Page 76 - Buried Pipe Design

52 Chapter Two

3. (P r ) (E r ′)
4. ( r ) (E r ′)/(D r )

Because soil is a complex material, it would be convenient if the same
soil could be placed and compacted in the same way in both model and
prototype. The results are E r ′ 1 and r 1. But now design conditions
3 and 4 are not met. From design condition 3, P r 1. Therefore, all
pressures P must be the same in the model as at corresponding points
in the prototype. For example, tire pressures must be the same in model
and prototype. The soil pressure must be the same at corresponding
depths in the model and prototype. But this is impossible for a small-
scale model if the soil has the same unit weight. One remedy is to test
the model in a long-arm centrifuge such that centrifugal force plus
gravity increases the effective unit weight of the soil in the model.
Another approximate remedy is to draw seepage stresses down through
the model (air or water if the soil is to be saturated) in order to increase
the effective unit weight of the model soil. For most minimum soil cover
studies, the effect of soil unit weight is negligible, so DC 4 is ignored.
From tests on the model, weight W can be observed when the buried pipe
is dented.
The prediction equation (PE) is the equation of pi terms on the left
sides of Eq. (2.21) for model and prototype becomes

2 2
(W/E′D ) (W/E′D ) m
If E r ′ 1, then the prediction equation is

W W m (D r ) 2

where D r is the length scale ratio of prototype to model. If the length
scale ratio is 5 (that is, 5:1 prototype to model), the load W on the pro-
totype that will dent the buried pipe is 25 times the load W m that dents
the model pipe.
In order to write a mathematical equation (model) for the phenom-
enon, enough tests must be made to provide graphs of data for 1
f( 2 ) with 3 held constant and for 1 f′( 3 ) with 2 held constant.
From the best-fit graphs plotted through the data, an equation of
combination can be written for 1 f( 2 , 3 ). This becomes a mathe-
matical model.
In fact, neglecting dead load, design condition 3 is met when tire
pressures are the same in model and prototype. Then the mathemati-
cal model is simply the equation of the best-fit graph of 1 f( 2 ). It
can be written in terms of the original fundamental variables.