Name: beam deflection and boundary conditions
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 23 s
Channel: JoVE

La curvatura della curva piana in un punto della curva può essere espressa utilizzando un'espressione che coinvolge la prima e la seconda derivata della curva.
La pendenza è insignificante per la curva elastica della trave, quindi l'equazione che governa la curva elastica è espressa come un'equazione differenziale lineare del secondo ordine, considerando la rigidità flessionale .
Se la rigidità flessionale varia lungo la trave, è espressa in funzione di x. Tuttavia, per una trave prismatica, la rigidità flessionale rimane costante.
L'integrazione di questa equazione fornisce l'angolo formato dalla tangente alla curva elastica in un punto con l'orizzontale.
Questo piccolo angolo, quando integrato, ci dà la deflessione del fascio in qualsiasi punto. Le condizioni al contorno dei vincoli esterni delle travi determinano le costanti nelle equazioni.
Supportato, sporgente e a sbalzo sono i tre tipi di travi che vengono principalmente considerati. Per le travi appoggiate e a sbalzo, la deflessione nei punti di appoggio è pari a zero.
Sia la deflessione che la pendenza nel punto di appoggio sono nulle per le travi a sbalzo. Queste condizioni consentono il calcolo delle costanti.

equazione della curva elastica

Equation of the Elastic Curve

Il concetto di curvatura nelle curve piane, cruciale nell'ingegneria strutturale, definisce quanto bruscamente una trave sotto carico si piega. Questa curvatura viene determinata utilizzando la prima e la seconda derivata della curva.

Consideriamo una trave a sbalzo con un carico concentrato all'estremità libera (ad esempio, un trampolino). Quando si analizza la deflessione della trave con piccole pendenze, la forma della curva elastica della trave diventa fondamentale. L'equazione che governa quest’analisi coinvolge il momento flettente e la rigidità flessionale della trave, che è un prodotto del modulo di elasticità e del momento di inerzia della sezione trasversale della trave.

<img alt="Equation 1" src="/files/ftp_upload/15667/15667_Equation_1.svg" style=" height: 32px;">

Per le travi prismatiche, dove la sezione-trasversale rimane costante, l'analisi si semplifica, rendendo costante la rigidezza flessionale lungo la lunghezza della trave. L'integrazione dell'equazione governante consente il calcolo dell'angolo formato dalla tangente alla curva in qualsiasi punto, che, dopo un'ulteriore integrazione, produce la deflessione della trave in quel punto.

<img alt="Equation 2" src="/files/ftp_upload/15667/15667_Equation_2.svg" style=" height: 28px;">

Le condizioni al contorno sui supporti della trave sono vitali per completare questi calcoli. Supportate, sporgenti e a sbalzo sono tipi comuni di travi, ciascuna con condizioni al contorno distinte. Ad esempio, la deflessione e la pendenza nel punto di supporto di una trave a sbalzo sono zero, il che è essenziale per calcolare le costanti delle equazioni di deflessione.

Prevedere con precisione la deflessione della trave è fondamentale per garantire la sicurezza e la funzionalità strutturale. Una deflessione eccessiva può causare cedimenti strutturali o problemi di funzionalità, sottolineando l'importanza di comprendere il comportamento delle travi sotto carico.

Equazione della curva elastica

La curvatura della curva piana in un punto della curva può essere espressa utilizzando un'espressione che coinvolge la prima e la seconda derivata della curva.
Le...

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

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 moment with flexural rigidity. Flexural rigidity is the product of the modulus of elasticity and the moment of inertia of the beam’s cross-section. For a prismatic beam, the cross-section stays constant, so the flexural rigidity is constant along the beam.

Integrating the governing equation gives the angle of the tangent to the curve at any point. A second integration gives the deflection at that point. Boundary conditions at the beam supports are needed to find the constants in these equations.

Supported, overhanging, and cantilever beams each have different boundary conditions. For a cantilever beam, the deflection and slope at the support point are zero. Predicting deflection is important because too much bending can lead to structural failure or serviceability problems.

Beam Deflection and Boundary Conditions

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 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 beam deflection, curvature, and boundary conditions work together in structural engineering, using cantilever and other beam types.

Beam Deflection and Boundary Conditions | 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 and Boundary Conditions

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