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

Belka o długości L poddana zmiennemu obciążeniu, które jest kombinacją rozkładu obciążenia parabolicznego i trapezowego wzdłuż osi x.
W obciążeniu parabolicznym należy uwzględnić element różniczkowy siły dR na długości dx. Obciążenie wynikowe uzyskuje się poprzez całkowanie elementu różnicowego na całej długości obciążenia. Znajduje się w środku ciężkości obszaru parabolicznego, a współrzędna x środka ciężkości jest określana za pomocą zasady momentu.
Obszar trapezu można podzielić na prostokątne i trójkątne, przy czym obciążenie wynikowe działa na ich centroidy. Te centroidy są umieszczone w połowie długości prostokąta i jednej trzeciej długości podstawy od pionowego boku trójkąta.
Obciążenie wypadkowe dla powierzchni trapezu uzyskuje się przez dodanie poszczególnych obciążeń wypadkowych i przechodzi przez środek ciężkości powierzchni, określony przez zasadę momentu.
Całkowite obciążenie rozłożone jest sumą obciążeń wypadkowych powierzchni trapezowych i parabolicznych, przechodzących przez środek ciężkości połączonego obszaru, określonych przez zasadę momentu.

<ol>
	<li>Piwo, F.P.; Johnston, E.R.; Mazurek, D.F.; Cromwell, PJ i Self, BP (2019). Mechanika wektorowa dla inżynierów \u2012 Statyka i dynamika. Nowy Jork: McGraw-Hill. Strona 264</li>
	<li>Meriam, J.L.; Kraige, L.G. i Bolton, J.N. (2020). Mechanika inżynierska \u2012 Statyka. Hoboken, New Jersey: John Wiley. Strona 273</li>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska \u2012 Statyka i dynamika. Hoboken, New Jersey: Pearson Prentice Hall. pp EM-2</li>
</ol>

obciążenie wzdłuż jednej osi

W inżynierii konstrukcyjnej analiza belki poddanej zmiennym obciążeniom jest istotnym aspektem w zrozumieniu zachowania i wydajności tych elementów strukturalnych. Często spotykanym scenariuszem jest belka poddana kombinacji różnych rozkładów obciążeń.

Rozważmy belkę o długości L, poddaną zmiennemu obciążeniu, będącemu kombinacją rozkładu parabolicznego i trapezoidalnego wzdłuż osi x. W tym przypadku istotne jest określenie obciążeń wynikowych, ich lokalizacji oraz środka ciężkości połączonego obszaru w celu przewidzenia reakcji belki pod tymi warunkami obciążenia.

Najpierw przeanalizujmy rozkład obciążenia parabolicznego. Przyjrzyjmy się elementowi różniczkowemu siły dR działającemu na małą długość dx. Aby określić siłę wypadkową dla obszaru parabolicznego, całkujemy element różniczkowy dR przez całą długość obciążenia. Lokalizacja tego obciążenia wynikowego znajduje się w centroidzie obszaru parabolicznego. Zasada momentów jest stosowana do znalezienia x-współrzędnej środka ciężkości, zgodnie z którą pierwszy moment obszaru względem osi jest równy iloczynowi pola obszaru i odległości jego środka ciężkości od osi.

Następnie analizowany jest rozkład obciążenia trapezoidalnego, dzieląc go na dwa obszary prostokątne i jeden trójkątny. Siły wypadkowe odpowiadające poszczególnym obszarom działają w ich centroidach. Dla obszaru prostokątnego centroid znajduje się w połowie długości prostokąta. Dla obszaru trójkątnego centroid jest położony w odległości jednej trzeciej długości podstawy od pionowego boku trójkąta.

Dodając poszczególne obciążenia wynikowe dla obszarów prostokątnego i trójkątnego, można określić obciążenie wynikowe dla całego obszaru trapezoidalnego. Punkt przyłożenia siły wypadkowej przechodzi przez centroid obszaru trapezoidalnego, który również można określić na podstawie zasady momentów.

Podobnie, dodając obciążenia wynikowe dla obszarów parabolicznego i trapezoidalnego, można określić łączne obciążenie wynikowe działające na belkę. Lokalizacja łącznego obciążenia wynikowego przechodzi przez centroid połączonego obszaru, który może być dalszy określony przy użyciu zasady momentu.

Obciążenie wzdłuż jednej osi

W inżynierii budowlanej analiza belek poddanych zmiennym obciążeniom jest krytycznym aspektem zrozumienia zachowania i wydajności tych ...

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Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Force System Resultants

wypadkowe układu sił

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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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load along a single axis

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

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Finding Resultant Loads on a Beam

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