Name: beam slope and deflection by singularity
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 19 s
Channel: JoVE

La deflessione di una trave in una struttura di copertura può essere determinata utilizzando il metodo dell'integrazione, a condizione che una singola funzione analitica possa rappresentare il momento flettente.
Tuttavia, se il carico della trave richiede più funzioni per rappresentare il momento flettente, sarebbero necessarie costanti ed equazioni aggiuntive, con conseguenti calcoli lunghi.
Questa complessità può essere semplificata utilizzando le funzioni di singolarità. Si consideri una trave prismatica supportata alle estremità che trasporta un carico eccentrico.
La funzione della forza di taglio di questa trave può essere rappresentata utilizzando un'appropriata funzione di singolarità. La funzione del momento flettente può essere derivata integrando questa funzione della forza di taglio.
La pendenza e la deflessione del fascio possono essere ottenute integrando questa funzione di momento. Le costanti nelle equazioni possono essere determinate dalle condizioni al contorno.
L'uso di funzioni di singolarità elimina la necessità di costanti ed equazioni aggiuntive, semplificando il calcolo. Di conseguenza, le funzioni di singolarità forniscono un modo efficiente per calcolare la pendenza e la deflessione di una trave in presenza di carichi complessi.

deflessione di una trave

Determinare con precisione la deflessione e la pendenza della trave in varie condizioni di carico nell'ingegneria strutturale è fondamentale per garantire la sicurezza e l'integrità strutturale. Le funzioni di singolarità offrono un approccio semplificato all'analisi delle travi, soprattutto quando più funzioni di carico complicano l'equazione del momento flettente.

Le funzioni di singolarità, descritte in una lezione precedente, sono potenti strumenti matematici che rappresentano le discontinuità all'interno di una funzione comunemente incontrata negli scenari di carico strutturale. Queste funzioni aiutano a esprimere in modo compatto le equazioni della forza di taglio e del momento flettente, anche sotto carichi complessi o multipli.

<img alt="Equation 1" src="/files/ftp_upload/15669/15669_Equation_1a.svg" style="height: 32px;">

<img alt="Equation 2" src="/files/ftp_upload/15669/15669_Equation_1b.svg" style="height: 32px;">

Per una trave prismatica, tipicamente uniforme su tutta la sua lunghezza e supportata su entrambe le estremità, un carico eccentrico presenta sfide specifiche. La forza di taglio in qualsiasi punto di tale trave può essere modellata utilizzando le funzioni di singolarità. Queste funzioni gestiscono facilmente le discontinuità introdotte dai carichi applicati in punti specifici o su determinati intervalli, come un carico rappresentato come una funzione a gradino nel diagramma della forza di taglio.

Il momento flettente, derivato integrando la funzione della forza di taglio, è fondamentale per valutare le prestazioni della trave. Questo passaggio influisce sulla distribuzione delle sollecitazioni della trave e sulla deflessione complessiva. La deflessione della trave viene determinata integrando due volte la funzione del momento flettente e applicando le condizioni al contorno della trave per risolvere le costanti di integrazione. L'utilizzo delle funzioni di singolarità per modellare le forze di taglio e i momenti flettenti elimina la necessità di molteplici costanti aggiuntive ed equazioni complesse, semplificando i calcoli e migliorando l'efficienza computazionale. Questo metodo consente una facile valutazione delle diverse condizioni di carico sulla deflessione della trave e sulla distribuzione delle sollecitazioni, essenziali per la progettazione e la manutenzione sicura dei sistemi strutturali.

Deflessione di una trave

La deflessione di una trave in una struttura di copertura può essere determinata utilizzando il metodo dell'integrazione, a condizione che una singola funzione analitica possa rappresentare il ...

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

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 beam example shown here, a prismatic beam, which means a beam with a uniform cross-section, is supported at both ends and carries an eccentric load, or a load placed off center.

The shear force at any point in the beam can be written with singularity functions. These functions handle the jump in the shear force diagram caused by the applied load. The bending moment is then found by integrating the shear force equation, and it is important because it affects the stress in the beam and the amount of deflection.

To find deflection, the bending moment equation is integrated twice. The beam boundary conditions are then used to solve for the integration constants. This approach keeps the equations compact, reduces the need for extra constants, and makes calculations more efficient for different loading conditions. It is a useful way to check beam performance and support safe structural design and maintenance.

Beam Slope and Deflection by Singularity

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 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 singularity functions simplify shear force, bending moment, slope, and beam deflection calculations for a prismatic beam with an eccentric load.

Beam Slope and Deflection by Singularity | 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 Slope and Deflection by Singularity

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