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

양쪽 끝이 핀으로 연결되어 있고 끝 P에 중심 축 방향 하중이 가해지는 컬럼 PQ를 생각해 보십시오. 이 하중이 임계 하중을 초과하면 좌굴이 발생합니다.
임계 하중을 계산하려면 기둥을 수직 보로 상상합니다. 이제, 자유단 P로부터 거리 x에 있는 빔의 탄성 곡선에 있는 점 O와 수직에서 처짐 y를 갖는 점을 고려하십시오.
점 O에서의 굽힘 모멘트는 거리 x에 대한 처짐의 2차 도함수로 쓸 수 있습니다. 항을 재정렬하면 2차 미분 방정식이 얻어지며, 이는 사인 및 코사인 함수로 해를 표현합니다.
첫 번째 경계 조건에서는 계수 B가 0이어야 합니다. 두 번째 조건은 계수 A 또는 사인 항이 0이어야 합니다.
사인 항을 0으로 설정하면 축 하중 표현식이 생성되며, 가장 낮은 값이 임계 하중이 됩니다. 이것이 오일러의 공식입니다.
오일러의 공식을 미분 방정식에 대입하면 좌굴 후의 탄성 곡선에 대한 방정식을 얻을 수 있습니다.

핀 끝 기둥에 대한 오일러 공식

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

구조 엔지니어링에서 압축 축 하중을 받는 기둥의 안정성은 좌굴로 설명되는 중요한 고려 사항입니다. 일반적인 예로는 양쪽 끝이 핀으로 연결되어 있고 한쪽 끝에서 중심 축 하중 F를 받고 다른 쪽 끝에서는 반력 F' = -F를 받는 기둥 PQ가 있습니다. 여기서, 작용하중이 임계하중을 초과하면 시스템이 불안정해지면서 좌굴이 발생한다는 것을 이해하는 것이 중요합니다.

임계 하중을 계산하려면 기둥 PQ를 수직 보로 생각해 보세요. 보의 탄성 곡선에 있는 점 O를 생각해 보세요. 자유 끝 P에서 거리 x만큼 떨어져 있습니다. 하중이 가해지면 점 O는 원래 수직 위치에서 거리 y만큼 편향됩니다. 이 시점에서 점 O의 굽힘 모멘트는 x에 대한 편향 y의 2차 도함수로 설명할 수 있으며 이는 응력 하에서 빔의 동작을 이해하는 방향의 피벗을 상징합니다.

<img alt="Equation 1" src="/files/ftp_upload/15676/15676_Equation_1v2.svg" style="height: 40px;">

여기서 f는 다음과 같이 정의됩니다.

<img alt="Equation 2" src="/files/ftp_upload/15676/15676_Equation_2.svg" style="height: 30px;">

이 방정식에는 사인 및 코사인 항을 갖는 일반 해가 있습니다. 시스템의 경계 값은 해의 계수를 제공합니다.

<img alt="Equation 3" src="/files/ftp_upload/15676/15676_Equation_3.svg" style="height: 24px;">

이 솔루션에서는 사인항이 0이어야 임계 부하에 대한 표현을 제공할 수 있습니다. 이 식은 오일러의 공식으로 알려져 있습니다.

<img alt="Equation 4" src="/files/ftp_upload/15676/15676_Equation_4v2.svg" style="height: 36px;">

오일러 공식을 다시 미분 방정식에 대입하면 좌굴 후 기둥의 탄성 곡선에 대한 식이 제공됩니다.

여기서 오일러의 공식은 하중이 가해지기 전에 기둥이 완벽하게 직선이고, 균질하며, 등방성이고, 축방향 하중이 수직축을 따라 완벽하게 적용된다는 가정에서 파생된다는 점에 유의하는 것이 중요합니다.

양쪽 끝이 핀으로 연결되어 있고 끝 P에 중심 축 방향 하중이 가해지는 컬럼 PQ를 생각해 보십시오. 이 하중이 임계 하중을 초과하면 좌굴이 발생합니다.
받는 사람...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Columns

컬럼

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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.

Critical Buckling Load in Pin-Ended Columns

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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 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-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...

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 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

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

Critical Buckling Load in Pin-Ended Columns

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