Gravity loading of massive concrete, stone, and soil structures often produce the largest components of stress for which the designer must provide. For slopes, retaining walls, and geologic formations, body force may, in fact, be the only load.

**Wedges with
Constant Body Forces **

Gravity loading of
massive concrete, stone, and soil structures often produce the largest components
of stress for which the designer must provide. For slopes, retaining walls, and
geologic formations, body force may, in fact, be the only load. Seismic loading
is often approximated by the assumption of a constant acceleration applied to
the structure and even when the acceleration is not constant, the assumption of
a constant peak value gives the engineer a worst-case estimate.

A solution can be
obtained in polar coordinates, but is more clearly pre-sented in the Cartesian
reference frame shown in Figure 8.10. Letting γ* _{x}*
and γ

This is a wonderfully
simple solution since it predicts a linear stress field. Therefore Ɐ^{2}(σ* _{x +}* σ

On the other hand, at β>= 90^{o} the analytic solution is perfectly reasonable and is used in the design of
concrete gravity dams. Figure 8.12 shows a comparison of the analytic solution
and experimental results for a gelatin model for a concrete dam shape in terms
of isochromatics and isoclinics. The agreement is excellent. The same is true
for a vertical cut (β=* *90^{o})
shown in Figure 8.13.

It would appear then that this “paradox”
is a problem with uniqueness, which, in retrospect, is not surprising. From our
discussion in Chapter 4 of the Laplace equation Ɐ^{2}(σ* _{x+ }*σ

skeptical of results where the stress
field does not either damp out as *r* ' or approaches the free
field.

Example 8.3

A concrete gravity dam with a vertical
upstream face is to be built so that, with a full reservoir, there will be no
tension and an adequate factor of safety against crushing ( f'C=1125
psi = allowable compressive stress).

Knowing from our
elasticity wedge solutions that the stresses will be distributed linearly over
every horizontal cross-section, and assuming γ* _{w}*
(water) = 62.4 lb/ft

i. the optimum angle

ii. the
maximum permissible height *z*_{max} *= **H*

Solve for the case of
no horizontal acceleration and also for the case of a hor-izontal acceleration *a _{x =}* p

Comment on your results knowing that
Shasta Dam in Northern California, (the highest gravity dam in the U.S.) is 580
feet high at an angle α = 36^{o} and was supposedly designed “at the limit” for such a concrete.

Shasta Dam is a good design. Actu-ally Shasta Dam looks more like the illustration at right. Concrete will take some tension and downward water force, Fw , will reduce or eliminate seis-mic tension at the heel. By bringing the downstream slope toward the vertical σt ' σz reducing crushing. Highest gravity dam is in Switzerland (no earthquakes) at z max > 900 ft. Reason- able since if σz = σy t, z max = fc′ /γc = 1080 ft with stronger concrete even greater heights are possible in non seismic zones.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Civil : Principles of Solid Mechanics : Wedges and the Half-Space : Wedges with Constant Body Forces |

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