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

考虑两根长度相等的刚性杆 AB 和 BC，它们相互连接，并在 B 点连接一个弹性系数为 k 的弹簧。若在 A 点施加一个垂直向下的载荷 F，该系统的临界载荷大小是多少？
由于施加的载荷，杆件发生微小角度偏转，导致弹簧伸长量为 X。
绘制两个杆件的受力分析图，并对水平方向应用力的平衡方程。
单独考虑杆AB时，在B点应用力矩平衡方程。求解该方程可得A点的支座反力。
同样，再次对 B 点处的力矩平衡进行分析，可得到 C 点处的支座反力。
将A点和B点的反力支座代入力平衡方程，可得弹簧总力与两个反力支座相平衡。此处弹簧的伸长量不能为零，从而得到该系统的相应临界载荷值。

欧拉列公式：解决问题

Euler&#39;s Formula to Columns: Problem Solving

<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>

考虑两根长度相等的刚性杆 AB 和 BC，彼此相互连接，并在 B 点连接一个弹性系数为 k 的弹簧。……的大小将是多少？

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

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

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

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Euler Buckling for Different Column Supports

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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...

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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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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

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Spring-Loaded Column Buckling Analysis

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