Name: moment-area method for cantilever deflection
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 15 s
Channel: JoVE

モーメント面積法は、集中荷重とモーメントの下でカンチレバービームに実装して、傾斜とたわみを特定できます。
このプロセスは、ビームの自由体線図を描画し、固定端での反力を計算することから始まります。
次に、曲げモーメント図を描画して、モーメントがゼロに等しい点を決定します。
次に、M/EI図を描画して、セグメントAOとOBに対応する領域を特定し、x軸に対する位置に基づいて正または負の符号を割り当てます。
最初のモーメント面積定理は接線間の角度を計算するために適用され、2 番目のモーメント面積定理は、A と B の間の総面積の B を通る垂直軸の周りの最初のモーメントに等しい接線偏差を決定するために使用されます。
最後に、Bでのたわみが決定され、これは接線方向の偏差に等しくなります。次に、ビームの偏向形状をスケッチして視覚的に理解します。

対称荷重を持つ梁

Beams with Symmetric Loadings

モーメント面積法は、さまざまな荷重下での梁の傾きとたわみを決定するために構造工学で使用される解析ツールです。自由端に集中荷重とモーメントがある片持ち梁を考えてみましょう。最初のステップは、固定端での反応を計算するための自由体図を構築することです。次に、曲げモーメントがゼロに等しい点に焦点を当てて、曲げモーメントが梁の長さに沿ってどのように変化するかを視覚化するために、曲げモーメント図がプロットされます。

次に、M/EI 図が描画されます。ここで、M は曲げモーメント、E は弾性率、I は梁断面の慣性モーメントです。この図は、曲げモーメントがゼロである点間のセグメントの曲線の下の領域を特定し、X 軸に対する位置に基づいて領域に正または負の符号を割り当てます。

第 1 モーメント面積定理を適用して、2 点間の M/EI 図の下の面積を積分することにより、梁上の任意の点での傾きを計算します。次に、第 2 モーメント面積定理を使用して梁のたわみを求め、端点を通過する垂直軸の周りの M/EI 図の下の面積の 1 次モーメントと同等とします。

最後に、梁のたわんだ形状がスケッチされ、解析結果が視覚的に表現され、加えられた荷重条件下での梁の曲がりが示されます。この方法により、片持ち梁の構造性能について正確な洞察が得られます。

モーメント面積法は、集中荷重とモーメントの下でカンチレバービームに実装して、傾斜とたわみを特定できます。
プロセス...

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 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, the M/EI diagram is constructed. In this diagram, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia of the beam cross-section. The areas under the curve are identified between the zero-moment points. Each area is given a positive or negative sign based on its position relative to the x-axis.

The first moment-area theorem is then used to find the slope at a point on the beam. It does this by using the area under the M/EI diagram between two points. The second moment-area theorem is used to find deflection. It equals the first moment of the area under the M/EI diagram about a vertical axis through the endpoint.

The final step is to sketch the deflected shape of the beam. This gives a clear picture of how the beam bends under the applied load and moment. The method provides precise insight into the structural behavior of cantilever beams.

Moment-Area Method for Cantilever Deflection

moment-area-method-for-cantilever-deflection

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

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...

beams with symmetric loadings

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

Use the moment-area method to analyze a cantilever beam with a load and end moment. Learn how free-body diagrams, M/EI diagrams, slope, and deflection are found.

Moment-Area Method for Cantilever Deflection | JoVE K12

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mechanical engineering

Moment-Area Method for Cantilever Deflection

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