Discussed herein are formulae for designing landslide-restraint piles with the horizontal subgrade reactions in consideration. With the intensity of the subgrade reaction
p assumed to be lineally proportional to the horizontal displacement of pile axis
y, that is,
p=
kh⋅δ⋅
ywhere δ is the pile diameter and
kh is the coefficient of horizontal subgrade reaction, there are still three different approaches: the first one assumes the force of landsliding
H as acting only along the slip surface and the pile length as being infinite both upwards and downwards from the slip surface; the second one assumes, with the loading condition as the same, the pile length as being infinite downwards only; and the third one assumes
H as distributing all along the pile above the slip surface while the root part of the pile being infinite.
Observations in large-scale shearing tests in situ and excavations of yielded piles in situ present us three markedly different curves of pile deflection as illustrated in Fig. 5. Any of piling formulae are to explain the differences between these three.
Fig. 1 shows a case in which the pile was so constructed as to yield only to the shearing force but actually ran to rupture by bending moment. Landsliderestraint piles are recommended to be due tested against the rupture by bending moment.
§1 presents the general solution to the elastically supported elastic pile under distributed load; §2.1 the solution particular for the case with additional lateral force
H0 and bending moment
M0 at the head; §2.2 that for the free-head pile and §2.3 that for the pile only under
H0 and
M0. §2.4 deals with pile of finite lengths both above and below the slip surface. As is seen in Fig. 6 or in Fig. 11 the point of
y2=0 is likely to deepen far below the slip surface with the value β
h decreased: the finite-length-pile formula presented in §2.4 will be warranted. Fig. 10 is the chart for easy calculation of
Mmax and based on exps. (11), (6), (7) and (8), the parameter being
n, the fourth root of the ratio of
kh values. Fig. 11 gives values for calculating resultant subgrade reaction and necessary length of pile below the slip surface. The meanings of symbols are:
EI: rigidity of pile, δ: diameter of pile
kh: coefficient of horizontal subgrade reaction
y: displacement of pile axis,
Es=
kh·δ·
y and β=4√
Es/4
EI: for the landsliding mass,
E
s, and β for the part below the slip surface
n=β/η,
h: depth to the slip surface
K=η/
Es H: resultant force of landsliding
η: load conversion factor
=2
H/
h2 for hydrostatically distributed load
i: deflection angle,
M: bending moment,
S: shear
ε, ν and ν: dimensionless factors of deflection, bending moment and shear, respectively.
ρ and λ
0: dimensionless factors of rusultant subgrade reaction and root length of pile, respectively.
Note: Suffix 0, 1 and 2 or head bar denote pile head, sliding mass and the root part of the pile, respectively. Numbers with asterisk refer to those in the literature.
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