With a horizontal component of body force, the behavior of the half-space is quite different. Although such body forces may arise in many ways (seepage being a common source in soils), acceleration, as for example in an earth-quake, is probably the most important.

**Uniform
Acceleration of the Half-space **

With a horizontal
component of body force, the behavior of the half-space is quite different.
Although such body forces may arise in many ways (seepage being a common source
in soils), acceleration, as for example in an earth-quake, is probably the most
important.

Consider the half-space of finite depth *H* of
material with unit weight Î³. From the previous
section, the geostatic field is:

If uniform horizontal* and vertical accelerations, *a _{x}*
and

and the field can be thought of as â€œone-dimensional
plane strainâ€œ (i.e., Ïƒ * _{x = }*Ïƒ

Essentially this dynamic (actually pseudostatic)
solution for the free field is the geostatic free field with a unit weight of
(1 â€' *k _{v}*)Î³ superimposed with
shear

introduced by the horizontal acceleration component.
However, considering the deformation, this is not â€œpure shear,â€œ but a case of
â€œsimple shearâ€œ since there must be an elastic rotation field that keeps
horizontal planes from rotating. From Equation (4.13) or by inspection

This simple result is remarkable. Both the geometric
field condition E* _{x}*
=

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Civil : Principles of Solid Mechanics : Linear Free Fields : Uniform Acceleration of the Half space |

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