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. 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

Question

Determine the moment of inertia [latex]I_{c}[/latex] 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 [latex]I_{c}[/latex] must be determined in order to calculate the stresses and deflections of this beam.

Accepted Answer

Find the moment of inertia [latex]I_{c}[/latex] 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:

[latex]A_{1}=3.0 in ^{2} \quad A_{2}=20.8 in ^{2} \quad A_{3}=8.82 in ^{2}[/latex]

[latex]\bar{y}_{1}=9.485 ext { in. } \quad \bar{y}_{2}=0 \quad \bar{y}_{3}=9.884 ext { in. } \quad \bar{c}=1.80 in.[/latex]

The moments of inertia of the three parts with respect to horizontal axes through their own centroids [latex]C_{1} , C_{2}[/latex], and [latex]C_{3}[/latex] are

[latex]I_{1}=\frac{b h^{3}}{12}=\frac{1}{12}(6.0 in.)(0.5 in.)^{3}=0.063 in ^{4}[/latex]

[latex]I_{2}=1170 in ^{4} \quad I_{3}=3.94 in ^{4}[/latex]

The moments of inertia [latex]I_{2} and I_{3}[/latex] 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:

[latex]\left(I_{c} ight)_{1}=I_{1}+A_{1}\left(\bar{y}_{1}+\bar{c} ight)^{2}=0.063 in ^{4}+\left(3.0 in ^{2} ight)(11.28 in.)^{2}=382 in ^{4}[/latex]

[latex]\left(I_{c} ight)_{2}=I_{2}+A_{2} \bar{c}^{2}=1170 in ^{4}+\left(20.8 in ^{2} ight)(1.80 ext { in. })^{2}=1240 in ^{4}[/latex]

[latex]\left(I_{c} ight)_{3}=I_{3}+A_{3}\left(\bar{y}_{3}-\bar{c} ight)^{2}=3.94 in ^{4}+\left(8.82 in ^{2} ight)(8.084 in.)^{2}=580 in ^{4}[/latex]

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:

[latex]I_{c}=\left(I_{c} ight)_{1}+\left(I_{c} ight)_{2}+\left(I_{c} ight)_{3}=2200 in ^{4}[/latex]

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
	Weight per Foot	Area	Depth	Web Thickness	Width	Thickness	I	S	r	I	S	r
	lb	in²	in.	in.	in.	in.	[latex] ext{in}^{4}[/latex]	in³	in.	[latex] ext{in}^{4}[/latex]	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
	Weight per Foot	Area	Depth	Web Thickness	Width	Average Thickness	I	S	r	I	S	r	c
	lb	in²	in.	in.	in.	in.	[latex]in^{4}[/latex]	in³	in.	[latex]in^{4}[/latex]	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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