A wooden beam is fabricated by gluing four dimension lumber boards, each 40-mm wide and 90-mm deep, to a 32 × 400 plywood web, as shown in Figure P9.45. Determine the maximum allowable shear force and the maximum allowable bending moment that this section can carry if the allowable bending stress

Question

A wooden beam is fabricated by gluing four dimension lumber boards, each 40-mm wide and 90-mm deep, to a 32 × 400 plywood web, as shown in Figure P9.45. Determine the maximum allowable shear force and the maximum allowable bending moment that this section can carry if the allowable bending stress is 6 MPa, the allowable shear stress in the plywood is 640 kPa, and the allowable shear stress in the glued joints is 250 kPa.

Accepted Answer

Moment of inertia [latex]I _{z}[/latex]:

[latex]I_{z}=\frac{(112 mm )(400 mm )^{3}}{12}-\frac{(80 mm )(220 mm )^{3}}{12}=526,346,667 mm ^{4}[/latex]

Maximum allowable bending moment:

[latex]\begin{aligned}&\sigma=\frac{M c}{I} \& herefore M_{\max }=\frac{\sigma I}{c}=\frac{\left(6 N / mm ^{2} ight)\left(526,346,667 mm ^{4} ight)}{200 mm }=15,790,400 N - mm =15.79 kN - m\end{aligned}[/latex]

Maximum allowable shear force:
Consider maximum shear stress, which occurs at the neutral axis:

[latex]\begin{aligned}&Q=(32 mm )(200 mm )(100 mm )+2(40 mm )(90 mm )(200 mm -90 mm / 2)=1,756,000 mm ^{3} \& au=\frac{V Q}{I t} \quad V=\frac{ au I t}{Q}=\frac{(0.640 N / mm )\left(526,346,667 mm ^{4} ight)(32 mm )}{1,756,000 mm ^{3}}=6,138.7 N =6.14 kN (a)\end{aligned}[/latex]

Consider shear stress in glue joints:

[latex]Q=(40 mm )(90 mm )(200 mm -90 mm / 2)=558,000 mm ^{3}[/latex]

The shear stress in the glue joints can be found from the shear flow across the glue joint divided by the width of the glue joint; thus,

[latex]\begin{aligned}& au_{ ext {glue }}=\frac{q}{t_{ ext {glue }}}=\frac{V Q / I}{t_{ ext {glue }}} \& herefore V=\frac{ au_{ ext {glue }} I t_{ ext {glue }}}{Q}=\frac{(0.250 N / mm )\left(526,346,667 mm ^{4} ight)(90 mm )}{558,000 mm ^{3}}=21,224 N =21.2 kN (b)\end{aligned}[/latex]

Compare results (a) and (b) to find that the maximum allowable shear force for the section is:

[latex] V_{\max }=6.14 kN[/latex]

Question 9.45: A wooden beam is fabricated by gluing four dimension lumber ...

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