How to Determine Vertical Stress on Soil Mass? (Newmark’s Chart) | Soil Engineering

Newmark gave a construction to determine the vertical stress at any point below a footing of any shape. Newmark’s influence chart is based on the concept of the Boussinesq theory for vertical stress below the center of a uniformly loaded circular area.

Construction of Newmark’s Influence Chart:

When a uniformly loaded circular area of radius r₁ is divided into 20 equal parts, the vertical stress at the center at any depth z due to load from each part (sector) will be equal to 1 /20 of that due full circular area. Thus,

If this vertical stress is assigned an arbitrary fixed value of say 0.005q, then –

Solving Eq. (8.35), for r_1, the radius of the loaded area that causes 1/20 of the total vertical stress is obtained as r₁ = 0.26975z. Thus, every 1/20 of the circle of radius 0.26975z would cause a vertical stress of 0.005q at a depth z below the center of the loaded area. For any given depth, where vertical stress is to be determined, say z = 1 m, a circle can be drawn with r₁ = 0.26975z = 0.26975 × 1 = 0.26975 m, as shown in Fig. 8.31.

A second circle is now drawn and divided into 20 equal sector parts with a radius r₂ such that each sector part would cause a vertical stress of 2 × 0.005q.

Solving Eq. (8.36) for r₂, we get r₂ = 0.4005z. In the two circles drawn so far, there are 40 parts, with each part exerting an equal vertical stress of 0.005q at depth z below the center of the loaded area. Similarly, the radius of the third circle can be determined from –

Using a similar procedure, a total of nine circles can be drawn, with each annular area divided into 20 equal parts, called unit areas, and each unit area exerting a vertical stress of 0.005q. The radii of these circles are given in Table 8.3. For the specific value of 0.005cq, the radius of the 10th circle would become infinity. Figure 8.32 shows Newmark’s influence chart with these nine circles.

The procedure used in constructing Newmark’s influence chart ensures that each part or unit area in the chart would exert a vertical stress of 0.005g at the center at depth z. The scale, to which the chart is drawn, is specified in the form of line AB, which is equal to the depth z to the scale of the chart.

Use of Newmark’s Influence Chart:

The plan of the loaded area (footing) is drawn on a tracing paper to such a scale that the length of line AB on the chart is equal to the depth where vertical stress is required. For example, if the vertical stress is required at a depth of 4 m and if the length of the line AB is 2 cm, then the scale to be adopted is 2 cm = 4 m, that is, 1:200.

The plan of the loaded area, drawn to this scale, is placed on the chart such that point P, where vertical stress is required, coincides with the center of the chart. For example, the loaded area placed on the chart as shown in Fig. 8.33(a) to determine vertical stress at the corner, and as in Fig. 8.33(b), to determine vertical stress at the center. The vertical stress at point P at the required depth is then given by –

σ_z = I_f q N …(8.38)

where I_f is the influence value, which is 0.005 in the present case; q is the pressure on the loaded area; and N is the number of unit areas of the Newmark’s chart enclosed by the loaded area.

In counting the number of unit areas enclosed by the loaded area (N), the unit areas partially enclosed should also be properly considered in terms of fractions such as 1/2, 1/3, 1/4, etc., and the total value of N should be computed.

If the vertical stress is required at a different depth, say z = 1 m, the plan of the loaded area is to be drawn to a new scale of 2 cm = 1 m, that is, 1:50, and the procedure is repeated.

Advantages and Limitations of Newmark’s Influence Chart:

The advantage of Newmark’s influence chart is that this chart can be conveniently used to estimate the vertical stress below loaded areas of any shape. The vertical stress can be determined at any point, either within or outside the loaded area. The chart also eliminates tedious computation to determine the Boussinesq influence factor I_B such as Eqs. (8.29) and (8.30).

However, the loaded area is to be plotted to a different scale each time the vertical stress is to be computed at a different depth. Further, there is a personal error involved in counting the number of unit areas partially covered by the loaded area. Also, with the availability of programmable calculators, Excel worksheets, and high-speed com­puters, the Newmark’s chart does not offer any special advantage. Thus, it has now become more or less obsolete.