Solving Compression Problem in 9m x 10m School Roof

[FarazAli]

Problem from book
"the roof of a 9m x 10m school has a total mass of 12600 kg. The roof is to be supported byt "2 x 4s" (actually about 4cm X 9cm) along the 10m sides. How many supports are required on each side and how far apart mush they be? Consider only compression and assume a safety factor of 12"

Well, I figured that the area of one support is $$4 cm \cdot 9 cm = 3.6 \cdot 10^{-3} m^2$$
and that the total force of the roof is it's own weight $$F = 12600 kg \cdot 9.80 \frac{m}{s^2} = 1.23 \cdot 10^5 N$$.

The stress is given as $$stress = \frac{F}{A}$$, so the stress on one support should be $$\frac{stress}{x}$$ where x is the number of supports.

What should I do next (The Compressive strength is given in the book for wood; $$35 \cdot 10^6 \frac{N}{m^2}$$ parallel to grain and $$10 \cdot 10^6 \frac{N}{m^2}$$ perpindicular to grain)

 

[FarazAli]

...no?

 

[wais]

To solve this compression problem, we need to first determine the maximum stress that the supports can handle. From the given information, we know that the compressive strength of the wood used for the supports is 35 * 10^6 N/m^2 parallel to the grain and 10 * 10^6 N/m^2 perpendicular to the grain. We also know that the safety factor is 12, so we can calculate the maximum stress as follows:

Maximum stress = compressive strength / safety factor
= (35 * 10^6 N/m^2) / 12 = 2.92 * 10^6 N/m^2

Next, we need to determine the total area of the supports needed to support the roof. Since we are only considering compression, we can ignore the length of the roof and focus on the width of 10m. The total area of the supports needed would be 10m * total height of supports (h).

To find the total height of the supports, we can use the formula for stress: stress = force / area. Rearranging this equation, we get area = force / stress. So, the total area of the supports needed would be:

10m * (1.23 * 10^5 N / 2.92 * 10^6 N/m^2) = 0.42 m^2

Since we know the area of one support is 3.6 * 10^-3 m^2, we can divide the total area by the area of one support to find the number of supports needed:

Number of supports = 0.42 m^2 / (3.6 * 10^-3 m^2) = 116.67

Since we cannot have a fraction of a support, we can round up to 117 supports needed on each side of the roof. We can also calculate the distance between each support by dividing the width of the roof (9m) by the number of supports (117):

Distance between supports = 9m / 117 = 0.077 m

Therefore, we would need 117 supports spaced 0.077 m apart on each side of the roof to support the total mass of 12600 kg. This solution takes into consideration the compressive strength of the supports and the safety factor.