# Distribution of contact forces on a square at rest

Consider a square (blue) on top of a platform (yellow), on which gravitational force (black) acts. We know that if the blue square is in equilibrium, contact forces from the platform (white) must act in the opposite way to gravity.

I would like to ask whether something can be said about the distribution of such forces on the boundary of the square?

If a force $$dF$$ acts on a small patch of "surface" around a point $$x$$, we must clearly have $$\int dF = F_g$$, i. e., the net force is the gravitational force. Also, we must have $$\int (a-x)dF = F_g \cdot r$$, which results from torque balance. Gravity produces a torque $$F_g \cdot r$$, which has to be balanced by the torques from the contact forces.

But is there something more that can be said about the distribution of forces? The two integrals by themselves give an infinite number of solutions. The force could be concentrated at one specific point, or spread out over the whole surface (which sounds more realistic)... Which is it and how exactly is it spread out?

• Your question ist not clear. is your square of homogeneous material, why should the border be different ? You talk with 3 colors, but five no picture Mar 18, 2022 at 21:42