Name: beam deflection under combined loads
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 20 s
Channel: JoVE

La méthode de superposition est utilisée en génie civil pour calculer la pente et la déflexion des poutres soumises à des charges multiples.
Lorsqu'une poutre subit diverses charges concentrées ou distribuées, il est souvent plus pratique de calculer séparément la pente et la déflexion causées par chaque charge.
Le principe de superposition est ensuite appliqué, en additionnant les valeurs de pente ou de déflexion correspondant à chaque charge pour déterminer l’effet total.
Considérons une poutre supportée soumise à des charges réparties et concentrées.
Tout d’abord, la pente et la déflexion causées par la charge concentrée sont déterminées, suivies de la déflexion due à la charge distribuée, à l’aide d’une équation de courbe élastique.
L’équation de la pente est obtenue en dérivant l’équation de déflexion. Les pentes et les déflexions causées par les charges réparties et concentrées sont ensuite combinées, ce qui donne la déflexion et l’inclinaison globales de la poutre.
La méthode de superposition simplifie les scénarios de charge complexes, en fournissant des calculs précis de la déflexion et de la pente des poutres, ce qui est essentiel pour une conception structurelle sûre et efficace.

méthode de superposition

La méthode de superposition est une technique déterminante en ingénierie des structures, utilisée pour analyser l'effet de charges multiples sur les poutres. Cette approche consiste à calculer séparément la déflexion et la pente de chaque charge sur une poutre, puis à additionner ces effets pour déterminer l'impact global. Elle n'est applicable que lorsque le matériau de la poutre reste dans sa limite élastique, garantissant que les déformations sont linéairement élastiques.

Lors de l'application de la méthode de superposition, chaque type de charge—qu'elle soit concentrée ou distribuée—est considérée indépendamment. Pour les charges concentrées, les déflexions et les pentes sont dérivées des formules standard de la théorie des poutres. Dans le cas de charges réparties, les déflexions sont généralement calculées à l'aide de l'équation de la courbe élastique, qui est intégrée pour dériver la pente et ensuite pour obtenir la déflexion.

<img alt="Equation 1" src="/files/ftp_upload/15670/15670_Equation_1v2.svg" style="text-align: center; height: 42px;">

La somme des déflexions et des pentes des charges individuelles donne l'effet total sur la poutre. Par exemple, si une poutre est soumise à une charge uniformément répartie et à une charge ponctuelle centrale, la déflexion globale est la somme des déflexions dues à chaque charge. Cette méthode simplifie les scénarios de chargement complexes qui prédisent avec précision le comportement des poutres sous plusieurs charges. En décomposant le problème en éléments plus simples, la méthode de superposition permet de concevoir des systèmes structurels à la fois sûrs et efficaces, garantissant le respect des normes de sécurité et des exigences de performance.

Méthode de superposition

La méthode de superposition est utilisée en génie civil pour calculer la pente et la déflexion des poutres soumises à des charges multiples.
Lorsqu’une poutre ...

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

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 result.

Each load is treated on its own. A concentrated load, or point load, uses standard beam theory formulas to find slope and deflection. A distributed load is handled with the elastic curve equation, which is integrated to find slope and then integrated again to find deflection.

The total beam response is the sum of the individual responses. For example, if a beam carries a uniform distributed load and a central point load, the overall deflection is the sum of the deflections from both loads. This approach simplifies complex loading problems and helps predict beam behavior for safe and effective structural design.

Beam Deflection Under Combined Loads

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 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 the method of superposition finds beam slope and deflection under combined point and distributed loads using elastic beam theory.

Beam Deflection Under Combined Loads | 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 Combined Loads

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