Name: beam load modeling with singularity functions
Uploaded: 2024-05-23T12:59:47.000Z
Duration: 1 min 18 s
Channel: JoVE

Consider a simply supported beam carrying a uniformly distributed load from its midpoint to its right-hand support.
To determine the bending moment in beams using singularity functions, the free-body diagram of the beam is drawn by replacing the distributed load with an equivalent concentrated one.
The moments are then summed about the right-hand support to obtain the total moment equation to determine the magnitude of the reaction force.
Next, the beam is cut at a point between the left-hand support and the mid-point. From its free-body diagram, the bending moment is expressed by a specific function by considering the interval.
Now, the beam is cut at a point between the mid-point and the right-hand support. The free-body diagram of this portion is drawn, replacing the distributed load with an equivalent concentrated load, and the bending moment is expressed by a different function by considering the interval.
These two functions are combined into a single expression representing the bending moment at any beam point. The expression within the angle brackets for different conditions is called the singularity function.

singulariteitsfuncties voor buigmoment

Singularity Functions for Bending Moment

<div>
 Singulariteitsfuncties vereenvoudigen de weergave van buigmomenten in liggers die zijn onderworpen aan discontinue belasting, waardoor het gebruik van een enkele wiskundige uitdrukking mogelijk is. Voor een ondersteunde balk AB, met uniforme belasting vanaf het middelpunt M tot het rechter uiteinde B, omvat de aanpak conceptuele 'sneden' op specifieke punten om het buigmoment in elk segment te bepalen. Door de balk op een punt tussen A en M door te snijden, wordt het buigmoment voor het segment voordat het middelpunt M wordt bereikt, weergegeven met behulp van een bepaalde functie.
 
 Een andere snede op een punt tussen M en B maakt het mogelijk dat het buigmoment voor het segment vanaf het middelpunt M tot het einde van de balk door een andere functie wordt beschreven. De sleutel tot het vereenvoudigen van de weergave is het combineren van deze functies in één enkele uitdrukking die zich aanpast op basis van de positie langs de balk.
 
 <img alt="Equation 1" src="/files/ftp_upload/15645/15645_Equation_1v2.svg" style="text-align: center; height: 42px;">
 Waarbij w0 de verdeelde belasting is die wordt uitgeoefend over de lengte van M tot het uiteinde van de balk. De uitdrukking wordt gevormd door de tweede functie alleen in berekeningen op te nemen voor posities voorbij middelpunt M, waarbij effectief gebruik wordt gemaakt van een voorwaardelijke benadering om de discontinuïteit te beheersen. Bovendien kan de verdeling van de belasting langs de balk en de resulterende schuifkracht ook worden weergegeven met behulp van singulariteitsfuncties. Deze methode, waarbij vaak de haakjes van Macaulay worden gebruikt voor representatie, stroomlijnt de berekening van buigmomenten in balken met variërende belastingsomstandigheden.
 
</div>

Singulariteitsfuncties voor buigmoment

Consider a simply supported beam carrying a uniformly distributed load from its midpoint to its right-hand support.
To determine the bending moment in ...

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Mechanical Engineering

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Analysis and Design of Beams for Bending

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beam load modeling with singularity functions

Singularity functions help model beam loads and bending moments when the loading changes along the beam. They make it possible to write the bending moment with one mathematical expression instead of separate equations for each section. In this method, Macaulay’s brackets are often used to show where a load starts or stops.

For a simply supported beam AB, the load is uniform from the midpoint M to the right support B. To find the bending moment, the beam is cut at different points. A cut between A and M gives the moment expression for the left segment before the midpoint. Another cut between M and B gives the moment expression for the right segment after the load begins.

These segment equations are then combined into a single form that changes with position along the beam. The second term is included only for points beyond midpoint M, so the formula handles the discontinuity in the load. The distributed load is written as w0 over the length from M to B, and the same singularity function idea can also be used to show the load pattern and the shear force.

Beam Load Modeling with Singularity Functions

beam-load-modeling-with-singularity-functions

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singularity functions for bending moment

Singularity functions help model beam loads and bending moments when the loading changes along the beam. They make it possible to write the bending moment with one mathematical expression instead of separate equations for each section. In this method, Macaulay’s brackets are often used to show where a load starts or stops. For a simply supported beam AB, the load is uniform from the midpoint M to the right support B. To find the bending moment, the beam is cut at different points. A cut...

key terms and concepts

Learn how singularity functions model a simply supported beam with a uniform load from the midpoint to one end, using one equation for bending moment and shear force.

Beam Load Modeling with Singularity Functions | JoVE K12

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Beam Load Modeling with Singularity Functions

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