Name: beam deflection and level points
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 17 s
Channel: JoVE

L’analyse d’une poutre supportée sous des charges asymétriques nécessite une tangente de référence avec une pente connue pour identifier le point de niveau.
La pente de la tangente de référence est calculée en déterminant le décalage tangentiel entre les extrémités.
La pente de la poutre en n’importe quel point peut être déterminée en utilisant le théorème de l’aire du premier moment, puis en écrivant une équation pour cette pente.
La distance verticale d’un point à la tangente de référence, également connue sous le nom de déviation tangentielle de ce point par rapport au support, peut être déterminée à l’aide du théorème de l’aire du deuxième moment.
Cet écart est égal à un certain segment et signifie la distance verticale entre le point et la tangente de référence. La déviation en un point indique sa distance verticale par rapport à une ligne horizontale.
Étant donné que cette déflexion est égale à la taille d’un segment spécifique, elle peut être définie comme la différence entre deux segments.
En étudiant des triangles similaires, l’un de ces segments peut être déterminé. Si l’on considère les conventions de signe pour les déviations et les déviations tangentielles, la déviation en un point est notée.

poutres avec chargements asymétriques

L'analyse d'une poutre supportée sous des charges asymétriques est essentielle en ingénierie des structures pour comprendre comment les poutres réagissent à des répartitions de forces variées. Cette analyse consiste à calculer la déflexion et à identifier les points où la pente de la poutre est nulle, points cruciaux pour garantir la stabilité et la fonctionnalité de la structure.

Le premier théorème du moment d’aire détermine la pente en tout point de la poutre. Ce théorème indique que le changement de pente entre deux points sur une poutre correspond à l'aire sous le diagramme des moments sur cet intervalle. La tangente de référence, qui permet de mesurer les écarts, est identifiée par sa pente connue, calculée à partir des décalages tangentiels entre les extrémités de la poutre.

Le deuxième théorème du moment d’aire calcule ensuite l'écart vertical de tout point par rapport à cette tangente de référence, appelé écart tangentiel. Cette mesure est essentielle pour comprendre le comportement en flexion de la poutre sous charge et identifier les points de déflexion maximum critiques pour les considérations de conception.

Enfin, après avoir déterminé la pente en un point souhaité à l'aide du premier théorème, le deuxième théorème mesure dans quelle mesure ce point s'écarte verticalement de la ligne de référence. Cet écart définit la déflexion maximale autorisée, garantissant que la poutre répond aux normes de sécurité et de fonctionnement en évitant les défaillances structurelles ou les déformations excessives. Grâce à de telles analyses, il est possible de concevoir des poutres qui maintiennent leur intégrité sous des charges opérationnelles et répondent aux exigences de sécurité.

Poutres avec chargements asymétriques

L’analyse d’une poutre supportée sous des charges asymétriques nécessite une tangente de référence avec une pente connue pour identifier le point de niveau.
La pente de la ...

Education

Enseignement

JoVE Core

Mechanical Engineering

Génie mécanique

Unit 1

Unité 1

Deflection of Beams

Flèche des poutres

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 that interval. A reference tangent is then chosen. This line has a known slope and is used to measure how the beam bends away from it.

The second moment-area theorem gives the vertical deviation from the reference tangent. This deviation is called the tangential deviation. It helps identify the level point, where the slope of the beam is zero, and it also shows the maximum deflection point.

Finding slope and tangential deviation is essential for checking structural stability and function. These values help engineers confirm that the beam stays within the maximum allowable deflection. That reduces the risk of failure or excessive deformation and supports safe design under operational loads.

Beam Deflection and Level Points

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 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...

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 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 moment-area theorems are used to find beam slope, tangential deviation, and level points for a supported beam under unsymmetrical loading.

Beam Deflection and Level Points | 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 and Level Points

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