Name: beam deflection from central load
Uploaded: 2024-05-23T13:00:13.000Z
Duration: 1 min 14 s
Channel: JoVE

Si consideri una trave PQ semplicemente supportata di lunghezza L, che porta un carico puntuale F al centro. Quale sarà la deflessione al centro della trave?
Qui, le reazioni ad entrambe le estremità del fascio sono uguali e ciascuna è la metà del carico centrale.
Considera il segmento PC e scegli un punto qualsiasi a una distanza x dall'estremità P. A questo punto, il momento dovuto alla reazione nel punto P è il carico nel punto P moltiplicato per la distanza. Una differenziazione parziale dell'equazione del momento rispetto al carico all'estremità P è la metà di x. 
Un'analisi simile può essere condotta per il segmento QC del fascio.
Utilizzando il teorema di Castigliano, la deflessione nel punto C è determinata dalle derivate parziali dell'energia di deformazione dovuta al carico applicato.
Questa equazione può essere semplificata considerando i due segmenti del fascio PC e QC.
L'esecuzione dell'integrazione su metà della lunghezza della trave fornisce un'espressione per la deflessione al centro della trave.

teorema di castigliano: risoluzione dei problemi

La deflessione di una trave semplicemente appoggiata che trasporta un carico concentrato centrale può essere analizzata utilizzando i principi della meccanica strutturale, in particolare applicando il teorema di Castigliano. Questo teorema mette in relazione lo spostamento nel punto di applicazione del carico con le derivate parziali dell'energia di deformazione nella struttura. La trave semplicemente appoggiata con un carico concentrato al centro presenta forze di reazione simmetriche sui supporti, ciascuno dei quali sopporta metà del carico. Da queste reazioni si ricava il momento flettente in qualsiasi punto lungo la trave, calcolato sulla distanza dal supporto più vicino.

Il teorema di Castigliano indica che la freccia nel punto in cui è applicato il carico è determinata differenziando l'energia di deformazione della trave rispetto al carico. L'energia di deformazione viene calcolata in base al momento flettente lungo la trave, integrato sulla sua lunghezza. Per questa trave, l'energia di deformazione dovuta alla flessione viene calcolata dal quadrato dell'espressione del momento flettente, integrato lungo metà della lunghezza della trave.

<img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="height: 106px;">

Poiché la trave è simmetrica, questo valore viene raddoppiato per tenere conto dell'intera trave e viene calcolata la deflessione al centro della trave. Dipende dall'entità del carico, dal cubo della lunghezza della trave ed è inversamente proporzionale al prodotto del momento d'inerzia per il modulo elastico della sezione trasversale della trave.

<img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="height: 42px;">

Teorema di Castigliano: Risoluzione dei problemi

Si consideri una trave PQ semplicemente supportata di lunghezza L, che porta un carico puntuale F al centro. Quale sarà la deflessione al centro della trave?
Qui, il ...

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Castigliano&#39;s Theorem: Problem Solving

beam deflection from central load

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam.

For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support.

Castigliano's theorem says the displacement at the load point is found by taking the partial derivative of the strain energy with respect to the load. Strain energy is the energy stored by the beam as it bends. It is calculated from the bending moment along the beam and integrated over the beam length.

Because the beam is symmetric, the strain energy is first found over half the beam and then doubled for the full length. The center deflection depends on the load, the cube of the beam length, and the beam's moment of inertia and elastic modulus. It is inversely proportional to the product of those two material and cross-section properties.

Beam Deflection from Central Load

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castigliano&#39;s theorem: problem solving

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam. For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support. Castigliano's theorem says the...

key terms and concepts

Learn how Castigliano's theorem is used to find the center deflection of a simply supported beam with a central point load using strain energy.

Beam Deflection from Central 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 from Central Load

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