Name: beam deflection under distributed load
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 16 s
Channel: JoVE

ビームが分布荷重を運ぶ場合、ビーム上の任意の点でのせん断力と曲げモーメントを微分形式で表すことができます。
3 次線形微分方程式は、式をさらに微分し、曲げ剛性を一定と仮定することによって作成されます。
もう一度微分すると、4 次線形微分方程式が得られます。この方程式は、基本的に、分布荷重を支持するときにビームが取る形状または弾性曲線を支配します。
4次線形微分方程式に曲げ剛性定数を掛けて4回積分すると、曲線の式が得られます。
梁の曲線を定義する方程式の定数は、支持体からの傾きやたわみ、片持ち梁の自由端でせん断力と曲げモーメントがヌルか、支持梁の両端でベンドモーメントのみがゼロかなどの端部条件によって決まります。
この手法は、荷重が分散している片持ち梁または支持梁の曲線を正確に計算します。ただし、張り出した梁の支持反作用はせん断不規則性を引き起こし、全長曲線の定義には明確な機能が必要です。

荷重分布からの弾性曲線

分布荷重下での梁の構造挙動は、そのような条件下で梁がどのように曲がり、反応するかを予測することに焦点を当てた工学解析にとって重要です。異なるタイプの梁 (片持ち梁、支持梁、張り出し梁など) は、分散荷重条件下では異なる動作をします。

すべての梁について、分布荷重に対する梁の反応の解析は、梁の荷重と、その結果生じるせん断力および曲げモーメントとの関係を理解することから始まります。最初に、この関係は微分方程式として表されます。

<img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="height: 32px;">

梁の柔軟性が一定であると仮定して、この式をさらに微分すると、梁のたわみ曲線、または荷重下で梁がどのように曲がるかを記述する、より複雑な微分方程式が得られます。

<img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="height: 40px;">

この複雑な方程式を複数回積分して、この曲線の実際の形状を決定します。各積分ステップでは、梁の端での支持や接続方法など、梁の境界条件によって定義する必要がある定数が導入されます。

<img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="height: 32px;">

境界条件は、梁がどのように支持されているかによって異なります。たとえば、片持ち梁は、一端が固定され、もう一端が自由になっているため、各端でたわみと力に関して異なる制約が課せられます。逆に、支持された梁は、主に支持点に焦点を合わせた状態になります。特に困難なのは、梁の一部が支持を越えて伸びる張り出し梁の解析です。これらのセグメントは独特の力と曲げモーメントを受けるため、全長に沿った梁の挙動を正確に記述するには明確な計算が必要です。この詳細な理解により、構造物の安全性と機能性が保証されます。

ビームが分布荷重を運ぶ場合、ビーム上の任意の点でのせん断力と曲げモーメントを微分形式で表すことができます。
3次...

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

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 differential equation for the deflection curve, which is the shape the beam takes as it bends.

Finding the actual curve requires integrating the equation several times. Each integration adds a constant that must be determined from boundary conditions. Boundary conditions depend on how the beam is supported or connected at its ends.

Different beam types need different boundary conditions. A cantilever beam is fixed at one end and free at the other, so its deflection and force limits are set differently at each end. A supported beam is handled mainly at the support points.

Overhanging beams are more complex because part of the beam extends beyond the supports. Those outer sections experience unique forces and bending moments. They need separate calculations so the behavior along the full beam length can be described accurately.

Beam Deflection Under Distributed Load

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 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

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 distributed loads create shear force, bending moment, and beam deflection. See how boundary conditions change for cantilever, supported, and overhanging beams.

Beam Deflection Under Distributed Load | 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 Under Distributed Load

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