Page 74 - Buried Pipe Design

50 Chapter Two

3. Pi terms are combinations of the FVs that meet the following
three requirements: (a) The number of pi terms must be at least
the number of FVs minus the number of BDs. (b) The pi terms
must all be dimensionless. (c) No subset of pi terms can be inter-
dependent. This is ensured if each pi term contains a fundamen-
tal variable not contained in any other pi term.
Pi terms can be written by inspection.

Example Problem 2.8 Write a set of pi terms for investigating the maximum
wheel load W that can pass over a buried flexible pipe without denting the
top of the pipe. See Fig. 2.28 for a graphical and Fig. 2.29 for the laboratory
test for the determination of the soil modulus E′. The use of the three pi-term
requirements yields the following:

FVs BDs
W wheel load F
EI wall stiffness FL
H height of soil cover L
P all pressures FL 2
D pipe diameter L
E′ soil modulus FL 2
soil unit weight FL 3
7 FVs 2 BDs 5 pi terms required
Here are the pi terms:
2
(W/E′D ) 1
3
(EI/D P) 2
(H/D) 3
(P/E′) 4
( D/E′) 5
This set of five pi terms, by inspection, is not the only possible set. If this
set is not convenient for investigating the phenomenon, a different set can
be written. For these pi terms, the maximum wheel load is given by the
mathematical function
1 f ( 2 , 3 , 4 , 5 ) (2.21)

This functional relationship of pi terms needs to be found. Principles of
physics provide one possibility while prototype studies that allow the writing
of empirical best-fit equations of graphs of data are another option. If small-
scale model studies are to be used, Eq. (2.21) must describe the performance
of both model and prototype. Therefore, the model must be designed such
that corresponding pi terms on the right side of Eq. (2.21) are equal for both
model and prototype. This can be accomplished, even for small-scale models,
because pi terms are dimensionless and therefore have no feel for size—or
any other dimension, for that matter. If the subscript m designates model, in
order to design the model, the design conditions (DCs) are ( m ) 2 ( 2 ), etc.: