PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS 1.81 REFERENCES 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Datsko, J., Material Properties and Manufacturing Process, John Wiley and Sons, New York, 1966 Datsko, J Material in Design and Manufacturing, Malloy, Ann Arbor, Michigan, 1977 ASM Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988 Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio, Vol 53, No 6, March 19, 1981 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986 Technical Editor Speaks, the International Nickel Company, New York, 1943 Shigley, J E., Mechanical Engineering Design, Metric Edition, McGraw-Hill Book Company, New York, 1986 Deutschman, A D., W J Michels, and C E Wilson, Machine Design—Theory and Practice, Macmillan Publishing Company, New York, 1975 Juvinall, R C., Fundaments of Machine Components Design, John Wiley and Sons, New York, 1983 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1981 and 1984 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983 SAE Handbook, 1981 Lessels, J M., Strength and Resistance of Metals, John Wiley and Sons, New York, 1954 Siegel, M J., V L Maleev, and J B Hartman, Mechanical Design of Machines, 4th edition, International Textbook Company, Scranton, Pennsylvania, 1965 Black, P H., and O Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1963 Niemann, G., Maschinenelemente, Springer-Verlag, Berlin, Erster Band, 1963 Faires, V M., Design of Machine Elements, 4th edition, Macmillan Company, New York, 1965 Nortman, C A., E S Ault, and I F Zarobsky, Fundamentals of Machine Design, Macmillan Company, New York, 1951 Spotts, M F., Design of Machine Elements, 5th edition, Prentice-Hall of India Private Ltd., New Delhi, 1978 Vallance, A., and V L Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951 Decker, K.-H., Maschinenelemente, Gestalting und Bereching, Carl Hanser Verlag, Munich, Germany, 1971 Decker, K.-H., and Kabus, B K., Maschinenelemente-Aufgaben, Carl Hanser Verlag, Munich, Germany, 1970 ISO and BIS standards Metals Handbook, Desk Edition, ASM International, Materials Park, Ohio, 1985 (formerly the American Society for Metals, Metals Park, Ohio, 1985) Edwards, Jr., K S., and R B McKee, Fundamentals of Mechanical Components Design, McGraw-Hill Book Company, New York, 1991 Shigley, J E., and C R Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Book Company, New York, 1996 Structural Alloys Handbook, Metals and Ceramics Information Center, Battelle Memorial Institute, Columbus, Ohio, 1985 Wood Handbook and U S Forest Products Laboratory SAE J1099, Technical Report of Fatigue Properties Ashton, J C., I Halpin, and P H Petit, Primer on Composite Materials-Analysis, Technomic Publishing Co., Inc., 750 Summer Street, Stanford, Conn 06901, 1969 Baumeister, T., E A Avallone, and T Baumeister III, Mark’s Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill Book Company, New York, 1978 Norton, Refractories, 3rd edition, Green and Stewart, ASTM Standards on Refractory Materials Handbook (Committee C-8) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website PROPERTIES OF ENGINEERING MATERIALS 1.82 CHAPTER ONE BIBLIOGRAPHY Black, P H., and O Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1983 Decker, K.-H., Maschinenelemente, Gestalting und Bereching, Carl Hanser Verlag, Munich, Germany, 1971 Decker, K.-H., and Kabus, B K., Maschinenelemente-Aufgaben, Carl Hanser Verlag, Munich, Germany, 1970 Deutschman, A D., W J Michels, and C E Wilson, Machine Design—Theory and Practice, Macmillan Publishing Company, New York, 1975 Faires, V M., Design of Machine Elements, 4th edition, McGraw-Hill Book Company, New York, 1965 Honger, O S (ed.), (ASME) Handbook for Metals Properties, McGraw-Hill Book Company, New York, 1954 ISO standards Juvinall, R C., Fundaments of Machine Components Design, John Wiley and Sons, New York, 1983 Lessels, J M., Strength and Resistance of Metals, John Wiley and Sons, New York, 1954 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962 Mark’s Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill Book Company, New York, 1978 Niemann, G., Maschinenelemente, Springer-Verlag, Berlin, Erster Band, 1963 Norman, C A., E S Ault, and I E Zarobsky, Fundamentals of Machine Design, McGraw-Hill Book Company, New York, 1951 SAE Handbook, 1981 Shigley, J E., Mechanical Engineering Design, Metric Edition, McGraw-Hill Book Company, New York, 1986 Siegel, M J., V L Maleev, and J B Hartman, Mechanical Design of Machines, 4th edition, International Textbook Company, Scranton, Pennsylvania, 1965 Spotts, M F., Design of Machine Elements, 5th edition, Prentice-Hall of India Private Ltd., New Delhi, 1978 Vallance, A., and V L Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Source: MACHINE DESIGN DATABOOK CHAPTER STATIC STRESSES IN MACHINE ELEMENTS SYMBOLS3;4;5 A Aw a b c D C1 F Fc Ft F Fcr e et E Ec G Mb Mt i I Ixx , Iyy J k k0 kt area of cross section, m2 (in2 ) area of web, m2 (in2 ) constant in Rankine’s formula radius of area of contact, m (in) bandwidth of contact, m (in) width of beam, m (in) distance from neutral surface to extreme fiber, m (in) diameter of shaft, m (in) constant in straight-line formula load, kN (lbf) compressive force, kN (lbf) tensile force, kN (lbf) shear force, kN (lbf) crushing load, kN (lbf) deformation, total, m (in) eccentricity, as of force equilibrium, m (in) unit volume change or volumetric strain thermal expansion, m (in) modulus of elasticity, direct (tension or compression), GPa (Mpsi) combined or equivalent modulus of elasticity in case of composite bars, GPa (Mpsi) modulus of rigidity, GPa (Mpsi) bending moment, N m (lbf ft) torque, torsional moment, N m (lbf ft) number of turns moment of inertia, area, m4 or cm4 (in4 ) mass moment of inertia, N s2 m (lbf s2 ft) moment of inertia of cross-sectional area around the respective principal axes, m4 or cm4 (in4 ) moment of inertia, polar, m4 or cm4 (in4 ) radius of gyration, m (in) polar radius of gyration, m (in) torsional spring constant, J/rad or N m/rad (lbf in/rad) 2.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS 2.2 CHAPTER TWO l l0 L n n0 l, m, n P T ÁT r q Q v V ÁV Z xy , yz , zx " "T "x , "y , "z b c sc cr e s t st x , y , z 1 , 2 , 3 y sy u su 0 00 length, m (in) length of rod, m (in) length, m (in) speed, rpm (revolutions per minute) coefficient of end condition speed, rps (revolutions per second) direction cosines (also with subscripts) power, kW (hp) pitch or threads per meter temperature, 8C (8F) temperature difference, 8C (8F) radius of the rod or bar subjected to torsion, m (in) (Fig 2-18) shear flow first moment of the cross-sectional area outside the section at which the shear flow is required velocity, m/s (ft/min or fpm) volume, m3 (in3 ) shear force, kN (lbf) volume change, m3 (in3 ) section modulus, m3 (in3 ) deformation of contact surfaces, m (in) coefficient of linear expansion, m/m/K or m/m/8C ðin=in=8F) shearing strain, rad/rad shearing strain components in xyz coordinates, rad/rad deformation or elongation, m (in) strain, mm/m (min/in) thermal strain, mm/m (min/in) strains in x, y, and z directions, mm/m (min/in) angular distortion, rad angle, deg angular twist, rad (deg) angle made by normal to plane nn with the x axis, deg bulk modulus of elasticity, GPa (Mpsi) Poisson’s ratio radius of curvature, m (in) stress, direct or normal, tensile or compressive (also with subscripts), MPa (psi) bearing pressure, MPa (psi) bending stress, MPa (psi) compressive stress (also with subscripts), MPa (psi) hydrostatic pressure, MPa (psi) compressive strength, MPa (psi) stress at crushing load, MPa (psi) elastic limit, MPa (psi) strength, MPa (psi) tensile stress, MPa (psi) tensile strength, MPa (psi) stress in x, y, and z directions, MPa (psi) principal stresses, MPa (psi) yield stress, MPa (psi) yield strength, MPa (psi) ultimate stress, MPa (psi) ultimate strength, MPa (psi) principal direct stress, MPa (psi) normal stress which will produce the maximum strain, MPa (psi) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS s xy , yz , zx ! 2.3 normal stress on the plane nn at any angle to x axis, MPa (psi) shear stress (also with subscripts), MPa (psi) shear strength, MPa (psi) shear stresses in xy, yz, and zx planes, respectively, MPa (psi) shear stress on the plane at any angle with x axis, MPa (psi) angular speed, rad/s Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context, are not given at this stage (Note: and with initial subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook.) Particular Formula SIMPLE STRESS AND STRAIN Ft ; A Fc A The stress in simple tension or compression (Fig 2-1a, 2-1b) t ¼ The total elongation of a member of length l (Fig 2-2a) ẳ Fl AE 2-2ị "ẳ ẳ l E 2-3ị c ẳ 2-1ị FIGURE 2-1 Strain, deformation per unit length Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS 2.4 CHAPTER TWO Particular Formula FIGURE 2-2 Young’s modulus or modulus of elasticity Eẳ " 2-4ị The shear stress (Fig 2-1c) ẳ F A 2-5ị Shear deformation due to torsion (Fig 2-18) ẳ L G 2-6ị Shear strain (Fig 2-2c) ẳ a ẳ G l 2-7ị The shear modulus or modulus of rigidity from Eq (2-7) Gẳ 2-8ị Poissons ratio ẳ lateral strain/axial strain ¼ Poisson’s ratio may be computed with sufficient accuracy from the relation ẳ E 2G 2-10ị The shear or torsional modulus or modulus of rigidity is also obtained from Eq (2-10) Gẳ E 21 ỵ ị 2-11ị The bearing stress (Fig 2-3c) b ẳ F bd2 2-12ị ẳ x cos2 ð2-13Þ "t "a ð2-9Þ STRESSES Unidirectional stress (Fig 2-4) The normal stress on the plane at any angle with x axis Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS Particular 2.5 Formula FIGURE 2-3 Knuckle joint for round rods FIGURE 2-4 A bar in uniaxial tension.3;4 The shear stress on the plane at any angle with x axis ¼ x sin 2 2-14ị Principal stresses 1 ẳ x and 2 ẳ 2-15ị Angles at which principal stresses act 1 ẳ 08 and 2 ẳ 908 2-16ị Maximum shear stress max ¼ Angles at which maximum shear stresses act 1 ¼ 458 and 2 ¼ 1358 x Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð2-17Þ ð2-18Þ STATIC STRESSES IN MACHINE ELEMENTS 2.6 CHAPTER TWO Particular The normal stress on the plane at an angle ỵ =2ị (Fig 2-4d) The shear stress on the plane at an angle ỵ =2ị (Fig 2-4d) Formula 0 ẳ x cos2 ỵ 2-19ị ¼ x cos2 cos ỵ ẳ x sin 2 2-20ị ẳ x sin ỵ 2 ẳ 0 and ẳ 2-21ị The normal stress on the plane at any angle ¼ xy sin 2 ð2-22Þ The shear stress on the plane at any angle ¼ xy cos 2 2-23ị The principal stress 1 ẳ xy and 2 ẳ xy 2-24ị Angles at which principal stresses act 1 ẳ 458 and 2 ẳ 1358 2-25ị Maximum shear stresses max ¼ xy ¼ ð2-26Þ Angles at which maximum shear stress act 1 ẳ and 2 ẳ 908 2-27ị FIGURE 2-5 An element in pure shear FIGURE 2-6 An element in biaxial tension Therefore from Eqs (2-13) and (2-19), (2-14), and (2-20) PURE SHEAR (FIG 2-5) BIAXIAL STRESSES (FIG 2-6) The normal stress on the plane at any angle ẳ x ỵ y x y ỵ cos 2 2 ð2-28Þ The shear stress on the plane at any angle ¼ x À y sin 2 ð2-29Þ The shear stress at ¼ ¼ ð2-30Þ The shear stress at ẳ 458 max ẳ x y ị=2 ð2-31Þ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS Particular 2.7 Formula BIAXIAL STRESSES COMBINED WITH SHEAR (FIG 2-7) The normal stress on the plane at any angle ẳ x ỵ y x y ỵ cos 2 ỵ xy sin 2 2 ð2-32Þ The shear stress in the plane at any angle ¼ x À y sin 2 À xy cos 2 2-33ị The maximum principal stress x ỵ y 1 ẳ ỵ The minimum principal stress 2 ¼ Angles at which principal stresses act 1;2 ¼ arctan x ỵ y x À y x À y !1=2 þ xy ð2-34Þ !1=2 2 þ xy 2 xy x À y ð2-35Þ ð2-36Þ where 1 and 2 are 1808 apart x À y Maximum shear stress max ¼ Angles at which maximum shear stress acts ¼ arctan The equation for the inclination of the principal planes in terms of the principal stress (Fig 2-8) tan ¼ σy θ x θ τxy ¼ 1 À 2 x À y 2 xy 1 x xy xy y xy !1=2 ỵ xy τxy n σx σx θ σx τxy σy (a) n τxy σθ θ τθ σy (b) FIGURE 2-7 An element in plane state of stress FIGURE 2-8 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð2-37Þ ð2-38Þ ð2-39Þ STATIC STRESSES IN MACHINE ELEMENTS 2.8 CHAPTER TWO Particular Formula MOHR’S CIRCLE Biaxial field combined with shear (Fig 2-9) Maximum principal stress 1 1 is the abscissa of point F Minimum principal stress 2 2 is the abscissa of point G Maximum shear stress max max is the ordinate of point H FIGURE 2-9 Mohr’s circle for biaxial state of stress TRIAXIAL STRESS (Figs 2-10 and 2-11) The normal stress on a plane nn, whose direction cosines are l, m, n ẳ x l ỵ y m2 ỵ z n2 The shear stress on a plane normal nn, whose direction cosines are l, m, n ¼ The principal stresses 1;2;3 ¼ x ; y ; z The cubic equation for general state of stress in three dimensions from the theory of elasticity 3 À x ỵ y ỵ z ị2 ỵ x y ỵ y z ỵ z x q 2 l ỵ 2 m2 ỵ 2 n2 x y z 2-40ị 2-41ị ð2-42Þ 2 À xy À yz À zx ị 2 x y z ỵ 2 xy yz zx À x zy À y zx À z xy ị ẳ0 The maximum shear stresses on planes parallel to x, y, and z which are designated as ð2-43Þ The three roots of this cubic equation give the magnitude of the principal stresses 1 , 2 , and 3 À 3 À 3 ; ð max Þ2 ¼ ; ð max Þ1 ¼ 2 2 max ị3 ẳ 2-44ị Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS 2.24 CHAPTER TWO Particular Formula The shear modulus G of isotropic material if the modulus of elasticity E is available G¼ The modulus of elasticity of the core material (Fig 2-31) Ef ¼ Em E 2ð1 À Þ ð2-130Þ À V 2=3 V 2=3 ỵ V 2-131ị where V ẳ Hh =Hị3 , Ef ẳ modulus of elasticity of foam, GPa (psi), Em ¼ modulus of elasticity of basic solid material, GPa (psi) Subscript f stands for foam/filament, m stands for matrix, and c stands for composite FIGURE 2-31 A unit cube foam subject to a tensile load The deflection for a beam panel according to Castigliano’s theorem ¼ FIGURE 2-32 Phantom load ð ð FIGURE 2-33 The deflection at midspan (Fig 2-32) L=2 @U @ ¼ @F @F @U @ ẳ ẳ @W @W Mb dx ỵ 2EI V dx 2GA Mb dx ỵ 2EI V dx 2GA 2-132ị W ẳ0 2-133aị 5FL3 FL ỵ 2-133bị 349EI 8GA where W is the phantom load (Fig 2-32) ¼ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS Particular The deflection per unit width for a sandwich panel at midspan (Fig 2-32) under quarter-point loading 2.25 Formula 5FL3 FL ỵ 349DB 8Dc B HH ỵ Hc ị where Dc ẳ Gcore 2Hc L=2 ẳ 2-134ị The deection per unit width for a sandwich panel at quarter panel (Fig 2-32) under quarter-point loading L=4 ẳ FL3 FL ỵ 96DB 8Dc B ð2-135Þ The deflection/unit width for a sandwich panel at center loading (Fig 2-33) L=2 ẳ FL3 FL ỵ 48DB 4Dc B ð2-136Þ The maximum normal stress (Fig 2-32) max F L  M FL ẳ ẳ ẳ BhHc h ỵ Hc =2ị Z 8BhHc H=2 ð2-137Þ FL 8BhH ð2-138Þ The minimum normal stress min ¼ The average stress often used in the composite panel design av ¼ The maximum shear stress in the core max ẳ V 2V ẳ ẵBH ỵ Hc ị=2 BH ỵ Hc ị 2-140ị The core shear strain core ẳ max Gcore 2-141ị FLL ỵ Hc ị FL % 16BhHc H 4BhH ỵ Hc ị 2-139ị FILAMENT REINFORCED STRUCTURES (Fig 2-34) The strain in the filament is same as the strain in the matrix of composite material if it has to have strain compatibility "m ¼ "f FIGURE 2-34 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð2-142Þ STATIC STRESSES IN MACHINE ELEMENTS 2.26 CHAPTER TWO Particular Formula The relation between stress in matrix and stress in lament m f ẳ Em Ef 2-143ị For equilibrium F ẳ m Am ỵ f Af ẳ c Ac 2-144ị f ẳ FEf Af Ef ỵ Am Em 2-145ị m ẳ FEm Af Ef ỵ Am Em 2-146ị Ec ẳ E f Af Am ỵ Af ị 2-147ị The stress in the filament The stress in the matrix The Young’s modulus of composite The Young’s modulus of chopped-up glass filaments in resin matrix but still oriented longitudinally with respect to load as proposed by Outerwater " Ec ịchpd-f Af ẳ Ef À Ac 4 yf D2 f Lpc # 2-148ị where ẳ applied tensile stress, MPa (psi) yf ¼ the strength of the fiber, MPa (psi) Df ¼ diameter of fiber, mm (in) pc ¼ uniform distance of one fiber from another on circumference, mm (in) L ¼ length of fiber, mm (in) Subscript chpd-f stands for chopped-up fiber The relation between m and f , which has to satisfy Eq (2-142) at any location on the curves, Fig 2-35 f m ¼ ðE0 Þm ðE0 Þf ð2-149Þ where E0 ¼ secant modulus, GPa (Mpsi) From Eq (2-144), the expression for c For structure with no filament, Af ¼ f Ac ¼ For structure with all filament, Am ¼ c ¼ u ịf Ac E0 ịm Am ỵ Af E0 ịf m ịmax Am ỵ Af u ịf c ẳ u ịf Ac u ịf Af ẳ u ịf Ac c ẳ 2-150ị m ịmax Am u 2-151ị 2-152ị ẳ m ịmax ẳ u ịm ð2-153Þ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS 2.27 Particular Formula FIGURE 2-35 Stress-strain data for system shown in Fig 2-34 FIGURE 2-36 Filament wound cylindrical pressure vessel FILAMENT BINDER COMPOSITE (Fig 2-36) Hoop stress for a closed end vessel/cylinder made of filaments winding ¼ pd 2h ð2-154Þ Longitudinal/axial stress for a closed end lament wound vessel/cylinder a ẳ pd 4h 2-155ị The force carried by a helical filament wound on a shell of width w subjected to internal pressure p in the -direction Fa ¼ so wh The force in helical filament wound on a shell of width w subjected to internal pressure p in the hoop direction F ¼ F sin The hoop stress in the vessel wall due to the pressure p ¼ The stress in the vessel wall in the longitudinal/axialdirection a ¼ 0 cos2 From Eq (2-154) to (2-159) the optimum winding angle for closed end cylinders tan2 ¼ The optimum winding angle for open end cylinders a cos2 ẳ cot2 ẳ ẳ sin2 2-156ị where so ẳ strength of the laments 2-157ị F ẳ 0 sin2 A a ð2-158Þ ð2-159Þ or % 558 2-160ị or ẳ 908 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð2-161Þ STATIC STRESSES IN MACHINE ELEMENTS 2.28 CHAPTER TWO Particular The stress in the hoop/circumferential direction for the filament wound cylinder/vessel consisting windings in longitudinal, hoop and helical directions to satisfy equilibrium condition Formula ẳ 0 h ỵ h h 2-162ị where 0 ¼ stress in the circumferential wound layer ¼ circumferential component of stress in the helical layer ht ¼ total thickness ẳ ỵ h ỵ h , h and h are the thicknesses in the preceding layers of filament windings The longitudinal stress for the case of winding under Eq (2-162) a ẳ 0a h0a ỵ a h h 2-163aị so h ỵ h sin2 ị h so a ẳ ỵ h cos2 Þ h where ¼ ð2-163bÞ ð2-163cÞ so ¼ uniform filament stress a ¼ longitudinal component of stress in helical layer From Eqs (2-159) and (2-158) ỵ a ẳ 0 sin2 ỵ cos2 ị ẳ 0 From Eqs (2-154) and (2-155) hẳ 2-164ị pd 4a 2-165ị pd 2h 2-154ị ¼ The sum of stresses and a þ a ¼ 3a ¼ 0 For the ideal vessel h¼ or a ¼ 0 3pd 40 ¼ h h cos2 2-166ị 2-167ị 2-168aị h ẳ À h sin2 ð2-168bÞ 2ha À h cos2 2-168cị h ẳ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS Particular The structural efficiency of the wound vessel/cylinder 2.29 Formula ¼ W Ven pi 2-169ị where W ẳ weight of the vessel, kN (lbf) Ven ¼ enclosed volume, m3 (in3 ) pi ¼ internal pressure, MPa (psi) FILAMENT-OVERLAY COMPOSITE T uw where T ¼ tension, kN (lbf) uw ¼ area of the element, m2 (in2 ) The stress in the wire which is wound on thin walled shell/cylinder with a wire of the same material (Fig 2-37) wr ¼ Under equilibrium condition over the length of shell L, the hoop stress ð Þsh ¼ À T wh ð2-170Þ ð2-171Þ FIGURE 2-37 Shell subjected to an internal pressure The tension in the wound wire on the shell under internal pressure Twr ¼ pd T ỵ 2h ỵ uị wu 2-172ị The tension in the shell under the above same condition Tcy ¼ pd T 2h ỵ uị wh 2-173ị The yielding of shell due to internal pressure, i.e., due to plastic flow of material of the shell 0 ịshy ẳ 2-174ị For the above same winding material under the tension equal to compression yield limits, the stress in the wire wr T ¼ Ày wh T h ¼ ¼ y uw u 2-175ị cy ẳ "sh Esh 2-176aị wr ¼ "wr Ewr If the vessel material is different from the winding material then stress in the wire and vessel ð2-176bÞ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS 2.30 CHAPTER TWO Particular Formula For uniform distribution of stress in the cylinder/shell and in the wire, strains are proportional to the mean radii "cy "cy cy Ewr r ¼ ¼ " rwr "wr wr Ecy From Eq (2-177), the stress in the cylinder and the wire sh ¼ cy ¼ wr wr ¼ cy Ecy "cy r Ewr "wr r " Ewr rwr " Ecy rcy ð2-177Þ ð2-178aÞ ð2-178bÞ where subscripts cy stands for cylinder, sh for shell " " and wr for winding rcy and rwr are mean radii of cylinder and winding respectively The total load on the cylinder and the winding cy 2Lhị ỵ wr 2Luị ẳ pdL ð2-179Þ From Eq (2-179), the stress in the cylinder (cy ) and the winding (wr ) pd cy ¼ Ewr "wr u r ỵh " Ecy rcy 2-180aị pd wr ẳ " Ecy rcy ỵu " Ewr rwr ð2-180bÞ The stress in the cylinder is the sum of results of Eqs (2-180a) and (2-171) pd T À Rcy ¼ " Ewr rwr u wh ỵh r Ecy "cy 2-181ị The resultant stress in the winding is the sum of results of Eq (2-180b) and (2-170) pd T ỵ Rwr ẳ " Ecy rcy h wu ỵu r Ewr "wr 2-182ị For advanced theory using Theory of elasticity and Plasticity construction on composite structures and materials Refer to advanced books and handbooks on composites, structures, handbooks and design data for reinforced plastics and materials For representative properties for fiber reinforcement Refer to Table 2-1 FORMULAS AND DATA FOR VARIOUS CROSS SECTIONS OF MACHINE ELEMENTS For further data on static stresses, properties and torsion of shafts of various cross-sections: shear, moments, and deflections of beams, strain rosettes, and singularity functions Refer to Tables 2-2 to 2-12 For summary of stress and strain formulas under various types of loads Refer to Table 2-13 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 0.4 0.4 0.4 0.2–0.4 0.5 0.001–0.01 0.2 0.2–0.5 0.8 5–10 127 102 13 0.025–0.25 5–13 20.5 Â10À3 in 10.2 10.2 10.2 102.0 Â10À3 mm 7.64 2.43 2.62 1.11 1.49 1.43–1.75 1.78 3.40 2.48 2.43 2.46 2.56 g/cm3 3100 4498 5510 2756 MPa 450 650 800 400 kpsi Tensile strength, st 0.283 0.090 0.097 0.041 0.052 2852–4184 689–2067 1378–2067 827 690 385–600 100–300 200–300 120 100 0.053–0.066 1723–3445 250–500 0.066 1240 180 0.126 2480 360 0.092 0.090 0.091 0.095 lb/in3 Density, 200 172 138–413 2.8 241–689 310 414 72.5 85 100 415 GPa 29 25 20–60 0.4 0.6 35–100 45 60 10.5 12.3 14.5 60 Mpsi Modulis of elasticity, Eg 30 3.7 45–50 45–50 54 6.5 81–90 81–90 1.5 6.4 2.2 2.8 5.0 2.7 11.5 4.0 2.8 lin/in8F 5.0 lm/m K Coefficient of thermal expansion, 381 381 22400–38080 13440 19488 6496 1680 W/(m2 K/m) 1.7 1.7 100–170 60 87 29 7.5 Btu/(ft2 h8F)/in) Thermal conductivity, K Courtesy: J E Ashton, J C Halpin, and P H Petit, Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., 750 Summer St., Stanford, Conn 06901, 1969 E glass S glass 970 S glass Boron on tungsten Graphite Beryllium Silicon carbide on tungsten Stainless steel Asbestos Aluminum Polyamide Polyester Fiber Typical fiber diameter TABLE 2-1 Representative properties for fiber reinforcement STATIC STRESSES IN MACHINE ELEMENTS Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 2.31 STATIC STRESSES IN MACHINE ELEMENTS 2.32 CHAPTER TWO TABLE 2-2 Torsion of shafts of various cross sections Polar section modulus, Polar radius of gyration, k0 Cross section Z0 ¼ J=c D pffiffiffi ¼ 0:354D D3 16 Angular deflection, In terms of torsional moment, Mt In terms of maximum stress, 2l D G 32l Mt D4 D at circumference ðD4 À D4 Þ 16D1 s q D2 ỵ D2 ẳ 0:354 D2 þ D2 B h A b b2 h 16 a p b2 ỵ h2 2l D1 G 32l Mt ðD4 À D4 Þ G at outer circumference 16b2 ỵ h2 ịl Mt G b3 h3 b2 ỵ h2 ịl G bh2 h>b 2b2 h at A b a h>b r b2 ỵ h2 12 p ẳ 0:289 b2 þ h2 mðb2 þ h2 Þl Mt G b3 h3 h ẳ1 b nb2 ỵ h2 ịl G bh2 at A c m ¼ 3:56 3:50 3:35 3:21 n ¼ 0:79 0:78 0:74 0:71 b3 20 46:2l Mt b4 G a 0:289b 2:31l b G at center of side 0:92b3 a 0:645b 0:967l Mt G b4 0:9l b G at center of side a This value is not true value of Z0 but is the value of Z0 for a circular section of equal strength and may be used for determining the maximum stress by the formula ¼ Mt =Z0 b At B, shear stress ¼ 10Mt =bh2 c At B, shear stress ¼ 9Mt =2bh2 Source: V L Maleev and J B Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS 2.33 TABLE 2-3 Shear stress in beams, caused by bending Shear stress at a distance y from neutral axis, , MPa (psi) Section 3F 1À 2bh 4F 1À 3r2 2y h ! Maximum shear stress, max , MPa (psi) 3F F ẳ 1:5 for y ẳ 0ị 2bh A ! y r 4F F ¼ 1:33 ðfor y ẳ 0ị A 3r2 p p ! F y y 1ỵ b b b 1:591 F A c for y ¼ 3F bc2 À ðb À aÞd 4a bc3 b aịd ! for y ẳ 0ị TABLE 2-4 The values of constants a in Eq (2-107) Yield stress in compression, yc Value of a for various end-fixity coefficients n 750 3000 1500 n  750 549 80 1600 6400 3200 n  1600 324 47 7500 30000 15000 n  7500 Material MPa kpsi Timber 49 Cast iron Mild steel TABLE 2-5 End condition coefficient n (Fig 2-25) Particular n TABLE 2-6 End-fixity coefficients for cast iron column to be used in Eq (2-104) One end fixed and the other end free Both ends rounded and guided or hinged One end fixed, and the other end rounded and guided or hinged Both ends fixed rigidly Both ends flat 0.25 End conditions C1 Maximum, l=k Round Fixed One fixed, one round 175 88 116 90 160 115 to Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STATIC STRESSES IN MACHINE ELEMENTS 2.34 CHAPTER TWO TABLE 2-7 Properties of cross sections Section Area, A Moment of inertia, I Distance to farthest point, c bh bh3 12 h bh2 0:289h ðH À cÞb b ðH À h3 Þ 12 H bðH À h3 Þ 3H sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H À h3 12ðH À hÞ BH À bh BH À bh3 12 H BH À bh3 6H sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi BH bh3 12BH bhị 6b2 ỵ 6bb0 þ b2 Þh3 36ð2b þ b0 Þ ð3b þ 2b0 ịh 32b ỵ b0 ị 6b2 ỵ 6bb0 ỵ b2 ịh2 123b ỵ b0 ị r I A D4 64 D D3 32 D 2b ỵ b0 h D2 D D2 ị Section modulus, Z ẳ I=c Radiusffiffiffiffiffiffiffiffiffi p of gyration, k ¼ I=A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 þ D2 ðD4 À D4 Þ 64 ẳ R4 R4 ị D1 ¼ R1 ab ba3 64 a ba2 32 a bh bh3 36 ÁÁÁ bh2 24 0:236h ðD4 À D4 Þ 32D1 ẳ q R2 ỵ R2 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Bending moment Mb , and maximum bending moment Max MbC ¼ ÀFa at C A to B : V ¼ B to C : V ¼ ÀF End supports, center load Cantilever, uniform load B to C : Mb ẳ ỵ Fl xị Max MbB ẳ ỵ1 FL at B A to B : V ẳ ỵ F B to C : V ¼ À1 F Max MbB ¼ À 1Wl at B A to B : Mb ẳ ỵ Fx W x l R1 ẳ ỵ1 F; R2 ẳ ỵ F 2 Vẳ W x l B to C : Mb ¼ Fx bị R2 ẳ ỵF Mb ẳ A to B : Mb ¼ V ¼ ÀF R2 ẳ ỵW ẳ wl Max MbB ẳ Fl at B R2 ẳ ỵF Cantilever, end load Cantilever, intermediate load Mb ¼ ÀFx Reactions R1 and R2 , vertical shear V Loading, support, and reference number TABLE 2-8 Shear, moment, and deflection formulas for beams ymax ¼ À F ð3l x À 4x3 Þ 48 EI Fl at B 48 EI A to B: y ¼ À Wl EI W x 4l x ỵ 3l ị 24 EIl ymax ¼ À y¼À F ð3a2 l À a3 ị EI F ẵx bị3 3a2 x bị ỵ 2a3 EI B to C: y ¼ À ymax ¼ À F a3 ỵ 3a2 l 3a2 xị EI A to B: y ¼ À Fl at A EI F ðx À 3l x ỵ 2l ị EI ymax ẳ yẳ Deflection y and maximum deflection STATIC STRESSES IN MACHINE ELEMENTS Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 2.35 A O 2.36 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website I W = wl B M2 One end fixed, one end supported, uniform load One end fixed, one end supported, center load End supports, uniform load End supports, intermediate load Loading, support, and reference number b a ; R2 ¼ þF l l ab at B l a ðl À xị l b x l Fbx ẵ2ll xị b2 À ðl À xÞ2 6EIl Wx l 2lx2 ỵ x3 ị 24 EIl Max ÀMbC ¼ 16 Fl at C B to C: V ¼ À 11 F 16 V¼W x À l M2 ¼ Wl R1 ¼ W; R2 ¼ W 8 Max ÀMbB ẳ Wl at B Max ỵMb ¼ 128 Wl at x ¼ l Mb ¼ Wð3 x À x2 Þ Wl at x ¼ 0:4215l El W ð3lx3 À 2x4 l xị 48 EIl ymax ẳ 0:0054 yẳ ymax ẳ 0:00932 Max ỵMbB ẳ 32 Fl at B A to B: V ẳ ỵ 16 F M2 ¼ 16 Fl Fl at x ¼ 0:4472l EI F ½5x3 À 16ðx À lÞ3 À 3l x 96 EI B to C: y ¼ A to B: Mb ¼ 16 Fx R1 ¼ 16 F; R2 ¼ 11 F 16 B to C: Mb ¼ Fð1 l À 11 xị 16 Wl Max y ẳ at x ¼ l 384 EI y¼À Faðl xị ẵ2lb b2 l xị2 B to C: y ¼ À 6EIl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fab ymax ẳ a ỵ 2bị 3aa ỵ 2bị at 27EIl q x ẳ 1aa ỵ 2bị when a > b A to B: y ¼ À Deflection y and maximum deflection F ð5x3 À 3l xÞ 96 EI Max Mb ẳ ỵ Wl at x ẳ l x2 Mb ¼ W x l Max MbB ẳ ỵF B to C: Mb ẳ ỵF A to B: Mb ẳ þF Bending moment Mb , and maximum bending moment A to B: y ¼ 2x V ¼ 1W l R2 ẳ ỵ W; R2 ẳ þ W 2 A to B: V ¼ þF b l a B to C: V ¼ ÀF l R1 ẳ ỵF Reactions R1 and R2 , vertical shear V TABLE 2-8 Shear, moment, and deflection formulas for beams (Cont.) STATIC STRESSES IN MACHINE ELEMENTS Bending moment Mb , and maximum bending moment A to B: V ¼ þ F Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Max ÀMb ¼ À 12 Wl at A and B Max ỵMb ẳ 24 Wl at x ¼ l 1 M1 ¼ 12 Wl; M2 ¼ 12 Wl 2x V ¼ 1W À l x2 Mb ¼ W x À À l l R1 ¼ W; R2 ¼ W 2 Max ÀMb ¼ ÀM2 when a > b B to C: V ¼ R1 À F ¼ R2 ab2 ỵ R1 a at B l2 Max Mb ẳ M1 when a < b Max ỵMb ẳ F ab2 ỵ R1 x Fx aị l2 ab2 þ R1 x l2 A to B: V ¼ R1 ab2 a2 b ; M2 ¼ ÀF l2 l B to C: Mb ẳ F Fa2 3b ỵ aị l3 R2 ¼ M1 ¼ ÀF A to B: Mb ẳ F Fb2 3a ỵ bị l3 R1 ẳ Source: J E Shigley, Mechanical Engineering Design, 3rd ed., McGraw-Hill Book Company, New York, 1977 11 Both ends fixed, uniform load 10 Both ends xed, intermediate load Max ỵMbB ẳ Fl at B M1 ¼ Fl; M2 ¼ Fl 8 Max ÀMbA;C ¼ À Fl at A and C B to C: Mb ¼ F3l 4xị R1 ẳ F; R2 ¼ F 2 Both ends fixed, center load B to C: V ¼ À F A to B: Mb ẳ F4x lị Reactions R1 and R2 , vertical shear V Loading, support, and reference number TABLE 2-8 Shear, moment, and deflection formulas for beams (Cont.) F a2 b3 2bl if a < b at x ¼ l À EI ð3b þ aÞ2 ð3b þ aÞ Wl at x ¼ l 384 EI Wx2 ð2lx À l x2 ị 24 EIl ymax ẳ yẳ ymax ẳ Fa2 l xị2 ẵ3b þ aÞðl À xÞ À 3bl EIl Fb2 x2 3ax ỵ bx 3alị EIl F a3 b2 2al if a > b at x ẳ EI 3a ỵ bị2 3a ỵ bị B to C: y ¼ ymax ¼ À F ð3lx2 À 4x3 Þ 48 EI Fl at B 192 EI A to B: y ¼ ymax ¼ À A to B: y ¼ À Deflection y and maximum deflection STATIC STRESSES IN MACHINE ELEMENTS 2.37 STATIC STRESSES IN MACHINE ELEMENTS 2.38 CHAPTER TWO TABLE 2-9 Some equations for use with the Castigliano method Type of load General energy equation U¼ Axial U¼ Bending Combined axial and bending U¼ Torsion U¼ Transverse shear U¼ Transverse shear (rectangular section) U¼ Open-coiled helical spring subjected to axial load F U¼ ðl ðl ðl ðl ðl ðl ðl Energy equation 2 General deflection equation ðl F ds 2AE U¼ F l Al ¼ 2AE 2E ¼ Mb l ds 2EI U¼ Mb l 2EI ¼ U¼ F l Mb l ỵ 2EA 2EI Sum of axial and bending load Mt2 ds 2GJ U¼ Mt2 l 2GJ ¼ V ds 2GA U¼ V2l 2 ¼ Al 2GA 2G ¼ 3V ds 5GA U¼ 3V l 5GA ẳ Uẳ Mt2 l Mb l ỵ 2GJ 2EI ẳ 2iFR3 sec F2 ds ỵ 2AE Mt2 ds ỵ 2GJ l l Mb ds 2EI Mb ds 2EI ẳ LFR2 cos2 sin2 ỵ GJ EI ! ðl ðl ðl ðl Fð@F=@QÞ ds AE Mb ð@Mb =@QÞ ds EI Mt ð@Mt =@QÞ ds GJ Vð@V=@GÞ ds GA 6Vð@V=@QÞ ds 5GA cos2 sin2 ỵ GJ EI ! D ẳ mean radius of coil ¼ helix angle of spring i ¼ number of coils or turns where R ¼ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ... 0.0 01? ??0. 01 0.2 0.2–0.5 0.8 5? ?10 12 7 10 2 13 0.025–0.25 5? ? 13 20.5 ? ?10 ? ?3 in 10 .2 10 .2 10 .2 10 2.0 ? ?10 ? ?3 mm 7.64 2. 43 2.62 1. 11 1.49 1. 43? ? ?1. 75 1. 78 3. 40 2.48 2. 43 2.46 2.56 g/cm3 31 0 0 4498 5 510 2756... ¼ 1: 046F 6 a ¼ 0:7 216 F 2 1 2 ỵ E1 E2 d1 d2 d1 ỵ d2 2 1 2 ỵ E1 E2 1 À d1 d2 ð2 -11 0Þ 31 = 3 7 ð2 -11 1Þ 31 = 3 7 ð2 -11 2Þ The maximum compressive stress cðmaxÞ 31 = 3 1 À d1... 2. 21 Formula 6 ¼ 1: 046F 2 1 2 ỵ E1 E2 d1 d2 d2 À d1 31 = 3 7 ð2 -11 4Þ Refer to Fig 2-28a " a ¼ 0:7 21 Fd1 2 1 2 ỵ E1 E2 #1= 3 2 -11 5ị 31 = 3 F cmaxị ẳ 0: 918 6 7 2 À 1 À 2 ỵ d1 E1