The stress in the column found in Example Problem 6–5 seems high for the SAE 1040 hot-rolled steel.
Redesign the column to achieve a design factor of at least 3.
The stress in the column found in Example Problem 6–5 seems high for the SAE 1040 hot-rolled steel.
Redesign the column to achieve a design factor of at least 3.
Objective: Redesign the eccentrically loaded column of Example Problem 6–5 to reduce the stress and achieve a design factor of at least 3 .
Given: Data from Example Problems 6-4 and 6-5.
Analysis: Use a larger diameter. Use Equation (6-15) to compute the required strength. Then compare that with the strength of SAE 1040 hot-rolled steel. Iterate until the stress is satisfactory.
s_{y}=\frac{N P_{a}}{A}\left[1+\frac{e c}{r^{2}} \sec \left(\frac{K L}{2 r} \sqrt{\frac{N P_{a}}{A E}}\right)\right] (6-15)
Results: Appendix 3 gives the value for the yield strength of SAE 1040 \mathrm{HR} to be 42000 psi. If we choose to retain the same material, the cross-sectional dimensions of the column must be increased to decrease the stress. Equation (6-15) can be used to evaluate a design alternative.
The objective is to find suitable values for A, c, and r for the cross section such that P_{a}=1075 \mathrm{lb} ; N=3 ; L_{e}=32 \mathrm{in} ; e=0.75 \mathrm{in}; and the value of the entire right side of the equation is less than 42000 psi. The original design had a circular cross section with a diameter of 0.75 in. Let’s try increasing the diameter to D=1.00 in. Then
\begin{aligned}A &=\pi D^{2} / 4=\pi(1.00 \mathrm{in})^{2} / 4=0.785 \mathrm{in}^{2} \\r &=D / 4=(1.00 \mathrm{in}) / 4=0.250 \mathrm{in} \\r^{2} &=(0.250 \mathrm{in})^{2}=0.0625 \mathrm{in}^{2} \\c &=D / 2=(1.00 \mathrm{in}) / 2=0.50 \mathrm{in}\end{aligned}
Now let’s call the right side of Equation (6-15) s_{y}^{\prime}. Then
\begin{aligned}&s_{y}^{\prime}=\frac{3(1075)}{0.785}\left[1+\frac{(0.75)(0.50)}{(0.0625)} \sec \left(\frac{32}{2(0.250)} \sqrt{\frac{(3)(1075)}{(0.785)\left(30 \times10^{6}\right)}}\right)\right] \\&s_{y}^{\prime}=37740 \mathrm{psi}=\text { requiredvalue of } s_{y}\end{aligned}
This is a satisfactory result because it is just slightly less than the value of s_{y} of 42000 psi for the steel.
Now we can evaluate the expected maximum deflection with the new design using Equation (6-16);
y_{\max }=e\left[\sec \left(\frac{K L}{2 r} \sqrt{\frac{P}{A E}}\right)-1\right] (6-16)
\begin{aligned}&y_{\max }=0.75\left[\sec \left(\frac{32}{2(0.250)} \sqrt{\left.\frac{1075}{(0.785)\left(30 \times 10^{6}\right.}\right)-1}\right]\right. \\&y_{\max }=0.076 \text { in }\end{aligned}
Comments: The diameter of 1.00 in is satisfactory. The maximum deflection for the column is 0.076 in.
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 |