Cantilever Beam: Forces and Bending Moments | 悬臂梁的受力分析与力矩

📚 Cantilever Beam: Forces and Bending Moments | 悬臂梁的受力分析与力矩

A cantilever beam is a structural element fixed at one end and free at the other. It is one of the most common idealised models used in physics and engineering to analyse how forces produce internal stress and bending. In this article, we will examine the force analysis and moments of a cantilever beam step by step.

悬臂梁是一端固定、另一端自由的构件。它是物理学和工程中最常见的理想模型之一，用于分析力如何产生内应力和弯曲。本文将逐步考察悬臂梁的受力分析与力矩。

1. Free-Body Diagram and Reaction Forces | 受力图与支反力

To analyse a cantilever beam, we first draw a free-body diagram. The fixed support exerts a vertical reaction force R and a reaction moment M_A on the beam. If a downward force F is applied at the free end, the beam is in translational and rotational equilibrium.

分析悬臂梁时，首先要画出受力图。固定端对梁施加一个竖直反力 R 和一个反力矩 M_A。若自由端作用有向下的力 F，则梁处于平移和转动平衡状态。

Applying equilibrium conditions gives the vertical reaction and the reaction moment:

应用平衡条件可得竖直反力和反力矩：

R = F, M_A = F · L

Here L is the length of the beam. The direction of M_A is opposite to the rotation caused by F about the fixed support.

其中 L 为梁的长度，M_A 的方向与 F 绕固定端产生的转动方向相反。

2. Internal Shear Force and Bending Moment | 内力剪力与弯矩

Forces inside a beam are described by shear force V and bending moment M. To find them, we cut the beam at a distance x from the fixed support and consider one side of the cut.

梁内部受力用剪力 V 和弯矩 M 描述。为求二者，可在距固定端 x 处截开梁，并考察截面任一侧的平衡。

For a cantilever with a single downward end force F, the shear force is constant along the beam:

对于只在自由端受一个向下力 F 的悬臂梁，剪力沿梁为常量：

V(x) = F

The bending moment at the section depends on the distance to the free end:

截面处的弯矩取决于该截面到自由端的距离：

M(x) = F · (L − x)

Here x is measured from the fixed support. The moment is largest at the support and zero at the free end.

这里 x 从固定端量起。弯矩在固定端最大，在自由端为零。

3. Shear Force and Bending Moment Diagrams | 剪力图与弯矩图

The shear force diagram for an end-loaded cantilever is a horizontal line of height F throughout the beam. The bending moment diagram is a straight line falling from F·L at the fixed end to zero at the free end.

端部受力的悬臂梁，剪力图是一条高度为 F 的水平线；弯矩图则是一条从固定端 F·L 线性下降到自由端零的斜直线。

At x = 0: V = F, M = F·L (maximum).

在 x = 0 处：V = F，M = F·L（最大值）。
At x = L: V = F, M = 0.

在 x = L 处：V = F，M = 0。

These diagrams immediately show where internal stress is greatest.

这些图能直观显示内应力最大的位置。

4. Maximum Bending Moment and Critical Section | 最大弯矩与危险截面

For a cantilever with a point load at the free end, the maximum bending moment occurs at the fixed support:

对于自由端受集中力的悬臂梁，最大弯矩出现在固定端：

M_max = F · L

The fixed support is therefore the critical section where failure is most likely to begin. In design, this section must be made strong enough to resist the maximum moment.

因此固定端是危险截面，最容易发生破坏。设计时必须保证该截面足以抵抗最大弯矩。

5. Bending Stress in the Beam | 梁的弯曲应力

A bending moment causes normal stress σ that varies linearly across the cross-section. At a distance y from the neutral axis, the bending stress is

弯矩使横截面上产生正应力 σ，并沿截面线性变化。距中性轴为 y 处，弯曲应力为

σ = M · y / I

Here I is the second moment of area of the cross-section. The maximum stress occurs at the top or bottom surface, where y = c:

其中 I 为截面惯性矩。最大应力出现在上表面或下表面，即 y = c 处：

σ_max = M · c / I = M / S

where S = I / c is the elastic section modulus. For a rectangular section of width b and height h, I = b·h³/12 and c = h/2.

式中 S = I / c 为抗弯截面系数。对于宽 b、高 h 的矩形截面，I = b·h³/12，c = h/2。

6. Shear Stress Distribution | 剪应力分布

Along with bending stress, a cantilever also experiences transverse shear stress τ due to the shear force V. The shear stress at a point is given by

除弯曲应力外，悬臂梁还因剪力 V 而产生横向剪应力 τ。任一点剪应力可由下式给出

τ = V · Q / (I · b)

where Q is the first moment of area above the point, b is the width at that level, and I is the second moment of area. For a rectangular cross-section, the shear stress varies parabolically and is maximum at the neutral axis:

其中 Q 为该点以上面积对中性轴的一次矩，b 为该处的截面宽度，I 为截面惯性矩。对矩形截面，剪应力呈抛物线分布，在中性轴处最大：

τ_max = 3·V / (2·A)

In most long, slender cantilevers, bending stress is the dominant concern, but short beams require shear stress checks.

对于细长悬臂梁，通常以弯曲应力为主；短梁则还需要校核剪应力。

7. Deflection of a Cantilever | 悬臂梁的挠度

The deflection of a cantilever under an end load F is obtained from the elastic beam equation. For a beam fixed at one end and loaded at the other, the vertical deflection at the free end is

悬臂梁在端部载荷 F 作用下的挠度可由弹性梁方程求得。对于一端固定、另一端受载的梁，自由端的竖向挠度为

δ = F · L³ / (3 · E · I)

Here E is Young’s modulus of the material and I is the second moment of area. The deflection is proportional to L³, so beam length strongly affects bending.

式中 E 为材料杨氏模量，I 为截面惯性矩。挠度与 L³ 成正比，因此梁长对弯曲影响极大。

For a uniformly distributed load w over the whole length, the maximum deflection is w·L⁴/(8·E·I).

若全梁承受均布载荷 w，则最大挠度为 w·L⁴/(8·E·I)。

8. Stiffness and Design Considerations | 刚度与设计考量

The stiffness of a cantilever is defined as the applied force divided by the resulting deflection:

悬臂梁的刚度定义为所施加的力与产生的挠度之比：

k = F / δ = 3·E·I / L³

To increase stiffness, engineers can choose a stiffer material (larger E), a shorter beam, or a cross-section with a large second moment of area I. I-beams and hollow sections are efficient because they place material far from the neutral axis.

为提高刚度，工程师可选择更刚的材料（更大 E）、缩短梁长，或采用惯性矩 I 较大的截面。工字梁和空心截面能高效利用材料，因为材料远离中性轴。

When designing a cantilever, both strength (maximum stress) and stiffness (maximum deflection) must be checked against allowable limits.

设计悬臂梁时，必须同时校核强度（最大应力）和刚度（最大挠度）是否满足许用要求。

9. Common Exam Problems and Solving Steps | 常见考题与解题步骤

Typical exam questions ask you to find reactions, draw diagrams, compute stresses, or calculate deflection. A reliable method is:

典型考题要求求解支反力、绘制内力图、计算应力或挠度。一个可靠的解题步骤是：

Draw a free-body diagram with all external forces and moments.

画出包含所有外力和外力矩的受力图。
Apply equilibrium to find reactions R and M_A.

利用平衡条件求支反力 R 和 M_A。
Cut the beam at a general position x and write expressions for V(x) and M(x).

在任意位置 x 处截开梁，写出 V(x) 和 M(x) 的表达式。
Identify maximum values and use formulas for stress or deflection.

找出最大值，并代入应力或挠度公式。
Check units and final directions.

检查单位与方向是否正确。

Always state whether x is measured from the fixed end or the free end before writing equations.

写方程前一定要说明 x 是从固定端还是从自由端量起。

10. Summary | 总结

A cantilever beam is a powerful physical model for understanding forces and moments in structures. The key results for a point load F at the free end of a length L are: reaction R = F, reaction moment M_A = F·L, maximum bending moment at the support, maximum bending stress σ_max = M·c/I, and free-end deflection δ = F·L³/(3·E·I).

悬臂梁是理解结构中力与力矩的重要物理模型。对于长度为 L、自由端受集中力 F 的情况，关键结果为：支反力 R = F，反力矩 M_A = F·L，最大弯矩在固定端，最大弯曲应力 σ_max = M·c/I，自由端挠度 δ = F·L³/(3·E·I)。

Mastering these ideas builds a foundation for advanced structural mechanics and practical engineering design.

掌握这些概念将为深入学习结构力学和实际工程设计奠定基础。

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Cantilever Beam: Forces and Bending Moments | 悬臂梁的受力分析与力矩