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

一长度为 L 的梁承受沿 x 轴方向变化的载荷，该载荷为抛物线形与梯形载荷分布的组合。
在抛物线荷载中，考虑长度 dx 上的力的微分单元 dR。通过将微分单元沿荷载长度积分，可得到合力。合力作用点位于抛物线面积的形心处，形心的 x 坐标可通过力矩原理确定。
梯形面积可划分为矩形和三角形区域，其合力作用于各自的形心处。这些形心分别位于矩形长度的一半处，以及三角形垂直边起算的底边长度的三分之一处。
梯形区域的合力通过将各个分力的合力相加得到，并通过该区域的形心，形心位置由力矩原理确定。
总分布荷载为梯形和抛物线区域合力的总和，其作用线通过由力矩原理确定的组合区域的形心。

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	<li>梅里亚姆，JL;Kraige， LG 和 Bolton， JN （2020）。工程力学 \u2012 静力学。新泽西州霍博肯:约翰·威利。第 273 页</li>
	<li>希伯勒，RC（2016 年）。工程力学 \u2012 静力学和动力学。 新泽西州霍博肯:Pearson Prentice Hall。pp EM-2 （英文</li>）
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单轴方向的载荷

在结构工程中，分析受不同加载的梁是能够理解这些结构元素所产生的行为和性能的关键方面。一种常见的情况是梁受到不同载荷分布的组合。 思考一个长度为L的梁受到变化载荷的作用，这是位于x轴上的抛物线和梯形载荷分布的组合。在这种情况下，必须确定合成荷载、它们的位置和组合面积的质心，以预测梁在这些荷载条件下的响应。 首先，检查抛物线载荷分布。思考一个作用在微小长度dx上的微分力dR。为了确定抛物线区域的合荷载，将微分元素dR在整个载荷长度上进行积分。该合力的位置处于抛物线区域的质心处。力矩原理用于求质心的 x 坐标，该原理指出了围绕轴的面积第一个弯曲的矩等于面积的乘积及其质心与轴之间的距离。 接下来，通过将其分为两个矩形和一个三角形区域来分析梯形载荷的分布。这些独立区域的合力在各自的质心中发挥作用。对于矩形区域，质心位于矩形长度的二分之一处。对于三角形区域，质心位于距离三角形垂直边中底长的三分之一处。 通过将独立矩形和三角形区域的合力进行相加，可以确定整个梯形区域的合力。合力的位置可以通过梯形区域的质心，也可以通过力矩原理来进行确定。 类似地，通过将抛物线和梯形区域的合力进行相加，可以以此来确定作用在梁上的总合力。总合力的位置通过组合区域的质心，并且可以通过力矩原理来进行进一步确定。

在结构工程中，对承受不同荷载的梁进行分析是理解这些构件行为与性能的关键方面。

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Force System Resultants

力系合力

KEY TERMS AND CONCEPTS

关键术语与概念

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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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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