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

モーメント-面積定理は、建物の床を支える梁上の任意の点でのたわみと傾きを決定するための弾性曲線の幾何学的特性を提供します。
ビームの長さに沿ってプロットすると、M/EI比は曲げモーメント図と同様の図を作成します。
モーメント-面積定理は、ビーム図から傾きと接線方向の偏差方程式を導き出します。最初のモーメント-面積定理は、点PとQの接線間の角度を、M/EI図の下のこれらの点間の面積に等しくします。
2 番目のモーメント-面積定理は、点 P の Q からの接線方向の偏差と、P を通る垂直軸に関する同じ領域の最初のモーメントに関連します。
この定理は、面積の積と、面積の重心からPを通る垂直軸までの距離の観点からも表現できます。
接線方向の偏差は、Q から P の接線までの垂直距離です。これは、M/EI ダイアグラムの下の面積に、その重心から Q を通る垂直軸までの距離を掛けて計算されます。

モーメント面積定理

モーメント面積定理は、特に建物の床支持材などの用途において、梁の曲げを解析する構造工学において重要です。この定理は、荷重下で梁がどのように変形するかを示す弾性曲線の幾何学的特性を利用して、たわみと傾斜の計算を簡素化します。


定理は 2 つの部分に分かれています。最初の部分は、梁の弾性曲線上の任意の二点間の接線の角度と、曲げモーメントM、弾性率E、断面二次モーメントIを用いてM/EIとしてプロットした曲線の下の面積とを結びつけます。この曲線の下の領域は、これら 2 点間で発生する回転の合計に直接対応します。 


定理の 2 番目の部分は、梁の曲げによって生じる任意の 2 点間の接線方向の偏差、または垂直方向の変位を扱います。この偏差は、これらの点の 1 つを通る垂直軸の周りの M/EI 曲線の下の領域の 1 次モーメントに相当し、元の位置からの梁セグメントの変位の尺度を提供すると述べています。これらの定理は、構造の安全性と荷重下での性能を確保するために不可欠な、梁に沿ったさまざまな点での傾斜とたわみを効率的に決定します。

モーメント-面積定理は、弾性曲線の幾何学的特性を提供して、支持するビーム上の任意の点でのたわみと傾きを決定します...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Deflection of Beams

梁のたわみ

KEY TERMS AND CONCEPTS

重要用語と概念

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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Beam Deflection Using Moment-Area Rules

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