Name: beam load analysis with moment diagrams
Uploaded: 2025-05-22T09:06:03.000Z
Duration: 1 min 30 s
Channel: JoVE

考虑一根在均布载荷作用下、由两个铰接支座支撑的外伸梁 AC，绘制其弯矩图。
绘制梁的受力图，并利用平衡方程计算各支座处的反作用力。
将该方法应用于梁。
接下来，考虑区域 AB 内任意一点距离 x。合力作用于该段的中点。通过在自由体图上应用平衡方程并代入数值，可得到力矩方程。
类似地，考虑区域 BC 内的任意一点。通过绘制受力图并利用平衡方程，可确定截面 BC 的弯矩方程。
将位置值代入力矩方程，计算得到弯矩值。
弯矩图展示了沿梁长度方向的弯矩值的图形表示。
通过回顾力矩方程并应用微分方法，可求得每一段的局部最大值。
通过抛物线曲线连接这些点，得到弯矩图。

<ol>
	<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 362 页。</li>
</ol>

弯矩图

Bending Moment Diagram

弯矩图是梁在负载下沿梁长度所经历的弯矩的图形表示方法。它是工程师和设计师用于分析结构并确保其能够承受外力的重要工具。下面列出了创建梁弯矩图的步骤。 确定反作用力和偶力矩：计算作用在梁上的所有反作用力和偶力矩。在某些情况下，当梁以一定角度进行倾斜时，这些力将会被分解为与梁垂直和平行的轴分量。 对梁进行分割并绘制受力分析图：考虑距离梁左端的任意距离。该任意距离可以沿着梁的长度延伸到不同区域中。在每个假定距离处，对梁进行剖析并创建其中一个段的受力分析图。根据既定的符号约定，以此来确保弯矩(M)的值为正值。 计算弯矩：通过力矩的平衡方程，可以得出关于分段末端的弯矩（M）。 绘制弯矩图：根据得出的沿梁长度的弯矩创建弯矩图（M与x）。如果通过计算得出的描述M的函数为正值，则在x轴上方来对这些值进行绘制。相反，如果值为负数，则在x轴下方对它们进行绘制。 按照这个步骤，你可以为任何梁构建准确的剪力和弯矩图。这些图表对于设计结构元件（例如货架臂）至关重要，这些结构元件可抵抗由施加的负载所引起的力和力矩。

弯矩图是梁在载荷作用下沿其长度方向所受弯矩的图形化表示。它是一种重要的……

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

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 act perpendicular and parallel to the beam axis. That gives the values needed for the next step.

Next, section the beam at an arbitrary distance from the left end and draw a free-body diagram for one part of the beam. The chosen distance can be moved to different regions along the beam length. On the cut section, show the bending moment, M, in the positive sense according to the sign convention.

Then apply the moment equilibrium equations to calculate the bending moment about the sectioned end. Use the results to plot the moment diagram, or M versus x, along the beam length. Positive values are drawn above the x-axis, and negative values are drawn below it. Following this method gives accurate shear and bending moment diagrams for beam design.

Beam Load Analysis with Moment Diagrams

sign convention

Beam Forces: Tension, Shear, and Bending

Beam analysis depends on the internal forces and moments inside a structure. These include normal force, shear force, and bending moment. Engineers use the method of sections to cut the beam and find these values. A sign convention helps show whether each result is positive or negative. Normal force acts perpendicular to the beam’s cross-section. It can pull the material in tension or push it in compression. Tension is taken as positive, while compression is taken as negative. This makes it...

normal and shear force

Beam Loads and Internal Forces

Beam loads create internal forces inside a structure. Weight, pressure, and other external forces can all act on a beam. These internal forces affect the beam’s stability and strength. Engineers study these forces to find their size and direction. A common approach is the method of sections. In this method, an imaginary cut is made at a chosen point on the beam, and the forces at that section are analyzed. One internal force is the normal force, also called the axial force. It acts...

bending and torsional moments

Forces That Bend and Twist Materials

Bending and torsional moments describe how materials respond to force. These ideas are important in structural engineering because they help explain how beams, rulers, and screws change shape under load. A moment depends on both the size of the force and the distance from the point of interest to the force’s line of action. A bending moment forms when an external force makes a structural element bend. If the element bends upward, the top side is in compression and the bottom side is in tension.

internal loadings in structural members: problem solving

internal-loadings-in-structural-members-problem-solving

Beam Internal Force Analysis at Point A

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...

beams

Beam Types and Support Conditions

Beams are key parts of structural engineering and construction. They support loads that act at different points along their length. Engineers use beam shape and support conditions to understand how a beam will perform. Beams can be classified by geometry. Straight beams are the most common and have a constant cross-section along their length. Tapered beams change in cross-section from one end to the other, while curved beams have a curved shape. Beam cross-section also matters. Common shapes...

shear diagram

Beam Shear Force Diagram

Beam shear force diagrams show how shear changes along the length of a supported beam. In beam mechanics, they help students track the internal shear force caused by perpendicular loads. The beam AB in this example is supported at both ends, so the first step is to draw a free-body diagram that shows the external forces and internal reactions. The reaction forces at supports A and B are found with equilibrium equations for force and moment. For the beam shown, the vertical reaction at A is 24...

bending moment diagram

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

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

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...

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

shear-and-bending-moment-diagram-problem-solving

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...

cable subjected to concentrated loads

Finding Tension in a Loaded Cable

Flexible cables can carry vertical point loads between two fixed supports. The cable is treated as flexible, inextensible, and nearly weightless. Under these conditions, it changes shape to form straight-line segments, and each segment carries a constant tensile force. To analyze the cable, a free-body diagram is drawn for the whole cable. This helps identify the reaction forces at the supports. In many cases, the number of unknown reaction components is greater than the available equilibrium...

cable subjected to a distributed load

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...

cable subjected to its own weight

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 to analyze beam loads by finding reaction forces, sectioning the beam, solving for bending moment, and plotting the moment diagram.

Beam Load Analysis with Moment Diagrams | 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

Beam Load Analysis with Moment Diagrams

Description

Transcript