I have a question about forces inside of walls.

When you concider the walls of a house, they have a certain density $\displaystyle \delta$ and they create a force downwards, which becomes bigger closer to the ground. Assume we don't need to care about the weight of the roof or that of the atmosphere. The force in each wall, will be

$\displaystyle \overbrace{\underbrace{w\cdot h\cdot t}_\texttt{total volume}\cdot \delta}^\texttt{total weight}\cdot g$

where w is the width of the wall, h is the heigth up to the top of the wall, and t is the thickness of the wall. So, independent of the width and the thickness of the wall, the pressure create by the wall above will be

$\displaystyle h\cdot\delta\cdot g$

Now to my question: Is the pressure uniform? That is, will the pressure be the same in all directions, vertically as horizontally? The pressure vertically will be $\displaystyle h\cdot\delta\cdot g$, since when the material gets squeezed from the top and the bottom, it gets compressed vertically, so it creates a pressure vertically since it wants to expand in that direction. Besides, it needs to support its own weigth. But what about horizontally, does it want to expand it that direction as well? How big will the pressure be in that direction? Near to the pressure vertically, or almost zero? Does it depend on the material of the wall? (wood/concrete/metal?)

Thanks in advance!