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

Quando una trave trasporta un carico distribuito, la forza di taglio e il momento flettente in qualsiasi punto della trave possono essere espressi in forma differenziale.
Un'equazione differenziale lineare del terzo ordine viene creata differenziando ulteriormente l'espressione e assumendo che la rigidità flessionale sia costante.
Differenziando ancora una volta, si ottiene un'equazione differenziale lineare del quarto ordine. Questa equazione governa essenzialmente la forma o la curva elastica che la trave assumerà quando sosterrà un carico distribuito.
Moltiplicando l'equazione differenziale lineare del quarto ordine per la costante di rigidità flessionale e integrandola per quattro volte, si può ottenere un'espressione per la curva.
Le costanti nell'equazione che definisce la curva della trave sono dettate dalle condizioni finali, come l'inclinazione o la deflessione dai supporti e se la forza di taglio e il momento flettente sono nulli all'estremità libera di una trave a sbalzo o solo il momento flettente è zero ad entrambe le estremità di una trave appoggiata.
Questa tecnica calcola con precisione le curve per travi a sbalzo o supportate con carichi distribuiti. Tuttavia, le reazioni di supporto delle travi sporgenti causano irregolarità di taglio, richiedendo funzioni distinte per la definizione della curva a lunghezza intera.

curva elastica dalla distribuzione del carico

Il comportamento strutturale delle travi sotto carichi distribuiti è fondamentale per l'analisi ingegneristica, che si concentra sulla previsione di come le travi si piegano e reagiscono in tali condizioni. Diversi tipi di travi (ad esempio, a sbalzo, appoggiate o sporgenti) si comportano diversamente in condizioni di carico distribuito.

Per tutte le travi, l'analisi della reazione della trave ai carichi distribuiti inizia comprendendo la relazione tra il carico della trave e le forze di taglio e i momenti flettenti risultanti. Inizialmente, questa relazione è espressa come un'equazione differenziale.

<img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="height: 32px;">

Un'ulteriore differenziazione di questa espressione, assumendo una flessibilità costante della trave, porta a un'equazione differenziale più complessa che descrive la curva di deflessione della trave o come si piegherà sotto il carico.

<img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="height: 40px;">

Questa complessa equazione viene integrata più volte per determinare la forma effettiva di questa curva. Ciascun passaggio di integrazione introduce una costante che deve essere definita dalle condizioni al contorno della trave, ad esempio il modo in cui la trave è supportata o collegata alle sue estremità.

<img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="height: 32px;">

Le condizioni al contorno variano in base al modo in cui è supportata una trave. Ad esempio, una trave a sbalzo, fissata a un'estremità e libera all'altra, avrà vincoli diversi per quanto riguarda la deflessione e la forza a ciascuna estremità. Al contrario, una trave supportata avrà condizioni focalizzate principalmente sui punti di supporto. L'analisi delle travi sporgenti è particolarmente impegnativa, dove parti della trave si estendono oltre i suoi supporti. Questi segmenti subiscono forze e momenti flettenti unici, che richiedono calcoli distinti per descrivere accuratamente il comportamento della trave lungo tutta la sua lunghezza. Questa comprensione dettagliata garantisce che le strutture siano sicure e funzionali.

Curva elastica dalla distribuzione del carico

Quando una trave trasporta un carico distribuito, la forza di taglio e il momento flettente in qualsiasi punto della trave possono essere espressi in forma differenziale.
Un terzo ordine ...

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

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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