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

点A、B、Cで支持され、外力を受ける長方形のプレートを考えてみましょう。
プレートに作用する力と結合モーメントは、点Aで作用する等価合力と結合モーメントに還元できます。
ここでは、点AとBを通る線はy軸に平行ではありません。プレートの平衡方程式は、代替の方程式セットを使用して記述できます。
最初の方程式は、合力が x 成分を持たず、y 軸に平行である場合に満たされます。
2 番目の方程式は、点 A での結果モーメントが 0 の場合に満たされます。
3 番目の方程式は、点 B でのモーメントがゼロであることを示しており、これはシステム上の合力の y 成分がゼロであることを意味します。
代替平衡方程式の 2 番目のセットは、点 A、B、C のモーメントの合計がゼロであることを示しています。これらの方程式は、これらの点が同一線上にない場合に満たされます。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。 ニュージャージー州ホーボーケン:ピアソンプレンティスホール。220ページ。</li>
</ol>

平衡方程式の代替集合

Alternative Sets of Equilibrium Equations

構造の挙動を分析する際、エンジニアはしばしば静力学の概念に頼ります。これは、全ての力とモーメントが相殺し、系にネットの移動や回転がない状態を指します。多くの場合、静力学は一連の標準方程式で記述することができますが、一部の状況では正確に系の挙動を記述するためには、代替の静力学の方程式が必要です。

そのような状況の例の一つは、図に示すように、点A、点B、点Cで支持されている長方形の板に外力が加わる場合です。

<img alt="Figure 1" src="/files/ftp_upload/14272/14272_PT_Figure_1v2.svg" style="text-align: center; height: 200px;" />

この場合、板に作用する力とカップルモーメントは点Aで作用する等效の合力と合モーメントに簡略化することができます。しかし、点Aと点Bを結ぶ直線はy軸に平行ではありません。つまり、標準の静力学方程式では板の挙動を正しく記述することはできません。代わりに、代替の静力学の方程式を使用する必要があります。

<img alt="Equation 1" src="/files/ftp_upload/14272/14272_Equation_1v2.svg" style="text-align: center; height: 55px;" />

第1の方程式は、板に作用する合力はx成分を持たず、y軸に平行である必要があることを述べています。これにより、板の水平方向の移動がないことを保証します。第2の方程式は、点Aにおける合モーメントがゼロであることを示しています。これにより、点Aを中心とした回転がないことを保証します。第3の方程式は、点Bにおけるモーメントがゼロであることを示しています。これは、系の合力のy成分がゼロであることを意味し、静力学のために必要です。これらの3つの方程式は、板の挙動を正確に記述し、平衡状態を保つことを保証します。

<img alt="Equation 2" src="/files/ftp_upload/14272/14272_Equation_2v2.svg" style="text-align: center; height: 55px;" />

特定の状況で使用されるもう一つの代替の静力学方程式は、点A、点B、および点Cにおけるモーメントの合計がゼロであることを述べています。

これらの方程式は、これらの点が直線上にない、つまりすべてが同じ直線上に存在しない場合に満足します。点が直線上に存在する場合、方程式は標準の静力学方程式に簡略化されます。

構造物の挙動を解析するとき、エンジニアはしばしば平衡の概念に頼ります。これは、すべての力と瞬間が...

alternative-sets-of-equilibrium-equations

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Equilibrium of a Rigid Body

剛体のつり合い

KEY TERMS AND CONCEPTS

主要用語と概念

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

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mechanical engineering

Equilibrium Conditions for a Supported Plate

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