Question 5.8: A temporary wood dam is constructed of horizontal planks A s...

Loading diagram. Each post is subjected to a triangularly distributed load produced by the water pressure acting against the planks. Consequently, the loading diagram for each post is triangular (Fig. 5-22c). The maximum intensity  q_0  of the load on the posts is equal to the water pressure at depth h times the spacing s of the posts:

q_0=\gamma hs                      (a)

in which  \gamma  is the specific weight of water. Note that  q_0  has units of force per unit distance,   \gamma  has units of force per unit volume, and both h and s have units of length.

Section modulus. Since each post is a cantilever beam, the maximum bending moment occurs at the base and is given by the following expression:

M_{max}=\frac{q_0h}{2}(\frac{h}{3})=\frac{\gamma h^{3}s}{6}                      (b)

Therefore, the required section modulus (Eq. 5-24) is

S=\frac{M_{max}}{σ_{allow}}=\frac{\gamma h^{3}s}{6σ_{allow}}                      (c)

For a beam of square cross section, the section modulus is  S=b^3/6  (see Eq. 5-18b)  S=\frac{bh^2}{6}.  Substituting this expression for S into Eq. (c), we get a formula for the cube of the minimum dimension b of the posts:

b^{3}=\frac{\gamma h^{3}s}{σ_{allow}}                      (d)

Numerical values. We now substitute numerical values into Eq. (d) and obtain

b^{3}=\frac{(9.81  kN/m^3)(2.0  m)^3(0.8  m)}{8.0  MPa}=0.007848  m^3=7.848\times 10^6  mm^3

from which

b=199  mm

Thus, the minimum required dimension b of the posts is 199 mm. Any larger dimension, such as 200 mm, will ensure that the actual bending stress is less than the allowable stress.