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Chapter 9

Q. 9.5

Consider three different materials and calculate the safety factor for the pinion and gear using the results from Example Problem 9–3 for the computed bending stresses. Design for a reliability of 0.999, fewer than one failure in 1000. The application is a drive for an industrial saw that will be fully utilized on a normal, one-shift, five-day per-week operation. The materials proposed are:
a. Ductile iron ASTM A536 type 80-55-06 quenched and tempered.
b. Though hardened steel pinion and gear: SAE 4340 OQT 1000.
c. SAE 8620 steel pinion and gear case hardened by carburizing to a hardness of Rockwell C 60-64 with a minimum effective case depth of 0.025 in.

Step-by-Step

Verified Solution

The results required from Example Problem 9–3 are summarized here:

\begin{array}{lllc}N_{P}=20 & N_{G}=70 & P_{d}=6 & F=2.00 \mathrm{in} \\n_{P}=1750 \mathrm{rpm} &n_{G}=500 \mathrm{rpm} & D_{P}=3.333 \mathrm{in} & D_{G}=11.667 \mathrm{in} \\S_{t P}=12376\mathrm{psi} & S_{t G}=9871 \mathrm{psi} & &\end{array}

The equation relating the bending stress number with the adjusted allowable bending stress number is:

s_{t}<s_{a t}^{\prime}=s_{a t} \frac{Y_{N}}{S F \cdot K_{R}}

Solving this equation for the safety factor, SF:

S F=\frac{s_{a t}}{s_{t}} \cdot \frac{Y_{N}}{K_{R}}

Design decisions will be made on the reliability and design life. From Table 9-11, we find K_{R}=1.25 for the desired reliability of 0.999 as specified in the problem. Because the saw will be fully utilized in an industrial environment, we choose a life of L=20000 hours, using Table 9-12 as a guide.

Compute the number of load cycles for the pinion and the gear using Equation (9-27) (N_c = (60)(L)(n)(q)). Each tooth sees one load cycle per revolution, q=1.

\begin{aligned}&N_{c P}=(60) \cdot L \cdot n_{p} \cdot q=60 \cdot 20000 \cdot 1750 \cdot 1=2.10\times 10^{9} \text { cycles } \\&N_{C G}=(60) \cdot L \cdot n_{G} \cdot q=60 \cdot 20000\cdot 500 \cdot 1=6.00 \times 10^{8} \text { cycles }\end{aligned}

From Figure 9-21, the bending strength stress cycle factor for the pinion and gear is:

\begin{aligned}&Y_{N P}=0.93 \\&Y_{N G}=0.95\end{aligned}

Part (a): Let’s look first at using ductile iron ASTM type A536 80-55-06. Using Table 9-10, the allowable bending stress number for this material is

s_{a t}=22000 \mathrm{psi}

Solving for the safety factor for the pinion:
S F=\frac{S_{a t}}{S_{tP}} \cdot \frac{Y_{N P}}{K_{R}}=\frac{22000 \mathrm{psi}}{12376\mathrm{psi}} \cdot \frac{0.93}{1.25}=1.3
Using the same equation for the safety factor of the gear:
S F=\frac{S_{a t}}{S_{t G}} \cdot \frac{Y_{N G}}{K_{R}}=\frac{22000 \mathrm{psi}}{9871 \mathrm{psi}} \cdot \frac{0.95}{1.25}=1.7
The minimum safety factor is for the pinion and it is within the range of 1.0 to 1.5, as recommended by AGMA, so this design is considered to be satisfactory for bending stress.

Part (b) considers a through hardened steel material SAE 4340 OQT 1000 for the pinion and gear. Using Appendix 3 the properties of the material are:
s_{u}=171000 \text { psi, } s_{y}=158000 \text { psi, HB }=363,16 \% \text { elongation }
Using Figure 9-18, along with the material Brinell hardness of \mathrm{HB}=363, the allowable bending stress number for the though-hardened steel gear is:
S_{a t}=41000 \mathrm{psi}
Solving for the safety factor SF of the pinion and the gear:
\begin{aligned}&S F_{p}=\frac{S_{a t}}{S_{t P}} \cdot \frac{Y_{N P}}{K_{R}}=\frac{41000\mathrm{psi}}{12376 \mathrm{psi}} \cdot \frac{0.93}{1.25}=2.5 \\&S F_{G}=\frac{S_{\mathrm{AT}}}{S_{\mathrm{tG}}} \cdot \frac{Y_{\mathrm{NG}}}{K_{R}}=\frac{41000 \mathrm{psi}}{9871 \mathrm{psi}} \cdot \frac{0.95}{1.25}=3.2\end{aligned}
The safety factor is well over 1.5 for both the pinion and the gear for bending stress with this through-hardened steel.

Part (c) considers a gear material, SAE 8620 case hardened by carburizing to Rockwell \mathrm{C} 60-64 with a minimum effective case depth of 0.025 in.
Using Table 9-9, the allowable bending stress number for both the pinion and the gear is:
s_{\text {at }}=54000 \mathrm{psi}
Solving for the safety factor of the pinion and the gear:
\begin{aligned}&S F_{p}=\frac{S_{a t}}{S_{t P}} \cdot \frac{Y_{N P}}{K_{R}}=\frac{54000 \mathrm{psi}}{12376 \mathrm{psi}} \cdot \frac{0.93}{1.25}=3.2 \\&S F_{G}=\frac{S_{a t}}{s_{t G}}\cdot \frac{Y_{N G}}{K_{R}}=\frac{54000 \mathrm{psi}}{9871 \mathrm{psi}} \cdot \frac{0.95}{1.25}=4.2\end{aligned}
The safety factor is well over 1.5 for both the pinion and the gear for bending stress with a carburized, case hardened steel material.

TABLE 9–11 Reliability Factor, K_R
Reliability K_R
0.90, one failure in 10 0.85
0.99, one failure in 100 1.00
0.999, one failure in 1000 1.25
0.9999, one failure in 10 000 1.5

 

TABLE 9–12 Recommended Design Life
Domestic appliances 1000–2000
Aircraft engines 1000–4000
Automotive 1500–5000
Agricultural equipment 3000–6000
Elevators, industrial fans, multipurpose gearing 8000–15 000
Electric motors, industrial blowers, general industrial machines 20 000–30 000
Pumps and compressors 40 000–60 000
Critical equipment in continuous 24-h operation 100 000–200 000
TABLE 9–10 Allowable Stress Numbers for Iron and Bronze Gears
Material designation Minimum hardness at surface (HB) Allowable bending
stress number, s_{at}
Allowable contact
stress number, s_{ac}
Gray cast iron, ASTM A48, as cast
Class 20 5 35 50 345
Class 30 174 8.5 59 65 448
Class 40 201 13 90 75 517
Ductile (nodular) iron, ASTM A536
60-40-18 annealed 140 22 152 77 530
80-55-06 quenched and tempered 179 22 152 77 530
100-70-03 quenched and tempered 229 27 186 92 634
120-90-02 quenched and tempered 269 31 214 103 710
Bronze, sand-cast, s_{u min} = 40 ksi (275 MPa) 5.7 39 30 207
Bronze, heat-treated,  s_{u min} = 90 ksi (620 MPa) 23.6 163 65 448

 

TABLE 9–9 Allowable Stress Numbers for Case-Hardened Grade 1 Steel Materials
Allowable bending
stress number, s_{at}
Allowable contact
stress number, s_{ac}
Hardness at surface (ksi) (Mpa) (ksi) (Mpa)
Flame- or induction-hardened
50 HRC 45 170 170 1172
54 HRC 45 175 175 1207
Carburized and case-hardened
55–64 HRC 55 379 180 1241

 

APPENDIX 3 Design Properties of Carbon and Alloy Steel
Material
designation
(SAE number)
Condition Tensile
strength
Yield
strength
Ductility
(percent
elongation
in 2 in)
Brinell
hardness (HB)
(ksi) (MPa) (ksi) (MPa)
1020 Hot-rolled 55 379 30 207 25 111
1020 Cold-drawn 61 420 51 352 15 122
1020 Annealed 60 414 43 296 38 121
1040 Hot-rolled 72 496 42 290 18 144
1040 Cold-drawn 80 552 71 490 12 160
1040 OQT 1300 88 607 61 421 33 183
1040 OQT 400 113 779 87 600 19 262
1050 Hot-rolled 90 620 49 338 15 180
1050 Cold-drawn 100 690 84 579 10 200
1050 OQT 1300 96 662 61 421 30 192
1050 OQT 400 143 986 110 758 10 321
1117 Hot-rolled 65 448 40 276 33 124
1117 Cold-drawn 80 552 65 448 20 138
1117 WQT 350 89 614 50 345 22 178
1137 Hot-rolled 88 607 48 331 15 176
1137 Cold-drawn 98 676 82 565 10 196
1137 OQT 1300 87 600 60 414 28 174
1137 OQT 400 157 1083 136 938 5 352
1144 Hot-rolled 94 648 51 352 15 188
1144 Cold-drawn 100 690 90 621 10 200
1144 OQT 1300 96 662 68 496 25 200
1144 OQT 400 127 876 91 627 16 277
1213 Hot-rolled 55 379 33 228 25 110
1213 Cold-drawn 75 517 58 340 10 150
12L13 Hot-rolled 57 393 34 234 22 114
12L13 Cold-drawn 70 483 60 414 10 140
1340 Annealed 102 703 63 434 26 207
1340 OQT 1300 100 690 75 517 25 235
1340 OQT 1000 144 993 132 910 17 363
1340 OQT 700 221 1520 197 1360 10 444
1340 OQT 400 285 1960 234 1610 8 578
3140 Annealed 95 655 67 462 25 187
3140 OQT 1300 115 792 94 648 23 233
3140 OQT 1000 152 1050 133 920 17 311
3140 OQT 700 220 1520 200 1380 13 461
3140 OQT 400 280 1930 248 1710 11 555
4130 Annealed 81 558 52 359 28 156
4130 WQT 1300 98 676 89 614 28 202
4130 WQT 1000 143 986 132 910 16 302
4130 WQT 700 208 1430 180 1240 13 415
4130 WQT 400 234 1610 197 1360 12 461
4140 Annealed 95 655 54 372 26 197
4140 OQT 1300 117 807 100 690 23 235
4140 OQT 1000 168 1160 152 1050 17 341
4140 OQT 700 231 1590 212 1460 13 461
4140 OQT 400 290 2000 251 1730 11 578
4150 Annealed 106 731 55 379 20 197
4150 OQT 1300 127 880 116 800 20 262
4150 OQT 1000 197 1360 181 1250 11 401
4150 OQT 700 247 1700 229 1580 10 495
4150 OQT 400 300 2070 248 1710 10 578
4340 Annealed 108 745 68 469 22 217
4340 OQT 1300 140 965 120 827 23 280
4340 OQT 1000 171 1180 158 1090 16 363
4340 OQT 700 230 1590 206 1420 12 461
4340 OQT 400 283 1950 228 1570 11 555
5140 Annealed 83 572 42 290 29 167
5140 OQT 1300 104 717 83 572 27 207
5140 OQT 1000 145 1000 130 896 18 302
5140 OQT 700 220 1520 200 1380 11 429
5140 OQT 400 276 1900 226 1560 7 534
5150 Annealed 98 676 52 359 22 197
5150 OQT 1300 116 800 102 700 22 241
5150 OQT 1000 160 1100 149 1030 15 321
5150 OQT 700 240 1650 220 1520 10 461
5150 OQT 400 312 2150 250 1720 8 601
5160 Annealed 105 724 40 276 17 197
5160 OQT 1300 115 793 100 690 23 229
5160 OQT 1000 170 1170 151 1040 14 341
5160 OQT 700 263 1810 237 1630 9 514
5160 OQT 400 322 2220 260 1790 4 627
6150 Annealed 96 662 59 407 23 197
6150 OQT 1300 118 814 107 738 21 241
6150 OQT 1000 183 1260 173 1190 12 375
6150 OQT 700 247 1700 223 1540 10 495
6150 OQT 400 315 2170 270 1860 7 601
8650 Annealed 104 717 56 386 22 212
8650 OQT 1300 122 841 113 779 21 255
8650 OQT 1000 176 1210 155 1070 14 363
8650 OQT 700 240 1650 222 1530 12 495
8650 OQT 400 282 1940 250 1720 11 555
8740 Annealed 100 690 60 414 22 201
8740 OQT 1300 119 820 100 690 25 241
8740 OQT 1000 175 1210 167 1150 15 363
8740 OQT 700 228 1570 212 1460 12 461
8740 OQT 400 290 2000 240 1650 10 578
9255 Annealed 113 780 71 490 22 229
9255 O&T 1300 130 896 102 703 21 262
9255 O&T 1000 181 1250 160 1100 14 352
9255 O&T 700 260 1790 240 1650 5 534
9255 O&T 400 310 2140 287 1980 2 601