Question D.5: Determine the moment of inertia Ic with respect to the horiz...
Determine the moment of inertia I_{c} with respect to the horizontal axis C–C through the centroid C of the beam cross section shown in Fig. D-16. (The position of the centroid C was determined previously in Example D-2 of Section D.2.)
Note: From beam theory (Chapter 5), axis C–C is the neutral axis for bending of this beam; therefore, the moment of inertia I_{c} must be determined in order to calculate the stresses and deflections of this beam.
Find the moment of inertia I_{c} with respect to axis C–C by applying the parallel-axis theorem to each individual part of the composite area. The area divides naturally into three parts: (1) the cover plate, (2) the wide-flange section, and (3) the channel section. The following areas and centroidal distances were obtained previously in Example D-2:
A_{1}=3.0 in ^{2} \quad A_{2}=20.8 in ^{2} \quad A_{3}=8.82 in ^{2}\bar{y}_{1}=9.485 \text { in. } \quad \bar{y}_{2}=0 \quad \bar{y}_{3}=9.884 \text { in. } \quad \bar{c}=1.80 in.
The moments of inertia of the three parts with respect to horizontal axes through their own centroids C_{1} , C_{2}, and C_{3} are
I_{1}=\frac{b h^{3}}{12}=\frac{1}{12}(6.0 in.)(0.5 in.)^{3}=0.063 in ^{4}I_{2}=1170 in ^{4} \quad I_{3}=3.94 in ^{4}
The moments of inertia I_{2} and I_{3} are obtained from Tables F-1(a) and F-3(a), respectively, of Appendix F.
Now use the parallel-axis theorem to calculate the moments of inertia about axis C–C for each of the three parts of the composite area:
\left(I_{c}\right)_{1}=I_{1}+A_{1}\left(\bar{y}_{1}+\bar{c}\right)^{2}=0.063 in ^{4}+\left(3.0 in ^{2}\right)(11.28 in.)^{2}=382 in ^{4}\left(I_{c}\right)_{2}=I_{2}+A_{2} \bar{c}^{2}=1170 in ^{4}+\left(20.8 in ^{2}\right)(1.80 \text { in. })^{2}=1240 in ^{4}
\left(I_{c}\right)_{3}=I_{3}+A_{3}\left(\bar{y}_{3}-\bar{c}\right)^{2}=3.94 in ^{4}+\left(8.82 in ^{2}\right)(8.084 in.)^{2}=580 in ^{4}
The sum of these individual moments of inertia gives the moment of inertia of the entire cross-sectional area about its centroidal axis C–C:
I_{c}=\left(I_{c}\right)_{1}+\left(I_{c}\right)_{2}+\left(I_{c}\right)_{3}=2200 in ^{4}This example shows how to calculate moments of inertia of composite areas by using the parallel-axis theorem.
| Table F-1(a) | ||||||||||||
| Properties of Wide-Flange Sections (W Shapes)—USCS Units (Abridged List) | ||||||||||||
| Designation | Weight per Foot |
Area | Depth | Web Thickness |
Flange | Axis 1–1 | Axis 2-2 | |||||
| Width | Thickness | I | S | r | I | S | r | |||||
| lb | in² | in. | in. | in. | in. | \text{in}^{4} | in³ | in. | \text{in}^{4} | in³ | in. | |
| W 30 × 211 | 211 | 62.2 | 30.9 | 0.775 | 15.1 | 1.32 | 10300 | 665 | 12.9 | 757 | 100 | 3.49 |
| W 30 × 132 | 132 | 38.9 | 30.3 | 0.615 | 10.5 | 1.00 | 5770 | 380 | 12.2 | 196 | 37.2 | 2.25 |
| W 24 × 162 | 162 | 47.7 | 25.0 | 0.705 | 13.0 | 1.22 | 5170 | 414 | 10.4 | 443 | 68.4 | 3.05 |
| W 24 × 94 | 94.0 | 27.7 | 24.3 | 0.515 | 9.07 | 0.875 | 2700 | 222 | 9.87 | 109 | 24.0 | 1.98 |
| W 18 × 119 | 119 | 35.1 | 19.0 | 0.655 | 11.3 | 1.06 | 2190 | 231 | 7.90 | 253 | 44.9 | 2.69 |
| W 18 × 71 | 71.0 | 20.8 | 18.5 | 0.495 | 7.64 | 0.810 | 1170 | 127 | 7.50 | 60.3 | 15.8 | 1.70 |
| W 16 × 100 | 100 | 29.5 | 17.0 | 0.585 | 10.4 | 0.985 | 1490 | 175 | 7.10 | 186 | 35.7 | 2.51 |
| W 16 × 77 | 77.0 | 22.6 | 16.5 | 0.455 | 10.3 | 0.760 | 1110 | 134 | 7.00 | 138 | 26.9 | 2.47 |
| W 16 × 57 | 57.0 | 16.8 | 16.4 | 0.430 | 7.12 | 0.715 | 758 | 92.2 | 6.72 | 43.1 | 12.1 | 1.60 |
| W 16 × 31 | 31.0 | 9.13 | 15.9 | 0.275 | 5.53 | 0.440 | 375 | 47.2 | 6.41 | 12.4 | 4.49 | 1.17 |
| W 14 × 120 | 120 | 35.3 | 14.5 | 0.590 | 14.7 | 0.940 | 1380 | 190 | 6.24 | 495 | 67.5 | 3.74 |
| W 14 × 82 | 82.0 | 24.0 | 14.3 | 0.510 | 10.1 | 0.855 | 881 | 123 | 6.05 | 148 | 29.3 | 2.48 |
| W 14 × 53 | 53.0 | 15.6 | 13.9 | 0.370 | 8.06 | 0.660 | 541 | 77.8 | 5.89 | 57.7 | 14.3 | 1.92 |
| W 14 × 26 | 26.0 | 7.69 | 13.9 | 0.255 | 5.03 | 0.420 | 245 | 35.3 | 5.65 | 8.91 | 3.55 | 1.08 |
| W 12 × 87 | 87.0 | 25.6 | 12.5 | 0.515 | 12.1 | 0.810 | 740 | 118 | 5.38 | 241 | 39.7 | 3.07 |
| W 12 × 50 | 50.0 | 14.6 | 12.2 | 0.370 | 8.08 | 0.640 | 391 | 64.2 | 5.18 | 56.3 | 13.9 | 1.96 |
| W 12 × 35 | 35.0 | 10.3 | 12.5 | 0.300 | 6.56 | 0.520 | 285 | 45.6 | 5.25 | 24.5 | 7.47 | 1.54 |
| W 12 × 14 | 14.0 | 4.16 | 11.9 | 0.200 | 3.97 | 0.225 | 88.6 | 14.9 | 4.62 | 2.36 | 1.19 | 0.753 |
| W 10 × 60 | 60.0 | 17.6 | 10.2 | 0.420 | 10.1 | 0.680 | 341 | 66.7 | 4.39 | 116 | 23.0 | 2.57 |
| W 10 × 45 | 45.0 | 13.3 | 10.1 | 0.350 | 8.02 | 0.620 | 248 | 49.1 | 4.32 | 53.4 | 13.3 | 2.01 |
| W 10 × 30 | 30.0 | 8.84 | 10.5 | 0.300 | 5.81 | 0.510 | 170 | 32.4 | 4.38 | 16.7 | 5.75 | 1.37 |
| W 10× 12 | 12.0 | 3.54 | 9.87 | 0.190 | 3.96 | 0.210 | 53.8 | 10.9 | 3.90 | 2.18 | 1.10 | 0.785 |
| W 8 × 35 | 35.0 | 10.3 | 8.12 | 0.310 | 8.02 | 0.495 | 127 | 31.2 | 3.51 | 42.6 | 10.6 | 2.03 |
| W 8 × 28 | 28.0 | 8.24 | 8.06 | 0.285 | 6.54 | 0.465 | 98.0 | 24.3 | 3.45 | 21.7 | 6.63 | 1.62 |
| W 8 × 21 | 21.0 | 6.16 | 8.28 | 0.250 | 5.27 | 0.400 | 75.3 | 18.2 | 3.49 | 9.77 | 3.71 | 1.26 |
| W 8 × 15 | 15.0 | 4.44 | 8.11 | 0.245 | 4.01 | 0.315 | 48.0 | 11.8 | 3.29 | 3.41 | 1.70 | 0.876 |
| Table F-3(a) | |||||||||||||
| Properties of Channel Sections (C Shapes)—USCS Units (Abridged List) | |||||||||||||
| Designation | Weight per Foot |
Area | Depth | Web Thickness |
Flange | Axis 1–1 | Axis 2–2 | ||||||
| Width | Average Thickness |
I | S | r | I | S | r | c | |||||
| lb | in² | in. | in. | in. | in. | in^{4} | in³ | in. | in^{4} | in³ | in. | in. | |
| C 15 × 50 | 50.0 | 14.7 | 15.0 | 0.716 | 3.72 | 0.650 | 404 | 53.8 | 5.24 | 11.0 | 3.77 | 0.865 | 0.799 |
| C 15 × 40 | 40.0 | 11.8 | 15.0 | 0.520 | 3.52 | 0.650 | 348 | 46.5 | 5.45 | 9.17 | 3.34 | 0.883 | 0.778 |
| C 15 × 33.9 | 33.9 | 10.0 | 15.0 | 0.400 | 3.40 | 0.650 | 315 | 42.0 | 5.62 | 8.07 | 3.09 | 0.901 | 0.788 |
| C 12 × 30 | 30.0 | 8.81 | 12.0 | 0.510 | 3.17 | 0.501 | 162 | 27.0 | 4.29 | 5.12 | 2.05 | 0.762 | 0.674 |
| C 12 × 25 | 25.0 | 7.34 | 12.0 | 0.387 | 3.05 | 0.501 | 144 | 24.0 | 4.43 | 4.45 | 1.87 | 0.779 | 0.674 |
| C 12 × 20.7 | 20.7 | 6.08 | 12.0 | 0.282 | 2.94 | 0.501 | 129 | 21.5 | 4.61 | 3.86 | 1.72 | 0.797 | 0.698 |
| C 10 × 30 | 30.0 | 8.81 | 10.0 | 0.673 | 3.03 | 0.436 | 103 | 20.7 | 3.42 | 3.93 | 1.65 | 0.668 | 0.649 |
| C 10 × 25 | 25.0 | 7.34 | 10.0 | 0.526 | 2.89 | 0.436 | 91.1 | 18.2 | 3.52 | 3.34 | 1.47 | 0.675 | 0.617 |
| C 10 × 20 | 20.0 | 5.87 | 10.0 | 0.379 | 2.74 | 0.436 | 78.9 | 15.8 | 3.66 | 2.80 | 1.31 | 0.690 | 0.606 |
| C 10 × 15.3 | 15.3 | 4.48 | 10.0 | 0.240 | 2.60 | 0.436 | 67.3 | 13.5 | 3.87 | 2.27 | 1.15 | 0.711 | 0.634 |
| C 8 × 18.7 | 18.7 | 5.51 | 8.00 | 0.487 | 2.53 | 0.390 | 43.9 | 11.0 | 2.82 | 1.97 | 1.01 | 0.598 | 0.565 |
| C 8 × 13.7 | 13.7 | 4.04 | 8.00 | 0.303 | 2.34 | 0.390 | 36.1 | 9.02 | 2.99 | 1.52 | 0.848 | 0.613 | 0.554 |
| C 8 × 11.5 | 11.5 | 3.37 | 8.00 | 0.220 | 2.26 | 0.390 | 32.5 | 8.14 | 3.11 | 1.31 | 0.775 | 0.623 | 0.572 |
| C 6 × 13 | 13.0 | 3.81 | 6.00 | 0.437 | 2.16 | 0.343 | 17.3 | 5.78 | 2.13 | 1.05 | 0.638 | 0.524 | 0.514 |
| C 6 × 10.5 | 10.5 | 3.08 | 6.00 | 0.314 | 2.03 | 0.343 | 15.1 | 5.04 | 2.22 | 0.860 | 0.561 | 0.529 | 0.500 |
| C 6 × 8.2 | 8.20 | 2.39 | 6.00 | 0.200 | 1.92 | 0.343 | 13.1 | 4.35 | 2.34 | 0.687 | 0.488 | 0.536 | 0.512 |
| C 4 × 7.2 | 7.20 | 2.13 | 4.00 | 0.321 | 1.72 | 0.296 | 4.58 | 2.29 | 1.47 | 0.425 | 0.337 | 0.447 | 0.459 |
| C 4 × 5.4 | 5.40 | 1.58 | 4.00 | 0.184 | 1.58 | 0.296 | 3.85 | 1.92 | 1.56 | 0.312 | 0.277 | 0.444 | 0.457 |
| Notes: 1. Axes 1–1 and 2–2 are principal centroidal axes. 2. The distance c is measured from the centroid to the back of the web. 3. For axis 2–2, the tabulated value of S is the smaller of the two section moduli for this axis. |
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