Question

The roof over a \({\bf{9}}{\bf{.0}}\;{\bf{m \times 10}}{\bf{.0}}\;{\bf{m}}\) room in a school has a total mass of 13,600 kg. The roof is to be supported by vertical wooden (actually about \({\bf{4}}{\bf{.0}}\;{\bf{cm \times 9}}{\bf{.0}}\;{\bf{cm}}\)) equally spaced along the 10.0-m sides. How many supports are required on each side, and how far apart must they be? Consider only compression, and assume a safety factor of 12.

Short answer

The number of supports required is \(12\), and the gap between each support is \(1.66\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {{\rm{gap}}}}} \right.} {{\rm{gap}}}}\).

Step-by-step solution

Understanding of compressive strength

The compressive strength may be described as the greatest compressive pressure that an object can sustain earlier to failure, divided by the object's cross-sectional area.

Given information

Given data:

The mass of the roof, \(m = 13600\;{\rm{kg}}\).

The area of the column per support, \(a = 4.0\;{\rm{cm}} \times 9.0\;{\rm{cm}}\).

The safety factor, \(N = 12\).

Evaluation of number of supports required

The number of supports required on each side can be calculated by using the following expression.

\(n = \frac{{mgN}}{{a{\sigma _{\rm{c}}}}}\)

Here, \({\sigma _{\rm{c}}}\) is the compressive stress on each column, and its value is \(35 \times {10^6}\;{\rm{Pa}}\).

Substitute the values in the above equation.

\(\begin{array}{c}n = \frac{{\left( {13600\;{\rm{kg}}} \right)\left( {9.8\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right)\left( {12} \right)}}{{\left[ {\left( {4.0\;{\rm{cm}} \times 9.0\;{\rm{cm}}} \right)\left( {\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 4}}}}\;{{\rm{m}}^{\rm{2}}}}}{{{\rm{1}}\;{\rm{c}}{{\rm{m}}^{\rm{2}}}}}} \right)} \right]\left( {35 \times {{10}^6}\;{\rm{Pa}}} \right)}}\\n = 12.69 \approx 12\end{array}\)

Thus, the number of supports required is \(12\).

Evaluation of gap between each support

Since there are to be more than 12 supports and have the same number of supports on each side, there will be 14 or seven supports on each side. That means there will be six support-to-support spans, each of which would be given by spacing:

\(\begin{array}{l}D = \frac{{10\;{\rm{m}}}}{{6\;{\rm{gaps}}}}\\D = 1.66\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {{\rm{gap}}}}} \right.} {{\rm{gap}}}}\end{array}\)

Thus, the gap between each support is \(1.66\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {{\rm{gap}}}}} \right.} {{\rm{gap}}}}\).