Name: finding resultant loads on a beam
Uploaded: 2023-09-22T11:49:22.000Z
Duration: 1 min 29 s
Channel: JoVE

Una trave di lunghezza L soggetta a un carico variabile, che è una combinazione di distribuzione del carico parabolico e trapezoidale lungo l'asse x.
Nel carico parabolico, si consideri un elemento differenziale di forza dR sulla lunghezza dx. Il carico risultante si ottiene integrando l'elemento differenziale per tutta la lunghezza del carico. Si trova al baricentro dell'area parabolica e la coordinata x del baricentro viene determinata utilizzando il principio del momento.
L'area trapezoidale può essere divisa in aree rettangolari e triangolari, con il carico risultante che agisce ai loro baridi. Questi centroidi sono posizionati a metà della lunghezza del rettangolo e a un terzo della lunghezza della base dal lato verticale del triangolo.
Il carico risultante per l'area trapezoidale si ottiene sommando i singoli carichi risultanti e passa attraverso il baricentro dell'area, determinato dal principio del momento.
Il carico distribuito totale è la somma dei carichi risultanti delle aree trapezoidale e parabolica, passanti per il baricentro dell'area combinata, determinata dal principio del momento.

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	<li>Birra, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. e Self, B.P. (2019). Meccanica vettoriale per ingegneri \u2012 Statica e dinamica. New York: McGraw-Hill. pagine 264</li>
	<li>Meriam, J.L.; Kraige, LG e Bolton, JN (2020). Ingegneria Meccanica \u2012 Statica. Hoboken, New Jersey: John Wiley. pagine 273</li>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica \u2012 Statica e Dinamica. Hoboken, New Jersey: Pearson Prentice Hall. pp EM-2</li>
</ol>

carico lungo un solo asse

In ingegneria strutturale, l'analisi delle travi soggette a carichi variabili è un aspetto critico per comprendere il comportamento e le prestazioni di questi elementi strutturali. Uno scenario comune riguarda una trave soggetta ad una combinazione di diverse distribuzioni di carico.

Consideriamo una trave di lunghezza L soggetta a un carico variabile, che è una combinazione di una distribuzione di carico parabolico e trapezoidale lungo l'asse x. In questo caso, è essenziale determinare i carichi risultanti, le loro posizioni e il centroide dell'area combinata per prevedere la risposta della trave in queste condizioni di carico.

Innanzitutto, esaminiamo la distribuzione di carico parabolico. Consideriamo un elemento differenziale di forza dR che agisce su una piccola lunghezza dx. Per determinare il carico risultante per l'area parabolica, l'elemento differenziale dR viene integrato lungo tutta la lunghezza del carico. La posizione di questo carico risultante è al centroide dell'area parabolica. Il principio del momento viene applicato per trovare la coordinata x del centroide, secondo il quale il primo momento dell'area rispetto ad un asse, è uguale al prodotto dell'area e della distanza del suo centroide dall'asse.

Successivamente, viene analizzata la distribuzione del carico trapezoidale, suddividendola in due regioni rettangolari e triangolari. I carichi risultanti per queste singole regioni agiscono nei loro rispettivi centroidi. Per l'area rettangolare, il centroide è posizionato a metà della lunghezza del rettangolo. Per l'area triangolare, il centroide è situato a un terzo della lunghezza della base dalla parte verticale del triangolo.

Aggiungendo i carichi risultanti individuali per le aree rettangolari e triangolari, è possibile determinare il carico risultante per l'intera area trapezoidale. La posizione del carico risultante, passa attraverso il centroide dell'area trapezoidale, che può essere determinato anche utilizzando il principio del momento.

Allo stesso modo, aggiungendo i carichi risultanti sia per le aree paraboliche che per quelle trapezoidali, è possibile determinare il carico risultante totale che agisce sulla trave. La posizione del carico risultante totale, passa attraverso il centroide dell'area combinata, che può essere ulteriormente determinato utilizzando il principio del momento.

Carico lungo un solo asse

Nell'ingegneria strutturale, l'analisi di travi soggette a carichi variabili è un aspetto critico per comprendere il comportamento e le prestazioni di queste travi ...

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JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Force System Resultants

Risultanti dei sistemi di forze

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

Load along a Single Axis

finding resultant loads on a beam

Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted.

The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the full length of the load. The resultant load for the parabolic area acts at the centroid of that area. The x-coordinate of the centroid is found with the moment principle, which says the first moment of an area about an axis equals the area times the distance to its centroid.

The trapezoidal load is then split into two simpler shapes. One part is rectangular, and the other is triangular. Their resultant loads act at their own centroids. For the rectangle, the centroid lies halfway along its length. For the triangle, the centroid is one-third of the base length from the vertical side.

After those parts are found, their loads are added to get the resultant load of the full trapezoidal area. That load passes through the centroid of the trapezoid, which can also be found with the moment principle. Finally, the resultant loads from the parabolic and trapezoidal areas are combined. Their sum gives the total load on the beam, and its location passes through the centroid of the combined area.

Finding Resultant Loads on a Beam

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Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted. The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the...

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key terms and concepts

Learn how parabolic and trapezoidal loads on a beam are broken into simpler areas, then combined to find the resultant force and centroid.

Finding Resultant Loads on a Beam | 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

Finding Resultant Loads on a Beam

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