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

Балка длины L подвергается переменной нагрузке, которая представляет собой комбинацию параболического и трапециевидного распределения нагрузки вдоль оси x.
В параболической нагрузке рассмотрим дифференциальный элемент силы dR по длине dx. Результирующая нагрузка получается путем интегрирования дифференциального элемента по длине нагружения. Он расположен в центре параболической области, а x-координата центроида определяется по моментному принципу.
Трапециевидная область может быть разделена на прямоугольные и треугольные области, результирующая нагрузка которых действует на их центроиды. Эти центроиды расположены на половине длины прямоугольника и на одной трети длины основания от вертикальной стороны треугольника.
Результирующая нагрузка для трапециевидной площади получается путем сложения отдельных результирующих нагрузок и проходит через центроид площади, определенный по моментному принципу.
Общая распределенная нагрузка представляет собой сумму результирующих нагрузок трапециевидной и параболической областей, проходящих через центроид объединенной области, определяемой по моментному принципу.

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	<li>Бир, Ф..; Джонстон, Э.Р.; Мазурек, Д.Ф; Кромвель,.Д. и Селф, Б.. (2019). Векторная механика для инженеров \u2012 Статика и динамика. Нью-Йорк: Макгроу-Хилл. С. 264</li>
	<li>Мериам, Дж.Л.; Крайге, Л.Г. и Болтон, Д.Н. (2020). Инженерная механика \u2012 Статика. Хобокен, Нью-Джерси: Джон Уайли. С. 273</li>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. ПП ЭМ-2</li>
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Загрузка вдоль одной оси

В строительной инженерии анализ балок, подверженных варьирующимся нагрузкам, является важным аспектом для понимания поведения и производительности этих конструктивных элементов. Часто встречается ситуация, когда балка подвергается комбинации различных распределений нагрузки.

Рассмотрим балку длиной L, подверженную варьирующейся нагрузке, которая представляет собой комбинацию параболического и трапециевидного распределения нагрузки вдоль оси x. В этом случае важно определить результирующие нагрузки, их расположение и центроид объединенной площади, чтобы предсказать реакцию балки на эти условия нагружения.

В первую очередь изучим параболическое распределение нагрузки. Рассмотрим дифференциальный элемент с силой dR, действующей на маленькую длину dx. Чтобы определить результирующую нагрузку для параболической области, дифференциальный элемент dR интегрируется по всей длине нагрузки. Расположение этой результирующей нагрузки находится в центроиде параболической области. С помощью принципа моментов определяется x-координата центроида, который утверждает, что первый момент площади относительно оси равен произведению площади и расстоянию ее центроида от оси.

Затем анализируется трапециевидное распределение нагрузки, разделяя его на две прямоугольные и треугольные области. Результирующие нагрузки для этих отдельных областей действуют в их соответствующих центроидах. Для прямоугольной области центроид находится на половине длины прямоугольника. Для треугольной области центроид находится на трети длины основания от вертикальной стороны треугольника.

Путем сложения отдельных результирующих нагрузок для прямоугольной и треугольной областей можно определить результирующую нагрузку для всей трапециевидной области. Расположение результирующей нагрузки проходит через центроид трапециевидной области, которая также может быть определена с использованием принципа моментов.

Аналогично, путем сложения результирующих нагрузок параболических и трапециевидных областей можно определить общую результирующую нагрузку, действующую на балку. Расположение общей результирующей нагрузки проходит через центроид объединенной площади, которая также может быть определена с использованием принципа моментов.

В проектировании конструкций анализ балок, подвергающихся различным нагрузкам, является критически важным аспектом для понимания поведения и производительности этих ...

Education

JoVE Core

Mechanical Engineering

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Unit 1

Раздел 1

Force System Resultants

Результирующие силы системы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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Resolving Forces in Moment Calculations

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Torque About an Axis in 3D Systems

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Equivalent Force and Moment Systems

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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

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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