A wooden mast 15 m high tapers linearly from 250 mm diameter at the base to 100 mm at the top. At what point will the mast break under a horizontal load applied at the top? If the maximum permissible stress in the wood is 35 N/mm^2, calculate the magnitude of the load that will cause failure.

Question

A wooden mast 15 m high tapers linearly from 250 mm diameter at the base to 100 mm at the top. At what point will the mast break under a horizontal load applied at the top? If the maximum permissible stress in the wood is 35 N/mm², calculate the magnitude of the load that will cause failure.

Accepted Answer

The mast is shown in Fig. S.9.4.
At any section a distance h m from the top of the mast the diameter is given by

[latex]d=100+\left(\frac{h}{15} ight) imes 150[/latex]

i.e.

d = 100+10hmm

The maximum direct stress due to bending at this section is, from Eq. (9.9)

[latex] \sigma_{x}=-\frac{M y}{I} [/latex] (9.9)

[latex]\sigma_{\max }=\frac{P h imes 10^{3} imes\left(\frac{d}{2} ight)}{\left(\frac{\pi d^{4}}{64} ight)}=\frac{P h imes 10^{3} imes 32}{\pi d^{3}}[/latex] (i)

This maximum direct stress will be greatest when [latex] d \sigma_{\max } / d h=0 [/latex]. Therefore, substituting for d in Eq. (i) and differentiating

[latex] 0=\left(\frac{32 imes 10^{3} P}{\pi} ight) \frac{(100+10 h)^{3}-3 h(100+10 h)^{2} imes 10}{(100+10 h)^{6}} [/latex]

i.e.

[latex] 100+10 h=30 h [/latex]

which gives

[latex] h=5 m [/latex]

i.e. the mast will break at a distance of 5 m from the top.
The diameter at this section is then 150 mm and, from Eq. (9.9)

[latex] 35=\frac{P imes 5 imes 10^{6} imes 75}{\left(\frac{\pi imes 150^{4}}{64} ight)} [/latex]

from which

P = 2320 N.

Question 9.4: A wooden mast 15 m high tapers linearly from 250 mm diameter...

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