Machine Design Databook Episode 1 part 4 pptx

TABLE 2-10

Mechanical and physical constants of some materials1;2

Modulus of

elasticity, E

Modulus of

rigidity, G

Poisson’s

Density,

a

,

Unit weight,  b

Material GPa Mpsi GPa Mpsi ratio,  Mg/m3 kfg/m3 kN/m3 lbf/in3 lbf/ft3

Aluminum 69 10.0 26 3.8 0.334 2.69 2,685 26.3 0.097 167

Aluminum cast 70 10.15 30 4.35 2,650 26.0 0.096 166

Aluminum (all alloys) 72 10.4 27 3.9 0.320 2.80 2,713 27.0 0.10 173

Beryllium copper 124 18.0 48 7.0 0.285 8.22 8,221 80.6 0.297 513

Carbon steel 206 30.0 79 11.5 0.292 7.81 7,806 76.6 0.282 487

Cast iron, gray 100 14.5 41 6.0 0.211 7.20 7,197 70.6 0.260 450

Malleable cast iron 170 24.6 90 13.0 7,200

Inconel 214 31.0 76 11.0 0.290 8.42 8,418 83.3 0.307 530

Magnesium alloy 45 6.5 16 2.4 0.350 1.80 1,799 17.6 0.065 117

Molybdenum 331 48.0 117 17.0 0.307 10.19 10,186 100.0 0.368 636

Monet metal 179 26.0 65 9.5 0.320 8.83 8,830 86.6 0.319 551

Nickel-silver 127 18.5 48 7.0 0.332 8.75 8,747 85.80 0.316 546

Nickel alloy 207 30 79 11.5 0.30 8.3 0.300 518

Nickel steel 207 30.0 79 11.5 0.291 7.75 7,751 76.0 0.280 484

Phosphor bronze 111 16.0 41 6.0 0.349 8.17 8,166 80.1 0.295 510

Steel (18-8), stainless 190 27.5 73 10.6 0.305 7.75 7,750 76.0 0.280 484

Titanium (pure) 103 15.0 4.47 4,470 43.9 0.16 279

Titanium alloy 114 16.5 43 6.2 0.33 6.6

Brass 106 15.5 40 5.8 0.324 8.55 8,553 83.9 0.309 534

Bronze 96 14.0 38 5.5 0.349 8.30 8,304 81.4

Bronze cast 80 11.6 35 5.0 8,200

Copper 121 17.5 46 6.6 0.326 8.90 8,913 87,4 0.322 556

Tungsten 345 50.0 138 20.0 18.82 18,822 184.6

Douglas fir 11 1.6 4 0.6 0.330 4.43 443 4.3 0.016 28

Glass 46 6.7 19 2.7 0.245 2.60 2,602 25.5 0.094 162

Lead 36 5.3 13 1.9 0.431 11.38 11,377 111.6 0.411 710

Concrete (compression) 14–28 2.0–4.0 2.35 2,353 23.1 147

Wrought iron 190 27.5 70 10.2 7,700

Zinc alloy 83 12 31 4.5 0.33 6.6 0.24 415

a  ¼ mass density: b ¼ weight density; w is also the symbol used for unit weight of materials.

Sources: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers,

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

1986.

STATIC STRESSES IN MACHINE ELEMENTS 2.39

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STATIC STRESSES IN MACHINE ELEMENTS

TABLE 2-11

Relations between strain rosette readings and principal stresses

Rosette type

Required solutions

Maximum normal

stress, max

E

1
2 ð"1 þ "2Þ E

2

 "1 þ "3

1
 þ 1

1 þ  E

"1 þ "2 þ "3

3ð1
Þ þ 1

1 þ  E

2

"1 þ "4

1
 þ 1

1 þ 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð"1
"3Þ

2 þ ½2"2
ð"1 þ "3Þ2

q  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

"1
"1 þ "2 þ "3

3

2

þ

"2
"3 ffiffiffi

3

p

2

s  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð"1
"4Þ

2 þ 4

3

ð"2
"3Þ

2

r 

Minimum normal

stress, min

E

1
2 ð"2 þ "1Þ E

2

 "1 þ "3

1

1

1 þ  E

"1 þ "2 þ "3

3ð1
Þ
1

1 þ  E

2

"1 þ "4

1

1

1 þ 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð"1
"3Þ

2 þ ½2"2
ð"1 þ "3Þ2

q  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

"1
"1 þ "2 þ "3

3

2

þ

"2
"3 ffiffiffi

3

p

2

s  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð"1
"4Þ

2 þ 4

3

ð"2
"3Þ

2

r 

Maximum shearing

stress,
max

E

2ð1 þ Þ

ð"1
"2Þ

 E

2ð1 þ Þ

E

1 þ 

E

2ð1 þ Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð"1
"3Þ

2 þ ½2"2
ð"1 þ "3Þ2

q  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

"1
"1 þ "2 þ "3

3

2

þ

"2
"3 ffiffiffi

3

p

2

s  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð"1
"4Þ

2 þ 4

3

ð"2
"3Þ

2

r 

Angle from gauge 1

axis to maximum

normal stress angle,

’p

0 1

2 tan
1

2"2
ð"1 þ "3Þ

"1
"3



1

2 tan
1

1

ffiffiffi

3

p ð"2
"3Þ

"1
"1 þ "2 þ "3

3

2

66

4

3

77

5

1

2 tan
1 2ð"2
"3Þ ffiffiffi

3

p ð"1
"4Þ

 Poisson’s ratio. The author has used  as symbol for Poisson’s ratio.

Source: Perry, C. C., and H. R. Lissner, The Strain Gage Primer, 2nd ed., McGraw-Hill Publishing Company, New York, p. 147, 1962.

2.40

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STATIC STRESSES IN MACHINE ELEMENTS

Table 2-12

Singularity functions

Function Graph of fnðxÞ Meaning

Concentrated moment hx
ai

2 ¼

 1 x ¼ a

0 x 6¼ a

ðx

1

hx
ai

2 dx ¼ hx
ai

1

Concentrated force hx
ai

1 ¼

 1 x ¼ a

0 x 6¼ a ðx

1

hx
ai

1 dx ¼ hx
ai

0

Unit step hx
ai

0 ¼

 0 x < a

1 x  a ðx

1

hx
ai

0 dx ¼ hx
ai

1

Rump hx
ai

1 ¼

 0 x < a

x
a x  a ðx

1

hx
ai

1 dx ¼ hx
ai

2

2

Parabolic hx
ai

2 ¼

 0 x < a

ðx
aÞ

2 x  a ðx

1

hx
ai

2 dx ¼ hx
ai

3

3

STATIC STRESSES IN MACHINE ELEMENTS 2.41

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STATIC STRESSES IN MACHINE ELEMENTS

TABLE 2-13

Summary of strain and stress equations due to different types of loads

Symbols: a ¼ major semi-axis of ellipse of area of contact, mm (in), and also radius of band of contact in case of spheres, mm (in); b ¼ minor semi-axis of ellipse of area of contact,

mm (in), and also half-bandwidth of rectangle contact between cylinders with parallel axis, mm (in); d1, d2 ¼ diameters of small and large spheres respectively, mm (in);

E1,

E2 ¼ moduli of elasticities of bodies in contact respectively, GPa (psi);

F

¼ load, kN (lbf);

F0 ¼ ð

F=‘ Þ ¼ load per unit length, kN/m (lbf/in); k1, k2 ¼ material constants for

small and large solid elastic bodies in contact;

L

¼ length of cylinder, m (in); pmax

¼ maximum pressure on surfaces of contact, MPa (psi); c max

¼ maximum contact

compressive stress, MPa (psi); ¼ normal stress, also with subscripts, MPa (psi);
¼ shear stress, also with subscripts MPa (psi); 1, 2 ¼ Poisson’s ratio of materials of small

and large elastic bodies in contact respectively;  ¼ approach distance along the line of action of the load between two points on the elastic bodies in contact, mm (in);

 ¼ hoop or circumferential or tangential stress, MPa (psi); a ¼ axial or longitudinal stress, MPa (psi); h ¼ thickness of cylinder/vessel/shell, mm (in). Meaning of other

symbols used in this Table are given under Symbols introduced at the beginning of this Chapter.

do ¼ d1d2

d1 þ d2

; d0

o ¼ d1d2

d1
d2

; k1 ¼ 1
2

1

E1

; k2 ¼ 1
2

2

E2

; ¼ b

a

; e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2
b2 p

a

Figure showing Type of

Applied stresses

Strain equations/Area

Maximum stress produced

loads loads Figure showing stress x y z  /Approach distance max max Principal stresses

1 Axial

load

F

A

000 "x ¼ "1 ¼

E ¼ x

E

"y ¼ "2 ¼
"1 ¼
"x;

"z ¼ "3 ¼
"1 ¼
"z ;

x ¼ max

x

2 ¼ max

2 1 ¼ E"1; 2 ¼ 0; 3 ¼ 0

2 Bending

load

sn

b ¼ Mbc

I

000 "x ¼ "1 ¼ bx

E ;

"y ¼ "2 ¼
"1;

"z ¼ "3 ¼
"1

bx

bx

2

3 Bending

and axial

load

R ¼

F

A

þ Mb

l

000 "x ¼ "1 ¼

E ;

"y ¼ "2 ¼
v"1;

"z ¼ "3 ¼
v"1

bx

bx

2

4 Torsion 000

Mtr

J ¼ Mt

Zp

¼

G ¼ r

L

1t ¼
1c ¼

at 458 to the shaft axis

2.42

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STATIC STRESSES IN MACHINE ELEMENTS