As explained in the previous lesson on circular stress and trusses, we will now begin explaining beams.
In mechanical design, beam analysis is one of the most important aspects and is frequently used.
It also represents a major event in the early to middle stages of studying mechanics of materials.
Many new concepts will be introduced, but you should be able to understand them if you have a solid grasp of the explanations given so far.
Please do your best to keep up.
What is a beam?
First, let me explain what a beam is. In Japanese houses, beams are found, and in mechanical design, ribs are sometimes treated as beams.
They can also be found on seesaws in parks and all sorts of other structures.
First, a typical beam isA cantilevered beam that supports a rod on one side.It is.
Imagine it bending (flexing) as shown in the following diagram.
Next is a typical exampleA double-supported beam that supports both ends of the rod.It is.
If you imagine this, I think it would be easy to see how it could bend and dent in the middle.
There are other types of beams that have specific names, but I want you to first thoroughly understand these two basic ones.
Many people might think this is all obvious, but when designing structures, these two patterns become intricately intertwined, making it difficult to understand.
Conversely,No matter how complex a structure is, if you carefully break it down piece by piece, it can almost always be divided into the two patterns I've described.
By considering these two decomposed patterns, we can gain a good understanding of the stress distribution and deformation of many structures.
Next, let's look at the relationship between the external and internal forces on a beam.
Shear force and bending moment acting on a beam
From here, we will consider shear force and bending moment by looking at a simple example of a cantilever beam.
An external force is applied to the tip of a cantilevered beam, with load P. The coordinate system is set with the tip of the cantilevered beam as the origin, the parallel direction is x, and the vertical direction is y. The directions are as shown in the diagram.
Now let's consider the case where the beam is cut at an arbitrary position x. A force acts on the cut surface due to the balance with the external forces.The force acting on the cross-section is horizontal to the cross-section, so it becomes a shear force.
Next, consider the moment at the same arbitrary position x as above.A moment is generated in the cut cross-section that counteracts the moment generated by the load P on the beam (the beam will deflect). (It does not rotate.) This is called the bending moment.
As shown in the example, shear force and bending moment are generated within any cross-section of a simple cantilevered beam.
Before we delve into the specifics of the shear force and bending moment that occur, let me explain one important point.
The important thing is the rule regarding the signs of shear force and bending moment that occur in any given cross-section.
Rules for the signs of shear force and bending moment occurring in the cross-section of a beam (Extremely important point!!)
In a previous article, I mentioned that the sign of shear force is basically positive. In physics, a moment is defined as positive when it is clockwise.
Let's explain the rules within the cross-section of a beam, which differ from the common sense described above.
First, the sign of the shear force generated in the cross-section of the beam is considered positive if the force is generated in a clockwise direction.
Since it's difficult to understand from words alone, let's illustrate it with a diagram.
The sign of the bending moment generated in the cross-section of the beam is considered positive when the cross-section is concave upwards.
Let's also illustrate the bending moment with a diagram.
This is a very important symbol, but it's different from common sense in physics and is complicated, so it's easy to make mistakes. Make sure you understand it thoroughly. I often make mistakes too.
I know I'm being persistent, but remember that shear force is positive when clockwise, and bending moment is positive when concave upwards.
Since we're at it, let's illustrate the signs of shear force and bending moment at an arbitrary cross-section using a cantilever beam as an example.
I believe I emphasized the importance of the sign in my explanation of stress.In the world of engineering, including physics, this correspondence is extremely important.
In some cases, having a good match in sign is often more important than having a specific value.During my student days, I underestimated the importance of symbols and failed many courses. Conversely, I remember seeing others making the same mistakes after becoming a designer, and it was quite a sight to behold.
This sense of correspondence is very important, so make sure you develop it.
Now that we have all the definitions, let's set up the equations relating forces.
Relationship between external and internal forces
Now, let's finally consider the shear force and bending moment of the cantilevered beam that we defined at the beginning.
First, let's consider the balance of forces.
Since the external force P generates a shear force Q, the following equation holds true. (Note the signs.)
Load Q = -P (shear force) Since the shear force is in the counterclockwise direction, -.
Next, let's consider the balance of moments in the cross-section.
The following equation holds true because it counteracts the moment generated by the external force, load P (pay attention to the sign).
$ Bending moment M = -Px (moment due to load) $ The moment is concave downwards -
Next, let's illustrate the shear force and bending moment defined above in a diagram.
First, here is a diagram of shear force.
The diagram above, which illustrates shear force, is called a shear force diagram (SFD). As you can see from the diagram, the shear force remains constant regardless of distance.
Next, let's look at the bending moment diagram.
A diagram illustrating bending moment is called a bending moment diagram (BMD). The diagram shows that the amount of bending moment changes with distance, reaching its maximum at the rightmost end of length l. The section where the bending moment is maximum is called a critical section, but I have never actually used this term in design practice.
While a simple beam like the example might seem obvious and you might not see the point in drawing a diagram, when complex loads are involved, bending stress can accumulate in unexpected places, so I recommend taking the time to draw a diagram.
We will explain more about deflection in detail later.The point of maximum bending moment is not necessarily where the deformation is greatest; rather, the point of maximum bending moment may be where the deformation is small or even zero.
This kind of intuition is important for design, so let's try to develop it.
Up to this point, we have explained using cantilevered beams, but next we will generalize a bit so that we can consider many other patterns.
Relationship between load, shear force, and bending moment
To generalize the relationship of forces in a beam, we set up the following example problem.
A small section dx is cut out of a certain beam, and an external force, a uniformly distributed load q(x) [N/mm] (a force similar to pressure), is applied to that section.
Let Q be the shear force in the left cross-section generated at that time, and let Q+dQ be the shear force in the right cross-section. Let M be the bending moment in the left cross-section and M+dM be the bending moment in the right cross-section.
Here, we've only tentatively placed the shear force and bending moment in a positive direction; the actual signs will be determined by calculation.
Let's look at the forces and moments acting on the cross-section of a section of the beam that has been cut off.
Here we set up the force relationships. Since we are looking at a general force equilibrium here, we will use the signs according to physics rules, not the beam sign rules.
$ (Q+dQ)-Q+q(x)dx=0 $
When you organize
$ \frac{dQ}{dx}=-q(x) $
In other words, differentiating the shear force Q with respect to distance x gives the uniformly distributed load -q(x). Put another way, the slope at which the shear force changes is the same as that of a uniformly distributed load.
Next, considering the balance of moments in the right cross-section, the following equation holds. Here, the sign also follows physics, with counterclockwise moments being positive.
The moment generated by q(x) is considered to occur when q(x)dx is in the middle of a small interval.
$ (M+dM)-M-Qdx-q(x)dx\frac{dx}{2}=0 $
Since dx and dx are multiplied by an infinitesimal amount, the result becomes even infinitesimally small, so we can consider it as 0 (for example, 0.01 × 0.01 = 0.0001).
$ \frac{dM}{dx}=Q $
Becomes
This means that differentiating the bending moment with respect to distance x gives the shear force Q. In other words, the slope at which the bending moment changes is the same as the slope at which the shear force Q changes.
I apologize for getting into the field of differential calculus in mathematics, but what we can say from $ \frac{dM}{dx}=Q$ is that when Q=0, it becomes the curvature value, that is, it is the point where the sign of the slope of the change in bending moment changes. Therefore, the bending moment is often at its maximum and minimum values at the coordinates where the shear force Q is 0.
This is something that's easy to overlook. For example, if you only look at the shear force in a simulation and it's 0, you might think everything is fine, but the bending moment could actually be at its maximum. Be careful, as this can be dangerous.
Since it might be difficult to understand from the text alone, let me explain using an example.
Solution example for a double-supported beam and maximum bending moment
Let's create an example.
A simple cantilevered beam has a length of l and is supported at both ends by platforms A and B. A uniformly distributed load q (N/mm^2) acts uniformly on the beam. (According to the law of action and reaction, reaction forces are generated at A and B.)
Here, the following equation holds true due to the balance of forces.
$ RA+RB=ql $
$ RA=RB=\frac{ql}{2} $
Next, looking at the beam cut at an arbitrary position x as explained earlier, it looks like this:
From this, we can derive the shear force Q (pay attention to the sign).
$ Q=RA-qx=q(\frac{l}{2}-x) $
Also, from the balance of moments in the right cross-section (pay attention to the sign)
$ M=RAx-qx\frac{x}{2}=\frac{q}{2}x(lx) $ (This is equivalent to integrating the shear force Q with respect to x)
When these are illustrated, SFD and BMD are as follows.
This shows that the bending moment is maximum when the shear force Q is 0. You can see that the bending moment is often maximum or minimum at the coordinate of the shear force Q mentioned above.
As will be explained in more detail when discussing deflection, a characteristic of beams is that the deformation is greatest at the point where the shear force is zero and the bending moment is at its maximum.
Conversely, areas where the deformation is zero are where the shear force is at its maximum, making them quite dangerous.
In other words, I will explain the details later.What is often said about high rigidity is that it doesn't deform much, but the shear force it generates is very high.
Being able to visualize the relationship between the maximum and minimum values of deformation, shear force, and bending moment in this way is extremely useful in practice, so try to grasp this concept by drawing SFD and BMD.
Critics often say that it's good because it's rigid, but that's wrong. The design that impresses professionals isA structure that deforms the members evenly to distribute shear force and bending moment as evenly as possible..
We only increase rigidity when dimensional changes would drastically alter performance.
A structure with unnecessarily high rigidity indicates either a low level of skill on the part of the designer or that they were too lazy to properly consider the design.
If you unnecessarily increase rigidity, the shear force will increase, so you need to increase the surface area to withstand it, which means it will become heavier. Of course, heavier weight will reduce performance, and to put it extremely, it will also increase cost. Balance is key.
It's important to grasp the concept and design accordingly.
Summary
I believe you now understand the basic relationship between external forces, shear force, and bending moment in a beam.
What's particularly important is that it's persistentIt matches.
This is something I absolutely want you to internalize as an intuition.
If you get the sign wrong, you'll end up with the opposite value to the true value when calculating the deformation amount, which will lead to disastrous results.
Furthermore, while the number of simulations will continue to increase, it is humans who will judge the results. Anyone can read numbers, but it is dangerous if you do not learn the meaning of the symbols, as you will not understand them otherwise.
Just like tension and compression in stress, a change in the sign of a beam can completely alter the behavior of the material.
I'll say it again: trendy AI and simulations only perform calculations; they don't give you the answer. It's you, a human, who has to judge the result. That's why the meaning of the calculations and the meaning of the correspondences are so important.
Next time, I will explain another very important property of cross-sections: the second moment of area.
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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