Elastic Settlement Analysis of Rigid Footings Relying On The “Characteristic Point” Concept

7 In the present paper, the problem of finding the location of the so-called “characteristic point” of flexible footings 8 is revisited. As known, the settlement at the characteristic point, is equal to the uniform settlement of the respective 9 rigid footing. The cases of infinitely long strips and circular footings are studied fully analytically. For the case of 10 rectangular footings, analytical results (for flexible footings) are compared with the respective numerical results 11 (for rigid footings) obtained from 3D finite element analysis (210 cases were examined). As shown, the location 12 of the characteristic point may greatly deviate from the well-known values reported in the literature, as it strongly 13 depends on the thickness and Poisson’s ratio value of the compressible medium. For rectangular footings this 14 location also depends on their aspect ratio, L / B . The location of the characteristic point with respect to the center 15 of footing for the various cases examined is given in tabular form. Strain influence area values ( A j = ρ j E s /qB ) are 16 also given for the convenient calculation of the settlement ( ρ j ) of footings, especially the rigid, rectangular ones; 17 q is the uniform surcharge of footing and E s the soil modulus.


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The problem of settlement of shallow foundations is among the more important ones in classical soil mechanics.

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During the last several decades a fairly large number of approaches has been proposed in the literature for rigid 24 rectangular footings, an exact solution of which is still not available. In this respect, solutions have been provided 25 by Borodachev (1976), Borodachev and Galin (1974), Brothers et al. (1977), Butterfield (Pantelidis, 2020a(Pantelidis, , 2020b. The symbols are 45 described in Figure 1.

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The increase in the normal stress parallel to the i-axis due to loading over a whole surface area can be found by

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The strain influence factor concept has been introduced by Schmertmann (1970; see also Pantelidis 2020).

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For uniformly loaded flexible footings, it stands that the surface loading q is equal to the contact pressure at any 61 point on the plan-view of footing. For rigid footings carrying load q (assuming that the latter is also distributed 62 uniformly over the plan-view of footing), the contact pressure distribution σ is case-defined. 63 The elastic settlement of footing j  extending from the foundation level (z=0) to depth H, finally, derives from 64 the integration of vertical strain (recall Equation 6) between these limits. Using the contact pressure distributions of Equation 10 (see Figure 2; also Kézdi and Rétháti, 1988)

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From Table 2 it is inferred that, the Ch  parameter strongly depends on both the Poisson's ratio and the thickness, 82 H, of the compressible medium. More specifically, as H tends to infinity, Ch  tends to Grasshoff's 0.74 value 83 (Grasshoff, 1955;Kany, 1974 where, R is the radius of footing (R=B/2) and r is the independent variable (see Figure 2). The integration limits 95 are shown in Table 1. The calculated strain influence area values for flexible circular footings for various λ values 96 (recall Equations 9 and 7; see also the integration limits in Table 1) was compared against the respected strain 97 influence area values for the rigid disk. This comparison led to the Ch  values of Table 3 for the location of the 98 characteristic point. Table 3 Table 4).

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Given that seven Poisson's ratio values were considered in the analysis of rectangular footings (namely, ν = 0,  Table 5.

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It is very interesting that for each footing, the Ch  -ν relationship presents minimum for ν other than the extreme 136 0 and 0.5 values (see example curves in Figure 4; also Table 5). In addition it is noted that the maximum and It is reminded that Grasshoff (1955) suggested that Ch  be equal to 0.74 for all cases. 140 The question, now, is if the Ch  parameter is the same in the two (horizontal) directions (Grasshoff's assumption). 141 The analysis carried out in the framework of the present work showed that Grasshoff's assumption is valid. An           Contact pressure distribution chart for exible footings, rigid strips and rigid disks.  Example ν -λCh curves. Figure 5