Hydraulic Excavator Bracket Stress Analysis (Mechanical Engineering)

A mechanical design engineer analyzes the localized contact interface on a cast-steel hydraulic pivot bracket for a heavy-duty mining excavator. The curved load-bearing shoulder of the bracket is modeled mathematically by the logarithmic surface:

where $x, y, z$ represent spatial coordinates in inches ($\text{in}$).

During heavy bucket digging, a hydraulic actuator piston pushes a spherical pin directly against the bracket shoulder at the localized contact coordinate $(x_0, y_0) = (1, e - 1)\text{ in}$.

1. Compute the baseline height $z_0 = f(1, e - 1)$ on the bracket shoulder.
2. Apply the multivariable chain rule to compute the partial derivatives $f_x(x, y)$ and $f_y(x, y)$.
3. Evaluate the partial derivatives at $(1, e - 1)$ and formulate the equation of the tangent plane to the bracket contact zone in both explicit ($z = \dots$) and general ($ax + by + cz = d$) forms.
4. If the hydraulic piston applies a normal compressive force of magnitude $F_N = 50\text{ kips}$, explain how finite element analysis (FEA) software uses the tangent plane's normal vector to resolve the force into axial and shear stresses along machine frame axes.