Select the most economical W12 shape column 18 feet in height to support an axial load of 600 k using A572 Grade 50 steel. Assume that the column is hinged at the top but fixed at the base.

Question

Accepted Answer

As a first guess in this trial-and-error process, try W12 × 96 (about the middle of the available “boxier” sections).

W12 × 96 (A = 28.2 in.², [latex]r_{x}=5.44^{\prime \prime}, r_{y}=3.09^{\prime \prime}[/latex] )

Compute ( [latex]K\ell / r[/latex]) critical:

[latex]\frac{K \ell}{r_{x}}=\frac{(0.80)\left(18^{\prime} imes 12 in . / ft . ight)}{5.44^{\prime \prime}}=31.8[/latex]

[latex]\frac{K \ell}{r_{y}}=\frac{(0.80)\left(18^{\prime} imes 12 in . / ft . ight)}{3.09}=55.9[/latex]

The larger slenderness is critical; therefore, use

[latex]K \ell / r=55.9[/latex]

Enter Table 9.2, and obtain the respective [latex]F_{ a }[/latex] :

[latex]F_{ a }=23.41 ksi[/latex]

[latex]P_{ ext {allowable }}=F_{ a } imes A[/latex]

[latex]P_{ ext {allowable }}=23.41 k / in. ^{2} imes 28.2 in .^{2}=660 k >600 k[/latex]

[latex]\% ext { stress }=\frac{P_{ ext {actual }}}{P_{ ext {allowable }}} imes 100 \%=\frac{600 k }{660 k } imes 100 \%=91 \%[/latex]

This selection is quite efficient and still a bit understressed. Therefore, use W12 × 96.

Table 9.2

Table C-50 Allowable Stress For compression Members of 50-ksi Specified Yield Stress [latex] ext { Steel }^{ a }[/latex]
[latex]\frac{K l}{r}[/latex]	[latex]\begin{gathered}F_{a} \ ext { (ksi) }\end{gathered}[/latex]	[latex]\frac{K l}{r}[/latex]	[latex]\begin{gathered}F_{a} \ ext { (ksi) }\end{gathered}[/latex]	[latex]\frac{K l}{r}[/latex]	[latex]\begin{gathered}F_{a} \ ext { (ksi) }\end{gathered}[/latex]	[latex]\frac{K l}{r}[/latex]	[latex]\begin{gathered}F_{a} \ ext { (ksi) }\end{gathered}[/latex]	[latex]\frac{K l}{r}[/latex]	[latex]\begin{gathered}F_{a} \ ext { (ksi) }\end{gathered}[/latex]
1	29.94	41	25.69	81	18.81	121	10.20	161	5.76
2	29.87	42	25.55	82	18.61	122	10.03	162	5.69
3	29.80	43	25.40	83	18.41	123	9.87	163	5.62
4	29.73	44	25.26	84	18.20	124	9.71	164	5.55
5	29.66	45	25.11	85	17.99	125	9.56	165	5.49
6	29.58	46	24.96	86	17.79	126	9.41	166	5.42
7	29.50	47	24.81	87	17.58	127	9.26	167	5.35
8	29.42	48	24.66	88	17.37	128	9.11	168	5.29
9	29.34	49	24.51	89	17.15	129	8.97	169	5.23
10	29.26	50	24.35	90	16.94	130	8.84	170	5.17
11	29.17	51	24.19	91	16.72	131	8.70	171	5.11
12	29.08	52	24.04	92	16.50	132	8.57	172	5.05
13	28.99	53	23.88	93	16.29	133	8.44	173	4.99
14	28.90	54	23.72	94	16.06	134	8.32	174	4.93
15	28.80	55	23.55	95	15.84	135	8.19	175	4.88
16	28.71	56	23.39	96	15.62	136	8.07	176	4.82
17	28.61	57	23.22	97	15.39	137	7.96	177	4.77
18	28.51	58	23.06	98	15.17	138	7.84	178	4.71
19	28.40	59	22.89	99	14.94	139	7.73	179	4.66
20	28.30	60	22.72	100	14.71	140	7.62	180	4.61
21	28.19	61	22.55	101	14.47	141	7.51	181	4.56
22	28.08	62	22.37	102	14.24	142	7.41	182	4.51
23	27.97	63	22.20	103	14.00	143	7.30	183	4.46
24	27.86	64	22.02	104	13.77	144	7.20	184	4.41
25	27.75	65	21.85	105	13.53	145	7.10	185	4.36
26	27.63	66	21.67	106	13.29	146	7.01	186	4.32
27	27.52	67	21.49	107	13.04	147	6.91	187	4.27
28	27.40	68	21.31	108	12.80	148	6.82	188	4.23
29	27.28	69	21.12	109	12.57	149	6.73	189	4.18
30	27.15	70	20.94	110	12.34	150	6.64	190	4.14
31	27.03	71	20.75	111	12.12	151	6.55	191	4.09
32	26.90	72	20.56	112	11.90	152	6.46	192	4.05
33	26.77	73	20.38	113	11.69	153	6.38	193	4.01
34	26.64	74	20.10	114	11.49	154	6.30	194	3.97
35	26.51	75	19.99	115	11.29	155	6.22	195	3.93
36	26.38	76	19.80	116	11.10	156	6.14	196	3.89
37	26.25	77	19.61	117	10.91	157	6.06	197	3.85
38	26.11	78	19.41	118	10.72	158	5.98	198	3.81
39	25.97	79	19.21	119	10.55	159	5.91	199	3.77
40	25.83	80	19.01	120	10.37	160	5.83	200	3.73
[latex]^ { a }[/latex] [latex] ext { When element width-to-thickness ratio exceeds noncompact section limits of Sect. B5.1. }[/latex] [latex] ext { see Appendix B5. }[/latex] [latex] ext { Note: } C _{ c }=126.1[/latex]

Question 9.10: Select the most economical W12 shape column 18 feet in heigh...

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