Name: spring-loaded column buckling analysis
Uploaded: 2024-05-23T13:00:07.000Z
Duration: 1 min 23 s
Channel: JoVE

考虑两个长度相等的刚性杆 AB 和 BC 相互连接，并且一个常数为 k 的弹簧连接到点 B。如果在 A 点垂直向下施加载荷 F，该系统的临界载荷的大小是多少？
由于施加的载荷，杆会偏转一个小角度，导致弹簧伸长 X。
绘制两个柱的自由体图，并将力平衡方程应用于水平方向。 
单独考虑杆 AB，力矩的平衡方程应用于点 B。求解该方程得到点 A 处的支座反力。
同样，在点 B 处施加力矩的平衡再次得到点 C 处的支座反力。
将点 A 和 B 处的反作用力支座代入力平衡方程中，得到平衡两个反作用力支座的总弹簧力。在这里，弹簧的伸长率不能为零，从而为给定系统产生相应的临界载荷值。

欧拉列公式：解决问题

<div>
 欧拉公式在结构工程中通常用来确定各种条件下柱的屈曲载荷。然而，在同时处理包含刚性元件和弹性元件（例如弹簧）的系统时，需要采用更精细的分析方法来确定其临界载荷。其中所描述的问题涉及到在枢轴点处连接的两根刚性杆，其中的一端附有弹簧，并且在另一端施加垂直荷载。
 
 该系统包括两个长度相等的垂直刚性杆 AB 和 BC，它们在支点 B 处进行连接，这类似于膝关节。 在 B 点附有一个常数为 k 的拉伸弹簧，同时产生水平作用。当在 A 点向下垂直施加荷载F 时，杆便会绕枢轴点 B 来进行旋转。并用弹簧来抵抗所产生的力，同时将 B 点拉回到其原始位置，从而会导致弹簧伸长 X。
 
 首先，为了能够分析该系统，则需要使用每根杆的自由体图，并应用静平衡原理来分析作用在杆上的力。轴向载荷 F 会导致系统在 B 点发生枢轴转动，从而在杆中产生水平力和垂直力。
 
 建立作用在系统中水平力的平衡方程，其中包括由于伸长量 X 所产生的弹簧力。弹簧力能够抵消位于 B 点的侧向运动。系统在 B 点处所发生的枢轴转动也会在两个方向上产生弯矩。位于杆 AB 中 B 点处的力矩平衡允许确定位于 A 点的反作用力。同时计算由施加的载荷 F 所产生的力矩，并使其等于由反作用力所产生的力矩。类似地，通过研究杆 BC 在 B 点周围的力矩，可以得出位于 C 点的反作用力。该步骤还包括计算弹簧力及其相对于 B 点的力臂。我们可以将位于 A 点和 C 点的反作用力代入到水平力的平衡方程中来计算总弹簧力。该力抵消了位于 A 点和 C 点的反作用力所产生的力矩。
 
 <img alt="Equation 1" src="/files/ftp_upload/15678/15678_Equation_1.svg" style="text-align: center; height: 12px;">
 <img alt="Equation 2" src="/files/ftp_upload/15678/15678_Equation_2v2.svg" style="text-align: center; height: 18px;">
 <img alt="Equation 3" src="/files/ftp_upload/15678/15678_Equation_3.svg" style="text-align: center; height: 12px;">
 系统的临界载荷是由弹簧的伸长量 X 与所施加的载荷 F 直接相关来进行确定的。当弹簧的伸长量（以及弹簧力）达到一个无法通过弹簧伸长来进行平衡的值时，该系统的稳定性就会受到影响，同时也会影响到其结构的完整性，从而导致屈曲。临界载荷的精确数学表达式可以从平衡方程中推导得出，其中包括弹簧常数 k、杆的长度以及屈曲开始时系统的几何形状。
 
 <img alt="Equation 4" src="/files/ftp_upload/15678/15678_Equation_4v2.svg" style="text-align: center; height: 42px;">
</div>

考虑两个长度相等的刚性杆 AB 和 BC 相互连接，并且一个常数为 k 的弹簧连接到点 B。的量级会是多少......

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

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Columns

KEY TERMS AND CONCEPTS

关键术语与概念

Euler&#39;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 forces, free-body diagrams are drawn for each bar. Static equilibrium is then applied to track the horizontal and vertical forces in the rods.

The axial load F creates force components in the bars and also produces a bending moment at the joint. The moment equilibrium around B for bar AB is used to find the reaction force at point A. The same method is applied to bar BC to determine the reaction force at point C. This step also includes the spring force and its moment arm relative to B.

After the reactions at A and C are found, they are used in the horizontal force balance to calculate the total spring force. That spring force counteracts the moments created by the reactions in the structure. The critical load is identified when the spring elongation, and therefore the spring force, can no longer be balanced by the assembly. At that point, buckling begins.

The final expression for the critical load depends on the spring constant k, the bar length, and the geometry of the system at the onset of buckling. This makes the problem a useful example of how a spring changes the stability of a column-like structure. It also shows how Euler’s formula can be adapted when rigid members and elastic parts work together.

Spring-Loaded Column Buckling Analysis

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-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

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

Use static equilibrium to analyze a spring-loaded rigid bar system, find reaction forces, and determine the critical load where buckling begins.

Spring-Loaded Column Buckling Analysis | 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.

mechanical engineering

Spring-Loaded Column Buckling Analysis

Description

Transcript