Finding reaction forces of a 3D plate fixed at 3 points

[Hadiza]

Homework Statement

I was hoping someone could help me determining the reaction forces of a plate supported at three points (see image attached). I was able to find the reaction with only the weight of the plate, but now there is a force with three vectors Px, Py and Pz acting at a point on the plate. So basically with the addition of forces in the x and y axis, I now have to find the reaction forces at all three supports in all 3 vectors (i.e. 9 variables). But with only 6 equations of equilibrium. I was able to find All reactions in the z axis; Az, Bz, Cz ( which by the way is not shown in the images attached). But I am stuck as the X and Y axis reactions can only be expressed interms of each other ( e.g., Ay = Cx × constant) as only eqn 3 (sum of Mz) relates them, so I am unable to solve it.
I know this actual system is not statistically indeterminate nor imbalanced as it is used in real life applications. So am I missing something important here. Pls help!!!

Relevant Equations

Sum of forces =0
Sum of moments =0

 

[Lnewqban]

Welcome!
Is the plate supported from below?

 

[Hadiza]

Welcome!
Is the plate supported from below?

Yes, the 3 supports are holding it from below indeed

 

[Lnewqban]

Then, is there some relevant vertical distance (hp) between the point of application of force P and the three suports to be considered?

 

[Hadiza]

Then, is there some relevant vertical distance (hp) between the point of application of force P and the three suports to be considered?

Yes there is.. i forgot to add it to the diagram, but I mentioned it in the note, the force P is applied at a certain distance hp above the plate

 

[Lnewqban]

Therefore, Px and Py each induce some moment about axes y and x respectively, besides the evident moments created by Pz.

 

[Hadiza]

Therefore, Px and Py each induce some moment about axes y and x respectively, besides the evident moments created by Pz.

Indeed..I figured that too..I included that in my solution as well.. eqn 1 and eqn 2 you can see the moments. Although I do see the Mx eqn is missing the Py× hp moment indeed...thanks for pointing that out.. Although, I do not believe that will help solve for the forces in X axis and Y xis I think

 

[Lnewqban]

Indeed..I figured that too..I included that in my solution as well.. eqn 1 and eqn 2 you can see the moments. Although I do see the Mx eqn is missing the Py× hp moment indeed...thanks for pointing that out.. Although, I do not believe that will help solve for the forces in X axis and Y xis I think

You could solve the projections of forces, plane by plane, I believe.
I will try to help with your practical problem later today.

 

[Hadiza]

You could solve the projections of forces, plane by plane, I believe.
I will try to help with your practical problem later today.

I see..though I am still not clear what you mean.. thank you for your time and help so far..in case I take sometime to respond to you later, then pls know it is night time here so I may have fallen asleep..

 

[erobz]

Sum the torques about an axis parallel to $y$ passing through the points $B,C$. See if that gets you any new info.

Then do the same thing for an axis passing through $A$ parallel to $y$.

 

[Lnewqban]

Taking Px as example, and if supports A, B and C can restrain movement in that direction, the x-reaction force at each support should be proportional to the y distance of each respect to point of application of Px.
In that situation, you have more points of support than needed, the x-y plane is indeterminate.

 

[Hadiza]

Sum the torques about an axis parallel to $y$ passing through the points $B,C$. See if that gets you any new info.

Then do the same thing for an axis passing through $A$ parallel to $y$.

Hello. Thanks for the reply. I did indeed try to do both of those things and the forces were as I mentioned in the forces can only be solved in terms of each other

 

[Hadiza]

Taking Px as example, and if supports A, B and C can restrain movement in that direction, the x-reaction force at each support should be proportional to the y distance of each respect to point of application of Px.
In that situation, you have more points of support than needed, the x-y plane is indeterminate.

I had a feeling it was like that. But then would that mean it's solution can only be computed based on approximate solution and not through the basic rules of classical mechanics?

 

[erobz]

I had a feeling it was like that. But then would that mean it's solution can only be computed based on approximate solution and not through the basic rules of classical mechanics?

If its statically indeterminate the reaction forces are calculated from forces associated with material deformations (for some basic 2D problems its manageable). However, the details of that I imagine to be quite complex for this problem. Perhaps @Lnewqban will shed some light on how you might proceed in that case. The method behind it is Finite Element Analysis, but that's about all I know for problems of this complexity.

Philosophically: Is it approximate?...Isn't everything? Maybe I'm going to be slapped for this, but I would argue that if the "basic laws" solve anything directly, it's because we've underestimated reality in that model.

 

[Hadiza]

If its statically indeterminate the reaction forces are calculated from forces associated with material deformations (for some basic 2D problems its manageable). However, the details of that I imagine to be quite complex for this problem. Perhaps @Lnewqban will shed some light on how you might proceed in that case. The method behind it is Finite Element Analysis, but that's about all I know for problems of this complexity.

Philosophically: Is it approximate?...Isn't everything? Maybe I'm going to be slapped for this, but I would argue that if the "basic laws" solve anything directly, it's because we've underestimated reality in that model.

I find your reply fascinating especially towards the end. In hind sight I would say they are called basic laws because they are just that, 'basic'. Atleast that is my belief.

For how this system is solved, I think you might be right about using FEA as a tool to solve it. Having asked a more experienced colleague of mine, who then confessed that the typical reactions in X and Y axis used to represent the system supports have always been simplifications and approximations as determining it is a complex mathematical process..this of course was after I posted the question here

 

[erobz]

But then would that mean it's solution can only be computed based on approximate solution and not through the basic rules of classical mechanics?

It's like at some point the math says "STOP, you've already tortured reality enough, I refuse to participate any further"!

For how this system is solved, I think you might be right about using FEA as a tool to solve it. Having asked a more experienced colleague of mine, who then confessed that the typical reactions in X and Y axis used to represent the system supports have always been simplifications and approximations as determining it is a complex mathematical process..this of course was after I posted the question here

Yeah, that sounds about right. Good luck in your future approximating!

 

[Lnewqban]

I had a feeling it was like that. But then would that mean it's solution can only be computed based on approximate solution and not through the basic rules of classical mechanics?

As the target is to design those supports, you could reduce the restriction to movement of one of them, which will make the system non-indeterminate.
You could also alternate the position of the sliding support to A, B and C, and analyze the worst case regarding loads on the supports.

If you need to be precise, perhaps these solved statically indeterminate problems could help you:
https://mathalino.com/reviewer/mechanics-and-strength-of-materials/statically-indeterminate-members

 

[Hadiza]

As the target is to design those supports, you could reduce the restriction to movement of one of them, which will make the system non-indeterminate.
You could also alternate the position of the sliding support to A, B and C, and analyze the worst case regarding loads on the supports.

If you need to be precise, perhaps these solved statically indeterminate problems could help you:
https://mathalino.com/reviewer/mechanics-and-strength-of-materials/statically-indeterminate-members

Wow. Thank you for all the help. Your insights and answers have really helped me not just with solving it, but actually be more confident in my analysis and conclusions of mathematical problems. It is good to see confirmation from someone as knowledgeable as yourself. Thank you for the links and your time

 

[BNC]

For transporting one component we needed to do similar calculation long back. Same is attached here (simpaly supported case).

 

[Albert Verburg]

In case there are only z-forces (This is a simpler situation than the described problem, but I wanted to write it down anyhow because it is a common situation and i didn't see a clear answer)
The method to calculate the 3 reaction forces is as follows:

Lets call the gravitational force Fz.
Then:
calculate the distance from the COG (center of gravity) to one of the hinge lines through 2 supports, say line BC. Lets call this distance p. Also calculate the distance from (in this case) A to line BC. And call this distance q. Then the reaction force in A is given by Fa = Fz*p/q.
Do the same for another hinge line. This gives you the second reaction force. The 3rd is easier and follows from the sum of vertical forces = 0: Fz = Fa + Fb + Fc