Solid pressure tells us how strongly a solid object presses on the surface beneath it.
The same weight can produce small pressure if it is spread over a large area, or large pressure if it is concentrated on a small area.
1. Average pressure
Pressure is force divided by contact area:
\[
\begin{aligned}
P &= \frac{F}{A}.
\end{aligned}
\]
For an object resting on a horizontal surface, the force is usually the weight:
\[
\begin{aligned}
W &= mg.
\end{aligned}
\]
Therefore, the average pressure is
\[
\begin{aligned}
P_{\mathrm{avg}}
&= \frac{W}{A} \\
&= \frac{mg}{A}.
\end{aligned}
\]
2. Rectangular block
For a block with base length \(L\) and base width \(W_b\), the contact area is
\[
\begin{aligned}
A &= L\cdot W_b.
\end{aligned}
\]
If the load is centered and the block is level, the pressure is approximately the same everywhere on the base:
\[
\begin{aligned}
p_{\mathrm{center}}
&\approx P_{\mathrm{avg}}.
\end{aligned}
\]
3. Cylinder and sphere contact
For a cylinder standing on a circular base of radius \(R\), the area is
\[
\begin{aligned}
A &= \pi R^2.
\end{aligned}
\]
For a sphere, the true contact region depends on deformation and material stiffness.
In this calculator, a sphere uses an effective contact patch radius \(a_c\):
\[
\begin{aligned}
A_{\mathrm{contact}} &= \pi a_c^2.
\end{aligned}
\]
4. Why tilted objects have higher pressure at one side
If the object is tilted, the line of action of the weight may not pass through the center of the base.
That shifted load is called an eccentric load.
For small tilt angles, the shift of the load can be estimated by
\[
\begin{aligned}
e_x &\approx h_{\mathrm{cm}}\tan(\theta_x), \\
e_y &\approx h_{\mathrm{cm}}\tan(\theta_y).
\end{aligned}
\]
Here \(h_{\mathrm{cm}}\) is the height of the center of mass above the base.
A larger center-of-mass height makes the same tilt angle produce a larger load shift.
5. Simple local pressure model
For a rectangular base, a simple linear pressure approximation is
\[
\begin{aligned}
p(x,y)
&= P_{\mathrm{avg}}
\left(
1+\frac{12e_xx}{L^2}
+\frac{12e_yy}{W_b^2}
\right).
\end{aligned}
\]
This formula is useful for learning because it clearly shows the idea:
if the load shifts toward one side, pressure increases there and decreases on the opposite side.
For a circular contact region of radius \(R_c\), the calculator uses
\[
\begin{aligned}
p(x,y)
&= P_{\mathrm{avg}}
\left(
1+\frac{4(e_xx+e_yy)}{R_c^2}
\right).
\end{aligned}
\]
6. Lift-off warning
If the model predicts \(p_{\min}<0\), it means the full-contact assumption has failed.
A surface can push up on an object, but it cannot pull the object downward.
\[
\begin{aligned}
p_{\min} &< 0
\quad\Longrightarrow\quad
\text{partial contact, rocking, or tipping may occur.}
\end{aligned}
\]
7. Worked example
A concrete block has mass \(800\ \mathrm{kg}\) and base dimensions \(0.8\ \mathrm{m}\times0.5\ \mathrm{m}\).
\[
\begin{aligned}
A &= L\cdot W_b \\
&= 0.8\cdot0.5 \\
&= 0.4\ \mathrm{m^2}.
\end{aligned}
\]
\[
\begin{aligned}
W &= mg \\
&= 800\cdot9.81 \\
&= 7848\ \mathrm{N}.
\end{aligned}
\]
\[
\begin{aligned}
P_{\mathrm{avg}}
&= \frac{W}{A} \\
&= \frac{7848}{0.4} \\
&= 19620\ \mathrm{Pa} \\
&= 19.6\ \mathrm{kPa}.
\end{aligned}
\]
If the block is level and the load is centered, the center pressure is about \(19.6\ \mathrm{kPa}\).
If the block tilts, one side or corner carries more of the load, so local pressure there becomes larger.
8. Formula summary
| Situation |
Formula |
Meaning |
| Average pressure |
\(P_{\mathrm{avg}}=W/A\) |
Weight divided by contact area |
| Weight |
\(W=mg\) |
Downward load due to gravity |
| Rectangular area |
\(A=L\cdot W_b\) |
Contact area for a rectangular base |
| Circular area |
\(A=\pi R^2\) |
Contact area for a circular base |
| Rectangular local pressure |
\(p=P_{\mathrm{avg}}(1+12e_xx/L^2+12e_yy/W_b^2)\) |
Simple tilt/load-shift approximation |
| Lift-off warning |
\(p_{\min}<0\) |
The full-contact model is no longer physically valid |
9. Assumptions
- The surface is horizontal and rigid.
- The object is treated as a rigid body.
- The pressure model is a simple educational approximation.
- For real contact stresses, material deformation and surface shape may be important.
Key idea: pressure increases when the same weight acts over a smaller area, and tilt shifts the load so one side or corner carries more pressure.
