The cross section of a steel beam is constructed of a W 18 × 71 wide-flange section with a 6 in. × 1/2 in. cover plate welded to the top flange and a C 10 × 30 channel section welded to the bottom flange (Fig. D-8).

Locate the centroid C of the cross-sectional area.

Question

The cross section of a steel beam is constructed of a W 18 × 71 wide-flange section with a 6 in. × 1/2 in. cover plate welded to the top flange and a C 10 × 30 channel section welded to the bottom flange (Fig. D-8).

Locate the centroid C of the cross-sectional area.

Accepted Answer

Denote the areas of the cover plate, the wide-flange section, and the channel section as areas [latex]A_{1} , A_{2} , and A_{3}[/latex], respectively. The centroids of these three areas are labeled [latex]C_{1} , C_{2} , and C_{3}[/latex], respectively, in Fig. D-8. Note that the composite area has an axis of symmetry, so all centroids lie on that axis. The three partial areas are

[latex]A_{1}=(6 in.)(0.5 in.)=3.0 in ^{2} \quad A_{2}=20.8 in ^{2} \quad A_{3}=8.82 in ^{2}[/latex]

in which the areas [latex]A_{2} and A_{3}[/latex] are obtained from Tables F-1(a) and F-3(a) of Appendix F.

Place the origin of the x and y axes at the centroid [latex]C_{2}[/latex] of the wide-flange section. Then the distances from the x axis to the centroids of the three areas are as follows:

[latex]\bar{y}_{1}=\frac{18.47 in.}{2}+\frac{0.5 in.}{2}=9.485 in[/latex].

[latex]\bar{y}_{2}=0 \quad \bar{y}_{3}=\frac{18.47 in.}{2}+0.649 in.=9.884 in[/latex].

in which the pertinent dimensions of the wide-flange and channel sections are obtained from Tables F-1 and F-3.

The area A and first moment [latex]Q_{x}[/latex] of the entire cross section are obtained from Eqs.(D-6a and b) as

[latex]A=\sum\limits_{i=1}^{n} A_{i}=A_{1}+A_{2}+A_{3}[/latex]

[latex]= 3.0 in ^{2}+20.8 in ^{2}+8.82 in ^{2}=32.62 in ^{2}[/latex]

[latex]Q_{x}=\sum\limits_{i=1}^{n} \bar{y}_{i} A_{i}=\bar{y}_{1} A_{1}+\bar{y}_{2} A_{2}+\bar{y}_{3} A_{3}[/latex]

[latex]= (9.485 in.)\left(3.0 in ^{2} ight)+0-(9.884 in.)\left(8.82 in ^{2} ight)=-58.72 in ^{3}[/latex]

Now find the coordinate [latex]\bar{y}[/latex] to the centroid C of the composite area from Eq.(D-7b):

[latex]\bar{y}=\frac{Q_{x}}{A}=\frac{-58.72 in ^{3}}{32.62 in ^{2}}=-1.80 in[/latex].

Since ȳ is positive in the positive direction of the y axis, the minus sign means that the centroid C of the composite area is located below the x axis, as shown in Fig.D-8. Thus, the distance [latex]\bar{c}[/latex] between the x axis and the centroid C is

[latex]\bar{c}=1.80 in.[/latex]

Note that the position of the reference axis (the x axis) is arbitrary; however, in this example it was placed through the centroid of the wide-flange section because it slightly simplifies the calculations.

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-1(b)
Properties of Wide-Flange Sections (W Shapes)—SI Units (Abridged List)
Designation	Mass per Meter	Area	Depth	Web Thickness	Flange		Axis 1–1			Axis 2-2
	Mass per Meter	Area	Depth	Web Thickness	Width	Thickness	I	S	r	I	S	r
	Kg	mm²	mm	mm	mm	mm	[latex]×10^{6}mm^{4}[/latex]	[latex]×10^{3}mm^{3}[/latex]	mm	[latex]×10^{6}mm^{4}[/latex]	[latex]×10^{3}mm^{3}[/latex]	mm
W 760 × 314	314	40100	785	19.7	384	33.5	4290	10900	328	315	1640	88.6
W 760 × 196	196	25100	770	15.6	267	25.4	2400	6230	310	81.6	610	57.2
W 610 × 241	241	30800	635	17.9	330	31.0	2150	6780	264	184	1120	77.5
W 610 × 140	140	17900	617	13.1	230	22.2	1120	3640	251	45.4	393	50.3
W 460 × 177	177	22600	483	16.6	287	26.9	912	3790	201	105	736	68.3
W 460 × 106	106	13400	470	12.6	194	20.6	487	2080	191	25.1	259	43.2
W 410 × 149	149	19000	432	14.9	264	25.0	620	2870	180	77.4	585	63.8
W 410 × 114	114	14600	419	11.6	262	19.3	462	2200	178	57.4	441	62.7
W 410 × 85	85.0	10800	417	10.9	181	18.2	316	1510	171	17.9	198	40.6
W 410 × 46.1	46.1	5890	404	6.99	140	11.2	156	773	163	5.16	73.6	29.7
W 360 × 179	179	22800	368	15.0	373	23.9	574	3110	158	206	1110	95.0
W 360 × 122	122	15500	363	13.0	257	21.7	367	2020	154	61.6	480	63.0
W 360 × 79	79.0	10100	353	9.40	205	16.8	225	1270	150	24.0	234	48.8
W 360 × 39	39.0	4960	353	6.48	128	10.7	102	578	144	3.71	58.2	27.4
W 310 × 129	129	16500	318	13.1	307	20.6	308	1930	137	100	651	78.0
W 310 × 74	74.0	9420	310	9.40	205	16.3	163	1050	132	23.4	228	49.8
W 310 × 52	52.0	6650	318	7.62	167	13.2	119	747	133	10.2	122	39.1
W 310 × 21	21.0	2680	302	5.08	101	5.72	36.9	244	117	0.982	19.5	19.1
W 250 × 89	89.0	11400	259	10.7	257	17.3	142	1090	112	48.3	377	65.3
W 250 × 67	67.0	8580	257	8.89	204	15.7	103	805	110	22.2	218	51.1
W 250 × 44.8	44.8	5700	267	7.62	148	13.0	70.8	531	111	6.95	94.2	34.8
W 250 × 17.9	17.9	2280	251	4.83	101	5.33	22.4	179	99.1	0.907	18.0	19.9
W 200 × 52	52.0	6650	206	7.87	204	12.6	52.9	511	89.2	17.7	174	51.6
W 200 × 41.7	41.7	5320	205	7.24	166	11.8	40.8	398	87.6	9.03	109	41.1
W 200 × 31.3	31.3	3970	210	6.35	134	10.2	31.3	298	88.6	4.07	60.8	32.0
W 200 × 22.5	22.5	2860	206	6.22	102	8.00	20.0	193	83.6	1.42	27.9	22.3

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.

Table F-3(b)
Properties of Channel Sections (C Shapes)—SI Units (Abridged List)
Designation	Mass per Meter	Area	Depth	Web Thickness	Flange		Axis 1–1			Axis 2–2
	Mass per Meter	Area	Depth	Web Thickness	Width	Thickness	I	S	r	I	S	r	c
	kg	mm²	mm	mm	mm	mm	[latex]×10^{6}mm^{4}[/latex]	[latex]×10³mm³[/latex]	mm	[latex]×10^{6}mm^{4}[/latex]	[latex]×10³mm³[/latex]	mm	mm
C 380 × 74	74.0	9480	381	18.2	94.5	16.5	168	882	133	4.58	61.8	22.0	20.3
C 380 × 60	60.0	7610	381	13.2	89.4	16.5	145	762	138	3.82	54.7	22.4	19.8
C 380 × 50.4	50.4	6450	381	10.2	86.4	16.5	131	688	143	3.36	50.6	22.9	20.0
C 310 × 45	45.0	5680	305	13.0	80.5	12.7	67.4	442	109	2.13	33.6	19.4	17.1
C 310 × 37	37.0	4740	305	9.83	77.5	12.7	59.9	393	113	1.85	30.6	19.8	17.1
C 310 × 30.8	30.8	3920	305	7.16	74.7	12.7	53.7	352	117	1.61	28.2	20.2	17.7
C 250 × 45	45.0	5680	254	17.1	77.0	11.1	42.9	339	86.9	1.64	27.0	17.0	16.5
C 250 × 37	37.0	4740	254	13.4	73.4	11.1	37.9	298	89.4	1.39	24.1	17.1	15.7
C 250 × 30	30.0	3790	254	9.63	69.6	11.1	32.8	259	93.0	1.17	21.5	17.5	15.4
C 250 × 22.8	22.8	2890	254	6.10	66.0	11.1	28.0	221	98.3	0.945	18.8	18.1	16.1
C 200 × 27.9	27.9	3550	203	12.4	64.3	9.91	18.3	180	71.6	0.820	16.6	15.2	14.4
C 200 × 20.5	20.5	2610	203	7.70	59.4	9.91	15.0	148	75.9	0.633	13.9	15.6	14.1
C 200 × 17.1	17.1	2170	203	5.59	57.4	9.91	13.5	133	79.0	0.545	12.7	15.8	14.5
C 150 × 19.3	19.3	2460	152	11.1	54.9	8.71	7.20	94.7	54.1	0.437	10.5	13.3	13.1
C 150 × 15.6	15.6	1990	152	7.98	51.6	8.71	6.29	82.6	56.4	0.358	9.19	13.4	12.7
C 150 × 12.2	12.2	1540	152	5.08	48.8	8.71	5.45	71.3	59.4	0.286	8.00	13.6	13.0
C 100 × 10.8	10.8	1370	102	8.15	43.7	7.52	1.91	37.5	37.3	0.177	5.52	11.4	11.7
C 100 × 8	8.00	1020	102	4.67	40.1	7.52	1.60	31.5	39.6	0.130	4.54	11.3	11.6
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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