A 22-foot-tall glu-lam column is required to support a roof load (including snow) of 40 k. Assuming $8 \frac{3}{4 }^{ \prime \prime }$ in one dimension (to match the beam width above), determine the minimum column size if the top and bottom are pin supported.
Select from the following sizes:

Question

A 22-foot-tall glu-lam column is required to support a roof load (including snow) of 40 k. Assuming [latex] 8 \frac{3}{4 }^{ \prime \prime }[/latex] in one dimension (to match the beam width above), determine the minimum column size if the top and bottom are pin supported.
Select from the following sizes:

[latex] 8 \frac{3}{4 }^{ \prime \prime }[/latex] × 9" (A = 78.75 in.² )

[latex]8 \frac{3}{4}^{\prime \prime} imes 1010 \frac{1}{2}^{\prime \prime}\left(A=91.88 in .^{2} ight)[/latex]

[latex] 8 \frac{3}{4 }^{ \prime \prime }[/latex]× 12" (A = 105.0 in.² )

Accepted Answer

Glu-lam column: [latex]\left(F_{ c }=1,650 psi , E=1.8 imes 10^{6} psi ight)[/latex]

Try [latex]8 \frac{3}{4}^{\prime \prime} imes 10 \frac{1}{2}^{\prime \prime}[/latex]

[latex]8 \frac{3}{4}^{\prime \prime} imes 10 \frac{1}{2}^{\prime \prime}\left(A=91.9 in .^{2} ight)[/latex]

[latex]\frac{\ell_{ e }}{d_{\min }}=\frac{22 imes 12}{8.75}=30.2<50 [/latex]

(max. slenderness ratio)

[latex]F_{ cE }=\frac{0.418 E}{\left(\ell_{ e } / d ight)^{2}}=\frac{0.418\left(1.8 imes 10^{6} ight)}{(30.2)^{2}}=825 psi[/latex]

[latex]F_{ c }^{*} \cong F_{ c } C_{ D }=1,650 psi (1.15)=1,898 psi[/latex]

[latex]\frac{F_{ cE }}{F_{ c }^{*}}=\frac{825}{1,898}=0.43[/latex]

From Table 9.3, [latex]C_{ p }=0.403[/latex]

[latex]F_{c}^{\prime}=F_{c}^{*}C_{p}=1,898(.403)=765 psi[/latex]

[latex]\begin{aligned}P_{ a } &=F_{ c }^{\prime} imes A=765 psi \left(91.9 in .^{2} ight)=70,300 \# \&>40,000 \#\end{aligned}[/latex]

Cycle again, trying a smaller, more economical section.

Try [latex]8 \frac{3}{4}^{\prime \prime} imes 9^{\prime \prime}[/latex].

[latex]8 \frac{3}{4}^{\prime \prime} imes 9^{\prime \prime}\left(A=78.8 in .^{2} ight)[/latex]

Because the critical dimension is still [latex] 8 \frac{3}{4}^{\prime \prime}[/latex] , the values for [latex]F_{ cE }, F^{*}_{ c } [/latex] , and [latex]F_{ c }^{\prime}[/latex] all remain the same as in trial 1. The only change that affects the capability of the column is the available cross-sectional area.

[latex] herefore P_{ a }=F_{ c }^{\prime} A=(765 psi )\left(78.8 in .^{2} ight)=60,300 \#[/latex]

[latex]P_{ a }=60.3 k >40 k[/latex]

[latex] ext { Use } 8 \frac{3}{4}^{\prime \prime} imes 9^{\prime \prime} [/latex] glu-lam section.

Table 9.3

[latex] ext { Column S+A1:L53tability Factor } C_{p}[/latex]
[latex]{ }^{"} C _{ p }^{"}[/latex] [latex]F _{ C }^{\prime}= C _{ p } \cdot F _{ c }^{*} \quad F _{ CE }=\frac{30 E }{(l / d )^{2}} ext { for sawn posts } F _{ CF }=\frac{.418 E }{(l / d )^{2}} ext { for Glu-lam posts }[/latex]
[latex]\frac{F_{C E}}{F_{C}^{*}}[/latex]	Sawn [latex]C_{p}[/latex]	Glu-Lam [latex]C_{p}[/latex]	[latex]\frac{F_{C E}}{F_{C}^{*}}[/latex]	Sawn [latex]C_{p}[/latex]	Glu-Lam [latex]C_{p}[/latex]	[latex]\frac{F_{C E}}{F_{C}^{*}}[/latex]	Sawn [latex]C_{p}[/latex]	Glu-Lam [latex]C_{p}[/latex]	[latex]\frac{F_{C E}}{F_{C}^{*}}[/latex]	Sawn [latex]C_{p}[/latex]	Glu-Lam [latex]C_{p}[/latex]
0.00	0.000	0.000	0.60	0.500	0.538	1.20	0.750	0.822	2.40	0.894	0.940
0.01	0.010	0.010	0.61	0.506	0.545	1.22	0.755	0.826	2.45	0.897	0.941
0.02	0.020	0.020	0.62	0.512	0.552	1.24	0.760	0.831	2.50	0.899	0.943
0.03	0.030	0.030	0.63	0.518	0.559	1.26	0.764	0.836	2.55	0.901	0.944
0.04	0.040	0.040	0.64	0.524	0.566	1.28	0.769	0.840	2.60	0.904	0.946
0.05	0.049	0.050	0.65	0.530	0.573	1.30	0.773	0.844	2.65	0.906	0.947
0.06	0.059	0.060	0.66	0.536	0.580	1.32	0.777	0.848	2.70	0.908	0.949
0.07	0.069	0.069	0.67	0.542	0.587	1.34	0.781	0.852	2.75	0.910	0.950
0.08	0.079	0.079	0.68	0.548	0.593	1.36	0.785	0.855	2.80	0.912	0.951
0.09	0.088	0.089	0.69	0.553	0.600	1.38	0.789	0.859	2.85	0.914	0.952
0.10	0.098	0.099	0.70	0.559	0.607	1.40	0.793	0.862	2.90	0.916	0.953
0.11	0.107	0.109	0.71	0.564	0.613	1.42	0.796	0.865	2.95	0.917	0.954
0.12	0.117	0.118	0.72	0.569	0.619	1.44	0.800	0.868	3.00	0.919	0.955
0.13	0.126	0.128	0.73	0.575	0.626	1.46	0.803	0.871	3.05	0.920	0.956
0.14	0.136	0.138	0.74	0.580	0.632	1.48	0.807	0.874	3.10	0.922	0.957
0.15	0.145	0.147	0.75	0.585	0.638	1.50	0.810	0.877	3.15	0.923	0.958
0.16	0.154	0.157	0.76	0.590	0.644	1.52	0.813	0.879	3.20	0.925	0.959
0.17	0.164	0.167	0.77	0.595	0.650	1.54	0.816	0.882	3.25	0.926	0.960
0.18	0.173	0.176	0.78	0.600	0.655	1.56	0.819	0.884	3.30	0.927	0.961
0.19	0.182	0.186	0.79	0.605	0.661	1.58	0.822	0.887	3.35	0.929	0.961
0.20	0.191	0.195	0.80	0.610	0.667	1.60	0.825	0.889	3.40	0.930	0.962
0.21	0.200	0.205	0.81	0.614	0.672	1.62	0.827	0.891	3.45	0.931	0.963
0.22	0.209	0.214	0.82	0.619	0.678	1.64	0.830	0.893	3.50	0.932	0.963
0.23	0.218	0.224	0.83	0.623	0.683	1.66	0.832	0.895	3.55	0.933	0.964
0.24	0.227	0.233	0.84	0.628	0.688	1.68	0.835	0.897	3.60	0.934	0.965
0.25	0.235	0.242	0.85	0.632	0.693	1.70	0.837	0.899	3.65	0.936	0.965
0.26	0.244	0.252	0.86	0.637	0.698	1.72	0.840	0.901	3.70	0.937	0.966
0.27	0.253	0.261	0.87	0.641	0.703	1.74	0.842	0.903	3.75	0.938	0.966
0.28	0.261	0.270	0.88	0.645	0.708	1.76	0.844	0.904	3.80	0.938	0.967
0.29	0.270	0.279	0.89	0.649	0.713	1.78	0.846	0.906	3.85	0.939	0.968
0.30	0.278	0.288	0.90	0.653	0.718	1.80	0.849	0.908	3.90	0.940	0.968
0.31	0.287	0.297	0.91	0.658	0.722	1.82	0.851	0.909	3.95	0.941	0.969
0.32	0.295	0.306	0.92	0.661	0.727	1.84	0.853	0.911	4.00	0.942	0.969
0.33	0.304	0.315	0.93	0.665	0.731	1.86	0.855	0.912	4.05	0.943	0.969
0.34	0.312	0.324	0.94	0.669	0.735	1.88	0.857	0.914	4.10	0.944	0.970
0.35	0.320	0.333	0.95	0.673	0.740	1.90	0.858	0.915	4.15	0.944	0.970
0.36	0.328	0.342	0.96	0.677	0.744	1.92	0.860	0.916	4.20	0.945	0.971
0.37	0.336	0.351	0.97	0.680	0.748	1.94	0.862	0.918	4.25	0.946	0.971
0.38	0.344	0.360	0.98	0.684	0.752	1.96	0.864	0.919	4.30	0.947	0.972
0.39	0.352	0.368	0.99	0.688	0.756	1.98	0.866	0.920	4.35	0.947	0.972
0.40	0.360	0.377	1.00	0.691	0.760	2.00	0.867	0.921	4.40	0.948	0.972
0.41	0.367	0.386	1.01	0.694	0.764	2.02	0.869	0.922	4.45	0.949	0.973
0.42	0.375	0.394	1.02	0.698	0.767	2.04	0.870	0.924	4.50	0.949	0.973
0.43	0.383	0.403	1.03	0.701	0.771	2.06	0.872	0.925	4.55	0.950	0.974
0.44	0.390	0.411	1.04	0.704	0.774	2.08	0.874	0.926	4.60	0.950	0.974
0.45	0.398	0.420	1.05	0.708	0.778	2.10	0.875	0.927	4.65	0.951	0.974
0.46	0.405	0.428	1.06	0.711	0.781	2.12	0.876	0.928	4.70	0.952	0.975
0.47	0.412	0.436	1.07	0.714	0.784	2.14	0.878	0.929	4.75	0.952	0.975
0.48	0.419	0.444	1.08	0.717	0.788	2.16	0.879	0.930	4.80	0.953	0.975
0.49	0.427	0.453	1.09	0.720	0.791	2.18	0.881	0.931	4.85	0.953	0.975
0.50	0.434	0.461	1.10	0.723	0.794	2.20	0.882	0.932	4.90	0.954	0.976
0.51	0.441	0.469	1.11	0.726	0.797	2.22	0.883	0.932	5.00	0.955	0.976
0.52	0.448	0.477	1.12	0.729	0.800	2.24	0.885	0.933	6.00	0.963	0.981
0.53	0.454	0.484	1.13	0.731	0.803	2.26	0.886	0.934	8.00	0.973	0.986
0.54	0.461	0.492	1.14	0.734	0.806	2.28	0.887	0.935	10.00	0.979	0.989
0.55	0.468	0.500	1.15	0.737	0.809	2.30	0.888	0.936	20.00	0.990	0.995
0.56	0.474	0.508	1.16	0.740	0.811	2.32	0.889	0.937	40.00	0.995	0.997
0.57	0.481	0.515	1.17	0.742	0.814	2.34	0.891	0.937	60.00	0.997	0.998
0.58	0.487	0.523	1.18	0.745	0.817	2.36	0.892	0.938	100.00	0.998	0.999
0.59	0.494	0.530	1.19	0.747	0.819	2.38	0.893	0.939	200.00	0.999	0.999

Question 9.16: A 22-foot-tall glu-lam column is required to support a roof ...

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