Page 33 - Buried Pipe Design

External Loads 11

Trench condition. The Marston load theory is based on the concept of a
prism of soil in the trench that imposes a load on the pipe, as shown in
Fig. 2.1. A trench (ditch) conduit as defined by Marston was a relatively
narrow ditch dug in undisturbed soil. Marston reasoned that settlement of
the backfill and pipe generates shearing or friction forces at the sides of the
trench. He also assumed that cohesion would be negligible since (1) con-
siderable time would have to elapse before cohesion could develop and (2)
the assumption of no cohesion would yield the maximum load on the pipe.
The vertical pressure V at the top of any differential volume element
B d (1) dh is balanced by an upward vertical force at the bottom of the
element V dV (see Fig. 2.1). The volume element is B d wide, dh tall,
and of unit length along the axis of the pipe and trench. The weight of
the elemental section is its volume times its unit weight, expressed as

w B d (dh)(1)
where (B d )(dh)(1) is volume of the element and is the specific weight
density.
The lateral pressure P L at the sides of the element at depth h is

active lateral unit pressure
P L (vertical unit pressure)
vertical unit pressure

or

V
P L K (Rankine’s ratio)
B d
The shearing forces per unit length F s on the sides of the differential
element, induced by these lateral pressures, are F s K(V/B d )( ′) dh
where ′ coefficient of friction. The vertical forces on the element are
summed and set equal to zero.
F v 0

Or, the upward vertical forces are equal to the downward vertical forces.
Thus, for equilibrium, vertical force at bottom shear force at sides
vertical force at top weight of the element, (dimensionally, force per
length), or
2K V
(V dV) dh V B d dh
B
d
⎛
KV ′ ⎞
dV = γ B − 2
dh ⎜ ⎝ d B d ⎟ ⎠ (2.1)