Question 12.5: Determine the moment of inertia Ic with respect to the horiz...
Determine the moment of inertia Ic with respect to the horizontal axis C–C through the centroid C of the beam cross section shown in Fig. 12-16. (The position of the centroid C was determined previously in Example 12-2 of Section 12.3.)
Note: From beam theory (Chapter 5), we know that axis C–C is the neu-tral axis for bending of this beam, and therefore the moment of inertia I_{c} must be determined in order to calculate the stresses and deflections of this beam.
We will determine 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 12-2:
\begin{gathered}A_{1}=37.5 \mathrm{~cm}^{2} \quad A_{2}=178 \mathrm{~cm}^{2} \quad A_{3}=75.8 \mathrm{~cm}^{2} \\\bar{y}_{1}=227.5 \mathrm{~mm} \quad \bar{y}_{2}=0 \quad \bar{y}_{3}=246 \mathrm{~mm}\quad \bar{c}=34.73 \mathrm{~mm}\end{gathered}The moments of inertia of the three parts with respect to horizontal axes through their own centroids C_{1} , C_{2} , and C_{1} are as follows:
\begin{gathered}I_{1}=\frac{b h^{3}}{12}=\frac{1}{12}(25 \mathrm{~cm})(1.5 \mathrm{~cm})^{3}=7.031 \mathrm{~cm}^{4} \\I_{2}=63720 \mathrm{~cm}^{4} \quad I_{3}=597 \mathrm{~cm}^{4}\end{gathered}The moments of inertia I_{2} and I_{3} are obtained from Tables E-1 and E-3, respectively, of Appendix E.
TABLE E-1
Properties of European Wide-Flange Beams
| Designation | Mass per meter |
Area of section |
Depth of section |
Width of section |
Thickness | Strong axis 1-1 | Weak axis 2-2 | |||||
| G | A | h | b | t_{w} | t_{t} | I_{1} | S_{1} | r_{1} | I_{2} | S_{2} | r_{2} | |
| Kg/m | {cm}^{2} | mm | mm | mm | mm | {cm}^{4} | {cm}^{3} | cm | {cm}^{4} | {cm}^{3} | cm | |
| HE 1000 B HE 900 B HE 700 B HE 650 b HE 600 B |
314 291 241 225 212 |
400 371.3 306.4 286.3 270 |
1000 900 700 650 600 |
300 300 300 300 300 |
19 18.5 17 16 15.5 |
36 35 32 31 30 |
644700 494100 256900 210600 171000 |
12890 10980 7340 6480 5701 |
40.15 36.48 28.96 27.12 25.17 |
16280 15820 14440 13980 13530 |
1085 1054 962.7 932.3 902 |
6.38 6.53 6.87 6.99 7.08 |
| HE 550 B HE 600 A HE 450 B HE 550 A HE 360 B HE 450 A |
199 178 171 166 142 140 |
254.1 226.5 218 211.8 180.6 178 |
550 590 450 540 360 440 |
300 300 300 300 300 300 |
15 13 14 12.5 12.5 11.5 |
29 25 26 24 22.5 21 |
136700 141200 79890 111900 43190 63720 |
4971 4787 3551 4146 2400 2896 |
23.2 24.97 19.14 22.99 15.46 18.92 |
13080 11270 11720 10820 10140 9465 |
871.8 751.4 781.4 721.3 676.1 631 |
7.17 7.05 7.33 7.15 7.49 7.29 |
| HE 340 B HE 320 B HE 360 A HE 340 A |
134 127 112 105 |
170.9 161.3 142.8 133.5 |
340 320 350 330 |
300 300 300 300 |
12 11.5 10 9.5 |
21.5 20.5 17.5 16.5 |
36660 30820 33090 27690 |
2156 1926 1891 1678 |
14.65 13.82 15.22 14.4 |
9690 9239 7887 7436 |
646 615.9 525.8 495.7 |
7.53 7.57 7.43 7.46 |
| HE 320 A HE 260 B HE 240 B HE 280 A HE 220 B HE 260 A HE 240 A |
97.6 93 83.2 76.4 71.5 68.2 60.3 |
124.4 118.4 106 97.26 91.04 86.82 76.84 |
310 260 240 270 220 250 230 |
300 260 240 280 220 260 240 |
9 10 10 8 9.5 7.5 7.5 |
15.5 17.5 17 13 16 12.5 12 |
22930 14920 11260 13670 8091 10450 7763 |
1479 1148 938.3 1013 735.5 836.4 675.1 |
13.58 11.22 10.31 11.86 9.43 10.97 10.05 |
6985 5135 3923 4763 2843 3668 2769 |
465.7 395 326.9 340.2 258.5 282.1 230.7 |
7.49 6.58 6.08 7 5.59 6.5 6 |
| HE 180 B HE 160 B HE 140 B HE 120 B HE 140 A |
51.2 42.6 33.7 26.7 24.7 |
65.25 54.25 42.96 34.01 31.42 |
180 160 140 120 133 |
180 160 140 120 140 |
8.5 8 7 6.5 5.5 |
14 13 12 11 8.5 |
3831 2492 1509 864.4 1033 |
425.7 311.5 215.6 144.1 155.4 |
7.66 6.78 5.93 5.04 5.73 |
1363 889.2 549.7 317.5 389.3 |
151.4 111.2 78.52 52.92 55.62 |
4.57 4.05 3.58 3.06 3.52 |
| HE 100 B HE 100 A |
20.4 16.7 |
26.4 21.24 |
100 96 |
100 100 |
6 5 |
10 8 |
449.5 349.2 |
89.91 72.76 |
4.16 4.06 |
167.3 133.8 |
33.45 26.76 |
2.53 2.51 |
| Note: Axes 1-1 and 2-2 are principal centroidal axes. | ||||||||||||
TABLE E-3
Properties of European Standard Channels
| Designation | Mass per meter |
Area of section |
Depth of section |
Width of section |
Thickness | Strong axis 1-1 | Weak axis 2-2 | ||||||
| G | A | h | b | t_{w} | t_{t} | I_{1} | S_{1} | r_{1} | I_{2} | S_{2} | r_{2} | c | |
| Kg/m | {cm}^{2} | mm | mm | mm | mm | {cm}^{4} | {cm}^{3} | cm | {cm}^{4} | {cm}^{3} | cm | cm | |
| UPN 400 | 71.8 | 91.5 | 400 | 110 | 14 | 18 | 20350 | 1020 | 14.9 | 846 | 102 | 3.04 | 2.65 |
| UPN 380 UPN 350 UPN 320 UPN 300 |
63.1 60.6 59.5 46.2 |
80.4 77.3 75.8 85.5 |
380 350 320 300 |
102 100 100 100 |
13.5 14 14 10 |
16 16 17.5 16 |
15760 12840 10870 8030 |
829 734 679 535 |
14 12.9 12.1 11.7 |
615 570 597 495 |
78.7 75 80.6 67.8 |
2.77 2.72 2.81 2.9 |
2.38 2.4 2.6 2.7 |
| UPN 280 UPN 260 UPN 240 UPN 220 UPN 200 |
41.8 37.9 33.2 29.4 25.3 |
53.3 48.3 42.3 37.4 32.2 |
280 260 240 220 200 |
96 90 85 80 75 |
10 10 9.5 9 8.5 |
15 14 13 12.5 11.5 |
6280 4820 3600 2690 1910 |
448 371 300 245 191 |
10.9 9.99 9.22 8.48 7.7 |
399 317 248 197 148 |
57.2 47.7 39.6 33.6 27 |
2.74 2.56 2.42 2.3 2.14 |
2.53 2.36 2.23 2.14 2.01 |
| UPN 180 UPN 160 UPN 140 UPN 120 UPN 100 |
22 18.8 16 13.4 10.6 |
28 24 20.4 17 13.5 |
180 160 140 120 100 |
70 65 60 55 50 |
8 7.5 7 7 6 |
11 10.5 10 9 8.5 |
1350 925 605 364 206 |
150 116 86.4 60.7 412 |
6.95 6.21 5.45 4.62 3.91 |
114 85.3 62.7 43.2 29.3 |
22.4 18.3 14.8 11.1 8.49 |
2.02 1.89 1.75 1.59 1.47 |
1.92 1.84 1.75 1.6 1.55 |
| UPN 80 | 8.64 | 11 | 80 | 45 | 6 | 8 | 106 | 26.5 | 3.1 | 19.4 | 6.36 | 1.33 | 1.45 |
| 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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Now we can 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
\begin{aligned}&\left(I_{c}\right)_{1}=I_{1}+A_{1}\left(\bar{y}_{1}+\bar{c}\right)^{2}=7.031\mathrm{~cm}^{4}+\left(37.5 \mathrm{~cm}^{2}\right)(26.22 \mathrm{~cm})^{2}=25790 \mathrm{~cm}^{4} \\&\left(I_{c}\right)_{2}=I_{2}+A_{2} \bar{c}^{2}=63720 \mathrm{~cm}^{4}+\left(178 \mathrm{~cm}^{2}\right)(34.73\mathrm{~cm})^{2}=65870 \mathrm{~cm}^{4} \\&\left(I_{c}\right)_{3}=I_{3}+A_{3}\left(\bar{y}_{3}-\bar{c}\right)^{2}=597 \mathrm{~cm}^{4}+\left(75.8 \mathrm{~cm}^{2}\right)(21.13 \mathrm{~cm})^{2}=34430\mathrm{~cm}^{4}\end{aligned}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}=1.261 \times 10^{5} \mathrm{~cm}^{4}This example shows how to calculate moments of inertia of composite areas by using the parallel-axis theorem.