Knowing that column AB (Fig. 10.31) has an effective length of 14 ft, and that it must

Question

Knowing that column AB (Fig. 10.31) has an effective length of [latex]14 \mathrm{ft}[/latex], and that it must safely carry a 32-kip load, design the column using a square glued laminated cross section. The adjusted modulus of elasticity for the wood is [latex]E=800 imes 10^{3} \mathrm{psi}[/latex], and the adjusted allowable stress for compression parallel to the grain is [latex]\sigma_{C}=1060[/latex] psi.

Accepted Answer

We note that [latex]c=0.90[/latex] for glued laminated wood columns. We must compute the value of [latex]\sigma_{C E}[/latex]. Using Eq. (10.49) we write

[latex]\sigma_{C E}=\frac{0.822 E}{(L / d)^{2}}=\frac{0.822\left(800 imes 10^{3} \mathrm{psi} ight)}{(168 \mathrm{in} . / d)^{2}}=23.299 d^{2} \mathrm{psi}[/latex]

We then use Eq. (10.48) to express the column stability factor in terms of [latex]d[/latex], with [latex]\left(\sigma_{C E} / \sigma_{C} ight)=\left(23.299 d^{2} / 1.060 imes 10^{3} ight)=21.98 imes 10^{-3} d^{2}[/latex],

[latex] C_{P} =\frac{1+\left(\sigma_{C E} / \sigma_{C} ight)}{2 c}-\sqrt{\left[\frac{1+\left(\sigma_{C E} / \sigma_{C} ight)}{2 c} ight]^{2}-\frac{\sigma_{C E} / \sigma_{C}}{c}} \ =\frac{1+21.98 imes 10^{-3} d^{2}}{2(0.90)}-\sqrt{\left[\frac{1+21.98 imes 10^{-3} d^{2}}{2(0.90)} ight]^{2}-\frac{21.98 imes 10^{-3} d^{2}}{0.90}} [/latex]

Since the column must carry [latex]32 \mathrm{kips}[/latex], which is equal to [latex]\sigma_{C} d^{2}[/latex], we use Eq. (10.47) to write

[latex]\sigma_{\mathrm{all}}=\frac{32 \mathrm{kips}}{d^{2}}=\sigma_{C} C_{P}=1.060 C_{P}[/latex]

Solving this equation for [latex]\mathrm{C}_{P}[/latex] and substituting the value obtained into the previous equation, we write

[latex]\frac{30.19}{d^{2}}=\frac{1+21.98 imes 10^{-3} d^{2}}{2(0.90)}-\sqrt{\left[\frac{1+21.98 imes 10^{-3} d^{2}}{2(0.90)} ight]^{2}-\frac{21.98 imes 10^{-3} d^{2}}{0.90}}[/latex]

Solving for [latex]d[/latex] by trial and error yields [latex]d=6.45 \mathrm{in}[/latex].

Question 10.03: Knowing that column AB (Fig. 10.31) has an effective length ...

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