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## Q. 9.6

Consider the same three materials used in Example Problem 9–5 and calculate the safety factor for the pinion and gear. Use the results from Example Problems 9–1, 9–2, and 9–3 for the computed contact stress and use the same reliability and number of load cycles as used in Example Problem 9–4.
a. Ductile iron ASTM A536 80-55-06 quenched and tempered
b. Though hardened steel pinion and gear: SAE 4340 OQT 1000
c. SAE 8620 steel, case hardened by carburizing to Rc 60-64 with a minimum effective case depth of 0.025 in.

## Verified Solution

The results required from Example Example Problems 9–1, 9–2, 9–3 are:
$\begin{array}{lllc}N_{P}=20 & N_{G}=70 & P_{d}=6 & F=2.00 \text { in } \\n_{P}=1750 \mathrm{rpm} & n_{G}=500 \mathrm{rpm} & D_{P}=3.333 \text { in } & D_{G}=11.667 \text { in } \\s_{C P}=s_{C G}=100769 \mathrm{psi} & K_{R}=1.25 & N_{C P}=2.1 \times 10^{9} \text { cycles } & N_{c G}=6.0 \times 10^{9} \text { cycles }\end{array}$
The equation relating the contact stress number with the adjusted allowable contact stress number is:
$s_{c}
Use this equation to solve for the safety factor, $SF$:
$S F=\frac{S_{a c}}{S_{c}} \cdot \frac{Z_{N}}{K_{R}}$
From Figure 9-22, we find pitting resistance stress cycle factors:
\begin{aligned}&Z_{N P}=0.88 \\&Z_{N G}=0.91\end{aligned}
Part (a) Let’s consider the first proposed material, ductile iron ASTM A536 type 80-55-06. Using Table 9-9, the allowable contact stress number for this material is
$s_{a c}=77000 \mathrm{psi}$
Solving for the safety factor $S F$ of the pinion and gear:
\begin{aligned}&S F_{p}=\frac{S_{a c}}{S_{c P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{77000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.88}{1.25}=0.54 \\&S F_{G}=\frac{S_{a c}}{S_{c G}} \cdot \frac{Z_{N G}}{K_{R}}=\frac{77000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.91}{1.25}=0.56\end{aligned}
Both safety factors are below the minimum value of 1.00, as recommended by AGMA and are therefore considered to be unsatisfactory for this application.

Both safety factors are below the minimum value of 1.00, as recommended by AGMA and are therefore
considered to be unsatisfactory for this application.
Part (b) considers a through hardened steel SAE 4340 OQT 1000. Using Appendix 3, the properties of the material are:
$s_u = 171 000$ psi, $s_y = 158 000$ psi, HB = 363, 17% elongation
Using Figure 9–19, along with a Brinell hardness of HB = 363 for though-hardened steel gears material, the allowable contact stress number is:

$s_ac = 146 000$ psi
The safety factors, $SF$, of the pinion and the gear are:

\begin{aligned}S F_{p} &=\frac{S_{a c}}{S_{C P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{146000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.88}{1.25}=1.0 \\S F_{G} &=\frac{S_{a c}}{S_{C G}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{146000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.91}{1.25}=1.1\end{aligned}
The safety factors for the pinion and gear are both within the range of 1.0 to 1.5, as recommended by AGMA, with the pinion’s factor at the lowest limit.
Part (c) considers SAE 8620 steel, case hardened by carburizing to a Rc 60-64, with a minimum effective case depth of 0.025 in. Using Table 9-9, the allowable contact stress number is:
$s_{a c}=180000 \mathrm{psi}$
The safety factors $S F$ for the pinion and the gear are:
\begin{aligned}S F &=\frac{S_{a c}}{S_{C P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{180000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.88}{1.25}=1.26 \\S F &=\frac{S_{a C}}{S_{C P}} \cdot \frac{Z_{N P}}{K_{R}}=\frac{180000 \mathrm{psi}}{100769 \mathrm{psi}} \cdot \frac{0.91}{1.25}=1.30\end{aligned}
The safety factors for both the pinion and gear are well within the range of 1.0 to 1.5, as recommended by AGMA.

 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