Question 5.12: A wood beam AB supporting two concentrated loads P (Fig. 5-3...
A wood beam AB supporting two concentrated loads P (Fig. 5-33a) has a rectangular cross section of width b = 100 mm and height h = 150 mm (Fig. 5-33b). The distance from each end of the beam to the nearest load is a = 0.5 m.
Determine the maximum permissible value P_{\max } of the loads if the allowable stress in bending is \sigma_{\text {allow }} = 11 MPa (for both tension and compression) and the allowable stress in horizontal shear is \tau_{\text {allow }} = 1.2 MPa. (Disregard the weight of the beam itself.)
Note: Wood beams are much weaker in horizontal shear (shear parallel to the longitudinal fibers in the wood) than in cross-grain shear (shear on the cross sections). Consequently, the allowable stress in horizontal shear is usually considered in design.
The maximum shear force occurs at the supports and the maximum bending moment occurs throughout the region between the loads. Their values are
V_{\max }=P \quad\quad M_{\max }=P aAlso, the section modulus S and cross-sectional area A are
S=\frac{b h^{2}}{6} \quad\quad A=b hThe maximum normal and shear stresses in the beam are obtained from the flexure and shear formulas (Eqs. 5-16 and 5-40):
\sigma_{1}=-\sigma_{2}=-\frac{M c}{I}=-\frac{M}{S} \quad \text { or } \quad \sigma_{\max }=\frac{M}{S} (5-16a,b)
\tau_{\max }=\frac{V h^{2}}{8 I}=\frac{3 V}{2 A} (5-40)
\sigma_{\max }=\frac{M_{\max }}{S}=\frac{6 P a}{b h^{2}} \quad\quad \tau_{\max }=\frac{3 V_{\max }}{2 A}=\frac{3 P}{2 b h}Therefore, the maximum permissible values of the load P in bending and shear, respectively, are
P_{\text {bending }}=\frac{\sigma_{\text {allow }} b h^{2}}{6 a} \quad\quad P_{\text {shear }}=\frac{2 \tau_{\text {allow }} b h}{3}Substituting numerical values into these formulas, we get
\begin{aligned}P_{\text {bending }} &=\frac{(11 MPa )(100 mm )(150 mm )^{2}}{6(0.5 m )}=8.25 kN \\\\P_{\text {shear }} &=\frac{2(1.2 MPa )(100 mm )(150 mm )}{3}=12.0 kN\end{aligned}Thus, the bending stress governs the design, and the maximum permissible load is
P_{\max }=8.25 kNA more complete analysis of this beam would require that the weight of the beam be taken into account, thus reducing the permissible load.
Notes: (1) In this example, the maximum normal stresses and maximum shear stresses do not occur at the same locations in the beam—the normal stress is maximum in the middle region of the beam at the top and bottom of the cross section, and the shear stress is maximum near the supports at the neutral axis of the cross section.
(2) For most beams, the bending stresses (not the shear stresses) control the allowable load, as in this example.
(3) Although wood is not a homogeneous material and often departs from linearly elastic behavior, we can still obtain approximate results from the flexure and shear formulas. These approximate results are usually adequate for designing wood beams.