As mentioned at the end of the previous discussion on beam stress, we will now explain the second moment of area.
The second moment of area is a very important concept when designing structures in mechanical engineering.
It's a rare thing; despite being very important, you don't see many people who actually understand its meaning.
The second moment of area for many cross-sections is formalized, and you can find the value by looking at a table, so you can manage even with a shallow understanding.
However, if you don't understand the underlying principles when designing something new, you'll only be able to reuse past ideas.
I hope you will understand this if you aspire to be a designer who can create.
Furthermore, while many textbooks define the second moment of area before explaining the bending stress of a beam, my experience suggests that explaining it in reverse is more convincing.
Therefore, let's first understand the second moment of area by looking at the stress in the beam.
Bending stress of a beam
In the previous explanation of stress in beams, I believe you understood that shear force and bending moment occur inside a beam.
Let's consider this problem by setting up an example again.
Assume a load P is applied to the middle of a simple cantilevered beam. If we take a small section from the middle of the beam, a shear force Q and a bending moment M are generated. The cross-section is a suitable rectangle.
You should be able to understand up to this point. Now, let's set aside shear force and focus only on bending moment.
When the extracted minute section is further divided in the middle, stress is generated within the cross-section as shown in the following figure.
The stresses occurring within the cross-section are caused by bending moments, and therefore, as shown in the following diagram, tensile stress and compressive stress exist.
Imagine bending a pair of chopsticks; they should snap cleanly from the curved end (they won't break from the inside). That bending force is the bending moment, and the stress generated at the broken cross-section is shown in the diagram above. A rod breaks due to tensile stress. If you're curious, try breaking a pair of chopsticks, even though it might feel wasteful.
The stress generated by a bending moment in this way is called bending stress.
One of the important properties of bending stress is that the tensile and compressive bending stresses in a cross-section cancel each other out, resulting in zero stress. This is because the source of the stress is bending, so the tensile and compressive stresses are the same.
The point at which bending stress changes from tension to compression is called the neutral axis (which is a line in three dimensions), and the surface where no bending stress occurs is called the neutral plane.
Next, let's consider bending stress in more detail.
Bending stress, bending moment, and first moment of area, second moment of area
Let's set up an example to illustrate this more concretely.
The example problem will be the same as the bending stress example problem above. However, since we will be deriving the equation, coordinates and dimensions will be added to the diagram. As considered above, we will set aside the shear force Q and focus on the bending moment M.
The extracted small section is deformed (bends). Let ρ (rho) be the radius of curvature of this deformation (you can think of it as the radius). The radius indicated by the radius of curvature ρ is the distance to the central plane of the beam (the blue line in the diagram), and the radius at any coordinate y is ρ + y.
The coordinate system uses the center of the beam (neutral axis) as the origin, with x representing the rightward direction and y representing the radial direction relative to the curve.
First, let's consider the distortion in the x-direction of a surface MM at an arbitrary distance y.
The length of the surface MM after deformation is (ρ + y)dθ, and the length before deformation is dx as shown in the figure, so the strain εx is as follows.
$ Strain εx=\frac{(ρ+y)dθ-dx}{dx} $
The strain equation has three variables, y, dθ, and dx, which makes it difficult to work with, so we will consider ways to reduce the number of variables. We will target dx as the variable to reduce.
Here we consider the important assumption of a beam: "A cross section that was a plane before deformation remains a plane perpendicular to the axis after deformation" (Bernoulli-Navie).
The hypothetical wording is difficult to understand, but to put it another way, it means "even if it deforms, the position of the deformed surface does not change and we consider it as remaining a plane." In reality, when a beam bends, the surface curves, but we are supposed to consider it as remaining a plane.
The diagram below illustrates this.
From the diagram, it can be assumed that the pre-deformation dx is the same as the length of the surface NN, so we can write dx = ρdθ (surface NN is the neutral surface).
Rewriting the distortion equation, we get the following:
$ εx=\frac{(ρ+y)dθ-dx}{dx} =\frac{(ρ+y)dθ-ρdθ}{ρdθ}=\frac{y}{ρ} $
If you can find the bending stress σx from the strain εx,
Bending stress σx = E (elastic modulus) εx = \frac{E}{ρ}y
become.
Next, we will consider one of the important properties of a beam's cross-section: that the sum of the bending stresses within the cross-section equals zero, using the bending stress σx.
The sum of stresses in a cross-section can be expressed using integration as follows (let the cross-sectional area be A):
$ \int_{A}σxdA=\frac{E}{ρ}\int_{A}ydA =0 $
In this equation, $ \int_{A}ydA$ is called the first moment of area.
Since E and ρ in the equation are constants and not zero, a bending stress of zero means that the first moment of area is zero. Furthermore, if the coordinate origin of the beam's cross-section is taken as the neutral axis, the first moment of area becomes zero.
Next, let's consider the moment around the origin o in the cross-section. Since the moment due to bending stress and the bending moment M are equal, the following equation holds (after all, bending stress is generated by bending stress).
$ \int_{A}σxdA × y=\frac{E}{ρ}\int_{A}y^2dA=M $
Here, $ \int_{A}y^2dA$ in the equation is called the second moment of area. The second moment of area is a variable used to determine the moment due to bending stress.
Since it's cumbersome to always write the second moment of area as $ \int_{A}y^2dA$, we'll use Iz (meaning the z-axis direction of the neutral axis).
Rewrite the bending moment equation using Iz.
$ \frac{E}{ρ}Iz=M $
From this equation, if we eliminate E and ρ using the bending stress σx = E (elastic modulus) εx = \frac{E}{ρ}y$
$ σx=\frac{M}{Iz}y $
This equation shows that a very important characteristic of a beam is that the bending stress generated in the cross-section of the beam is determined by the bending moment and the second moment of area (it does not depend on the elastic modulus E or the radius of curvature ρ).
In other words, the bending stress within the cross-section of a beam is small when the second moment of area is large , and large when the second moment of area is small.
If we denote the upper and lower surfaces of the beam as h1 and h2 respectively, the upper surface is under compression and the lower surface is under tension, so the resulting stress can be expressed as follows.
$ σ1=-\frac{M}{Iz}h1, σ2=\frac{M}{Iz}h2 $
here
$ Z1=\frac{Iz}{h1}, Z2=\frac{Iz}{h2} $
Therefore, the bending stress can be expressed as follows.
$ σ1 = -\frac{M}{Z1} (compression), σ2 = \frac{M}{Z2} (tension) $
Z1 and Z2 are called section moduli.
This allows us to derive the first moment of area and the second moment of area.
Summary
Up to this point, we have derived the first moment of area and the second moment of area by utilizing the properties of the bending stress generated in the cross-section of the beam.
In my own way, I keep each of them in mind as follows:
The first moment of area is defined as "the sum of the bending stresses due to the bending moment acting on a small section of the cross- section of a deflected beam being zero, and the formula is first-order ."
The second moment of area is "the balance of bending stress moments due to bending moments acting on a small section of the cross- section of a deflected beam ; the formula is quadratic ."
The first moment of area and the second moment of area can be summarized as follows:
$ \int_{A}ydA$ is called the first moment of area.
- The first moment of area becomes 0 when the origin of the beam's cross-section is taken as the neutral axis.
$ \int_{A}y^2dA$ is called the second moment of area.
The bending stress in the cross-section of a beam can be determined by the bending moment and the second moment of area.
• A larger second moment of area results in lower bending stress; conversely, a smaller second moment of area results in higher bending stress.
While we're on the subject, the section modulus Z is also quite useful, so please be sure to remember it.
$ Z1=\frac{Iz}{h1}, Z2=\frac{Iz}{h2} $
One practical example of the effects of the second moment of area that I have experienced is when cracks appeared on the surface of a rib even though only a small load was applied to the structure.
The ribs had a small section modulus, so the reinforcement was completely ineffective.
Even if you add reinforcements like ribs with good intentions, if the section modulus is poor, it's not only pointless, but it can also lead to higher bending stresses than before, potentially triggering failure. Be careful.
Next time, we will explain how to actually calculate the second moment of area and its mathematical properties.
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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