Consider a rigid rectangular planar plate of width and height lying horizontal on four supports at its corners in and with a concentrated vertical load at , with , .
How can I determine the reactions ?
This is a hyperstatic problem (the object has redundant constraints) and as such the equilibrium equations make an underdetermined system.
Force equilibrium along Z:
Torque equilibrium around X at y=0:
Torque equilibrium around Y at x=0:
Any other equilibrium equation I could write wouldn’t increase the rank of the system.
Such problems are usually solved by considering the deformation of the object and requiring a null displacement at the constraints.
I said that the plate is rigid, so it will stay planar even when the force is applied. How can I use this information to add a fourth equation?
Suppose the supports are elastic with a large constant , so we do have small displacements . If the corners lie on a plane, and neglecting for simplicity the slant of the plane, which is small if the displacements are small, it must be
This is my fourth equation, whence the solution
The reverse problem is easier. If I know , I can directly obtain from the force equilibrium, from the equilibrium around Y and from the equilibrium around X: