Determining deflection lines is one of the key steps in structural design. Whether you are a student preparing for a strength of materials colloquium or an engineer verifying the stiffness of a component, you need to know how a beam „works” under load.

In this article, we'll walk the full path: from classical theory Integration of the differential equation of the deflection line, through practical drawing of rotation angle diagrams, to examples for simply supported beam and bracket

And if you value your time and want to avoid tedious hand calculations, at the end of the article you will find my proprietary beam calculator, which will perform these operations for you in a few seconds. Let's get started!

1. Theoretical basis: Differential equation of the deflection line
2. Analytical method: Step-by-step integration
3. Example 1 : Freely supported beam
4. Example 2: cantilever beam
5. Quick verification: Use the online beam calculator

1 Theory and Methodology: Where does deflection come from?

Calculation of beam deflection is one of the most important elements of checking the Serviceability Limit State (SGU). In order to understand this process, we need to go to the foundation of the differential equation of the beam deflection line.

The basic relationship that binds bending moment vs. deflection, is described by the formula:

Where:

• $EI$- that bending stiffness of the beam (E - Young's modulus, I - moment of inertia of the section).
• $y”(x)$ - is the second derivative of deflection (curvature).
• $M(x)$ - is a function of the bending moment in a given section.

We use analytical method for calculating deflections involves scaling this equation twice, which allows you to go from internal forces to the actual deformation of the component.

2. integrating the deflection line equation step by step

This part of beam deflection calculations generally causes the most problems because of the integrals, which are not liked by everyone. In the case of moment equations for beams, they are generally simple functions to integrate, so there is nothing to worry about. Integration of the deflection line equation We perform in two stages:

1. First integration: Allows you to get a function of the angles of rotation of the sections $\theta(x)$
2. Second integration: Allows you to determine the deflection function $y(x)$, which is the deflection line being sought.

During the calculation, integration constants C1 and C2 appear. To determine them, we need to define the so-called. initial conditions, resulting from the way the beam is supported (e.g., no deflection at the support).

3. example 1: A simply supported beam - deflection and calculations

This is the most common case in the construction industry. A simply supported beam and its deflection with concentrated or uniform load is a classic of exam tasks.

Below you will find an example solution for calculating the deflection of the beam. For the example solutions I used beam calculator Which I recommend to you.

In our example, there is a 2m-long beam supported at two ends and loaded with a continuous load q and a concentrated force F.

To calculate the deflection we will need the bending moment, so first we need to solve the beam by determining the reaction and bending moment equations. For more on this, see this entry.

Calculation of reactions in supports.

Calculation of bending moment in compartments.

Once we have determined the equations for the bending moment for our example beam, we can move on to integration and from determining the deflection line. To do this, we use the equations for the bending moment for each compartment calculated in the previous step and integrate them twice. The first equation gives us the solution for the angle of deflection the second for the deflection. And in this way we get 4 integration constants C1, C2, C3, C4. two constants for each compartment.

In the next step, we need to calculate the integration constants. To do this, we will use the initial conditions. The rule is that we need as many conditions as there are integration constants- In our case 4. At the locations of the supports, we are sure that there will be no deflection of the beam, so y(x) at the locations of the supports is taken equal to zero. In addition, we know that at the junction of two compartments we must have continuity of deflection and deflection angle, so we have two additional equations.

Having the boundary conditions, we proceed to calculate the integration constants, by substituting the appropriate values into the equations. This is now pure mathematics.

The results obtained for the integration constants

After substituting the integration constants into the equations, we get the final form of the equations:

Once we have the equations in this form, substituting in place of x numbers in the range of 0 to 2 m, we get the deflection arrow and deflection angle of our beam along its length and can draw them in the form of a graph.

4. example 2: cantilever beam - deflection and calculations

In the case of bracket, deflection calculations They look slightly different because of the restraint. W cantilever beam The greatest deflection and the largest angle of rotation occur at the free end itself.

Below you will find an example solution for calculating the deflection of the beam. For the example solutions I used beam calculator Which I recommend to you.

In our example, there is a beam of length L restrained on the left side at point A and loaded with a concentrated force F=5qL at the other end.

To calculate the deflection we will need the bending moment, so first we need to solve the beam by determining the reaction and bending moment equations. For more on this, see this entry.

Calculation of reactions in supports.

Calculation of bending moment in compartments.

Once we have determined the equations for the bending moment for our example beam, we can move on to integration and from determining the deflection line. To do this, we use the equation for the bending moment and integrate it twice. The first equation gives us the solution for the angle of deflection the second for the deflection. And in this way we get 2 integration constants C1 and C2.

In the next step, we need to calculate the integration constants. Key to this scheme is that at the point of restraint, both the deflection and the angle of rotation are zero. With this rotation and deflection angle diagram starts from zero values at the wall and rises sharply toward the end of the beam.

After substituting the integration constants into the equations, we get the final form of the equations:

Once we have the equations in this form, substituting in place of x the numbers from 0 to L will give us the deflection arrow and deflection angle of our beam along its length, and we can draw them in graph form.

5 Check your results: Online beam calculator

Manual integration of bending moment is one of the most important skills in studying strength of materials. It's how you really begin to understand how a beam works. The problem is that in more complex tasks, it's very easy to make a small mistake - a mark by the moment, a poorly written boundary condition or a mistake in integration.

That's why I created a beam deflection calculator that allows you to quickly verify your calculations - especially with more difficult examples.

This tool can help you:

Check your results
Compare the solution calculated by hand with the result from the computational model and make sure that your integrals and boundary conditions are correct.

Better understand the behavior of the beam
Automatically generated plots of shear forces, bending moments and deflection lines help you see what's really going on in the structure.

Deal with more difficult tasks
Support for various load and support schemes makes the tool great for more challenging examples from homework or projects.

If you want to make sure that your calculations are correct -. check them out in seconds.

Don't let one small mistake in integration ruin the whole task. Verify your calculations and learn strength of materials much faster.