The roof over a 9.0-m ×10.0-m room in a school has a total mass of 13,600 kg . The roof is to be

The roof over a 9.0-m ×10.0-m room in a school has a total...

The roof over a $9.0-\mathrm{m} \times 10.0-\mathrm{m}$ room in a school has a total mass of $13,600 \mathrm{kg} .$ The roof is to be supported by vertical wooden $" 2 \times 4 \mathrm{s}^{\prime \prime}$ (actually about 4.0 $\mathrm{cm} \times 9.0 \mathrm{cm} )$ 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 $12 .$

The roof over a $9.0-\mathrm{m} \times 10.0-\mathrm{m}$ room in a school has a total
mass of $13,600 \mathrm{kg} .$ The roof is to be supported by vertical wooden $" 2 \times 4 \mathrm{s}^{\prime \prime}$ (actually about 4.0 $\mathrm{cm} \times 9.0 \mathrm{cm} )$ 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 $12 .$

Key Concepts

Compressive Stress and Material Strength

Compressive stress refers to the internal forces per unit area within a material when it is subjected to compressive loads. In the context of vertical supports, it is essential that the material's compressive strength exceeds the maximum expected load (after accounting for the safety factor) to prevent failure. This concept is fundamental in material selection and design, ensuring that each support can safely handle the compressive forces from the structure.

Structural Support Spacing and Layout

Determining the proper spacing of support elements is crucial in structural design to achieve uniform load distribution and maintain structural integrity. The spacing is calculated based on the total load, the number of supports, and their individual load capacities. Proper layout minimizes the risk of local overloading and ensures that the loads are effectively transferred from the roof to the foundations.

Safety Factor in Structural Design

The safety factor is a critical design parameter that provides a margin of safety beyond the calculated load-bearing requirements. It is used to accommodate uncertainties in load estimations, material imperfections, and future degradation, ensuring that the structure can support loads significantly higher than the expected maximum. A safety factor of 12, as mentioned in the problem, means that the supports must be designed to withstand forces 12 times greater than the actual load.

Load Distribution and Equilibrium

This concept involves understanding how forces (such as weight) are distributed evenly across multiple supports in a structure. In structural engineering, the total load from a distributed mass, like a roof, must be allocated among the supports such that the sum of the forces from each support equals the applied load. This ensures that no single support is overloaded, which is fundamental in the analysis of static equilibrium.

Transcript

00:02 Force so we know that stress is equal to the force which is here the load force over area and here area means area and this is equal to compressive strength over the safety factor and since the load force only comes from the weight of the roof.

00:41 So we substitute weight here in place of force.

00:46 So weight is mass times gravity and area is basically the multiplication of the dimensions which are zero point.

01:00 Let's just write it area for now.

01:02 We can substitute the values later and okay one more thing that area is area of all the supports which means that the total area will be equal to number of supports let's call it it's denoted by this hash sign times area for one support so this is the total area that is equal to the term here this means that number of supports is equal to the weight which is m g times 60 factor over compressive strength times unit a sorry area per support which is equal to a 1 now let's plug the values into this equation so we have m g which is mass times gravity so mass is 1 .36 time 10 to the power 4 kg gravity is 9 .8 meter per second square safety factor is given to be 12 here the compressive strength is 35 times 10 to the power 6 newton per meter square and then we have area for one support which is multiple application of the dimensions which is equal to 0 .04 meter times 0 .09 meter.

02:56 Make sure you convert the units into meters to be consistent with the units.

03:03 Now we solve this to find the number of supports which comes out to be 12 .69.

03:12 Now this is strange because a number should not be a fraction or a there should not be any decimal point here.

03:23 So since there are these are more than 12 supports and to have since there are to be more than 12 supports and to have the same same number of supports on each side, so there will be 14 supports.

03:53 This means that there are seven supports on each side...

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