Name: beam deflection and boundary conditions
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 23 s
Channel: JoVE

曲線上の点における平面曲線の曲率は、曲線の 1 次導関数と 2 次導関数を含む式を使用して表すことができます。
傾きはビームの弾性曲線にとって重要ではないため、弾性曲線の支配方程式は、曲げ剛性を考慮した 2 次線形微分方程式として表されます。
曲げ剛性がビームに沿って変化する場合、それはxの関数として表されます。ただし、プリズムビームの場合、曲げ剛性は一定のままです。
この方程式の積分により、水平のある点で弾性曲線の接線によって形成される角度が得られます。
この小さな角度が統合されると、任意の点でビームがたわむことができます。ビームサポートの境界条件によって、方程式の定数が決まります。
支持梁、張り出し梁、片持ち梁の3種類が主に考慮されます。支持梁と張り出し梁の場合、支持点での変形はゼロです。
支持点での変形と傾斜は、片持ち梁ではどちらもゼロです。これらの条件により、定数の計算が可能になります。

弾性曲線の方程式

Equation of the Elastic Curve

構造工学において重要な平面曲線の曲率の概念は、荷重下で梁がどの程度鋭く曲がるかを定義します。この曲率は、曲線の一次導関数と二次導関数を使用して決定されます。

自由端に点荷重がかかる片持ち梁 (飛び込み台など) を考えてみましょう。小さな勾配での梁のたわみを解析する場合、梁の弾性曲線の形状が重要になります。この解析の支配方程式には、曲げモーメントと梁の曲げ剛性が含まれます。曲げ剛性は、梁断面の弾性率と慣性モーメントの積です。

<img alt="Equation 1" src="/files/ftp_upload/15667/15667_Equation_1.svg" style=" height: 32px;">

角柱状梁の場合、断面が一定のままであるため、解析が簡素化され、梁の長さに沿って曲げ剛性が一定になります。支配方程式を積分すると、任意の点での曲線の接線によって形成される角度を計算でき、さらに積分すると、その点での梁のたわみが得られます。

<img alt="Equation 2" src="/files/ftp_upload/15667/15667_Equation_2.svg" style=" height: 28px;">

これらの計算を完了するには、梁の支持点における境界条件が不可欠です。支持梁、張り出し梁、片持ち梁は一般的なタイプの梁であり、それぞれに異なる境界条件があります。たとえば、片持ち梁の支持点でのたわみと傾きはゼロですが、これはたわみ方程式の定数を計算するために不可欠です。

梁のたわみを正確に予測することは、構造の安全性と機能性を確保するために非常に重要です。過度のたわみは構造上の破損や保守性の問題を引き起こす可能性があり、荷重下の梁の動作を理解することの重要性が強調されます。

曲線上の点における平面曲線の曲率は、曲線の 1 次導関数と 2 次導関数を含む式を使用して表すことができます。
ザ。。。

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

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 moment with flexural rigidity. Flexural rigidity is the product of the modulus of elasticity and the moment of inertia of the beam’s cross-section. For a prismatic beam, the cross-section stays constant, so the flexural rigidity is constant along the beam.

Integrating the governing equation gives the angle of the tangent to the curve at any point. A second integration gives the deflection at that point. Boundary conditions at the beam supports are needed to find the constants in these equations.

Supported, overhanging, and cantilever beams each have different boundary conditions. For a cantilever beam, the deflection and slope at the support point are zero. Predicting deflection is important because too much bending can lead to structural failure or serviceability problems.

Beam Deflection and Boundary Conditions

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 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

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 beam deflection, curvature, and boundary conditions work together in structural engineering, using cantilever and other beam types.

Beam Deflection and Boundary Conditions | 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 and Boundary Conditions

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