Name: beam deflection from loads and supports
Uploaded: 2024-05-23T13:00:01.000Z
Duration: 1 min 15 s
Channel: JoVE

Un élément crucial dans la conception des poutres est l’identification et la mesure de la déflexion. Une compréhension complète des flèches est nécessaire lorsqu’il s’agit de poutres indéterminées.
Considérons une poutre en porte-à-faux qui supporte deux charges concentrées. Les réactions au niveau des supports sont calculées en analysant le diagramme du corps libre de la poutre.
À l’aide du diagramme de corps libre et des supports de réaction, un diagramme de moment de flexion est construit, qui décrit la façon dont la poutre se plie.
Le diagramme du moment de flexion montre que la poutre s’incurve vers le haut entre le début et un point spécifique en raison d’un moment de flexion positif, et qu’elle se courbe vers le bas de ce point jusqu’à la fin de la poutre en raison d’un moment de flexion négatif.
La plus grande valeur de la courbe ou le plus petit rayon de courbure se produit au niveau du support où le moment de flexion est le plus élevé.
Pour une analyse détaillée et la conception d'une poutre, il est crucial de disposer de données précises sur la déflexion et la pente de la poutre en différents points, en mettant l'accent sur la compréhension de la déflexion maximale.

déformation d'une poutre sous chargement transversal

Deformation of a Beam under Transverse Loading

Comprendre la déflexion des poutres, en particulier pour les poutres indéterminées avec des segments en porte-à-faux et de multiples charges concentrées, est primordial pour garantir l'intégrité structurelle et la fonctionnalité. Le processus commence par la construction d’un diagramme de corps libre précis, qui permet d’identifier les forces et les moments agissant sur la poutre. Ce diagramme est essentiel pour visualiser la façon dont les moments de flexion varient sur la longueur de la poutre, influençant sa courbure.

Les informations tirées du diagramme des moments de flexion s'étendent à l'analyse de la courbure de la poutre. La courbure de la poutre est en corrélation avec les moments de flexion le long de la poutre et est inversement liée à la rigidité et à la section transversale de la poutre. Des moments de flexion plus élevés entraînent une plus grande courbure, ce qui est essentiel pour comprendre le profil de déflexion de la poutre.

L'accent est particulièrement mis sur la déflexion maximale, car une déflexion excessive peut entraîner des problèmes structurels, voire une défaillance ou rupture. Cette déflexion maximale se produit généralement là où la courbure est la plus grande, souvent à proximité des supports ou directement sous les points où les charges sont appliquées. L'identification de cette déflexion maximale est cruciale pour garantir que les conceptions de poutres sont conformes aux normes de sécurité et répondent aux exigences fonctionnelles. En effectuant des analyses approfondies à l’aide de diagrammes de corps libres et de moments de flexion, on peut concevoir efficacement des structures sûres et adaptées à des conditions de charge et à des exigences architecturales spécifiques.

Déformation d'une poutre sous chargement transversal

Un élément crucial dans la conception des poutres est l’identification et la mesure de la déflexion. Une compréhension complète des déflexions est nécessaire ...

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

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

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 will curve. Beam curvature is related to the bending moment and is affected by the beam’s stiffness and cross-section. When the bending moment is higher, the curvature is greater, so the deflection profile becomes easier to evaluate.

The main goal is to find the maximum deflection. Too much deflection can cause structural problems or even failure. This largest deflection often appears where curvature is greatest, such as near supports or directly under applied loads. Using free-body and bending moment diagrams helps engineers design beams that stay safe and meet functional requirements under specific loading conditions.

Beam Deflection from Loads and Supports

deformation of a beam under transverse loading

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

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 free-body diagrams and bending moment diagrams help predict beam deflection, identify maximum deflection, and check structural safety under load.

Beam Deflection from Loads and Supports | 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 from Loads and Supports

Description

Transcript

Tags