Name: forces that bend and twist materials
Uploaded: 2025-05-22T09:06:03.000Z
Duration: 1 min 20 s
Channel: JoVE

弯矩定义为结构元件在受到外力作用时产生的反作用力，导致该元件发生弯曲。
当一个结构构件发生弯曲时，其顶部会产生压应力，底部会产生拉应力，从而形成一对力偶，该力偶决定了构件的弯矩。
尺子一端固定时的弯曲、跳水板以及撑杆跳高是弯矩的一些实例。
当沿物体的横轴施加扭转力时，会产生扭转力矩，导致该截面发生扭曲。
拧衣服、拧螺丝和扭转弹簧是扭转力矩的几个例子。
作用在梁横截面上的内力和内力偶矩可以用其 x、y 和 z 分量来表示。
扭转力矩 My 沿 y 方向表示，而 Mx 和 Mz 分别表示沿 x 和 z 方向的弯矩分量。

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	<li>Hibbeler， R.C. （2016） 工程力学:静力学和动力学 新泽西州霍博肯:Pearson Prentice Hall，第 344-345 页。</li>
	<li>梅里亚姆，JL;Kraige， LG 和 Bolton， JN （2020）。工程力学 \u2012 静力学。新泽西州霍博肯:约翰·威利。第 276 页。</li>
</ol>

弯曲和扭转力矩

Bending and Torsional Moments

弯曲和扭转力矩是结构工程中的两个基本概念。它们在了解材料和结构在不同加载条件下的行为方面起着重要作用。 当结构元素受到外部力的作用时，则会使其产生反应，同时使元素发生弯曲。当结构元素向上弯曲时，顶部产生压缩正方向的力，底部产生拉伸正方向的力，从而产生一对能够决定元素弯矩的力矩。该力矩是通过考虑到作用点与力作用线之间的距离和力的大小来进行计算的。 固定在一端的尺子所发生的弯曲是弯矩的一个例子。当在不支撑端附近向下施加垂直于尺子的力时，它会使尺子发生弯曲。施加的力在固定端产生一个力矩，从而会导致尺子发生弯曲或变形。在这种情况下，尺子将同时承受压力和拉力。尺子的上侧，是远离施加力的一侧，将会处于拉伸状态。尺子的下侧，是朝向施加力的一侧，将会处于压缩状态。 扭转力矩是当对一个物体沿着其横向或纵向的轴施加扭转力时所发生的，从而致截面扭曲。这种发生扭转的力在材料内产生内部阻力，并阻止其施加的力。扭转力矩是由扭转力和作用点与力作用线之间距离的乘积来计算得出的。 在使用螺丝刀或电动扳手将螺丝拧入材料时，会对螺丝施加扭转力矩，因为螺丝在转动的过程中会遇到材料所产生的阻力。螺丝刀或电动扳手所产生的扭转力会使螺丝发生转动，而材料的阻力则会产生反作用力，从而产生扭转力矩。

弯曲矩和扭转矩是结构工程中的两个基本概念。它们在理解……的行为方面起着重要作用。

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Internal Forces

内力

KEY TERMS AND CONCEPTS

关键术语与概念

forces that bend and twist materials

Bending and torsional moments describe how materials respond to force. These ideas are important in structural engineering because they help explain how beams, rulers, and screws change shape under load. A moment depends on both the size of the force and the distance from the point of interest to the force’s line of action.

A bending moment forms when an external force makes a structural element bend. If the element bends upward, the top side is in compression and the bottom side is in tension. These opposite forces create a couple that determines the bending moment. A ruler fixed at one end shows this effect when a downward force is applied near the free end.

In that ruler example, the force acts perpendicular to the ruler and causes it to deform around the fixed end. The side facing away from the applied force is in tension, and the side facing toward the force is in compression. This makes the bending moment easy to see in a simple everyday object.

A torsional moment happens when a twisting force is applied along a body’s transverse or longitudinal axis. The material resists that twist, so an internal force develops inside it. A screwdriver or power drill turning a screw shows torsion in action. As the screw rotates and meets resistance from the material, the twisting force and the opposing resistance create the torsional moment.

Forces That Bend and Twist Materials

sign convention

Beam Forces: Tension, Shear, and Bending

Beam analysis depends on the internal forces and moments inside a structure. These include normal force, shear force, and bending moment. Engineers use the method of sections to cut the beam and find these values. A sign convention helps show whether each result is positive or negative. Normal force acts perpendicular to the beam’s cross-section. It can pull the material in tension or push it in compression. Tension is taken as positive, while compression is taken as negative. This makes it...

normal and shear force

Beam Loads and Internal Forces

Beam loads create internal forces inside a structure. Weight, pressure, and other external forces can all act on a beam. These internal forces affect the beam’s stability and strength. Engineers study these forces to find their size and direction. A common approach is the method of sections. In this method, an imaginary cut is made at a chosen point on the beam, and the forces at that section are analyzed. One internal force is the normal force, also called the axial force. It acts...

bending and torsional moments

Bending and torsional moments describe how materials respond to force. These ideas are important in structural engineering because they help explain how beams, rulers, and screws change shape under load. A moment depends on both the size of the force and the distance from the point of interest to the force’s line of action. A bending moment forms when an external force makes a structural element bend. If the element bends upward, the top side is in compression and the bottom side is in tension.

internal loadings in structural members: problem solving

internal-loadings-in-structural-members-problem-solving

Beam Internal Force Analysis at Point A

Beam internal force analysis shows how normal force, shear force, and bending moment act inside a structural member. These internal loadings help engineers check whether a beam can support the external forces applied to it. The example uses a beam OC that weighs 5 kN and is inclined at 53.13° with the horizontal. The beam is supported at both ends. The goal is to find the internal loadings at point A. The first step is to find the reaction force at point C. This is done by taking the moment...

beams

Beam Types and Support Conditions

Beams are key parts of structural engineering and construction. They support loads that act at different points along their length. Engineers use beam shape and support conditions to understand how a beam will perform. Beams can be classified by geometry. Straight beams are the most common and have a constant cross-section along their length. Tapered beams change in cross-section from one end to the other, while curved beams have a curved shape. Beam cross-section also matters. Common shapes...

shear diagram

Beam Shear Force Diagram

Beam shear force diagrams show how shear changes along the length of a supported beam. In beam mechanics, they help students track the internal shear force caused by perpendicular loads. The beam AB in this example is supported at both ends, so the first step is to draw a free-body diagram that shows the external forces and internal reactions. The reaction forces at supports A and B are found with equilibrium equations for force and moment. For the beam shown, the vertical reaction at A is 24...

bending moment diagram

Beam Load Analysis with Moment Diagrams

Beam load analysis uses a bending moment diagram to show how bending changes along the length of a beam. This diagram helps engineers and designers check whether a structure can handle applied forces. It is also useful for parts such as shelving arms, which must resist forces and moments from a load. To build the diagram, start by finding the reactive forces and any couple moments on the beam. In some cases, especially when a beam is inclined, these forces must be resolved into components that...

relation between the distributed load and shear

relation-between-the-distributed-load-and-shear

Shear Change Under Distributed Loading

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment. To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear...

relation between the shear and bending moment

relation-between-the-shear-and-bending-moment

Shear and Moment Diagrams in Beam Loads

Shear and moment diagrams help show how a beam responds to loads. A beam can carry a distributed load, concentrated loads, and a couple moment. These loads create internal shear forces and bending moments. The beam is examined by isolating a small elemental section and drawing a free-body diagram. For that section to stay in equilibrium, the moment on the right side must be slightly larger than the moment on the left side. The distributed load also has a resultant force that acts at a...

shear and bending moment diagram: problem solving

shear-and-bending-moment-diagram-problem-solving

Beam Load Analysis with Shear Diagrams

Beam load analysis uses shear and bending moment diagrams to track the internal forces in a beam. These diagrams are important when a beam carries concentrated loads and a distributed load. They help show how the beam responds so structures can be designed safely and efficiently. The first step is to draw a free-body diagram of the whole beam. A free-body diagram shows the concentrated loads, the distributed load, and the reaction forces at the supports. Use the equilibrium equations, where...

cable subjected to concentrated loads

Finding Tension in a Loaded Cable

Flexible cables can carry vertical point loads between two fixed supports. The cable is treated as flexible, inextensible, and nearly weightless. Under these conditions, it changes shape to form straight-line segments, and each segment carries a constant tensile force. To analyze the cable, a free-body diagram is drawn for the whole cable. This helps identify the reaction forces at the supports. In many cases, the number of unknown reaction components is greater than the available equilibrium...

cable subjected to a distributed load

Finding Cable Shape Under Distributed Load

Cable shape under a distributed load is found by analyzing the forces along a small piece of the cable. This method is important in suspension bridge design because the main cables must carry changing loads safely. The load causes the tensile force in the cable to change, and it also creates cable deformation. The analysis starts with a free-body diagram of a short cable segment. The diagram shows the distributed load and the change in tensile force across that segment. Equilibrium equations...

cable subjected to its own weight

Cable Sag Under Its Own Weight

Overhead power transmission cables sag under their own weight, and that shape affects how they carry electricity across long distances. To study this behavior, the cable is analyzed with a generalized loading function. This function treats the load along the cable's arc length instead of its projected length, which gives a more accurate picture of the cable's response. The analysis then focuses on a small cable segment. A free-body diagram is drawn to show the forces acting on that segment.

cable: problem solving

Cable Tension Under Uniform Loading

Cable tension under uniform loading is found by analyzing a cable fixed between two supports. The problem starts with the support conditions and the load on the cable. From there, the cable’s shape must be described with an equation that matches the curve formed by the load. The next step is to integrate the shape equation to get the cable’s shape function. This makes it possible to solve for constants in the equation. The boundary condition at the origin gives one constant. The slope, which...

key terms and concepts

Explore how bending and torsional moments act on materials, from a ruler bending at one end to a screw twisting under a screwdriver.

Forces That Bend and Twist Materials | 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

Forces That Bend and Twist Materials

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