If, rather than surface traction, self weight is the loading, geostatic fields occur.

**Geostatic Fields
**

If, rather than surface
traction, self weight is the loading, geostatic fields occur. We shall see in
the next chapter that a general solution for arbitrary geometry is very
difficult (actually unknown for some simple and important cases), but for the
half-space and the uniaxial case, a solution is straightfor-ward.

Considering first the half-space where *z* is
again taken as the depth coordinate, vertical equilibrium and the geometric
condition that E_{x =E }* _{y}* = 0 everywhere lead to the solution:

which, if *v=1/*_{2}
, is again isotropic. Since the only variable is *z*, the half-space can
be considered of finite depth, *H*, supported on a rigid base as shown in
Figure 5.3. The vertical displacement is therefore:

Determining the support condition necessary to
achieve the linear geo-static stress field in the uniaxial case provides a
simple demonstration of the physical importance of elastic rotations and the
efficiency of determining them as a prelude to calculating the displacement
field. Consider the block shown in Figure 5.4 standing under its own self weight
and free now to

since from symmetry at *x = *0, Ï‰* _{xz =}* 0 and the integration constant is zero. Similarly following the same procedure,

Thus to achieve a uniform contact pressure, â€'Î³*H*,
at *z =* *H*, the supporting surface (Figure 5.4) must
be a smooth parabolic dish satisfying the displace-ment boundary condition
dictated by the required vertical displacement. If, however, the supporting
surface were flat, this geostatic solution, by St. Venantâ€™s principle, would
still be correct some distance from the bottom since the actual distribution of
the contact stress must, by vertical equilibrium, be statically equivalent to a
uniform distribution.

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Civil : Principles of Solid Mechanics : Linear Free Fields : Geostatic Fields |

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