Name: beam deflection under distributed load
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 16 s
Channel: JoVE

When a beam carries a distributed load, the shear force and bending moment at any point on the beam can be expressed in a differential form.
A third-order linear differential equation is created by further differentiating the expression and assuming the flexural rigidity to be constant.
Differentiating once again, results in a fourth-order linear differential equation. This equation essentially governs the shape or elastic curve the beam will take when supporting a distributed load.
By multiplying the fourth-order linear differential equation by the flexural rigidity constant and integrating it four times, an expression for the curve can be obtained.
The constants in the equation defining the beam&#39;s curve are dictated by end conditions, such as tilt or deflection from supports and whether the shear force and bending moment are null at a cantilever beam&#39;s free end or only the bending moment is zero at both ends of a supported beam.
This technique accurately calculates curves for cantilevered or supported beams with distributed loads. However, overhanging beams&#39; support reactions cause shear irregularities, requiring distinct functions for the full-length curve definition.

elastische curve van de belastingsverdeling

<div>
 Het structurele gedrag van balken onder verdeelde belastingen is van cruciaal belang voor technische analyses, die zich richten op het voorspellen van hoe balken onder dergelijke omstandigheden buigen en reageren. Verschillende soorten balken (bijvoorbeeld vrijdragend, ondersteund of overhangend) gedragen zich verschillend onder verdeelde belastingsomstandigheden.
 
 Voor alle balken begint de analyse van de reactie van de balk op verdeelde belastingen met het begrijpen van de relatie tussen de belasting van een balk en de resulterende schuifkrachten en buigmomenten. Aanvankelijk wordt deze relatie uitgedrukt als een differentiaalvergelijking.
 
 <img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="text-align: center; height: 32px;">
 Verdere differentiatie van deze uitdrukking, uitgaande van een constante bundelflexibiliteit, leidt tot een complexere differentiaalvergelijking die de doorbuigingscurve van de bundel beschrijft of hoe deze onder de belasting zal buigen.
 
 <img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="text-align: center; height: 40px;">
 Deze complexe vergelijking wordt meerdere keren geïntegreerd om de werkelijke vorm van deze curve te bepalen. Elke integratiestap introduceert een constante die moet worden gedefinieerd door de randvoorwaarden van de balk, zoals hoe de balk aan de uiteinden wordt ondersteund of verbonden.
 
 <img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="text-align: center; height: 32px;">
 Randvoorwaarden variëren afhankelijk van hoe een balk wordt ondersteund. Een vrijdragende balk, vast aan het ene uiteinde en vrij aan het andere uiteinde, zal bijvoorbeeld aan elk uiteinde verschillende beperkingen hebben met betrekking tot doorbuiging en kracht. Omgekeerd zal bij een ondersteunde balk de omstandigheden primair gericht zijn op de steunpunten. Bijzonder uitdagend is de analyse van overhangende balken, waarbij delen van de balk buiten de steunen uitsteken. Deze segmenten ondergaan unieke krachten en buigmomenten, waardoor specifieke berekeningen nodig zijn om het gedrag van de balk over de gehele lengte nauwkeurig te beschrijven. Dit gedetailleerde begrip zorgt ervoor dat constructies veilig en functioneel zijn.
 
</div>

When a beam carries a distributed load, the shear force and bending moment at any point on the beam can be expressed in a differential form.
A third-order ...

Education

JoVE Core

Mechanical Engineering

Werktuigbouwkunde

Unit 1

Deflection of Beams

Doorbuiging van balken

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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 differential equation for the deflection curve, which is the shape the beam takes as it bends.

Finding the actual curve requires integrating the equation several times. Each integration adds a constant that must be determined from boundary conditions. Boundary conditions depend on how the beam is supported or connected at its ends.

Different beam types need different boundary conditions. A cantilever beam is fixed at one end and free at the other, so its deflection and force limits are set differently at each end. A supported beam is handled mainly at the support points.

Overhanging beams are more complex because part of the beam extends beyond the supports. Those outer sections experience unique forces and bending moments. They need separate calculations so the behavior along the full beam length can be described accurately.

Beam Deflection Under Distributed Load

deformation of a beam under transverse loading

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

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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 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...

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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

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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 distributed loads create shear force, bending moment, and beam deflection. See how boundary conditions change for cantilever, supported, and overhanging beams.

Beam Deflection Under Distributed Load | 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 Deflection Under Distributed Load

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