Question 5.16: A wood box beam (Fig. 5-44) is constructed of two boards, ea...
A wood box beam (Fig. 5-44) is constructed of two boards, each 40 × 180 mm in cross section, that serve as flanges and two plywood webs, each 15 mm thick. The total height of the beam is 280 mm. The plywood is fastened to the flanges by wood screws having an allowable load in shear of F = 800 N each.
If the shear force V acting on the cross section is 10.5 kN, determine the maximum permissible longitudinal spacing s of the screws (Fig. 5-44b).
Shear flow. The horizontal shear force transmitted between the upper flange and the two webs can be found from the shear-flow formula ƒ = VQ/I, in which Q is the first moment of the cross-sectional area of the flange. To find this first moment, we multiply the area A_{f} of the flange by the distance d_{f} from its centroid to the neutral axis:
\begin{gathered}A_{ f }=40 mm \times 180 mm =7200 mm ^{2} \quad d_{ f }=120 mm \\Q=A_{ f } d_{ f }=\left(7200 mm ^{2}\right)(120 mm )=864 \times 10^{3} mm ^{3} \end{gathered}The moment of inertia of the entire cross-sectional area about the neutral axis is equal to the moment of inertia of the outer rectangle minus the moment of inertia of the “hole” (the inner rectangle):
I=\frac{1}{12}(210 mm )(280 mm )^{3}-\frac{1}{12}(180 mm )(200 mm )^{3}=264.2 \times 10^{6} mm ^{4}Substituting V, Q, and I into the shear-flow formula (Eq. 5-52), we obtain
f=\frac{V Q}{I}=\frac{(10,500 N )\left(864 \times 10^{3} mm ^{3}\right)}{264.2 \times 10^{6} mm ^{4}}=34.3 N / mmwhich is the horizontal shear force per millimeter of length that must be transmitted between the flange and the two webs.
Spacing of screws. Since the longitudinal spacing of the screws is s, and since there are two lines of screws (one on each side of the flange), it follows that the load capacity of the screws is 2F per distance s along the beam. Therefore, the capacity of the screws per unit distance along the beam is 2F/s. Equating 2F/s to the shear flow ƒ and solving for the spacing s, we get
This value of s is the maximum permissible spacing of the screws, based upon the allowable load per screw. Any spacing greater than 46.6 mm would overload the screws. For convenience in fabrication, and to be on the safe side, a spacing such as s = 45 mm would be selected.