Name: column bending under off-center load
Uploaded: 2024-05-23T13:00:07.000Z
Duration: 1 min 16 s
Channel: JoVE

偏心荷重Fを受ける柱を考えてみましょう。この偏心荷重は、偏心荷重Fと偏心荷重によって生成されるカップルモーメントの組み合わせに相当します。 
列PRの断面の自由体図を描画し、適切な座標系を選択すると、点Rでのカップルモーメントが得られます。
弾性曲線の微分方程式の点Rでのカップルモーメントを代入して解くと、弾性曲線の方程式が得られます。解の係数は、境界値条件を使用して計算できます。
カラムの半分の長さでは、カラムの最大たわみを決定できます。最大たわみの方程式は、割線項が無限大になると無限大になります。
無限変形基準は、臨界荷重条件の式を与えます。これを式で最大変形に代入すると、臨界荷重に関して最大変形が得られます。 
柱の最大応力は、曲げモーメントが柱の中点の横断面で最大になる場合に発生します。

偏心荷重

偏心荷重は、構造工学と力学の研究、特に柱の安定性と応力分布を分析する場合に重要な概念です。力が重心軸に沿って加えられて均一な圧縮が生じる中心荷重とは異なり、偏心荷重は力が中心から外れたときに発生します。この偏心した適用により、直接的な圧縮応力だけでなく曲げ応力も導入され、荷重下での柱の挙動に大きな影響を与えます。偏心荷重は、概念的に 2 つの成分、つまり柱の軸に沿って作用する中心荷重 (F) と荷重の偏心によって生じる偶モーメントに分解できます。この偶モーメントの大きさは、加えられる荷重が柱の重心軸からどれだけ離れているかによって決まり、軸方向の圧縮に加えて曲げが生じるため、柱の応答を理解する上で極めて重要です。

このシナリオを分析的に調査するには、偏心荷重がかかる柱を考えてみましょう。柱の断面 PR が選択され、その自由体図が描画されます。 これは、柱に作用する力とモーメントを視覚化するのに役立ちます。適切な座標系を選択すると、特定の点 (点 R など) での偶モーメントを決定できます。これは、その後の柱の動作の数学的モデリングに不可欠です。次のステップでは、この偶モーメントを柱の弾性曲線を支配する微分方程式に組み込みます。この微分方程式の解は、加えられた荷重の下で柱がどのように曲がるかを記述する弾性曲線の方程式を与えます。境界値条件を適用することで、解の係数を決定し、モデルの精度をさらに高めることができます。

この解析の重要な点は、柱の最大たわみを特定することです。通常、このたわみは柱の中間点で発生します。 この最大たわみは、加えられた荷重の下で柱がどの程度曲がるかを示すため、柱の安定性を評価する上で極めて重要です。この偏向の方程式は興味深い現象を示しています。方程式内の正割項が無限大になるにつれて偏向が無限大に近づくことを示唆しています。この状態は、それを超えると柱が安定性を失い、座屈が発生するしきい値を示します。無限たわみ基準から導出される臨界荷重条件は、エンジニアが柱が安全な動作制限内で設計されていることを確認するための基本です。臨界荷重条件を最大たわみの式に代入すると、臨界荷重の観点から最大たわみを表す式を導き出すことができます。この関係は、過度の変形や破損を生じることなく偏心荷重に耐えられる柱を設計する上で極めて重要です。

偏心荷重Fを受ける柱を考えてみましょう。この偏心荷重は、中心荷重Fとカップルモーメント...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Columns

カラム

KEY TERMS AND CONCEPTS

主要用語と概念

Eccentric Loading

column bending under off-center load

Eccentric loading shows how a column bends when a force is applied off-center. In structural engineering, this matters because the load no longer acts along the centroidal axis, so the column sees both direct compression and bending stress.

The off-center force can be treated as two parts. One part is a centric load F that acts along the column’s axis. The other part is a couple moment created by the eccentricity, or distance, between the load and the centroidal axis.

To study the column mathematically, a section PR is selected and a free-body diagram is drawn. This diagram helps show the forces and moments on the section. A coordinate system is then chosen so the couple moment at a point such as R can be found and used in the model.

That moment is added to the differential equation for the elastic curve of the column. Solving the equation gives the elastic curve, which describes the column’s shape under load. Boundary value conditions are then used to determine the coefficients in the solution.

The analysis also focuses on maximum deflection, which usually occurs at the midpoint of the column. This value shows how much the column bends under the applied load. The deflection expression includes a secant term, and the deflection becomes infinite when that term becomes infinite.

That infinite-deflection condition marks the loss of stability and the start of buckling. The corresponding critical loading condition is important for safe column design. Substituting that condition into the maximum-deflection expression gives a relationship between maximum deflection and critical loading.

Column Bending Under Off-Center Load

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euler&#39;s formula for pin-ended columns

Critical Buckling Load in Pin-Ended Columns

Critical buckling load in pin-ended columns is a key idea in structural engineering. It explains when a column under compressive axial load becomes unstable and starts to buckle. A pin-connected column with a centric load F at one end has a reaction force of F' = -F at the other end. When the applied load rises above the critical load, buckling begins. To find the critical load, the column PQ can be treated like a vertical beam. Point O is chosen on the elastic curve of the beam at a distance...

euler&#39;s formula to columns with other end conditions

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Euler Buckling for Different Column Supports

Euler buckling explains how a column can fail when it is compressed. In structural engineering, it helps predict the critical load, which is the load level where a column is likely to buckle. The formula uses the modulus of elasticity, the moment of inertia of the cross-section, and the column length. The same idea also helps describe critical stress. Critical stress is found by dividing the critical load from Euler's formula by the cross-sectional area of the column. This leads to the...

euler&#39;s formula to columns: problem solving

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Spring-Loaded Column Buckling Analysis

Spring-loaded column buckling analysis uses static equilibrium to find the critical load in a rigid bar system. The setup has two equal-length rigid bars, AB and BC, joined at pivot point B. A tension spring with constant k is attached to B and pulls horizontally. A vertical load F is applied at point A, and the bars rotate around the pivot like a knee joint. As point B moves, the spring stretches by an amount X. The spring then pushes back against the sideways motion of B. To study the...

eccentric loading

Eccentric loading shows how a column bends when a force is applied off-center. In structural engineering, this matters because the load no longer acts along the centroidal axis, so the column sees both direct compression and bending stress. The off-center force can be treated as two parts. One part is a centric load F that acts along the column’s axis. The other part is a couple moment created by the eccentricity, or distance, between the load and the centroidal axis. To study the column...

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Column Buckling and Load Design Basics

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design of columns under an eccentric load

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Eccentric Load Stresses in Structural Columns

Structural columns under eccentric load must be designed for both axial force and bending stress. An eccentric load is an off-center load, and it creates a non-uniform stress pattern across the column. Engineers must analyze this stress distribution carefully to reduce the risk of failure. The stress in an eccentrically loaded column can be treated as the sum of two parts. One part comes from an equivalent centric load, which is a load applied through the center of the column. The other part...

key terms and concepts

Learn how an off-center force creates compression and bending in a column, how the elastic curve is modeled, and why infinite deflection signals buckling.

Column Bending Under Off-Center 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

Column Bending Under Off-Center Load

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