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

Lorsqu’une poutre supporte une charge répartie, la force de cisaillement et le moment de flexion en tout point de la poutre peuvent être exprimés sous une forme différentielle.
Une équation différentielle linéaire du troisième ordre est créée en dérivant davantage l’expression et en supposant que la rigidité en flexion est constante.
En dérivant une fois de plus, on obtient une équation différentielle linéaire du quatrième ordre. Cette équation régit essentiellement la forme ou la courbe élastique que prendra la poutre lorsqu’elle supportera une charge répartie.
En multipliant l’équation différentielle linéaire du quatrième ordre par la constante de rigidité en flexion et en l’intégrant quatre fois, on peut obtenir une expression de la courbe.
Les constantes de l'équation définissant la courbe de la poutre sont dictées par des conditions finales, telles que l'inclinaison ou la déflexion des supports et si la force de cisaillement et le moment de flexion sont nuls à l'extrémité libre d'une poutre en porte-à-faux ou si seul le moment de flexion est nul aux deux extrémités d'une poutre soutenue.
Cette technique calcule avec précision les courbes des poutres en porte-à-faux ou supportées avec des charges réparties. Cependant, les réactions de support des poutres en porte-à-faux provoquent des irrégularités de cisaillement, nécessitant des fonctions distinctes pour la définition de la courbe sur toute la longueur.

courbe élastique à partir de la répartition de la charge

Le comportement structurel des poutres sous charges réparties est essentiel pour l'analyse technique, qui se concentre sur la prévision de la façon dont les poutres se plient et réagissent dans de telles conditions. Différents types de poutres (par exemple, en porte-à-faux, supportées ou en console) se comportent différemment dans des conditions de charge répartie.

Pour toutes les poutres, l'analyse de la réaction de la poutre aux charges réparties commence par la compréhension de la relation entre la charge d'une poutre et les forces de cisaillement et les moments de flexion qui en résultent. Initialement, cette relation est exprimée sous forme d’équation différentielle.

<img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="text-align: center; height: 32px;">

Une différenciation plus poussée de cette expression, en supposant une flexibilité constante de la poutre, conduit à une équation différentielle plus complexe qui décrit la courbe de déflexion de la poutre ou la manière dont elle se pliera sous la charge.

<img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="text-align: center; height: 40px;">

Cette équation complexe est intégrée plusieurs fois pour déterminer la forme réelle de cette courbe. Chaque étape d'intégration introduit une constante qui doit être définie par les conditions aux limites de la poutre, telles que la manière dont la poutre est supportée ou connectée à ses extrémités.

<img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="text-align: center; height: 32px;">

Les conditions aux limites varient en fonction de la manière dont une poutre est supportée. Par exemple, une poutre en porte-à-faux, fixe à une extrémité et libre à l'autre, aura des contraintes différentes concernant la déflexion et la force à chaque extrémité. À l’inverse, une poutre supportée aura des conditions concentrées principalement sur les points d’appui. L’analyse des poutres en porte-à-faux est particulièrement difficile, car certaines parties de la poutre s’étendent au-delà de ses supports. Ces segments subissent des forces et des moments de flexion uniques, nécessitant des calculs distincts pour décrire avec précision le comportement de la poutre sur toute sa longueur. Cette compréhension détaillée garantit que les structures sont sûres et fonctionnelles.

Courbe élastique à partir de la répartition de la charge

Lorsqu’une poutre supporte une charge répartie, la force de cisaillement et le moment de flexion en tout point de la poutre peuvent être exprimés sous une forme différentielle.
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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

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

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 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 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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