13'JSRlP IOU F5D ~~=2ZSN

I ZZ$III~=IOL6 I., "7r,-;r,,, .. 1l.. i

I

J I"""I

V '0(~J I V

" I '"I III fl..

i/'1 J9J7S

(&.4'/11) I0

1

/01/

-IZIJ (N._) aI

lf3, IN. /'1 " 2/)6e,/) r I.DOC'I) = 3J fJOL4'JN

rv~" r- ZDIlC') to 600(8) C ~'1(j()/.IJ-I~(Nor SHow'" )

¥IXJU! C .... • ,//~"'"rolC'" C.187.r

~,= ,e~=, 11'1.::..4 zss. ;t.II

F~ ,- SlIM r.=2IlIlf " ..,.} 11'(.3:p.j,,,. .:tt. "(L8) 0

I 'IV i (SHE/fA. =CE.to)

(N) 0 I -ZP·7I I II I -'Is,Z.I

rso"~I I

M(11:'-'> (L"/II)°1 01

2/

y(, ) o f---+--- .....

'a Co I• ,.,,, 8oIJU'I1 1/,,= V I L.u rU8

'I1O~5 (kt) °1I . I,

I I"1 ,_'110 (.11.<1 .... ) 0 !6 - -,21$

M.;' zs.l/tl(. %2~)· g, 2?SJN''''''r» I, ZS All t,3QII ..j " O,J7SM.,..

/,1.05111./ N4T IOHlW NIII a 1./StU• (() ~ZilO(,)r yau la·/1/.

... If) - )ColID) • ~l4'IJI/'I,r ~."'''' c .~O"" ,S-S ....

l1.. o.Z1Shl· ... 1.,&2.I{3ItAi

(u. )Ol'-----~

FUo-

., ~J/(l.S_)>> 1/2.$)H. __

.......... ,., : 1/ 2.S» '" .""',..

~==~~:ir':--F."'1SDK

liB,

-1/ lSI)

I

o r----4--.-':l~-_:::'!

I

J '''R'-I."'f I . I

M :(!JI.".,) 0 l'-'::-~---I~-IJSY

~.

Combined Nannal Stresses

GJ ~A. Z,0(60):: "00,...-

s. bJ./10" 2.o(.f0)~ ~ U~J3 -- ')

OA • .b-? &1·(10) = l,?-H +-(pu ....)'" ........A S ,... - ~3~J _, ..:~

0;. '"' j,i'l T- 60..18" 'Z.t)? .....

,of

IJ.~

".."'-l

r V!'I I(.. ,.o :0;&. -

A c:

_____ 1_",

1'7(1..1.,..) 0 ...... ~

;·,,.F..n----..<4- • 10......

I1521t-9" '/.......O.$": EP.i!6.6'

. [:1'1("0" /8.. CS) - j4y{{l}

A,r/140 ,"I. : C1• "", t.rI/:MLIJ

......." 'ZAr" Z,I'M<4 "'CrAT" 6 Oil nt' 0 I" tl~"....

(T:::: - A. _ t::)... .. -dO! _.su- ;:... pit't 5 l.Y' 1.81 (Q"')

"r IS ON atm.., u·lSAnl..,«» -A,+- d. _ ~ S1 '_f'D ''I psI

,4 S - 2..1 1••' - (T'f!N'Y-)

"".J.n

15_-----------IIIIJ!!!!!!""-----~

.4r A cnr)(1"= F~ /II ,o/,,! YJoIII!¥OII'." ..j

T .,..S•.$I£U1~' CZ,)(tDJ7I..... ''~r=tJ.J 01 +-ZD.t.y C'ZIl.9YMP __ ~Kr/I"

AT 8 I"'f -"!Jo(244)'e,,,,,,III.-, • __ I'tiJ!$Sv ... " AXI.4t. ~ril4SJ IJ s:... I#~".'" - "5 1..elfQD S. ~. S(o,oN.-- _ '{In....""~=' ritA

d" 2".9~11I/,....... • tA I. z«~Q'l.A ...; ,if; ...!'IJi!El. c 3$_ / />'If"'" (.41 A· 3'- ~5":.T • ~ = '1324""';

e I ~ /Js.rA .:no,..,- ....a;; = OSOIN ~ oj- 114000 N· .... _ -= UJ.331'11'1V QII::

710........ '1310 --'itT e) "...,.'I]. (S'a)" 21 J"«uv._.... ., S=' ~ z.ISOO ./, %1 ,..,_~ -Ii'· = 4(/07.7) :/7,1.........

cT - 2,.91' J ~ JII . .

err A" 11__ ; /:...2';.., .....; s- ZlU'~ ./Dla--._J,~A:./l'ClI) =36/>..........

ac .. '" 11 • Z;nIt> .#-.... = 2(/.''''''1'10..- "II(1~O,.,.... /1110__ '

,

F, F

A

! Is~,16](2 ,LJy HotulAIl2r:r-r. roee. A-J'f.I'~S~oll. t:/2IIs.; J«'77I1-J I~ CSe;,...,

A - 3.59 I IV ""

S· "',60 /IV os

A, It: (1" ... -ff r ~4;" -6.928 t.a -f 10110(") '-"'IN

3. Sf ""~ '1.60IN3

c:r,. ,. .-1910PS; -/-2o9loI'ri'/IJ'ff(O(lSj

<'tT§ : (1"'" 'f- = 20 87Q Jt>s; /1 0 "'.:--jL.4--jI3.--.."I(La-f'T)

tJN Barra", Of' s~C!,.,o'"

V(&sJ 0 +- -+-----1.~---l-"~ Ii

'5l:l "~ i3~= V stI C,..&o" ., l/?d JI1 jBF'= 117,IS"

CF = 2.$/,..1 S"Y~ .. 69.DS'IJI

~""B"'0=-Cy (1M) +Fy(m.lr)Cr" fy(m.,¥,I.,)'-17Z.6L.S(Ox- CIt/"a-1! • I'IJ'L~~""e_0 .. /In.(3'1.ttr)-By (7,.1)

8, = :r"L.4&r. C;r+Fx" 1'I3'l+/Ji3" 29z.z.u,

roe bJ(II,,"~Tu6E: (AM A/&-S-)A= ~S91A1"; S,=7,3/'/li1

.JIlJ"T ItBoVE" C:CTa -Fit ;. ~=:!!!L~ ~'1qfJl.

A 5 i'J7 7•.ft.

a:» 58// Pr/ r~"s"",J liST /SELo" c.:t:r = ~ - M = -~ _ '1'1'18'

A S ".sf 7.oJ'a: = - b 7Z.11'!>i COI'1PRrr~'s/'"

",' 4-. fJ= 6~ :,B.S'I,1. •'\

.... Q,..9·r%,,:9=39.8·Fy" F ~lNe.. /lS"l.J.JJF;r • F uS • a /383. ..6'y

B;-_-.J~ J..L"., ...3~'I:.::.01"':-...;F/I. Fy"

1 JIst.11~r----';

--rh -bmI b I

I If",.A:b "l..; s~bo/,(j = - £. -..J!!L ., -.!.a -~As,;, b¥"(j :: - 60 _ /Yvo Jj «r t4Uv..,,..

h" Jo:7 fill I.I ~"'''J.l..(T Z -I'IYO/. s > I ~NY" l,-"y""Y""'o""j.:r"·,-w-/# • ~= :: =0,'193/""

,. ..... 0/ '/Z /1114L8//N"

T/ly e= OSlO>", =- ~IAI :.4 =o.z.s-/~".)· s= b,ii =D.tJZ<18 IN~cr" -&OLl$ _ %110GIS-IN .:: - z.y" -11$2.0 =//7*">~; C!'(.~~~~~_3 -

16'8.1 FED FRoM PHoSI.EI1 j·'!2: 21S11J ..... - 1...

PA~r,.~ s-c sas 3'1.37~N'-"'" I'f,_..",. i3,eI:l/....~78.7rN 0::

~= ..E.. 'l-L!:1. = Z2$'N ~ ... 3'1,J7S"/II'- ....,A S 2nD...... 20833 ~ A =so "#' zs;Do/?ff ....,. -

~ t1.D'9~ /,8'1 - 1,98111'''' relllJlw s- 5ll~ I' ZD833/?ff~I.su 1_" a..~r.n Sl/~PAC.E lJ~""""'A/ oS AwD C.

J5

"'- Zl'o (.IS-III

p-------------------F~'" U.oS. 3 -'13: /'I'" COllr,.NT ~ 931SDII'"""Fx. S1'O'" F~,., A n C

~*"'-li -~A 5T£3~-~#~---.ll:4==~tH~~--+=1~Q"'"

z. ....A- '1.5' .: 6'3ZJ"'_

s- zft • ')fJ.3/3_'_ -.suo-!1 _'1~?SDN._- .. '".st.ZS'"__ 70313""

<;;...... -o,ogq -I, 313 =7.Y~ZI'f'''' C,ftltw,s.s"..,ON TO' • 'I_eMotE ,,.rw~N Ii A1tIi? d

f'1t4" '~OI$3-$0: PO/AIr 4 IS IIJII /TtU "RI/<.«"" II-TTllcJl6's

ro ~/t$Hr tJF 4; /I'I=1I8/t,/J'/1I ~ (T = 'o/S ;'126QbS~ r/le.JI~s.~ =4·'112;J , ') ~o ~....~ ,..,3.s~1tO~31.J D:V3Z#/;r"{1L{~.Y'1,)AJ. "/'69m :1I$I:D=/,1r.",,1

e/l.Qt. TlI ",.pr OF.: ,fl=/.'1ll.l4.JN; F~''''14DLIJ ~& trO*',,'1II"''''o:» br.i!L '106 + /tJ1lz S$17O~ •• '.J2,'...?

/I s - z:w ".S2.. = ", + Z0711 =" ~"!c4D'r"Stress Concentrations

63.1 LEpr '*'~:;ttY '= .7~.I"(I=~,.Sl~.=z..18 ApP. ,4-ls-3 (A)iT;, - :l.I, '~l.J>o) '" 39 ,so list" ~

f!M> - -n";I.s .MtDOIS: J/'v - AV~.'>'o • d.Z1-,i:pl..VOeIi-:: z''IoUuli! .: Z-9 7~ PsI

f/oVD-.VO'»,s)Rlf,lfn .ylv= ,r~ 1", • D.3' -.k. • l.300::, - /:!oUlM) .: 317(10 PSI

•'I,-.SfJ(;s;)

..:::::Il>4 = Vl../J. = I. '10 J J:.c: 1, u-I\./J = 1,.51lAo .: 4,0$" AP" A-Iro!

~:::: €,zS')(.J0lllO)AI .. ",.YM'","ao)~ ""..iI> •

0h = Z.IJe!I.ZS'· /.60 ~J4e-.2.ZS : Oi; .. -t.~:U~O)U; = 32 i'/S'O PsiA..!.I. O.O~2S""I7.D'IgjAPP>lS"-f I. JjYft

ef'w = 1.38/z. ..... ()." -K.=/.39 API! IS-3 -conr« eo

"M .. 14 CWIll'l • k~ 6"""11' .:: (/,3V('J{I2.0fJO)(Z •••) e. ~9680Psi(W1-J5}C [@ •• ,y-e·3SPJQ.1?

Problems of a General Nature r:fl.SJ SEen •.; of- rfll t c-o

0- e ~7.)4.JLo C z.~,..._ e

1:; Iv/AN) f) j n,~l t

-1.,! IMO~I

(Jut.~) ! ~.7~

o

~ 1'1, = 0 = /l.slt;J LY."Ito) - tel!~' $",...)f?¥ z: hs)i'l.0Y2.s'= ;;)0.0--1.; rf'c =' ZvAtftl-ll. >"t.J r :J.}"It AI ~

f1Mtf'l. ~ 18. ,S-AtN ""')I/D'!<I /oJ_ ....

7/Y' /1'l

h ",Ax' It!. 7"-;(/0 'H.__~Ecr/.;v 110ptiLIU=S: tl!4~/Jo··......U

'15= In.3 ..........~ ~AIi!El'l tJf' A;6 =tzo,... -) ". '100.....,....

1Pl?oP.u.P cesssSee TIO,", OF CO

�III------------v

~LIZ. ,I' L/2. I

d_'y, 'y,'Yo '1/. Yo't. I(l ,}I

"" :' Ie:# " ¥ ,~

'hi,/1..-"'l ~IoX

0

",o""/z- V ~~VV f>"'"0 ,.II o

~.·l'f"'/Y FlIo :t~2oo

Lr""AI/s,.",oS. toll~,I' -roD AN-(;~e.,-,_

I"V r-.'/./

/ -i,jI

/II

1111

0.'1'"O.IIS,

0.110

80a

8fJ /1.tJ:r{_"" )

FRoM "'~Ot5. 70 l M = 0.60AN.,.., IJ ....OeILLoft-o. I1pl'. 4/s-~ r(1'"'" k~M : 5N~~ (..,.1_<11)t: ~,,1-L~~)(l,,) '!1J11", ...'J

5N~ 6..w 6 (¥,,) J-t-

J/..w = ~¥o =a.6ZS; 4=/'2S"&= (;.2.,>XtJ.,O.l",._) J.3# JoJ........

'1031,... ... 1.. '"b' .- ~/86 npa..

a() 1'0

28

11Z,.

I,

~/~ "'0 IOU m=-!A'] I)

a '..0-

lzp&01 .. . ¥,

t:-S_,., z.p

tlI'

V ~

0'- L, ~~L~-

-pL .310""" ..yZ"'l-- J.--

~

1 IL"O

M t?VI~0 14 if '" 0 ~

11,-ZPt.l!fir:: ZP1,/II" .Z80'~id"')1'/'I ~= )'11'.,1.1 PS .. t:A.% -S,. 8W~'-m"....'>c' 6fZYJ':.,lU

Sa ·Jf1')~/1JY" .

SI!= !!!!!.2)I1Z"-

,II",...,... "'~ 1.8 I

PARr

PIllar A-T" 0.

t

p-------------....

Pili 0 r A-r A - c/. :. (p.7s1l..••; =0.111,</1AT FJJI.-l./f.v;,'. 5= 6- b

M _ 5"2.00 U'I" :/iJ 'Idd ~$/OJ, = 7 - D.SO ltV' , [ ]

Ar 13: Icte= J,~/ S,vn-<=0.,318m:J f'1l.eU.

tTi =- Jet; f1 =- /.ZS {IS(6) =.5/ft) pr/ CSia.5 0.378

.3/l.tJ

=- /. 2S (:JIl-D) z: /tJ 3'2-t> 1'5/oJJ8

PIVorA-.,..l. •

.4.,.. I"-UI.-CIlVM: 5 = IJ.so IN rCip = n 0 = 2&0 L./!·I'" : 5"600 pst'

S O.SOIN J

TUJ

+r.JN

• c

30

o

CHAPTER 4COMBINED STRESSES AND MOHR'S CIRCLE

z.•

o z rAT· rs: (:710) ,. fSO J ¢J,~n.s' &o.Jz¢r. 90'· ~4,.= JlS'; ¢Jr· ~.S' cCIAJ

"",:loItS.·

z~ = rO-:' fY6z.r) " 2.$'.6' ; ¢,.;:/2.e· CIAJz'¢r- 90'· 2.fAr~ 6Y-Y',; ¢r".3l.Z.' CC.4;

~ =-'IU/lJf

I+

31

o z.¢~=I~I ey,Q)-=z""j'Prn,y'a:",,lfJ,.. 90'~2¢r= IZ.6"!· j'~' 63.f' '"-IV<+z -J{~ .,.,~ ?OI'$t'

&Z'o-.(]y~ S'.3·z,fJ,= /~ol~9_/2J.?fJ. f/Ir.II.~·ccw~a-l'o

r;,~= i__-r-_,"-y 1/I!iiJ

-2164"= ro:..' ~Yvo).. 2£""'/10" /J.J' CW

~1:Si tli aloS;JJ<jj"'£ -0 Ptllf6",.IC/IfA/t

11l p~"I"'.. N· IVI/CS; ,..,.¥,sotntr,A""~--tt--~+--....,~....",,~_-.,....:..

T -r:.",.(,2. V/lsi__-t-.... -roW.,...v"

1'-

IT! OKal.oJtsi; r~./'laltsi.;r,.,=ldICS:a;= ''Il.'l.sJ<sl J c1'i.·/6.?,r.csi' ~ -09" T~'(l.Y,~,", /S.Y3' j J

l.;~,/10:-9- I"/'~'J' ¢r=6'o,e'cwdj.=l¥o 4'

-

~.OJ''o f1t;~_It."""7"11 PI4~

r"'A'" '71., Jl.si.4T''fS· ndo; 11/10 <lli

CAoaZ.o

Tz·

32..

2.¢r ..."10';' ¢,..::. ¥so C'C'fAJ

(1",. cr,,,-"oJ<SJ

c1X~#i2.1,.., i

8. z.t1,:: T_-' "Y,o) =.$1.14; tP,.' ~7°e'"24- 10'- 2pl,.=- '11.M"; fIJr= ".30 ac...., .<r,'_Y~ d"",,=YoM5.0-,.

9. zP.,..o rO:(Gtl/s-j .. sa.O',j Pr=2.,°CI;JZp,.. 9,·-Z.(J,c .3~0.;¢ra"· CCI.J

11"'~"4.tT&.a

- W;;~;...;lfo......j_*iil.l,.,..-II.-..J......!t:r~ .

-I~HII:-o__ -I-~~~--.--h~~~.l.-

/0. 2¢r= To.:! ''''h6S):' ,3,J·7°J ~ ~ /L8So CW

2167"'" 90'_ 2fJ,." $"6•.3'./ ~r.Z9.IS· eGW.rye. -80"'~ ~ c-IIJ.311A. "-""....I!S""If~

"'tZlDI'l"""__ -i~.";;"~:""_""""'_"*-'~~~-~

/I. ~¢tr:/1Jo".... (""ltS)=:J1.4'J ¢cr=lS".a"CW2.;"." q,'-2?,. =- sa.Vo; ire Z.9.z.·C'c-w~-8eM'" di .....,I.".,,... dA,' -1Si'l4

I. CT,(=i $/NlilPo.

23

17';= /1~foS3.t = 1'8.2.""t'.. jl1i.,/If"-SJ.Z ·61./J",f4,.Z (Jrr= ra:! (otO/)S) = Ve-S',' fJrr=Z¢"CW

cry: eUMt'... 0&.6/.,,,,11. 'i" 0 p,It'.'" D,,~LIMTO PA~

-r:.Altz f1VJI1"--/I'r '1$' ro tI;

1tN"4)

13.

9-''';' ('"/")=JI." TZ(,a-a ItO'-9 .. IVc9.V';cj,.=7!tt' Ca-J

11i= IJ~+ '.3 al'1l.JM 1'", j "i .1I$"-74.J =JI.1l'1fo.cr,.= ISol1"- dJ rO ~'Qf"IC_

TO 'AI'''''1'""", -'1$:7 ",,, ...A.,. 'IS" n iiiIf..., "i

If.

/5. Z¢,.= 90'" ~r= liS' ec.w<ry= ... -9'I'I#\"

1-r-t-----~iI-""".Jr...,.o

J't

z tilT" rAN -I (ID'YJsO) = z r;.7;'t/J"..- /'1.9 'cW q; ·SJ+-'IDJ~)ls:! Mt'...Pr =)0.,- CCr,J Oi. - 5" -Yll3 • -:35JMP ....

--+

. /00

OA., ;: '1";1 C;:&DN.. / 'iS3nll

//

2fJ,,'" 180'-"1'_" f'YLS): IUo; tilT'" (,I'eN2~$ '10°-fit" "(lyzr):= )z.'; ~ .. " °cw

tri.2-sz..1.

2 tfltr:=o: ; tllT-o·2 ¢r- .. fldo. f1.r- ¥,sO c cw~

OJ=1M' = ZlO""A...in... 8'"t= -1l.OI"!P ...

"'1',=0GIVEN ~ PH/Nellt'll.ST!lEa eLEM/!"Nr

~r ..Sl)M'"

N Ax //"1'''' !:116f411..J'rREU ELEME",r

0".. "-+---tr--.l...--ir-~O"y=

o

.---OJ " d"r .. ¥DJt.s1O""z. ;: t1'j ;: OI<.S;

..z'tr=dj ;tr"b';2~'" 10' ~ ~r=~· ccwor= ..

---@--;~--_--':f-*~+~-""'Z.lai1';1;:· t7Av =2IJICS;

GiVEN'~ PIJNCJM~ JlflflClMUM S/frAR.ST"L6~ IFLFMENr ,srRIi'SS FLEMeKr

r

zz: :z tPtr = 'lb' ; ¢rr = 'l.$"CtI'I;l¢r=Oo ; 4-0·

a;=oy=o;=;=,

q;= ~= 'Io;a/az." -1'Q'= -'IoksJ

23. ,2~"TAN.,tYJ4S)=2'.,,: ~(f'.1P.9.ccwZ4-:r 'fo'1-Z~tr= 119.,v'j tJ,- S'I.'.eCIVerr= 'S+-JL,J = 'Il..S~sl •Gl,: 6S-.1U :-1.7."-y .. -Z$/q" l7'L )

,=:t:!:;.~y=-i,/(5ia:...

o;;v8 ~~.TN.' ('f'htJ);63.V·; rJoJ'31.,.ce...,,;14,.. 90'1-lAr= /S3.y': ~"';7&.7.t:e1NOJ .. ss:« '1'1.7 = 7'1.U:s.;Ol = lS-'I'/.7= -9.7J:si

"'y

36

l.zs' ';UJ'tT:' 711#-' ('% )::31. 7l; ¢tr:' IS;' 0 cw

2f).r-= '1o'-2.'tr=S8.l7' ;¢r=Z"," ccwa; = ~ +12..9=2.3.9,a/ J &..= 1/-12.'/= -I. '1ks-:

u-"'''o---.,;.

--i:::~=LUS:.~~=6.8~1

O&.·-I.,~

U. dl. ¢o-'" J&'- 711H" fe'3,.stJ) =1'I1.~;¢~ 7/.6·CW

Z.¢r '"lPtr "'9D'",S:J.2S-; /JrsU•.'· CfII ~ ""'-0;. -SOO+~'c!b = 'f180 fS/1T"a. = -Soo - '1480= -sBC I'si

~·Jl.SD'sl.

trr=

~1"'"".. 19

11j(:.

'1'Yy

27. 0; ...z ....+-1l.9- J.2a 1'1""-G"c.'" 200 -/28 = 7ZM'o-<1'", ., 07"..,...... ot/z =16 11./"11'....2~lT= TAN-t F'YJoo) =36.b(','¢tr"I?Jocwa; =/oo/'IA..

37

p---------------a;» Zoo +' 'I ,. 2. 6'/ ""PII- t'0i = 2llO-611' = 1361'11'0.-d}:' a'/Jt1/JJf:: 0"":/, :a/,j2.MI'",2~tr= 1JIH-< (lrSfJ) =18.,,'; ~tT:aI').Jcw

rry: ISfI/'fP., Oi.~/J'I'1?!Jio.!:...t-----_tt:;:;___:;:oF~_ttr::.l...::.".cT,·

0; '"0 ; t1) := - S8[j-l8tJ= -8hBhl' ...

02- -S81!J+ZBo",,-30fJJ.J1.

1"fVfM· Oih:= 'I3YU..:If)o-= /80 - TitN-f {'zr2S2.) =/.$f.SO~·17.J·ccw

CTy=~a:

0;"o

eJ/=O ; 03= -18U-ISD=-J31.SJIA..11'"z.:: -IS?S 1-ISlJ= -37.S.JlI' ...

ilrlll'l' = q-~ = 16B.BA./ ...

:J.¢tr = /80 - r,w /'%J1.S):: 1.s6. 'I •fJtr :: 78.212

CC IN' OJ ..-331$-+F:::::::::==-=~4=-_--r~-!I--

_rq

Oi

38

tr.=o

us/!" D::: D.stlo III 1""6((:SHAFT ABC. SrlZ~ GtEII1I;NToN SoTT/JM.

FRtJM FIG>. 3-11: l'1iJ = 2.S2. LINJI. ; riI·2tJ1) LINN.

~ = 17'/)~.J.1:: 1rfO.sIJI_;//Zl ~ €J.tJl2Z7IJJ'"8 = .Me = 2SZ. LINN. = Zo5.J8 ~ =Gif

i! ",t)IU 71N iZ!, • 1l'D~~:..2r. O.OZ.V.sI//II~-ra = -r., = 2..0 La·/JI· = /1.2L.S~'

~e IJ.Dl'lSY/N'

R= r;,,~"//0 Z'9z+IZZU'l..: IS"'" '03: -r0; = /IU.'" r /.r96'" 2." z rs- hiOi .. /026' -IS"'''~ = -56" ,~.

2 ¢. .. 1AN-I UZZa")= '19.97 ..~ LlOU,f •

¢rr - 2.'1-.98" ccw24- =Z~(T +-9II'-I.1'1.'n°fJr-= 69,98"

Oi.

(JSE o-/.,stJO/!J. F=~S".HAPrA8c• .rrtewa IrLEIOfDllr tJN4.rr.".,.

rtr"", SOLclnlW Foil. PR,tJUl'f J-3D" Fl6.J 73' :He='l:s7l.U./6;~="~DOlQ./il.jZ .. ffDYi~ =- /J.J.1I.1/IJ:!; 4" 1'11l/A: = 'Isn.Uo/N/a.:J.J/J/"lc a s(){) I'J/ ""dj(

r, effO~/' ='22-0.66Z7111.:J/'h= 741~,= 6J'OO/O.6t.2.7 -96.SCJl's/,f..T;.,AIf = Vt5,Dfj ....,..f'-SS s:= 118 70 l"SiOJ - 6'100 T 11870 .. 18 770 I'SIdL ~ 6"100-// ell)::: -'/'I70l's;2 rJo- = 1M'" t:",;:): .5Y,/{6 0; tfJo-=2'1.l"

•2¢,..= 2PtT~"tJ·"/W."'·;4=7l..Z.

"'it~'"10 U71s-1;,,,,-7$,,,,~

+

ITi

Use 0- Z. ZS'm. rOlf: SIIAtt:rACt:.. : .sreess eLCAfENr ON I1Dm,."

rlOH S'/UvnMI ~ ffoUG"1'I 2-3/, FIt; .1-.J~/'18~86'1oa.,w.j rc'l~IJ~C .: lrD:&J. - /.118 IN:' OJ - "'Ie-= 8l~(U.lNN//.llaIN] :: 71U ;Os/ rt!llS/I"J _

tp."'D'#' ::;lZ:: ';'.J.J 7/1/; 74" -r;/b = ~ tlJ'III/z.2Jh~=171~;/i'cr;.".l/- VI111+-JI631> ... Je&,po<· rCj =.38'3 +-.11&7· 77JO'Sftrz. "1~61-3361 .. -'1,s;Z~IT''"rA-.J-' fl%ea) «z.cs:~ = j,jjO C lI>J

24- .. '1o'-Z;r =87$'4- • '1.1.1.1 0 CCw

386J

us« 0:> r".o.- ...nut SHtPr 1+6: SrRt:u cur.NeHr ON8.TTIM.

Ro,., .J"'CJTlW 1'01 Pte4/JU", 3-3l..;FI6< J~8: /'f;4 "·I,'-S'~b."",·7,f .o,J7~1.J;,...t = /Tl)Y.h ..a Z1Z._!; .!-%::.2SbN.....J' /O~ :-Sl.97 ",r'a. ~0I11'-'?6tl'1Q#'t .11'1/1, -li - 'p. J /Z27l.",.,,,,,.., ,

,. - 2: 5lfY,..,... J' 7; .. .L= 37S',;{._ • /0 __ ~/s:.Z8;ff1:..-;:-:::--=--:r==;::::==:::;:::-- __ ~i,. 2i'.5Ylf........ ~ ,.../i''''1,:iAY:: VIS:l.B -, 24'181. ~~S1"""tJj = -U.,/8.,. 34.s1 • +'.01",PI..:T"L· ~2.6.Y8-se.s» - -.s7.as",~..G( = Tao. (IS.Z%,.\\5 )"10.0 •2 ¢tr = /80 "-ri :: ISO'\2l~:: -ZS-" c c ..,2al,..:: 900rl( = flo'Ill,...:: 60° CIA)

'10

.5UI!MCE FLEI'1EH1"1{s-r,.,"ll;; lPS9l £+ UfJI/"·:ds82 P.s;0; = -Z93Y +- )18'Z.= 3?8'si01 = -l94Y - Z.182.= -{,J&6~"

4i

D '" '1.00 IN j A" 7f'0n•IZ.s7 h.I'1.; i:p:: tr%,.12..$'71Nez(J= Z-_ -7SaIlU.I = -S"9h8's:

A - IZ.s?IN"

_L:. 200CfJt.'·IN = 1S9Z.PS;,- e,l /1. .S711J J

1 iT"I= -S'l68'~1*'

1;;&/$9,0Oi -s'/~8 di~+:r~---::;~....M..=...."-tr- zrJtr:: ta.."t-::;~

2t!a=Zlfj.08'¢ltr '" IV. DY· cf/J

2¢,., ..,9D·-ZPtr= 61.'IL·

¢r = 30/" •ccu

D=1.o_ ' A" ~~ 3/,/__ ".::..-==.:.;=~ I 'I

(1;- J:.. :1kJoiJN ... IIY. bill'.. 11FAI.[I,#I• A 'IN_''

l!,.:..!L!!.. ~ IS1/_""S/l, J

7:- 7/t '" YSO¥'," • Itl_ --286.'I8,41f14.'-1 , 1511...... 3 /Jot

r "l= OAv,". /I'(.t,1'o: S7.J ~+- o,K ~£ t---tr-- 1?"1'"",..:: {n.3 ~ z.ac.~~ 1irJ- ~4I:: 2'1Z.Zf1 1'4-

C7i z 5"?,j 'toZ9J.~. 3'19.5MA.0-•• 57.S-Z'l.l::-G3'1.'1I'lPo..2tP~= r~/f8'·5jS7'1) ;;78,7'; ¢0'=J9.~'2¢r. '70- Zilla-'" 11,'"; ¢,.=,s;65·

F•

1-r-J4X!·1';.A'=2'l..l "'''''

29'r//cr.· ,.

'lOU-

\ OJ-

\\

'-1/

.,..

CHAPTER 5DESIGN FOR DIFFERENT TYPES OF LOADING

Stress Ratio

0-= Fh-! :A=: TTlI~_,··,)t..:78,5'lhI"'~

trH;4J< =- 3 S"ttJ AI/ L = '11,1. 11IA./1",sYM/Of

(7M'" =- S£JSAI//1:::' 6.31 hI',,"

r"" '" OS-a> rS-{)())j2 ~2"poAi

0-....::2t'oo~ '" Z$,,s n I'D..

a-a. :: O'"nA)(' - 0;;,: 'IY'-6 - j,5;S '" /1./4 111'",-R "" 0"'1'170': ., 6.37/ - a 1'13MltJl /~J'.(f; -_. __

STIf:ESJ

tJ 1---------:7)"1",,:':"1::-

Icr: ~ ; A = (;o......Jao__ )s: 3a:J,..,,,,, ~O";,/W = 2IJ 11'1.' AI'/., ~ = 66,7/1 I'D../~D'''''_0"'11":: -8,4X103% .: -26,7/1'4-F.....=1J.o -8)/z. :::.&J2,;0""_ '" 6"XI',l ~:::. :20.0111'4-

0-a.- "" t7)14J;. -0-... ::66..7 -Jo.. ;:;¥(,7~PA.R :: 0-""10/ _ -;, 7/ 'I

/trI'1Ar- ·/~6.7=-O, tJ

(j= '/A :A '"{o,'1o IN )t..:: 0./, I.,Z.

an.,.. = 860lB/ .. .: .s:J?S" Pt ./".'6//1 J

tr, = -ILo L8/ - _ 7."-A '11,,., /fJ./'IN .....- ...,IV'S,

r_ ::(abO -/~}i= 37tJ LS

er_ c::. 370l4/A:::. 23/3 PSt'

(J'.. ., 0-" A,,( - /7"_ '" S:i7S - 231],::. 30' i psiR '" crl"llN / - 7>p /P. /lTnAJt'" /531'>=-0./'11

()= /A; A::. lTDY\, = TT(O,J'S)/Y.:o.t/OJlnv'"O"MIfI'- J840U/0,1I1Y11..r:: /, 2.'17/'s,'

trmlN= /SlJUh,H.YIN~= IJSBI'S/F..., :: (LeOo+ISO)/l ==97r-LlJCT :: 97S"L8/A::. tl8z8Ps;tr '"&-M4,or- 0-.....=/I.Z'7-,tn.9:: 7'17ops;k''' crhl/~ _ IJS8/ _

tJ"'1t4.t.- //G ZJ/l- O.OB)

o h'----~_.,f_--T;:;;;

0'".... I-+--.....Jr--+---'---\--

------------------tr::: F'il ' 14.= ffO_';, IT0.0,..-.): 7. ...

/,', 'I " - .",,,""-erMA,.-;:' 780N ,1,.117,..",,1 ~ /10.3 n~1A.cr;, I/O =: 360% = SO. 9 )11"'4-

Fm = (f80~3'l»/z =: .so» Na":..... :::: f7o,vM =: .IJo.6 1'11"'4-

tr4- = Ilb.l - 80,' =- 2'1,? /'f1"A,R = SO. 'Y//I.J ::: d. '162W 8~1't"': cr = .a: : 5 = 1.5'I/,IJ:J FaI(J' y

PO/NrC HAS n,.,.,r """ ANi)

L"Ulc.~sr O-ev.£r. ~ rl'lWl./J·,N ~ qJSII'~:MAr /"SYj;r.)

a;",,, = 38'/0""" s: 2'19YI'S; 18•• ;----,1.Sl/ IAI J If

0-... =('1ZSI t-L,/9' ): S'Zz..,s,·2.

0-...:: o;..A:r- 0-,.....- 3' ,1/"-)v., =- 'USl-ftll.-= 'Ilfl':>1 ./;':':::;,.1_,- __R= tr",%-_=Z'I%S-I=O.Z£.7

M

o:_ ."'\

0;. 1-+__ +__+-"_+_

. 01-----------::=n...r1')( 1)(:.tv rv liE

16'00£6 / tJbOl.-a

o-:~ :5=3.0'IIN3F'LSY~1.7

nh4t< =(500l~)(&J1"')= 30Q:l'Lts·'A/O="',

1'1",,# = (sOOI.l)OI/N)= S~LI!.·/t'I· nl

)1_ c: CSo~ ~5aIlY2.=/7$, L$·/,..1'14. = 30 Doo -17S/JIJ = 12 SObL-B'h'"

trhAJf =: 31NDtJL,.'N = '1S{,SI'5,'$. Of/N'J.

IY",,,, " S'OO%.Oy • /6VS-'s;,'".."" = 17SJl~. cY ~ 57511"5'-

tr_" /2.sD~oY" '11/l I".sl

j(: ~ = /"v,z =0./67cYnAX qIJ6B

:-:::-..r".......:...:...._..L..=::-_ ..0

:J:q=======:J. I_~.'----' -/1/10

[1J I3EAn!

c.

M:~o. .A a Co o

17;. f-+----'r--+-..L..--'r-5nlE55

o f-----------:T<"'I""N'"'I!:=-

£ [-I -Go -~--J l=:t$<=Qo ====:j

c;:IL-' 1 ~{D~Sll<l----'V .

/'1 '-- -5000 ~,.. t.=r1HI'"~a.lN)

0;.. I--+--~--f_ .......~r--STResS

<>...------'7

-30000= J1,~111'11tJ<

''IS01tl IVoN SEAM £bS.J0 .171''-'! . US,... 5e-£ntJ#/.l1r ..... .2.7S,.. !~ 16 c1 1-A e lSS\SS'l /7-.",,=eJj(.1I 14~&"" ~8sa,.i ~.,

! f-;;..~--lT,

8fiI;V I .s.~;l6<lJ't ..V

01b b

0

J-~v .5= /11'10 _,.,i1 :""'../S5A1._=/"'ft

cr=. 11 &NiJlilG/'1: /,/S;o#·_':=. S, - 1'1'LI""".........·0 D!

.s,tenu; IS A SuPP.tlTED Cfhvr'Lflrt<1l..- CJ4J~(0) Itf'P AIIf-3.

OEFLI:z." o.J PRoPDf>n ouJfL 7D F=OI.CE. BEND/N';' /'t IltE"''- P/l4PIC T7o...l/'£ n l'iJ'UI',EJEFc:WCfh'" Ar: Lal>o 17:

1, :-Nt- : -IJ a b '3£+6)·-,s ILEI LJ L~ J

-J.- .r=~=: {).OfO/lkAo."lZlZZ,:Z2Z2Z:121zaZ;2~" L /1.

r"·O--j O.&D 5= SltJ~D.JO"",,,"J

t:: L07'f!0" A//_,.. z.SoLye AJI{ P:

F = 11.EIL 3..".. = (1z){lD7x!l)(d.oQol{lfalN/'o AI4-~b z (3Ltb) osJ'fzsf[x,/oj+ zsJ

p:: '1(".'18 JlIYa M"

Fa ~ ~ : 0.25".,,.., j P, ="11,7&fa-z f) =11.7N: I'il'l',.,. = O. '10","_, fif. .16'.7 AIMDI1ENr;1 : _,...._

- -P4.b U. ) - -Pf/s}as>MJf- 2L ~ rt. - 'L{Va)'- {<tn-yo} =-7.6/7 p MAIOl'7I/",

h8 = Po. Lh (j, -ec) == P{JJ» 'as) Zs:« z.(YO>] .. 61¥1'ill' :l('Io)3 l' = r

SUM'" JI(lf

1i~/8.7.v; I1AL= 7.617(11'.2) -= IY<t.rN........; eJ;,t!!..= 1'IL.5#--. _ ¥7Snp4.".r;tA>ID - /11#'" 8' s 11.'.14_"r t -. • "It, s '1,1 AI.",,,,,,) 6;1,::: Z97 111'11. "O"'MIN

Ir,... "(y7S~2'il J/.z. :::386 Nt9r...IT.. : 'I7r-386= 8'/.0 nt9L

f( ~ .-",.,/ fIM,= 79/''17S = O,6l.S

JM.::: I rrH·,.. 10,........:::1d7.7f7p..0"'""".4,,= s IYYIJ~"':S /)If

O"""'N = ".,~ = JS,.Xlo'",._ = lo.'1hfJ/1..S I'IIJ~"""" 1

0',.. =un., rro,v)!;!., S'I.lI1/.... 51'/(£55

U-.. = tr,.,."-tr,,, : 1II7,I-Sf.I: '18./,111'~

R: o-~~ .....= IO,~I7.' :0,"47

I> f---if------~f------i: .R..

Endurance Strength

/t' f/ 10'10CD; SA: lJol<~i ) 'AO'" 15~S" (F/<f. s-v /for /l.OLLO?Pl</~nt¥~,,_)0. 7S-/lvON! - t! s=. 0.90 lfd:;. 50 q} : t:_ =;'0, C~r: /.o .err. 8ENru",~R= '19 Yo / CI!= O,~I

5_ I", .r,.. Cr c.: tsr c~ .:(Js,KS,)to.9/)f/-o)(/,oJ[ P.K/) = /8.t ~~,.

ItIS/S)60 Ol/r/30{)j S....=79J-M ......;S ...:3acnpc. /'1AcHINElJ

l.o -- PIA. ~ t!~= II. "Ie· ~ C .. , I. 0 •C s r =. I. O' I?:::. "'IPI--e.Je=O.1?1• I J

$",' - (~oo)(".'fO)(J")(/.O){O.K1) =U'f ""A..

ItIS/ '1/30 W (Jr /30D ; .5-0(= 6761'11"'.... " 5.....=:. zeo HPJ:<.. I1/tCHIIJa>; 1>e=.ZB.D""'"

20"6of/ecr--Cc=".tJ7; C,..-/.o, Csc""/-a.; R"99:to-CJt"'(J,1?1

S 1'1 i ~ ~ L O.!J7 -)( j,o) ( 1.0) ( tJ. fI) =:. /83 11t7A.-

At5/ 301 Sm"C/Las ~:n.: S-'<=/SlJJ<.s/; S'",= Sz..~;/ M'+CH/A/t;.7J

O.60/#Il/A - (!s e (1.93; C .... =. /.0 / t!sr -:'O.so @YllttJ/ Ii? ="'1..,,r-C'1l. ,t). ,}S'

S ....' ; (s;;l)(o.93){ 1.0)((JJfO)(~.?') = ~9.o K(/

Itsrn 1f2-'1Z. "1M..: S..u =- 6J "'5// 5...e-Ji/.oK.s; I'7ACHINE2:J/).)J;r"" -1:.=1-0 J ~-" /.0; Ccr-=1.11;~=.9q % -t!L.(J,EI

S_ (= f;l t(o){I· oJ( /'01./. d)(".rl)::19.VKsi

Design and Analysis

Problems 15 - 18 are open-ended design problems for which there is no unique answer. The GeneralDesign Procedure from Section 5-10 should be used. The loading and support conditions should becompared with the cases described in Section 5-9 to determine the appropriate design stress. A designfactor should be specified using the guidelines in Section 5-i When needed. the endurance streng1hshould be computed from Equation 5-4 in Section 5-'I.

15. The link is subjected to a fluctuating normal stress. Use Case G from Section 5-9. See also thesolution for Problem 1.

16. The rod is subjected to a fluctuating normal stress. Use Case G from Section 5-9. See also lhesolution for Problem 4.

17. The strut is SUbjectedto a fluctuating normal stress. Use Case G from Section 5-9. See also thesolution for Problem 2.

18. The latch part is subjected to a fluctuating normal stress. Use Case G from Section 5-9. See alsothe solution for Problem 5.

r

9 $;o,-U 171110/,rL'Cnl~llNn LDItO

af>, '"' 5'1./ nr:...; 0-...= '18.6111'1-

De J/(f;./V' f7~LE"" - Ho UNf OvEr >OLU T1IJN.SutSGot:rTl'AIJ": Ke:1' 60 """,,,,,-"",pTH Foil S€1't-r A-f7I1LICAt'/M/.

coes /O~~ IfI~HBt.. ~rfZE;J~ rH 1'1A-rEIUALJ r#/AiNETl- S7DeJ;: rdltl'lerJ .INTO

CHANNEL 'l:H"'~F- --=jf. Cov~ /OE~ T/tf'E1ZJII/Gr CRo,.> seeLnolV

DEPTH - f)EEIf'8'- Ar c seo r Zess DCl:lP AI~ Srll"~tJIlT.J W;f&2E nOM(;.rr/ S S /I1I1-LL GI-.

DA-m IM£ sAnE M .IN PItDe. '7 Wlfex~ A IV .$"/)( 7.7 s ~ lSa4M

ivAJ PILCtP'S~I1. FLlJt1VIffJNIf 5TI2IK.J • IIS~ CArt: q. "tr =2.Z",.},

.L ~ Kr ~ . ~ .$r#s/.N = S +- s.:' 'SPECIFtASTNt'13t.S.78:L S;=3~'7 '" ~

4"-, - • cs; A95=O.lJ~A'r'~ o.DS{;J.,:J-'IIJ1.);;O,/I:J/lJ~:a,.0766Pe.5,., =2oi:{;"j ;PII.. s--t1), 0. =t/o.IIJ/O.07" = I.ur ~CJ=tJ.e6

C_"",f.o; Cs.,=J,o; LeT 12:-0.91- CIl=d.8/.5jOOb(.86)6.sIYzoj= /1.'1 I<-S;: 0-...= 5"7.5"7f7S;; P#,=/'//IZ.PS,· -1"/totl.7.

-L. =- 5757/ls/+- Id)(////l)1's/ = 0.)9>11/ 58 l\9O f7s, / oJ 9tJo'Sl ; AI:;:l S'3cltrO< Y/rLPt J

_ ~t .. '060 _

N - ,l:tC~.. t-O"__) - 1('111.:1.1-,>757) ~3.,S'/'I .. 2.53 IS .1"'flsI'A''('Otlt IF no UNVSlIlI'CONilJdJDKJ olt. UI{(~llr",,"T'/Of PltrA I1X/ST'.

1:z~·1 D!/-r/j. ~J4-IIte- J't-J f'R.D6. 6. ru)c17I1H1"'1fi A/OIl.I'lAL srl{Er.s I cMe 6-.ArO: (T""" '" 810<1rs) J t1I>- "" /2'17 P$!'I A:JIlfsr,., ~Do &MoeBf 5",,-5BiCflj 5'/-=-'16/1:5/lin' 0-_ '" slfZJ. f?s'- ) cr ... - ~'I'U I"s;'· 5,.." ;i.e"'s; j R>O.~7-. e(l: o. 8/ J' Cs FfIA. II,,(/;<}lr rt.($t!

I'I.,.~ J.5 9 1M'j Ijq!'; (J.Of(J.s9):O.180,4' ://.07"".,; p~: IS;J//I-7'(}: d.8Y; 5::' o (tr.·8V){.8/){zo) ,1.7.6.1'5;

ArC: -L '" S9Z'l- or 6.0)( .N24l = .3" ' I IV:::- 2.8" Lc#""", N° '/-- _ '{,"2 01<.JY ,8000 IJ ~o 0 ;>, , . c- . (,1ft-V· 1(3'lZ.~~:f9n) 7

SH{J(}LQs« $Anrp.1c.rolt. Y If" /..I)IIOS AilE: NOT flPPtI~ WIT]f-IM I'l4CTA-,vO

IF OlffYr ~ J(NO/;J,.) NI17f (o";l"Io£.<J~£. 1"0 A-cHI,,"Vl: AIfIl,hEll.,v)(OULP V.Jc AIil'tl.LoY 57!CL WItH I1I~HOL5At,sy'. ,4/S/ '1/30 J911/"'IIYo hlt£ AIlIf;LA~Le /10/ PII't: I'ftvO SII,tfpET_ ("""LD At-oS"CHO'SE I'!slflfl"t= WIT?f .57/-.s¥/N3 TD J)cot_~r~ ~f1(ESS.

PISTON Roo. ~/d P$'-Z'l.. Dlh=O. bOll'i. A:: ""h=". 29311'1'"FLlurVllnll6 Lo sa. FI'1A"=500LtJ n;-NJ.; rAo" = -¥'ot13 CIi#ll'. c"'St' &'

,c"" = ~SOO-ro')!l:: sru.l!>; r .. =- ./5lJo-SSb-= ss» i.1!>cr,... =- f''''k =- SS/J<%.l.-81J/\/'z 1193's;; t'A- = F<;/R= q5lJl%.;g3//,.~=~3S'!'s/A,S", ~8o /AI or1300; tf:.., =-9,~~,.J' )'y= 8'tI.l::5'·) S,..~ 37.Ksi FIt>.r~)t~= tI,'l3 ; c.". =/.o..i t! s r: '=0.80 f4'tII'zJ ; R=~1% -CL=O,il5:',..' =: 6. ,3){/.o){().8tJ)(o,lJ) 3,~!oi : n,3 Its'!

-L :; 0-.... +- fk-rc){ 0-...) ='- 19113'5<' of- (j,()Ja3S7'~)H $')M. 5 .... ' '100<>0 I's; tt.300 P3-1

S/lFE SlIr Hlid/. ~IfOIJt-() At-fl> CnEt.K. Foill. Kt /oN

Ir 1200 OM. 1.1RIfl?VlS'? 1'0 a. SO/tv. 1'1-=0. /ej{ ,AI ..o--,...,=- 2801";'; cr~="'&31I'S/i N=Jf.J5" L5~.VJcO""(},S7J/N.

BRl'rT'LF MA-rERlltL -STATU. LOtk) -c~.U: AZ: AI:: ..r~,,1tr~OMAX" Kt:': = (z..6"L Y7S'DQ).4) = /2 oSS psi Co,,",PtESS/iJAi

,4 '11' N.lJfJ 11f.) 9'1A./d = O.2S'N/iI.-4I/II" tJdJ6l..S"; l>/d -= S:1J/)'..yf'.MI"''' J.z.r: 1iI~ K(;=2.02

u~ ~ • ItPp' !S-JN = S""'.. = /7V ""fJ r",~ • II. b !4J(/,fi. L.,,./:J

0-_ /Z. as::.- 4':1/

f!~/rn.E M If17ntJltL -.IT'lH7c Lolto -CAS.AI: N- S~t:/o-M"J(P"'A1f = Kt-'" .. a..I2.){ 12 tJfJlJjJ.) = J'Il8 Ps/ • [k't: S"t't_A.l P1!o6.. IJ

A ". (I/... ,,,J ~/y ~J'I - S .. t: _ 'ID IJIJD I'Sf .. .2.0.1

trMN l"l.t9~i

B'€/T7"L IT MI'1rE1l.Iil'tL - .i/AJt'/A'- s:rltEsJ -CJ'f.rtr A.:lI1S{' MID,fIEO /IlOfftl- IC';, ~ FfJlJND Fob,.. nIHil.. C~CL.8MEffiOO S7Z~.u Iin.E~£NT' IN F/£L1i'r AItEA

AK/AL ClJ""IlE,u/yE' SrllEU FIlDAr J'1lDS. l o;.,,,,r'''-/26S'S,:;s;.= orCON-rINIEtJ -NerT PI/fic.

'17

\17.1 CON17A/Vea ,... ()7'"OA.SlON: -r s /(t r : r~ ~. O,(J61.~; ~ ';/,lS' J )(~; cs»

CP Sa: fWo6.I M" /S-I-rtJl!S/DA/11 OJ {'J .lJrp# _: Ir 7.100/ " ...., 1I' /6 3n.•~ IN

r= (;..$0 }{ZP 000,1, 'IN.l//.J.,S'I/JIJ= zser Ps;F~QHN/HIItaeae: E-JZJ87"~ 602/J'' = {/Idl i"ro •

()""j - tJA,+ K = - bola f-'''SJ = 'Isr ,~iTENS/OW

{flo. = ~., -~#-i,02B-b'lB1 =-li.Slll'si {)D"'~,,

DiJC71L.tFMoIfr2IlfL.- Sr,4·m ..LoIfQ -~ A,sEG :

A/s/ /I)7CD .. sf· Sbr/>1P ...0; = S;;/N· StS I'1PAoh.: ISB/III ...

III MIDDl.lF (Jf' ~HAPT;f"l:3J1.S,KIfI~_""",R/fQ'D S" ..i!1.,. 137 •.sxIOJ ,v.-. _/ J

Ci J88NI ... ",,~ - ."r__

Sa. 17'O~.'h: D»f'3"L£'/11' ~Z.6.3".,_lISE' ~PSR.f4lEI:I 1/11'1160"2<9-'"

(rII1Zt-G M-l)fY1

: F/~D :r WM/UI WOVLOBF S;fF~ . F'll.d=z~ 'AlN"''''''» ,~-,I--+__ ~TtJ'3 17(2D)~ . I:.'"S::. h1. = /3~:7eS:l{,.,.,,,,,3 ,,-1.1;(

11 • "d S.w~8dN/"""",..y.;BS:'f1'fttMl.3)::'/'11, 6S7J N·_, /'I70? AN.",,-8 IJT iI1:= Z.7K : X".J!:L = /'h/l- .AAI·,..."",= Sll.'l_ ..... ~1fJi)

Z., 2. ?RAJ~E (1.-5.........-

20 '1o

OJ (Jt... ", ..

-'I. A

IS A./t/ll/c(JL 501-1117.1'1'-/ rH &~"P t'l.Altf'

Pr, It 1'1'" rr;, ysn'>;) cr.. -/Z GilrL IM~ 0 It e {2SUI (lsi

Pr "'.f 1$ ~A"u£.! 1'61"'-L.mlE t! Af ~fl. ~oo'$1 (rcAL/")N:: OA';IoA = ~. 60

vI!IN SM~-11Jl -Af--1'/0 -0:;~

~0'/ =-IZ.()~

-l- II O'i=o1-:• :?r,ry=

T 2.187 .

IT

t5'·'''NIZS1"='

fl.t6AJ IV z-1---- ....

(JtAJ) 01---+----11-__ -+-2.1

IU_JU........- 5S~

~ . tza ...... -f.s.> -I . I-, 1 - T 1

'-II Iw...... f R OP11."II~ II ~ltvs~. 0,.. STIt1'1C'- srR.IJrU

s ee: ':14-. 1'5"-Z8 ,IINP flt<.ofJ. 28 /fN.P 2.C1. t<~:.z.() F~KEYSe. 110001te-ilUtE

CAse II: ~ '" let' fr"')I'lA" .,. (?-.... ) .....A ... I (t''''')..,-1X = Y'CP :;:I'/:~r::~Js s), Ssp., (1-...) ...."...= O-~:l.

0-...= )1,., := !'1a. _~ I rHEN tr.) .1'1.. -r~-J...-/r~~s ~...l-,- - bl' ..) .. "4, Cf' f

...:L:: K~ M.... r " [Kt:.Mc... TN ~,. S'r:' + 2-,. SSAA • !<.E(i)II &1' elY 5~j" + ";$;AIl

AUI 1137 CD I 5...e, 67'''''1' .. ) !>","'.)70 nl'bo-AJ5U/'fE 0=50,...""-'; {}=,o.&/, C~"'O.BI 3s-: "0.5775:' = (cJ.sn)(481)(a 8/){Z70) = ItlZl"f{Jo.... 1'1<L=.3nS-)<.!tSSA'" o.7J!> .... • O.r;l676) ~ ~07I'11Po- J P,;'oII. ;:;.::;""'

lCe:Q '0 cy;~~.0 jOn. n/, ) + /50 '110 ::2(} 7'/0 /)0 ""/ = f'rt)Y/~1 10;1.. S07

D"'IN:: /6' rr ~U7Z72.GNol = <f7.3_,...: Si?f3:£/"-Y £)"'so.o _________ . :lL-Loc 1417 0AI o~.HE' I"/lc/"'l..:zo,...,,.., ro S/1".,,,,,:rp = !T1)1/b'" /7'(:].0);&6_/$7/.-._.1 :501..1/£ {<Q@ FoIZ.,na.-n.....= 55:" Lr(l_ .:I:..- J

Kt LN oSs:""-Poll d =20 ........ :, C's~6.qO;5S"':" =(o.96·!d.8/)(;l70){O,517) '=/1'/,,<11'4-t1 = 11'f/\ft--'" [1,,;7/ ..... "_ /s-O)uo~A'·_""'l =/.1. CfBS/Y· ... -~ d. [3 .Y07 N/ __ >- J

PI(CM '£.l>8. 2-Z8, M =~701))/):K-X - MG./ '" I.l. '105H''''''''' = /f.BI /10/)<'> I/E/t.y St01~'-'-

",,,,~ - /.:1701 :7 7£\:) IV

X 1..1'1"1<.01'/f1IPf)LG()~Btl/(VNI7 7'll sree.!?EPV/{;N IS K~G1V1Il~o, COl'/s/£J£12. LAiZGFE" dL:lf{ MA rcl2/;rL

0-

F

~ Sy:::o 30I<.S"; 1; =< aSs/A r ()•.r(3.)/.1 = s~· =$D"opr/CASfF C-,2: CDI'14/1\16D '$r1l.¢S!-SrAT7G ~#"tD, /11A){. .. If~"',z sm~il ,..",rn,o,I+S3"",0 ~X'A(.. CI","'~SAJ.I IS ~nAu..· 771trlt//I1AK 'i"oc"VII.JA 7'" F/2(W'r A"'r:l It~ - 4M(J;AI/ill ,1IfrNOJNq 1+";1:1 r»L$/olll. (/,;IS

1S!f'IIJ"'+U;,.,-r TDIl9,//c ~S'T7f(JQ -CH. '1J Gr;. «-a "'Ntl }'-t¥.r::. f;.ooL.d){/e/N) :::r 3""oU1_ NOTe": 8'JVDJ,IJI~'D"" n uOJ..S

L.<lIfD IS '11/'-1 I#'OIJO U 'J'" ""'I'1-1 '" ('IPOI.4XISIN) '"' 721)0 ~"/N A'~ I'\"r /+O,,.,.,.te..6wr ,-'UNr.

7-e ~ 1/11,.rToz, =v ]2.D0 '&0.,. S'co""',: ~(JSOU,,"1"MI1J'.7"4/e, :: ~ : ~'" c:,p~';-'"8IJSrJuwv "'/',6/ JIt'

rpOD~//J?Evr z~ e,.p = f).eas /N'-F.e1JJ'1 IfPP.A-I""'Vs~ lh"SeN,IID

S=-./.<%/....3; Z... Z,J.). /~; A'" /. 70 ""I:&' h, ...

Clf~ ~:.. f?:Er- f- r:=7[;.7028/;../ +-(16'l5)" == 3903 I!'S; oe'1;;

o-~-di- oe'" -....!1 - 'tJOul -72db - +'Oll 7 •s ,,' - - = - 02.8 P:V"tV-"'" ." 1,9' /,70I - - IJ oDO

'2:,'" Z ./~ .= /'f~,ri

f3Zlt.:::J FLlJCTU!f,IA/ti SH94tt 'sr(/lFu:

l&t-:::-...--.J....::;::--~~--...,.'/70$

30 t---....lo...~:...-__ ~L-_

[i]FLUCTUlrnAJ6 JlOlihlhL .fTlt.~S: F,"On~Lh.

eM", tr: L~~ +- &: d"... ~ 211 2'/ /S'I.N 5..... S... I F'f .s»-9 ~~r-....::l:""'-L-.::.!--r~

Sy= ~i;~",=7S74; ;N=3;S,.~28kSiA-sS',,,,.. I"IIrCHINIItJ ~V~I'ItCE /tHO Cs·o.q,l~~s,.,' .(~.9}(D.81){28f,!S;) ~20.YKS' ~20fOOI'Si (~: ~---+----i~'" .a: gccC"11 .. '!f.oD '-'·IN.

'I 'I

~,.. • tt1Ao => '18tJlH ..4 -IN

5_b~·A"'J."I

~= fL..,. ~A 5 f-o-t

~ = i£9L +- ~ =t.2!!.+Z881Oo~ ~/£ b~ 63

~ = ~~ "SOoe 28eus .~" ~3

® ~ + 1.11'00 ZHaoo.L .. 6'" b:$ -L.:? - d,02.0 (J,38Y. /'.+'a. 1.11.o J /.796-H -r-: ~ - + - ...-

15000 2.{},"'.:dJ - b L -;;r- . 6 j - J.... «~T~ IN lIol-l''n'1f. ,," /.t 7S-"''''l.: 6;:: ,!!/V"'-79b) = /.75/111 ~g ..,.6'O'N,1

" - PflEf'tiIZ'Z/!PI?t:CI.fEfJ",C, Faa. ""1.80m r~vIlII.E. ~·o.80S l!bh~o.FOfVb"'~tJ..34Fj; srr«

Pe: o..gag (f.~O) ~ /,ys'IIN • 7J1E"; C, :( t.I'sY)-a.1I ~ O.8Yl C.3

s":,, Co.8v)(O.et)(Z81<SI)" /9.oS!<Si 0 /9DSO rl'si

EGlVIlT/OtV (£) Blfco,..es l

l----4" -'7...I I

~Ii I10

Q-t'-----.;:~---

.L: _ ~ .,.28~ooJI - J,"- ),3

75'000I : 0,020 + 1.8'"IV U.80)' aeo)3

leeooT ~,

/90~0

:: 1J3'3/ )

_ 0.02.0 ~ 0.3£111 1.5/2 _ o.o-uJ /.89'_.".. P +- 1-' - j,'" I-;;r-N::3.02 01:.. SI'ECIFf b~J.eo/'"

Sf

h Ol/t. r Ott: CLIE -/"lEAN Sr~

"t:KnA-I- :.1...,=R.

{}S~ .b :: I.6'0 oj

F Woe IV # a/, a '"1'... ~N = 'Y--:r.... 2.9D..4.. <>:: '/,//~= -:r: - 1"')" • /.,s,.". _A tro« - rz. Yl' - 11IStloYo HfI.' sr' 2901'1P.. _

(1"= F!.4_ SCOt>/l2)~_~-3i1.7J"1"' ... : IV= JY/.,... a'+J6c.1A..) 102..0 IIR. : IV· ZC?,4y.7 :s.. 91.h) 8~a;rIOCf)j N- 1010,/.1"'= 30.8 Au... HI'H -/2GPIIC/EC) ()IJUH.C /12.0,.)J '0-'10-/8: N= 3"'~f'-7'" 8. "13J) Aw,.,. ~I>I-i" : tV= 2?V.JSl,7 = 7.9Se) TI-'Ih-'1V: II" 82.7/.11/.7 =' v.a~) PVC : /'I G4sn> OAIn-NrN.~sr'it£lvG,7]J ; AI = ~~7 =l.le. 7. L",J-'}) fH~N(LJ' ,... .. '" J AI= ~hY.7~j,3t1Y"CM11Se

~ ~-::. [2l~oe:-~.o.n/N"=//aS9PS":h=~k #'1tHotjI8$i-3.37

---_...

I~laocr« ..€ .4MTWlIAL-Sn;ADY LoItO- tA.!E C/: ,D1f"ss./'1,/S,/I..

;£.F.. or " ;< /S"ODJI,- z. F,c S"" I(U'

F/l.. /$"oojz. f!"'''' 'zI.r; -/o/" ~fl~. S",/..... "'~OClCl/3~ /'f"oo~s; =- nr/"A - F". '" /D6/U 0 = 0,'1stJ""'-.PP7f' :~=tJ..J// ...N.

d; I~ODO~" us. fj-, IN.

~

0 B- 2B99oft). F~·/S'o"/z.6;AI/,r).z.8'i&H.IJ: As c.:; : /v_ = o.z.o'7,",~;D=D,S'13 ...1'/,

1/s, ~/,,IN.Iff I Ii'E'EI'tTJRJ es...,/t.r#r1 A:lIltL. LaJ+O: Cz '" a•• 7 ; 4r - 0.8 AXML L.~ell" 0.8/CJ+.16"D f tV= S_' /.:r:: r az • let: P/'A - J,~.,.ncN}

/ ~/ / ("'A"j" ",Az " ' - It'!,,,)'' _'" / S"!J I'!I".../1.'/.•• /I~/P - D.lt" ke";' 6 --;r---Did • l"zi'/" = /, Z.D. S'/M

s.....//601"1"'" --s..= ¥()0tt11' ... ( t=1".r-8);S_·=@·.'l1Xt1.aX~D'){O.81)·Z5J .,1'0.-

I'll = z.~1/S3= ),{,~ LOIN -IIS6' ~9f- D,I~.ItI(Ol:'1:';;'N4~",~.tML.1'121 RE~n:-rJ Ri!YtTlUet1 SIfS'tIt.~rIlESr: CltStr EI:Cs:o.f}l ...C(=(J.lll

1"- -l- - soox",sM-,., :a 31.. (,1'11'.. : ~4sr § : S...."//3;<'$;; 5'....=300...,"'....,.... , JT{.sv.-...)~~ CfJ~.Av-iY (FJt,.S"-e)~'= (O.r)(D.8J)(O.gI)(J~)~'1e.'fMfp", 0 0 5'5~~ o.ss:N= sr"'/rl'l,u:' Q8,'I/u., := 5.02. e5

I JJ31 KIFP£,Arr;t} -oNF () 11l1it.T7&AJ~lfntlt STIlr::U ~ F~"c.TV"TlH6 .s°HEItI'L SrI(E'S

_, ,,;, /!:.t-'1a..eMir'-#: EQ. $-/.0: IV - s;-r So I

Cp. 1tD7'~ # 2)lsYY __ 'S ,,... S...

1it • 1£.. '110,1(/& 'SI,' .....7z.ysow ....._, - Iii. '3u"".f'a.S~~O.7~f,oA~O.7~(/l3/lSi) -= 8'f,8 1"$1 It U'SI'1P-."tef;· ,. 5'8'1111/...

.l..' == /t..30 of- 0.0)(1".10) =o,l"N' .-.;=5')7 (.r'r,..' FlUJh fif"'. ¥Z)N ~8'f 'lB." 0 ,

53

F

•

-r-- ,...-= Be.o ,-a ·,N :0 7003' psi/'4~ 2,. Tr' -"'01"') y"" 3_===-ILeQ'D S'y" IV' 7/../f1t'/04" =' 3(?",,:r~..r~ «z on,,'/;/~I.IJMINln1U2.V -i'l HM 'Y = ~1()(J(J ps)

~ FLUc.rlJltnJl~ s~~ SrACSS: CArer H;;:;.... _ ~l OOD(A') = ('.3""0(/~4) =-/ Z3?.rL4'/u,

~ .s-60 rr:7A .. , ...... T;.,lflt'/Z. • ~/ 9S t.4·N/ : ~ ..1'"...... a,SSM.~M'5"" 'O,7SfZOfj) :!5l~i ; s~"ZIJ8~si - s"':::. 6'1.1:'$; (1=1(. s-a)ss-...'= iJ.f77)(ti.fl2X0,8!J(6'I) ~ ~'IS'kSi = zY.>«J,-'

J. = r, + n, = .L.[.IA..~T~]:~ ...}lrr", r ra. '7N z, s,.... e'<;r ...• ~ S"~ t:r...' LsI.... .ss.... ]l!,.- 3 [G.-u + ~/n ]=0.877/11/3; D=)h~,7/T:- /.(S/~ VS~O=1.7$'AI.

ISh'I1Ot1 2'15fX:J '/1

1'161 SrEIfOY LDItO- I!.As~ CZ: /I. /J•s:Jy/r.M/flf.

f= f a 28:IUtz"'''''''''t= 6.z.z. N.- " 6Z.~oOJ N,-...,.."IS ~/J

'&, .. ,.,. (O~-.I,,).: !rC/(o'l-3o'f)"",..3: dS90_3/6 A " (/fIJ)

r"'Nr = Ye-, '=6Z2Xlo'3;Y.",,_ ~3 .1Z.'11f1,,"':OAse cJ../#-.s-f"".,_N" o.rs, /r-"'<4/r .. Sy= tV 1",.. .. ~ •.i: 3 {7Z-%..r;' Y'3yrtt.fh.frt.!/ /D"'O COl-O Otlt-4wlV #"t Soy=- ~"?"I"fl" .....

As}u,01e c, :0.82.CR·o.BIN=~

If'?l rLUCTUATlII6 S/'It:Nl. Smesr -C IISI: II: ~ 8diJ. $-Z .

'Ill

v.s' -1--f-..I.J\-P. IS:+"'-~""""~~~Vc~

rLUGTlJltrtW(, "'Dt(M~H. STIlIA5: t:ASc 6-: E@.S-ZoPRoo/.E....,.s jf 8J ¥tf 1 ~} ,s-I,

J. (J';. .-OtS5~: N .~~t~&.S'IJ. $"_'

A- (iv-cl) to.. (is. -as.) (q,.r.) "O,St)~N 14f.: - /00.0 •...- ;$ _lD_,rtO"Ao. zSDo .. St'!J6 ,s;

1I.,rJllU•• ..y/.so ;:t'J,3~J -kr:~'2.3.r (I+f'AA-JJ"-~ rI'tX", ...

~,. IO,,lLSi; S_=- '/f'~i -s::. =JT:J~s.~-5...'"'b.J)(o.8J)(¥J)=Z7.9/C';L.. ZO0_ , ?:J:r(S'oo-)N 11e>.... 2.7Qoo _o.S'l1 _/'1=/.6'1 (j..o ....)

A= 1T(l.)7'y= Z8.Z'J/"t_'"

d;.= ~ =: IOZfN ""3' .z.s'I'J,...It z.,P.'Z.7 __ '"

d;.. • .f!:.. ZZSN - 7,'l(;~1?.....A ZS.Z?--'"

1\..~=II •.v~Ad.D832k_2..oo (fl'~IJ-/S'-I)l)~ = '1/& '= /.ft J-h. ;~iz.s-+-~"~~7.f') = 0.122.j N = 8/8

Z. I auL.l1 UJe ", /L

Ito 1It1""f[1U",t..

/'.z:J'If--'--r,----:rf--,-..,..--l.tIlJ" +-+-~-f--+-~•3 t-£----.....~--l-L.-

lOye:'7?1,.,' ..../ ...... P;-/,.,,&.S.., .. J/(J""'~. c..:.l",' •p.~(O.;;(;IQ} • ~ "" I'A..

L-AxJlf ..

Aj=3·/1"

s,= IZ.r I-tS/S",04 • I"'of J<,S)

$N" $"1Jt.Sis:...·~.Bfl.·8/)CSI) t: 33·05'.{$Ic... ('l

FI<IJIOf PIi?44 3-6'1 J d.W-w. I~~SD f?f; /JlCLV&>/Att kir : (!,lIseAft#- ~Yc:r-/26'Q'o. 5"'t - #(1"'" .30~ &.sD)= lf99SDP~"'/1.f~~~tr..stJ ~/II!W

CA-S~ <rr: L:z !!J:J-r J.4. ~ : N'075' J7f.f-r.+ J),fI/ZtJCr Sl'-VT7OAJIJ NlrN $'..... S-'

POSS/I'cr IlEC/lIISS" /ZInt :5'.11.11-'<1S-' hiLr III11.NNtJw..l. Ill-Sa D~rA

Foe. ~"O/lIlA~'E Alil. nrl1A1JlliM I'tll.tr#.r DIA4CT%.Y 1'I1111/W'HIL6H,.~".A" All o-&T1"'*~ we WIt-I- VSG Flk s-9 IIMfI .1lN" O/St:-/,/SJ/''''FfIL ~~ n d6T*1N S...'. ,4LSD ;Yor F~1"t I~I/I'IIJ Pll#tI'-"~S ' -1: SM:A/ • 7""#/J pGIUft/1J :$1(./,/77.'" rdt:/. S....• I"f':7DL /II~7SI'IM'-'" .Stnee171ltl ~ .&/A/AI.. oN Of .... 8£ C()~PVTED.. rs;l.f'-F ...

31.3t----:1"""t"~.;_,.......:___r_FA",- ff!D6141'1 3 -6S' Ki:-z.z.s zs.Ji. II / \ Y / \ I~A ' "" ./ \../ \ /A & 11"(11)/f- '}(l1"""" z·

a:...::.£A. zS' ./.r~."It~ 3.S".S7I"1P ...II 7.7""......,r~.F... £,.IS"XIO AI • ~z.a""'4.It - 7.'-~

.L. ~~ }ted... » 3s~7 .~.2S){7.U)_ lUI/II $)(. .s;..' S'-,< r 3M.1t.j - S',A.l

rHsIoI .sM.~N{ 101,) ~ l{/ol J=303 'll'l1I..:TPrriSDA' f'y:z'",.,I' ...·5A~) JFIfb", ':1e.S-(J: S....=: 120/1111'.. (e:rT1"'~~ \ .

5'... ' - €J. 'i)(O.Bt) (fl-O) = 77.81'116-

.J..... '" .iUi) •.rJ r 2.ZS"("'),~ = d.3/~ -,N': 3./9 o« =="=,,:=Sr:J:=:II=7V '1,77.8 _

4

INS6" CflSfi At BffCltVSli/lltfHI4!. Sr/24IV4 rH /)fIQl~ /A4NS /fIlE 1"1'/1_ ,s1lZ1TT£r.

F~"f Pb e. 3-67 , ~.32tfS'tJjJS; IAlCLVPIAU$.te. ; N~ s..;y~R&$D. S .....,.~ No- '" scsz 6'SD)~ 9tP.s:sDP~ _p.s. <$Mor 100 -70 ~3~

j='~'" fl2, •• z -6(;, a:= II' ~4'fJPsi //.I~LVDIJI~,.I:", : CASE D: A/~ ~/.,-

.R.e-410 s;..' - N r= ;Z (~go)= q1 j 60 IJ'sA.~,.elU!lll$ ilJ Flt!i• s-s, 7;;'4 /.J yaY 1!/.,v.;Vo P~c:..~-+1SIt/AL/ZeD ..r/q/lJ r4'i!" 116"',g~",..

L 01kJ IJ" ;;~"l!1'tr1!:i:1 -()NE" OILEc...nl1111-j(UUIONItr- ~HEI'IP.. srl2~~ I

FLIJC77IIrT7AIf. SHEmc S'Tl2tUJ - CASE }/ : T,:, - r;.. =- IIXJU'.I'YJ. = .sOU·IN.

A, rltLer' % =- o.SZlA).,o= /'67,' AIeI= fJ.fJ15/o.Jo:;.o.083:14 =/.'IS"~p: rrJ#6 = 1/"tJJ.:JcU#& =- a.oo.sz a 1N7 /lPP.IS'-/

"1"..... =1"".= '= SDLt/·IM- :1'1J1fJsi~I" ().IJ'G'~N3

A,s/ 81'/0 oar /O/JO: S,& /67~; j S-... :IASxsi; 1/fD/ S ... :::6tJIU/ FIG..s-fJ).sS"... ~ ().S7?'':' :(O.s7J)(~%f)(~~)6oest c Ze .o¥ ,,,"i.i Sr,4 = D.7rS....»o.trtrrr) >=/31.3 KSi

..L = 7'"" to ke r.... & 9'/31 ~ /.ys-{ql/JI) :: 0 'fee. ,,..; = 2N S'4:a $_' /3/300 zs ov» .' J .05

ST~A(J)fLDltfl -BRITn.F I11An.; CAse- A-/AT M /()OI.E - B~"ntI EfiJl t ifill) D :

r'" .b.H~ -(o.7S.Y2.U'J~ =IJ.~/NJ

0== "'h~'Z2.SlJU"/lYI).6JS~ =3SS"61's/liT 51EP- PtMA,r E;z - bJ, /6 ·(o.75){"'l-~l76=fU'1S]IJl3

.....;.. = O-2D;7.JS"= D./6 1j( t: ,. /,Sr" ~

/(~ = 2.~ZS~/·8tJ .+1'1'. /S"-z· M 0

t:J;,N" At~ M = /.sr(';.soo):: II'IdDtP.r1 UoIII'i- 4.19.n

/1/' SMt/o-_ = '100fD/1/1IfJ::: 3,3b

Re-fJot1B1 tJllt:D'~Ell1o~ I CIISE & t !:(;J.!:-ZO[)'::O.SDIN. : A.: 111~Yv; ()./~~IN 20 ; AU,"'E" Jt,,-I.b0-.... ,. 0;.. =% .. .J.SDiJL.a/,./~'~N""?619P.f1UT" rrVAL I AIU lOYDCo : f"y-7/ ~IJ $" sa"K.s/

S'" Z]O K.51 {F/~S~ J ~5".... I =(clI.l(~sr)s =Ul.BI)(tJ.,)(3oJ= 1'1.'1~I...L;: ~ Kt:' tr... 76J'I r J,D('),s.,J - 0 ""'q , H- Z .,II !:JA, r 5...' - 80000 1'1 '100 - , 'fJ J - .,0

S7

A I'3 'I '3 Iz 1~ 7JlJL',B C- D1~=7!lJ

I I

I t ! -I I ,

- - -i> r-.7I I

7SO

7!P

V 0La

58

RE~E1lT1J reoes, ~/J6Z. : .AI&lA·.a¥D,,;/&,,6;O'~Il=d.20%, =' 3.~.o ~/.SlJ J' k~·I.'IJIF a...:o !?'.DdH4 At 6tVEI'J j "n= ZI{/K)U1.//f h'r II) It'"_-~= 3an. PsI_,_: ~t- I<t;: tn- 3c-& + J,;Jf3oSl.) -0.218' K#I(.2./.-N $».- S...• 901110 ZI f(DU J

DIt'tE#rlDN ~ ""II.rrlH: I2EPlfc.B:J Td rrt!'TN~ «.so kATe..a.,' :0 a..}I. 1(.21/'1.50.: e,O{(). '13') == 7·11' '" j t.er a» 7.Z~IN.

r/t'EltJ M -= (7.Z5')!JoDl ~ Z'7~. [6'/#; a:» A'A - 2,711 -I"J/

_I • 2..1.1/- of- e lQ2tz 7'7 /J = a 2/6 • h'.: ¥. 63 . .o.K.# Soup" 2/906 • J"'"'-_'--';;;""

REP€A-rEtJ -ONE D/LEcl/oN: Ft..IICT71.+-nN~ ~Ttf~: eMf: Cr, t?GJs-zoA/sl /oYO /I~: :s,= YZkS;;SA = ?1.K.Si,;.s:. = 23n; F/6-S-tl#llrll4u..BD.C& -j.O Dut"c1' 'fI§trll.oJjf:lt~(J.'8t,'Cre~tl.lJOMAG L~IIOs;.,':::~,gJ(b.elilJKs;) ~ 1'1.'1/<:,ja.) AT P/AIHOl.G: d = ().U,II. OM.,.LIr= /.aJlN.4".- =- tl.U1 .t't-::: '1.30 -CUIA!F81At:'" /s-.3a: :0 F .: 25rJOLII =/.1JJ.31's;WtlA t.r-d)t UrI-4.U't".U)/N"'

..L = 0.:.. +- & tZ_ = /.u:n ~ (/I..3o}(I.U33)::r:03 ' h'=tl.2':f~AI S.(.(, ..s;.• 7 Z 4JJIl /'1 9IJD ) / ND' CAn:-:1

FAIt.lllt.c.b)A-r Flt.L.ZrS: ~ & O'~ll'" t).OZj ~::Z-o/f." »z.oKt::irla -()t-PCHH-r. /Iofl' /$'"-2.

tr,,1.. ~ 1._ ~"1 = /,~ IS;J't - (j,1lO){J).z&~'"

...L: 10000.,.. (ie,JrJDD4G)=2.6'1 • K =0.3 j!.Z ~/fI{'l/te.H 7U)oo JrI 9.,0 )

II1(1J(OVl:MtFNrr.: /'J/IIC12EA-JE' rH/CKN~J) 2.)/IKLE14.urrll.i.ef"RAP/i.tJJ.l) IJ.rE S'n,/tI6e-1t NI#'1l51t/A/....Jr.)/JJC~E1UlZ f'/~/fau:ni'I!F-OIl.-CHANG~ t11I1NAle1tIJ~A"'I..YJ"1fr l'7JIl.ce< 77J me f'A,f!'fro ,E:UI'1JJ1lfrE HDL.E -o~- t"1A.t::r "/M:r~/c.KSt. Arl11E !ltJLE;S !7I6A/IN I"1IPOI.I: ()F" 1Jre ,J1tl.r: I'IA'~ IH/f~ Be- /i!6HOI/5lJ //V .2.o0/N.5:t=c.n6..J NEI'tJt. /IUPDLE" t1P'ALr TIl OPPJ". ApPS7 ;tM17!!:/VAi..ELJEwHsz.e.

F 2QIIl+-f-+Ilirl--l-lL--,-(JJ.)

---------~/661 rWCT7JMIIJ6 NORA/At,.rre.e.rS - USC- ~ - eo. S: ZOo ._

(.jUI /0'10 CD: S-y:: 7Ikr/;.s....:: 8otii;.s:. :::20.KS~ r/I.S71. ('/l.-MIW=/.O, C~:tJ.8()1I){1I#': .s:'= O.!Jfo.81)(1o), 17 ~oo~Ir;., ==~r.8?-J.D)j~ .:: 13.9JAI{f.°UjIf.¥YIAlj"JJ.2n..; 2f.<!F~= 2'/,8 -11,' == /P.9 J." (.oo/'/4'(,AI)= 2'1Jat..6'1;=tJ,:J7.riN. i ..JAr==/.nJ/~ j d =O.6ZSiN

J• ~ i?'117 ~ 1-.....,...

Itr PiAl l1oL.erJ:4- =-2.8S AI'P./s~.

a:» .1=... '" SIU ::-tJn'l'Ji J.OI- _-~-.... t:W-cOr VoSD-d.t.zsXa..J7SJ 0' _

a;. '" Fe. .:: 2~.so = 1f67 ,,~/" ..Jt d.so-().u~J(P.J'!J

1.== 0-... ~ COt'0-.. == tl.s-l.Yr e.8S"){7'/61).:: 1,J;l' #={j.BJ1/1 s:l.I. !"", I 80000 I'J'I~ , FAk1J2c./'1-r "'IDOL" Hou:: .Ke-=2..ZS; .J'Mrtrlll~t1I/AAI- ~re£1fI5,

..L - 9.ny.,. ~.ZSJl?J"61)::- ()FJ66 • AI= I. IS" LiJCAI•.AI - &l O{)() ;'1'Ito • . J _

I'/fLr /'lInT B~ I?FOE!/t!fA'ED.

[671 /?cPt:"I'tTl:O .€evetl.It:1J LOH-k-NJ1IN'6 -C4.r~D: N==S"'/~AA'AlJ/ /..!Yo oor /300: Sy=S/7fl1Pa.j 5./.4::6foI1lP",; 5..,= cSt)/"'o.; CIt~O.8/Cs» o.'1~ c.r=/.o, C... =l.o ; S: .:~. 9~)(().tI)(ZSl»= /981'11'0.. ¥. WI/('~Vl!'I.f{rD)R.. = ¥".N (lsIl/.y~~2.slJ1/I CsAr B: ~"o.808'i:6.o.8'J8j,,fit ...~AI(/.SlI/l'Q,d::'/$Ao' pp,~,eoe(tz).po ...... , ISll ,IrIO

Cs"'-O.98 "cerg: s= s~~.: Il?6 '" 288",,110,"J ISDN/1..1 = 2..:1//2.0= 0./&;7: "4 == 2y't. '""~l V IJlJ 1---(-,Kt-- /ss: .4?1'. /S.-z. f!I) ol---t--t-t-...,a;.,~J( '" K". 1'1 = V-z)(ZZsiJo ~ _ /21 / ...A -2.SD,s .Z18 ' _r....IV '" s..,/ :: /'1If/ _/ ""

~~~~~~~~~~~~/- 'bY u~.4rc: 5z'/'A~= IUZoJ'x=aoo",,,,,J 1'14'"_~37SX){/.o) ~''1 i: ) ~,...-J

&:lo :: • n '4. f- /Ire. 0 ~ •-:---:------AT 0: F/i?OM /9,.p' /S-3 - "'/_= /~" =- /l.lo -£t=/,Yo0"",,,,, ~ 6;0, oUr == E.(lS-OdO)(2~) ._

~'-J J)(t) !2o'-IY')rn) =: '11,fd11'4..J~J(-K.:-dM~ ..

O"NIlI' "'k!t' .:?N,.. -=d.'10)('I7'&}=66.l.I1, .. t(,A-rtl.)1'11111..,,,,.., AI ~ I.,., "fr- 11 fi:.ow)

§J ,="'...";''''N,J; 50':K--_, IlJtI2JJ) >.?''J''~'f>9.(M/)M,r"""",. Su2"'1lZ),::. ~/S()() 0'1"'4. OItZ;WI!;:S; r... _ \ = '1ft,.,,"'"1'14,; ,f Lrt6j .58J , ...

n PP3' 4IS/13'10 OQr 700 IfItJ s"... " ZUksi Olt !.FloJ'r''''

60

~ .FL.uavArtJl6 )JOTlMAL. ~7"R~sS - CAsE 6-: F"".. =.100UIJ F"AJi"" ?aUBPm .. (:foo+-100)/1. =S4o£s ~ F~" 7'0""96 -£o()u t-"",IIN F11, ":It.urr: ""IJ .. O,"%U" 0,0'1-8;%.. z'/{./S=/.ta

,l(t;'= 2,2tJ /lI'p' /5-1 Or ,.;.10" F1-+--- ......s- ffcl~z.= 1T(/,ZSJ/]. =O.I'lZJA/' ~;:c":r;~~~fV D~----.J..

0;.. '"~= 9F_.. l{{stJo) 1#'" ..b/flo If/ ~;;::6~',q o~------,";5 S a.19l-IN' M.. u~ .N ) , cs=o,q,

O""a.. '" ~= ~= T< ZIID LINN,; '117l fSI '- ~F5· 5 ",/~z//I~

Ihll 10.5Zl II£': S" '=' '!'1~s";S"..=9ok!.; .SAl =Z61<Si (F165""tI H/{C/lI.1lf!r); s.,,'={I,'?5)(M·I) (z...6)<10, o,l::Si

.J... cr... of'" I<trO"_ e /0'110 +@.ll1)(I//ZZJ =a..n5" i/=/.1'f' U~N S J.... 90000 ).tUiIiO '.:.:.......;.....;.;",.

Se~fl!Ds,6"j (INC~SE' 1/ TO 3.4:0 O~1//6HGIL BJ'VSlN6 £A(((frS'l- A...8~Sr 'AlSll!lE' IMP~ollr;f1EAJr IAIDUL.OeC Ki;' =/.0 411""{At(~IF/l.. •

..J...: cr_ + k~ tr.. = /1)'930 t.t;·,,){t,!/7z) =0.32'1 ' /y =;'08 OKN S'" S.. I '10 OfJO .20 IJ/JO ) ~..:..::.:...;:..

or10 1ub too Ui'

leO

'I 0 f.::=:::t-l-,.+-

5c~ peal. 6'1: /NCIlEIU~ J'lA-reJl.JAL...S7'R.~trrH TO G-~ N Z3.0.fRY AISI /31"0 QQT /000: Sy=13t,4(~; 5..=IW~i (11M II'-~) 17:'"t!l.4V(;.

~e /'IACHINSI StlHAa:; S_=Sl>lai j s..!.. %ftJ,fs){If"eJl(~h' 38,S-oKSi

...L:: 0;.. ~ ka- ... = /o'/Jo +-rzJo){'II1~) =0,'31/ • N= 3.;2.j, ()J::H so ,f'", I NOlo 3B soo ) :;.:.........::;;.:;,;;~

l?t:tPt:/rn;o RE'V~LSS7 .S'TI!.ES.!: CASE D: Oe.r/litol Fa/! .N~.3. O.

SPE'C-IFV A1/H7:~/AI..; ~=s..IA1·cr,..~~!+r8: t110::= 6tiOU'IIlI5 -lTd'ih.:: n-{/."0~ :: 0,Oft2.I/l3hr F/~1 Aoh = O.Il~o ::17,0'; % .. j.J~~ =-/..JI:JKt:= 1..0 ~ IS-ILer o;;a (J;. .. I<~'" ;:'2,D(fea:»£O." =IZ2Z31's!

'"'" 5 00!1I8l /~~REG'O5"~':: /II tr ""b.DY/}. 22.3;= 36 b70'S,'

CIL'O.~JCs::=IUiB) S"~:. $"",'j@.88)(O.BI) ~ S-I S'60f~;F~o('of FJ6.S-a J R£G/O SA = Ii$' O()/)/Ps! (I'VICHlJJ£lJ,f'tI~A(l,)ONE POSSJtlLE SoLV110ll1 !lISI 31'1' oIJr Ie«> J 5 .....=-JS;2.ooc(Sl.; l7t.el.IMI&.

Gl

Ii o:..--....:....~

I

O~~/6N 5FCm~ /fr'B' FDIl.N ~ 3.O.£,fIPPi :0 = BoJ{<{,Jr) -tlJz.u)

I?, = BCD (t,l.7J'1/z.zs-= /6B'ILSl1uLn,ur SQLUT1'~.sPosr/~LE;L ErlUmT71 :::2,IJD/Nj 5AM"~S 01JfSf..ra::#1MI If 01--++-----4Dn/6J1 ,.,It & ~/oSD. -CIIJ~ D: /1/= s;,7(J'/tIN = s:.46H_ -J""

HUT trllllll__ =.L!!- = ~ fI1 o~-+-1----..,iS ""~, ",It ..

rtfE" N _ SO,..' All ....K~{6,..) 1, "'8"~"89(/~)

Ir = /.;'1 !f~"'J~&.o)1,.saY6Y2n])7~ (J.""I( IA. I'lll/. -UP 1.6.1"

V - .>... j, '/li8 91J1)v.Z. DO ) :JL er A ~ 6,5D IN ; HA = .;I..Dy'os = '1.00 : FtJe. it.,.=1f,$D ) ~ = fJ.z.'I esr-

me-N ~ - tI,2'lJ a AV/{MD) =- 0.17..IN~RoB~IIfJ 71 -83 - A-RIF: DE".!/&,N re» B/.EAJ F~ fAJltiUl "11JEZF

ARE' MIlL""£" !tJLV71D,a IfJ.!SIJlI.F:.

CHAPTER 6COLUMNS

1\..= l>/y= O.7r;/ya Q.!6'81/1J: xL/A.- Ao(Jz.2 '0./88 "/71.;. - 'tz.o(J()PJ/ : Cc ·f21T1-e- = 2q~(3e,!fl!'J% 1/9-LoM!i GWJ'IAI

~ ~~M ~~Ac ITDn ~ d.'I'/.z.Hi~

1'<:-.", tr"'SA .::' ?"'(J~~'06K4rY'Z) ... '1y'7.$t..8(1<1./",)'" (/71) L

/(1./".- 1.'(,1$'>,#.1". ?,./J <CC -S/l~lZr-.fbN..,S"6iIJF"tJt&1r1~

Pc: :A~I'r/_ ::tfl:'L/A.t] =(. y,/zX'J'z.-'i_ tI~e)'Z;7=/f'~?.3 UIe L' Jtql.~ If J;!81.($M'06)

-1.= Ool8S",..,; kL~./7/; Sy;: Z/()(J~n;; e-- /OM ipS;Cc. ·1~1rL[/DJU' ~ Q?-£o,,"~ ....~ ".0 I '" """"''';

.Pelt ... ;;r'&.e" ". ;r:lph,,4)(.p. Wz.) = ///12. 1..4~'/A.)a. {/7U z.

k '/A. • {tit69(az0....es =- III <.Cc._ &'!rJ~,:r-vOlW.r,,.; ,.tJbIV4Ap~ -(d.Wlo)(J;!lO'O)[i- !UI»(I'I)Z. I: /tJ.5"''' 1..8

tlqa.(SOiV'· )

&'QMIlIi' I ~ - J/i;i. - ".6¥~_od4e/.., .$('1,.,& /IS /24I1NO _P~.I.PCIL" '1'173 ce.

[ZJ A,,= o.s...,/~=- '.1'1'/ /'¥ .

. 1<1./;'" = I.o($'.S");0.,yy-s8.1S)'. /1'11100 ,s/ - (jt .. $"1 (FI"'J.-'lJ L.'Nif ell'.Pu. ".foe-A • r'LCJDJt"")(d.S) = ¥l. 6",rL/J

e''1~y' ($'1.9) z.

A.a- (7J) ~ d ~:Vy.. f (p.'o/"+~.ui)J/t-- t1..rZ.,,,,,,.:L:I..2SFr(!JF1= 7SINA- rr(Dl.~1.)lr.#.s/IJ'/N&.: I../Il.= '/%o$.z.,. NZ.S1 = "3lJ , ,on; J c; =/t''' (~4.'-.5)~) PIN"''!) EA;()S: -""Ii\." 1,,11(4'o1J= 1+'2;><::' UN" -EIli.41C.

pc...= rrI.C:JoJlAJ')(II1S/tJ6! = 1'/18U,tV;).&.

b) F/)(El1-~/)(1!IJ t K¥A.~iJ.6S"(+'.y=fZ,..~ '1-,j#,er-JDH~S''''P.:.e. = (tJ.S'//lJ}(JDDII~ [r- 301JIlo~9'l.-3) ] =/1. soo LB

t'17' "(3')('11')c) F/)(n-PII'oW-: -"4/","" ".8C/1'1.)= /I 4.~c.-.J/I'~r-J'H"'s,,u

p~ -(".S7I1J1(:r#l".J D - 10 ... (1) ....- If' 30c44y,"(JDIf'l1j

d) FJX6'o/i¥tn!": J:.L/4.. = ZoIC ~/'/1J= Z?B;>Cc. - UlNI$!'CA.- .,.,....!3Dlllf')(P,s;,,)

~Y8) ~ :" /7aJ/.4

.),Y. 3"/.t.~/ : Cc. tC/f'~/4 b-f.J: ffSS"{/ ... ~ Q)Lv",,1lJ J! L,N~

I). 1,'tIS'//'fIO"'~t+r PIt .... ,) vJtrl:)~/4'# :k¥.-t.=I.33 <-Ceo -JO)/~'N~ ~j~,(j). 6"7'0) _ gSQl} r '/(!II_)(SOj l. .. /-y.rA"/ Vf~/.ro.w.l> - ".(30 IIIlI) tr ~J.IfI/)') ,M) M flWTltU' 7'\ 'TI'I' srre;

AWM. "Ui'l-rr : .))'. 'Il.ooo/'r i ; ~ ..6" I AUf/ME; L~lII'ijCU.""''';

P 2 €oyO S$7Jo) ~~/.90/H.-v.sE".Zl.l.<lO~"'/,t.. ... t;y-(J,sD".!rt(/OM ') r

J<.lI~-(I.9J($'Y~ -/00 ~C'. ~.,. /) ,.I<.

1

fOil" SoON: c!"41.(6 -1)

P~ - /'0 .. ASy [!- ~';,.e:-:~t]-tte ~. A - A.ry Q:.Lj'Z.5'y '/~£" 11&

:. TrD(',-zer.) _ 2tn'~SK@tC.y ~ CI"~~

.1tL = ""'0"'U--It. 'j _ f.,Q:J S'...@J 3-

sy ¥ 71'" 6" ""'It...),rr/) ~t/-It.>.J _ Np t- (-Ito":) SOy CAe,,) 1-

f Sy l7"trUt-Il."L)

o F./(NP .,. '/oGJ-tL"'5SY{A) 1. _7~~U"r)lu-£ ...) 1r'7!' Cl1"/1t....JO 0f?'.1j

D =[- oVNP +- ~S"'r ilt'L) &. J~2.n-SyC!-tf..'-; "'''1:" "rtt.~J

66

l'~r-r-Ozo()

J~~R-~OO=dJI,J. eD

. (IJ;VPf'<.t.fl-] u-!+-$jV/lt,. CdW,",N IS t.~v,,:FtCtfl1 P~oB,/1-: S'=e ff1.e

j(.L." t:,6SC'I/)= '1M, IAI C~/XIFD GNlAS)

S'" [12CSJ( J.s~/J'j'l/.l,}1L] ~*;,./.1/23 .w -()SE ,r-4fl>D/,<J".. (I' )(/0 ')

CHGO:: .A.'" ¥)'ir" /,so/J7i. = (},I/33 : J<LA,. =w.~.+'.lJ ... '16.1FfJIl ~o6l- r6 J sy=: YO/tS/: .c.e1J/'1 PI4 (1-6) t'c = 7D Ulltrr ~k

I<• .II) /(JD ,. tJ.li: (j-~~)- O..$"'1D'I

ASS-If,*,6 ~u6"; Fila,", t~~,J3 ~''fo= [6¥ 1/11' (J4) :z.]~..[6S'OX~DD)I'lM')"" & /. 3/ ''I

l ~J'T.G (t-'-'!J f111J"~Ib.x/D~d..my)• -.I 1.use D-/,50/N ~a: 0.8/)=/,20/,<,:",· ~ -=-d.'I8,1r:&:f!7~N"

A.. ,W67qH'r ctJ""~()AJ: Wr. P.2,p'A'r/4vlf(. 'n) ~ Eli.

Sfill~ 1 A • .s2._~.S()) 2... Z,l,S"/~ ...rllS6' , A- -F(t>"'-J":} -JF (J,SD~-/'2j"')=O,6$6/N~

Us /NT .: Z.2S/"'616" 3 ,Sf! - nair I'tVcJ>I ""1&6 GPFICIIEAn

lIt.. I ASS"M" Ct1~ .. ", II t.11N<i-:etp.u.-'1J=f"I'..v1' (j<LYI.]?'

/) TrZ 6

D .. 'Y(M,,){Sl7' ~D"1.~ I. ~'/II .

1rJ(30~G") J =vs~P=I.~7JIN.

CH6C.l' Ii.- 0/" • a.rrs")C¥4.- ~V'.J2S"=I'~o LoNt!r. ~

Ce :::= Go F"/~.~ -5).;' S'J'= /$7)4/Now: CItIZG" WDII~O /I""" 7'D Z~ VoIer1 If-reONNae:n'NJ n E~IC6'-1I'M' UN.

117.1 Mill..71l1l.tr D~~AS-N~ POS~Jet.4". aJ"sIPe~ ~ Tt/66-IlD/I,,1} 4~S"~IIAi&E.

C/tEfII'u ,..,tf'f'Eil.lA,- ""'/fy I1LSO If£' USS,

·"e.1MVC.T)PL6" DE&h;lIIS ~Dt;SltJLE

F,s' F"/~""9- 90flO .. ISS9aut)....3lJ'

1/9.11'111'-"I'Ltr D~/t:AIt ,.000J/II,trFj = 9uo := 3:3.s:""" III. 'D-.A'" <TO to

b7

a ll.o4CtrD CIJLIJIff"': A,.ed,08; D=0. ?!"/~Jc ".D/z.:o,.1m~It. ",.Dh.. ",18~/~ 11.~~ ~.~~ /,,(I~; Aap.WZAI; ~£ .. rJ'1J~IN~.'-I/: __L f (. a.c:)]. CP/fo6.,-11C. - N S,'1A+ /-1--,0 ...... ~~

C. =: - ffJ.II'I?JO''IY)') trio (P'IJ:.~:::a.'~))(¥YJ3)7::-B"/S7

c • SOyAPOle So '12.'110(4 'IY'L)(Lty1.,):: 9. 2UXID '& /'I" (3J a.

p. (U[-C-81S1) - Y~f;"8-'S1.-~:""!'L""--f-:(9:-'Z-U-.x-IO-;:')1" //99 ~

C~(JIr#l:) C.OLVI"IAIf4.- ",oy",.,;c ..(/J'Ja. "II, ZS".,NI' p~.l'Z'7SL6 1:40")~ I4= ~.ll'Y/AI;.c.. • ()'/)2d'/JY~~'A &(/,~""""~$/=/'1 !IiIfJJls/

C', = -f'fr'/ IJ''.J{(J,JD)~ (Ir €;~;~,UJ)(1'U1S'~a_S"/2'ZD

C'1.:' (I901lIl)(IJ..r,){ 4fZ' 1SJ. & t'.z'j/ X/DS .(.,) ... ,

;0= ~.S[-C-S/Z.Z4) ~ lJf·s'n.Z4)~--Y(~21'/M~7=/o.rn.LIJ

(J. RcDIt:6D Ct t./lI"IN : a =OolS'/H; ~,. 0% .. lJ,kJ",,~ R:1t

... 1'1geu (.", ... ,..,/1,.- (/,.s-.l9;-r;.A-"'.".,uo/,q':; 1f=<, •.rIIJ,..I,q"'; Sy= 4001JOPS;

c. ...f[ftooooYloS,II4J +-{I+ rrff:·1~1(7'1U;J~-.;'(,7101- (3

QO"1::06)(J";:l .I.~?,x~'/

p .. .?'S{(..S&?'V -1(-S(,7?J"-'I(,<z.."x,.1]. /~nL8

68

~Alrl.lc. COLIIMN: L~'IZm.jS=J,2SIJI.;C.~·I),&2S/N. 0-A... *,2-'" 8.J6IIN.; .h.:t- ().!JOIIV:t; A =Sz. /cS'l.JI",~e=O,60IN !....L=,S/fz. ::O.ZD1~/A"';.sy""J30OOl'si Fo~ltt.. 6D63-n. I:; 1 ~D

_ Il..SO[ 6·60 ¥o,6JSj .t I.J2. LaS/)FRo'" e(;}.[6-/2J OH",,- 0"'Yz. - 1.563 I+- 0.130 S~C(2.(D.3IJJVO..5bJ:-==~YIl~'1A/O---;-;")

0;,.. =3'1$8 PsI' - )E&.('-I'I) = 1'/'1..t= O.60[see.h.~~3W)Ib.~/'N#')-0=t1.1J9ISIN.

1211 ECC~NrJ!" CtJl.lJl1N: L=~.1._· S,zDDMoA-f;1'-3a.s".11I1c.Io.!2lJA/3-11'/SQi,VO ; j).=~DIN,C=o.,& =1.7SIAlf:'''-,.,.)='IJ'..s- .....-Ii - /,/6IN.(,zS;Y)#Z'l. ~l._ j 1J.1<t:868"..._; II c2..2JIN'/!S;Y'j-ln,_ '1.

I = 3.02/'1" f2,s, tj '1)=J,2S'1Jt/O'.- ...'!~=/S/)"......;IIS/ /oZOHIl.: E :< 20761' __=207~.Vp .. =l.61D.oM'~.'=2D7DD.N/-'1. ~

edl-l2.): a:...= oz. = ?D.SDO fit- ~SlJJ(?'Is)~J~ .UOSDO"" ,1. /.".19 L' 868 ~ztz1,Y,j V(I'IJ9X247t:D#J

o;,A1t = 2/Z//If fQ. : 8fJ r:or, = 207A111?d..J TdstI.r71lirJ.lf n,HIIN.

1!tA¥ - PS6)[Sec! ) -;J = ZS; 'I /lit .... IF ~TL Don N,r Y/SU1~

1281 eCC~N7TlJC. COLIIHN: L sr/Jj,JS'"J.N;C=OJO)N;S-/J,2SlJIN 0-A = S1==o.tJ6z.S'1l/1- ;,h- 7'1ii .:r ~ =-.1.172"LIJI)",'L-o,I)()S"Zhr'- 5

P:~.rL8· e=2BIC/6'1Si: e= f/,.,.='O./Zf.lHo '\1 I s I£f¥({dlJ: ~M'. ,,'::;£1+ ~-::::~l.S)5e-c({;;;':~)~:~1S1z8Kd)11=6{,8S"~/c6J((-;'!): "1',..4f =to.Jo)[s~q )-tl=tJ,'utS'//I.

o eCCE/l/r~1C. COWNAI ~L = 'IOJ"'; ~=O.!:lJ.lH"P='JS'/JIb La ~-jF11-'1-rl?_IIPp.lt.-3; 1/- 3•.s9.1Ni 11.= I.S1.IH; h. ....2,Z8/IIJ~= %.~Z.dO'N. I 'f ,-lI.rr,., A2I/2.: 5'y::SlJolJIP.r/;e=.]oXIO ('S;

Le:TN=3 1T71~K4i:: s~ ./6667's/LET rr: " I?/GHr.f~ a': F:~ (G-/?) = 7.soo0[1'" h..sa]fl.,I)"s;_.1 '16 111/00»7

....." ",. s...s? 1.,2.9 '9. -2lJSJ)'iiUlfl1JOJ '=a3l.;Q]YtI's).?Qd U~5AFE' IN.Fe: I.xt 1I?L Re~b: c;/='nJer/CEil/rIAL. LLvto: L=I/..o Frel..""/A'")~/91..1/'f; ArT,., 113'.JSjl:36d

F~'>l1 ItflP ~'3/ A= 5:5'1 ;" .... I'y,,9.1.J1AI'I; A= 1~"=1,Z8; sc, (J.sl(l1zt 19. 7) A h~ •FtrOH FIt{, • oS (<#:: /2S; JH~"COLlu.AI- .1oH,,/SIJAIFblZ_1.4. LEr A/ =3/'4:: Pc.~: f5.~'I){J,0") ~_ 8'60/l6){//1.7)"'1 =- 37sbb L6

AI .J [.-n" b'o 11/") J

£9

CENT"lf!ALLO~D: 51'«./1"{ It 5'"~ TLla,: Sy=J'ocoPslj ~/'I=S

A~rll"'tr C&l..lJI't" S'fJPPu:a YLo7U-'Z. LtJAO: '-=S:SOOOU¥.,l. -2:lSl1JLBASSu""G C(JLII""JI If LOAl~ : ~Q. 6-8: .I.or HPa-t'KLJ L. C,::.IJo

lT~F1i,ulliOl, F 1Xt;t)-P, JI~6DC(JL~"'IJJ K.:: s.e: KL.~(iJ.IJX/8'..s7''rJ 'F':'"I?U, 111.),

( )( ~ ~4~4x- 3 2~SllO /n.6) • 8.7"IIN'1'USES,..TII_" V~Y"YL".r=/Z,ll1/"

H (3DlIO') •Qg Il=r, rv81!!' "~fJ/-lv,Z:y=II,71"'_L/~Hrrt

13'1 I CeHTVtL LOAD: C-5,-J(f sren.clMt.'IV6l.: A -a. (,Y/N'; 12,=t1,Y91/11;#=-3K,-/JJ. .. 1.(J{hJ!p,'I") "'2.Z,: AJ'r... IfJ' -s,=.s~0C0('S;J Ce o/:JD -LoN'8t'1L.Fa.= P<:Jt/Ai= rr~eA = /I' ~(JOX/t')U.N) _ '1U;UJ- )/(J:V ...)" 1(2UJ&. --:....:..:==-

~ CC"NTRALLoArJ: SIf,f(E" AJ Jy UCCPr rl¥e:/ swar, k~tJ. 'SI<LI/L.:=(~·6sJ{/ll.)/O. 'td9 =lrA.9 - LON" COLIJ,.,A/P. IT"! lox,,') 1..)./'1)

A· ~U'f~"J L = /I?SlJL8

~ CCc.GHT7tIc. LIJItO : e=?( FRD" /fpP./, -o.'l7a'N, I9ND ~<VII!' G'Q('-/J): tti .. .rY4- ~J60~=120oo~iOJ'' .&. r/+ @'Y)8lo.'I)8)S'~ 1.//2. Jr--U~I':!""A)~-ll

.;J,6fI L: /J.",.,).... <:(.2(0.'1") tl,6'1YJ.,(Ii'J/J/3'( IfFR.+rJOl!: foR. PIL2 WOO L, •Od ~ II 1z.o~"

70

ECCENTRIC COLUMN ANALYSIS Data from:RI5.~ii;;;,l;Solves Equation 6-13 for design stress and Equation 6-14 for maximum deflection

.-~'dBt!iifQi\.'V4i:!l!QI!i!l":!i1'~~"Qi..:::_-.:i~_$!' Use consistent U.S. Customary units.

FINAL RESULTS

Computed Values:

Eq. Length, L. = KL = 126.0 in

Column const., C c = 113.5

Argument of sec = 0.769 for strengthValue of secant = 1.3913

Argument of sec = 0.444 for deflectionValue of secant = 1.1073

Slender. ratio, KLJr= 90.6

Column is: short

Req'd yield strength = 46,929 psiMust be less than actual yield strength:

s = 46,000 psiMax Deflection, y.... = 0.322 In

'It

c

ECCENTRIC COLUMN ANALYSIS Data from: F~~'-Solves Equation 6-13 for design stress and Equation 6-14 for maximum deflection

, .--d.T~jL]jjll~lnr·,_.~~!, Use consistent U.S. Customary units

Com utlld Values:NOTE: This solution considers the eccentricload with bending about the horizontal axis.

Eq. Length, L. = KL = 40.0 In

Column const., C c = 70.2

Argument of sec = 0.855 for strengthValue of secant = 1.5236

Argument of sec = 0.494 for deflectionValue of secant = 1.1355

Slender. ratio, KUr = 92.4

Column is: longFINAL RESULTS

Req'd yield strength = 39,964 psiMust be less than actual yield strvngtl/:

5 = 40,000 psiMax Deflection, y.... = 0.237 In

See also Solution 38B for buckling aboutthe thinner vertical axis.

COLUMN ANALYSIS PROGRAM

Eq. Length, L~ = KL = 40.0 in

Refer to Figure 6-4 for analysis logic\":j!c!lQ!!iEfQl1t'!!m,,:JlJi'$:m' It$', Use consistent U.S. Customary units.

Computed Values:

Column const., Cc = 70.2

NOTE: Cross section properties taken withrespect to the vertical axis because the load iscentral to that axis. But buckling is expectedabout the axis through the thin (0.40 in) section.

Slender. ratio, KLJr= 347.8

Column is: long

Critical Buckling Load = 4I91b

Allowable Load = 183 IbThis value governs the design, not soIutlon 3BA

AIYl1L.YJ/J" As A S'rli;q-II, rrr: C£lVritAl.L Y Lo/ht:laJ CUt..V,AfN

7?tA'T reHOT m IJv~ LII AriJlIl 1'rtnv .4XI.r I t:= a,va III

h.::: -L ~ ~,"'III $ i/J.//o/Nvn rtrF~n t:iUtGlt- l''''ol.l'1t1LA Wl1I11V:>:! P.. " /6.1 ".

73

-------------iJ!I

ECCENTRIC COLUMN ANALYSIS Data from: Pfti6renf.~SolvesEquation6-13 for designstress and Equation 6-14 for maximum deflection

,. i'it8r,d~iI!!I?Silil""'''~''!''r .. , ...,'''''Xes", I Use consistent 51M . .. ". '~'" ".... • ".".•"J... ..'tV. ...A.. etneunit[

Com uted Values:

Eq. Length, L. = KL = 750.0 mm

Column const., C c = 63.9

Argument of sec =Value of secant = 0.811 for strenglb

1.4512

Argument of sec =Value of secant =

0.468 fordefledj~1.1205

Slender. ratio, KUr = 102.9

Column is; longFINAL RESULTS

Req'd yield strength .. 389 MPIMust be less than actual yield strength:

.. 9" MPIMax Deflection, y.... .. 2.41 mm

Piston rod Is safe for p•.. 6200 N.

Eq. Length, L. = KL = 156.0 In

COLUMN ANALYSIS PROGRAMRefer to Figure 6-4 for analysis logic.l!i!4.~:,",i!!¥I!,r!fI*~!ii~inj·_-':~";:~.$'~ Use consistent U.S. Customary units.

Com u18d Values:NOTE: Analysis of straight pipe. See alsoSolution 40B for crooked pipe.

Column const., C c = 128.3

Slender. ratio, KUr = 198.2

Column is: long

Critical Buckling Load = 8,101 IbStraight Pipe

Allowable Load = 2,700 IbSee also Solution 40B for crooked pi

75"

Solves Equation6-11 for AllowableLoadCROOKED COLUMN ANALYSIS

Use consistent u.s. Customaryun~:

Com uted Values:

Eq. Length, L. = KL = 156.0 In

Column const., C c = 128.3

Euler buckling load = 8101 Ib

C, in Eqn. 6-11 = ·22074C 2 in Eqn. 6-11 = 3.483E+07

Slender. ratio. KUr = 198.2

Column is: 1000gStnJlght Column

Critical Bucklin Load = 8,101 Ib

This value governs the use of the I •

Crooked ColumnAllowable Load. 1,711 Ib

See solution for slrel ht pipe; Problem 4OA.

V-8ELrs

ooo[1J

~

~

~

~

~

~

CHAPTER 7BELT DRIVES AND CHAIN DRIVES

a , 2YoO.IN; J)z,~ n.'1S"/N,J [)/ =-,s;ZS.lN.1· ~V Ba.r

E4l 7 -3 I .. f!'U-.s:zsjz.t.: '" 2(~'+/.r? C!J.,r+s:'ZSj~'/( :7tP.'D/A!USc L:: 7$1'1 -SnfHOIftlO LeH& 1H 2.#)

.4terIfAt.. a FIt.4I'1 EfiJ· 1-yt L=7S"8=' ¥(7!1 -6, 2.1CI.J.,r~s;zs) ;::;/:J9,Y_ 1'J9.Y or Vb9.11/- :17.(/3--9 r-e: U) OJ..

Co - - = ;<~.tJtJ/Ai/~° r-: -I /:1,I1,s- -.s; z.s:'1 ~.:"'IS, 0:: /J't' - 2....... Z( .a..2.o) J. /S7, ,- cE~' 7 ..S',;

is....:: IltJ°+Z.s;;.."'f J:l.4.1"'~2..rJ .. 2.4;.,94 ~.-n •.,)".t._.II'J ] (' ~. 7 !I

e:= '0,0 IN.J /)1,-" ,77,7":' j /)r'8'.Y ;~; sr 84'l..'alh"II'v76D z:» /,a,Z/N. (,I/tF t...==/;?o/,N

~I = /£0./4; 92,= 199.' °c.~/'IY/H. ~ 1};).= 'f.8/N;J)I=/3.8/N; BY&z.-r~/'1 'UTln7 L = 'T(,9, 9 /,</. tJJ~ L = ¥S'lL/H.

~. ~ / Gl.i'" <!"r2.S"A,./C 17m e.>fv urMII)(. If; • 2Y6S Pr/....,w2. /"'I.lI4J)t ~ /3-

IV"; - ~ UJ.::: tf. f/N Jf. U6(J ~£v )4 2Jr(lM If. /pr -.2-S'S'J pr/I'fl"t'1- ;tIN ~"'~

;1Ii .. (?, w, ~ l3.r",,, ffl U?l/)t. ~ ",Iff = 3i'l3 Pr/"""Ni1- ,or..-v IZtrV I z.//V.

/(A-110'" 13·'iS/Sou-- 2..uj P == ~ 2S"Af> ; c!.,; =0.9$1; C.. -/.tJ3CO~~ I'D .....l!iIL =. C6 CL. p ... ~tO.D3X,"2.J") ,~QS' Iv'

11

--_ ......---------V-BIFL.r5

iIe~770:: ;1.7.7/8./f"" .3.3 fJ; P::~r+ /",14 -/6.'1' I.,.; ('.-.9$"j ~.II~rC¢1UBc.Te?7 IIflW$l.-= 4. C(. p...~9SJ(~AS")(/~,7') ..../~.7l-;"p

I!.AT14c. 1'1.8/ /3.8 ~6.aJ; p= ~3'/..,3 ~~::-.'111' ~ ~: j.fJ9~TlW 'fJWllt.a a. ~/'::>~9()1')(/."9){iI.9)=z 'l'1.j,{,

78

7'1

V-BtfLr .s

J)t::r/~AJ: S.F. '= /.'1 ; /)~.l'd""El!.a 1,1'('14) =:S"~'<'I'-.sV 047-i1!A-T/0" /t>Oi]/ 6"S'O~ A.,7.:J ; o, ::IU Y06D)/trt/SW =/d. Z,/h.PoL O. '/0.'1./"'0' P. &. 27.'HI; /f1z.=/S"4D ,,/D·Y,z..~7 =oSZ-ll.I';er ~~rrrtJ PDWtr~cU,,?i,6Y /p1SV~,'" "N F/60 /3-vo /1-, AS"6' q"2'7.'7<~< //t' ,; us« <: :l: 3~ /N.; Le 133,'/#, -/lItr /.PIn/A.At..,.VftL ¢= 3S""IN. 5 9,,, /S7..t-' /&t.? ZO'l.~~ C .. ",9t.~C,.-f,(J/CcU. lowlTll.=: '-,;J{J,")(U,1)=Z-3., h/lJ!EZ.rNo. 0('& B~ =: SO/Z3.9 ""'z..3r - VStr 386't-rS

D~/~ItJ S.t".:::1.1': J)~"'6W_ -/,V(Z/))=Ub -$V Bez.:rM'no = /~/ ~9S- = /.34 .; D,:t./2.(Y,eY1l'(/2.S'I) = /1.,Z./NFot. D,;,: /O.S,uAJ /h.&ltJI.90IV; /11a, -/2.S"D"/'·'S0",,r= ~9.r.,~ ~J!A7'El1 '''IIII"st. % 14.1',(" er ,INn;lLIlI-AfloN ,,, RIf./3J:1 A-r/z.nIVA/r,9r<C c es.r ; /It, a~ZO/H ; I. :l:87,l,,,---l1se- i.-90"",AC1'"lI.tt.. C. -=.'2,J.lIIIN ; ~/-/S7.'I'; sz.-za-,,'; c!9 -.9'1J•C..~/.4?C(J«~, fOIlFRLa '9'f){J.47)V(),y)=--/IJ. YI,(I' /4~~IID,OP Be-t,"r.S.: ze/lM" == 2-61_ez..rs - 1/.!~3 6S"L1'\

D~/';f/= ~r.= J.D ~I1UUN~);/)6.t. p==.;J.,4(/N) =2001, -oS"lI' eEL#"/l.;t-rro- e'Jo/6UW/..19; ~/,.,j2{ViXJDVp-(e}u· /7.'_FoL D• .:r/I),KIN; DJ..-1Y.901A1;~,l..=> 8'h /o.~'f'" = ~3/'1'1'1 ~~lrrm d'C1NE£,=:/',H • .,'-II.)7,(,; 1'1.9ca <. n'/5u,~C ?r."8/~1.':; /3"'IN _USS-t..a/32.Nt; f+.CTVI'tL.c .. IIS,7e//f9,:: n'l,'l' ; Eft.:: /6's.I' ; C~. "ar ; CLa lilJ/Cou. Powcf2=: ~9n)(MI)(It,37) :: /f.33 ;., /41T1...TNo. OF- 86c..T3 = '200/1'.13 - /O.? - ~IF II 8EL.~

,g,'1 eV IJ6l.T: F'JI. fl, ",/7,8,1'1; D2.=Z'/.8~N';t*'1~c,~y.'I~1'1'1 ~IM'~ PdW6(l. ~ 6:.<, ; Z-flJ<:C< IZU; {)U' C:lo.~1'/4J.:: 16.3111-/)JFL.=/6QJlr~ J'TC.1'tIALc: = '1&,'1.3/,</Q,:: /7/,Y' ; (h.-/tll,6· J' C~=. 98'; C~... 9YCau, 'o..,~«.= C9'.X.91f)C")· 60.e,{'/4~Alp. OF- 1l~L.1S" zao/'.{o.'· .1,3 - V:tE ,,"~EL~

A'O£.L€& 'JIll-IN~lCltlf-I}I MI, 1"'4: 1'/7Uf".J: /% "'"/!r/N.

12i':1 ~;fIrJ/'I K"(Ji {,O; PITCHa. ~8 = YYA-Y

~.s"n+nc /..411#" IZ.s(J t.4: AV~. rD'iJllL STI2ENG!1f=-k)W-12SDfJLJ.

()Jtr A NI~ If) aNN" (I.DDIN. I'nu),; r.J. =/¥SllQ ~~Lr 71,

l..41r" 4,,) G'f1e1+ CItAJ"- z.sDO U; r.:r, = /OW=2SlJloLIJ111'£ A/d,J'2D c,,1f.4t/'l /!/7,p'l.I'J7C#; r:,s. =J1/,tNJd'4

§J FA-17GrG OF t-I"'J' p~; /~"lk.r ~F ~LLot.1 0# CI'¥f!4t:A!d~fJ+j t;""/.lJ"'~errw""; ~/NJ""'#tJ ~/.UH/MIi~

rtfttUG" 7-6 ': (fInN Nd. 60 ~/A' .. ZO T6'E1H ,/.fDIl.I'''I

I!ArEtl (7"wl:ll. • ,21.,q "I" (/NTE1U'4I.1W1Do.b .?TY'''' Iii tUf6 (&fTJI)SF~YJe, MeTD~ =I.Z Foil #YQ#f,Qvt..u PtIll y~bl!",f/~III P4Mi#t. I.+ntrll,::' ;J.I,6'1/""Z ""/&9'H'-

a S7Zt!I1.wJ: FAc:ml. = 'Z'.S'J'o~M.. ~NI,. 2.r(lI.f/6) = ¥J;:J...HI'

rAB"'7 -5"! ~/1It:"ItJ M1.S!o CIt~~,/'2.ne 1'71" 860//.PM/<Am"¥I Pt1Wl!11t- f,·'If HI- €~ltl'lwm,'!JoTY~r8t.1Itl~ (e19~)S. ~:: .I.Z ~ /)f!'S"/t'f! '.~a.M77H'; = '¥. y~/,Z, "':? 7o"w>: ,

131.1 IfSTMA/QS: P~17J'. 3.3 j I'OIllIlt:. Itf'1lNG, ::>3.3(:).10)- /;2.,.2/A(I

13Zj i,JfISLr 1-7 ; ($IWIt/ ;fI4.8() CH,..... ; 31. r~ ; IliOU""RAnI1 tI.wm.- '78.67-' ,,~,v~c.M7(101j TYII"s-, t t.ViSr {PIt- .f'7tZiI'A",j

, or -I,z..; ~~,J/~N'4wa..'8.'9/A2-=~S'7.1p1~3·1ZSnANOl· FACTTJIl-=1.7' A

• " ' • J '6WEft. HArJlI~:: {j,1 )&5'.S'71., s111,£1..'r;;J Ml.6Cl ~~. M~'-'M ",.~ L ..' '''" &.".5'0'; ... ~ 3""'-VI~t:~ 7'-'p. Y~I/V-0.1.1'"///,

t:. 3~""/M.s-""I'nu_ '!8P~t.::: 2CfR)'" ~t7r t- @ 7.r;)z ~ IZ,9.1 hrr:#1ii;SUSI!!" /2.8 '1 Y?1' (~8)---';";'':''::':':':~=Ltil'1Jl1IV /Y,~; L:: 128(d.';)$) III:.9/M.

eo

c-tfZf- f~"""1r ~2.I/- '1~"J1'Z._ 8;~~O1-]~716'J PJ~~.J

C. =: 11-7. (,9PneHeS~1,o/~JIf!1nc-II-Zi.6S"/N.-

roli. J.,-/ZS PIT&htU j t:, FUJOr 0(1. ~IJ

0.-1UM - .nJ.;rr fza- <IZl;/J:r _ ~~~!l Z.]=Y7,'tZ-'ITC#ItSa. - '17,'I1."~ 16' 'Q/~/l'pCH-f.r;,S"7/.tr,-N"VD CHNIiI NI:'II"Ha.:7'S"~· c.~ZlIPf.L-I'. YU'F.C(7J'D;/II. : c» ZY/IJ.r= 'J'6'hr:HJ

. ~H/ (!rs-w ~1..= '2( '1~) r z..T- Y7I"'-ble.>::' /Zlf,6 M'UI'VJ -VS~/ZF PI1"ChIl$/ZI/(o,s)=Q...,.,.

De-SltillJ~UN;~, =3/()b'I'I; ""'J.=/MIl",,;II~.~·3IY/6Il~/.9PC,. - /4'; /)6&ljfN PiJ",.'-'· /-s-(zsJ -:!~...rJ" ./J,fr 3GnAND,s ~,.R.+77JY4=0 7~.r/z,r·/S;(JA.',NW..$'rtIANCAlb, #0 elM.,,,,; 1'7liilf'M M'17iD ..,p;,-t' I/'r 31#~.,)·TY'GS,'L4II16"N&.-N,~bno=-/S(/'").a<;'1 - ~ 'J'6.4?HM.2,.:a",e" " .III'/N1." 3/0 ,.1$"/2'''-/14..7 'N'" u.l>. '" /.I'/~ et'0~~ 'I. 810 I"".; 1Jz.=/.IJyt.. .. ~n/29)='I.Z-;, 11'1·Use- c.. ~ YD ~,7Vf6$. wrr» P-/'IJOIN FoR.. No. 80 CHA-IH

~J.9ij~· (Z9-/S") _

L= zCVo)+ z- '+' f'dJ.(¥()) -,10)',1 'InHe!-J'Jrlo~~neh6,J-L

c.- VUD;l. - ,29:£"..- r~,;..n;<f_ 6'C7ff~r)J.]=p.?/iY, I'l7emtS= CJ' ,/8" C =-'J1.9( IN.

Problems 38-42 are design problems for chain drives for which there are no

unique solutions. The general procedure is illustrated above for one possible

solution for Problem 38. This and the other design problems are shown on the

following pages using the spreadsheet from the CD-ROM that is included with

this Solutions Manual. Data for design power from Tables 7-5,7~, or 7-7must

be used to ensure that the selected chain has sufficient capacity,

8/

�_r------------

CHAIN DRIVE DESIGNInitial'n ut Data:

Actual chain length:Computed actual center distance:

Actual center distance:Angle of wrap-Driver sprocket:Angle of wrap-Driven sprocket:

102.00 in39.938 pitches39.938 in _ ~173.8 degrees rSIl.@Id,be' ifl'lt~teNlfaifj20!m~.188.4 degrees

CHAIN DRIVE DESIGNInitial In ut Data:

Actual output speed:Pitch diameter-Driver sprocket:Pitch diameter-Driven s rocket:

Com

------------ .. 00

Initial In ut Data:CHAIN DRIVE DESIGN

from vendor

11.95 hp rating at 2200 rpm

Actual output speed:Pitch diameter-Dnver sprocket:Pilch diameter-Driven s rocket:

8S

CHAIN DRIVE DESIGNInitial In ut Data:

from vendor

CHAPTER 8KINEMATICS OF GEARS

Gear Geometry

[] N~'/'/./ P,,:: /2.. 4. D-Al/P.r='I'I/n.-;;.'6?/II'

b. Pc. ~ "./P.l .. ".//~. ().26/SIH'C. ""'. 2S.!VP" .. 2s.VII:L=-:l.//7_ ....d. ~ - ;'.11 ~ "..".,.e. a. = '/,od" '//~. ~.0833"N.

1c.1 jr/= 3Y,. fJ".rrz.,y4. D= "~hl'= /.I1I71N.IJ. ~ s I1'/ZY'" IM3"'1fc. AOI- U·'f/ZY - /'158_d. /1>1- ~{)i1 __

~. t:l.= Y~Y"O.o'fl1llV.

~ ,,=: SIS' j 1'.. =-z

a.. 0= i'~ - ZZ.$'IIO /H.

6. Pc= 11'h,... /'5711#'C. . ttof:: ~1S'.1'/;.:: /).,1 /'1'0""

d. "'" = }J..-__eo a. '" IIi- = f).sIa IN

[fJ AI: /8; 'pJ= 8

a. D".I~:: 2.2SDIN.

I.. Pc. = 11'J, .. ". 39 2.'2 IN.C, 1M = .:z~r/, '" 1,/7S"""-d. 1M; 3d)",..-~. a..= '14= t'./UI#·

~ /1/ = l2 d' P", so j.1S'

a.. D = l.Z//.7S' = /:J..S?/ IN.b. Pc.= ""/ItU'" 1,7'!S"IH.C. /)00: .:zS.V/t.,!'=I'(.SJ'I "" .....d. t"t'" / (, "'l "'"e. (1." 'I/o?!' = 0,S1 N 111,

.;. J '" I.~ j P" .. /..,l.S'"/I.:lT: "./D'/~Ih •1.' c'" CJ.2SjP",- o.zsJ/.:l.Oo47.D8"..,~ • .,{.,= a-H•• 2.U/~·o.n7r/k

.J,.. ~ .. = .:l4.= '<;'" • ""~::"'/66'1IN.J. to .. "7J.P,,· "ht'-'I • "./.1IIN.1. D." f!V~l;J;'pJ. '/~.,.'" 3.833/N•

1.2.00~. J,- I!J .,. <J.«J'Z. = #.6S'ZPm.~, C ... o.•...,~ +-0,00Z. = eM>/ o3/~A, /1,.".. (1.+ ... 1I.0'lJ7~II,O SOlo:: 0.0931 IN.

J.. hI<" ZQ.,.:' ~y:: c. D433 /N.

t. t .. "'/,lC.a.y) = t!)oO'S'// .....1c.. D... fti~1J/PJ" 3'/V/ = !-StJD/N.

;, b '"' /.~ ....= o. r.U"/.(J,3'" c- 0.U'/• .: tJ.t:1.S'/N.A. ~~- ,2.0I8"/..L': /.I;l.r/~.

A., i" =0 -'Ij.. ... 1.0'. 1,(/'

f' 1:' =. 17"/.>.l~)~ (J.?9S"IN·I.

, '111:.4,. /).'" ;l...:o 23.SDI~.

r;. 6.: 1.,u"le '" (J.1Qb311J.

'J: a.... '·U'/ea. O.O~J?>,I./A. At:: J~/8" O,'Z'8t3IJ1.j. /.A." "/8 ..0.ZiS'o "'"j.. t:. 7""/"l(\,,)= C>'l'i"3IN.k. D. = Zo'/a'" Z.SOO IN,

sr.. b: }·U'/J.1~= ().?/ '(3/11

~'. c..=" '2S1M!': 0./'1 2~IN.'" h.~;JS·It.1r:a /, Z.8S1 ,/'I.oA • A. Jr.. ;./,. 7SO::: f,I 't29 IN.

~. t" .. rr'/~U.1S').. o.8?7(,NI~. D,.:::l.'f /1'7 $"'=: )J. 71'11/>1.

g] H"Zo; p"~6Y f.. " ::1.1.O~GY NlooO<':: ".dZ~111J

';. e." D. 1.""Iw ;.I1.1J1JZ. ::- tJ.()OS"IIII.f) .. ZYGY= O.31Z~/'I. ~.A ~:: I(.~. Q.036Y IN.a..

A". " "'-/IY=' 0.OJ/3IMfc" Tr/,y.=- b.()Y'l111l. •11. A..t:. tr/.l (, l') • o» Z'IS-INC. /lit .. ~Y/6'f. 0.3'1'"",,_ 1:.Da., "u/,W· /J.Y/38/N.d. ".,,. D.'10m" ... Jl.

e. a.. ,/,,,-D.US-. IN.

I!J N= 180 j PJ" 80

a... /)= '''''/eo = Z.ZS"OIN.

b. P~- tr/8• :: O.o3'lJ INC. tttt_.t :,.y180:: ~.311....""-J. t1tt:& 0.3" "" ""e. a.. = I/ID • O.O{ZS'IN.

~ N- 28 j Pd "'/8

a.. D:: "1/1=-/'SS'eIN.6. Pc. .... 11'/1( .... 0.17"1$"111'e.•. t1'\ = __s. y/It::. I/r/I,..,~d. """:: /.S-".,.,;<'(

e. a.. '" 'I;e ::O. IJ$"So IN.!II11=28 iPd= 20

a... D:. U/z.., = /.'100 .lH.I.. ~c:: 1/''12.. :=O'!S71IN.C. "" = J.E·"/~::./,).? __d• """:: I,j. S-"'" ""~. a. .. '/-'0 • O,OSOIN.

~. b= I'~rn ,.".u'/..:= 0.0170111.c .. CI.20·/r•~0.002. -= (!).()() '{SIll't J.... ... a-+l:>:: iJ.IJ7l1SII'1.

L. AJ<,= ~/ID :: O.D1.S"~ :: "'/,1(10):11 C.C{'u, 1,<1.i: D. ~ I az.lto 4 a 21./,<1.

fo. 6 = I- ~/8 = 0.061'1 IN.} C" (l.U/le,. O.Cl39/"'.It.. J... 1; = 'J..~//8. o,nJ 'N

..(. ..<...... :J./IJ' - 0.//11 I~'

11,' C'- ""1.1C/tJ" a.o,/?3IN.J. D... 30/I'i Co /,.6')/,11.

f. J,,, /,.20% • .,..iJ,~,.· tJ.~(,~iJlI"

c,. o.ZCOI:. +- 0.001. :: 0.0/2./1"t A'f;· a.-+J,:::: 0, I/Jo t) /",.11k" ·:V~~ O./HIJ IN •

t:'- 11'/.1..0<» _ CJ.~?S$"I""D.;: 10Jzo • i.S()D It'!.

•A..

88

a.. [)= "., IJ ,. /.zst'Ir) '"S~.u __b. (lL'" tr""""1rc,l.m ....:1.'121__c. Pol • ~·lV......:lsoj1,;.r224,3d, Pd· 2.0e. et .....,...,.. /.~,..,-

a.. 0."", ",.. l'JlI') = 211>".,,,,,b. fl.= trll1J" 31. '}D""' .....C. P". lS·"/I~. ;".111

J. '--;I.e. GL • "" .. O· ""-

4.. D._H· 20[22.)- '/'10_",b. 'c"",(024)= (,;/..83_-C. Pd· ,1.r.~/.a." /,;1.7d, Id-/·).$"e. a.. .. ,.,....=- U""_

a.b.

/) .. /1tf 1'/ ... ::l.0 .oD ""-

Pc· tr_ • 3,IY/1ff-'I'll - cU; '11 .. ;l.$", fI,,,. ~'Ia. -_ .. h'O~-

..£ b =/,).S' (~. /.;'r(t..:lr). ;,$"'&J ".,-'to ce().~_·o.2J"(4.zsJ- 0,313__A. A.. ·2.u_=zu-t".ar)·Z.PI3_-

A. All. • :/- e;1.(I.;.s) = ':;•.$'10""-.; t- 1"'../;..= 11'''''h, .. /.163_-i DI.""lHt-'-J=I.~CY1J=- S"!.1t'""...

~. ~ .. j,;.st 1:1.) .. I~."I ""_l' c» 6.l-r (/ ..) ~ $. J Q -

F... I. Z.Z.:s'C/~ = Z'7.o.,..,-....i.. 1t ;1._.:= ;l.lJ.oo_-f t:= '11""11.)/.,,; I<t-f!s>..........A. 'fJo .. ~(AI~'" 1~{,,1.o)-~i'o__

q., b "/,:u"(iliU - ZS",DO",.,-

". C:'" o,2S"C~)".. S,D4,..,-~• ),~ .. Z 2S"C26) .. '/.;; DO ,...,...

A. A.... ~- .. :U.ao)'" 'iiJ.14".,-.i. t: .. 17'C1A~:z.=.1/,'12,....-~. /)." ';',( •.;1.,/).. '110 .....-

; _ 1,,l.r(l) ~ /';1.S'",., .....

c;: O,U(I)= 0.7..""""At" ~.~Ct): ;;.;.S'"".,~A A: " .:l. CI)::> .2. 00 .-,-t:- 1T,I)/~=/.S')I.- .....

I). III: (/) ( ;.).) .. ;2.".O";1t1;!'<\.c.d.e.

a.b.cde.

/'I" IRO 3' ptI"'O.'

O· 4,'1{14'0): 7.:l.£l 0".,.-.Pe. 1T'lo.·t) Co /.~,,..... ......

,. - ~. ~/ rJo'/ -: 63.S"'4· "I(1. '=- t>"I = ().If0 ~h"

b .. ".utd.")· O.ro"".......c... o.ZSt().v)" 0.10,..,--Itt" .2..M"CM}= o.9DO ,;/"1""1

h- ~lD.V)- ().fo.--t"C fTCo"")/J. - O.62.~/>"11"'\D.- o.V(lYl.)c?Z.8o,..., ........

89

�1/= 28 ;..-=/'S'

;; /:.= j,Zr(/.s)'" hB'7.r __0" /1>111:tJ""{Z8)= I/Z,IOAIf'- ~,Ca"zstJ.r):: (),3?S"",,_a,

ha 2.U"ao» :J,J 75""""...1.-11" If"<:=' frCu') = ¥- 71"" ..... A.b.• I.e.-=" "J..{;.r)" !.~6....-.Pd· 2.l;'0.r= /6,93 ..c.Co

i- t;- 7r{I,r¥;L" ~3 6_"",-d. ~. It.Do= /..5-C.30)- 7:5"~I)D""''''''e. It. ...__ .. /.tb Aft...... lJ..

N- 28 j /1It co f),e

b - I.U( 0.8)'" /'00 __~ .. tJ,U(IJJ1):. IJ.z.o"",_

~. It • .. Z. 'Z. ~ (o.t).. /. to""'_• I...• ~t.1J.8)= /,60_--..4. ,.

e» 11"'t",,.'/..;l = j, ZS?,,-.0." f).I( 30)" Z'I.DO..,.,-.

l'I:: ().SClt') .. ZZ.yo ....._A.. '"D. I .. '" tr (o.n '=' 2.1>[ not_c. f_. ;$'-+'/".,. 31.?rd. 101 z: 3z.~. a.:: "":=. d.'o .........

jle.j IJAcKLh.rH - .reF Po 389.

119.) t:Q'ff2. ~O"'" P~oIJ.1: PrJ·/2..:/!I'/CKl.AD/ o.oos ... tJ.#Dq",I

606111{Fl&1"I P,fo,.ll:..,..:: /2 : BIfCt:t..NH (),S2 7DtJ,al. __/}E'I'4"NII'/~ ON Ce-N11i1! /}1.r~Nt:1".

Veloc:ity Ratio

c = .//, +"i... It ~''f - S'./l.S" IN.14'~ :L'V>

b. 'II~.#4/"",. z: 6,1'1,. ... :I.$"S"~C. m./i. ht.~ (("~.,) .. 2Y.me~ ...)= &!P ,/Zp,.,c/. 1Vf:=. 7r d hi,. s: tr AI... ;/'f", :=. 7>"£"~}(zYn)::

rz; J :L.P.i I ~t.8J

§JA.c.d.

90

d. tVe-- (i1.i)(/t)(llftJ)19 q,/ 1

tV"t" ~, 'U'l_/J

~ a.. C - (jts-f3'){'t)/;..a /'Z",..-...p. V~. vs:I.u· I.U-

c. ;'Of". j,fl) ey~- IZo fl.''"'a. e· 6 'rlS;) (/2,.)!J. • 36',--b. V~.,.. 3¥/j=- ;l,t'IJ> >C!. ~.; _ ~ Btl J~, '" ';'1)() ,Q.I''''

Errors In statements lor~i.Z9 - 32I'/4h"'" /tA/O t'iiMt'- CA..n- HI1Vr i)ll"l"'lI~Etvr "ITCH5~

Too Ptlt.oJ )ilaffl 10/ r'JfrZ ~/v/f)AI.I.,.rSf/"'I"'1i '2<J'~.D.thOrn

/I'J T'SLPtnf.ti",c.tF woos-a OCCVIe..

~~;,~r= Z';S' 6 /N : /). ~"'''''f)T" 15. lAS6'l1 ill PI"''' e."" /II,+- "'''c.- ...7.P_

Housing Dimensions

f/o~$/"'t; 1"1~srC£.C'llft. A1JO~JlOV'" C/~"I!!" ()F' ItU- 1(/!1'tICS 11)1 t:!,/D/W/.UI1t;

4.= ~..,,,_ ~6' .. "./z.r/N.; /)o~= f4.""l-)/~""/e=e.U/NY • s.zr/N + 2,(I,/t)· 8.'/./1/.X = J +D + 20. -r2.Cll"O):O C:l£. + Nt. + ~ to ae» = Z.ZSHI.Of+,ZS'roZO

X" 10.700 1/oJ. P" Pol f.l

(1,

�--------------Do.;= WCliT1..}lP.. = zr;V6Y= 3,1.381/11': Y=3.938~tJ,z ='f,/3$/N.YX== cl+O t-zp,+ZlD.()= ~+~r~ ~a,lD= 1j,761,w=)(

6'/ 6'1 61'

O.t; '"' Vv~r7.)lt\= $0(0.1):0 lfo.a_ .... I y= YtJ.()rZ('Lfot-J= r1f-,"""1)( "" J t-D t- 2A- +-Z ll.) = '"' IY', f Aft H,f + 2."" +-Z( 1,)X = o,'(lr) r ue'!t) t-z.(tJ.t)+- 'to. 0<9.1/"""'-...r

D.t4~ 97('{)=-/e'fltI- :Y·lltrZC2.J=/9Z.",_X:: oJ rD.,.. 2t1- +- 2.C'2-) = ~ ;\1'''''''''#4 +-lh.. + 'I"X = I'IYt-/i'Or.9+Y=3Jf:."..-.=)(

Gear Trains - Analysis

IV ~ - lie Alo Alt: -~2. .MJ.d m ,/--. 0_._ ::-_ ,~. _ = -/S;,.r::" '/JJ/~ ~

AJ,t Nt;. IfJIT " " 2.y •/11.... 111",/rv ... /7S'O;<'Yf/S;7£j::::'l//iel'l'1 e~"J

TY = - ~ IV~ Ale- - 6,g 6•.- '-:: -.;:<. - -6'. .y", '"'~"h.. III,. /fI.4 AID 2.2. z.s- -. ,fir

At_= /I1,../r/,/ '"' /7SOlttl'''f/-8.V07 =.-2D~~'h CI:.W

'TV:::+ Ds • ..!!.., ~ . .MJi.:: z.lJ1r, Z.l?r, Z.2.ftI.~=IZ,JJ.90.. De. OF 1fJ~ /,J.S'l) I,n$ /,.$'1JO /9

f),.. AI.../~,. 20//6 = ;'UO/N/ 1'11".,..'"' ""',.... J7sp = IflY tf(.M ~IJ D • N,/'If!_ 34/'/1 = 2.371" rv 1.2,/39

IJIf' ~ /II~/IJJ~ 18/12 .. =: I.Sla

ry- of. II- ~. ~ =.£t. /.9 ra 09~A Ale 81J' -;;-.. ,

hI_ ... h1/III/TV '"' 17~A.D 'I =' If! 'Iyy ~t*1 Cw

Helical GearsEJ HEl...lcAL ~E"Afl. f;, ...8, ¢c" /'Ih·, 1V::.~.rre-1?I, F=2.00/N#ELlJ{ AN61-C"" 1/1a 30 o.Clt?CPI.Att.. ~/TUt «er o/~.: 77"/0''= O.39Z7JIJ.NaRltfltL. C,lGIJLAIl 1717r.1+·P#v .. fJ' CN rr-~.J9Z7)CIJ(?IJV· O.:J'IO/N.

NDKMlfL. DIIf""e:rHrt- 1"1 n:H- ~aI '" !?J/GDS1'- 'Ycosf!o'): ·9.Z.1ll

A;rIJ1l,..PI1Z.H:: ~= I'd:".., » B~.')= o.6BO/N

P/Tt.H DIAMw1"F7t.. = D,.= )I//~.. Y.5"/a a 5. 6L1;"/1V.

IV o.e1'1I'tL fill G~SV.t." /+)1/~ I.t;' # ¢... = r""-I [tzJ,... Pot c.,.o...,]¢...= ,-.-1 [t--.I17',.r) c.ro.tJo~}:=12.'2·

FII"", = Z.o(Jn~/O,I.&J'Ii= 2.9'/ -",.;rIAL I'ITCH5J IAI FAeFWIO";'

Ht:LlCAL Gel'll JV:: fB J ~",,J'" /:1.) ¢...alD~ F=I.S01N J JI'= ~.fJ'. 71'/1"11- Stir p.! .. P_ CQJ y= 1.2.·c"sf.',r)=8.'I8S

p.= tT/".'I8S# O..3i"i11'l : ;'... = p- e-1'= %D&' ffdz=,o.Z.6/~/i'.

F',.,.. ~~. ,~_ 1).3'O'N _ 0..110'/01 .. 0. '" N/6 ~ f<s/~ /t-.T b-Y - .~ /1,,- 1&'I4rc,1';6S""1"f

tP~· r--I p;.""J-"""~'(C-1.D~]= 21.20

- e- Y LG=I/.r J .FIIS-= /.SOO/Ii/0.3'bIN = ¥.f)$"AK//h.. fll'rutlti IN FA-c~ WID7»

HELIcAL GIFItIi. 1I::.1~,!?J" &.J (A -llf1.,·, If·r~ F =/.OO/N

.1::-# lT/~.. %...tJ..5236IJJ. ; ~ = p c.o."If. f·e-.!Ir):= 0,.170 //01.

f",,,:: PtI/CAsy• r~{/S"a,6.,/,fS'; f'x= J:t;M.'/'= ,':r-if = O.SZ;'.IN .

.!2. -= ~~ ... .1~ = '.1J0f) /11. ; ¢,.. ...rr..-'[i:..rlt: e-yJ = If).U·r/px= 1.00 '%..5Z3'"~ :. /,'1/ AOItt.. ~/rcllt:J,/1/ F~U Mom (t..ow)

I"+'! 1IB-ICAL Ger/tfL )1'72.; P...d =2'f J' tP.. = 11/'1.' j r'O.z.$/NJ f= 'loS·.1":# "";'4.1 Blrr!!t. : e..d CI4 1'...z« we 'If" --/6.91

r- ~1.97 =- O.18$'1 IJII. j 1"'", = p.C<rS"'. p./,gs';/J/·!D.I,/r"(},/Jp,?/M

~ ...'/t<- t...o.IMf/I:;-¥,s'='tJ./t9S7J1lI.; Dr;,:%» ~/.9'J= '/.2n.lp.

¢ .1D, -t(ti- fJ#IJ]= 10-4 [G-liYS) 7,:20 ()-. FL. o,zS/ _.J. 0- e- Y 0-. e-'IS- J '; /~. /11Jas-./ •.1fLgw

Set?'" I'.lZO(!,lt:rI I.i9 tJ'" N£«r PII6E Poll. roaM/lt4-S A-NO Sjl~64U

BEVEL GEAR GEOMETRY

'~~fJ~:~f.f.~WgllfM~ili~f@~:~!~jMtnf:~I~;fJH~:;GIVEN DATAN.oD'llileui)i' .fllt41mfh'Iii'Diiinrell:al itee'mt§JiL~~·COMPUTED VALUESGear ratioPitch diameter: PinionPitch diameter: GearPitch cone angle: PinionP~ch cone angle: GearOuter cone distance

COMPUTED VALUESGear ratioPitch diameter: PinionPitch diameter: GearPitch cone angle: PinionPitch cone angle: GearOuter cone distance

3.0002.500 in7.500 in18.435 degrees71.565 degrees3.953 in

2.0002.500 in5.000 in

26.565 degrees63.435 degrees2.795 in

Nominal face width 1.186 inMaximum face width (a) 1.318 inMaximum face width (b) 1.667 inf!ff1!l'!IE~~9~t~~:~[il{1JFfW:W!]~~~!!\'!!I'i]:I\WiijJ!!il1!f

Nominal face widthMaximum face width (a)Maximum face width (b)!l1Ir?OfFace widtlJ

0.839 in0.932 in1.000 in

" O.llOQ i~~ ~Mean cone distanceRatio Am/AoMean circular pitchmean wor1ling depthClearanceMean whole depthmean addendum factorGear mean addendumPinion mean addendumGear mean dedendumPinion mean dedendumGear dedendum anglePinion dedendum angleGear outer addendumPinion outer addendumGear outside diameterPinion outside diameter

3.328 in0.8420.441 in0.281 in0.035 in0.316 in0.2420.068 in0.213 in0.248 in0.103 in4.257 degrees1.774 degrees0.087 in0.259 in7.555 in2.992 in

Mean cone distanceRatio Am/AoMean circular pitchmean wor1ling depthClearanceMean whole depthmean addendum factorGear mean addendumPinion mean addendumGear mean dedendumPinion mean dedendumGeardedendum anglePinion dedendum angleGear outer addendumPinion outer addendumGear outside diameterPinion outside diameter

2.345 in0.8390.264 in0.168 in0.021 in0.189 in0.2830.047 in0.120 in0.141 in0.068 in3.450 degreeS1.670 degrees0.061 in0.148 in5.054 in2.764 in

9'1

Given: N» = 18; NG = 72; Pd = 12; 20° pressure angle.Computed values:

Gear ratio mG = NdNp = 72/18 = 4.000

Pitch diameter: Pinion d = Np/Pd = 18/12 = 1.500 inPitch diameter: Gear D = Nc/Pd = 72/12 = 6.000 in

Pitch cone angle: Pinion r= tan-1(NplNc) = tan" (18/72) = 14.03°Pitch cone angle: Gear F = tan-1(NdNp) = tan" (72/18) = 75.96°

Outer cone distance Ao = 0.5D/sin(I) = 0.5(6.00 in)/sin(75.96°) = 3.092 in

Face width must be specified: F = 0.800 in Based on the following guidelines:

Nominal face width: Fnom = 0.30 Ao = 0.30(3.092 in) = 0.928 inMaximum face width: Fmax = A.l3 = (3.092 in)/3 = 1.031 in

or F_ = 10/Pd = 10/12 = 0.833in

Mean cone distance Am = Am(]= Ao - 0.5F = 3.092 in - 0.5(0.80 in) = 2.692 in

Ratio (A",IA,J = (2.692/3.092) = 0.871 [This ratio occurs in several following calculations]

Mean circular pitch pm = (wPd)(A",IA,J = (w12)(0.871) = 0.228 in

Mean working depth h = (2.00/P,J(A",IA,J = (2.00/12)(0.871) = 0.145 in

Clearance c=0.125h=0.125(0.145 in)=0.018 in

Gear outside diameter

hm = h + c = 0.145 in + 0.018 in = 0.163 inCJ = 0.210 + 0.290/(mc)2 = 0.210 + 0.290/(4.00)2 = 0.228

ao = CJ h = (0.228)(0.145 in) = 0.033 inap=h-aG =0.145 in -0.033in=0.1l2in

bG=hm-aG =0.163 in-0.033 in=0.130inb» = h« -ap = 0.163 in - 0.112 in = 0.051 in

&:;= tan-1(bdAmaJ = tan-1(0.130/2.692) = 2.76°8p = tan-1(bP/AmaJ= tan-1(0.05112.692) = 1.09°

aoG= ao + 0.5FtanopaoG = (0.033 in) + (0.5)(0.80 in)tan(1.09°) = 0.0406 in

aoP= ap + O.5F tan&:;aop = (0.112 in) + (0.5)(0.80 in)tan(2.76°) = 0.1313 in

Do = D + 2aoG cos TDo = 6.000 in + 2(0.0406 in)cos(75.96°) = 6.020 in

do=d+2aoPcosrdo = 1.500 in + 2(0.1313 in)cos(14.04°) = 1.755 in

9~

Mean whole depth

Mean addendum factor

Gear mean addendumPinion mean addendum

Gear mean dedendumPinion mean dedendum

Gear dedendum anglePinion dedendum angle

Gear outer addendum

Pinion outer addendum

Pinion outside diameter

u-'--- -------rBEVEL GEAR GEOMETRY

-~

COMPUTED VALUESGear ratiopnch diameter. Pinionpnch diameter. Gearpnch cone angle: PinionPitch cone angle: GearOuter cone distance

COMPUTED VALUESGear ratiopnch diameter: PinionPitch diameter. Gearpnch cone angle: Pinionpnch cone angle: GearOuter cone distance

4.0001.500 in6.000 in14.036 degrees75.964 degrees3.092 in

4.0000.500 in2.000 in

14.036 degrees75.964 degrees1.031 in

Nominal face width 0.928 inMaximum face width (a) 1.031 inMaximum face width (b) 0.833 in

1191i,g;r;~~~~~~:jj;4l~~g,!~~f1:'II!!mr'H;,~t,illfiii

Nominal face widthMaximum face width (a)Maximum face width (b)!!\!~ce width ti'

0.309 in0.344 in0.313 inl!l~jn

Mean cone distanceRatio Am/AoMean circular pitchmean working depthClearanceMean whole depthmean addendum factorGear mean addendumPinion mean addendumGear mean dedendumPinion mean dedendumGear dedendum anglePinion dedendum angleGear outer addendumPinion outer addendumGear outside diameterPinion outside diameter

2.692 in0.8710.228 in0.145 in0.018 in0.163 in0.2280.033 in0.112 in0.130 in0.051 in2.767 degrees1.090 degrees0.041 in0.131 in6.020 in1.755 in

Mean cone distanceRatio Am/AoMean circular pitchmean working depthClearanceMean whole depthmean addendum factorGear mean addendumPinion mean addendumGear mean dedendumPinion mean dedendumGear dedendum anglePinion dedendum angleGear outer addendumPinion outer addendumGear outside diameterPinion outside diameter

0.881 in0.8540.084 in0.053 in0.007 in0.060 in0.2280.012 in0.041 in0.048 in0.019 in3.113 degrees1.227 degrees0.015 in0.049 in2.007 in0.596 in

BEVEL GEAR GEOMETRY

'!l~;d~'~~f0~~~g,E"J"l!ilP~ftf!{'t'6;~i7},t,*!F;2GIVEN DATAfiif

&SlI~:~'COMPUTED VALUESGear ratioPitch diameter. PinionPitch diameter: GearPitch cone angle: PinionPitch cone angle: GearOuter cone distance

COMPUTED VALUESGear ratioPitch diameter: PinionPitch diameter: GearPitch cone angle: PinionPitch cone angle: GearOuter cone distance

3.0002.000 in6.000 in

18.435 degrees71.565 degrees

3.162 in

3.0000.250 in0.750 in

18.435 degrees71.565 degrees

0.395 in

Nominal face width 0.949 inMaximum face wid1h (a) 1.054 inMaximum face width (b) 1.250 in!B~lflT1!ll~iilW!i:!!h~)!filrl~W,~j];r~:l1';Q@,~fiir,~r;11l1::111)1:

Nominal face wid1h 0.119 inMaximum face width (a) 0.132 inMaximum face width (b) 0.208 in!fl!!f!~iiffi'@~IM~!li:1l!r,'i!1!1:liii19f;~~~lmiil!1111:111',i;

2.662 in0.8420.331 in0.210 in0.026 in0.237 in0.2420.051 in0.159 in0.186 in0.077 in3.992 degrees1.663 degrees0.065 in0.194 in6.041 in2.369 in

Mean cone distanceRatio Am/AoMean circular pitchmean working depthClearanceMean whole depthmean addendum factorGear mean addendumPinion mean addendumGear mean dedendumPinion mean dedendumGear dedendum anglePinion dedendum angleGear outer addendumPinion outer addendumGear outside diameterPinion outside diameter

0.333 in0.8420.055 in0.035 in0.004 in0.039 in0.2420.008 in0.027 in0.031 in0.013 in5.316 degrees2.217 degrees0.011 in0.032 in0.757 in0.311 in

Mean cone distanceRatio Am/AoMean circular pitchmean working depthClearanceMean whole depthmean addendum factorGear mean addendumPinion mean addendumGear mean dedendumPinion mean dedendumGear dedendum anglePinion dedendum angleGear outer addendumPinion outer addendumGear outside diameterPinion outside diameter

'17

Wormgearing

WOlMGFlffi/Jl6: Dw=/.lSo'N,:IAlw"/, !?J.::/p.i¢ ......./y,s,Nt; .. '10; F- o, 62S/K. -CS/N6LE" 'T1f1?G/fD

LEAl? =ANItl. I'lrc.H~ OeCdJAl.P/nJ+ - J/pcJ = %.:: tJ.YY2hv'.~/1NGtC= A= rs: k~)= r....L,"~~)-_~_.r_7·,(IDPe-NO"", -a..= Yt&. Z&=O.lO{)/N.j lJFDENO""". ~~S1:a/iS"1/N

WtJbr DllrJ'/OE" fhA •... D,w= DIN~la..:: /.l.rtJ rJ.U./I.J" /. '1SD/N.WtJil"! ROfJr OIA,'" D"w" J)w-l6 '"1.l.m-Z{P,IIS7) 6I.tJ/8'/N,

C;~1h2P/f"CHO/A, • DG.. H6k .. YOllo.::-Y.t¥Jd/N.

CeHf7T/t. OU rtf/KeF = C .: ~(f +b",,)h = f!.,Od rczs» )/2..=2.6ZS'/JI.VeuJc.lTY RA-T7o=V,= 1Y4"w1u"" -¥Ofi.:: ~

NOTE: On the following two pages are the results of Problems 52-57 giving pertinentgeometric properties of worms and wonngears and their velocity ratios. The detailedcalculations follow the pattern illustrated above for Problem 52. The equations come fromSection 8-10, Equations 8-33 10 8-38.

Compare the results to discern how variations in geometry such as diametral pitch and thenumber of threads in the worm affect the overall results. This is especially pertinenllaProblem 53 in Whichthree different designs for wOrm!wonngear sets provide the samevelocity ratio. The single threaded worm produces the smallest center distance and overall~jze of the reducer. But note, also, that it has the smallest lead angle. The lead angleIncreases as the number of threads is increased. On the positive side, the small lead anglemakes ~e redue:erself-lacking. On the negative side, the small lead angle results in lowermechanical effiCiencyas will be shown in Chapter 10, Section 10-11. The designer muslbalance these advantages and disadvantages for each application.

'18

WORMGEARING PROBLEM: 52 WORMGEARING PROBLEM: 53AINPUTDATA INPUT DATA

Worm pitch diameter = 1.250 in Worm pitch diameter = 1.000 inDiametral pitch = 10 Diametrat pitch = 12

No. of worm threads = 1 No. of worm threads = 1No. of gear teeth = 40 No. of gear teeth = 20Face width of gear = 0.625 in Face width of gear = 0.500 in

COMPUTED RESULTS COMPUTED RESULTSCircular pitch of gear = 0.3142 in Circular pitch of gear = 0.2618 inAxial pitch of worm = 0.3142 in Axial pitch of worm = 0.2618 inLead of the worm = 0.3142 in Lead of the worm = 0.2618 in

Lead angle = 4.574 deg Lead angle = 4.764 degAddendum = 0.100 in Addendum = 0.083 inDedendum = 0.116 in Dedendum = 0.096 in

Worm outside diameter = 1.450 in Worm outside diameter = 1.167 inWorm root diameter = 1.019 in Worm root diameter = 0.807 inGear pitch diameter = 4.000 in Gear pitch diameter = 1.667 in

Center distance = 2.625 in Center distance = 1.333 inVelocity ratio = 40.00 Velocity ratio = 20.00

WORMGEARING PROBLEM: 53B WORMGEARING PROBLEM: 53CINPUTDATA INPUT DATA

Worm pitch diameter = 1.000 in Worm pitch diameter = 1.000 inDlametral pitch = 12 Diametral pitch = 12

No. of worm threads = 2 No. of worm threads = 4No. of gear teeth = 40 No. of gear teeth = 80Face width of gear = 0.500 in Face width of gear = 0.500 in

COMPUTED RESULTS COMPUTED RESULTSCircular pitch of gear = 0.2618 in Circular pitch of gear = 0.2618 inAxial pitch of worm = 0.2618 in Axial pitch of worm = 0.2618 inLead of the worm = 0.5236 in Lead of the worm = 1.0472 in

Lead angle = 9.462 deg Lead angle = 18.435 degAddendum = 0.083 in Addendum = 0.083 inDedendum= 0.096 in Dedendum = 0.096 in

Worm outside diameter = 1.167 in Worm outside diameter = 1.167 inWorm root diameter = 0.807 in Worm root diameter = 0.807 inc:oearpitch diameter = 3.333 in Gear pitch diameter = 6.667 inCenter distance = 2.167 in Center distance = 3.833 inVelocity ratio = 20.00 Velocity ratio = 20.00

/00

Gear Trains - Analysis

FOil PI2DItL~'" 58

IlllTltrt.s C.L()tJ:WI!~.

~ rI!A,N VItLIle: .:-rJl ~ /TII'~4 )'I1I,-3YSOfU'/'t

TV ~ -A1JJ , AID, JlF , niH • Nr : -..stl:. "" .f!.£ .£l!. .JB :-//1.1Mt N~ Nt'" /Vii Nil /8 17 z» 18 Ib

"'1.# !!!.!.... ~ 3V$f)I!.,.", -/7.~L.eP/'t CDV~fL~LDCKw,s IFrv - 1""./

G~MH ISMIDlli1f-. /r: T)av h'lrAI"FFd rne: ,V lJyr

Cfl!1t1t.EJ nre: PN(EC r/oi/ OF f}tE ot/rf't/r Sd/lFj",

/11,-=- JZ Zvo te..Prl ~ rlNO A'ls-: TV=- At'/",5'

TV= !!.!. H P tt.£.. .!iJ:L a ~ ~ ~?].. = $'0 000N". N c. /'(1' H& 12. J'Z I Z

/H~".t!!.!. = 11. ZOOR'I'1:= tJ.'10t.71t.I'M-rv 30000

/7lt, =-6 8to ~/'t ; FIN/) ;'Vii; -r v» '-:1....,1'TV =- Ns ~.Itr = '18 '/8 60:. YOEKACTLY

NA Ne I'I~ /{, /8 /Z

,of,,: ~ s: 6/3'10.; hi!?"'" EXACTLY7'11 Yo

AI'" 2B7y/?PI't; F/.vo ~" : TY=- ;t1~y

IY= ;Yo ;Vb N~ -=.i££ 90 8S- '" 5666.7Kh II, I'IE J Z. Z4

""'1/ ~ /1'1, =- Z137S-RpM :: 0.01)1'1 .el'MTV SUb,7

/0/

Gear Trains - Kinematic Design

VELOCITYRATIOFORGEARS PROBLEM 62DESIREDVR = 3.1416;: tr

NP NG NGAct VR-Act DIFF =Des VR - VR Act

16 50.27 50 3.1250 0.0165917 53.41 53 3.1176 0.0239518 56.55 57 3.1667 0.0250719 59.69 60 3.1579 0.0163020 62.83 63 3.1500 0.00841

xxi 211 65.97 166 I 3.1429 0.00126 XX22 69.12 69 3.1364 0.0052323 72.26 72 3.1304 0.0111624 75.40 75 3.1250 0.01659

Min dill = 0.00126

VELOCITYRATIOFORGEARS PROBLEM 64DESIREDVR= 6.1644 em

NP NG NGAct VR·Act DIFF =Des VR - VR Act

16 98.63 99 6.1875 0.0230917 104.80 105 6.1765 0.01206

XX 1181 110.96 1111 I 6.1667 0.00225 XX19 117.12 117 6.1579 0.0065220 123.29 123 6.1500 0.0144121 129.45 129 6.1429 0.0215622 135.62 136 6.1818 0.0174023 141.78 142 6.1739 0.00950XX 1241 147.95 1148 I 6.1667 0.00225 XX.,-",D l!(Wit\.

SDL<ln,",s Min dlff = 0.00225

VELOCITY RATIO FOR GEARS PROBLEItI63DESIRED VR = 1.7321 " v'j

NP NG NG VR DIFF=Actual Actual DesVR·VRk1

16 27.71 28 1.7500 0.0179517 29.44 29 1.7059 0.0261718 31.18 31 1.7222 0.0098319 32.91 33 1.7368 0.0047920 34.64 35 1.7500 0.0179521 36.37 3R 1 714~ 1).01m

XXI221 38.11 138 I 1.7273 0.00478 XX23 39.84 40 1.7391 0.0070824 41.57 42 1.7500 0.01795

Min dlff = 0.00478

VELOCITY RATIO FOR GEARS PROBLEItI65DESIREDVR = 7.42

NP NG NG VR DIFF=Actual Actual Des VR •VRAct

16 118.72 119 7.4375 0.0175017 126.14 126 7.4118 0.0082418 133.56 134 7.4444 O.02AU

XX 1191140.98 1141 I 7.4211 0.00105 XX

20 148.40 148 7.4000 0.0200021 155.82 156 7.4286 0.0085722 163.24 163 7.4091 0.0109123 170.66 171 7.4348 0.0147824 178.08 178 7.4167 0.00333

Min dlff" 0.00105 -

IOz.

-" .. or =~=: /9•• =-Z/.~Sf4'?A-rv 83.S] O,e

Pe,j /GN : rrf IN ==3360 R/?,., flMT: ""a.r =- /,2 1?P"" &K/k:r: Nltllt= /$1)--

ao·".a ~~. F.f!D'" TA~'E"S1-~ LET #",,,=17 FOIZ/Vo I/V1?IlFEIlEIVC£

Vlln4~ P.f:1'lP;hlt.= ISJ,1J= ,g. iZ: 2fl:1"u VIl,..~t$.8z)"'~ '17.1:3"""J = bB~.IlL= .J3m/n. -= 2.8oeKAC.r : lI~e '3PAIIlS S/~/t.A1l. TO F,6- 8-11 "V 1elrr

FA-crtJlVl.II,: l. 280 Z·2·.s;Z ·7 ;ZJ'4.1!.~ID"'~"'E 8·J'S'=2Soz.Jt'O V(U=-tf; NA=17 Jl/e=13'

70 ~Z. JlI tJfl2~? : /Vc=/J,.t'VD=//9

7 1Ifl.3=S: h;;;-17,NF=8S

Tv == /1b. 119.ft ~z ~o ~1C,ff.nY1/ /'I /7

COTHE1lD.rU.//J P,SS/~L6)

DES/GAl: MIA, ~ ISo.1lI"M ft1.Jr=:J. Ill"t'f .!!:KAC'" ~D kcJo.TV: 16'~:. 90()E%A-C1": VJe- t:AcrOll//IIt!r : A',l1....e/SO"2. "00 ,cAeTlJItS:" AIrE: z- Z· .s- S •.3- 3

Z'lSO £:r:t: Tl'lilLc e-b ,coll.l/IIrenJ=EltcNU: DArA

s: ZZ5 r o«. lo'F.P rset» ose "'""./: /6 DIl 175' 'Is J'tIo"'liIIAL VP"NL PEl!. PAtio: /Sl)/n & ,9.8Z run r ,."ALL.J ., rWiJ P,flltS I (8.91. )~. 77. B S""A-LL

J THR«: l'tl"u ~I/J.8t) r:;-6lJ 7 .s h '" LLF'olhl. ""1-IICS: (8.8l-)"- 666/ eeo'o .

,..r-

,..:;,8ctJ~ ~ /"fOqr=.. ... r- 'x"IN

1\-/ ",0 ,oF f-I/

~~ d r-. .. 6r"g/ ~ -: ~

--

'- IJ/::

VlfJ=&.) Y~2-=-t..J I :3=-.5".1 1Ij('1=~

~ Hit ~ II. I 6 N/! 'It Is-N. =- 91. r AlF-80 r#e;/~ L~ NA~/' l,-ND~'Jb I AI/f-=8I1f

T7481E 8-' S'Avl N' IAI rr:-IZFEIl.,,,vCl: F4IZ.JY,=16 /F JY.. .c/Ol.

., (, 96 .9'£ 80 =- '1/S() E;(~c:r" /6 /~ /~

D.f:s/&-Al: "'_ = /8""U,.. EJ(4l.r; ZI ~A'1ovrL 22..: N,..,....r/$b /20·~'O-

IV",,,. = 180'YZI.f" = 83.? TV,..".:: ISDYar!JI.8 TV,.,.,,= /~ <-8S.7

rteD", 774~l-E ,g.,- ,NO/N~FFnE"NCc 1.J''fH #172. /7 Ft>£.ZtJ'/iA,1"El!:1'U

\1,('",,,,, ~iLPI'fJ«'= /5lJ~7:: 11.8Z·~J14LL : n.r.f'ltIeJ JlR""",=(I1.gz;)';. 77.'1 /-HI

LAyovr Ar IN FI(;,. 8-3" IN ~-m/"tr; R&Pfld7r1A/.~ _·V ~-~~,IlY £"6>IIAt.. I!EPlI£TT • .,; RAno ~ Vel=V/u'::', ~j~V~.7~'f.J")

LeT .IV"" AIL" = lYe=- 17 : ce r Vi</: 5"1 Vltl e 'I: 1)/~" Vlt3= ~·7/zD ='1-/9

NF .. (j7)('1.tf) = 7/.t -=:> .jf'cc/fY /IIF =7/

F/NA< 'TV= 86 .68 ,~ 83.f117 17 17

ID3

~ CI<2OcS/t: AI! /ti,,., =-'12fJfJet'I1I:A'Acrt-Y : n.D..c Iffour t.17.r~,,,,,: fbSI1/TP nlrV"'N~ '1200 :: 5//./! iVNo .. = i'lOO ::J/,.9R: 711,.,...:y.z.. =3Z1,A8n.t> IJ,2,) I~,('

FI!D11 Pft46 68 1 PAc/ltJ ee« '0. LA>/Ol/-r IN F/~ 8-~1 P/200lTtEJ AI

)v'-:{,lIrrlllr!' TV, tJse IPlBl. 11'1AAlY PAIIl ,

r/ly 1l~/D~1tL RAno MEf}lfJO. N,,,,,,IIIAl- V~ :.13/7-' 6,1lL ~ PA,1.- -rev i/1l1"'7; VA-l:l.: T1fEN V~3 = 3/!/yz:::-7,sr: us~IIIl.J-:',>D

PIIV"IL. rv: IJ~/'VLZ' 1M:!= (7.f6X7· JiV = .J/F tJK

LETNII",tJc"N~:lIr= /8/V11~ 7'/4=/26: H, ~6'18=-/08 ~IVp=-7,'lQrJt/fJ

HNAL TV= /zs: loB /8 /J) ""36- 78'18' 18'~ -

ill: /Yo. No .ss: ,NFN.. Ale. /Y~ Nr

"t1ovr ~ /Of,,. i/ I(}()-- - -,- = 13,1) ~,~ etA!III ..IS

r-: -,

C nnIn" r+ lrF>--

(3->- - ~ccZ l- I,. s:

lo'-L..: .... e-

,It-/)-

L-

w... , OLI!1l

.1:-eI-

10'1

Pest4tH: .M,.,,"17Sb; I'I{,.( hto,x-LISO

TTlNOI't" I7SO~I{J} = /1.82-ter ve,» N~N..:> 7~82'1.161RellPtliIh. llA-no :: 11.8Z/'/,lt7:.-z.BJ7ur/ll~ "18: NfPllICZ.8r?)= 51.46 :!;~IN"" 18) #4=75; Ale:>19 ,Nt> <=$7

f'YlDtrr= 17SlJJ( ~iIt ~ "1'Ia.2~"M Ole.

SK~,,!'t SA-ME A-I IW WIf]fDNl.Y t'/.'rD PA-/'Jt·SC11tae fl.t5c1L~ USep I~ P~L eo. "-7'1, J

OESI6N! mill '" 8SlJ IlRtIt • /fo c ~fr L '1'/ : V.Ft:.:l p~/tlS,rvNO,. = 6'.it,A,.l.= Z().Z/I. LEY V/(, c .N~,,~ llJta= 'l.5'D

RES'I/1ulk.. ~o=vjtz= 2D,Z';l'¥,J1>- '1.~:Nc=19,Hpc81Motlr =' 8sD I/. ~ ~ /8 ... '//.98 I."., oK

81 81[riffle" ,Urt/Ul rAeJlN fkJ8LJi"'1 tt-7S]

Oe-UGIIJ I UJc TWO PJfld.s; /I1/N =3tJo{)APt'f: .$'fa lit '*'Dur<sr;z.TI'",o.. '"' Jooyzo ",s;ysvf: l£n/~/ "Vt.. "Jiyr~f' C' 2--"'>LET II,,~/S-/ .N.. : 1S-(Z.~t.S)=3.s;6~93,S"; urJlr::>/s-~#p:>3r-

IS IS" - S3l,hi. !l"r ". ~ooo" 3S 'I- U - tJlP '" ~

£!"HEle' JUJlJtn LlJf:D 'JI/ PIlO(8LI:T" Q-76.]

IO~

1/(o I

IF

I--+-I--+G"

OOtH

A" T?7J.iV ~ SIJ/Y,-n 01'/ w,N", HIlI(rPl t'I nJD71Ilcer IVA~lVe=,vc""IVC?'"/,(. Lcr IVa =-;Vo~ 95, Ltrr ,vF="~8/

V~,C lJ"'t= q~6 c5".'137SJ< Vlll:.8Ih'=-S;/HZ> _VI.,'l--V,(l. 'I-- Y~l :: /7,f.1f7. Rl:.fN,I/!tl- f{A'f)o=- '/°0/171.1'7 - S;,W3L~ NH:: 8/ . Vlq'" 11I3= 90, ~S;()62'>n rA-L rv:. (!/R.Y7)(.!>~tJ6Zr);" tlff;].s-FiNth. MDur:= 36M /'JoJ,y.: 3. "18 If ~,., OJ:'

17~.sltiN l A,/I =- .J60o :1.o<C/11our<S;o~"',-ffrVND,., = J"'o/v..;:, 90D: lIS~ nvo P~"lS IJp W()~I'I/WIl"'ti~M-J#4 ~ N& = / ; /Vo ',11.. " J0 II"

/Dr.

17ql DESIGN: mlAl='i2001¥1"I, l3,o</f1Do~/LrliSt! c.",SINI8J /lIEU'. !-JUTltIvD~1"It::EW'Z- ,...> IN f'tU&F1"I 17,LET vlt-z ~ til>. WOt?M 6~ DLln=-, Ale == /, /110" SO

-rv /l/1J/1 = '1l-ool/.?,zf:= 3/6.9&. V~I = J/1.Q8/ sa == (,,3Y

cer: '" ~-p'&, N jJ:-18(t.JY) == 11'1·,=;>f/SF//Y= ,AlB

FINAl DII.,.PVr SP4ErJ = Y.lOO>c ;/~ X ~ -: 13.26~'" 0,1;

180.1 De.r16A1: -nll/= SToO/(I'M /5 .• c /"1our~/'/.OAf''''

VOSS rAJ" woaM 6el'WL ()~Ivt:.r hs l/'l P46tEM 7'.TVNUM;; SSDO/fJ,f: '107.S'-.LET V~I= 20. 711£)/ vlt-z.=- ~17'Y/io=20.3?

r~.,Nc=:S. ,lv'p=3C20.37)~61.II~ USE61

IY '" e 3 ) /118 = 60 I /'It 0= ~.ND " ~ /

:s 3FI IVA-t O,n"VT 51"S!Jf? =5>00 1'-;0 I< 6I == 12..s;L IV',", CJ.I:

C:4A1A1.So use: No= «z: rHJ:#.-.1ou,.:; 13,.)0 fl,,,AI p - ~o . 17fEAI .... aer: '" /3.7!r /(,,..

/07

CHAPTER 9SPUR GEAR DESIGN

Forces on Spur Gear Teeth

(iIV£I'I~ ¢=20,) 1'= 7.fJ.'J /H1'~/?SlJUM..INP~2o",Ntf='7Z.~~'=IZA) _ M . JVp "/750£1'11 • .1.2. =' 9&.IIel',.,.... /11,; - " Nli 7l.

h) Y~~-l!i" A/6/#'p ~ 72-/Zo =- :1.6'':>0

c) PI"" JV~ = Zr}z:: I.D£7//W i .94= .N~ ~ 7};Z:: 6.6IJ()/,N

J) C ~ '/vP f-/VG = 20 r7Z =- 3. 83.s NJz I'J 2C/Z-)

't!) I'V"t" IT 01'Alp/;2 = T/{/.1.67){/7.1lJ ~~ =- 7£Y h;('.w.

f) r; '" b3 OIJO tP) - 6.3lJooC7.r) = ZJOL.&'/N1""'1" /7SD

r& =- 6300,/}>;>.. L.~c>ool7.r) = 'Ill US-//IIAlt; ¥ ~6-/

,.) INt - rp .: 270 U,/,N :. 3.2Y LISD,/Z /.6.t7"N/7

CJ£ t./b" .3.3 O<Jo I:.PJ =- 13060L7. sJ -= 3ZYut.l'V"e 7£y

J..) Nil =- hJ~ b.-¢j={-ny..Jt}t;.... ZtJ'= //8LI3

.-i) IN'N :: ~ / ~, =- 32$/U/ =' 3yS-US/Co! 20'

A S/M/tAtt. nC"r'HoO /s UJcO FUR ?tI('I!(..~J 2-6.

5/'tJ2c/t05..11€Er 5OLu17IM/J AR.E SHO",uJ ON :ntE' P/JU4W1N0

FA ees . 1"I1~ Sdc,tJno.,) t:,1L I'tlO&u:,. I 1(' AL~0 ShOw,",

Fall. CD,.",al/fllTot</ To 7?fl; 5tk./1174., :sl'lowv ~ove,

loa

Ij----'Ff';;o:;:rc:::e::s:-:o::n:-:S;:p::u~r:-:G:;:e::a:':r~T~e~et::h:-----

RESULTS:a Gear speed =b VR=mG=c pinion PO =

gear PO =d center distance = C =e pilch line speed =f torque on pinion shaft =

torque on gear shaft =g tangential force =h radial force =I normal force =

Problem 2Chapter 9

486.13.6001.6676.0003.833764270972324118345

rpm

inininfIIminIb inIb inIbIbIb

RESULTS:a Gear speed = 304.4 rpmb VR=mG= 3.778c pinion PO = 3.600 in

gear PO = 13.600 ind center distance = C = 8.600 ine pitch line speed = 1084 fIIminf torque on pinion shaft = 2739 Ib in

torque on gear shaft = 10348 Ib ing tangential force = 1522 Ibh radial force = 5541bI nonnal force = 1620lb

/0'1

Forces on Spur Gear Teeth

Problem 3 "Chapter 9

RESULTS:a Gear speed = 752.7 rpm

b VR=mG= 4.583

c pinion PO = 1.000 ingear PO = 4.583 in

d center distance = C = 2.792 ine pitch line speed = 903 ftIminf torque on pinion shaft = 13.70 Ib in

torque on gear shaft = 62.77 Ibing tangential force = 27.40 Ibh radial force = 9.97 IbI normal force = 29.16 Ib

Problem 4Chapter 9

RESULTS:a Gear speed = 486.1 rpmb VR=mG= 3.600c pinion PO = 1.667 in

gear PO = 6.000 ind center distance = C = 3.833 ine pitch line speed = 764 ftImin

torque on pinion shaft = 270lbintorque on gear shaft = 972 Ib in

g tangential force = 3241bh radial force = 151 IbI normal force = 3581b

//0

Forces on Spur Gear Teeth

Problem 5. Chapler9.

RESULTS:a Gear speed = 304.4 rpm

b VR= mG= 3.778

c pinion PO = 3.600 ingear PO = 13.600 in

d center distance = C = 8.600 ine pitch line speed = 1084 ftIminf torque on pinion shaft = 2739 Ibin

torque on gear shaft = 10348 Ibing tangential force = 1522 Ib

h radial force = 710 IbI normal force = 1680 Ib

RESULTS:a ·Gearspeed=b VR=mG=c pinion PO =

gear PO =d center distance = C =e pitch line speed =

torque on pinion shaft =torque on gear shaft =

g tangential force :h radial force =I normal force =

752.7 'rpm4.5831.000 in4.583 in2.792 in903 ftImin

13.70 Ib in62.77 Ib in27.401b12.781b30.241b

ILl

Gear Manufacture and Quality

7. See Section 9-4. Fonn milling, shaping, and hobbing.

For Problems 8 - 16 refer to Table 9-2 for recommended quality numbers. Some judgment is requiredbased on more detailed knowledge of the application. The total composite tolerance (fCT) is estimatedfrom Table 9-1. Rough interpolation has been used. If more precise data are required, refer to AGMAStandard 2000-88 Gear Classification and Inspection Handbook.

8. Grain harvester. Use Q = 5. For Pd = 8, Np = 40, No = 100: TCT p = 0.0130 in; TCT 0 = 0.0150 in.9. Printing press: Q = 10. For Pd = 20, Np = 40, No = 100: TCT p = 0.0015 in; TCT G = 0.0017 in.10. Auto transmission: Q = 10. For Pd = 8, Np= 40, No = 100: TCTp = 0.0047 in; TCT 0 = 0.0055 in.11. Gyroscope: Q = 14. For Pd = 32, Np = 40, No = 100: TCT p = 0.00031 in; TCT 0 = 0.00035 in.

12. Using Q = 10 as compared with Q = 5, with all other data being equal, requires tooth accuracyapproximately three times greater.

13. Quality number is increasing because of the application. The size of the teeth is also decreasingbecause of the increasing numerical value of the diametral pitch. This results in a dramatic increaseof the accuracy required in the tooth fonn as represented by the TCT values. For the pinion the TeTmoves from 0.0130 in to 0.0047 in to 0.00031 in. Similar reductions for the gear.

For Problems 14, 15, and 16, use the lower part of Table 9-2. For precision machine tool, the qualitynumber choice is related to the pitch line speed of the gears, with a higher quality number for fasterspeeds because the dynamic effects become more pronounced. See solutions for Problems 1,2, and3for pitch line speeds.

14. Pitch line speed = 764 fIImin (From Prob. 1) Use Q = 8. For Pd = 12, Np = 20, No = 72: TCTp =0.0037 in; TCT0 = 0.0044 in. (Rough interpolation)

15. Pitch line speed = 1064 fIImin (From Prob. 2) Use Q = 10. For Pd = 5, Np = 18, No = 68: TCTp =0.0037 in; TCT G = 0.0039 in. (Rough interpolation)

16. Pitch line speed = 764 fIImin (From Prob. 3) Use Q = 10. For Pd = 24, Np = 24, No = 110: TeT p=0.0013 In; TCT 0 = 0.0016 in. (Rough interpolation)

Gear Materials

Answers for Problems 17 - 25 are found in Section 9-7. Only brief statements are given here.

17. Bending stre~s are created by the tang~ntial force on the gear teeth acting in a manner similar toth.at on a.cantilever. The maximum bending stress occurs in the root of the tooth where it blendswith the mvolute tooth fonn. High levels of contact stress (sometimes called Hertz stress) occur inthe face of the t~eth near the pitch line as forces are exerted between the pinion and the gear teeth.The probable failure mode is pitting of the tooth surface.

18. ~e~:t:;t~h=~~drdanrdeSssglVo'fethallOWatbl~bending stress numbers and allowable contact stress numberse ma enal of the teeth.

19. Gear steels are typically medium carbon pi . IIhardening using a quenching d t . am or a oy steels that are heat treated by through-3140 4140 4340 6150 t an empenng process. Examples are given in Section 9-7. 1040,

I I • • e c.

20. The AGMA recommends hardnesses from HB 180 to HB 400.21. Grade 1 steel is typical commercial qualit G d' .

quality controls on the alloy content and y. ra es 2 and 3 require progressively more stnngenthigher grades. See also AGMA Sta d rdcle2aOnlinessof the materials. Costs also increase for the

n a 04-B89 Gear Materials and Heat Treatment.

II~

22. Grades 2 and 3 may be specified for high-speed aerospace applications, turbine engine drivensystems, ship propulsion drives, and high-capacity industrial gears such as those in steel rolling mills.

23. Flame or induction hardening, carburizing, and nitriding.

24. AGMA Standard 2001-C95 (or latest revision).

25. Gray cast iron, ductile (nodular) iron, and bronze. See Table 9-4.26. Allowable bending stress numbers are obtained from Figure 9-10 or by computing values from the

equations given in the figure. Use Figure 9-/1 for allowable contact stress number.

a) Grade 1, 200 HB: Sat = 28 200 psi; Sao=93 500 psi

b) Grade 1, 300 HB: Sat= 26 000 psi; Sac=125 700 psi

c} Grade 1, 400 HB: Sat = 23 700 psi; Sac=157 900 psi

d) Grade 1, 450 HB: Using HB > 400 in not recommended.

e) Grade 2, 200 HB: Sat = 36 800 psi; Sac=104 100 psi

f) Grade 2,300 HB: Sat = 47 000 psi; Sac=139 000 psi

g) Grade 2, 400 HB: Sat= 57 200 psi; Sac=173 900 psi

27. Grade 1, 300 HB; Grade 2,192 HB from Figure 9-/0.

28. 55-64 HRC. See Table 9-3.29. From Appendix 5: AIS11020, AISI 4118, AISI 8620 as examples.

30. 50-54 HRC with materials having good hardenability. See Tabte 9-3.

31. AIS14140. AISI 4340, AISI 6150. All must have good hardenability.

32. 83.5 on the HR15N hardness scale used for thin-case materials.

33. Data are found in Tables 9-3, 9-4, and 9-11,a} Sat= 45 000 psi; sao=170 000 psi

b} Sat= 45 000 psi; Sao=175 000 psi

c} Sat = 55 000 psi; Sac=180 000 psi

d} Sat= 65 000 psi; Sac=225 000 psi

e) Sat = 55 000 psi; Sao= 180000 psi

f) Sat= 39 000 psi; sao=155 000 psi

g} Sat = 51 500 psi; Sac=168000 psi

h} Sat = 63 500 psi; Sac=216 000 psi

i} Sat = 5 000 psi; sao=50 000 psi

j} Sat = 13 000 psi; Sao=75 000 psi

k} Sat = 27 000 psi; sao=92 000 psi

I} Sat = 5 700 psi; sao=30 000 psi

m} Sat = 23 600 psi; sao=65 000 psi

n} Sat = 12000 psi; Notlisted

0) Sal= 9 000 psi; Not listed

34. Depth = 0.027 in. Figure 9-'1:1..35. Depth = 0.90 mm. Figure 9- JJ...

1/3

� ~-~M"'~<sesoII II II,. CD-., .,

It £~.-:s"-,, II- - H:--

~l~~$..~o-~.-_~...to_ ,-'- _.0 _._ _

•e.;;:

JI'f

II II II II- - II IIII) II) lI) 01II II II II

2222j j Iiun1111s H s0000

'"-~:'t: '.::':l

','dF',dJd;~Il~~~"l.5

E Ee- .5 .5 .5 :i! .eN 0

~ §~8 "'ai -e- -e-

N '"~ l"i tD .o~ ~:::~;;::;;;:. .~.-,:':i~~~-:g :g

1')1')0. '--o,:,,:CS" 0". -~ ::"ii ;~-:;-=-il~:':H

:.:';::c:''''~.:,:,,,'·~:::-.., ..,-~.__ .,.

I

1/6'

lib

8Nv

.c;'.::::., ...

~~:i;"1:i"':""i';:"i:·

111

:glia~... ..,~c:Sod• H H.. ",-.~ ~

II 11 II II- - l,i II

CI) " CIJ lIJ" JI II II

22 U

1II1fill11110000