LITKOUHI, S. & POSKITT, T. J. (1980). GCotechnique 30, No. 1, 77-86

Damping constants for pile driveability calculations

S. LITKOUHI* and T. J. POSKITT*

A discussion of the form of damping law which should be used in the wave equation analysis is given. It

is concluded from published data and the results of laboratory tests on small piles that damping depends

on velocity in a highly non-linear manner. A power

law is proposed to take account of this and a labora-

tory test facility for measuring the constants is des-

cribed. Failure to take proper account of non-linearity

in the damping law can give misleading results in the

interpretation of field data by the wave equation. For clays side damping constants are found to be greater

than point damping values.

INTRODUCTION

Cet article de la forme de la loi d’amortissement qui devrait Ctre utilisCe dans l’analyse par Equation d’onde. D’aprts les conclusions tirtes des donnCes publiCes et de rCsultats d’essais en laboratoire de pieux de petites dimensions, l’amortissement est fonction de la vitesse d’une maniere fortement non-IinCaire. L’article propose une loi de force pour en tenir compte et decrit une installation pour essais en laboratoire permettant de mesurer les constantes. S’il n’est pas correctement tenu compte de la non-linearit dans la loi d’amortissement, l’interprttation des donnt?es in situ par 1’8quation d’onde peut donner lieu ?+ des resultats trompeurs. Dans le cas des argiles, on trouve pour les constantes d’amortissement lateral des valeurs supirieures aux valeurs d’amortissement B la pointe.

The use of the wave equation method for predicting the driveability of foundation piles for off- shore structures is now well established in offshore geotechnical practice. In view of the im- portance of this method it is surprising that some of the soil parameters used in this method such as damping or quake are not better understood. For example, in a paper by Heerema (1979) it is shown that for clays, skin friction is very strongly velocity dependent at low velocities and relatively insensitive at high velocities. In most wave equation treatments which use Smith’s (1960) original formulation the skin friction is assumed to be proportional to velocity over the whole range of velocities a pile is subjected to during driving. The consequences of this when interpreting field data by means of the wave equation are important.

In wave equation work the word damping is used to indicate the gain in strength which soils show under fast rates of loading, i.e. it is a viscous parameter. It is of particular importance in fine-grained soils and is associated with the layers of water which are bound to the soil particles by molecular forces. These layers surround the particles and prevent mineral-to-mineral contact. Relative movement between particles takes place within the layers and due to the molecular forces gives rise to a strong viscous resistance. For clay soils the resistance at fast rates of loading may be several times greater than when the soil is loaded slowly.

The importance of rate-dependent phenomena in soil mechanics was first appreciated by Taylor (1942) in his classical studies of secondary consolidation. Taylor showed that clay skeletons had non-linear viscous laws. However, in order to obtain analytical solutions he assumed a linear law (Newtonian viscosity) and obtained the solutions given in his Theories A and B (see line 1, Table 1).

Smith (1960) also assumed linear viscous resistance in his numerical treatments of the wave equation (line 2, Table 1). Here the side resistance on the mth discrete element (R,) during driving

Discussion on this Paper closes 1 June, 1980. For further details see inside back cover. * Queen Mary College, London.

78 S. LITKOUHI AND T. J. POSKITT

Table 1. Viscosity laws in current use

Originator

Taylor (1942)

Smith (1960)

Marayama and Shibata (1964)

Barden ( 1965)

Gibson and Coyle (1968)

Dayal and Allen (1975)

Heerema (1979)

Viscosity Law

7j = structural viscosity of soil skeleton

R, = R,,(l +J’J’,,,) J’ = side damping constant

E = f170 sinh(ar/r,) 70 = T at zero time a and /3 are constants given by

rate process theory

7 = &?)I’” 6 and n are empirical constants

P,/Ps = 1 +JVN J = damping constant

q&c = 1 +KL Jog( Vi K)

qco = unit dynamic cone resistance

q. = unit dynamic cone resis- tance at lowest penetration velocity

V, = lowest penetration velocity (1.3 mm/s)

7 = a2 +a3 V”.2 for clay UZ, 3 = constants which depend

upon the shear strength

Test considtions

Secondary consolidation (oedometer)

Pile penetration

Creep deformation of clay

Secondary consolidation (oedometer)

Triaxial tests

Constant rate of penetration test

Simple shear

+ = shear stress, E =z strain rate, V,,, = pile wall velocity, b = rate of change of void rtaio, V = axial velocity of specimen.

is considered to be increased by an amount (RJ’V,,,) due to the velocity V,,,. R,, is the soil resistance on the mth element at the time of driving.

In general, the non-linear nature of the viscosity of soil cannot be ignored. For example, in

consolidation theory this appears as either a sinh law (Murayama and Shibata, 1964) or a power law (Barden, 1965) (lines 3 and 4 of Table 1). The magnitudes of a and /I or b and n are generally such that the laws give strongly non-linear behaviour. It should be noted that the sinh law is invariant with respect to changes in the signs of z and 8. In general this is not the case with the

power law unless n = 1, 3, 5, 7, etc. It therefore appears that unless n is odd the power law will introduce arbitrary directional properties which the real soil does not possess. Experimental evidence given by Barden suggests n is about 5.

Soils tested to failure in the triaxial machine at fast rates of loading also indicate highly non- linear behaviour (Gibson and Coyle, 1968). The experimental data given by these authors were fitted with a power law (line 5, Table 1). For sands an optimum value of N = 0.2 was obtained and for clays 0.18. N was found to be relatively insensitive to soil type.’

Dayal and Allen (1975) have described tests in which a penetrometer was driven at constant velocity into samples of either clay or sand. The penetrometer enabled cone and sleeve resistance to be measured separately. Test data for unit cone resistance were plotted in the form of qc,,/qc

’ Pd is the dynamic end load on the sample and P, the end load when the sample is strained very slowly.

DAMPING CONSTANTS FOR PILE DRIVEABILITY CALCULATIONS 79

versus log V/V, and the authors established the empirical relationship given in line 6 of Table 1. Unit sleeve friction was dealt with similarly.

Figures 8 and 10 given by Dayal and Allen show marked curvature over a velocity range 0.13-762.0 cm/s and data given in their Table 1 show K, to be strongly dependent upon velocity. If the data for medium stiff clay given by Dayal and Allen in their Fig. 10 are replotted on Fig. 4 of this Paper, it is found that over the full range of penetration velocities a reasonably straight line is obtained.

Heerema (1979) has described a test arrangement in which a flat metal plate is moved relative to the surface of a sample of soil in simple shear. The plate oscillates backwards and forwards on the soil surface and is kept in contact with a known normal pressure. Shear stress and velocity are recorded continuously. This arrangement indicates negligible viscosity for sand. However, for clays significant velocity dependence was observed, and a power law was fitted to the date (line 7, Table 1).

An examination of Table 1 shows that the majority of experimental data has been correlated with the empirical power law (including n = 1). This has convenience in analytical work and when used in the form proposed by Gibson and Coyle it allows the concept of Smith’s damping constants J or J’ to be retained. A power in the region of n = 5 (N = 0.2) seems to fit most observations well. The test data discussed cover a range of rates of deformation from the very slow encountered in consolidation processes to the very fast which occur during pile driving. The similarity of the results obtained by the different investigators using different experimental techniques gives confidence that the observations reflect the true non-linear behaviour of the soil rather than some combined characteristic of the type of test and the soil. They confirm the non- linear viscous behaviour of the soil skeleton.

For numerical computations using the wave equation the form of law used is immaterial provided N and J are obtained experimentally. However, when correlating N and J with soil characteristics the empirical nature of the power law may present problems. It may then be more advantageous to use the sinh law. This can be written in the form

P,/P, = l+a sinh-’ /?V

which is analogous to that given by Gibson and Coyle for the power law. c( and p might then be related to basic soil parameters through rate process theory.

In experimental work on soils at fast rates of deformation it is difficult to decide what pro- portion of the recorded changes in soil strength are due to viscous effects and what are due to changes in effective stress. Changes in effective stress can be assessed if pore pressures are monitored. In consolidation work this can be done with confidence; however, in piling work where pile wall velocities may reach values of 3-4 m/s in a time interval of 0.0005 s, the measure- ments are difficult. Pending advances in experimental technique, it is probably best to base the form of viscosity laws on those which apply for consolidation processes on the assumption that at the faster rates of deformation which occur during pile driving, the structural forms of the laws are still valid. If this is done, then the physical constants such as ./ and N must be obtained experimentally taking care to simulate field conditions.

In the following work a power law will be used since this enables a direct comparison to be made with the conventional Smith treatment.

MEASUREMENTS OF DAMPING CONSTANTS

The object of the tests was to measure the point and side damping constants for small piles driven at constant velocity into samples of soil. Since there are considerable practical difficulties

80 S. LITKOUHI AND T. J. POSKITT

Cylinder

Linear displacement transducer

Load cell

Pile

So11 sample container

Fig. 1. Model pile test rig

in measuring point and side forces simultaneously at speeds corresponding to field penetration velocities it was decided to measure the tip resistance using a cone of identical geometry to the pile point. The resistance so obtained could then be subtracted from the total resistance of the pile to give the side resistance. The procedure results in some error in tip resistance since the cavity above the conical tip has no lateral support. Estimates of side resistance are less affected

by this error since the magnitude of the point resistance is less than 10 % of the side resistance. The merit of using small piles is that the mode of deformation is similar to what might be

expected in the field. The piles were 10 mm dia. and 260 mm long with a 120” cone angle at the end. The conical tips were 10 mm base dia. and had a 120” cone angle. For the range of fine- grained soils tested these dimensions are thought to be large enough to avoid scale effects.

The equipment for driving the small piles is shown in Fig. 1. It consists of a cylinder and piston

with a stroke of 300 mm. The piston is driven by a high-pressure nitrogen/oil accumulator. The piston can be rapidly accelerated or decelerated without undue shock using high-speed switching valves. Over the majority of the stroke the piston is maintained at a constant velocity using commercially available flow control valves. This covers the range of speeds at which piles in the field are known to move following impact from a hammer.

A linear displacement transducer attached to the piston enables the displacement-time record of the pile to be recorded. This provides an immediate visual check on the velocity of the pile as it penetrates the soil. The piston is able to drive against a force of 2 kN without discernible fluctuation in speed. This makes the equipment suitable for all clays and soft rocks.

A load cell is mounted on the end of the piston and gives a continuous record of load against time as the pile penetrates. For convenience of interpretation load and displacement signals are recorded simultaneously on a UV recorder. Full details of the apparatus are given by Litkouhi (1979).

By analogy with Gibson and Coyle (1968) (line 5, Table 1) the dynamic point resistance is given by

and the side resistance by (R,/R,) point = 1 +JVN . . . . . . . . (1)

(RJR,) side = 1 +J’P . . . . . . . . (2)

where R, = dynamic resistance (point or side)

R, = soil resistance (point or side) when the pile is pushed in at 0.3 mm/s

DAMPING CONSTANTS FOR PILE DRIVEABILITY CALCULATIONS 81

Table 2. Properties of soils tested

Material LL: % PL: %

London clay 70 27 Forties clay 38 20 Magnus clay 31 17

The speed at which R, is measured should be sufficiently slow for viscous effects to be negligible but not so slow that pore pressures will have significantly dissipated before penetration is complete. In the tests it was taken to be 0.3 mm/s since this was the slowest speed at which the equipment would operate.’

The pore pressures generated as the pile penetrates are assumed to depend on the soil deforma- tion and not the rate of deformation, i.e. it is independent of V. Provided the time taken to insert the pile is small compared with the time taken for the clay to reconsolidate this assumption will be justified. For fine-grained soils of low permeability the assumption is reasonable.

Soils tested

Most of the tests to date have been on remoulded samples of London, Forties and Magnus clay. Forties clay is a normally consolidated or lightly overconsolidated silty clay and Magnus clay is a very stiff silty clay with shell fragments and scattered gravel. Some test data on un- disturbed Magnus clay are also given. The properties of these materials are given in Table 2.

Undisturbed samples of soil were taken with thin-walled tubes 254 mm dia. and 254 mm long. The tubes were mounted directly in the test rig.

Test results

The test procedure consisted of pushing the conical tip and the pile into the sample at 0.3 mm/s in order to obtain R,. Five speeds were then selected and the conical tip and the pile were pushed in at these in order to give R,.

The minimum spacing between pile cavities was taken as five pile dia. in order to reduce the influence of disturbance. For the same reason the test zone was kept at least five pile dia. from the sampler wall in order to avoid the effects of any disturbance during sampling.

For most of the tests the depth to which the conical tip or the small pile penetrated was of the order 150-200 mm. Over this range the properties of the soil are reasonably constant. The resistance of the conical tip is virtually independent of the depth of penetration while the side resistance increases almost uniformly with depth. Interpretation of the load cell results is therefore straightforward.

Typical plots of RJR, for London clay are shown in Figs 2 and 3. The curves are strongly non-linear and this has been observed in all the soils tested.

For comparing data it is more convenient to write equations (1) and (2) in the form

log[(R,/R,)-l]=logJ+Nlog V . . . . . . . . (3)

with a similar expression for equation (2). Plotting (RJR,) - 1 and Van log scales gives a straight line of slope N. The intersection of this

line with the ordinate through V = 1 gives the value ofJ. Average data for London, Forties and

2 This exceeds the recommended speed at which a constant rate of penetration pile test should be carried out. Hence R, is not the capacity of the pile as measured by a standard load test.

82

25b

Case LPl

- Best fit

S. LITKOUHI AND T. J. POSKITT

I Case LP3

2.5 .

Case LP2 Case LP4 +,Smith

2.0 VI

15.

l.OW pASmith

0

0 50 100 150 200

Velocity: cm/s

4 n

/+ 0 50 100 150 200

Velocity: cm/s

Fig. 2. Point tests on London clay

Magnus clay are given on Fig. 4. Shown on the same graph are results given by Gibson and Coyle (1968), Dayal and Allen (1975) and Heerema (1979). It will be observed that N and N’ are less than unity which confirms the non-linear trend of all the data.

An alternative method of comparing test data such as shown in Figs 2 and 3 is to compute J and N, and J’ and N’, using the method of least squares. The same general method can be used to computeJandJ’ corresponding to Smith’s law. The results of 38 tests are given in Tables 3-5.

Tube tests

The tables show that in general for Smith’s formula J is less than J’. Since this is contrary to what is generally assumed, it was felt desirable to confirm this with an ad hoc set of tests based upon a different testing technique.

The procedure adopted was based upon the experimental observation that when an open- ended tube is pushed at constant velocity into a sample of soil it will after some penetration plug. This occurs when the force between the soil plug and the inside wall of the tube exceeds the force on the base of the plug, i.e.

nDL,r(l +J’V) = &cLPp(l +.W) . . . . . . . (4)

where D = internal diameter of the tube

L, = plug length

r = shear stress between soil plug and inside pile wall

p = pressure at the base of the soil plug

From equation (4) 42 l+JV -.-L,=- DP

*+J,v . . . . . . . . . . (5)

DAMPING CONSTANTS FOR PILE DRIVEABILITY CALCULATIONS 83

4.0

1 Case LS4 (12)

4.0 r Case LS4 (17)

Smith

Velocity: cm/s

Fig. 3. Side tests on London clay

Table 3. Result for London clay

,

Case LS5 (17)

Velocity: cm/s

Test No.

LPI LP2

LP3 LP4 LPS

LP6 LP7

I- Point tests Side tests

Best fit

I: s/cm

0.29 0.20

-- Smith Test N r: s/cm No.

0.22 0.0067 0.22 OW46

0.25 0.22 oxMI57

0.24 0.22 00x4 0.29 0.22 0.0069 0.27 0.19 0.0054

0.27 0.21 0.0059

0.27 0.16 0.0046 0.28 0.21 0.0060

0.27 0.19 0~0053

LSl (12) LSl (17) LS2 (12) LS2 (17)

0.69 0.64 0.47 0.48

0.57

N’ --

0.23 0.21 0.17 0.21

0.21

0,017 0.014 of)09 0.011

0.013

LS3 (12) 0.77 0.24 0.018 LS3 (17) 0.89 0.19 0.017 LS4 (12) 0.82 0.15 0.012 LS4 (17) 0.72 0.19 0.013 LSS (12) 1.18 0.17 0.021 LS5 (17) 1.13 0.18 0.021 LS6 (12) I.27 0.07 0.014 LS6 (17) 1.14 0.11 0,016

0.99 0.16 0,017 Average

LS7 (12) LS7 (17) LS8 (12) LS8 (17)

0.49 0.20 0.011 0.43 0.25 0.011 0.53 0.13 0.008 0.56 0.08 0.007

0.50 0.17 0.009

.-

.I

-

Best fit -__ I’: s/cm

-

J

-

Smith Average, I’: s/cm Cu: kN/m’

15

Average

35

60 320

Average

Rem. cons. pressure:

kN/m2

70

210

84 S. LITKOUHI AND T. J. POSKITT

Table 4. Results for Forties clay

-

-

Average, C,: kN/mZ

Rem. cons. pressure:

kN/m2

5

45

70

320

Average, C,: kN/m2

Rem. cons. pressure:

kN/m*

5 70

40 320

80 Undis- turbed

Test No.

FPl

Point tests --

Best fit Smith

J: s/cm N J: s/cm

0.36 0.17 0.0061

0.36 0.17 OX)061

0.12 0.37 oGx9

0.12 0.37 oxtO

Table 5. Results for Magnus clay

FP2

Side tests

Best fit

I’: s/cm N’ ____~

0.54 0.15 0.56 0.24

0.55 0.20

0.22 0.37 0.39 0.15

0.31 0.26

___

Test No.

FSl (12) FSl (17)

FS2 (12) FS2 (17)

Smith I’: s/cm

0.009 0,014

0,012

0,010 0,006

0.008

*

-

Average

Average

Point tests

Best fit

N

0.33

I I: s/cm

0.22

Smith I: s/cm

0.0087

0.22 0.33 0.0087

0.20 0.28 0.0059

0.20 0.28 0.0059

0.08 0.37 0.0036

0.08 0.37 0.0036

Side tests

Best fit Test No.

MP1

MP2

MP3

I’: s/cm N’ Smith

r’ : s/cm

0.27 0.44 0.017 0.19 0.46 0.015

0.23 0.45 0,016

0.47 0.35 0,020 0.56 0.38 0.028

0.51 0.36 0.024

0.09 0.45 0.006 0.06 0.69 0.012

0.08 0.57 0.009

Test No.

10.0.

MS1 (8) MS1 (13)

M2S (8) MS2 (13)

MS3 (8) MS3 (13)

Average

Average

Average

Heather clay g,, = 230

Kontichclayu,,= 184 Claymore

\ i

Heatherclayo,= 150

O,lL 1.0 10

8 100 1000

Velocity: cm/s

Fig. 4. Non-hear trend of data

DAMPING CONSTANTS FOR PILE DRIVEABILITY CALCULATIONS 85

--. LT3 .n,

4 a

OOb---- 10 15 Velocity: cm/s

Fig. 5. L, against velocity of penetration

Equation (5) provides a simple qualitative method of determining the relative magnitudes of J and J’. If the tube is pushed in at various velocities and a graph plotted of L, against V then if this has

positive slope then J>J’

zero slope then J=J’

negative slope then Jc J’

The results of four tests on a tube 19 mm OD, 1 mm WT and 260 mm long driven at velocities of 0.3, 51 and 142 mm/s into a sample of remoulded London clay, c, = 35 kN/m’, are given in Fig. 5. In general, the slope is negative indicating that J< J’ and confirming the results given in Tables 3, 4 and 5.

CONCLUSIONS

(a) The viscous resistance of a clay soil is non-linear.

(b) A power law of the form R, = R,( 1 + JVN)

fits many soil types over the range of velocities encountered during most pile-driving operations.

(c) N for point and side resistances lies in the region of 0.2.

(d) J point is less than J’ side.

(e) Measurements of J and J’ for new soil types is necessary.

(f) Correlation of field data using conventional wave equation programmes should take account of the highly non-linear viscous resistance of the soil.

ACKNOWLEDGEMENTS

The Authors are grateful to BP Trading Ltd for their support of this work and to Mr W. J. Rigden of BP and Dr R. Hobbs of Lloyds Register of Shipping for their valuable comments on the paper.

REFERENCES

Barden, L. (I 965). Consolidation of clay with nonlinear viscosity. GPorechnique 15, No. 4, 345-362. Dayal, U. & Allen, J. H. (1975). The effect of penetration rate on the strength of remoulded clay and sand

samples. Can. Geotech. J. 12, 336-348.

86 S. LITKOUHI AND T. J. POSKITT

Gibson, G. C. & Coyle, H. M. (1968). Soil damping constants related to common soil properties in sands and clays. Report No. 125-1, Texas Transportation Institute, Texas A and M University.

Heerema, E. P. (1979). Relationships between wall friction displacement velocity and horizontal stress in clay and in sand for pile driveability analysis. Crowd Engineering January 1979.

Litkouhi, S. (1979). The behaviour offoundation piles daring driving. PhD thesis, London University. Marayama, S. & Shibata, T. (1964). Flow and stress relaxation of clays. Sytnp. Rheol. Soil Mechs., Grenoble.

99. Internation Union of Theoretical and Applied Mechanics. Smith, E. A. L. (1960). Pile driving analysis by the wave equation. Trans. Am. Sot. Civ. Engrs, Paper No. 3306,

127, Part 1. Taylor, D. W. (1942). Research on consolidation of clays. Massachusetts Institute of Technology, Department

of Civil and Sanitary Engineering, Serial 82.