Name: shear change under distributed loading
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 23 s
Channel: JoVE

考虑一个承受分布载荷、两个集中载荷以及一个力偶矩的梁。建立剪力与分布载荷之间的关系。
首先，考虑梁上不受任何集中载荷或集中力偶矩作用的基本段，并绘制该段的受力体图。
为了保持平衡，应使作用在截面右侧的剪力增加一个微小且有限的量。
分布载荷的合力作用点位于该段右端点的某一分数距离处。
利用竖向力的平衡方程，可得到剪力与载荷之间的关系。
接下来，将方程两边同时除以 Δx 并令 Δx 趋近于零，即可确定剪力的斜率，该斜率等于分布载荷的集度。
最后，重新整理方程并对两个任意点之间的微元段上的分布载荷进行积分，可得到剪力变化与载荷曲线下的面积之间的关系。

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	<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。第 370-371 页。</li>
	<li>比尔，F.P.;约翰斯顿，ER;马祖克，DF;克伦威尔，PJ 和塞尔夫，BP（2019 年）。工程师的矢量力学 \u2012 静力学和动力学。纽约:麦格劳-希尔。第 391-392 页。</li>
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分散荷载与剪力之间的关系

Relation Between the Distributed Load and Shear

了解结构分析中分散荷载与剪力之间的关系对于分析承受各种载荷条件的梁至关重要。假设梁会受到分散荷载、两个集中荷载和一对力矩的情况。 <img alt="Figure 1" src="/files/ftp_upload/14304/14304_Figure_1v2.svg" style="text-align: center; height: 300px;"> 在给定情况下剪力与分散荷载之间的联系可以按照给定的过程来进行确定。首先，假设在一段梁上没有任何集中的荷载或力矩的微小截面。我们可以绘制一个力的图来分析作用在该截面上的力。在该图中将会包括作用在梁上沿着其长度的分散荷载、右侧截面上的剪力V(x)以及为保持平衡而添加的增量剪力dV。 为了保持垂直方向的平衡，作用在截面右侧的剪力应当增加一个较小的有限量 ΔV。分布载荷的合力 w(x)Δx 作用在距离右侧一小部段离的部分。 利用垂直力的平衡方程，可以得到剪力与荷载之间的以下关系： <img alt="Equation 1" src="/files/ftp_upload/14304/14304_Equation_1v2.svg" style="text-align: center; height: 13px;"> 接下来，我们将等式的两端除以Δx并让Δx趋近于零： <img alt="Equation 2" src="/files/ftp_upload/14304/14304_Equation_2v2.svg" style="text-align: center; height: 32px;"> 这个方程表明了剪力的斜率等于分散荷载的强度。 最后，可以对方程进行重新排列并对两个任意点Q和R之间微小截面上的分散荷载进行积分: <img alt="Equation 3" src="/files/ftp_upload/14304/14304_Equation_3v2.svg" style="text-align: center; height: 50px;"> 该积分方程演示了剪切力的变化与载荷曲线下面积之间的关系。 Q 点和 R 点之间的剪力差等于这两点之间分布载荷曲线下的面积。

理解结构分析中分布载荷与剪力之间的关系，对于分析承受各种载荷的梁至关重要。

relation-between-the-distributed-load-and-shear

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Internal Forces

内力

KEY TERMS AND CONCEPTS

关键术语与概念

shear change under distributed loading

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment.

To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear force dV needed to keep the section in equilibrium.

The vertical forces must balance. For that reason, the shear force on the right-hand side changes by a small amount, ΔV. The distributed load over the small length Δx creates a resultant force of w(x)Δx. That force acts at a fractional distance from the right end of the section.

From the equilibrium equation for vertical forces, a relation between shear and load is obtained. When both sides are divided by Δx and Δx is allowed to approach zero, the slope of the shear force curve equals the distributed load intensity. This means the load level at a point controls how quickly shear changes along the beam.

The equation can also be rearranged and integrated over the elemental section between two points, Q and R. This integral form shows that the change in shear force equals the area under the distributed load curve between those points. In other words, the difference in shear force from Q to R is found by measuring the area under the load curve.

Shear Change Under Distributed Loading

sign convention

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Beam internal force analysis shows how normal force, shear force, and bending moment act inside a structural member. These internal loadings help engineers check whether a beam can support the external forces applied to it. The example uses a beam OC that weighs 5 kN and is inclined at 53.13° with the horizontal. The beam is supported at both ends. The goal is to find the internal loadings at point A. The first step is to find the reaction force at point C. This is done by taking the moment...

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Beam Types and Support Conditions

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Beam Shear Force Diagram

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Beam Load Analysis with Moment Diagrams

Beam load analysis uses a bending moment diagram to show how bending changes along the length of a beam. This diagram helps engineers and designers check whether a structure can handle applied forces. It is also useful for parts such as shelving arms, which must resist forces and moments from a load. To build the diagram, start by finding the reactive forces and any couple moments on the beam. In some cases, especially when a beam is inclined, these forces must be resolved into components that...

relation between the distributed load and shear

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment. To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear...

relation between the shear and bending moment

relation-between-the-shear-and-bending-moment

Shear and Moment Diagrams in Beam Loads

Shear and moment diagrams help show how a beam responds to loads. A beam can carry a distributed load, concentrated loads, and a couple moment. These loads create internal shear forces and bending moments. The beam is examined by isolating a small elemental section and drawing a free-body diagram. For that section to stay in equilibrium, the moment on the right side must be slightly larger than the moment on the left side. The distributed load also has a resultant force that acts at a...

shear and bending moment diagram: problem solving

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Beam Load Analysis with Shear Diagrams

Beam load analysis uses shear and bending moment diagrams to track the internal forces in a beam. These diagrams are important when a beam carries concentrated loads and a distributed load. They help show how the beam responds so structures can be designed safely and efficiently. The first step is to draw a free-body diagram of the whole beam. A free-body diagram shows the concentrated loads, the distributed load, and the reaction forces at the supports. Use the equilibrium equations, where...

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Finding Cable Shape Under Distributed Load

Cable shape under a distributed load is found by analyzing the forces along a small piece of the cable. This method is important in suspension bridge design because the main cables must carry changing loads safely. The load causes the tensile force in the cable to change, and it also creates cable deformation. The analysis starts with a free-body diagram of a short cable segment. The diagram shows the distributed load and the change in tensile force across that segment. Equilibrium equations...

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Cable Sag Under Its Own Weight

Overhead power transmission cables sag under their own weight, and that shape affects how they carry electricity across long distances. To study this behavior, the cable is analyzed with a generalized loading function. This function treats the load along the cable's arc length instead of its projected length, which gives a more accurate picture of the cable's response. The analysis then focuses on a small cable segment. A free-body diagram is drawn to show the forces acting on that segment.

cable: problem solving

Cable Tension Under Uniform Loading

Cable tension under uniform loading is found by analyzing a cable fixed between two supports. The problem starts with the support conditions and the load on the cable. From there, the cable’s shape must be described with an equation that matches the curve formed by the load. The next step is to integrate the shape equation to get the cable’s shape function. This makes it possible to solve for constants in the equation. The boundary condition at the origin gives one constant. The slope, which...

key terms and concepts

Learn how distributed load affects shear force in beams, from a free-body diagram to the slope relation and the area under the load curve.

Shear Change Under Distributed Loading | 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

Shear Change Under Distributed Loading

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