Name: beam deflection using moment-area rules
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 17 s
Channel: JoVE

Le théorème du moment et de l’aire fournit les propriétés géométriques d’une courbe élastique pour déterminer la déflexion et la pente en tout point de la poutre supportant un étage de bâtiment.
Lorsqu'il est tracé le long de la longueur de la poutre, le rapport M/EI crée un diagramme similaire au diagramme des moments de flexion.
Le théorème moment-aire dérive l’équation de la pente et de la déviation tangentielle à partir du diagramme de faisceau. Le premier théorème moment-aire assimile l’angle entre les tangentes aux points P et Q à l’aire entre ces points sous le diagramme M/EI.
Le deuxième théorème de l’aire de moment relie la déviation tangentielle du point P de Q au premier moment de la même aire concernant l’axe vertical passant par P.
Ce théorème est également exprimable en fonction du produit de l’aire et de la distance entre le centroïde de l’aire et l’axe vertical passant par P.
L’écart tangentiel est la distance verticale entre Q et la tangente en P. Il est calculé en multipliant l'aire sous le diagramme M/EI par la distance entre son centroïde et l'axe vertical passant par Q.

théorèmes des moments d’aires

Le théorème des moments d’aires est crucial en ingénierie structurelle pour analyser la flexion des poutres, en particulier dans des applications telles que les supports de plancher de bâtiment. Ce théorème utilise les propriétés géométriques de la courbe élastique, qui représente la façon dont une poutre se déforme sous charge, pour simplifier les calculs de déflexions et de pentes.

Le théorème est divisé en deux parties. La première partie relie l'angle entre les tangentes en deux points quelconques de la courbe élastique de la poutre à l'aire sous une courbe dérivée en traçant la quantité M/EI (où M est le moment de flexion, E est le module d'élasticité et I est le moment d'inertie) contre la déviation de la poutre sur sa longueur. L'aire sous cette courbe correspond directement à la rotation totale se produisant entre ces deux points.

La deuxième partie du théorème concerne la déviation tangentielle – ou le déplacement vertical – entre deux points quelconques résultant de la flexion de la poutre. Il indique que cet écart est équivalent au premier moment d'aire sous la courbe M/EI autour d'un axe vertical passant par l'un de ces points, fournissant une mesure du déplacement du segment de poutre par rapport à sa position d'origine. Ces théorèmes déterminent efficacement la pente et la déflexion en différents points le long d'une poutre, essentiels pour garantir la sécurité structurelle et les performances sous charge.

Théorèmes des moments d’aires

Le théorème moment-aire fournit les propriétés géométriques d’une courbe élastique pour déterminer la déflexion et la pente en tout point de la poutre supportant une ...

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

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 between two points on the elastic curve.

The second theorem deals with tangential deviation, or vertical displacement, between two points. It says this distance equals the first moment of the area under the M/EI curve about a vertical axis through one of the points. Together, the two theorems make it easier to find slope and deflection at different points along a beam. That information helps engineers check structural safety and performance under load.

Beam Deflection Using Moment-Area Rules

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

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 moment-area theorems use the elastic curve and M/EI diagram to find beam slope, rotation, and deflection in structural engineering.

Beam Deflection Using Moment-Area Rules | JoVE K12

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mechanical engineering

Beam Deflection Using Moment-Area Rules

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