A steel column is constructed from a W 250 × 89 wide-flange section (Fig. 11-38). Assume that the column has pin supports and may buckle in any direction. Also, assume that the steel has a modulus of elasticity of E = 200 GPa and a yield stress of $σ_{Y} = 250 MPa$ .

(a) If the length of the column is L = 6.5 m, what is the allowable axial load?

(b) If the column is subjected to an axial load P = 890 kN, what is the maximum permissible length?

Question

A steel column is constructed from a W 250 × 89 wide-flange section (Fig. 11-38). Assume that the column has pin supports and may buckle in any direction. Also, assume that the steel has a modulus of elasticity of E = 200 GPa and a yield stress of $σ_{Y} = 250 MPa$ .

(a) If the length of the column is L = 6.5 m, what is the allowable axial load?

(b) If the column is subjected to an axial load P = 890 kN, what is the maximum permissible length?

(a) If the length of the column is L = 6.5 m, what is the allowable axial load?

(b) If the column is subjected to an axial load P = 890 kN, what is the maximum permissible length?

Accepted Answer

Use a four-step problem-solving approach. Combine steps as needed for an efficient solution.

1, 2. Conceptualize, Categorize: Use the AISC formulas of Eqs. (11-82) through (11-86) when analyzing this column. Since the column has pin supports, the effective-length factor K = 1. Also, since the column will buckle about the weak axis of bending, use the smaller radius of gyration r = 65.3 mm, as obtained from Table F-1(b), Appendix F. The critical slenderness ratio [Eq. (11-83)] is

[latex]\left(\frac{K L}{r} ight)_{ c }=4.71 \sqrt{\frac{E}{\sigma_{ Y }}}=4.71 \sqrt{\frac{200 GPa }{250 MPa }}=133.2[/latex] (a)

[latex]\sigma_{ e }=\frac{\pi^{2} E}{\left(\frac{K L}{r} ight)^{2}}[/latex] (11-82)

[latex]\sigma_{ ext {allow }}=\frac{\sigma_{ cr }}{1.67}[/latex] (11-86)

Part (a): Allowable axial load.

3, 4. Analyze, Finalize: If the length L = 6.5 m, the slenderness ratio of the column is

[latex]\frac{L}{r}=\frac{6.5 m }{65.3 mm }=99.5[/latex]

which is less than the critical ratio of Eq. (a), so obtain the critical stress using Eq. (11-84) with

[latex]\frac{K L}{r} \leq 4.71 \sqrt{\frac{E}{\sigma_{ Y }}}[/latex] (11-84)

[latex]\sigma_{ cr }=\left(0.658^{\sigma_{Y} / \sigma_{ e }} ight) \sigma_{ Y }=\left(0.658^{1.254} ight) 250 MPa =147.9 MPa[/latex]

where

[latex]\sigma_{Y} / \sigma_{e}=250 MPa / 199.4 MPa =1.254[/latex]

and

[latex]\sigma_{ e }=\frac{\pi^{2} E}{\left(\frac{K L}{r} ight)^{2}}=\frac{\pi^{2}(200 GPa )}{99.5^{2}}=199.4 MPa[/latex]

The allowable stress is

[latex]\sigma_{ ext {allow }}=\frac{\sigma_{ cr }}{1.67}=\frac{147.9 MPa }{1.67}=88.6 MPa[/latex]

Since the cross-sectional area of the column is A = 11,400 mm² [from Table F-1(b)], the allowable axial load is

[latex]P_{ ext {allow }}=\sigma_{ ext {allow }} A=88.6 MPa \left(11,400 mm ^{2} ight)=1010 kN[/latex]

Part (b): Maximum permissable length.

To determine the maximum length when the axial load P = 890 kN, begin with an estimated value of the length and then use a trial-and-error procedure. Note that when the load P = 890 kN, the maximum length is greater than 6.5 m (because a length of 6.5 m corresponds to an axial load of 1010 kN). Therefore, as a trial value, assume L = 7 m. The corresponding slenderness ratio is

[latex]\frac{L}{r}=\frac{7000 mm }{65.3 mm }=107.2[/latex]

which is less than the critical ratio in Eq. (a). First, use Eq. (11-82) to find the elastic buckling stress as

[latex]\sigma_{ e }=\frac{\pi^{2} E}{\left(\frac{K L}{r} ight)^{2}}=\frac{\pi^{2}(200 GPa )}{107.2^{2}}=171.8 MPa[/latex]

Then use Eqs. (11-84) and (11-86) to find the allowable stress:

[latex]\sigma_{ ext {allow }}=\frac{\sigma_{ cr }}{1.67}=\frac{\left(0.658^{250 MPa / 171.8 MPa } ight) 250 MPa }{1.67}=81.4 MPa[/latex]

The allowable load is

[latex]P_{ ext {allow }}=\sigma_{ ext {allow }} A=81.4 MPa \left(11,400 mm ^{2} ight)=928 kN[/latex]

which is greater than the given load of 890 kN, so the permissible length is greater than 7 m. With further trials, the permissible length is found to be

[latex]L=7.2 m \quad P_{ ext {allow }}=896 kN[/latex]

[latex]L=7.3 m \quad P_{ ext {allow }}=880 kN[/latex]

Interpolate between these results to find that the maximum permissible length is approximately 7.24 m, as confirmed by

[latex]\frac{L}{r}=\frac{7240 mm }{65.3 mm }=110.9 \quad ext { so } \quad \sigma_{ e }=\frac{\pi^{2}(200 GPa )}{110.9^{2}}=160.5 MPa[/latex]

and

[latex]\sigma_{ ext {allow }}=\frac{\left(0.658^{250 MPa / 160.5 MPa } ight)250 MPa }{1.67}=78 MPa[/latex]

resulting in

[latex]P_{ ext {allow }}=(78 MPa )\left(11,400 mm ^{2} ight)=889.2 kN[/latex]

Hence, the maximum permissible length of the column is 7.24 m.

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

Question 11.6: A steel column is constructed from a W 250 × 89 wide-flange ...

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