Vertical Stress Under Circular Load Calculation

This section explains the formula for calculating the vertical stress ($\sigma_z$) at a specific depth ($z$) directly beneath the center of a uniformly loaded circular area on the surface of a homogeneous soil layer. Understanding this distribution is crucial in geotechnical engineering for foundation design.

Identifying the Correct Formula

The problem asks for the vertical stress ($\sigma_z$) at a depth ($z$) below the center of a circular load with radius ($a$) and uniform pressure ($p$). This is a standard problem in soil mechanics, often solved using principles derived from elasticity theory, specifically Boussinesq's analysis for point loads, adapted for a distributed circular load.

The accepted formula for the vertical stress ($\sigma_z$) directly under the center of a uniformly loaded circular area is:

σz = p \(\left[ {1 - \frac{{{z^3}}}{{{{\left( {{a^2} + {z^2}} \right)}^{3/2}}}}} \right]\)

Where:

• $p$ = uniform pressure applied over the circular area
• $a$ = radius of the circular loaded area
• $z$ = depth below the surface at which the stress is calculated

Analyzing the Options

We need to match the derived formula with the given options:

Comparing the standard formula with the options, Option 2 is the correct representation of the vertical stress under the center of the circular load.

Conclusion on Stress Distribution

The vertical stress ($\sigma_z$) at depth ($z$) beneath the center of a circular load of radius ($a$) and uniform pressure ($p$) is correctly given by the expression in Option 2.