Name: beam deflection using moment-area rules
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 17 s
Channel: JoVE

面积矩定理提供了弹性曲线的几何特性，可用于确定承受建筑物楼面荷载的梁上任意点的挠度和转角。
沿梁的长度方向绘制时，M/EI 的比值会形成一个类似于弯矩图的图形。
面积矩定理从梁的受力图推导出斜率和切向挠度方程。第一面积矩定理指出，P点和Q点处切线之间的夹角等于这两点之间M/EI图下的面积。
第二面积矩定理将点 P 相对于点 Q 的切向偏差与同一面积对通过点 P 的垂直轴的一次矩相关联。
该定理也可以用面积与其形心到通过点 P 的垂直轴的距离的乘积来表示。
切向偏差是从点 Q 到 P 点切线的垂直距离。它通过将 M/EI 图中曲线下的面积乘以该面积形心到通过点 Q 的垂直轴之间的距离来计算。

矩面积定理

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 矩面积定理是结构工程中用来分析梁弯曲的关键定理，尤其是在建筑楼板支撑等应用中。该定理能够利用弹性曲线的几何特性（描述梁在载荷下如何发生变形）来简化对挠度和斜度的计算。
 
 该定理分为两部分。第一部分是将位于梁弹性曲线上任意两点切线之间的夹角与通过M/EI （其中 M 是弯矩，E 是弹性模量，I 是惯性矩）在曲线下方所围成的面积与沿梁长度方向的挠度联系起来。该曲线下方的面积直接对应于这两点之间发生的总旋转量。
 
 定理的第二部分解决了由梁弯曲所产生的在任意两点之间的切向偏差或垂直位移。该定理指出，该偏差相当于 M/EI 曲线下面积绕通过这些点之一在垂直轴上的一阶矩，从而提供了梁段相对于其原始位置的位移量。这些定理有效地确定了位于梁上各个点的坡度和挠度，这对于确保在荷载下的结构安全和性能是至关重要的。
 
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弯矩-面积定理提供了弹性曲线的几何特性，可用于确定承受载荷的梁上任意点的挠度和转角。

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Deflection of Beams

梁的挠度

KEY TERMS AND CONCEPTS

关键术语与概念

Moment-Area Theorems

beam deflection using moment-area rules

The moment-area theorems help describe beam bending in structural engineering. They are often used for building floor supports and other beams under load. The theorems use the elastic curve, which is the shape a beam takes as it deforms.

The first theorem links slope change to area on an M/EI diagram. Here, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia. When M/EI is plotted along the beam length, the area under the curve gives the total rotation between two points on the elastic curve.

The second theorem deals with tangential deviation, or vertical displacement, between two points. It says this distance equals the first moment of the area under the M/EI curve about a vertical axis through one of the points. Together, the two theorems make it easier to find slope and deflection at different points along a beam. That information helps engineers check structural safety and performance under load.

Beam Deflection Using Moment-Area Rules

deformation of a beam under transverse loading

deformation-of-a-beam-under-transverse-loading

Beam Deflection from Loads and Supports

Beam deflection depends on the loads on a beam and the support conditions at its ends. For indeterminate beams, especially those with overhanging parts and multiple concentrated loads, careful analysis is needed to predict how the beam will bend. The first step is to draw an accurate free-body diagram. A free-body diagram shows the forces and moments acting on the beam. It helps reveal how the bending moment changes along the beam length. The bending moment diagram also shows how the beam...

equation of the elastic curve

Beam Deflection and Boundary Conditions

Beam deflection and boundary conditions help describe how a beam bends under load. In structural engineering, the curvature of a plane curve shows how sharply the beam bends. That curvature is found from the curve’s first and second derivatives. A cantilever beam with a point load at its free end is a common example. A diving board works like this type of beam. When slopes are small, the beam’s elastic curve becomes the main shape used in the analysis. The governing equation connects bending...

elastic curve from the load distribution

Beam Deflection Under Distributed Load

Beam deflection under distributed load is a key part of beam analysis in engineering. It helps predict how a beam bends and reacts when a load is spread along its length. Engineers use this relationship to judge whether a structure will stay safe and functional. The analysis starts with the link between load, shear force, and bending moment. These quantities can be written in differential form. When beam flexibility is treated as constant, further differentiation leads to a higher-order...

deflection of a beam

Beam Slope and Deflection by Singularity

Beam slope and deflection can be found with singularity functions when beam loading is hard to write as one simple equation. This method is useful in structural engineering because it helps describe beams under several loads without breaking the problem into many separate cases. Singularity functions are mathematical tools for showing discontinuities in a load, shear force, or bending moment equation. They are especially helpful when a load acts at a single point or over a set interval. In the...

method of superposition

Beam Deflection Under Combined Loads

Beam deflection under combined loads can be found with the method of superposition. This structural engineering method is used to analyze how multiple loads affect a beam. It works by finding the slope and deflection from each load separately, then adding those results together. The method applies only when the beam stays within its elastic limit. In that range, the material deforms in a linearly elastic way. That makes it valid to combine the effects of separate loads without changing the...

moment-area theorems

The moment-area theorems help describe beam bending in structural engineering. They are often used for building floor supports and other beams under load. The theorems use the elastic curve, which is the shape a beam takes as it deforms. The first theorem links slope change to area on an M/EI diagram. Here, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia. When M/EI is plotted along the beam length, the area under the curve gives the total rotation...

beams with symmetric loadings

Moment-Area Method for Cantilever Deflection

The moment-area method is used in structural engineering to find the slope and deflection of a beam under load. It is especially useful for a cantilever beam with a concentrated load and a moment at the free end. The analysis begins with a free-body diagram. This diagram helps calculate the reactions at the fixed end. After that, the bending moment diagram is drawn to show how the bending moment changes along the beam. Points where the bending moment is zero are important in this step. Next,...

beams with unsymmetric loadings

Beam Deflection and Level Points

Beam deflection and level points can be found for a supported beam under unsymmetrical loadings using moment-area theorems. This type of analysis helps show how a beam bends when forces are spread unevenly along it. It is an important part of structural engineering because it links load patterns to beam response. The first moment-area theorem is used to find the slope at any point on the beam. It states that the change in slope between two points equals the area under the moment diagram over...

maximum deflection

Locating the Beam’s Highest Deflection

Maximum deflection in a supported beam depends on where the load is placed. With an unsymmetrical load, such as a train moving across a bridge, the beam may bend most at a point that is not the midpoint. Finding that point is important for judging stress and deflection in the structure. The point of maximum deflection is called point O. At point O, the tangent to the deflection curve is horizontal. That means the slope is zero there. To locate point O, the slope at any point X along the beam...

key terms and concepts

Learn how the moment-area theorems use the elastic curve and M/EI diagram to find beam slope, rotation, and deflection in structural engineering.

Beam Deflection Using Moment-Area Rules | 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 Deflection Using Moment-Area Rules

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