Name: beam slope and deflection by singularity
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 19 s
Channel: JoVE

屋根構造内の梁のたわみは、単一の解析関数で曲げモーメントを表せる場合、積分法を使用して決定できます。
ただし、ビームの荷重が曲げモーメントを表すために複数の関数を必要とする場合、追加の定数と方程式が必要になり、計算が長くなります。
この複雑さは、特異点関数を使用して簡略化できます。偏心荷重を支えている端部で支持された角柱状のビームを考えてみましょう。
このビームのせん断力関数は、適切な特異点関数を使用して表すことができます。曲げモーメント関数は、このせん断力関数を積分することで導出できます。
ビームの傾きとたわみは、このモーメント関数を積分することで求めることができます。方程式の定数は、境界条件から決定できます。
特異点関数を使用すると、追加の定数や方程式が不要になり、計算が簡単になります。その結果、特異点関数は、複雑な荷重下でのビームの傾きとたわみを効率的に計算する方法を提供します。

梁のたわみ

構造工学におけるさまざまな荷重条件下で梁のたわみと傾斜を正確に決定することは、安全性と構造的完全性を確保するために非常に重要です。特異点関数は、特に複数の荷重関数が曲げモーメント方程式を複雑にする場合に、梁を解析するための効率的なアプローチを提供します。

前のレッスンで説明した特異点関数は、構造荷重シナリオで一般的に発生する関数内の不連続性を表す強力な数学ツールです。これらの関数は、複雑な荷重や複数の荷重がかかっている場合でも、せん断力と曲げモーメントの方程式をコンパクトに表現するのに役立ちます。

<img alt="Equation 1" src="/files/ftp_upload/15669/15669_Equation_1a.svg" style="height: 32px;">
<img alt="Equation 2" src="/files/ftp_upload/15669/15669_Equation_1b.svg" style="height: 32px;">

通常、長さ方向に均一で両端が支持されている角柱梁の場合、偏心荷重が特有の課題を引き起こします。このような梁の任意の点におけるせん断力は、特異点関数を使用してモデル化できます。これらの関数は、せん断力図でステップ関数として表される荷重など、特定の点または特定の間隔で適用される荷重によって生じる不連続性を簡単に処理します。

せん断力関数を積分することによって導出される曲げモーメントは、梁の性能を評価するために重要です。このステップは、梁の応力分布と全体的なたわみに影響を与えます。梁のたわみは、曲げモーメント関数を 2 回積分し、梁の境界条件を適用して積分定数を求めることによって決定されます。特異点関数を使用してせん断力と曲げモーメントをモデル化すると、複数の追加の定数や複雑な方程式が不要になり、計算が簡素化され、計算効率が向上します。この方法により、構造システムの安全な設計とメンテナンスに不可欠な、梁のたわみと応力分布に関するさまざまな荷重条件を簡単に評価できます。

屋根構造内の梁のたわみは、単一の解析関数が...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Deflection of Beams

KEY TERMS AND CONCEPTS

重要用語と概念

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 beam example shown here, a prismatic beam, which means a beam with a uniform cross-section, is supported at both ends and carries an eccentric load, or a load placed off center.

The shear force at any point in the beam can be written with singularity functions. These functions handle the jump in the shear force diagram caused by the applied load. The bending moment is then found by integrating the shear force equation, and it is important because it affects the stress in the beam and the amount of deflection.

To find deflection, the bending moment equation is integrated twice. The beam boundary conditions are then used to solve for the integration constants. This approach keeps the equations compact, reduces the need for extra constants, and makes calculations more efficient for different loading conditions. It is a useful way to check beam performance and support safe structural design and maintenance.

Beam Slope and Deflection by Singularity

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 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 singularity functions simplify shear force, bending moment, slope, and beam deflection calculations for a prismatic beam with an eccentric load.

Beam Slope and Deflection by Singularity | 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 Slope and Deflection by Singularity

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