A beam is a supporting element of a building structure with different support conditions, most often it is supported at two points. In private construction, wood and metal are most often used as beams, less often reinforced concrete beams.
Beams act as the basis of ceilings (floor, ceiling, balconies) and roofs, and of course, every owner of his house wants any such structure in his house to be reliable and durable.
I have a very good friend who has been working as a carpenter for four decades, who constantly recommends installing beams that have a cross-sectional height of √2 times the width. How so and what is this, at first glance, a new rule ?!
Of course not, this is far from a new rule, it is applied everywhere and let's take a closer look ...
All of us at least once, but have heard from builders that the maximum strength of a beam is obtained if one rule is observed: the optimal transverse the section of a rectangular beam should be made up of the aspect ratio of 7: 5 - professionals in their field say that such a beam has the maximum durability. But is it?
There is nothing complicated here, and in order to understand this, you need to remember the basics of physics. The strength of any beam directly depends on its cross section and is calculated by the formula: K * A * H², in which A and H are the width and height of the beam, respectively, and TO - coefficient taking into account the length of the beam and the material.
For example, we have a need to get a wooden beam from a round log that would have the best bearing capacity.
This carpenter drew a rectangle for me, in which the diagonal is equal to the diameter of the log:
Then there will be some mathematical calculations, they can be skipped to the "Conclusion" section.
The cross-section of the beam is divided by the diagonal into two right-angled triangles, in which the leg AC (height) is calculated as follows by the Pythagorean theorem:
AC² = AB² - BC², and accordingly AC = √ (4R²-X²).
Now, let's substitute this into the above strength formula for strength:
Strength = k * X * (4R²-X²)
I used my school knowledge and, having opened the brackets, depicted this very function of strength in the form of a graph of a function on a coordinate grid:
The graph shows us how the strength of the beam structure changes depending on the size of the diagonal and the width of the beam (X or leg BC).
And now we need to find the projection of the peak point of the graph on the axis, this is done using our favorite derivative, which is expressed by the limit of the ratio of the function increment to the argument increment.
We find X, at the value of which our derivative of the function would vanish:
X =2R√3 / 3
Knowing the width of the beam (X) at the peak of the strength function, we find the height of the beam by substituting the value into the Pythagorean formula:
AC = √ (4R²-X²). Substitute X and get:
h = 2R√6 / 3
Conclusion
Look, our beam width turned out to be 2R√3 / 3, and the height of this beam is 2R√6 / 3. If we divide one by the other, then we get the ratio of exactly √2 and this value of the ratio of the two sides of the beam characterizes the highest point on the strength graph!
In other words, the beam with the maximum strength must have a cross-section in which its height is √2 times greater than its width.
And what does the aspect ratio of 7: 5 have to do with it? Given that the square root of two, this is a simple mathematical fraction 7/5. It's just that the √2 value is easier to operate on than calculating the 5th and 7th parts.
I believe that every builder working with lumber should have an idea of where this aspect ratio comes from!
The ratio of 7: 5 have beams:
Thank you for your time and I hope it was interesting!