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

점 A, B 및 C에서 지지되고 외부 힘을 받는 직사각형 플레이트를 고려하십시오.
플레이트에 작용하는 힘과 커플 모멘트는 점 A에서 작용하는 동등한 합력과 결과 커플 모멘트로 감소할 수 있습니다.
여기서 점 A와 B를 통과하는 선은 y축과 평행하지 않습니다. 플레이트의 평형 방정식은 대체 방정식 세트를 사용하여 설명할 수 있습니다.
첫 번째 방정식은 결과 힘에 x 성분이 없고 y축과 평행한 경우 충족됩니다.
두 번째 방정식은 점 A의 결과 모멘트가 0인 경우 충족됩니다.
세 번째 방정식은 점 B의 모멘트가 0임을 나타내며, 이는 시스템에 가해지는 힘의 y 성분이 0임을 의미합니다.
두 번째 대립 평형 방정식 집합은 점 A, B 및 C에서 모멘트의 합이 0이라고 명시합니다. 이러한 방정식은 이러한 점이 동일선상에 있지 않을 때 충족됩니다.

<ol>
	<li>히벨러, R.C. (2016). Engineering Mechanics \u2012 정역학 및 동역학. 뉴저지 호보켄: 피어슨 프렌티스 홀. 220쪽.</li>
</ol>

평형 방정식의 대체 집합

Alternative Sets of Equilibrium Equations

구조물의 동작을 분석할 때 공학자들은 종종 평형의 개념에 의존합니다. 이것은 시스템에 작용하는 모든 힘과 모멘트가 서로 균형을 이루어 알짜 운동이나 회전이 일어나지 않는 상태를 말합니다. 많은 경우 평형은 표준 방정식의 집합으로 설명할 수 있습니다. 그러나 어떤 상황에서는 시스템의 거동을 정확하게 설명하기 위해 평형 방정식의 대체 집합을 사용해야 합니다.

이러한 상황의 한 예는 그림과 같이 A, B, C 지점에 지지되고 외력을 받는 직사각형 플레이트에서 관찰할 수 있습니다.

<img alt="Figure 1" src="/files/ftp_upload/14272/14272_PT_Figure_1v2.svg" style="text-align: center; height: 200px;" />

이 경우 판에 작용하는 힘과 우력 모멘트는 동등한 합력과 점 A에서 작용하는 합우력 모멘트로 감소할 수 있습니다. 그러나 점 A및 B을 지나는 선은 y-축과 평행하지 않습니다. 이는 표준 평형 방정식이 판의 거동을 설명하기에 충분하지 않다는 것을 의미합니다. 대신에 평형 방정식의 다른 집합을 사용해야 합니다.

<img alt="Equation 1" src="/files/ftp_upload/14272/14272_Equation_1v2.svg" style="text-align: center; height: 55px;" />

첫 번째 방정식은 판에 작용하는 결과력이 x-성분이 없어야 하고 y-축과 평행해야 한다는 것을 나타냅니다. 이것은 판의 수평 이동이 없음을 보장합니다. 두 번째 방정식은 A 지점에서의 결과 모멘트가 0이어야 한다는 것을 의미합니다. 이것은 판이 A 지점 주변에서 회전하는 것이 없음을 보장합니다. 세 번째 방정식은 B 지점의 모멘트가 0이어야 한다는 것을 의미합니다. 이것은 시스템에 작용하는 결과력의 y-성분이 평형에 필요한 0임을 의미합니다. 이 세 방정식은 함께 판의 거동을 정확하게 설명하고, 판이 평형 상태에 있는지 확인합니다. 

특정 상황에서 사용할 수 있는 또 다른 대안적인 평형 방정식 세트는 A, B, C 지점의 모멘트의 합이 0이어야 한다는 것입니다.

<img alt="Equation 2" src="/files/ftp_upload/14272/14272_Equation_2v2.svg" style="text-align: center; height: 55px;" />

이 방정식들은 점들이 동일선상에 있지 않을 때 만족되며, 이는 점들이 모두 동일선상에 있지 않다는 것을 의미합니다. 점들이 동일선상에 있는 경우, 방정식들은 표준 평형 방정식 집합으로 줄어듭니다.

구조의 거동을 분석할 때 엔지니어는 종종 평형의 개념에 의존합니다. 이것은 모든 힘과 순간이 ...

alternative-sets-of-equilibrium-equations

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Equilibrium of a Rigid Body

강체의 평형

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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