Name: euler buckling for different column supports
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 15 s
Channel: JoVE

オイラーのピンエンドコラムの公式は、弾性率、断面の慣性モーメント、およびコラムの長さを使用して臨界荷重を表します。
対応する臨界応力は、オイラーの公式を断面積で割ることによって表されます。ここで、断面の回転半径に対する柱の長さは、柱の細長率として定義されます。 
オイラーの公式は、柱の耐荷重能力を測定する有効座屈長を計算することにより、異なる端部接続を持つ柱に拡張できます。
カラムの有効長は、カラムの長さと経験定数kの積として表されます。
一方の端が固定され、もう一方の端が空いている列の場合、k の値は 2 です。一方、両端が固定されている列の場合、kは0.5です。
一方の端が固定され、もう一方の端がピンで固定されている列の場合、k は 0.7 です。
これらのkの値を利用すると、オイラーの公式を異なる端接続を持つ列に拡張できます。

他の端条件を持つ柱に対するオイラーの公式

オイラーの公式は構造工学の分野で非常に重要であり、ピン端柱の臨界荷重条件を理解するための基礎となります。この式は、弾性率、断面の慣性モーメント、および柱の長さを結び付け、柱が座屈しやすい臨界荷重を正確に計算します。

<img alt="Equation 1" src="/files/ftp_upload/15677/15677_Equation_1.svg" style="height: 38px;">

オイラーの臨界荷重の意味をさらに分析するには、臨界応力の概念を検討します。これは、オイラーの公式から得られる臨界荷重を柱の断面積で割ることによって計算されます。これにより、応力分布の理解が簡素化され、細長比の概念が導入されます。細長比は Le/r, で表されます。 ここで、 Le は後述する有効座屈長さ、r は柱の長さとその断面の回転半径の比です。

<img alt="Equation 2" src="/files/ftp_upload/15677/15677_Equation_2.svg" style="height: 38px;">

オイラーの洞察はピン端の柱を超えて広がり、有効座屈長さ Le の概念を通じてさまざまな構造構成について議論しています。この概念は、実験定数 k を導入することにより、オイラーの式をさまざまな端部条件を持つ柱に適合させます。この定数は、式 Le = Lk によって端部接続に基づいて柱の有効長を調整します。たとえば、一方の端が固定され、もう一方の端が自由になっている柱の k 値は 2 であり、安定性の低下を反映しています。逆に、両端が固定された柱の k 値は 0.5 であり、座屈に対する耐性の向上を反映しています。 k の値は、一方の端が固定され、もう一方の端が固定されている柱の場合は 0.7 など、他の端の条件によってさらに変化し、オイラーの公式を普遍的に適用できます。オイラーの公式のこの適応性により、エンジニアは幅広い構造シナリオの臨界荷重条件を予測でき、より安全で弾力性のある構造を設計できるようになります。

オイラーのピンエンドコラムの公式は、弾性率、断面の慣性モーメント、および長さを使用して臨界荷重を表します...

euler-s-formula-to-columns-with-other-end-conditions

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

Euler&#39;s Formula to Columns with Other End Conditions

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 slenderness ratio, written as L e /r. Here, L e is the effective buckling length, and r is the radius of gyration of the cross-section.

Euler's method also works for columns with different end conditions. Engineers use an empirical constant, k, to adjust the effective length of the column with the formula L e = Lk. A column with one end fixed and the other free has k = 2, so it is less stable. A column with both ends fixed has k = 0.5, so it resists buckling better. A column with one end fixed and the other pinned has k = 0.7.

These end-condition values let Euler's formula apply to many structural cases. That makes it useful for predicting critical loading conditions in a wide range of columns. It also helps engineers design safer and more resilient structures.

Euler Buckling for Different Column Supports

euler-buckling-for-different-column-supports

stability of structures

Critical Load and Structure Stability

Structural stability is an important idea in mechanical engineering. It helps engineers design buildings, vehicles, and machines that stay safe and efficient under load. A simple model for studying this behavior uses two rods joined at a pivot point O with a torsional spring, which is a spring that resists twisting, at the pivot. The setup is similar to a scissor jack used to lift a car. In the example, the linkage arms are vertical, which matches the case where the vehicle has been raised as...

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

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

euler-s-formula-to-columns-problem-solving

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

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...

design of columns under a centric load

Column Buckling and Load Design Basics

Column design under a centric load is a key part of structural engineering. It helps engineers keep buildings and other structures stable and safe. The main tools used here are Euler’s formula and Secant’s formula. Euler’s formula is used when a column is assumed to be perfect, straight, and uniform. It also applies when the material stays within its elastic limit. The critical load from Euler’s formula depends on the material’s modulus of elasticity and the column’s shape. This formula works...

design of columns under an eccentric load

design-of-columns-under-an-eccentric-load

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 Euler buckling predicts critical load, critical stress, and slenderness ratio for columns with fixed, pinned, or free ends.

Euler Buckling for Different Column Supports | 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。

Euler Buckling for Different Column Supports

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