For the beam shown in Figure 3–16, the load F due to the pipe is 12 000 lb. The distances are a = 4 ft and b = 6 ft. Determine the required section modulus for the beam to limit the stress due to bending to 30 000 psi, the recommended design stress for a typical structural steel in static bending. Then specify the lightest suitable steel beam.
Objective:
Compute the required section modulus S for the beam in Figure 3–16.
Given:
The layout and the loading pattern are shown in Figure 3–16.
Lengths: Overall length =L=10 \mathrm{ft}; a=4 \mathrm{ft} ; b=6 \mathrm{ft}.
Load =F=12000 \mathrm{lb}.
Design stress =\sigma_{d}=30000 \mathrm{psi}.
Analysis:
Use Equation (3–25) (S=M / \sigma_{d}) to compute the required section modulus S. Compute the maximum bending moment that occurs at the point of application of the load using the formula shown in Part (b) of Figure 3–16.
Results:
\begin{aligned}&M_{\max }=R_{1} a=\frac{F b a}{a+b}=\frac{(12000 \mathrm{lb})(6 \mathrm{ft})(4 \mathrm{ft})}{(6 \mathrm{ft}+4 \mathrm{ft})}=28800 \mathrm{lb} \cdot \mathrm{ft} \\&S=\frac{M}{\sigma_{d}}=\frac{28800 \mathrm{lb} \cdot \mathrm{ft}}{30000 \mathrm{lb} /\mathrm{in}^{2}} \frac{12 \mathrm{in}}{\mathrm{ft}}=11.5 \mathrm{in}^{3}\end{aligned}Comments:
A steel beam section can now be selected from Tables A15-9 and A15-10 that has at least this value for the section modulus. The lightest section, typically preferred, is the W8 \times 15 wide-flange shape with S=11.8 \mathrm{in}^{3}.
| TABLE 15–9 I-Beam Shapes: Steel Wide Flange Shapes, W-Shapes: 4 in to 24 in Depth | |||||||||
| *Section properties | |||||||||
| Designation | Area A (in ^{2} ) |
Depth d (in) |
Web thickness t_{W} (in) |
Flange | \text { Axis } X-X | \text { Axis } Y-Y | |||
| Width, B (in) |
Thickness, t_{f} (in) |
\begin{gathered}I_{x} \\\left(\mathrm{in}^{4}\right)\end{gathered} | S_{x}\text { (in }^{3} \text { ) } | \begin{gathered}I_{y} \\\left(\mathrm{in}^{4}\right)\end{gathered} | S_{y}\text { (in }^{3} \text { ) } | ||||
| \begin{aligned}&\text { W24×76 } \\&\text { W24×68 } \\&\text { W21×73 } \\&\text {W21×57} \\&\text { W18×55 } \\&\text { W18×40 } \\&\text { W14×43 } \\&\text { W14×26 } \\&\text {W12×30 } \\&\text { W12×16 } \\&\text { W10×15 } \\&\text { W10×12 } \\&\text { W8×15 } \\&\text { W8×10 } \\&\text { W6 } \times 15 \\&\text { W6 } \times 12 \\&\text { W5×19 } \\&\text { W5×16 } \\&\text { W4×13 }\end{aligned} | \begin{gathered}22.4 \\20.1 \\21.5 \\16.7 \\16.2 \\11.8 \\12.6 \\7.69 \\8.79 \\4.71 \\4.41\\3.54 \\4.44 \\2.96 \\4.43 \\3.55 \\5.54 \\4.68 \\3.83\end{gathered} | \begin{aligned}23.92 \\23.73 \\21.24 \\21.06 \\18.11 \\17.90 \\13.66 \\13.91 \\12.34 \\11.99 \\9.99 \\9.87 \\8.11 \\7.89 \\5.99 \\6.03 \\5.15 \\5.01 \\4.16\end{aligned} | \begin{aligned}&0.440 \\&0.415 \\&0.455 \\&0.405 \\&0.390 \\&0.315 \\&0.305 \\&0.255 \\&0.260 \\&0.220 \\&0.230 \\&0.190 \\&0.245 \\&0.170 \\&0.230 \\&0.230 \\&0.270 \\&0.240 \\&0.280\end{aligned} | \begin{aligned}&8.990 \\&8.965 \\&8.295 \\&6.555 \\&7.530 \\&6.015 \\&7.995 \\&5.025 \\&6.520 \\&3.990 \\&4.000 \\&3.960 \\&4.015 \\&3.940 \\&5.990 \\&4.000 \\&5.030 \\&5.000 \\&4.060\end{aligned} | \begin{aligned}&0.680 \\&0.585 \\&0.740 \\&0.650 \\&0.630 \\&0.525 \\&0.530 \\&0.420 \\&0.440 \\&0.265 \\&0.270 \\&0.210 \\&0.315 \\&0.205 \\&0.260 \\&0.280 \\&0.430 \\&0.360 \\&0.345\end{aligned} | \begin{aligned}2100 \\1830 \\1600 \\1170 \\890 \\612 \\428 \\245 \\238 \\103 \\69.8 \\53.8 \\48.0 \\30.8 \\29.1 \\22.1 \\26.2 \\21.3 \\11.3\end{aligned} | \begin{gathered}176 \\154 \\151 \\111 \\98.3 \\68.4 \\62.7 \\35.3 \\38.6 \\17.1 \\13.8 \\10.9 \\11.8 \\7.81 \\9.72 \\7.31 \\10.2 \\8.51 \\5.46\end{gathered} | \begin{gathered}82.5 \\70.4 \\70.6 \\30.6 \\44.9 \\19.1 \\45.2 \\8.91 \\20.3 \\2.82 \\2.89 \\2.18 \\3.41 \\2.09 \\9.32 \\2.99 \\9.13 \\7.51 \\3.86\end{gathered} | \begin{gathered}18.4 \\15.7 \\17.0 \\9.35 \\11.9 \\6.35 \\11.3 \\3.54 \\6.24 \\1.41 \\1.45 \\1.10 \\1.70 \\1.06 \\3.11 \\1.50 \\3.63 \\3.00 \\1.90\end{gathered} |
| Note: Example designation: W14×43. 14 = nominal depth (in); 43 = weight per unit length (lb/ft). *I = moment of inertia; S = section modulus. Sources for data for additional sizes: Central Steel & Wire Co., multiple locations. Reliance Steel & Aluminum Co./Earl M. Jorgensen Co., multiple locations. |
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| TABLE 15–10 I-Beam Shapes: Steel, American Standard Shapes, S-Shapes: 3 in to 24 in Depth | |||||||||
| *Section properties | |||||||||
| Designation | Area A (in ^{2} ) |
Depth d (in) |
Web thickness t_{W} (in) |
Flange | \text { Axis } X-X | \text { Axis } Y-Y | |||
| Width, B (in) |
Average Thickness, t_{f} (in) |
\begin{gathered}I_{x} \\\left(\mathrm{in}^{4}\right)\end{gathered} | S_{x}\text { (in }^{3} \text { ) } | \begin{gathered}I_{y} \\\left(\mathrm{in}^{4}\right)\end{gathered} | S_{y}\text { (in }^{3} \text { ) } | ||||
| \begin{aligned}&\mathrm{S} 24 \times 90 \\&\mathrm{~S} 20 \times 96 \\&\mathrm{~S} 20 \times 75 \\&\mathrm{~S} 20 \times 66 \\&\mathrm{~S} 18 \times 70 \\&\mathrm{~S} 15 \times 50 \\&\mathrm{~S} 12 \times 50 \\&\mathrm{~S} 12 \times 35 \\&\mathrm{~S} 10 \times 35 \\&\mathrm{~S} 10 \times 25.4 \\&\mathrm{~S} 8 \times 23 \\&\mathrm{~s} 8 \times 18.4 \\&\mathrm{~S} 7\times 20 \\&\mathrm{~S} 6 \times 12.5 \\&\mathrm{~S} 5 \times 10 \\&\mathrm{~S} 4 \times 7.7 \\&\mathrm{~S} 3 \times 5.7\end{aligned} | \begin{aligned}&26.5 \\&28.2 \\&22.0 \\&19.4 \\&20.6 \\&14.7 \\&14.7 \\&10.3 \\&10.3 \\&7.46 \\&6.77 \\&5.41 \\&5.88 \\&3.67 \\&2.94 \\&2.26 \\&1.67\end{aligned} | \begin{aligned}24.00 \\20.30 \\20.00 \\20.00 \\18.00 \\15.00 \\12.00 \\12.00 \\10.00 \\10.00 \\8.00 \\8.00 \\7.00 \\6.00 \\5.00 \\4.00 \\3.00\end{aligned} | \begin{aligned}&0.625 \\&0.800 \\&0.635 \\&0.505 \\&0.711 \\&0.550 \\&0.687 \\&0.428 \\&0.594 \\&0.311 \\&0.441 \\&0.271 \\&0.450 \\&0.232 \\&0.214 \\&0.193 \\&0.170\end{aligned} | \begin{aligned}&7.125 \\&7.200 \\&6.385 \\&6.255 \\&6.251 \\&5.640 \\&5.477 \\&5.078 \\&4.944 \\&4.661 \\&4.171 \\&4.001 \\&3.860 \\&3.332 \\&3.004 \\&2.663 \\&2.330\end{aligned} | \begin{aligned}&0.870 \\&0.920 \\&0.795 \\&0.795 \\&0.691 \\&0.622 \\&0.659 \\&0.544 \\&0.491 \\&0.491 \\&0.426 \\&0.426 \\&0.392 \\&0.359 \\&0.326 \\&0.293 \\&0.260\end{aligned} | \begin{aligned}2250 \\1670 \\1280 \\1190 \\926 \\486 \\305 \\229 \\147 \\124 \\64.9 \\57.6 \\42.4 \\22.1 \\12.3 \\6.08 \\2.52\end{aligned} | \begin{aligned}&187 \\&165 \\&128 \\&119 \\&103 \\&64.8 \\&50.8 \\&38.2 \\&29.4 \\&24.7 \\&16.2 \\&14.4 \\&12.1 \\&7.37 \\&4.92 \\&3.04 \\&1.68\end{aligned} | \begin{aligned}&44.9 \\&50.2 \\&29.8 \\&27.7 \\&24.1 \\&15.7 \\&15.7 \\&9.87 \\&8.36 \\&6.79 \\&4.31 \\&3.73 \\&3.17 \\&1.82 \\&1.22 \\&0.764 \\&0.455\end{aligned} | \begin{aligned}&12.6 \\&13.9 \\&9.32 \\&8.85 \\&7.72 \\&5.57 \\&5.74 \\&3.89 \\&3.38 \\&2.91 \\&2.07 \\&1.86 \\&1.64 \\&1.09 \\&0.809 \\&0.574 \\&0.390\end{aligned} |
| Note: Example designation: S10×35. 10 = nominal depth (in); 35 = weight per unit length (lb/ft). *I = moment of inertia; S = section modulus. Sources for data for additional sizes: Central Steel & Wire Co., multiple locations. Reliance Steel & Aluminum Co./Earl M. Jorgensen Co., multiple locations. |
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