Name: critical buckling load in pin-ended columns
Uploaded: 2024-05-23T13:00:07.000Z
Duration: 1 min 21 s
Channel: JoVE

両端でピン接続されたカラムPQと、端Pに中心軸荷重が加えられている場合、この荷重が臨界荷重を超えると座屈が発生します。
臨界荷重を計算するには、柱を垂直梁として想定します。ここで、自由端Pからの距離xにあり、垂直からのたわみyを持つビームの弾性曲線上の点Oを考えます。
点Oでの曲げモーメントは、距離xに対するそのたわみの2次導関数として書くことができます。項を並べ替えると、2次微分方程式が得られ、正弦関数と余弦関数で解が表されます。
最初の境界条件では、係数 B が 0 である必要があります。2 番目の条件では、係数 A または正弦項が 0 である必要があります。
正弦項を 0 にすると、軸方向の荷重式が得られ、最小値が臨界荷重になります。これがオイラーの式です。
微分方程式にオイラーの公式を代入すると、座屈後の弾性曲線の方程式が得られます。

ピン端の柱に関するオイラーの公式

Euler&#39;s Formula for Pin-Ended Columns

構造工学では、座屈と呼ばれる圧縮軸方向の荷重下での柱の安定性は重要な考慮事項です。典型的な例には、両端がピン接続され、一端に中心軸荷重 F が加えられ、もう一端に F' = -F の反力がかかる柱 PQ が含まれます。ここで、適用される荷重が臨界荷重を超えると、装置が不安定になり座屈が発生することを理解することが重要です。

臨界荷重を計算するには、柱 PQ を垂直梁として想定します。梁の弾性曲線上で、自由端 P から距離 x の位置にある点 O を考えます。荷重を加えると、点 O は元の垂直位置から距離 y だけたわみます。この時点で、点 O での曲げモーメントは、x に対するたわみ y の 二次導関数によって説明でき、応力下の梁の挙動を理解するための軸を象徴します。

<img alt="Equation 1" src="/files/ftp_upload/15676/15676_Equation_1v2.svg" style="text-align: center; height: 40px;">

ここで、f は次のように定義されます。

<img alt="Equation 2" src="/files/ftp_upload/15676/15676_Equation_2.svg" style="text-align: center; height: 30px;">

この方程式には、正弦項と余弦項を含む一般解があります。装置の境界値は、解の係数を与えます。

<img alt="Equation 3" src="/files/ftp_upload/15676/15676_Equation_3.svg" style="text-align: center; height: 24px;">

この解決策では、正弦項がゼロであることが必要であり、臨界荷重の式が得られます。この式はオイラーの公式として知られています。

<img alt="Equation 4" src="/files/ftp_upload/15676/15676_Equation_4v2.svg" style="text-align: center; height: 36px;">

オイラーの公式を微分方程式に再度代入すると、座屈後の柱の弾性曲線の式が得られます。

ここで、オイラーの公式は、荷重を受ける前は柱が完全に真っ直ぐで均質かつ等方性であり、軸荷重が垂直軸に沿って完全に適用されるという仮定に基づいて導出されていることに注意することが重要です。

両端でピン接続されたカラムPQと、端Pに中心軸荷重が加えられている場合、この荷重が臨界荷重を超えると座屈が発生します。
宛先。。。

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

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 x from the free end P. When the load is applied, point O deflects by a distance y from its original vertical position. The bending moment at point O is then related to the second derivative of y with respect to x, which helps describe how the beam changes shape under stress.

The resulting equation has a general solution with sine and cosine terms. The boundary values of the system determine the coefficients in that solution. The sine term must be zero, and that condition gives the expression for the critical load. This expression is known as Euler's formula.

After Euler's formula is substituted back into the differential equation, the elastic curve of the column after buckling can be found. The formula is based on several assumptions. Before loading, the column must be perfectly straight, homogeneous, and isotropic. The axial load must also be applied perfectly along the vertical axis.

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key terms and concepts

Learn how pin-ended columns buckle under compressive axial load and how Euler's formula gives the critical load and elastic curve after buckling.

Critical Buckling Load in Pin-Ended Columns | JoVE K12

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Critical Buckling Load in Pin-Ended Columns

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