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Tony's Intertwingled Memex • Reference Manual

Triangular Foundation Slab Soil Bearing Pressure (Civil & Construction Engineering)

Formulas & Definitions

A geotechnical structural engineer at the UNLV Howard R. Hughes College of Engineering is designing a reinforced concrete mat foundation for a heavy construction crane tower on an alluvial Las Vegas soil subgrade. The footprint of the triangular foundation slab forms a closed and bounded region DD in the horizontal plane with vertices at:

V1(0,0),V2(4,0),V3(0,3)V_1(0, 0), \quad V_2(4, 0), \quad V_3(0, 3)
where spatial dimensions xx and yy are measured in meters (m\text{m}).

Under severe eccentric overturning wind loads and tower deadweight, the contact soil bearing pressure distribution beneath the slab is modeled by the multivariable quadratic function:

P(x,y)=x2+y2−2x−4y+20(in kilopascals, kPa)P(x, y) = x^2 + y^2 - 2x - 4y + 20 \quad (\text{in kilopascals, kPa})

To prevent differential settlement and localized punch-through failure, the engineer must find the global maximum and global minimum soil bearing pressures across the entire slab footprint.

  1. State why the Extreme Value Theorem guarantees the existence of absolute extrema for P(x,y)P(x, y) on this domain.
  2. Perform Tier 1 (Interior Search): Find all critical points of P(x,y)P(x, y) and determine whether they lie strictly within the open interior of the triangular slab.
  3. Perform Tier 2 (Boundary Edge Search): Parameterize each of the three boundary edges of the triangle and locate all 1D critical points along the perimeter.
  4. Perform Tier 3 (Vertex Check): Evaluate the bearing pressure at each corner vertex of the foundation.
  5. Assemble the Master Comparison Table and state the absolute minimum and absolute maximum soil bearing pressures, identifying their precise physical coordinates.