Deflection, in structural engineering terms, means the movement of a beam or node from its original position. It happens due to the forces and loads being applied to the body. Deflection also referred to as displacement, which can occur from externally applied loads or from the weight of the body structure itself. It can occur in beams, trusses, frames and basically any other body structure. In this article, we will discuss the beam deflection formula with examples. Let us learn it!
If you want to learn more, please visit our website OLIVET.
Deflection is the degree to which a particular structural element can be displaced with the help of a considerable amount of load. It can also be referred to as the angle or distance. The distance of deflection of a member under a load is directly related to the slope of the deflected shape of the body under that load. It can be computed by integrating the function which is used to describe the slope of the member under that load.
The Beam is a long piece of a body that is capable to hold the load by resisting the bending. The deflection of the beam towards in a particular direction when force is applied to it is known as Beam deflection.
The beam can be bent or moved away from its original position. This distance at each point along the member is the representation of the deflection. There are mainly four variables, which can determine the magnitude of the beam deflections. These include:
There is a variety of range of beam deflection equations that can be used to calculate a basic value for deflection in different types of beams. Generally, we calculate deflection by taking the double integral of the Bending Moment Equation means M(x) divided by the product of E and I (i.e. Young's Modulus and Moment of Inertia).
The unit of deflection, or displacement, will be a length unit and normally we measure it in a millimetre. This number defines the distance in which the beam can be deflected from its original position.
Cantilever beams are the special types of beams that are constrained by only one given support. These types of objects would naturally deflect more due to having support at one end only. To calculate the deflection of the cantilever beam we can use the below equation:
D= \( \frac{WL^3}{3EI} \)
Where,
D Beam deflection W Force at one end L Length of beam E Young's Modulus I Moment of InertiaQ: Calculate the deflection of a cantilever beam of length 2 meter which has support at one end only. Young's modulus of the metal is \( 200\times 10^9\) and the moment of inertia is 50 Kg m². At the end force applied is 300 N.
Solution:
Given values are,
E= \( 200\times 10^9 Nm^{-2} \)
I = 50 kgm²
L = 2 m
W = 300 N
Now applying the formula,
D= \(\frac{WL^3}{3EI}\)
Substituting the values,
D = \( \frac{300 \times 2^3}{3\times 200 \times 50} \)
D = \( \frac{}{} \)
So,
D = 0.08 m
Therefore value of beam deflection will be 0.08 m i.e. 8 cm.
Share with friends
What is deflection? Deflection, in structural engineering terms, refers to the movement of a beam or node from its original position due to the forces and loads being applied to the member. It is also known as displacement and can occur from externally applied loads or from the weight of the structure itself, and the force of gravity to which this applies.
Deflection can occur in beams, trusses, frames, and basically any other structure. To define deflection, let's take a simple cantilevered beam deflection that has a person with weight (W) standing at the end:
The force of this person standing at the end will cause the beam to bend and deflect from its natural position. In below diagram, the blue beam is the original position, and the dotted line simulates the cantilever beam deflection:
As you can see, the beam has bent or moved away from the original position. This distance at each point along the member is the meaning or definition of deflection.
For more information, please visit Beam Rotators.
There are generally 4 main variables that determine how much beam deflections. These include:
The unit of deflection, or displacement, is a length unit and is normally taken as mm (for metric) and in (for imperial). This number defines the distance in which the beam has deflected from the original position. Since deflection is a short distance measurement (beams should generally only deflect by small amounts), it is commonly expressed in units of length, such as inches or millimeters.
In addition to this, there are other parameters used to measure deflection such as angles, but these units such as radians or degrees are not commonly used. Additionally, it is important to note that the allowable deflection for a structure is often specified as a percentage of the span length (L/X), this is a measure that allows comparison between different beams regardless of their length and its unitless.
The equations are easy to apply and allow engineers to make simple and quick calculations for deflection. If you're unsure about what deflection actually is, click here for a deflection definition Below is a concise table that shows how to calculate the maximum deflection in a beam. Don't want to hand calculate these? SkyCiv offers a free beam deflection calculator to help with your needs! For more powerful structural analysis software, sign up for a free SkyCiv Account and get instant access to all the free versions of our cloud structural analysis software!
Use the below formula to calculate the max displacement in beams. Click the 'check answer' button to open up our free beam calculator. These beam displacement equations are perfect for quick hand calculations and quick designs.
The deflection of a beam is calculated based on a variety of factors, including materials, the moment of inertia of a section, the force applied, and the distance from support. These can be simplified into simple deflection formula for quick back of the envelope calculations. Here is a simple table of the different loading scenarios and their corresponding max beam deflection formulas:
Cantilever beams are special types of beams that are constrained by only one support, as seen in the above example. These members would naturally deflect more as they are only supported at one end.
It is important to consider that the deflection of a cantilever beam is also affected by the position of the load and the location of the fixed support. The farther the load is from the fixed support, the greater the deflection will be. And also, the longer the cantilever the greater the deflection will be, due to the fact that the load is applied to a longer distance.
It's also important to note that there is a maximum allowable deflection for structures, this value is usually established by building codes and standards, it varies with the type of structure and the purpose of the structure, and the deflection calculated should be within this limit to ensure safety and functionality of the structure.
Reference Max Deflection BMD Shape
Another example of deflection is the deflection of a simply supported beam. These beams are supported at both ends, so the deflection of a beam is generally left and follows a much different shape from that of the cantilever. Under a uniform distributed load (for instance the self-weight), the beam will deflect smoothly and toward the midpoint.
Reference Max Deflection BMD Shape
Reference Max Deflection BMD Shape
Let's consider a simple supported beam with a span of L = 10 m, a uniform load of w = 10,000 N/m, and the following material properties: Young's modulus, E = 200 GPa, the moment of inertia, I = 0. m^4.
The deflection of the beam can be calculated using the equation, taken from our above table of equations:
\(δ = (5wl^4)/(384EI) \)
\(δ = (5 * 10,000 * 10^4)/(384 * 200 *10^9 * 0.) \)
\(δ = 0. m = 4.34mm \)
So the deflection of the beam is 0. m or 4.34 mm. It's always important as an engineer to verify your result, so let's plug the same numbers into SkyCiv's Free Beam Calculator:
It's worth mentioning that this is a very simple example and in practice, there are various other factors that need to be considered like the effects of temperature change, live load, self-weight and many other things, in a real life scenario, a structural engineer would take these factors into account while calculating the deflection of a beam.
SkyCiv Beam Analysis Software allows users to analyze beam structures easily and accurately. You can get an analysis of your beam member, including reactions, shear force, bending moment, deflection, and stresses in a matter of seconds.
If you want to give it a try first, our FREE Beam Deflection Calculator is a great way to start, or simply sign up for free today!
Contact us to discuss your requirements of Benchtop Welding Positioner. Our experienced sales team can help you identify the options that best suit your needs.