Determining static equilibrium this way?

[Termotanque]

Suppose that one has a heavy rigid beam supported by two vertical rods at two different points A and B, and wants to calculate the reaction R_A, R_B made by the rods.

I found the reactions first by taking moments with respect to A, and then with respect to B. In both cases I included the moment of the weight. The solutions were correct.

I want to know if this is just a particular case, or if in general it is true that if the sum of the moments with respect to n (possibly 2) different points A, B, ... are all null, then the rigid body is at equilibrium.

Sorry for not using pretty Tex, but it's not rendering properly.

 

[tiny-tim]

HI Termotanque!

(try using the X₂ icon just above the Reply box )

I want to know if this is just a particular case, or if in general it is true that if the sum of the moments with respect to n (possibly 2) different points A, B, ... are all null, then the rigid body is at equilibrium.

Yes and no.

Yes, all the moments have to be zero for there to be equilibrium, but if there's more than 2 points (for a rod) or more than 3 points (for a plate), then no, there's insufficient information to find the reactions, and you need to know something about the flexibility of the rod or plate also.

For example, with an ordinary four-legged table, you can remove any leg and the table will still stand … if you know nothing about the flexibility of the table, you can't tell how much weight is being supported on each of the four legs.

 

[Termotanque]

I'm actually only referring to statically determinate systems. One which one could find the reactions (like the one I described above). I wasn't asking if it's possible to find the reactions, but if it's correct to do it just by calculating a finite number of moments with respect to interesting points, and not calculating the sum of external forces.

Even though my example was 2 dimensional, I want to know the answer for a more general 3 dimensional statically determinate system.

 

[tiny-tim]

(just got up :zzz: …)

Yes, taking moments about each point of support (plus one equation for the vertical components of force) will always solve the problem if the number of supports is small enough to be statically determinate.