Name: beam slope and deflection by singularity
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 19 s
Channel: JoVE

The deflection of a beam in a roof structure can be determined using the integration method, provided that a single analytical function can represent the bending moment.
However, if the loading of the beam requires multiple functions to represent the bending moment, additional constants and equations would be necessary, leading to lengthy calculations.
This complexity can be simplified using singularity functions. Consider a prismatic beam supported at the ends carrying an eccentric load.
The shear force function of this beam can be represented using an appropriate singularity function. The bending moment function can be derived by integrating this shear force function.
The beam&#39;s slope and deflection can be obtained by integrating this moment function. The constants in the equations can be determined from the boundary conditions.
Using singularity functions eliminates the need for additional constants and equations, simplifying the computation. As a result, singularity functions provide an efficient way to calculate the slope and deflection of a beam under complex loadings.

doorbuiging van een balk

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 Het nauwkeurig bepalen van de doorbuiging en helling van de balk onder verschillende belastingsomstandigheden in de bouwtechniek is van cruciaal belang voor het waarborgen van de veiligheid en structurele integriteit. Singulariteitsfuncties bieden een gestroomlijnde aanpak voor het analyseren van balken, vooral wanneer meerdere belastingsfuncties de vergelijking van het buigmoment compliceren.
 
 Singulariteitsfuncties, beschreven in een eerdere les, zijn krachtige wiskundige hulpmiddelen die discontinuïteiten vertegenwoordigen binnen een functie die vaak voorkomt in structurele belastingscenario's. Deze functies helpen de vergelijkingen van de schuifkracht en het buigmoment compact uit te drukken, zelfs onder complexe of meervoudige belastingen.
 
 <img alt="Equation 1" src="/files/ftp_upload/15669/15669_Equation_1a.svg" style="text-align: center; height: 32px;">
 <img alt="Equation 2" src="/files/ftp_upload/15669/15669_Equation_1b.svg" style="text-align: center; height: 32px;">
 Voor een prismatische balk, die doorgaans uniform is over de lengte en aan beide uiteinden wordt ondersteund, brengt een excentrische belasting specifieke uitdagingen met zich mee. De schuifkracht op elk punt in een dergelijke balk kan worden gemodelleerd met behulp van singulariteitsfuncties. Deze functies kunnen gemakkelijk omgaan met de discontinuïteiten die worden geïntroduceerd door belastingen die op specifieke punten of over bepaalde intervallen worden uitgeoefend, zoals een belasting die wordt weergegeven als een stapfunctie in het schuifkrachtdiagram.
 
 Het buigmoment, afgeleid door integratie van de schuifkrachtfunctie, is van cruciaal belang voor het beoordelen van de prestaties van de balk. Deze stap heeft invloed op de spanningsverdeling en de algehele doorbuiging van de balk. De doorbuiging van de straal wordt bepaald door de buigmomentfunctie tweemaal te integreren en de randvoorwaarden van de straal toe te passen om de integratieconstanten op te lossen. Het gebruik van singulariteitsfuncties om schuifkrachten en buigmomenten te modelleren elimineert de noodzaak voor meerdere extra constanten en complexe vergelijkingen, waardoor berekeningen worden vereenvoudigd en de rekenefficiëntie wordt verbeterd. Deze methode maakt een eenvoudige evaluatie mogelijk van verschillende belastingsomstandigheden op de doorbuiging van de balk en de spanningsverdeling, essentieel voor het veilige ontwerp en onderhoud van structurele systemen.
 
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Doorbuiging van een Balk

The deflection of a beam in a roof structure can be determined using the integration method, provided that a single analytical function can represent the ...

Education

JoVE Core

Mechanical Engineering

Werktuigbouwkunde

Unit 1

Deflection of Beams

Doorbuiging van balken

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

Deflection of a Beam

beam slope and deflection by singularity

Beam slope and deflection can be found with singularity functions when beam loading is hard to write as one simple equation. This method is useful in structural engineering because it helps describe beams under several loads without breaking the problem into many separate cases.

Singularity functions are mathematical tools for showing discontinuities in a load, shear force, or bending moment equation. They are especially helpful when a load acts at a single point or over a set interval. In the beam example shown here, a prismatic beam, which means a beam with a uniform cross-section, is supported at both ends and carries an eccentric load, or a load placed off center.

The shear force at any point in the beam can be written with singularity functions. These functions handle the jump in the shear force diagram caused by the applied load. The bending moment is then found by integrating the shear force equation, and it is important because it affects the stress in the beam and the amount of deflection.

To find deflection, the bending moment equation is integrated twice. The beam boundary conditions are then used to solve for the integration constants. This approach keeps the equations compact, reduces the need for extra constants, and makes calculations more efficient for different loading conditions. It is a useful way to check beam performance and support safe structural design and maintenance.

Beam Slope and Deflection by Singularity

deformation of a beam under transverse loading

deformation-of-a-beam-under-transverse-loading

Beam Deflection from Loads and Supports

Beam deflection depends on the loads on a beam and the support conditions at its ends. For indeterminate beams, especially those with overhanging parts and multiple concentrated loads, careful analysis is needed to predict how the beam will bend. The first step is to draw an accurate free-body diagram. A free-body diagram shows the forces and moments acting on the beam. It helps reveal how the bending moment changes along the beam length. The bending moment diagram also shows how the beam...

equation of the elastic curve

Beam Deflection and Boundary Conditions

Beam deflection and boundary conditions help describe how a beam bends under load. In structural engineering, the curvature of a plane curve shows how sharply the beam bends. That curvature is found from the curve’s first and second derivatives. A cantilever beam with a point load at its free end is a common example. A diving board works like this type of beam. When slopes are small, the beam’s elastic curve becomes the main shape used in the analysis. The governing equation connects bending...

elastic curve from the load distribution

Beam Deflection Under Distributed Load

Beam deflection under distributed load is a key part of beam analysis in engineering. It helps predict how a beam bends and reacts when a load is spread along its length. Engineers use this relationship to judge whether a structure will stay safe and functional. The analysis starts with the link between load, shear force, and bending moment. These quantities can be written in differential form. When beam flexibility is treated as constant, further differentiation leads to a higher-order...

deflection of a beam

Beam slope and deflection can be found with singularity functions when beam loading is hard to write as one simple equation. This method is useful in structural engineering because it helps describe beams under several loads without breaking the problem into many separate cases. Singularity functions are mathematical tools for showing discontinuities in a load, shear force, or bending moment equation. They are especially helpful when a load acts at a single point or over a set interval. In the...

method of superposition

Beam Deflection Under Combined Loads

Beam deflection under combined loads can be found with the method of superposition. This structural engineering method is used to analyze how multiple loads affect a beam. It works by finding the slope and deflection from each load separately, then adding those results together. The method applies only when the beam stays within its elastic limit. In that range, the material deforms in a linearly elastic way. That makes it valid to combine the effects of separate loads without changing the...

moment-area theorems

Beam Deflection Using Moment-Area Rules

The moment-area theorems help describe beam bending in structural engineering. They are often used for building floor supports and other beams under load. The theorems use the elastic curve, which is the shape a beam takes as it deforms. The first theorem links slope change to area on an M/EI diagram. Here, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia. When M/EI is plotted along the beam length, the area under the curve gives the total rotation...

beams with symmetric loadings

Moment-Area Method for Cantilever Deflection

The moment-area method is used in structural engineering to find the slope and deflection of a beam under load. It is especially useful for a cantilever beam with a concentrated load and a moment at the free end. The analysis begins with a free-body diagram. This diagram helps calculate the reactions at the fixed end. After that, the bending moment diagram is drawn to show how the bending moment changes along the beam. Points where the bending moment is zero are important in this step. Next,...

beams with unsymmetric loadings

Beam Deflection and Level Points

Beam deflection and level points can be found for a supported beam under unsymmetrical loadings using moment-area theorems. This type of analysis helps show how a beam bends when forces are spread unevenly along it. It is an important part of structural engineering because it links load patterns to beam response. The first moment-area theorem is used to find the slope at any point on the beam. It states that the change in slope between two points equals the area under the moment diagram over...

maximum deflection

Locating the Beam’s Highest Deflection

Maximum deflection in a supported beam depends on where the load is placed. With an unsymmetrical load, such as a train moving across a bridge, the beam may bend most at a point that is not the midpoint. Finding that point is important for judging stress and deflection in the structure. The point of maximum deflection is called point O. At point O, the tangent to the deflection curve is horizontal. That means the slope is zero there. To locate point O, the slope at any point X along the beam...

key terms and concepts

Learn how singularity functions simplify shear force, bending moment, slope, and beam deflection calculations for a prismatic beam with an eccentric load.

Beam Slope and Deflection by Singularity | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Beam Slope and Deflection by Singularity

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