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

Il teorema momento-area fornisce le proprietà geometriche di una curva elastica per determinare la deflessione e la pendenza in qualsiasi punto della trave che sostiene un pavimento di un edificio.
Quando viene tracciato lungo la lunghezza della trave, il rapporto M/EI crea un diagramma simile al diagramma del momento flettente.
Il teorema dell'area momento deriva l'equazione della pendenza e della deviazione tangenziale dal diagramma delle travi. Il primo teorema dell'area del momento equipara l'angolo tra le tangenti nei punti P e Q all'area tra questi punti sotto il diagramma M/EI.
Il secondo teorema momento-area mette in relazione la deviazione tangenziale del punto P da Q al primo momento della stessa area riguardante l'asse verticale passante per P.
Questo teorema è anche esprimibile in termini di prodotto dell'area e della distanza dal baricentro dell'area all'asse verticale attraverso P.
La deviazione tangenziale è la distanza verticale da Q alla tangente in P. Viene calcolato moltiplicando l'area sotto il diagramma M/EI per la distanza dal suo baricentro all'asse verticale attraverso Q.

teoremi dell'area-momento

Il Teorema Momento-Area è cruciale nell'ingegneria strutturale per analizzare la flessione delle travi, in particolare in applicazioni come i supporti dei solai degli edifici. Questo teorema utilizza le proprietà geometriche della curva elastica, che descrive come una trave si deforma sotto carico, per semplificare i calcoli di deflessioni e pendenze.

Il teorema è diviso in due parti. La prima parte collega l'angolo tra le tangenti in due punti qualsiasi della curva elastica della trave all'area sotto una curva derivata tracciando la quantità M/EI (dove M è il momento flettente, E è il modulo di elasticità e I è il momento d'inerzia) contro la deflessione della trave lungo la sua lunghezza. L'area sotto questa curva corrisponde direttamente alla rotazione totale che avviene tra questi due punti.

La seconda parte del teorema affronta la deviazione tangenziale –o lo spostamento verticale– tra due punti qualsiasi risultante dalla flessione della trave. Si afferma che questa deviazione è equivalente al momento primo dell'area sotto la curva M/EI attorno ad un asse verticale passante per uno di questi punti, fornendo una misura dello spostamento del segmento di trave dalla sua posizione originale. Questi teoremi determinano in modo efficiente la pendenza e la deflessione in vari punti lungo una trave, essenziali per garantire la sicurezza strutturale e le prestazioni sotto carico.

Teoremi dell'area-momento

Il teorema momento-area fornisce le proprietà geometriche di una curva elastica per determinare la deflessione e la pendenza in qualsiasi punto della trave che supporta un ...

Education

Didattica

JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Deflection of Beams

Deflessione delle travi

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

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 Using Moment-Area Rules

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