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

Consider a rectangular plate supported at points A, B, and C and subjected to external forces.
The forces and couple moments acting on the plate can be reduced to the equivalent resultant force and resultant couple moment acting at point A.
Here the line passing through points A and B is not parallel to the y-axis. The equations of equilibrium of the plate can be described using the alternative set of equations.
The first equation is satisfied if the resultant force has no x-component and is parallel to the y-axis.
The second equation is satisfied if the resultant moment at point A is zero.
The third equation states that the moment at point B is zero, which implies that the y-component of the resultant force on the system is zero.
A second set of alternative equilibrium equations states that the sum of moments at points A, B and C is zero. These equations are satisfied when these points are not collinear.

alternative sets of equilibrium equations

Alternative Sets of Equilibrium Equations

When analyzing the behavior of structures, engineers often rely on the concept of equilibrium. This refers to the state where all forces and moments acting on a system balance each other, resulting in no net movement or rotation. In many cases, equilibrium can be described by a set of standard equations. However, in some situations, alternative sets of equilibrium equations must be used to describe the system's behavior accurately.
One example of such a situation can be observed in a rectangular plate supported at points A, B, and C and subjected to external forces, as shown in the figure.
<img alt="Static equilibrium diagram showing forces F1-F4 on a beam; ΣFx=0, ΣMA=0 analysis." src="/files/ftp_upload/14272/14272_PT_Figure_1v2.svg" style="text-align: center; height: 200px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium diagram; ΣFx=0, MA=0; forces on surface, illustrating torque balance concepts.">
In this case, the force and couple moment acting on the plate can be reduced to the equivalent resultant force and resultant couple moment acting at point A. However, the line passing through points A and B is not parallel to the y-axis. This means that standard equilibrium equations will not suffice to describe the plate's behavior. Instead, alternative sets of equilibrium equations must be used.
<img alt="Static equilibrium equations ΣFx=0, ΣMA=0, ΣMB=0, ΣMR=0; mechanical force diagram." src="/files/ftp_upload/14272/14272_Equation_1v2.svg" style="text-align: center; height: 55px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equations; ΣFx=0, (MR)A=ΣMA=0, ΣMB=0; mechanical analysis diagram.">
The first equation states that the resultant force acting on the plate must have no x-component and be parallel to the y-axis. This ensures that there is no horizontal movement of the plate. The second equation states that the resultant moment at point A must be zero. This ensures that there is no rotation of the plate around point A. The third equation states that the moment at point B must be zero. This implies that the y-component of the resultant force on the system is zero, which is necessary for equilibrium. Together, these three equations accurately describe the behavior of the plate and ensure that it remains in a state of equilibrium.
Another set of alternative equilibrium equations that can be used in certain situations states that the sum of the moments at points A, B, and C must be zero.
<img alt="Static equilibrium equations; ΣMA=0, ΣMB=0, ΣMC=0; diagram for mechanical analysis study." src="/files/ftp_upload/14272/14272_Equation_2v2.svg" style="text-align: center; height: 55px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equations: ΣMA=0, ΣMB=0, ΣMC=0, diagram of rotational balance analysis.">
This set of equations is satisfied when these points are not collinear, meaning they do not all fall on the same line. In cases where the points are collinear, the equations reduce to the standard set of equilibrium equations.

When analyzing the behavior of structures, engineers often rely on the concept of equilibrium. This refers to the state where all forces and moments ...

alternative-sets-of-equilibrium-equations

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Mechanical Engineering

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Equilibrium of a Rigid Body

KEY TERMS AND CONCEPTS

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

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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...

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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.

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Equilibrium Conditions for a Supported Plate

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