Machine Design Databook Episode 1 part 3 ppsx

REFERENCES
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 Metric
Units), 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 Pub-
lishing Company, New York, 1975.
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 Co-opera-
tive Society, Bangalore, India, 1962.
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 Metric
Units), 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, International
Textbook 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, New
York, 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 American
Society for Metals, Metals Park, Ohio, 1985).
27. Edwards, Jr., K. S., and R. B. McKee, Fundamentals of Mechanical Components Design, McGraw-Hill Book
Company, New York, 1991.
28. Shigley, J. E., and C. R. Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Book
Company, New York, 1996.
29. Structural Alloys Handbook, Metals and Ceramics Information Center, Battelle Memorial Institute, Colum-
bus, Ohio, 1985.
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).
PROPERTIES OF ENGINEERING MATERIALS 1.81
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PROPERTIES OF ENGINEERING MATERIALS
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 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-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 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.
1.82 CHAPTER ONE
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PROPERTIES OF ENGINEERING MATERIALS
CHAPTER
2
STATIC STRESSES IN MACHINE

ELEMENTS
SYMBOLS
3;4;5
A area of cross section, m
2
(in
2
)
A
w
area of web, m
2
(in
2
)
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)
C
1
constant in straight-line formula
F load, kN (lbf)
F
c
compressive force, kN (lbf)
F
t

tensile force, kN (lbf)
F

shear force, kN (lbf)
F
cr
crushing load, kN (lbf)
e deformation, total, m (in)
eccentricity, as of force equilibrium, m (in)
unit volume change or volumetric strain
e
t
thermal expansion, m (in)
E modulus of elasticity, direct (tension or compression), GPa
(Mpsi)
E
c
combined or equivalent modulus of elasticity in case of
composite bars, GPa (Mpsi)
G modulus of rigidity, GPa (Mpsi)
M
b
bending moment, N m (lbf ft)
M
t
torque, torsional moment, N m (lbf ft)
i number of turns
I moment of inertia, area, m
4
or cm

4
(in
4
)
mass moment of inertia, N s
2
m (lbf s
2
ft)
I
xx
, I
yy
moment of inertia of cross-sectional area around the respective
principal axes, m
4
or cm
4
(in
4
)
J moment of inertia, polar, m
4
or cm
4
(in
4
)
k radius of gyration, m (in)
k

0
polar radius of gyration, m (in)
k
t
torsional spring constant, J/rad or N m/rad (lbf in/rad)
2.1
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Source: MACHINE DESIGN DATABOOK
l length, m (in)
l
0
length of rod, m (in)
L length, m (in)
n speed, rpm (revolutions per minute)
coefficient of end condition
n
0
speed, rps (revolutions per second)
l, m, n direction cosines (also with subscripts)
P power, kW (hp)
pitch or threads per meter
T temperature, 8C(8F)
ÁT temperature difference, 8C(8F)
r radius of the rod or bar subjected to torsion, m (in) (Fig. 2-18)
q shear flow
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, m
3
(in
3
)
shear force, kN (lbf)
ÁV volume change, m
3
(in
3
)
Z section modulus, m
3
(in
3
)
 deformation of contact surfaces, m (in)
coefficient of linear expansion, m/m/K or m/m/8 C 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)
 Poisson’s ratio
 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)
2.2 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS


normal stress on the plane nn at any angle  to x axis, MPa (psi)
 shear stress (also with subscripts), 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)
! 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.)
SIMPLE STRESS AND STRAIN
The stress in simple tension or compression (Fig. 2-1a,
2-1b)
The total elongation of a member of length l
(Fig. 2-2a)
FIGURE 2-1
Strain, deformation per unit length

t
¼
F
t
A
; 
c
¼

F
c
A
ð2-1Þ
 ¼
Fl
AE
ð2-2Þ
" ¼

l
¼

E
ð2-3Þ
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.3
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STATIC STRESSES IN MACHINE ELEMENTS
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
axis
E ¼

"
ð2-4Þ
 ¼
F

A
ð2-5Þ
 ¼
L
G
ð2-6Þ
 ¼

G
¼
a
l
ð2-7Þ
G ¼



ð2-8Þ
 ¼ lateral strain/axial strain ¼
"
t
"
a
ð2-9Þ
 ¼
E
2G
À 1 ð2-10Þ
G ¼
E
2ð1 þÞ
ð2-11Þ

b
¼
F
bd
2
ð2-12Þ


¼ 
x
cos
2

 ð2-13Þ
Particular Formula
2.4 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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


¼

x
2
sin 2 ð2-14Þ

1
¼ 
x
and 
2
¼ 0 ð2-15Þ


1
¼ 08 and 
2
¼ 908 ð2-16Þ

max
¼

x
2
ð2-17Þ

1
¼ 458 and 
2
¼ 1358 ð2-18Þ
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.5
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STATIC STRESSES IN MACHINE ELEMENTS
The normal stress on the plane at an angle  þð=2Þ
(Fig. 2-4d)
The shear stress on the plane at an angle  þð=2Þ
(Fig. 2-4d)
Therefore from Eqs. (2-13) and (2-19), (2-14), and
(2-20)
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

0

¼ 
x
cos
2

 þ

2

¼ 
x

cos
2
 ð2-19Þ

0

¼ 
x
sin

 þ

2

cos

 þ

2

¼
1
2

x
sin 2 ð2-20Þ


¼ 
0


and 

¼À
0

ð2-21Þ


¼ 
xy
sin 2 ð2-22Þ


¼ 
xy
cos 2 ð2-23Þ

1
¼ 
xy
and 
2
¼À
xy
ð2-24Þ

1
¼ 458 and 
2

¼ 1358 ð2-25Þ

max
¼ 
xy
¼  ð2-26Þ

1
¼ 0 and 
2
¼ 908 ð2-27Þ
FIGURE 2-6 An element in biaxial tension.


¼

x
þ 
y
2
þ

x
À 
y
2
cos 2 ð2-28Þ


¼


x
À 
y
2
sin 2 ð2-29Þ


¼ 0 ð2-30Þ

max
¼ð
x
À 
y
Þ=2 ð2-31Þ
Particular Formula
2.6 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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)
x
σ
y
σ
y
σ
x
τ
xy
σ
θ
τ
θ
τ
xy
τ
xy
τ
xy
σ
x
y
θ
θ
n
n
(a) (b)

σ
y
σ
x
τ
xy
τ
xy
θ
θ
FIGURE 2-7 An element in plane state of stress.


¼

x
þ 
y
2
þ

x
À 
y
2
cos 2 þ
xy
sin 2 ð2-32Þ



¼

x
À 
y
2
sin 2 À
xy
cos 2 ð2-33Þ

1
¼

x
þ 
y
2
þ


x
À 
y
2

2
þ 
2
xy
!

1=2
ð2-34Þ

2
¼

x
þ 
y
2
À


x
À 
y
2

2
þ 
2
xy
!
1=2
ð2-35Þ

1;2
¼
1
2

arctan
2
xy

x
À 
y
ð2-36Þ
where 
1
and 
2
are 1808 apart

max
¼


x
À 
y
2

2
þ 
2
xy
!
1=2
¼


1
À 
2
2
ð2-37Þ
 ¼
1
2
arctan

x
À 
y
2
xy
ð2-38Þ
tan  ¼

1
À 
x

xy
ð2-39Þ
FIGURE 2-8
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.7
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STATIC STRESSES IN MACHINE ELEMENTS
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

1
is the abscissa of point F

2
is the abscissa of point G

max

is the ordinate of point H


¼ 
x
l
2
þ 
y
m
2
þ 
z
n
2
ð2-40Þ


¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
x
l
2
þ 
2
y
m
2

þ 
2
z
n
2
q
ð2-41Þ

1;2;3
¼ 
x
;
y
;
z
ð2-42Þ

3
Àð
x
þ 
y
þ 
z
Þ
2
þð
x

y

þ 
y

z
þ 
z

x
À 
2
xy
À 
2
yz
À 
2
zx
Þ
Àð
x

y

z
þ 2
xy

yz

zx

À 
x

2
zy
À 
y

2
zx
À 
z

2
xy
Þ
¼ 0 ð2-43Þ
The three roots of this cubic equation give the magni-
tude of the principal stresses 
1
, 
2
, and 
3
.
ð
max
Þ
1
¼


2
À 
3
2
; ð
max
Þ
2
¼

1
À 
3
2
;
ð
max
Þ
3
¼

1
À 
2
2
ð2-44Þ
Particular Formula
2.8 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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.

t
¼
1
3
ð
1
þ 
2
þ 
3
Þ¼
1
3
ð
x
þ 
y
þ 

z
Þð2-45Þ
or 
t
is the abscissa of point T

t
¼
1
3
½ð
x
À 
y
Þ
2
þð
y
À 
z
Þ
2
þð
z
À 
x
Þ
2
ð2-46aÞ
þ 6ð

2
xy
þ 
2
yz
þ 
2
zx
Þ
1=2
¼
1
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½ð
1
À 
2
Þ
2
þð
2
À 
3
Þ
2
þð
3
À 
1

Þ
2

q
or 
t
is the ordinate of point T
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.9
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STATIC STRESSES IN MACHINE ELEMENTS
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
"
1
¼

1
E
; "
2
¼À

1
E
; "
3
¼À

1
E
ð2-47Þ

1
¼ E"
1
ð2-47aÞ
ÁV
V
¼
ð1 À2Þ

1
E
¼ "
1
ð1 À 2 Þð2-48Þ
"
1
¼
1
E
ð
1
À 
2
Þð2-49Þ
"
2
¼
1
E
ð
2
À 
1
Þð2-50Þ
"
3
¼À

E

ð
1
þ 
2
Þð2-51Þ

1
¼
E
1 À
2
ð"
1
þ "
2
Þð2-52Þ

2
¼
E
1 À
2
ð"
2
þ "
1
Þð2-53Þ

3
¼ 0 ð2-53aÞ

ÁV
V
þ
ð1 À 2 Þ
E
ð
1
þ 
2
Þð2-54Þ
"
1
¼
1
E
½
1
À ð
2
þ 
3
Þ ð2-55Þ
"
2
¼
1
E
½
2
À ð

3
þ 
1
Þ ð2-56Þ
"
3
¼
1
E
½
3
À ð
1
þ 
2
Þ ð2-57Þ

1
¼
E
ð1 À À 2
2
Þ
½ð1 À Þ"
1
þ ð"
2
þ "
3
Þ ð2-58Þ


2
¼
E
ð1 À À 2
2
Þ
½ð1 À Þ"
2
þ ð"
3
þ "
1
Þ ð2-59Þ

3
¼
E
ð1 À À 2
2
Þ
½ð1 À Þ"
3
þ ð"
1
þ "
2
Þ ð2-60Þ
Particular Formula
2.10 CHAPTER TWO

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STATIC STRESSES IN MACHINE ELEMENTS
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 
c
acting 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 L
a
of the bar
e ¼
dV
V
¼
ð1 À2Þ
E
ð
x
þ 

y
þ 
z
Þð2-61aÞ
¼
ð1 À 2 Þ
E
ð
1
þ 
2
þ 
3
Þð2-61bÞ
ÁV
V
¼
À3ð1 À2Þ
c
E
¼À

c

ð2-62Þ
 ¼
E
3ð1 À2Þ
ð2-63Þ
E ¼

9KG
ð3K þ GÞ
ð2-63aÞ
R
a
¼
FL
b
L
a
þ L
b
¼
FL
b
L
ð2-64aÞ
R
B
¼
FL
a
L
a
þ L
b
¼
FL
a
L

ð2-64bÞ

a
¼
R
A
L
a
AE
¼
FL
a
L
b
LAE
ð2-65Þ
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.11
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STATIC STRESSES IN MACHINE ELEMENTS
The shortening of right portion L
b
of 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 F
cb
developed 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

b
¼À
R
A
L
a
AE
¼À
FL
a

L
b
LAE
ð2-66Þ
FIGURE 2-14
"
T
¼ ðÁTÞð2-67Þ
 ¼ LðÁTÞð2-68Þ
F
cb
¼ AEðÁTÞð2-69Þ

cT
¼
F
cb
A
¼ÀEðÁTÞð2-70Þ

s
L
E
s
þ

c
L
E
c

¼ð
c
À 
s
ÞLðÁTÞð2-71Þ

c
A
c
¼ 
s
A
s
ð2-72Þ

t
L
E
t
þ

s
L
E
s
¼ [number of turns ðiÞ
Âðthreads/meterÞ or pitch ðPÞ
¼ iP ð2-73Þ

s

A
s
¼ 
t
A
t
ð2-74Þ
Particular Formula
2.12 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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
(Fig. 2-17)
FIGURE 2-17
FIGURE 2-16
F ¼
X
E
i
A
i


i
L
i
¼ 
X
E
i
A
i
L
i
ð2-75Þ
 ¼
F
P
ðE
i
A
i
=L
i
Þ
ð2-76Þ
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.13
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STATIC STRESSES IN MACHINE ELEMENTS
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
external load F
POWER
The relation between power, torque and speed
F
1
¼
ðE
1
A
1
=L
1
Þ
P
ðEA=LÞ
F ð2-77Þ
F
i

¼
E
i
A
i

L
i
ð2-78Þ
E
c
¼
E
1
A
1
þ E
2
A
2
þ E
3
A
3
þÁÁÁþE
n
A
n
A
1

þ A
2
þ A
3
þÁÁÁþA
n
ð2-79aÞ
¼
P
E
i
A
i
P
i ¼1;2;:::;n
A
i
ð2-79bÞ
 ¼
F
P
i ¼1;2;3;:::
A
i
ð2-80Þ
" ¼
F
E
c
P

i ¼1;2;3;:::
A
i
¼

L
ð2-81Þ
 ¼
FL
E
c
P
i ¼1;2;3;:::;n
A
i
¼ "L ð2-82Þ
P ¼ M
t
! ð2-83Þ
where M
t
in N m (lbf ft), ! in rad/s (rad/min), and
P in W (hp)
¼
M
t
n
0
159
SI ð2-84aÞ

where M
t
in kN m, n
0
in rps, and P in kW
¼
M
t
n
9550
SI ð2-84bÞ
where M
t
in kN m, n in rpm, and P in kW
¼
M
t
n
63030
USCS ð2-84cÞ
where M
t
in lbf in, n in rpm, and P in hp
Particular Formula
2.14 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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 M
t
The torsional spring constant
FIGURE 2-18 Cylindrical bar subjected to torque.
P ¼
F
1000
SI ð2-85aÞ
where F in newtons (N),  in m/s, and P in kW
¼
F
33000
USCS ð2-85bÞ
where F in lbf,  in fpm (feet per minute), and P in
hp (horsepower)
M
t
J
¼
G
L
¼


ð2-86Þ

M
t
¼
159P
n
0
SI ð2-87aÞ
where M
t
in kN m, n
0
in rps, and P in kW
¼
9550P
n
SI ð2-87bÞ
where M
t
in kN m, n in rpm, and P in kW
¼
63030P
n
USCS ð2-87cÞ
where M
t
in lbf in, n in rpm, and P in hp

max
¼
16M

t
D
3
ð2-88Þ
k
t
¼
M
t

¼
GJ
L
ð2-89Þ
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.15
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STATIC STRESSES IN MACHINE ELEMENTS
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.

M
b
I
¼

b
c
¼
E

ð2-90Þ

b
¼
M
b
c
I
ð2-91Þ
 ¼
V
Ib
ð
c
y
0
ydA ð2-92Þ
q ¼
VQ
I

ð2-93Þ
Particular Formula
2.16 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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
(Figs. 2-20 and 2-21)
FIGURE 2-21 Element cut out from a beam subjected to
shear stress.
For a solid circular section beam, the maximum shear
stress
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
Q ¼
ð
c
y
0
ydA ð2-94Þ

max
¼
3V
2A
ð2-95Þ

max
¼
4V
3A
ð2-96Þ

max
¼
2V
A
ð2-97Þ

max
¼
V
A

w
ð2-98Þ
where A
w
is the area of the web

max
¼
F
A
þ
M
b
Z
and 
min
¼
F
A
À
M
b
Z
ð2-99Þ

z
¼Æ
F
A
Æ

M
bx
e
y
I
xx
Æ
M
by
e
x
I
yy
ð2-100Þ
F
cr
¼
n
2
EA
ðl=kÞ
2
¼
n
2
EI
l
2
ð2-101Þ
Particular Formula

STATIC STRESSES IN MACHINE ELEMENTS
2.17
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STATIC STRESSES IN MACHINE ELEMENTS
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.
Particular Formula
2.18 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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
F

cr
¼ A
y
1 À

y
4n
2
E

l
k

2
"#
ð2-102Þ
F
cr
¼ A 
y
À
2
y
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
y
=3nEÞ
p


l
k

"#
ð2-103Þ
F
cr
¼ A

 ÀC
1
l
k

ð2-104Þ

c
¼
F
A
1 þ

e
n
2
E

l
k


2
"#
ð2-105Þ

c
¼
F
A
1 þ

e
n
2
E

l
k

2
þ
ce
k
2
"#
ð2-106Þ

c
¼
F
cr

A
1 þ a

l
k

2
"#
ð2-107Þ
F
cr
A
¼

y
1 þ
ec
k
2
sec
l
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðF
cr
=4AEÞ
p
ð2-108Þ
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS

2.19
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STATIC STRESSES IN MACHINE ELEMENTS
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
a ¼ 0: 721 F
1 À
2
1
E
1
þ
1 À 
2
2
E
2


1
d
1
þ
1
d
2

2
6
6
6
4
3
7
7
7
5
1=3
ð2-109Þ

cðmaxÞ
¼ 0:918 F

1
d
1
þ
1
d

2

2

1 À 
2
1
E
1
þ
1 À
2
2
E
2

2
2
6
6
6
6
4
3
7
7
7
7
5
1=3

ð2-110Þ
 ¼ 1:04 F
2

1 À
2
1
E
1
þ
1 À
2
2
E
2

2

d
1
d
2
d
1
þ d
2

2
6
6

6
4
3
7
7
7
5
1=3
ð2-111Þ
a ¼ 0: 721 F

1 À
2
1
E
1
þ
1 À
2
2
E
2


1
d
1
À
1
d

2

2
6
6
6
4
3
7
7
7
5
1=3
ð2-112Þ

cðmaxÞ
¼ 0:918 F

1
d
1
À
1
d
2

2

1 À
2

1
E
1
þ
1 À 
2
2
E
2

2
2
6
6
6
6
4
3
7
7
7
7
5
1=3
ð2-113Þ
Particular Formula
2.20 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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 sym-
metry of cylinders
 ¼ 1:04 F
2

1 À
2
1
E
1
þ

1 À 
2
2
E
2

2

d
1
d
2
d
2
À d
1

2
6
6
6
4
3
7
7
7
5
1=3
ð2-114Þ
Refer to Fig. 2-28a.

a ¼ 0: 721 Fd
1

1 À
2
1
E
1
þ
1 À 
2
2
E
2

"#
1=3
ð2-115Þ

cðmaxÞ
¼ 0:918
F
d
2
1

1 À 
2
1
E

1
þ
1 À
2
2
E
2

2
2
6
4
3
7
5
1=3
ð2-116Þ
where d ¼ d
1
(Fig. 2-27c).
2b ¼ 1: 6
F
L

1 À
2
1
E
1
þ

1 À 
2
2
E
2


1
d
1
þ
1
d
2

2
6
6
6
4
3
7
7
7
5
1=2
ð2-117Þ

cðmaxÞ
¼ 0:798

F
L

1
d
1
þ
1
d
2


1 À
2
1
E
1
þ
1 À
2
2
E
2

2
6
6
6
4
3

7
7
7
5
1=2
ð2-118Þ
2b ¼ 1: 6
F
L

1 À
2
1
E
1
þ
1 À 
2
2
E
2


1
d
1
À
1
d
2


2
6
6
6
4
3
7
7
7
5
1=2
ð2-119Þ

cðmaxÞ
¼ 0:798
F
L

1
d
1
À
1
d
2


1 À
2

1
E
1
þ
1 À
2
2
E
2

2
6
6
6
4
3
7
7
7
5
1=2
ð2-120Þ
Refer to Fig. 2-28b.
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.21
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STATIC STRESSES IN MACHINE ELEMENTS

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
For sphere
For cylinders
The depth from contact surface to the point of the
maximum shear
FIGURE 2-28 Distribution of pressure over bandwidth of contact and stresses in contact zone along line of symmetry of
spheres and cylinders for  ¼ 0:3.
2b ¼ 1: 6
Fd
1
L

1 À
2
1
E
1
þ
1 À
2
2
E
2


"#
1=2
ð2-121Þ

cðmaxÞ
¼ 0:798
F
Ld
1
1

1 À
2
1
E
1
þ
1 À 
2
2
E
2

2
6
4
3
7
5
1=2

ð2-122Þ
where d ¼ d
1
(Fig. 2-27c).
Ád
1
¼
4F
L

1 À
2
1
E

1
3
þ log
e
2d
1
b

ð2-123Þ

max
¼ 0:31
cðmaxÞ
ð2-123aÞ


max
¼ 0:295
cðmaxÞ
ð2-123bÞ
h ¼ 0: 786b ð2-123cÞ
Particular Formula
2.22 CHAPTER TWO
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STATIC STRESSES IN MACHINE ELEMENTS
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
ðH
c
=hÞ > 5
The shear modulus of the core material as per Jones
and Hersch
Refer to Fig. 2-29.

Refer to Fig. 2-30.
FIGURE 2-30 Honeycomb.
I ¼ 2

Bh
3
12

þ 2Bh

H
c
þ h
2

2
ð2-124Þ
I ¼ BhH
c

h þ
H
c
2

ð2-125Þ
D ¼ EI ð2-126Þ
where E ¼ modulus of elasticity of the facing metal
I is given by Eq. (2-125).
D ¼

EðH
3
À H
3
c
Þ
12ð1 À
2
Þ
ð2-127Þ
D ¼
EhðH þH
c
Þ
2
8ð1 À
2
Þ
ð2-128Þ
G
core
¼
1:5FL
c
BðH þH
c
Þ
2
ð11
4

À 8
2
Þ
ð2-129Þ
where 
4
and 
2
¼ deflection at quarter-span and
midspan respectively
F ¼ force over a support span L
c
Particular Formula
STATIC STRESSES IN MACHINE ELEMENTS
2.23
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STATIC STRESSES IN MACHINE ELEMENTS