Force-couple system equivalence: 3 moment equations

In summary, to find the equivalent force in a force-couple system, you can use either the resultant force or moment equations, as long as you consider the direction of the forces and moments to determine the line of action.
  • #1
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In force-couple system equivalence, We can show that two systems are equivalent if the resultant force in both are equal and that it has the same line of action.

So Rx1 = Rx2, Ry1= Ry2, Ma1 = Ma2

Now my book says we can use another form of equation: We can choose 3 point and find the moment about them and use it to find the equivalent force, as long as a, b and c are not on the same line

I fail to see why should this work.
Ma = x Ry -y Rx
Mb = x Ry - y Rx
Mc = x Ry - y Rx

THe x and ys in each equation is different, How do you solve such a system to find Ry and Rx and the line of action? You can also use 2 moments and 1 resultant force equation in x or y direction
 
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  • #2
, but how do you use this to find the line of action?It is true that if you have three point, you can find the moment around them and use this to find the equivalent force. However, in order to find the line of action, you need to consider the direction of the forces and moments. To do this, you need to consider the cross-product of the two vectors (the forces and moments) to determine the direction of the vector along which the force acts. For example, let us consider two points, A and B. The vector of force at point A is given by F_A, and the vector of moment at point B is given by M_B. The line of action of the resultant force is then given by the cross product between these two vectors, i.e. the force at A × the moment at B = [F_A x M_B]. This gives the direction of the line along which the resultant force is applied. Once you have determined the direction of the line of action, you can then use the equations for forces and moments to find the magnitude of the force and the moment about that line.
 

What is a force-couple system?

A force-couple system is a combination of forces and moments acting on a rigid body. The forces act through the center of mass, while the moments act around the center of mass.

What is the principle of force-couple system equivalence?

The principle of force-couple system equivalence states that any force-couple system can be replaced by an equivalent force-couple system that produces the same external effects on a rigid body.

What are the 3 moment equations in a force-couple system?

The 3 moment equations in a force-couple system are the sum of moments equals zero, the sum of forces in the x-direction equals zero, and the sum of forces in the y-direction equals zero.

How are force-couple systems used in structural analysis?

Force-couple systems are used in structural analysis to determine the external forces and moments acting on a structure, and to predict how the structure will respond to these forces.

Can force-couple systems be applied to both 2D and 3D problems?

Yes, force-couple systems can be applied to both 2D and 3D problems. The 3 moment equations can be used to solve for the unknown forces and moments in either 2D or 3D space.

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