Name: column bending under off-center load
Uploaded: 2024-05-23T13:00:07.000Z
Duration: 1 min 16 s
Channel: JoVE

考虑一个承受偏心载荷 F 的柱体。该偏心载荷等效于一个轴向载荷 F 与由于偏心载荷产生的力偶矩的组合。&#160;
绘制柱体 PR 段的受力分析图，并选择合适的坐标系，可得到点 R 处的力偶矩。
将作用在点 R 处的力偶矩代入弹性曲线的微分方程并求解，可得到弹性曲线方程。通过边界条件可计算该解的系数。
在柱体长度的一半位置，可确定柱体的最大挠度。当正割项趋于无穷大时，最大挠度的方程也随之趋于无穷大。
无限挠度准则给出了临界载荷条件的表达式。将该表达式代入最大挠度的公式中，可得到以临界载荷表示的最大挠度。&#160;
柱中的最大应力出现在柱中点横向横截面上弯矩最大的位置。

偏心荷载

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 偏心荷载是结构工程和力学研究中的一个重要概念，特别是在分析柱的稳定性和应力分布时。与中心载荷不同的是，中心载荷是力沿质心轴施加，从而导致其均匀压缩，而偏心载荷则会在偏离中心时所施加的力。这种偏心荷载不仅引入了直接压应力，而且还引入了弯曲应力，从而能够显著影响柱在荷载作用下的行为。偏心荷载在概念上可以分为两个部分：沿支柱轴线作用的中心荷载（F）和由载荷偏心引起的力偶力矩。该力偶力矩的大小取决于所施加的荷载与柱的质心轴之间的距离，并且这对于理解柱的状态是至关重要的，因为除了会导致出现轴向压缩之外，它还会导致弯曲。
 
 为了能够分析这种情况，则需要对偏心荷载下的色谱柱进行分析。选择柱的截面 PR 并绘制其自由体图，这有助于直观的观察到作用在其上方的力和力矩。选择适当的坐标系使我们能够确定给定点（例如 R 点）处的力偶力矩，这与随后对柱的状态进行数学建模是不可或缺的。下一步则将这个偶矩合并到能够控制柱弹性曲线的微分方程中。该微分方程的解给出了弹性曲线方程，该方程描述了柱在施加荷载作用的情况下是如何进行弯曲的。通过应用边界值条件，可以确定解的系数，从而能够进一步提高模型的准确性。
 
 该分析的一个关键方面是确定柱的最大挠度，该最大挠度通常位于其中点的位置。该最大挠度对于评估柱的稳定性是至关重要的，因为它表明了柱在施加荷载下弯曲的程度。这种偏转的方程指向了一个有趣的现象：当方程中的正割项变得无穷大时，偏转则会接近于无穷大。这种情况标志着柱将会失去稳定性并达到屈曲的临界值。从无穷挠度准则中推导出的临界荷载条件对于工程师确保支柱设计能够在安全运行的范围之内是至关重要的。通过将临界荷载条件代入到最大挠度表达式中，我们可以用推导出的临界载荷来表示最大挠度的方程。这种关系对于设计能够承受偏心荷载而不会发生过度变形或失效的柱是至关重要的。
 
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考虑一个承受偏心载荷 F 的柱体。该偏心载荷等效于一个轴向载荷 F 与一个力偶矩的组合……

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

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 mathematically, a section PR is selected and a free-body diagram is drawn. This diagram helps show the forces and moments on the section. A coordinate system is then chosen so the couple moment at a point such as R can be found and used in the model.

That moment is added to the differential equation for the elastic curve of the column. Solving the equation gives the elastic curve, which describes the column’s shape under load. Boundary value conditions are then used to determine the coefficients in the solution.

The analysis also focuses on maximum deflection, which usually occurs at the midpoint of the column. This value shows how much the column bends under the applied load. The deflection expression includes a secant term, and the deflection becomes infinite when that term becomes infinite.

That infinite-deflection condition marks the loss of stability and the start of buckling. The corresponding critical loading condition is important for safe column design. Substituting that condition into the maximum-deflection expression gives a relationship between maximum deflection and critical loading.

Column Bending Under Off-Center Load

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

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

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 an off-center force creates compression and bending in a column, how the elastic curve is modeled, and why infinite deflection signals buckling.

Column Bending Under Off-Center Load | 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

Column Bending Under Off-Center Load

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