Hillside Foundation Slope Stability (Civil & Construction Engineering) Solution

A geotechnical and civil engineering firm assesses hillside terrain stability prior to constructing a reinforced concrete retaining wall and foundation slab for a commercial development in Henderson, Nevada.

A drone-based LiDAR survey generates a continuous digital elevation model $E(x, y)$ (in meters, $\text{m}$) across the building pad:

1. State how cross-sectional vertical cuts $y = y_0$ and $x = x_0$ define directional slopes on the terrain surface.
2. Determine the algebraic expressions for the directional slopes $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$.
3. Evaluate the slopes $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$ at the proposed foundation center location $(x, y) = (4, 10)\text{ m}$.
4. State whether the ground slopes uphill or downhill when walking due east ($+x$) and due north ($+y$) from the center location, and state the angle of inclination along each cardinal direction in degrees ($\theta = \arctan(m)$).

Step-by-Step Solution:

1. State the Cross-Sectional Trace Principle:

   • Slicing the 3D elevation surface with a plane $y = y_0$ isolates an east-west cross-section where longitude $y$ is constant; the derivative $\frac{\partial E}{\partial x}$ is the instantaneous slope $\frac{\Delta E}{\Delta x}$ along this cut.
   • Slicing with a plane $x = x_0$ isolates a north-south cross-section where latitude $x$ is constant; the derivative $\frac{\partial E}{\partial y}$ is the instantaneous slope $\frac{\Delta E}{\Delta y}$ along this cut. (Defined in Geometric Interpretation of Partial Derivatives.)

2. Compute the Algebraic Slope Expressions:

   • Differentiating with respect to $x$ (holding $y$ constant):
     $$\frac{\partial E}{\partial x} = \frac{\partial}{\partial x}(50) - 0.05 \frac{\partial}{\partial x}(x^3) + 0.1y \frac{\partial}{\partial x}(x) + 0.02 \frac{\partial}{\partial x}(y^2)$$
     $$\frac{\partial E}{\partial x} = 0 - 0.05(3x^2) + 0.1y(1) + 0 = -0.15x^2 + 0.1y$$
   • Differentiating with respect to $y$ (holding $x$ constant):
     $$\frac{\partial E}{\partial y} = \frac{\partial}{\partial y}(50) - 0.05 \frac{\partial}{\partial y}(x^3) + 0.1x \frac{\partial}{\partial y}(y) + 0.02 \frac{\partial}{\partial y}(y^2)$$
     $$\frac{\partial E}{\partial y} = 0 - 0 + 0.1x(1) + 0.02(2y) = 0.1x + 0.04y$$

3. Evaluate at $(x, y) = (4, 10)\text{ m}$:

   • For the east-west slope:
     $$\left.\frac{\partial E}{\partial x}\right|_{(4, 10)} = -0.15(4)^2 + 0.1(10) = -0.15(16) + 1.0 = -2.40 + 1.0 = -1.40\text{ m/m}$$
   • For the north-south slope:
     $$\left.\frac{\partial E}{\partial y}\right|_{(4, 10)} = 0.1(4) + 0.04(10) = 0.40 + 0.40 = 0.80\text{ m/m}$$

4. Interpret Cardinal Behavior and Angles of Inclination:

   • Eastward Direction ($+x$): Since $\frac{\partial E}{\partial x} = -1.40 < 0$, walking due east is walking downhill at a $-140\%$ grade.
     $$\theta_x = \arctan(-1.40) \approx -54.46^\circ$$
     (A steep drop of $54.5^\circ$ below the horizontal).
   • Northward Direction ($+y$): Since $\frac{\partial E}{\partial y} = 0.80 > 0$, walking due north is walking uphill at an $+80\%$ grade.
     $$\theta_y = \arctan(0.80) \approx 38.66^\circ$$
     (A climb of $38.7^\circ$ above the horizontal).
   • Engineering Conclusion: The steep drop to the east mandates an engineered retaining wall system to prevent catastrophic landslide slippage along the cut slope.