Knowing that column AB (Fig. 16.25) has an effective length of 14 ft, 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 E = 800 × 10³ psi, and the adjusted allowable stress for compression

Question

Knowing that column AB (Fig. 16.25) has an effective length of 14 ft, 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 × 10^{³}[/latex] psi, and the adjusted allowable stress for compression parallel to the grain is [latex]σ_{C} = 1060[/latex] psi.

Accepted Answer

We note that c = 0.90 for glued laminated wood columns. We must compute the value of [latex]σ_{CE}[/latex]. Using Eq. (16.33) we write

[latex]σ_{CE} =\frac{0.822E}{(L/d)^{2}} = \frac{0.822(800 × 10^{3} psi)}{(168 in./d)^{2}} = 23.299d^{2} psi[/latex]

We then use Eq. (16.32) to express the column stability factor in terms of d, with [latex](σ_{CE} /σ_{C}) = (23.299d^{2}/1.060 × 10^{3}) = 21.98 × 10^{-3} d^{2}[/latex],

[latex]C_{P} = \frac{1 + (σ_{CE} /σ_{C})}{2c} - \sqrt{\left[\frac{1 + (σ_{CE} /σ_{C})}{2c}\right]^{2} - \frac{σ_{CE} /σ_{C}}{c}}[/latex]

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

Since the column must carry 32 kips, which is equal to [latex]σ_{C}d^{2}[/latex], we use Eq.(16.31) to write

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

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

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

Solving for d by trial and error yields d = 6.45 in.

Question 16.3: Knowing that column AB (Fig. 16.25) has an effective length ...

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