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

step 1 This time we have a roof. We have a roof that's rectangular. And, you know, assume that the dist

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

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

Load Distribution Analysis

This concept involves analyzing how loads are distributed throughout a structure, from the applied loads at the roof or flooring to the forces experienced by each support element. By conducting a load distribution analysis, engineers can design support systems that adequately handle stresses and prevent localized failures.

Support Spacing

Support spacing involves determining the optimal distance between structural supports to ensure that loads are distributed effectively while minimizing material usage and ensuring safety. Proper spacing is crucial to avoid excessive load on any single support and to maintain the structural integrity of the entire system.

Safety Factor

A safety factor is a design parameter that provides a margin of safety in engineering calculations. It accounts for uncertainties in material properties, loading conditions, and potential flaws in construction, ensuring that structures remain safe under loads that exceed the minimum calculated values.

Compressive Stress

Compressive stress refers to the force per unit area experienced by a material subjected to a load that tends to shorten or compress it. In the context of vertical supports, it is essential to calculate the stress to ensure that the material’s capacity is not exceeded, preventing failure or buckling under the load.

Uniformly Distributed Load

This concept involves spreading a load evenly across a surface or a set of supports. In structural engineering, understanding how a load from a roof or floor is uniformly distributed helps determine the load that each individual support must carry, which is crucial for safe and efficient design.

Transcript

00:01 This time we have a roof.

00:05 We have a roof that's rectangular.

00:11 And, you know, assume that the distance, the length of the roof happens to be given us 10 meters.

00:24 The longer side is 10 meters.

00:28 This side is 10 meters.

00:33 This side right here happens to be 10 meters.

00:36 And then the short side, the width, is equivalent to 9 .0 meters.

00:45 And so the roof itself is supported by columns.

00:54 And these columns are spread out.

01:02 So these columns are completely spread out.

01:06 And each one of them is a 2 by 4.

01:10 Each one of these columns is what we call a 2 by 4, but that simply means that they happen to be 4 centimeters by 9 centimeters.

01:25 We can always transform these units into meters right away by saying if this is 4 centimeters, we multiply it by 1 of 100 centimeters.

01:37 And then this one gives me 0 .04 meters and the same thing is going to happen to the second unit where we take the 9 centimeters and multiply it by 1 meter over 100 centimeters, unit conversion, and that gives us 0 .09 meters.

02:00 So that's the information we've given.

02:03 And the other piece of information is that the safety factor for each one of these columns is 12 units and the weight itself or the mass itself of the roof m is equivalent to 13 ,600 kilograms.

02:30 That's pretty heavy.

02:32 So given that piece of information, we want to find the number of supports so and you know how many supports are these if we count them spread out that's the first thing we want to find out and then the second thing we need to find out is this gap right here in between the supports so you know we can call that gap we can call it x so those are the two pieces of information we're looking for.

03:18 If we're going to find the number of supports, we use n is equivalent to wn, capital, n, of a times sigma c, and we need to define all these units.

03:38 W is the weight of the roof, mg.

03:44 N is 12.

03:46 That's the safety factor.

03:49 And then the a is the area of the supports.

03:54 Remember, each one of them has a length of 4 centimeters and a width of 9 centimeters.

04:03 So that area in mid -squared will be 0 .04.

04:09 Meters times 0 .09 meters.

04:16 That's the area of itself and we also have the compressive force.

04:25 You know, this is the weight...