Beam deflection explained for beginners (deflection curve, amount of deflection, and deflection angle)

I've explained beams over the past three installments. This time, as a culmination of that, I'd like to calculate the deflection of a beam.

I want to read it together

Shear force and bending moment (SFD, BMD) of beams explained for beginners As explained in the previous lesson on annular stress and trusses, we will now begin our discussion of beams. Beam analysis is one of the most important aspects of mechanical design and is frequently used. Furthermore, it is covered in the first half of material mechanics…

Determining the deflection of this beam is crucial, as without it, the deformation of the mechanical structure cannot be calculated.

While we have certainly explained the deformation of round bars due to twisting, tension, and compression, this alone is insufficient to determine the deformation of mechanical structures. This is because these explanations do not take into account the deformation caused by bending moments within the object.

Mechanical structures are composed of various shapes that are intricately intertwined, but when broken down piece by piece, they can be separated into combinations of several types of simple beams.

Therefore, if we know the deflection of the beams that will be explained here, we can find the amount of deformation of each beam and add them up to find the total deformation of the entire mechanical structure.

This process is often similar to what is done in computer-aided engineering (CAE) calculations.

I think you understand its importance by now. I will now explain further, and I would like you to understand it.

It also involves some mathematical techniques, but it's not difficult, so don't get confused by the terminology.

Deflection curve

Before even considering flexibility, you must first establish rules, regulations, and standards; nothing can begin without them.

First, let me explain the basic rules.

As usual, I'll provide an example problem.

Consider a simple cantilevered beam that is deformed by a load P applied in the middle. Let the coordinate system be x horizontally and y in the direction of deformation (downward).

Here the deformed beamThe neutral axis (center line) is the deflection curve or elastic curve.It is called.

Furthermore, at any locationThe downward displacement y is called deflection, and the angle at which the tangent to the deflection curve at that point intersects with the neutral axis of the original beam is called the deflection angle.

Here's something important to note, and I'll reiterate this point as always: in the diagram, the sign for deflection is positive when it's downwards (the direction of deformation), and the deflection angle is positive when it's clockwise.

This is very important.

These are the only rules, so please be sure to keep them in mind.

Next, we will determine the deflection and angle of deflection of the beam.

Find the deflection curve

First, please recall the example problem we just discussed.

Since the deflection cannot be determined all at once, we will use our usual method of cutting out a small section and considering it. This time, let's cut out the part indicated by the yellow dashed line in the diagram.

Here is the extracted diagram. Various symbols have already been added, but let's look at them one by one. The settings, including the coordinates, remain the same.

First, let's look at the symbol ρ (rho). This represents the radius of curvature. Simply put, the shape of the deformation is considered as an arc, and ρ represents the radius of that arc.

Next, regarding the bending moment M, this was explained in the previous-to-last lesson.

In the explanation so far, both bending moment M and shear stress have been generated inside the beam, but in the deflection curve, the deformation due to bending moment is dominant and the deformation due to shear force is small, so we will ignore it.

Later, I'll explain why deformation due to shear force is small and can be ignored, as there are cases where it cannot be ignored. For now, just assume it doesn't exist.

Next, let's consider the length ds between points NN'. Here, the length ds is an arc and can be found by multiplying the radius by the angle. In other words, it can be found by multiplying the radius of curvature ρ by the angle N-O'-N' (-dθ).

The reason why the angle has a negative sign is that as the length ds increases, the angle increases. When the length ds increases, if we observe the movement of the point, point N' moves further and further away from N.The angle increases counterclockwise. Yes, since we've decided that clockwise is positive, the angle dθ is negative.

It's a bit complicated, so please look at the diagram and visualize in your mind that points N' and T' are moving away from the original points N and T. If you do that, you should see that the angle N' - O' - N increases in a counterclockwise direction.

Let's put the diagram back together and start formulating the equation from here.

First, we set up an equation relating the length NN'(ds) to the angle dθ at point O'.

$ ds=ρ(-dθ),\frac{1}{ρ}=-\frac{dθ}{ds} $

On the other hand, the slope of the deflection curve (the slope of the neutral axis) is (the angle at which the tangent to the arc N-N' intersects the neutral axis before deformation).

$ \frac{dy}{dx}=tan(θ) $

Since the angle θ is quite small here, the following equation holds true.

$ ds = dx (since the angle is shallow, we can ignore dy), θ ≈ tan(θ) = \frac{dy}{dx} $

Therefore, the relationship between the curvature half-ρ and dx and dy is as follows:

$ \frac{1}{ρ}=-\frac{dθ}{ds} =-\frac{d}{dx}(\frac{dy}{dx}) $

$ \frac{1}{ρ} =-\frac{d^2y}{dx^2} $

Now, let's recall what the second moment of area is. The bending moment M and the radius of curvature ρ are related by the following equation.

Iz is the second moment of area, and E is the elastic modulus of the material.

$ \frac{E}{ρ}Iz=M, \frac{1}{ρ}=\frac{M}{EIz} $

Therefore, the equation for deflection is as follows:

$ \frac{d^2y}{dx^2}=-\frac{M}{EI} $

As defined initially, y is the amount of deflection and x is the coordinate, so y=f(x) represents the relationship between the amount of deflection, and therefore $ \frac{d^2y}{dx^2} is the equation obtained by differentiating y=f(x) twice.In other words, it becomes a differential equation for deflection (where I is the second moment of area along the z-axis).

The important point here is that the second moment of area is in the denominator; in other words, the larger the second moment of area, the greater the deflection, and therefore the smaller the angle of deflection.

While the input bending moment and the elastic modulus of the material are difficult for the designer to control, the second moment of area can be freely determined.

The challenge here lies in devising a shape that maximizes the second moment of area within a limited space.

For a review of the second moment of area, please click here.

I want to read it together

First and Second Moment of Area Explained for Beginners 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 is very…

Next, let's actually calculate the deflection and the angle of deflection.

Typical methods for solving deflection problems (multiple integral method)

Now we will actually calculate the deflection. First, let's solve it using the common multiple integral method.

The name "multiple integral method" sounds impressive, but it simply involves integrating twice back into the differential equation for deflection, which itself involves differentiating the deflection amount twice.

First, let's try integrating the differential equation for deflection, $ \frac{d^2y}{dx^2}=-\frac{M}{EI}$, with respect to x once.

$ \frac{dy}{dx}=θ=-\int\frac{M}{EI}dx+C1 (where C1 is the integration constant)$

This equation takes the form of the first derivative of the deflection amount y=f(x) and represents the slope of the deflection curve. Physically, the slope of the deflection curve represents the angle of deflection.

Yes, we can find the angle of deflection. Here, C1 is the integration constant, and it's a value that appears when the range of integration isn't defined.

Furthermore, if we integrate with respect to x once more (the integral of C1 with respect to x is C1x),

$ y=\intθdx=-\int\int\frac{M}{EI}dxdx+C1x+C2 $

Yes, this allows us to determine the amount of deflection. However, the integration constants C1 and C2 get in the way here. These are very important, so let me explain them.

The meaning of boundary conditions (the most basic concept of integration)

First, regarding the concept of integration, integrating a function f(x) with respect to x is equivalent to finding the area of ​​the following:

In reality, integration simply calculates the area from the values ​​dx and dy of an infinitesimal interval and adds them together.

You understand this, right?

Here's the problem: if we only have the relationship between dx and dy, we can't find the value of y when x=0.

Even if we know the relationship between two local intervals x and y, we cannot determine the absolute value or the actual value without a reference point (in many cases, the value at x=0 is used as the reference point).

The integration constant is a way to set C as a reference point if you don't know where to start. (This is a rough explanation that would probably get me in trouble with a math teacher.)

Conversely, if you know C, you can find the absolute value of the integral (although this depends on the type of integration constant, the most basic case is that the value of y at x=0 is the value of the integration constant).

That's right, unlike the world of mathematics, in the world of engineering, all you need to do is measure things.

Now, please recall the beam in the example problem.

You can see that both ends of the beam are physically fixed.No matter what you do, the angle of deflection and the amount of deflection will be zero at the edges.

Therefore, in the example, C1 and C2 become 0.

These integral constants C1, C2, etc., which are physically determined by differential equations in engineering, are called boundary conditions.

This is important (not just about deflection). However, deciding this is difficult..

Differential equations appear frequently not only in mechanical engineering but also in other fields such as electrical engineering.

All of these have boundary conditions, and if you get them wrong, everything goes wrong.

Even the great genius physicist Einstein added the ether term, which in engineering terms corresponds to boundary conditions, to his special theory of relativity. Later, he realized his mistake and said that the biggest regret of his life was this error in boundary conditions. That's how difficult and important it is.

Furthermore, the computer simulations I will explain later simply solve a large number of differential equations at high speed. The boundary conditions are entered by humans.

Therefore, no matter how advanced AI becomes, calculations cannot be performed without human judgment.

Even if boundary conditions are automatically entered, the AI ​​cannot determine whether they are correct.

Ultimately, it's all about the human element. If there's enough demand, I might offer another course in industrial mathematics. This isn't mathematics for exams, but for practical application.

Summary

Summary

At this point, you're almost ready to start design work.

After learning about how things fail, such as buckling and fatigue failure, you'll be able to modify drawings and design simple parts.

To be honest, I think it's more important to thoroughly understand the fundamentals of mechanics of materials than to study drafting.

Minor errors in drawing notation or rules can be corrected, but if the strength calculations using material mechanics are wrong, even if the drawings themselves are correct, they become worthless, leading to delays in the development plan, the need to redo the test plan, delays in the start of production, production stoppages, and if the error is discovered in the market, it could result in free replacement of the corrected product, or in the worst case, a recall.

This is an extreme example, but in the worst-case scenario, the impact could be this significant.

Incidentally, there is no strength calculation that does not use mechanics of materials, and furthermore, there are no (and should not be) machine design drawings that are created without using mechanics of materials.

That concludes this section.

Next time, we'll use the differential equation for beams to numerically solve for the deflection and angle of deflection of typical beams. If we have time, I'd also like to introduce two more solution methods.

I want to read it together

Beam deflection calculations for beginners: cantilever beams, double-sided cantilever beams In the previous lesson, we explained the basics of how to determine the deflection of a beam. This time, we will actually calculate the deflection of some typical beams. There are two basic types of beams: cantilever beams and simply supported beams (double-supported beams)...

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.

Basic Mechanics of Materials (Revised Edition)

created by Rinker

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