Name: equilibrium conditions for a supported plate
Uploaded: 2025-05-22T09:06:02.000Z
Duration: 1 min 31 s
Channel: JoVE

考虑一块在 A、B 和 C 三点处被支撑并受到外力作用的矩形板。
作用在平板上的力和力偶矩可以简化为作用在点 A 处的等效合力和合力偶矩。
此处，通过点 A 和 B 的直线不平行于 y 轴。平板的平衡方程可用另一组方程来描述。
如果合力在 x 方向上无分量且与 y 轴平行，则第一个方程成立。
如果点 A 处的合力矩为零，则第二个方程成立。
第三个方程表明点 B 处的力矩为零，这意味着系统上合力的 y 分量为零。
另一组替代的平衡方程指出，点 A、B 和 C 处的力矩之和为零。当这些点不共线时，这些方程成立。

<ol>
	<li>希伯勒，RC（2016 年）。工程力学 \u2012 静力学和动力学。 新泽西州霍博肯:Pearson Prentice Hall。第 220 页。</li>
</ol>

平衡方程的替代组合

Alternative Sets of Equilibrium Equations

在分析结构的行为时，工程师经常会依赖平衡的概念。这是指作用在系统中的所有力和力矩保持相互平衡的状态，从而导致其不会发生净运动或旋转。在许多情况下，平衡可以通过一组标准的方程来进行描述。然而，在某些情况下，必须使用替代的平衡方程组来准确的描述系统的行为。 这种情况的一个例子可以在A点、B点和C点支撑的矩形板中所观察到，并以此来考虑受到外部的作用力，如图所示。 <img alt="图1" src="/files/ftp_upload/14272/14272_PT_Figure_1v2.svg" style="text-align: center; height: 200px;"> 在这种情况下，作用在板上的力和力矩可以简化为在A点处发挥作用的等效合力和合力矩。然而，通过A点和B点的直线与y轴是不平行的。这意味着标准的平衡方程不能足够准确地描述板的行为。而是必须使用其中所替代的平衡方程组。 <img alt="方程1" src="/files/ftp_upload/14272/14272_Equation_1v2.svg" style="text-align: center; height: 55px;"> 第一个方程表明了作用在板上的合力必须没有x分量并且平行于y轴。这确保了板不会产生水平移动。第二个方程说明了A点的合力矩必须为零。这确保了板围绕点A没有旋转。第三个方程说明了B点的力矩必须为零。这意味着系统的合力位于y轴方向的分量为零，这是用来保持平衡所必需的。这三个方程共同准确地描述了板的行为并确保它能够保持在平衡状态。 在某些情况下，可以使用另一组方程来替代平衡方程，该方程要求A点、B点和C点处的力矩之和为零。 <img alt="方程2" src="/files/ftp_upload/14272/14272_Equation_2v2.svg" style="text-align: center; height: 55px;"> 当这些点不共线时，这组方程将会被满足，即它们不在同一条直线上。在这些点发生共线的情况下，方程将会简化为标准的平衡方程组。

在分析结构的受力行为时，工程师通常依赖于平衡的概念。这指的是所有力和力矩处于……的状态。

alternative-sets-of-equilibrium-equations

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Equilibrium of a Rigid Body

刚体的平衡

KEY TERMS AND CONCEPTS

关键术语与概念

equilibrium conditions for a supported plate

Equilibrium conditions for a supported plate are used to describe when forces and moments balance. In this state, the plate has no net movement and no rotation. Engineers use these equations to study how structures behave under external loads.

In many cases, the standard equilibrium equations are enough. But some systems need an alternative set of equations to match the geometry of the support points. A rectangular plate supported at points A, B, and C is one example.

For this plate, the applied force and couple moment can be reduced to an equivalent resultant force and resultant couple moment at point A. However, the line through points A and B is not parallel to the y-axis. Because of that, the standard equations do not fully describe the plate’s behavior.

The alternative set requires the resultant force on the plate to have no x-component. It must be parallel to the y-axis, which prevents horizontal motion. It also requires the resultant moment at point A to be zero, so the plate does not rotate around point A. A third equation sets the moment at point B to zero, which means the y-component of the resultant force must also be zero for equilibrium.

Another alternative set states that the sum of the moments at points A, B, and C must be zero. This works when the three points are not collinear, meaning they do not lie on the same line. If the points are collinear, these equations reduce to the standard equilibrium equations.

Equilibrium Conditions for a Supported Plate

equilibrium-conditions-for-a-supported-plate

conditions of equilibrium

Rigid Body Force Balance in Equilibrium

Rigid body equilibrium depends on a balance of external forces and moments. A rigid body is in equilibrium when it has no translational motion and no rotational motion. This happens when the net force and the net couple acting on the body are both zero. Internal forces are not used when checking equilibrium. These forces act in equal and opposite pairs inside the body, so they cancel each other. Because of that, they do not affect the overall motion or change in motion of the rigid body. The...

support reactions

Types of Structural Supports

Types of structural supports control how a coplanar force system moves and reacts under load. A coplanar force system is a set of forces that all lie in the same plane. The supports create reactions at the point of contact. These reactions help keep a structure stable under external forces. Supports work by stopping translation with an equal and opposite force. They can also stop rotation by applying a couple moment in the opposite direction. In structural design, these reactions are important...

internal forces and center of gravity

Forces Inside a Body and Balance

Internal forces and the center of gravity help explain how objects stay in balance. These mechanics ideas matter for structures, machines, and vehicles that must remain stable under different loads and conditions. Internal forces are created inside a body when its particles interact. They can appear as tension, compression, or shear forces. These forces act in collinear pairs with equal magnitude and opposite direction. Between connected parts, they cancel each other and help keep the body in...

alternative sets of equilibrium equations

Equilibrium conditions for a supported plate are used to describe when forces and moments balance. In this state, the plate has no net movement and no rotation. Engineers use these equations to study how structures behave under external loads. In many cases, the standard equilibrium equations are enough. But some systems need an alternative set of equations to match the geometry of the support points. A rectangular plate supported at points A, B, and C is one example. For this plate, the...

two force member

Two-Force Body Equilibrium

A two-force body is a rigid body acted on by only two external forces. In equilibrium, those two forces must have the same magnitude, the same line of action, and opposite directions. This balance is what keeps the body at rest. For a plate with forces F1 and F2 applied at points A and B, equilibrium means the sum of moments about any axis is zero. If moments are summed about point A, the moment from F2 must be zero. That means the line of action of F2 must pass through point A. The same idea...

three force member

Solving a Three-Force Member Example

A three-force member is a rigid body acted on by three forces at three points. The forces must have concurrent lines of action. If the forces are parallel, their lines of action stay parallel instead. A dumpster provides the example used here. It is connected to a pin support at point A and to a pin on a hydraulic cylinder at point B. The hydraulic cylinder acts as a two-force member in equilibrium. The dumpster weight acts through the center of gravity, and the support reaction acts at point...

equations of equilibrium in three dimensions

equations-of-equilibrium-in-three-dimensions

Checking Force Balance in 3D Structures

Checking force balance in 3D structures is a key step when a system is at rest. The vector and scalar equations of equilibrium are used to test whether a body is stable and will not collapse or fall apart. These equations give a clear way to study the forces acting on a body. The vector equations of equilibrium require the vector sum of all external forces on a body to equal zero. They also require the vector sum of the couple moments and the moments of all forces about a point to equal zero.

support reactions in three dimensions

3D Support Types for Stability and Motion

Support reactions in three dimensions keep structures stable and in equilibrium. They stop unwanted translation and rotation, so a design can handle external forces safely and work as intended. These reactions are important in many structures and systems. A ball and socket joint is one support that allows free rotation about any axis. It does not provide a moment that resists turning. At the same time, it restricts motion in space with three reaction forces along the three dimensions. A...

constraints and statical determinacy

Constraints in Structural Equilibrium

Constraints in structural engineering control how a body can move and help keep it in mechanical equilibrium. A constraint is a restriction on motion. When the right number of constraints is used, the system can stay balanced with the fewest supports needed. A system with the proper number of supports is called statically determinate. In that case, the unknown reaction forces can be found from equations of equilibrium. These systems are the easiest to analyze and design because the forces can...

key terms and concepts

Learn how a supported plate can be analyzed with alternative equilibrium equations, including force balance and zero moments at points A, B, and C.

Equilibrium Conditions for a Supported Plate | 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

Equilibrium Conditions for a Supported Plate

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