Vertical Stress Below Arbitrarily Shaped Areas
Newmark (1942) developed a chart to determine the increase in vertical stress due to a uniformly loaded area of any shape. The chart consists of concentric circles divided by radial lines (Figure ). The area of each segment represents an equal proportion of the applied surface stress at a depth z below the surface. If there are 10 concentric circles and 20 radial lines, the stress on each circle is qs/10 and on each segment is qs/(10 3 20). The radius-to-depth ratio of the fi rst (inner) circle is found by setting Dsz 5 0.1qs in Equation , that is,
from which r/z 5 0.27. For the other circles, substitute the appropriate value for Dsz; for example, for the second circle Dsz 5 0.2qs, and fi nd r/z. The chart is normalized to the depth; that is, all dimensions are scaled by a factor initially determined for the depth. Every chart should show a scale and an infl uence factor IN. The infl uence factor for Figure is 0.001. The procedure for using Newmark’s chart is as follows:
1. Set the scale, shown on the chart, equal to the depth at which the increase in vertical stress is required. We will call this the depth scale.
Newmark’s chart for increase
in vertical stress.
2. Identify the point below the loaded area where the stress is required. Let us say this point is A.
3. Plot the loaded area, scaling its plan dimension using the depth scale with point A at the center of the chart.
4. Count the number of segments (Ns) covered by the scaled loaded area. If certain segments are not fully covered, you can estimate what fraction is covered.
5. Calculate the increase in vertical stress as Dsz 5 qsINNs