RATZ
Version 4-2
LOAD TRANSFER ANALYSIS OF
AXIALLY LOADED PILES
M. F. Randolph
February 2003
RATZ
Version 4-2
LOAD TRANSFER ANALYSIS OF
AXIALLY LOADED PILES
M. F. Randolph
February 2003
The accuracy of this program has been checked, and it is believed that, within the
limitations of the analytical model, results obtained with the program are correct.
However, the author accepts no responsibility for the relevance of the results to a
particular engineering problem.
Technical support in relation to operation of the program, or in respect of engineering
assistance, may be obtained from the author, who reserves the right to make a charge for
such assistance.
Address:
Centre for Offshore Foundation Systems
The University of Western Australia
Perth
Western Australia 6009
Telephone:
+61 8 9380 3075
Email:
mark.randolph@uwa.edu.au
February 2003
RATZ Manual Version 4-2
M.F. Randolph
CONTENTS
Page No.
PART A: GENERAL DESCRIPTION
1
INTRODUCTION ............................................................................................1
2
LOAD TRANSFER CURVES .........................................................................2
2.1
Monotonic Loading...............................................................................2
2.1.1
2.2
Hyperbolic Pre-peak Load Transfer Curve ...............................3
Cyclic Loading ......................................................................................4
2.2.1
Standard Yield Criterion ...........................................................4
2.2.2
Alternative Yield Criterion A ...................................................5
2.2.3
Alternative Yield Criterion B ...................................................6
2.3
Cyclic Residual Shaft Friction ..............................................................6
2.4
Simulating Creep Effects ......................................................................6
2.5
Group Effects ........................................................................................7
2.6
Pile Base Response ...............................................................................8
2.6.1
Hyperbolic Base Response .......................................................8
3
EXPLICIT TIME INTEGRATION ..................................................................9
4
EXTENSIONS TO PROGRAM.......................................................................10
5
4.1
Residual Stresses ...................................................................................10
4.2
External Soil Movement .......................................................................10
4.3
Thermal Strains .....................................................................................11
4.4
Measured Test Data ..............................................................................11
EXAMPLE ANALYSES..................................................................................11
5.1
Bored Pile in Interbedded Clays and Sands ..........................................11
5.2
Driven Precast Concrete Pile ................................................................11
5.3
Rock-socketed Pile................................................................................12
5.4
Offshore Drilled and Grouted Pile ........................................................12
5.5
Monotonic Loading...............................................................................13
5.6
Cyclic Loading ......................................................................................13
PART B: PROGRAM DOCUMENTATION
6
STRUCTURE OF PROGRAM ........................................................................15
6.1
7
Overview of Program Operation ...........................................................15
PROGRAM INPUT ..........................................................................................16
7.1
Data Input..............................................................................................16
7.2
Data Items .............................................................................................16
7.2.1
Problem Title ............................................................................16
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7.3
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M.F. Randolph
7.2.2
Pile Segement Properties ..........................................................16
7.2.3
Pile Details and Base Response ................................................16
7.2.4
Soil Details................................................................................17
7.2.5
Miscellaneous Parameters.........................................................17
7.2.6
Soil Profile ................................................................................17
7.2.7
Loading Details .........................................................................18
7.2.8
Residual Loads ..........................................................................18
7.2.9
Downdrag..................................................................................19
7.2.10
Thermal Strains .......................................................................19
Measured Load Test Data .....................................................................20
PROGRAM OUTPUT ......................................................................................20
8.1
Printed Output .......................................................................................20
8.2
Graphical Output ...................................................................................20
9
REFERENCES .................................................................................................21
10
FIGURE TITLES..............................................................................................23
FIGURES .....................................................................................................................24
February 2003
RATZ Manual Version 4-2
M.F. Randolph
PART A: GENERAL DESCRIPTION
1
INTRODUCTION
The overall load deformation response of an axially loaded pile depends on the axial
compressibility of the pile as well as on the shape of the load transfer curves for the soil
around the pile. These curves may be thought of as the integrated (in a radial direction)
effects of shear strains in the soil, leading to relationships between the shear stress applied
at the pile wall and the resulting local displacement of the soil. Unless the pile is very
compressible, changes in vertical stress induced in the soil will be small compared with
the induced shear stresses. This justifies the load transfer approach, where the soil
continuum is idealised as a number of separate horizontal layers, each with its own load
transfer curve, (see Figure 1).
The compression or extension of the pile may be calculated from consideration of the
variation of axial load, P, down the length of the pile. For a pile of radius, ro, and
cross-sectional stiffness, (EA)p, the variation of axial load is given by
dP
 2ro  o
dz
(1)
where o is the shear stress at the pile wall. The axial strain in the pile is
dw
P
  z  
EA p
dz
(2)
Combining these two equations gives the governing equation for axial compression of
d2w
dz
2

2ro

EA p o
(3)
Traditionally, this equation has been solved, in conjunction with appropriate load transfer
curves relating o and w, by dividing the pile into elements, and adopting a finite
difference approximation for the second derivative of w. For non-linear load transfer
curves, this approach requires iteration at each increment of axial load. For an adequate
discretisation of the pile, the computing effort required becomes considerable,
particularly if the effects of cyclic loading are to be investigated.
An alternative approach, which avoids the assembly and solution of any system of
equations, is the so-called explicit time approach described, for example, by Cundall and
Strack (1979). The technique involves the introduction of time as a variable (artificially
in the case of straightforward static loading) and following the load applied to the pile
head as it propagates down the pile, causing local accelerations, velocities and thus
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displacements of the pile elements. By using a sufficiently large number of load
increments, and time steps which are small in comparison with the time taken for a shock
wave to travel across each pile element, any loading path for the pile may be followed in
a stable fashion. The approach is described in more detail later.
2
LOAD TRANSFER CURVES
In choosing appropriate load transfer curves for the soil, it is convenient to divide the
response of the soil into two distinct phases (for example, Kraft et al (1981)). Initially,
the soil will deform as a continuum, with shear strains induced according to the applied
shear stress at the pile wall. For a given shear stress, these shear strains will give rise to
displacements of the soil around the pile that are proportional to the pile radius - even
for non-linear stress-strain response of the soil. For this stage of loading, it is therefore
helpful to consider relationships between o and w/ro (that is the normalised pile
displacement).
As the shear stress level reaches the maximum value that the interface between pile and
soil can withstand, a rupture surface will develop close to the pile. Thereafter, the shear
stress level may be expected to vary with the absolute displacement of the pile relative to
the soil, and it becomes more logical to consider relationships between o and the absolute
displacement, w. Figure 2 shows this two stage approach.
2.1
Monotonic Loading
In the program, a general form of load transfer curve has been adopted, although specific
curves could be incorporated with minor modifications. The form of curve has no precise
theoretical basis, but allows sufficient flexibility to model most aspects of soil response,
for example non-linear, work hardening behaviour prior to peak, followed by
strain-softening after peak. There are three main stages to the curve (see Figure 2):
(1)
A linear stage where = kw/ro, which extends from zero shear stress up to a
fraction  (xi) of the peak shear stress, p.
(2)
A parabolic stage, with initial gradient, k, and final gradient of zero when o = p.
(3)
A strain-softening stage, where the current value of shaft friction is related to the
absolute pile displacement by



 o   p  1.1  p   r 1  exp  2.4(w / w res ) 

(4)
where r is the residual value of shaft friction, reached after an additional
displacement of wres, with w being the post-peak displacement. The parameter
 (eta) controls the shape of the strain softening curve.
The value of the initial gradient, k, may be related to the shear modulus, G, for the soil
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(Randolph and Wroth, 1978; Baguelin and Frank, 1979). The ratio k/G will generally lie
in the range 0.2 - 0.3, with the smaller value corresponding to more slender piles.
Following Randolph and Wroth (1978), the relationship between k and G may be
expressed as
k
G

(5)
The parameter  (zeta) is given by
  n2.51    / ro   n rm / ro 
(6)
where  is the ratio of the average shear modulus over the depth of penetration of the
pile, to that at the bottom of the pile shaft ( = G /G in the notation of Randolph and
Wroth). The other parameters in equation (6) are Poisson’s ratio for the soil, , and the
pile length, l. For typical pile geometries, the value of  is about 4, giving k/G = 0.25.
Figure 3(a) shows the pre-peak load transfer curve for the case of  = 4, G/s = 400 and
 = 0 (parabolic from origin). The peak shaft friction is mobilised at a displacement of 1
% of the pile diameter, which agrees with recommendations for load transfer curves in
the API RP2A guidelines.
The parameter, , which defines where the load transfer curve becomes non-linear, may
be taken anywhere in the range 0 to 1. A value of zero corresponds to a parabolic load
transfer curve right from the origin, while a value of unity corresponds to a linear load
transfer curve right up to failure. For the strain softening part of the load transfer curve,
typical values for the parameter  are about unity, with lower values giving more abrupt,
and higher values more gradual, strain softening. Figure 3(b) shows examples of load
transfer curves for values of  = 0.8, 1 and 1.2, for the case of r/p = 0.6 and wres taken
as 4 % of the pile diameter.
2.1.1 Hyperbolic Pre-peak Load Transfer Curve
An alternative form of pre-peak load transfer curve has been implemented for monotonic
loading only, based on a hyperbolic stress-strain model for the soil. The basic hyperbolic
model gives a secant shear modulus which decreases linearly with stress ratio, as
G sec  G i 1  R f ( /  f ) 
(7)
where f is the failure shear stress, Gi is the initial tangent shear modulus and Rf is a
parameter which controls the degree of non-linearity, and is generally taken in the range
0.8 - 1 for most soil types. This relationship may be integrated in order to derive an
equivalent load transfer curve for a pile (Randolph, 1977; Kraft et al, 1981). The final
form of load transfer curve may be expressed as
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w  *
M.F. Randolph
 o ro
Gi
(8)
where the parameter * now varies with the shear stress level according to

r / r  
 *  n  m o
where   R f o

p
 1  
(9)
Figure 3(a) shows a comparison of the pre-peak load transfer curve obtained using the
hyperbolic stress-strain response, compared with the parabolic curve. It may be seen that
a value of Rf = 0.95 gives a response which matches the parabolic curve very closely.
Lower values of Rf give curves which are more linear, and may be matched using the
linear-parabolic model, with increasing  values.
2.2
Cyclic Loading
In order to simulate degradation of load transfer under the action of cyclic loading,
particular attention has been paid to the manner in which unloading and reloading is
modelled. Consider a monotonic loading path ABCD in Figure 2, where the current
failure stress at point C is f. On unloading and reverse shearing (path CE), the failure
shear stress is assumed to be the same magnitude, that is -f. Initially, unloading will be
elastic. However, at some point yield will occur. If there has been no previous history
of unloading, then yield is taken to occur at the same point as for loading, that is at a shear
stress of -p. The yield point for reloading (and also for unloading where previous
unloading has occurred) is calculated from the detailed path followed.
2.2.1 Standard Yield Criterion
In the standard yield criterion, the yield point is assumed to move down the unloading
path at a rate of 0.5(1 - ) times the rate of the current stress-displacement point. Thus if
the pile reverses direction such that the mobilised shear stress decreases by , then the
yield stress reduces by 0.5(1 - ). In practice, this is achieved by defining a yield point
in relation to the minimum shear stress, min, and the peak shaft friction, p, as

 y   min  0.51     p   min

(10)
The form of this expression automatically ensures that the yield stress is always greater
than p (the initial yield point). A check is made to ensure that the yield stress never
increases during unloading.
Once yield occurs, the plastic (non-recoverable) component of displacement is treated as
equivalent to post-peak monotonic displacement. This leads to gradual degradation of
the shaft friction from peak to residual, even though there may be no net accumulation of
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displacement. The particular algorithm for calculating the yield point implies that, under
one-way cyclic loading between zero and a maximum value of max, degradation to
residual shaft friction will ultimately occur if max exceeds the elastic limit given by
 max  0.51    p
(11)
Under symmetric two way loading between limits of ±max, the corresponding elastic
limit is
1   
 max  
p
3   
(12)
For cyclic loading that exceeds the limits given by equations (7) and (8), the rate of
degradation is a function of the shape of the strain softening part of the monotonic loading
curve, together with the extent by which the cyclic shear stress exceeds the above limits.
The expressions (11) and (12) allow the safe cyclic range to be varied by means of the
parameter . For  = 1, the cyclic range extends right up to the peak shaft friction, while
for  = 0, the safe cyclic amplitude varies linearly with the mean shear stress, from ± p/3
for symmetric two-way cyclic loading (mean shear stress of zero), through ± p/4 for
one-way cyclic loading (mean shear stress equals cyclic amplitude) down towards zero
as the mean shear stress approaches the peak shaft friction. This form of variation of the
safe cyclic range is similar to that proposed for metals by Goodman, and is shown in
Figure 4. In the more conventional cyclic stability diagram shown in Figure 5, the yield
criterion becomes a straight line (see examples for  = 0 and  = 0.333).
2.2.2 Alternative Yield Criterion A
The safe cyclic zone corresponding to  = 0 (between the upper dashed and lower solid
lines in Figure 4, or beneath the lower straight line in Figure 5) was considered to be
potentially optimistic for the calcareous sediments of the North West Shelf, and possibly
for other compressible or loose soils which are susceptible to volume decrease under
cyclic loading. As such, the algorithm in RATZ has been modified to allow the safe
cyclic zone to be reduced (in the extreme case, to nothing). In place of equation (10), the
yield stress is taken as

 y   min    p   min

(13)
with an overriding criterion that the yield stress should not be less than p. Clearly, if 
is taken as zero, yield occurs immediately, and the safe cyclic range disappears.
The cyclic loading algorithm in RATZ may be used to establish theoretical fatigue curves,
showing the number of cycles to failure for cyclic loading of varying amplitude and
different mean shear stress. Figure 6 shows such fatigue curves for the original RATZ
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M.F. Randolph
algorithm, for two-way (zero mean stress) cyclic loading and one-way (zero minimum
stress) cyclic loading. Similar curves are shown in Figure 7 for the modified algorithm,
taking  = 0.1. The shape of the fatigue curves depend on the various load transfer
parameters for the soil, and also on the radius of the pile. These parameters are
summarised in Table 1.
The fatigue curves shown in Figures 6 and 7 are compared with data from laboratory tests
on intact samples of calcarenite. The data are reasonably consistent with the theoretical
fatigue curves.
2.2.3 Alternative Yield Criterion B
Both the above yield give linear threshold contours in the cyclic stability diagram shown
in Figure 5. In practice, however, threshold contours from actual pile tests have tended
to be convex, as shown by the two parabolic curves (Poulos, 1988). Although such an
approach is less conservative than the linear thresholds discussed above, an alternative
yield criterion has been implemented in order to match measured data. This is referred
to as the ‘parabolic’ yield criterion in the main menu of the program.
2.3
Cyclic Residual Shaft Friction
In granular soils, particularly lightly cemented sands, it is often found that the shear
transfer during two-way displacement limited cyclic loading is very low. This may be
modelled within the program by nominating a ‘cyclic residual shaft friction’, which limits
the shear transfer within a limited zone. The manner in which this operates is shown in
Figure 8. Initially the two markers that control the limit of the cyclic residual zone are
positioned together at the origin. Before failure, the two markers move together, staying
at the point where an elastic unloading response would intersect the displacement axis.
After failure, the upper limit marker (the circle) continues to track the current position of
the shear stress:displacement point, while the lower limit stays fixed. On unloading, the
path BCDE is followed, with the shear resistance staying at the residual value within the
‘failure’ zone between the two markers. If the element is reloaded, from point F, the path
FGHIJ is followed. The shear transfer remains at the cyclic residual until the upper
marker is reached, after which the shear transfer reverts back to the main curve. Note,
however, that the shaft friction at point I will be lower than at B, due to the plastic
displacement (and accompanying degradation) that occurs during the cycle HCFGH.
2.4
Simulating Creep Effects
In most geotechnical constructions, the effects of creep are small, amounting to perhaps
10 to 20 % additional movement compared with consolidation effects. The case of an
axially loaded pile can be rather different, especially for long slender piles, since the shear
stress level adjacent to the pile may be close to failure over the upper part of the pile. The
creep can then become more significant, and can lead to creep rupture, whereby small
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movements accumulate and lead to a rupture surface at the pile wall, with peak shaft
friction values lower than in a short term load test. Figure 9(a) shows a possible change
in the form of the load transfer curve due to the effects of creep. At high shear stress
levels, the displacements are increased in a localised band of soil adjacent to the pile, and
the original peak value of shaft friction is never achieved.
The explicit time approach used in the analysis is essentially a strain controlled approach.
At each load increment, the current displacement of the pile determines the shear stress
mobilised at the pile wall. Implementation of creep effects is thus most easily
accomplished by a stress relaxation process, whereby the load transfer curve is shifted by
a small amount over each time increment, as shown in Figure 9(b). In the program, the
degree of movement is based on a creep rate law of the type proposed by Singh and
Mitchell (1968)
m
 
dw
t 
 A o  exp  o
dt
 t 
 f



(14)
There are three constants in this equation, A, and m. In addition, the time to is the time
associated with a standard rate of testing (at which the values of A and are determined).
In order to express the creep in a non-dimensional form, equation (14) has been re-cast
in the form
m
 
 t 
w  w *   exp  o
 t 
 f



(15)
where w* is the displacement to mobilise peak shaft friction (point B in Figure 2). Typical
values for the constants  and m are 6 - 8 and 0.75 - 1.2 respectively (Singh and Mitchell
(1968)). A parametric study indicates that values of  should generally lie in the range 0
(no creep) to 0.01. The effects of different rates of loading are best investigated by
varying the value of , thus avoiding the need for analyses with an excessive number of
loading increments. In the program, creep is assumed to occur only between the yield
point (o ≥ p) and failure.
2.5
Group Effects
Pile group effects may be assessed by analysing a single pile with an appropriately
modified load transfer curve. In the program, this is accomplished following the
procedure suggested by Focht and Koch (1973) for laterally loaded piles. The user
specifies a ‘group settlement ratio’, Rs, by which the elastic part of the load transfer curve
is factored, to allow for interaction between piles in a group. The settlement ratio may
be estimated following Randolph (1979), whereby the parameter is replaced by
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r
*  n  m
 ro
M.F. Randolph

r
   n  m
 si




(16)
The summation is taken over all remaining piles in the group, spaced at si from the pile
in question, with the parameter rm being the maximum radius of influence of the pile,
defined in equation (6). The settlement ratio, Rs, is then taken as the ratio */.
It should be emphasised that only the elastic component of the load transfer curve is
factored in the manner described above. The plastic component of the load transfer curve
for single pile response is then added on to the factored elastic component in order to
obtain the ‘group’ load transfer curve. Figure 10 illustrates this process.
2.6
Pile Base Response
The load-displacement response of the pile base has been taken as parabolic in shape,
with the vertex of the parabola at the ultimate base capacity. Unloading and reloading
has been assumed elastic, subject to the condition of no tension being developed at the
pile base. The response under monotonic loading is as shown by curve OACD in Figure
11. The base capacity is assumed to remain constant, independent of the magnitude of
base displacement. For displacements outside the range of the initial parabola, the
parabola is ‘dragged’ along to the right (increasing displacement) or left (decreasing
displacement) without changing shape. The final position of the parabolic curve is shown
dashed in Figure 11.
Small unload-reload loops are elastic, with a gradient parallel to the initial gradient of the
parabola (see AB in Figure 11). Larger loops are governed by the zero tension condition
and the current position of the parabola (see DEFGD in Figure 11).
2.6.1 Hyperbolic Base Response
A hyperbolic base response has been implemented, assuming a secant stiffness that
decreases linearly with increasing load level. To be consistent with the parabolic base
model, where a limiting end-bearing pressure of qbf is mobilised at a base displacement
of wbf, the initial gradient is taken as
k bi  2
q bf
w bf
(17)
The secant stiffness at any pressure, qb, is then given by

k b sec  k bi 1 

8
qb
q bf

q
  2 bf
w bf


q
1  b
 q bf



(18)
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EXPLICIT TIME INTEGRATION
As discussed earlier, pile-soil interaction is calculated using an explicit time integration
procedure to solve the non-linear problem. The effect of pile compression or extension
is incorporated by first dividing the pile into a number of elements, each of mass m. Each
element is connected by a spring of stiffness, K, which represents the pile stiffness. For
a pile of embedded length, l, divided into n segments, the value of the spring stiffness
may be calculated as
K
EA p
 / n 
(19)
Each element of the pile is connected to the soil by a non-linear spring, representing the
load transfer curve at that particular depth (see Figure 1). At any stage during the
analysis, a net force will act on any given pile element, which will cause the element to
accelerate, leading to changes in its velocity, v, and displacement, w. For an element of
mass, m, with out of balance force, F, the changes in velocity and displacement over a
time period, t, may be calculated as
t
m
(20)
w t  t  w t  v t  t / 2 t
(21)
v t  t / 2  v t t / 2  Ft
and
The form of these equations assumes that the force, Ft, is the average value of the force
over the time period t-t/2 to t+t/2. Similarly, vt+t/2 is the average velocity over the
period t to t+t.
Each increment of load may effectively be seen as involving two scans of all the pile
elements. In the first scan, the forces acting on each pile element are evaluated, and the
net force calculated. In the second scan, the element velocities and displacements are
updated according to equations (20) and (21). There are three components to the force
acting on each pile element. The internal force on the ith element in the pile is given by
Fa i  K w i1  w i   K w i1  w i 
(22)
while the force from the static component of the soil displacement is
Fb i
 2ro  / n  o i
(23)
In addition, stability of the solution is maintained by incorporating a damping force given
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by
Fc i
 2ro  / n k ' v i
(24)
The quantity k' is taken from solutions for the dynamic response of soil around a pile,
and is related to the shear modulus, G, of the soil and the soil density, , by (Novak et al,
1978); Simons and Randolph, 1985)
k '  D f G
(25)
where the quantity Df is a factor specified by the user, ranging between zero and unity.
The net force on the pile element is the algebraic sum of the three forces given above.
For a pile loaded at a constant strain rate, the inclusion of a damping force such as that in
equation (25) will give rise to a spurious excess force at the head of the pile, generally of
a relatively small magnitude. This is allowed for within the program by subtracting the
additional force prior to outputting the load displacement relationship for the pile (see
later for discussion of program output).
4
EXTENSIONS TO PROGRAM
In addition to straightforward monotonic and cyclic loading of the pile, a number of
additional situations may be analysed. These are recent extensions, and include (a)
external soil movement, causing downdrag, (b) thermal strains in the pile, (c) residual
loads developed during pile installation, and (d) matching of a measured loaddisplacement response. These extensions are discussed briefly below.
4.1
Residual Stresses
A more common cause of residual stresses within the pile are those set up during pile
installation, generally in the form of a compressive load locked in at the base of the pile,
and negative values of shear stress between pile and soil, particularly towards the lower
end of the pile. The compressive base load may arise either from pile driving (especially
for full-displacement piles) or from a base grouting operation on a cast-in-situ pile. A set
of residual loads along the pile shaft is input prior to starting the analysis.
4.2
External Soil Movement
At any stage during the loading history of the pile, the effect of external soil movement
(either upward, causing heave, or downward, causing downdrag and negative friction)
may be analysed. Essentially, the zero points (and other key markers) on the load transfer
curves are shifted to reflect the soil movement at that depth. The shear stress at the pile
shaft is then a function of the relative movement between soil and pile, rather than the
absolute pile movement.
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Thermal Strains
In some circumstances, curing of the concrete or grout in a large diameter pile can lead
to very high temperatures within the pile. As the concrete sets, the pile starts to cool
down and significant shrinkage may occur. This will give rise to residual stresses
between the pile and the soil. The effect of such thermal strains may be modelled by
specifying, at the initial loading stage, a distribution of thermal strains down the length
of the pile. The effect of these strains is then computed prior to starting the main analysis.
It should be noted that, for long piles, thermal strains may lead to actual slip between pile
and soil.
4.4
Measured Test Data
In the interpretation of pile load tests, it is useful to be able to compare the predicted and
measured responses at the pile head. Test data giving the measure load-settlement
response may be input and compared with the computed response, allowing iteration of
the soil and pile parameters until a close match is achieved. This process is comparable
to that of matching stress-wave data from dynamic tests.
5
EXAMPLE ANALYSES
Several example problems are presented here in order to illustrate use of the program.
The first example concerns a bored pile, for which a measured load-settlement response
is also available. Other examples include a relatively stubby rock-socketed pile with a
strain-softening shaft response, a driven precast pile with significant residual stresses, a
downdrag example and a slender offshore pile subjected to monotonic and cyclic loading.
5.1
Bored Pile in Interbedded Clays and Sands
Figure 12 shows a fit to the measured load test on a bored pile, cast under bentonite. the
pile was 800 mm in diameter, by 20 m long. The response at small displacements is
shown in the upper diagram for clarity. The ultimate shaft capacity has been modelled
as 4.3 MN (using a hyperbolic soil model), while the base capacity has been taken as 3.5
MN (7 MPa), again with a hyperbolic response. The form of the hyperbolic base model
is such that full mobilisation of the base capacity will require a base movement of 200 300 mm. As such, the total ‘deduced’ capacity of 7.8 MN exceeds the practical capacity
of about 7 MN, mobilised at a settlement of 80 mm. The average tangent shear modulus
for the shaft response is Gi = 200 MPa (G/p = 2350) and for the base is 68 MPa (Gib/qbf
= 9.7).
5.2
Driven Precast Concrete Pile
The second example is a precast concrete pile, 300 mm square, that was driven 34 m
through layered alluvial deposits to found in a dense sandy gravel layer. Figure 13 shows
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a comparison of the computed response with and without allowance for residual stresses.
In this instance, the residual stresses have been obtained from the back-analysis of a
dynamic test on the pile. The pile response under working load would be significantly
stiffer due to the presence of residual stresses in the pile.
5.3
Rock-socketed Pile
The third example is a hypothetical example of a 10 m long by 1.4 m diameter
rock-socketed pile, embedded in siltstone with an unconfined compression strength of
about 5 MPa. Peak and residual shaft friction of 1000 kPa and 600 kPa have been
adopted, with a displacement of 30 mm required to degrade from peak to residual (wres
= 30 mm,  = 1). The shear modulus of the siltstone was taken as G = 700 MPa, with
the yield parameter  as 0.5. The base capacity was taken as 25 MPa, mobilised at a
displacement of 140 mm (10 % of the pile diameter).
The ‘ideal’ capacity of the pile may be calculated from the peak shaft friction and base
capacity as 44.0 MN (shaft) plus 38.5 MN (base), giving a total of 82.5 MN. In practice,
strain-softening of the shaft response will occur before the base resistance has built up,
and the actual capacity should be based on the residual shaft friction of 600 kPa. This
leads to a lower capacity of 26.4 MN plus 38.5 MN, which equals 64.9 MN.
The actual response of the rock-socket is shown in Figure 14. The load-displacement and
shear stress-displacement responses are plotted at depths of 0, 4.8 and 9.8 m. The bottom
curve effectively gives the response at the pile base. It may be seen that a peak load of
44.2 MN is reached at a displacement of about 8 mm, after which the load decreases, due
to strain-softening of the shaft response. Subsequently, the load increases again towards
the final capacity of 64.9 MN.
The form of the response emphasises the need to choose working loads carefully.
Clearly, there is potential for large differential settlements between piles if the working
load is taken in the region of 40 MN.
5.4
Offshore Drilled and Grouted Pile
The final example is an offshore drilled and grouted pile embedded in calcarenite which
shows significant strain-softening and degradation under cyclic loading. The pile
dimensions are:
Embedded length = 70 m;
Outer radius = 1.0 m;
Wall thickness = 60 mm;
Young’s modulus of steel = 210 GPa.
The soil properties for the shaft resistance are tabulated below. The base resistance of
the pile has been ignored in this example.
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Shear modulus of soil = 500 MPa;
Parameter  = 4;
Yield point is at  = 0;
Peak shaft friction = 400 kPa;
Residual shaft friction = 40 kPa;
Displacement to residual = 1000 mm;
Strain-softening parameter = 0.7
Cyclic residual shaft friction = 4 kPa.
5.5
Monotonic Loading
Figure 15 shows the pile response to monotonic loading, with different amounts of creep.
Although the theoretical capacity (assuming a rigid pile) is 176 MN, the actual capacity
under static loading (with no creep) is only 151 MN, due to effects of progressive failure.
As the value of the creep parameter, , is increased (which may be viewed as equivalent
to decreasing the loading rate for a constant degree of creep), the peak load decreases.
Thus peak loads of 150 MN, 132 MN and 87 MN are achieved for  values of 0.001, 0.01
and 0.1 respectively. Within the framework of assumptions in the program, the
theoretical lower limit to the pile capacity corresponds to the residual shaft friction of 40
kPa, which would give a capacity of 18 MN.
The phenomenon of progressive failure of the pile may be seen clearly in Figure 16,
which shows the axial pile load and shear stress mobilisation at 3 depths down the pile
(for zero creep). At peak load, the shaft friction in the upper pile elements has degraded
to about 70 % of the maximum value.
5.6
Cyclic Loading
The pile response under a typical design storm sequence has been analysed with both the
original and the modified yield algorithm. The yield parameter  has been reset to 0.1
for the modified yield algorithm, since a value of zero would imply continuous
degradation under cyclic loading of any amplitude, which is considered unrealistic. The
results of the analyses are shown in Figures 17 and 18. The storm starts with large
numbers of load cycles of low amplitude, builds up to the maximum design loading with
a single cycle of 62.5 MN ± 23.5 MN, and then decays symmetrically. In the example,
about 1000 load cycles have been applied in 25 stages, although in a real design it may
be necessary to analyse a somewhat greater number. As an indication of the speed of
analysis, the example took about 2 minutes to compute on a 1 GHz (Pentium 3)
microcomputer.
Some interesting effects may be observed from Figures 17 and 18:
•
during the early cyclic loading, the mean shear stress in the upper part of the pile
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reduces, tending towards symmetric two-way loading, even though the overall
cyclic loading is one-way;
•
the element shown (at a depth of 9 m) reaches failure on the first cycle and then
again under the peak design loading; the effect of the very low cyclic residual shear
stress may be noted, from the exaggerated ‘S’ shape to the response;
•
little further displacement of the pile occurs during the second half of the storm
sequence, although the upper soil elements continue to degrade;
•
the final ‘post-cyclic’ capacity of the pile is reduced to 136 MN using the original
yield algorithm, and 132 MN using the modified algorithm, compared with the
original monotonic capacity of 151 MN;
•
comparing Figures 17 and 18, it is clear that significantly greater displacements
accumulate with the modified yield algorithm, although the decrease in post-cyclic
capacity is only small.
The effect of cyclic loading on pile capacity will vary with the relative stiffness of the
pile. Piles that are less compressible will show less concentration of load transfer at the
head of the pile. As such, cyclic loading at typical design levels will tend to cause less
degradation. This has been discussed by Randolph (1983). In the example shown here,
for a very compressible pile, the upper five elements all degrade to residual shaft friction.
However, the final capacity shows only moderate reduction, since the effects of
progressive failure during the final monotonic loading are reduced.
The program outputs the status of each of the soil elements at the end of each loading
stage. This information may be used to follow the pattern of degradation through a cyclic
loading sequence. Figure 19 shows results from the analysis using the modified yield
algorithm. The upper part of the Figure shows the distribution of current shaft friction
down the pile at two stages during the storm sequence, while the lower part shows the
variation of pile capacity with each stage of loading. The ‘current’ capacity after each
stage has been estimated by linear interpolation, from the monotonic capacities before
and after cyclic loading, and the integrated shaft friction (the ideal capacity) at the end of
each loading stage.
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PART B - PROGRAM DOCUMENTATION
6
6.1
STRUCTURE OF PROGRAM
Overview of Program Operation
The program comprises an Excel workbook, which provides the framework for data input
and output of results. The workbook calls a dynamic link library (DLL) compiled from
Fortran code, which performs the main numerical computations. In order for Excel to
find the DLL file, the directory must be set to that containing it. The simplest way of
achieving this is to keep a copy of this file in the same directory as the RATZ workbook,
and to open the workbook file by going through the full ‘File: Open’ procedure (not just
double-clicking on the file name). This resets the directory in which Excel will look for
the DLL to the current one.
The program performs an analysis of the axial response of a pile under monotonic or
cyclic loading. The pile, which may be made up in a number of segments of varying
properties, is treated as an elastic bar, able to deform one-dimensionally along its length.
The pile is partly (or wholly) embedded in soil which may consist of a number of different
layers, with homogeneous or heterogeneous properties in each layer. The loading may
be specified in a number of different stages, with each stage being defined in terms of
monotonic or cyclic variation of load or displacement.
The main operations involved in running the program, with optional items shown in
square brackets, are as follows:
(1)
Input or edit data (see ‘Data’ worksheet)
(2)
Run analysis (press ‘Run’ button, or type Ctrl-r).
(3)
View graphical output
The main output from the program is graphical. However, tabulated output is also
provided, reflecting the input data and summarizing results at intervals defined by the
user. In addition, output is provided giving the status of every soil element at the end of
each loading stage (see‘Degradation’ worksheet). This latter feature is primarily used for
detailed analysis of degradation under cyclic loading.
In addition to the main input worksheet (‘Data’), two additional worksheets for data input
are provided. The first of these (‘Xtradata’) gives information on (a) residual loads in the
pile; (b) downdrag soil movement; and (c) thermal strains. The second additional data
worksheet (‘Testdata’) allows tabulation of the results of a load test, so that measured and
computed response can be compared.
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PROGRAM INPUT
Data Input
Data input is on the worksheets ‘Data’, ‘Xtradata’ and ‘Testdata’ must be confined to the
areas with yellow background. Standard Excel techniques, including formulae may be
used. It is important within the array areas for pile or soil properties, or for the loading
stages, not to leave any gaps between successive entries. The system of units assumed
by the program comprises forces in kN and lengths in m (and hence modulus or shear
stress in kPa), although in principle other consistent sets of units could be used.
7.2
Data Items
7.2.1 Problem Title
The title may comprise any sequence of alphanumeric characters or punctuation marks.
7.2.2 Pile Segement Properties
The pile is considered to be made up of a number of segments (maximum: 5). The
segment properties are the segment length, the outer and inner pile diameters and the
Young’s modulus of the pile material. For non-circular, or H-section piles, the
‘diameters’ should be chosen so as to match both the gross cross-sectional area of the pile
(for example, the encompassing rectangle for an H-section pile) and also the actual
cross-sectional area of the pile section.
7.2.3 Pile Details and Base Response
The pile penetration into the soil must be specified, and also the number of elements that
the embedded part of the pile is to be divided into. Typically, pile elements of one or two
diameters in length should be sufficient for an accurate analysis.
The pile base may be assigned a diameter, db, that is different from the shaft diameter.
The base response of the soil is taken as parabolic. The user specifies the ultimate base
pressure, qbf and the pile displacement, wbf, at which this is mobilised. The properties
of a parabola are such that the initial gradient of the pile response will be 2qbf/wbf, which
implies a shear modulus for the soil in the region of the pile base of
G b  0.25(1  )d b
q bf
w bf
(27)
The user should check whether this represents a reasonable modulus, consistent with
values expected at the base of the pile.
The base response may be switched to the hyperbolic model by specifying the pile
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displacement for full mobilisation of end-bearing pressure in ‘mm’ rather than ‘m’.
(There are few occasions where this could lead to any confusion!) Essentially, the initial
tangent shear modulus may be taken from Equation 27, and the secant shear modulus is
then assumed to decrease linearly to zero as qb approaches qbf.
7.2.4 Soil Details
The global load transfer parameters and creep parameters have been discussed already,
in Section 2 of this manual.
7.2.5 Miscellaneous Parameters
The damping factor should be set in the range 0.1 - 0.3 in order to ensure numerical
stability. The strain-softening switch is set to 0 for degradation from peak to residual shaft
friction that is linear with displacement, or 1 for the exponential function given in
Equation 4.
7.2.6 Soil Profile
The soil properties are specified in terms of a number of different layers, with the load
transfer parameters for each soil element down the pile obtained by interpolation within
each layer. A fully homogeneous deposit is indicated by specifying only the first line of
data in this input area.
Up to seven soil properties must be specified for each layer. The key parameters that
determine the shape of the load transfer curve are: the shear modulus, G, the yield
threshold,  (xi), the peak shaft friction, p, the residual shaft friction ratio, r/p, and the
displacement to residual, wres. In addition, for non-linear strain-softening, a shape
factor, (eta), is specified that determines the exponential shape. Finally, for cyclic
loading only, the cyclic residual shaft friction ratio,cr/p must be specified (see section
2).
The absolute value of the yield ratio, , may be taken anywhere in the range 0 to 1
(inclusive), depending on what degree of non-linearity is required before peak shaft
friction is reached. This parameter also controls the cyclic response of the soil, through
the yield algorithms described in section 2.2.
In order to switch to the hyperbolic shaft model, the parameter  should be set to a number
between 1 and 2 (any number higher than 2 will be reduced to 2). The program then
assumes a value for Rf of
Rf =  - 1
(26)
Note that the hyperbolic model may only be used for monotonic loading.
Values of peak and residual shaft friction should be assessed from normal design
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procedures, while the two parameters that control the strain softening part of the curve,
wres and  should be adjusted to give the required shape (see Equation 4). Generally, a
value of wres in the range 30 - 100 mm appears to be appropriate for monotonic loading,
together with a value for  of around unity. For cyclic loading, where the residual shaft
friction can be very low, it may be appropriate to adopt a much larger value of wres (see
example in section 5.4).
7.2.7 Loading Details
Multiple stage loading may be specified (up to 100 stages), with each stage either
displacement or load controlled, and either monotonic or cyclic. In practice,
displacement control is more useful for monotonic loading, while load control is
generally more appropriate for cyclic loading. The tabulated output from each load stage
may be either slim-line (pile head and ground surface response only) or full (tabulated
load, displacement and shear stress down the embedded section of the pile). When
choosing the number of output stages, it should be remembered that graphical output
occurs at ten times the frequency of tabular output. Thus, for monotonic loading, 10 - 20
output stages should be sufficient to give good definition to the plotted load-displacement
curve. For cyclic loading it will generally be sufficient to plot 10 points per cycle (so
only one tabular output per cycle).
The size of load or displacement increment should generally be chosen so as to give
displacement increments that are no greater than 0.01 % of the pile diameter. For very
stiff piles, or where rapid strain softening occurs, increments as low as 0.001 % of the
diameter may be required in order to retain stability.
Under monotonic loading, the overall number of increments is determined by the required
total pile displacement (or final load). Under cyclic loading, the maximum load (or
displacement) and the minimum load (or displacement) are specified, with the number of
increments per cycle being chosen according to the incremental criterion given above. In
general, 200 - 500 load increments will be needed per cycle.
7.2.8 Residual Loads
A profile of residual loads may be specified in the ‘Xtradata’ worksheet. The loads are
specified as a series of ordered pairs (one pair per line) giving (a) the depth (in m), and
(b) the magnitude of the residual force (in kN). The program interpolates between each
given depth in order to establish the residual force at any intermediate depth. Thus, a
simple triangular distribution of residual force in a pile of embedded length 20 m, could
be specified as:
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where 1500 (kN) is the magnitude of the residual force at the pile base.
The program checks that there is sufficient shaft capacity to equilibrate the requested
residual force, and if not adjusts the residual force accordingly.
7.2.9 Downdrag
In addition to specifying ‘monotonic’ or ‘cyclic’ control, a third option is provided which
is ‘downdrag’. If this option is selected for a particular loading stage (but only once in
any analysis), the profile of soil movements specified on the ‘Xtradata’ worksheet is
applied over the specified number of increments.
The form of the soil movement profile is a series of ordered pairs (one pair per line)
giving (a) the depth (in m), and (b) the magnitude of the soil movement (in m, downwards
positive). The program interpolates between each given depth in order to establish the
soil movement at any intermediate depth. Thus, a simple triangular distribution of soil
movement from 0.1 m at the soil surface to zero at a depth of 25 m, could be specified
by:
0
0.1
25
0
7.2.10 Thermal Strains
Analysis of the effects of thermal strain may be achieved by specifying a profile of
thermal strains in the ‘Xtradata’ worksheet. Ordered pairs of (a) depth (in m) and (b)
thermal strain (contractive strain taken as positive) are specified, with the depth measured
from the top of the pile (independent of the penetration of the pile). The program then
interpolates between the given depths in order to evaluate the strain at each pile element,
which are then listed in the output file (where the stated strain corresponds to the average
strain in the section of pile immediately above each node). Note that only the embedded
section of the pile is divided into elements. Hence, for a free-standing length of pile of
length X, embedded length of , the thermal strain for element 1 is between the top of the
pile and a distance X + 0.5/N from the top. The thermal strain for the ith element is
between X + (i-1.5)/N and X + (i-0.5)/N from the top of the pile.
A thermal strain loading stage is activated automatically if zero loads or displacement
(under the monotonic section if loading switch set to monotonic, or cyclic section if
loading switch set to cyclic) are specified. The program then picks up the thermal strains
from the ‘Xtradata’ worksheet and applies them over the specified number of increments.
During a thermal strain loading stage, specifying ‘Load’ control allows free movement
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of the pile head, under constant load, while specifying ‘Displacement’ control fixes the
top of the pile (zero change in displacement) and lead to a build up of pile head load due
to thermal strains.
7.3
Measured Load Test Data
Data from a pile load test may be entered on the ‘Testdata’ worksheet, and plotted to
allow comparison with the computed pile response, facilitating the back-analysis of pile
load tests. The form of the data is a series of ordered pairs giving (a) the displacement
(in m) and (b) the load (in kN).
8
8.1
PROGRAM OUTPUT
Printed Output
Tabulated output is provided on the ‘Output’ worksheet, and is reasonably self
explanatory, with the exception of the two profiles of load given in the full output. The
profile (P) is calculated from the internal pile strains, with the component of force due to
damping in the soil subtracted. The profile (S) is calculated from the cumulative shear
stress acting on the pile. Agreement between the two profiles of load is an indication of
the accuracy and relative stability of the numerical solution. In general, the two profiles
should not differ by more than 5 % near the pile head, and up to 10 % near the pile base
(where the loads will be smaller).
In addition to the main output, the status of each soil element at the end of each loading
stage is provided in the ‘Degradation’ worksheet. This information is extremely useful
in following the pattern of soil degradation during a cyclic loading sequence (see section
5.4, and Figure 19). The tabulated data may be presented graphically by modifying one
of the plots in the ‘Userarea’ worksheet.
8.2
Graphical Output
The main output from the program is graphical, and the user may adjust the form of the
graphs to suit their preferences. All available data are tabulated in the ‘Plotoutput’
worksheet, with pile loads, displacements and shaft friction specified at the pile head and
up to 5 levels down the pile.
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REFERENCES
1.
Baguelin F. and Frank R.A. (1979), Theoretical studies of piles using the finite
element method, Proc. Int. Conf. on Num. Methods in Offshore Piling, ICE,
London, 83-92.
2.
Cundall P.A. and Strack O.D.L. (1979), A discrete numerical model for granular
assemblies, Geotechnique, 29(1):47-66.
3.
Focht J.A and Koch K.J. (1973), Rational analysis of the lateral performance of
offshore pile groups, Proc. 5th Offshore Technology Conf., Houston, 2, Paper OTC
1896, 701-708.
4.
Kraft L.M., Ray R.P. and Kagawa T. (1981), Theoretical t-z curves, J. of Geot.
Engng Div., ASCE, 107(11):1543-1562.
5.
Poulos H.G. (1988), Cyclic stability diagram for axially loaded piles, J. Geot. Eng.
Div., ASCE, 114(8):877-895.
6.
Randolph M.F. (1977), A Theoretical Study of the Performance of Piles, PhD
Thesis, University of Cambridge.
7.
Randolph M.F. and Wroth C.P. (1978), Analysis of deformation of vertically
loaded piles, J. of Geot. Eng. Div., ASCE, 104(GT12):1465-1488.
8.
Randolph M.F. (1979), Discussion in Conf. on Numerical Methods in Offshore
Piling, ICE, London, 197.
9.
Randolph M. F. (1983), Design considerations for offshore piles, Proc. Conf. on
Geotechnical Practice in Offshore Engineering, Austin, 422-439.
10.
Randolph M.F. (1994), Design methods for pile groups and piled rafts, State-ofthe-art Lecture, Proc. 13th Int. Conf. on Soil Mech. and Found. Eng., New Delhi,
5:61-82.
11.
Singh A. and Mitchell J.K. (1968), General stress-strain-time function for soils, J.
Soil Mech. and Found. Eng. Div., ASCE, 94(SM1):21-46.
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TABLE 1
LOAD TRANSFER PARAMETERS ADOPTED FOR FATIGUE CURVES
Original Algorithm Modified Algorithm
Shear modulus, G (MPa)
200 or 1000
200 or 1000
Load transfer parameter, 
4
4
Yield parameter, 
0
0.10
Residual shaft friction (%)
18
10
1000
1500
0.7
0.65
Cyclic residual friction (%)
1
1
Fatigue threshold (%)
33
10
Radius of pile, ro (m)
1.15
1.15
Disp. to residual (mm)
Shape parameter, 
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10 FIGURE TITLES
Figure 1
Idealisation of pile in load transfer analysis
Figure 2
Details of load transfer curve
Figure 3
Example load transfer curves
Figure 4
Goodman diagram showing linear projection to allow for stress bias
Figure 5
Cyclic stability diagram
Figure 6
Fatigue curves for original yield algorithm
(a) Shear modulus of G = 200 MPa
(b) Shear modulus of G = 1000 MPa
Figure 7
Fatigue curves for modified yield algorithm
(a) Shear modulus of G = 200 MPa
(b) Shear modulus of G = 1000 MPa
Figure 8
Modelling low cyclic residual shaft friction
Figure 9
Modelling effects of creep
(a) Effects of soil creep
(b) Displacement of load transfer curve due to creep
Figure 10
Modelling group effects by factoring load transfer curve
Figure 11
Response of pile base
Figure 12
Measured and simulated response of bored pile
(a) Small displacements
(b) Overall response
Figure 13
Effect of residual stresses on pile head and base response
Figure 14
Rock-socket example: 10 m long by 1.4 m diameter
Figure 15
Load-displacement response of offshore pile with varying amounts of
creep
Figure 16
Offshore drilled and grouted pile: 70 m long by 2 m diameter
Figure 17
Offshore drilled and grouted pile: cyclic loading, original algorithm
Figure 18
Offshore drilled and grouted pile: cyclic loading, modified algorithm
Figure 19
Shaft friction and pile capacity during cyclic loading
(a) Profiles of failure shaft friction at different stages
(b) Progress of degradation through storm
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FIGURES
P
P
Lumped mass, m
Spring stiffness, k

w
Load transfer curves
along pile shaft
Load transfer at
pile base
(a) Actual pile
(b) Idealisation of pile
Figure 1 Idealisation of pile in load transfer analysis
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Shear stress, o
wres
B
p
C
r
parabolic
D
f
p
linear
A
Normalised
displacement
w/ro
wo
yield
f
E
Figure 2 Details of load transfer curve
25
Displacement, w
Proportion of peak shaft friction  / p
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1
0.9
0.8
0.7
0.6
0.5
Hyperbolic: Rf = 0.8
0.4
0.3
Hyperbolic: Rf = 0.95
0.2
RATZ Parabola: xi = 0
0.1
API Guidelines
0
0
0.2
0.4
0.6
0.8
Displacement/pile diameter (%)
1
Proportion of peak shaft friction  / p
(a) Pre-peak response
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
RATZ: xi = 0; eta = 1.3
0.2
RATZ: xi = 0; eta = 1
0.1
RATZ: xi = 0; eta = 0.7
0
0
1
2
3
4
5
Displacement/pile diameter (%)
(b) Post-peak response
Figure 3 Example load transfer curves
26
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RATZ Manual Version 4-2
M.F. Randolph
max /p and min/p
1
Pr
n
tio
ec
oj ine
l
1-way
loading
0.33
2-way
loading
max
min
0.5
cyclic
0.25
mean
0
1 min/p
-0.33
Figure 4 Goodman diagram showing linear projection to allow for stress bias
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February 2003
RATZ Manual Version 4-2
Limiting condition
Normal algorithm: xi = 0
Normal algorithm: xi = 0.333
Revised algorithm: xi = 0
Revised algorithm: xi = 0.333
1
0.9
0.8
0.7
 cyclic / p
M.F. Randolph
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
/
 me an p
Figure 5 Cyclic stability diagram
28
0.8
1
February 2003
RATZ Manual Version 4-2
M.F. Randolph
1
1-way Triaxial
1-way CNS
2-way CNS
Resonant Column
Samples not failed
RATZ: 2-way
 cyclic  f
0.8
0.6
0.4
0.2
RATZ: 1-way
0
0
1
2
3
4
5
6
7
8
7
8
Log(No. of cycles)
(a) Shear modulus of G = 200 MPa
1
1-way Triaxial
1-way CNS
2-way CNS
Resonant Column
Samples not failed
RATZ: 2-way
 cyclic /  f
0.8
0.6
0.4
0.2
RATZ: 1-way
0
0
1
2
3
4
5
6
Log(No. of cycles)
(b) Shear modulus of G = 1000 MPa
Figure 6 Fatigue curves for original yield algorithm
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February 2003
RATZ Manual Version 4-2
M.F. Randolph
1
1-way Triaxial
1-way CNS
2-way CNS
Resonant Column
Samples not failed
RATZ: 2-way
 cyclic /  f
0.8
0.6
0.4
0.2
RATZ: 1-way
0
0
1
2
3
4
5
6
7
8
7
8
Log(No. of cycles)
(a) Shear modulus of G = 200 MPa
1
1-way Triaxial
1-way CNS
2-way CNS
Resonant Column
Samples not failed
RATZ: 2-way
 cyclic /  f
0.8
0.6
0.4
0.2
RATZ: 1-way
0
0
1
2
3
4
5
6
Log(No. of cycles)
(b) Shear modulus of G = 1000 MPa
Figure 7 Fatigue curves for modified yield algorithm
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February 2003
RATZ Manual Version 4-2
Figure 8 Modelling low cyclic residual shaft friction
31
M.F. Randolph
February 2003
RATZ Manual Version 4-2
M.F. Randolph
Shear stress, o
without creep
with creep
Displacement, w
(a) Effect of soil creep
Shear stress, o
displaced
curve
1
2
w
Displacement, w
(b) Displacement of load transfer curve due to creep
Figure 9 Modelling effects of creep
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February 2003
RATZ Manual Version 4-2
A
Shear stress
O
B
C
M.F. Randolph
D
Group pile
Single pile
OC = R s OA
CD = AB
Displacement
Figure 10 Modelling group effects by factoring load transfer curve
Base pressure, qb
C
qbf
D
G
A
2
B
E
F
O
wbf
Base displacement, wb
Figure 11 Response of pile base
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February 2003
RATZ Manual Version 4-2
M.F. Randolph
6
Load (MN)
5
4
3
2
Measured
RAT Z
1
0
0
2
4
6
8
10
80
100
Displacement (mm)
(a) Small displacements
8
7
Load (MN)
6
5
4
3
Measured
2
RAT Z
1
0
0
20
40
60
Displacement (mm)
(b) Overall response
Figure 12 Measured and simulated response of bored pile
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February 2003
RATZ Manual Version 4-2
M.F. Randolph
4000
Pile head
Residual stresses
Load (kN)
3000
No residual stresses
Residual stresses
2000
1000
Pile base
No residual stresses
0
0
20
40
60
80
100
120
Displacement (mm)
Figure 13 Effect of residual stresses on pile head and base response
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February 2003
RATZ Manual Version 4-2
70000
M.F. Randolph
Pile head
z = 4.8 m
60000
Pile head load (kN)
z = 9.8 m
50000
40000
30000
20000
10000
0
0
0.02
0.04
0.06
0.08
0.1
Pile head displacement (m)
(a) Load-displacement response
Mobilised shaft friction (kPa)
1200
1000
z = 4.8 m
z = 9.8 m
800
600
400
200
0
0
0.02
0.04
0.06
0.08
0.1
Pile head displacement (m)
(b) Load transfer response
Figure 14 Rock-socket example: 10 m long by 1.4 m diameter
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February 2003
RATZ Manual Version 4-2
M.F. Randolph
160
140
Pile head load (MN)
120
100
80
60
Beta = 0
Beta = 0.001
40
Beta = 0.01
Beta = 0.1
20
0
0
20
40
60
80
Pile head displacement (mm)
Figure 15 Load-displacement response of offshore pile
with varying amounts of creep
37
100
February 2003
RATZ Manual Version 4-2
Pile head
z = 9.5 m
160000
140000
Pile head load (kN)
M.F. Randolph
z = 29.5 m
z = 49.5 m
120000
100000
80000
60000
40000
20000
0
0
0.02
0.04
0.06
0.08
0.1
Pile head displacement (m)
(a) Load-displacement response
Mobilised shaft friction (kPa)
450
400
350
300
250
z = 9.5 m
200
z = 29.5 m
z = 49.5 m
150
100
50
0
0
0.02
0.04
0.06
0.08
0.1
Pile head displacement (m)
(b) Load transfer response
Figure 16 Offshore drilled and grouted pile: 70 m long by 2 m diameter
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February 2003
RATZ Manual Version 4-2
Pile head
z = 9.5 m
160000
140000
Pile head load (kN)
M.F. Randolph
z = 29.5 m
z = 49.5 m
120000
100000
80000
60000
40000
20000
0
0
0.02
0.04
0.06
0.08
0.1
Pile head displacement (m)
(a) Load-displacement response
Mobilised shaft friction (kPa)
500
400
300
200
100
0
-100
0
0.02
0.04
0.06
-200
0.08
0.1
z = 9.5 m
z = 29.5 m
z = 49.5 m
-300
Pile head displacement (m)
(b) Load transfer response
Figure 17 Offshore drilled and grouted pile: cyclic loading, original algorithm
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February 2003
RATZ Manual Version 4-2
140000
Pile head
z = 9.5 m
120000
Pile head load (kN)
M.F. Randolph
z = 29.5 m
z = 49.5 m
100000
80000
60000
40000
20000
0
0
0.02
0.04
0.06
0.08
0.1
Pile head displacement (m)
(a) Load-displacement response
Mobilised shaft friction (kPa)
500
400
300
200
100
0
-100
0
0.02
0.04
0.06
-200
0.08
0.1
z = 9.5 m
z = 29.5 m
z = 49.5 m
-300
Pile head displacement (m)
(b) Load transfer response
Figure 18 Offshore drilled and grouted pile: cyclic loading, modified algorithm
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February 2003
RATZ Manual Version 4-2
M.F. Randolph
Shaft friction (kPa)
0
100
200
300
400
500
0
10
Depth (m)
20
30
40
50
Peak
Max. storm
60
End storm
Residual
70
(a) Profiles of failure shaft friction at different stages
Ideal and actual pile capacity (MN)
180
170
160
Ideal capacity (integrated failure shaft friction)
150
140
130
Actual capacity (allowing for strain softening)
120
0
5
10
15
20
Loading stage
25
30
(b) Progress of degradation through storm
Figure 19 Shaft friction and pile capacity during cyclic loading
41