Newmark's influence chart
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Newmark's Influence Chart is an illustration used to determine the vertical pressure at any point below a uniformly loaded flexible area of soil of any shape. This method, like others, was derived by integration of Boussinesq's equation for a point load.[1]
Newmark obtained values of R/z that corresponded to various pressure ratios by using the equation (R/z)=√(1-(〖∆σ〗_z/q)^(-2/3)-1), where R = the radial distance away from the point at which the load is applied, z = the vertical depth below the applied load, 〖∆σ〗_z = the stress at the point of interest a depth of z below the surface, and q = the load per unit area applied at the surface.[1] Using the pressure ratios obtained from the equation above, he was able to form the influence chart.
Application
The chart is constructed by drawing concentric circles. The circles are divided by equally spaced radial lines. The radii of the circles are equal to the R/z values corresponding to F U K〖∆σ〗_z/q = 0, 0.1, 0.2,...,1. There are nine circles shown since when 〖∆σ〗_z/q = 0, R/z = 0 also. The unit length for plotting the circles is AB.[1]
When solving a vertical stress problem using Newmark's influence chart, the influence value (IV) must be taken into account. It is proportional to the number of elements in the chart and is given by 1/N, N being the total number of elements in the chart. For example, a typical chart consists of 200 elements; therefore, the influence value is 0.005.[1] The procedure for obtaining the vertical pressure at any point below a loaded area is as follows:
- Verify the depth z below the uniformly loaded area where the stress increase is to be obtained.
- Plot the plan of the loaded area with a scale of z equal to the unit length of the chart (AB).
- Place the plan on the influence chart in such a manner that the point below which the stress is to be determined in located at the center of the chart.
- Count the number of elements (M) of the chart enclosed by the plan of the loaded area.
The formula used to solve for the increase in pressure at the point being considered is 〖∆σ〗_z = (IV)qM, where IV = influence value, q = pressure on the loaded area, and M = number of elements enclosed by loaded area.[1]