2085
Technical Committee 207 /
Comité technique 207
In the light of these considerations, the influence of the steel
sockets on the global pull-out capacity can be said to be
twofold. First, the sockets directly sustain a portion
socket
F
of
the external load owing to their shear/flexural strength:
henceforth, this will be referred to as the “direct effect”. Besides
this, an “indirect effect” stems from the formation of a global
failure mechanism with a remarkable increase in the radial
stress around the tie rod and, therefore, in the mobilizable shear
force
lateral
. For any future design purpose, the necessity of a
reliable estimation of both
F
socket
F
and
is self-evident.
lateral
F
Figure
7. Radial stress contour plot at the onset of failure.
3 A DESIGN-ORIENTED ANALYTICAL MODEL
In this section a design-oriented analytical model is defined
to estimate each resisting component contributing to the total
pull-out capacity of the anchor. The accuracy of the model has
been also assessed by comparing the analytical results with the
outcomes from full-size FEM analyses, i.e. with no embedment
surcharge.
Apparently, the near presence of the tie rod prevents the
sockets from being interpreted as isolated deeply buried pipes,
so that the force exerted by the soil on each socket (and
viceversa) cannot be evaluated via the well-known concept of
``uplift coefficient'' (or even ``break-out coefficient'' - see for
instance, Rowe and Davis (1982a,b), Merifield and Sloan
(2006), White
et al.
(2008). In contrast, it has been numerically
observed that the soil between a single socket and the tie rod
behaves as if it was rigidly connected to the anchor, giving rise
to a sort of ``corkscrew mechanism''.
Figure 8. Simplified static scheme for the anchoring system.
To evaluate at the same time both
socket
F
and
lateral
, the
simplified vertical wedge mechanism in Figure 7 (left picture) is
considered (the shear force
int
coincides with
lateral
). For the
sake of simplicity, the two half-sockets are assumed to be
positioned at the same elevation, so that a cylindrical reference
frame
can be setup to describe an axisymmetric
stress/strain state within the soil wedge (axisymmetric
conditions are indeed expected beyond a given distance over the
sockets). The right picture in Figure 8 illustrates the reference
static scheme, i.e. the forces acting both on the tie rod and the
soil wedge (the rod weight is neglected and boundary reaction
forces are not visualized).
F
T
F
, ,
z
From the analysis of all numerical results the following
conclusions have been drawn:
1. except for local disturbances next to the sockets, all
the stress components are almost linearly
distributed along the vertical
z
-axis;
2. the vertical direct strain component
z
is much less
than the other two (
and
), so that
0
z
can
be assumed;
3. the normal force
ext
N
in Figure 8 can be
approximately evaluated by assuming a
P
k
linear
distribution for the radial stress
r
along the outer
side of the soil wedge (
P
k
stands for the passive
earth pressure coefficient);
4. the inner and the outer
r
distributions can be
linearly related through a dimensionless constant
;
5. the failure distributions of the mobilized friction
angle
arctan /
mob
z
along the inner and
outer sides of the soil wedge exhibit a mean value
less than the soil friction angle
and
approximately equal to:
lim
cos cos
tan
1 sin sin
mob
(1)
where
is the soil dilatancy angle.
Relationship (1) stems from the fact that, during the pull-out
process, the aforementioned soil wedge undergoes a sort of
``axisymmetric simple shear loading''. In other words, the
loading conditions of the soil elements around the tie rod are
highly constrained and similar to those a soil specimen
experiences within a so-called simple shear apparatus: as was
recently discussed by di Prisco and Pisanò (2011), this implies
the material dilatancy to significantly affect the limit shear
stress.
The above considerations lead to the formulation of the
following system of equations:
int
int
int
int
int
tan
tan
socket
ext
mob
ext
ext
mob
T F T
T N
T N
N N
(2)
with the unknowns
int
N
,
int
T
(=
la
),
and
teral
F
ext
T
socket
(
ext
F N
is
simply obtained by integrating the
P
k
-
r
distribution along the
outer surface of the soil wedge). While the equations in system
(2) hold at any stage of the loading process,
lim
mob
mob
is to be
set at failure – i.e. when the limit frictional capacity is attained
along the sides of the soil wedge.
For system (2) to be solved, the determination of the
coefficient
introduced in the previous assumption 4 is
required. For this purpose, an original procedure has then been
conceived: this is based on the solution of a classical rock
engineering problem, concerning the determination of the
elastic stress state around a circular cavity (Jeager
et al.
2007).
For this purpose, an axisymmetric boundary value problem has
been posed by assuming that: (i) the soil wedge in Figure 8 is
internally elastic; (ii) the direct strain
z
is nil; and (iii) the
vertical stress gradients are much lower than those along the
radial direction. While the problem formulation and the solution
strategy can be found in di Prisco and Pisanò (2012), the
obtained
expression is reported here:
1 2
2
4
2 2
s
c c
R L z
R
(3)
where:
