Machine Design Databook 2010 Part 2 pptx

L length, m inn speed, rpm revolutions per minute coefficient of end condition n0 speed, rps revolutions per second l, m, n direction cosines also with subscripts Q first moment of the cros

1 Datsko, J., Material Properties and Manufacturing Process, John Wiley and Sons, New York, 1966.

2 Datsko, J Material in Design and Manufacturing, Malloy, Ann Arbor, Michigan, 1977

3 ASM Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988

4 Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio, Vol 53, No 6, March 19,1981

5 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers,Bangalore, India, 1986

6 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary MetricUnits), Suma Publishers, Bangalore, India, 1986

7 Technical Editor Speaks, the International Nickel Company, New York, 1943

8 Shigley, J E., Mechanical Engineering Design, Metric Edition, McGraw-Hill Book Company, New York,1986

9 Deutschman, A D., W J Michels, and C E Wilson, Machine Design—Theory and Practice, Macmillan lishing Company, New York, 1975

Pub-10 Juvinall, R C., Fundaments of Machine Components Design, John Wiley and Sons, New York, 1983

11 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College tive Society, Bangalore, India, 1962

Co-opera-12 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers,Bangalore, India, 1981 and 1984

13 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary MetricUnits), Suma Publishers, Bangalore, India, 1983

14 SAE Handbook, 1981

15 Lessels, J M., Strength and Resistance of Metals, John Wiley and Sons, New York, 1954

16 Siegel, M J., V L Maleev, and J B Hartman, Mechanical Design of Machines, 4th edition, InternationalTextbook Company, Scranton, Pennsylvania, 1965

17 Black, P H., and O Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1963

18 Niemann, G., Maschinenelemente, Springer-Verlag, Berlin, Erster Band, 1963

19 Faires, V M., Design of Machine Elements, 4th edition, Macmillan Company, New York, 1965

20 Nortman, C A., E S Ault, and I F Zarobsky, Fundamentals of Machine Design, Macmillan Company, NewYork, 1951

21 Spotts, M F., Design of Machine Elements, 5th edition, Prentice-Hall of India Private Ltd., New Delhi, 1978

22 Vallance, A., and V L Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York,1951

23 Decker, K.-H., Maschinenelemente, Gestalting und Bereching, Carl Hanser Verlag, Munich, Germany, 1971

24 Decker, K.-H., and Kabus, B K., Maschinenelemente-Aufgaben, Carl Hanser Verlag, Munich, Germany,1970

25 ISO and BIS standards

26 Metals Handbook, Desk Edition, ASM International, Materials Park, Ohio, 1985 (formerly the AmericanSociety for Metals, Metals Park, Ohio, 1985)

27 Edwards, Jr., K S., and R B McKee, Fundamentals of Mechanical Components Design, McGraw-Hill BookCompany, New York, 1991

28 Shigley, J E., and C R Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill BookCompany, New York, 1996

29 Structural Alloys Handbook, Metals and Ceramics Information Center, Battelle Memorial Institute, bus, Ohio, 1985

Colum-30 Wood Handbook and U S Forest Products Laboratory

31 SAE J1099, Technical Report of Fatigue Properties

32 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

33 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

34 Norton, Refractories, 3rd edition, Green and Stewart, ASTM Standards on Refractory Materials Handbook(Committee C-8)

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 Publish-ing 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-operativeSociety, 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 Text-book 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

A area of cross section, m2(in2)

Aw area of web, m2(in2)

a constant in Rankine’s formula

b radius of area of contact, m (in)

bandwidth of contact, m (in)

width of beam, m (in)

c distance from neutral surface to extreme fiber, m (in)

D diameter of shaft, m (in)

C1 constant in straight-line formula

e deformation, total, m (in)

eccentricity, as of force equilibrium, m (in)

unit volume change or volumetric strain

et thermal expansion, m (in)

E modulus of elasticity, direct (tension or compression), GPa

(Mpsi)

Ec combined or equivalent modulus of elasticity in case of

composite bars, GPa (Mpsi)

G modulus of rigidity, GPa (Mpsi)

Mb bending moment, N m (lbf ft)

Mt torque, torsional moment, N m (lbf ft)

I moment of inertia, area, m4or cm4(in4)

mass moment of inertia, N s2m (lbf s2ft)

Ixx, Iyy moment of inertia of cross-sectional area around the respective

principal axes, m4or cm4(in4)

J moment of inertia, polar, m4or cm4(in4)

k radius of gyration, m (in)

k0 polar radius of gyration, m (in)

kt torsional spring constant, J/rad or N m/rad (lbf in/rad)

L length, m (in)

n speed, rpm (revolutions per minute)

coefficient of end condition

n0 speed, rps (revolutions per second)

l, m, n direction cosines (also with subscripts)

Q first moment of the cross-sectional area outside the section at

which the shear flow is required

v velocity, m/s (ft/min or fpm)

V volume, m3(in3)

shear force, kN (lbf)

V volume change, m3(in3)

Z 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

xy,yz,zx shearing strain components in xyz coordinates, rad/rad

 deformation or elongation, m (in)

" strain,mm/m (min/in)

"T thermal strain,mm/m (min/in)

"x,"y,"z 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)

 radius of curvature, m (in)

stress, direct or normal, tensile or compressive (also with

subscripts), MPa (psi)

b bearing pressure, MPa (psi)

bending stress, MPa (psi)

c compressive stress (also with subscripts), MPa (psi)

hydrostatic pressure, MPa (psi)

sc compressive strength, MPa (psi)

cr stress at crushing load, MPa (psi)

e elastic limit, MPa (psi)

s strength, MPa (psi)

t tensile stress, MPa (psi)

st tensile strength, MPa (psi)

x, y, z stress in x, y, and z directions, MPa (psi)

1, 2, 3 principal stresses, MPa (psi)

y yield stress, MPa (psi)

sy yield strength, MPa (psi)

u ultimate stress, MPa (psi)

su ultimate strength, MPa (psi)

0 principal direct stress, MPa (psi)

00 normal stress which will produce the maximum strain, MPa (psi)

s shear strength, MPa (psi)

xy,
yz,
zx shear stresses in xy, yz, and zx planes, respectively, MPa (psi)

 shear stress on the plane at any angle with x axis, MPa (psi)

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.)

SIMPLE STRESS AND STRAIN

The stress in simple tension or compression (Fig 2-1a,

FIGURE 2-2

Young’s modulus or modulus of elasticity

The shear stress (Fig 2-1c)

Shear deformation due to torsion (Fig 2-18)

Shear strain (Fig 2-2c)

The shear modulus or modulus of rigidity from Eq

(2-7)

Poisson’s ratio

Poisson’s ratio may be computed with sufficient

accuracy from the relation

The shear or torsional modulus or modulus of rigidity

is also obtained from Eq (2-10)

The bearing stress (Fig 2-3c)

STRESSES

Unidirectional stress (Fig 2-4)

The normal stress on the plane at any angle with x

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

Principal stresses

Angles at which principal stresses act

Maximum shear stress

Angles at which maximum shear stresses act

The normal stress on the plane at an angle

PURE SHEAR (FIG 2-5)

The normal stress on the plane at any angle

The shear stress on the plane at any angle

The principal stress

Angles at which principal stresses act

Maximum shear stresses

Angles at which maximum shear stress act

FIGURE 2-5 An element in pure shear.

BIAXIAL STRESSES (FIG 2-6)

The normal stress on the plane at any angle

The shear stress on the plane at any angle

The shear stress
at ¼ 0

The shear stress
at ¼ 458

0¼ xcos2

 þ2

cos

 þ2

BIAXIAL STRESSES COMBINED WITH

SHEAR (FIG 2-7)

The normal stress on the plane at any angle

The shear stress in the plane at any angle

The maximum principal stress

The minimum principal stress

Angles at which principal stresses act

Maximum shear stress

Angles at which maximum shear stress acts

The equation for the inclination of the principal

planes in terms of the principal stress (Fig 2-8)

FIGURE 2-7 An element in plane state of stress.

2

þ
2 xy

2

þ
2 xy

MOHR’S CIRCLE

Biaxial field combined with shear (Fig 2-9)

Maximum principal stress 1

Minimum principal stress 2

Maximum shear stress
max

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

The shear stress on a plane normal nn, whose

direc-tion cosines are l, m, n

The principal stresses

The cubic equation for general state of stress in three

dimensions from the theory of elasticity

The maximum shear stresses on planes parallel to x, y,

and z which are designated as

1is the abscissa of point F

2is the abscissa of point G

maxis the ordinate of point H

MOHR’S CIRCLE

Triaxial field (Figs 2-10 and 2-11)

Normal stress at point (Fig 2-11b) on one octahedral

plane

Shear stress at point T (Fig 2-11b) on an octahedral

plane

FIGURE 2-10 An element in triaxial state of stress.

FIGURE 2-11 Mohr’s circle for triaxial octahedral stress state.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½ð 1
2Þ2þ ð 2
3Þ2þ ð 3
1Þ2q

or
tis the ordinate of point T

STRESS-STRAIN RELATIONS

Uniaxial field

Strain in principal direction 1

The principal stress

The unit volume change in uniaxial stress

Biaxial field

Strain in principal direction 1

Strain in principal direction 2

Strain in principal direction 3

The principal stresses in terms of principal strains in a

biaxial stress field

The unit volume change in biaxial stress

Triaxial field

Strain in principal direction 1

Strain in principal direction 2

Strain in principal direction 3

The principal stresses in terms of principal strains in

triaxial stress field

The unit volume change or volumetric strain in terms

of principal stresses for the general case of triaxial

stress (Fig 2-12)

FIGURE 2-12 Uniform hydrostatic pressure.

The volumetric strain due to uniform hydrostatic

pressure cacting on an element (Fig 2-12)

The bulk modulus of elasticity

The relationship between E, G and K

STATISTICALLY INDETERMINATE

MEMBERS (Fig 2-13)

The reactions at supports of a constant cross-section

bar due to load F acting on it as shown in Fig 2-13

The elongation of left portion Laof the bar

The shortening of right portion Lbof the bar

FIGURE 2-13

THERMAL STRESS AND STRAIN

The normal strain due to free expansion of a bar or

machine member when it is heated

The free linear deformation due to temperature change

The compressive force Fcbdeveloped in the bar fixed

at both ends due to increase in temperature (Fig 2-14)

The compressive stress induced in the member due to

thermal expansion (Fig 2-14)

The relation between the extension of one member to

the compression of another member in case of rigidly

joined compound bars of the same length L made of

different materials subjected to same temperature

(Fig 2-15)

The forces acting on each member due to temperature

change in the compound bar

The relation between compression of the tube to the

extension of the threaded member due to tightening

of the nut on the threaded member (Fig 2-16)

The forces acting on tube and threaded member due

to tightening of the nut

FIGURE 2-15

COMPOUND BARS

The total load in the case of compound bars or

col-umns or wires consisting of i members, each having

different length and area of cross section and each

made of different material subjected to an external

load as shown in Fig 2-17

An expression for common compression of each bar

The load on first bar (Fig 2-17)

The load on ith bar (Fig 2-17)

EQUIVALENT OR COMBINED MODULUS

OF ELASTICITY OF COMPOUND BARS

The equivalent or combined modulus of elasticity of a

compound bar consisting of i members, each having a

different length and area of cross section and each

being made of different material

The stress in the equivalent bar due to external load F

The strain in the equivalent bar due to external load F

The common extension or compression due to

Another expression for power in terms of force F

acting at velocity v

TORSION (FIG 2-18)

The general equation for torsion (Fig 2-18)

Torque

The maximum shear stress at the maximum radius r

of the solid shaft (Fig 2-18) subjected to torque Mt

The torsional spring constant

FIGURE 2-18 Cylindrical bar subjected to torque.

BENDING (FIG 2-19)

The general formula for bending (Fig 2-19)

FIGURE 2-19 Bending of beam.

The maximum values of tensile and compressive

bending stresses

The shear stresses developed in bending of a beam

(Fig 2-20)

The shear flow

FIGURE 2-20 Beam subjected to shear stress.

The first moment of the cross-sectional area outside

the section at which the shear flow is required

The maximum shear stress for a rectangular section

For a hollow circular section beam, the expression for

maximum shear stress

An appropriate expression for
max for structural

beams, columns and joists used in structural

indus-tries

ECCENTRIC LOADING

The maximum and minimum stresses which are

induced at points of outer fibers on either side of a

machine member loaded eccentrically (Figs 2-22

and 2-23)

The resultant stress at any point of the cross section of

an eccentrically loaded member (Fig 2-24)

COLUMN FORMULAS (Fig 2-25)

Euler’s formula (Fig 2-26) for critical load

FIGURE 2-22 Eccentric loading.

FIGURE 2-23 Eccentrically loaded machine member FIGURE 2-24

FIGURE 2-25 Column-end conditions (i) One end is fixed and other is free (ii) Both ends are rounded and guided or hinged (iii) One end is fixed and other is rounded and guided or hinged (iv) Fixed ends.

Johnson’s parabolic formula (Fig 2-26) for critical

load

FIGURE 2-26 Variation of critical stress with slenderness

ratio.

Straight-line formula for critical load

Straight-line formula for short column of brittle

material for critical load

Ritter’s formula for induced stress

Ritter’s formula for eccentrically loaded column (Fig

2-23) for combined induced stress

Rankine’s formula for induced stress

The critical unit load from secant formula for a

round-ended column

Fcr¼ A y 1
y

2E

lk

lk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðFcr=4AEÞ

HERTZ CONTACT STRESS

Contact of spherical surfaces

Sphere on a sphere (Fig 2-27a)

The radius of circular area of contact

FIGURE 2-27 Hertz contact stress.

The maximum compressive stress

Combined deformation of both bodies in contact

along the axis of load

Spherical surface in contact with a spherical socket

(Fig 2-27b)

The radius of circular area of contact

The maximum compressive stress

377

1=3

ð2-109Þ

cðmaxÞ¼ 0:918 F

1

377

377

377

1=3

ð2-112Þ

cðmaxÞ¼ 0:918 F

1

377

1=3

ð2-113Þ

Combined deformation of both bodies in contact

along axis of load

Distribution of pressure over band of width of contact

and stresses in contact zone along the line of

sym-metry of spheres

Sphere on a flat surface (Fig 2-27c)

The radius of circular area of contact

The maximum compressive stress

Contact of cylindrical surfaces

Cylindrical surface on cylindrical surface, axis parallel

(Fig 2-27a and Fig 2-28b)

The width of band of contact

The maximum compressive stress

Cylindrical surface in contact with a circular groove

(Fig 2-27b)

The width of band of contact

The maximum compressive stress

Distribution of pressure over band of width of contact

and stresses in contact zone along the line of

377

37

377

377

377

377

1=2

ð2-120Þ

Refer to Fig 2-28b

Cylindrical surface in contact with a flat surface

(Fig 2-27c):

The width of band of contact

The maximum compressive stress

Deformation of cylinder between two plates

The maximum shear stress occurs below contact

surface for ductile materials

37

1=2

ð2-122Þ

where d ¼ d1(Fig 2-27c)

d1¼4FL

DESIGN OF MACHINE ELEMENTS AND

STRUCTURES MADE OF COMPOSITE

Honeycomb composite

For the components of composite materials which

give high strength–weight ratio combined with

rigidity

For sandwich construction of honeycomb structure

FIGURE 2-29 Sandwich fabricated panel.

The moment of inertia of sandwich panel, Fig 2-30

Simplified Eq (2-124) after neglecting powers of h

The flexural rigidity

The flexural rigidity of sandwich plate/panel

The flexural rigidity of sandwich construction for



þ 2Bh

Hcþ h2

F ¼ force over a support span Lc

The shear modulus G of isotropic material if the

modulus of elasticity E is available

The modulus of elasticity of the core material (Fig

2-31)

FIGURE 2-31 A unit cube foam subject to a tensile load.

The deflection for a beam panel according to

Casti-gliano’s theorem

FIGURE 2-32 Phantom load.

The deflection at midspan (Fig 2-32)

, Ef ¼ modulus of elasticity offoam, GPa (psi), Em¼ modulus of elasticity

of basic solid material, GPa (psi) Subscript

f stands for foam/filament, m stands formatrix, and c stands for composite

 ¼@U@F ¼@F@

ð

Mb2dx2EI þ

ð

V2dx2GA

ð

V2dx2GA



W ¼ 0

ð2-133aÞ

¼ 5FL3349EIþ FL

where W is the phantom load (Fig 2-32)

The deflection per unit width for a sandwich panel at

midspan (Fig 2-32) under quarter-point loading

The deflection per unit width for a sandwich panel at

quarter panel (Fig 2-32) under quarter-point loading

The deflection/unit width for a sandwich panel at

center loading (Fig 2-33)

The maximum normal stress (Fig 2-32)

The minimum normal stress

The average stress often used in the composite panel

design

The maximum shear stress in the core

The core shear strain

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

L=2¼ 5FL3349DBþ FL

8DcB

where Dc¼ Gcore

HðH þ HcÞ2Hc



ð2-134Þ

L=4¼ FL396DBþ FL

2L4



BhHcðh þ Hc=2ÞH=2

 ¼ FL8BhHc

ð2-137Þ min¼ FL

core¼
max

FIGURE 2-34

The relation between stress in matrix and stress in

filament

For equilibrium

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

The relation between mand f, which has to satisfy

Eq (2-142) at any location on the curves, Fig 2-35

From Eq (2-144), the expression for c

For structure with all filament, Am¼ 0

For structure with no filament, Af ¼ 0

4



yf

where ¼ applied tensile stress, MPa (psi)

yf ¼ the strength of the fiber, MPa (psi)

Df ¼ diameter of fiber, mm (in)

pc¼ uniform distance of one fiber fromanother on circumference, mm (in)

L ¼ length of fiber, mm (in)Subscript chpd-f stands for chopped-up fiber m

FIGURE 2-35 Stress-strain data for system shown in

The force carried by a helical filament wound on a

shell of width w subjected to internal pressure p in

the-direction

The force in helical filament wound on a shell of width

wsubjected to internal pressure p in the hoop

direc-tion

The hoop stress in the vessel wall due to the pressure p

The stress in the vessel wall in the

longitudinal/axial-direction

From Eq (2-154) to (2-159) the optimum winding

angle for closed end cylinders

The optimum winding angle for open end cylinders

FIGURE 2-36 Filament wound cylindrical pressure vessel.

¼

cos2sin2¼ cot

2 or  ¼ 908 ð2-161Þ

The stress in the hoop/circumferential direction for

the filament wound cylinder/vessel consisting

wind-ings in longitudinal, hoop and helical directions to

satisfy equilibrium condition

The longitudinal stress for the case of winding under

Eq (2-162)

From Eqs (2-159) and (2-158)

From Eqs (2-154) and (2-155)

The sum of stresses and a

For the ideal vessel

ha, h and h are the thicknesses in the preceding

layers of filament windings

a¼ 0ah0aþ ah

where ¼

soh

The structural efficiency of the wound vessel/cylinder

FILAMENT-OVERLAY COMPOSITE

The stress in the wire which is wound on thin walled

shell/cylinder with a wire of the same material

(Fig 2-37)

Under equilibrium condition over the length of shell

L, the hoop stress

The tension in the wound wire on the shell under

internal pressure

The tension in the shell under the above same

condi-tion

The yielding of shell due to internal pressure, i.e., due

to plastic flow of material of the shell

For the above same winding material under the

ten-sion equal to compresten-sion yield limits, the stress in

the wire

If the vessel material is different from the winding

material then stress in the wire and vessel

Venpi

ð2-169Þwhere W ¼ weight of the vessel, kN (lbf)

Ven¼ enclosed volume, m3(in3)

pi¼ internal pressure, MPa (psi)



ð2-175Þ

For uniform distribution of stress in the cylinder/shell

and in the wire, strains are proportional to the mean

radii

From Eq (2-177), the stress in the cylinder and the

wire

The total load on the cylinder and the winding

From Eq (2-179), the stress in the cylinder ( cy) and

For advanced theory using Theory of elasticity and

Plasticity construction on composite structures and

materials

For representative properties for fiber reinforcement

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

For summary of stress and strain formulas under

various types of loads

com-Refer to Table 2-1

Refer to Tables 2-2 to 2-12

Refer to Table 2-13

Polar section

modulus,

Angular deflection,  Cross section Z 0 ¼ J=c Polar radius of gyration, k 0 In terms of torsional moment, M t In terms of maximum stress, 

G

at outer circumference B

G

G

at center of side

Shear stress at a distance y from neutral axis,

, MPa (psi) Maximum shear stress, max , MPa (psi)

3F 2bh



1

2y h

3 2 ¼ 1:33F

A ðfor y ¼ 0Þ

F ffiffiffi 2 p

b 2



1 þy

ffiffiffi 2 p

b
4

y b

 2 

1 :591FA

for y ¼4c



3F 4a



bc2
ðb
aÞd 2

bc 3
ðb
aÞd 3

 ðfor y ¼ 0Þ

TABLE 2-4

The values of constantsa in Eq (2-107)

Yield stress in compression,  yc Value ofa for various end-fixity coefficients

One end fixed and the other end free 0.25

Both ends rounded and guided or hinged 1

One end fixed, and the other end rounded and

guided or hinged

2

TABLE 2-6End-fixity coefficients for cast iron column to be used in

Section Area, A Moment of inertia, I

Distance

to farthest point, c Section modulus,Z ¼ I=c Radius of gyration,k ¼pffiffiffiffiffiffiffiffiffiI=A

2b þ b 0

3

32

D 4

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R 2 þ R 2

q 2

3

64

a 2

2

32

a 4

Type of load General energy equation Energy equation General deflection equation

GJ þsin2EI

 where R ¼D2¼ mean radius of coil

 ¼ helix angle of spring

i ¼ number of coils or turns