Force in the legs of a hexagonal table

[guv]

Homework Statement:

Imagine a table shaped as a regular hexagon with length $a$ as shown in the attached figure. The table itself is massless. The table has 6 legs supporting it at each vertex on a leveled ground. Table itself is also horizontal.

Use the bottom left of the hexagon as the origin with side $a$ pointing to the right. Now suppose a force F is applied at a point $(x,y)$ on the table, what would be the force in each leg supporting the table?

The problem can be dimensionalized so the length $a$ is not really important, one can pretend $a=1$

Relevant Equations:

$$\vec F_{net} = 0$$
$$\vec \tau = 0$$

We know that the net force on the table must be zero
$$\sum F_i = F$$

We know that the components of the torque with respect to the origin is also 0.
$$\sum \tau_x = 0$$
$$\sum \tau_y = 0$$
$$\sum \tau_z = 0$$

But the problem becomes insufficiently constrained that there are only 4 equations while there are 6 potential unknowns other than the trivial symmetric cases. How to solve the 6 different tension force in each leg for arbitrary position $(x,y)$ where the force $F$ is applied?

Thanks

 

[haruspex]

Homework Statement:: Imagine a table shaped as a regular hexagon with length $a$ as shown in the attached figure. The table itself is massless. The table has 6 legs supporting it at each vertex on a leveled ground. Table itself is also horizontal.

Use the bottom left of the hexagon as the origin with side $a$ pointing to the right. Now suppose a force F is applied at a point $(x,y)$ on the table, what would be the force in each leg supporting the table?

The problem can be dimensionalized so the length $a$ is not really important, one can pretend $a=1$
Relevant Equations:: $$\vec F_{net} = 0$$
$$\vec \tau = 0$$

We know that the net force on the table must be zero
$$\sum F_i = F$$

We know that the components of the torque with respect to the origin is also 0.
$$\sum \tau_x = 0$$
$$\sum \tau_y = 0$$
$$\sum \tau_z = 0$$

But the problem becomes insufficiently constrained that there are only 4 equations while there are 6 potential unknowns other than the trivial symmetric cases. How to solve the 6 different tension force in each leg for arbitrary position $(x,y)$ where the force $F$ is applied?

Thanks

The only way to get an answer to such a problem is to make assumptions about elasticity. Easiest would be to take the table as perfectly rigid and allow the floor to be compressed slightly, with equal spring constant and independently at each leg.

 

[Lnewqban]

Please, see:
https://en.wikipedia.org/wiki/Statically_indeterminate

https://www.mathalino.com/reviewer/...of-materials/statically-indeterminate-members