In this post we want to share our knowledge in structural engineering and make a simple pre-dimensioning introduction available to anyone interested in building his/her own structure but is too afraid of the technical knowledge involved.
We are going to show how actually is all a matter of controlling only five parameters easily understandable by anyone:
- Length (of the beam): L
- Height (of the beam): h
- Width (of the beam): b
- Area Load (per squared meter): q
- Separation (to the neighbor beam): s
The main concept in structural design is the notion of tension, or force applied per unit area, commonly called sigma in the jargon. Tension occurs when we stretch a cable, squeeze a can, bend a stick or, in general whenever we apply a load to an object.
The origin of this tension in a beam is normally due to the effect of loads, and generally the least favorable situation for it to happen is when the beam is simply supported in its ends. In such case the tension reaches its maximum values in the center of the beam (half of the length) and all of it is assumed to be due to the bending stress. There are also shear, axial and torsion stresses, but we want to keep it simple…
So, notion of tension-stress being introduced (as you can see we are all familiar with it), we can go ahead and introduce the idea of characteristic tension of a material, i.e. the amount of tension a material can withstand before breaking. In posh structural jargon it is know as ultimate tensile strength.
There is not much to say about it without beginning a dissertation: it is generally given in MegaPascales (MPa), and each material has its particular range of values:
- Wood: 35-50 MPa
- Bamboo: 30-100 Mpa
- Steel: 250-400 Mpa
- Concrete (without reinforcement): 10-40 Mpa.
These values are normally uncertain unless for steel and generally it is needed to verify before construction, not to mention that they are different depending on the type of stress, but that also is another story for another occasion..
Now, to get to the point, let us examine the easiest possible example and try to understand those five parameters we mentioned before.
In the figure above, we have represented a floor leaning over two beams that has a superficial load applied with a value q (in kg per squared meter). The separation between beams is the distance s (in meters), and each beam has dimensions length, height, width (L x h x b), all of them in meters.
As our example is a simple one, the section of the beam is a square, so its moment of inertia, or the amount of resistance its area opposes to rotate about its center, can be calculated as:
If we don’t do this dimensionality would be violated and we would have a big mess. Anyway, it is very very simple to do:
As we introduced before, our simply supported beam is only affected by bending stresses. For our analysis of whether it will hold or not the load q given its dimensions L, b and h, we only need to know the maximum value of the bending moment M, which is the one that will affect the maximum value of bending stress. Luckily for us, for its computation there exists the simple formula:
…And now we have all the ingredients to obtain the maximum stress in the beam due to the bending efforts:
With this formula we can estimate the maximum value of tension of any given beam of rectangular shape b x h and length L, simply supported on its extremes under a load q…