In the previous pages, we have explained the mechanics of materials related to beams.
This time, we will introduce some typical second moment of area values necessary for determining the deflection of a beam.
Furthermore, simply listing formulas is not helpful, so I will explain them using examples I have actually encountered.
I think it could be useful as a quick reference guide during the design process, so please feel free to use it if you find it helpful.
However, while the ability to calculate the second moment of area solely through manual calculation isn't essential, not understanding its meaning is crucial.
If you've forgotten or are seeing this for the first time, please refer to this to understand the meaning.
Typical second moment of area of a cross-section
From here, we will introduce the second moment of area of some representative cross-sections. By combining the cross-sections we will introduce, you should be able to cover 99% of the cross-sections actually used.
square cross section
The second moment of area I of a rectangular cross-section with width b and height h.
$ Area A=bh $
The second moment of area I = \frac{bh^3}{12}
This is the basic form, and everything starts from here.
triangular cross section
The second moment of area of a triangular cross-section with height h and base b.
$ Area A=\frac{bh}{2} $
The second moment of area I = \frac{b}{h^3}36
Triangular cross-sections are not used on their own, but rather in combination with other shapes.
Circular cross-section (cross-section of the axis)
The second moment of area I of a circular cross-section with diameter d
$ Area A=πd^2 $
The second moment of area I = πd⁴/6⁴
A typical example of its use is on a load-bearing shaft.
In the case of engines, which is my area of expertise, there are many such parts, including piston pins and crank pins.
Hollow circular cross-section (cross-section of a hollow shaft)
The second moment of area I of a hollow circular cross-section with an outer diameter of d1 and an inner diameter (hollow diameter) of d2.
Area A = π(d2^2 - πd1^2)
The second moment of area I = \frac{π(d2^4-d1^4)}{64}
In fact, most machine shafts that need to be as lightweight as possible are hollow shafts. I've hardly ever seen a shaft made of just a simple round bar.
Rhombus cross-section (square bars such as aluminum booklets)
The second moment of area I of a rhombic cross-section where all sides have equal length a.
Area A = a^2
The second moment of area I = a⁴/1²
This is just a piece of lumber that you can easily buy at a home improvement store. In mechanical design, the shape of the rib tip is made rhombic to increase the second moment of area.
As I will explain later, when making parts by casting, this method results in a shape that is very moldable and the mold lasts a long time.
I-shaped cross section (perhaps common in architecture)
The second moment of area I of an I-shaped cross section with thickness b, thinness t, total height h, and height c of the thickest part.
$ Area A=hb-(bt)(h-2c) $
The second moment of area I = \frac{bh^3}{12} - \frac{(bt)(h-2c)^2}{12}
This is another type of lumber commonly sold at home improvement stores. In actual mechanical design, it's used for things like car frames. In engines, my area of expertise, it's sometimes used as the cross-sectional shape for connecting rods. It's a cross-section that's generally used for rigid members.
Double-cut circular cross-section (for joints and keys)
It's difficult to explain in words, but the second moment of area I of a cross section with radius d and notch angle α (please see the diagram)
$ Area A=a^2π-a^2(α-sinα) $
The second moment of area I = a⁴/2⁴(6π-12α+8sin 2α-sin 4α)
This is a very important cross-section.
Its primary use is as a coupling when connecting two shafts (Oldham coupling). This is a fundamental mechanism for transmitting rotational force.
This cross-section is an important component used in various mechanical elements besides joints, such as shaft locks.
Important mechanical elements will be introduced on a separate, specialized page.
Regular hexagonal cross-section (honeycomb structure)
The second moment of area I of a regular hexagonal cross-section with side length a.
$ Area A=\frac{3\sqrt{3}}{2}t^2 $
The second moment of area I = \frac{5\sqrt{3}}{16}a^4
This is a cross-section of the famous honeycomb structure. I often use this cross-sectional shape for ribs where a large thickness can be achieved.
When using this shape for casting, it can be difficult to accurately form narrow ribs, so careful consideration is needed before using it. Otherwise, it will just end up as a simple rectangular cross-section with rounded corners, rendering it useless.
Summary
I think this method can be used to determine the second moment of area for most cross-sections, except for very unusual ones.
Even seemingly complex cross-sectional shapes, when broken down piece by piece, are almost always combinations of the shapes described so far.
If you know the combination, and you know the area and second moment of area of each part, you can use the parallel axis theorem to add or subtract them to find even complex cross-sections.
In reality, the axis passing through the centroid is often not the neutral point of the beam, but this can be easily determined using the parallel axis theorem.
Those with time on their hands, especially students, should try solving the problem themselves using the parallel axis theorem; this should deepen their understanding.
These days, simulations are often used to determine the optimal cross-section, but since humans are the only ones who can judge what kind of cross-section is best during the layout and sketching stages, it's still worthwhile to know.
Next time, I will introduce a quick reference chart for typical needle deflections.
To those who found this article helpful in understanding design:
While there is basically no textbook covering this content, and it is my own original work, I will introduce the textbook that I have been using since I was a student.
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