Computers and Geotechnics 9 (1 990) 291-305

PILES UNDER AXIAL AND TORSIONAL LOADS

Michael Georgiadis and Sofia Saflekou Department of Civil Engineering

Aristotle University of Thessaloniki Greece

ABSTRACT

An analysis of axially and torsionally loaded piles is presented in which the pile is treated as an elastic bar supported on two series of interacting non-linear axial and torsional springs. The characteristics of these springs depend on the soil properties and the diameter of the pile as well as on the interaction between the axial and torsional response. Predicted pile responses ape compared to the results of model tests conducted on piles installed in soft clay bed and loaded with combined axial and torsional loads. Both experi- mental results and theoretical predictions show that a torque applied to the pile head affects significantly the settlement and the axial bearing capacity of the pile.

INTRODUCTION

Pile foundations are normally designed ~o support axial and la<eral loa4s

which result from the weight of the structure and any acting sea wave or seismic

forces. The analysis of pile response for these types of loading can be per-

formed with several methods which ar~ either considering the soil as an elastic

/ elasto-plastic half-space [I-3] or, more often in practice, as a series of non

linear springs [4-9].

Although axial and lateral are the main forms of loading, the foundation

piles of some types of structures, especially offshore structures, are liable to

receive significant torsional loads which are caused by the eccentricity of the

applied lateral forces. Methods for analyzing the pile response under torsional

loading have been presented in references [10], [11] and [12]. All methods of

analysis consider that axial, lateral and torsional loads act independently,

assuming that there is no interaction. The influence of torque on axial pile

response and the influence of axial load on torsional pile response were dis-

cussed by Georgiadis [13]. It was shown, using a simple elastoplastic spring

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292

model, that the interaction between axial and torsional pile responses contri-

butes significantly to the global pile response, resulting in large~ ~ axial

displacements and rotatiens.

This paper presents the results of an experimental investigation in which

model piles in clay were subjected to combined axial and torsional loads to

study the effect of torsion on axial pile response. The experimental results

were compared to the results of a numerical analysis in which the soil was ~rea-

ted as two series of interacting non-linear springs (~xial and torsional). Fi-

nally, the numerical analysis was applied to the case of a typical full size

offshore pile.

NUMERICAL ANALYSIS

The analysis was conducted with the numerical model presented in reference

[13] but with some essential changes which were made in order to account for

the non-linearity of the soil and to improve the interaction between axial and

torsional springs. The pile is considered as an elastic bar of length L, which

is divided by "n" nodes into (n-l) segments of length i= L/(n-1). The soil is

treated as two series of non-linear springs (one axial and one torsional) which

are attached to the pile nodes (Figure I). Another spring is connected to the

pile base to represent the end bearing resistance. A torsional base spring is

not introduced in the analysis because its contribution to the response of or-

dinary piles is expected to be minor 110].

(i) Spring Characteristics

The first step in the analysis is to establish the load-displacement rela-

tionships of the axial springs and the torque-rotation relationships of the

torsional springs. The load-displacement relationship of an axial spring is

characterized by an initial stiffness K a non-linear part and an ultimate load oa

P at which the spring fails (Figure 2a). The initial stiffness K was deter- u oa

mined using an analytical formula proposed by Randolph and Wroth [3] for axially

loaded rigid piles in an elastic half-space :

z = (tr/Gs)in(rm/r) (I)

where z is the axial displacement, t the pile shaft load per unit area, G s the

shear modulus of the soil, r the pile radius, r the radius of the zone of in- m

fluence beyond which the shear stress becomes negligible rm= 2.5L(I"Ms) and M s

the Poisson's ratio of the soil. The values of G and ~ of the clay bed uti- s s

293

s

L-!I~ f

hll ~-U

U

d

T

7L-

FIGURE I. Numerical pile model

lized in this study were 2 MN/m ~ and 0.5 respectively.

stiffness K of a pile segment of length 1 becomes : oa

2~lG P s

K = --- = .......... oa z ln(rm/r)

where P is the shaft resistance of the pile segment.

Using Eq. 1 t h e spring

(2)

The ultimate axial spring resistance P U

P = 2~rlt U U

of a pile segment of !en~h 1 is :

(3)

294

p ~ Koa (a)

P = Pu(1- e -Koa

z i

z/Pu)

z

(b) T /o: . . . .

. . . . . T = Tu(I-e u~-///' -KotO/Tu >

v

FIGURE 2. Idealized axial and torsional spring characteristics

295

where t is the ultimate shaft resistance per unit area. Usingtheconventional u total stress approach, the value of t can be determined with she relationship u

t = cuz . For soft clays a=1 [141 and therefore the ultimate shaft resistance u u

is equal to the undrained shear strength c . It is noted that the same approach u

can be applied to sands provided that the unit shaft resistance is determined

approprietly.

Having determined the values of K oa

soil springs can be described by the equation (Figua-e 2a) :

p = Pu(l-e-Koaz/Pu )

It will he shown later that equation 4 proved to give very good

pile response with respect to the experimental results.

and Pu' the non-linearity of the axial

(4)

predictions of

The torque-rotation relationship of the torsional springs is characterized

by an initial stiffness, Kot, a non-linear part and an ultimate torque Tu, at

which the spring fails (Figure 2b). The initial s%iffness Kot was determined

with the following equation derived by Randolph [11] for rigid piles in an

elastic half-space :

T = 4r~r21G ~ (5) s

where T is the torque applied to a pile segment of length 1 s2~d ~ is the resul-

ting rotation.

This equation can be written as :

Kot : (T/S) : 4rLr21O s (6)

The ultimate torque T u of a torsional spring is a function of the ultimate unit

shaft resistance t u and can be computed with the equation :

T u = 2Kr21t u (7)

The non-linearity of the torsional springs was modeled with the equation :

T = Tu(1-e-K°t~/tu ) (8)

296

The same type of exponential relationship was also used to describe

linear response of the axial end bearing spring, i.e. :

Q : Qu(1-e-KobZb~Qu )

where Q is the end bearing resistance, z the base settlement, Kob c

stiffness of the load-settlement curve and Qu the ultimate bearing

the pile base.

t he non-

(9)

the initial

capacity of

The stiffness Kob was determined using the following rigid punch equation

[15]:

q(1-v ) s

Z. ---- ......... n 0 4rO

s

(10)

where n is a factor introduced to allow for the depth of the pile base below the

surface. Typical values of this factor range between 0.5 and 0.78 [7].

Equation 10 can be written :

4rG __Q ....... s ....

K°b : Zb ( 1-V )n s

(11)

The ultimate capacity of the base spring introduced in Eq.9 was computed with

the equation :

Qu = 9Curer2 (12)

(ii) Numerical Procedure

The analysis was performed with the Transfer Matrix Method [16,17]

lowing four stages :

fol-

STAGE I: Assuming that both axial and torsional springs are linear with stiff-

nesses K and K respectively, the Transfer Matrix Method is applied to oa ot '

determine the axial and torsional pile responses for a set of loads acting on

the pile head. The computation of the axial force, torque, axial displacement

and rotation of each node is performed considering that there is no interaction

between axial and torsional loads.

297

STAGE 2: An iterative scheme is utilized using secant stiffnesses to account

for the non-linearity of the soil springs. The displacements and rotations

determined in the previous stage are used in conjuction with Eqs. 4, 8 and 9 to

establish new linear spring stiffnesses Kai and Kti for each node and the Trans-

fer Matrix analysis is repeated until finally the error which results from the

soil non-linearity become negligible.

STAGE 3: The interaction between axial and torsional pile responses is intro-

duced into the analysis following the procedure illustrated in Figure 3. The

axial force and torque of each node, computed in Stage 2, are combined to deter-

mine the total pile skin friction of the node :

R i = (P~ + (Ti/r)2)0"5 (13)

This combined friction cannot exceed the ultimate skin friction R of u and therefore the ultimate axial spring resistance P and the ultimate

U

torque T u are reduced according to the following relationships :

the node

spring

and

Pui = Ru(Pi/Ri)

Tui = Ru(Ti/R i)

I~)

15)

where :

R U

= (T /r) = 2T~rlt = Pu u u 16)

STAGE 4: All previous stages are repeated with the new axial load - axial

displacement and torque-rotation relationships (Figures 3c and 3d) until finally

the values of P . and T . get stabilized. Ul Ul

EXPERIMENTAL INVESTIGATION

A number of model tests were performed on 500mm long aluminiumc!osed ended

piles of 19mm outside diameter and 1.5mm wall thickness. All piles were in-

stalled in a soft medium plasticity clay bed (700r~m wide x 1000mm long x 800mm

deep), in which WL=42% , Wp=24%, w=26% and Cu=8 kN/m 2 (measured throu~q uncon-

fined compression and shear box tests). Four of the tests were performed with

only axial loads (no torsion) while in four other tests a torque of 1500 N.~m

was continuously applied to the pile head during the axial loading. The axial

298

(a)

i

• ,L3-FTi/r

:3

P

Pu

Pui

(b)

"-#~ I \ 1 \ I \ b \

'

I \

'~lr Tui/r Tu/r T/r"

P (c)

~U . . . .

v

z

L

Li

(d)

y

FIGURE 3. Interaction between axial and torsional springs

299

i : : ~ w a s a p p ! i c d ;~'. ' : n , - ~ ' ~ - l i s p l < e _ m e : : t ='~:," - f -~t__~--. ~ . b mm :::[:~, i . e . < h e

[ : - i e h e a i wus i i s p i u ~ ' e d v = : ' - i z ~ _ ± i Z RI - _ h i ; ; " , , r e zs i : - .C %he t~[Lp<~:';~ i ~ _ 'hEwn i f !

?'£gur: 4, while thr bad hl?:'e:=se was s~ntirluoazly :'ec,"iDi. 'the to:-'.ue w;~s

applied to the pile with a dev'c~' , c0r:sis<in!; o f "< st:~i iisk u, tb:~.c!i~d to th{~

p i l e h e a d a n d a pal_," c f s t , i w-~-<; a n ! i ~ u l w c i g b u : - w % i c k c r , ~ t ~ ' d : ~ , p u r e memo: i t ,

Figure 5 shows th:~t ~.e measured u!tim;~t~ p-!e cay~mity ~'anges b:.tween ~-3:~

and 27] N. The results :~'f the -~,y~.bin<i '.v<ial to'--siena! pile testsar< presented

in Figure ,., <, showin S tiv:,~ tke '~_~;~',~,-~..~_.= %zi~.L pile cap,~city is significantly

,_~=~ce,a and ran6es between i00 and ~I_ ~ N ':atke other h'~nd th~ ~pplied torque

res ~lted in increasing the settlement of tie pile, e. 6. for ~ :~<ial load of

]00 N the torque incre~sed the pile head sett!e~:ent hy 50]~ from about 0.0~ m~.<

to 0.12 ram.

Some additional tests which were attempted with a much larger torque of

~,_0 N.mm resulted in pile f<~i!u~-e u~d~,r a very small axial load o~ about 20 N

FIGURE L. Experimental apparatus

300

3 0 0

2 5 0

2 0 0

v

r,, 1 5 0 < 0 /

_ J <

< 100

5 0

, , , ' "

t ,"

0 '" 0.0

v V v I V

/ ° / ; .. ! • X

I h .>" o/:~ ..../

? ..................................... 2::;- ? .... : ..... x , . . n r e d i c f e d .'°

x • o v m e a s u r e d

- - - - ~ W ~ re f . (4),(7)

i A ,"" j ,

. . . . . . . . . . . . r e f . ( 5 )

ref . (9)

0 . 2 5 0 . 5 0 0:175

P I L E - H E A D S E T T L E M E N T ( m m )

1,00 1.25

:GUBE 5. Z e u . ; ; r e d and predicted model pile re ponse for axial loading

301

300

250

200

A

z

a ,,~ 150

9 .J <

< 1 0 0

50

o

0 VII

FIGURE 6.

¢

0 0.0

0

X X

• ¥

X • 0 V

predicted

meosured

0.25 O.:iO 0.~5

PILE-HEAD SETTLEMENT (mm)

Measured and predicted model pile response for combined axial/torsional loading

1.~ )0 1.2 j=

302

which is equivalent to the end bearing capacity of the pile.

COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS

The analysis described earlier was applied to compute the response of the

model piles used in the experimental study. The axial load vs. pile head set-

tlement relationships determined for purly axial and for combined axial/torsio-

nal loading are presented in Figures 5 and 6, respectively, together with the

experimental results. These figures demonstrate that the pile response is sig-

nificantly affected by the 1500 N.mm torque applied to the pile head. They also

prove that the experimental and theoretical load-settlement relationships are

in very good agreement for both purely axial and axial/torsional loading condi-

tions. Pile responses determined using some other load-displacement relation-

ships [4,5,7 and 9] provided results which were not so satisfactory (Figure 5).

The pile diameter independent relationships of Coyle and Reese [5] and Vijayver-

giya [9] gave very large pile displacements while the Kraft et.al [7]/ Armaleh

and Desai [41 relationship gave pile displacements similar, but somewhat lower,

to those obtained experimentally and using Eq.4.

Measured ultimate pile capacities for purely axial loading r<nge between

230 and 270 N while the predicted capacity is 250 N. Similarly good is the

agreement for the combined axial/torsional loading where the measured pile capa-

cities range between 160 and 210 N while the numerical analysis gave 180 N.

TYPICAL APPLICATION

In order to study in more detail the effect of torsion on the axial pile

response, a typical full size offshore steel pipe pile in a uniform soft clay

deposit of c =20 kN/m 2 is analyzed. It is 50-meter long, 2.]-meterdiameterand u

has a wall thickness of 28mm.

The results are presented in Figure 7 in the form of axial load vs. pile

settlement curves for different values of the pile head torque. It is noticed

that the axial pile capacity can vary from 600 kN (end bearing capacity)to 7200

kN depending on the applied torque. Similarly important is the effect of an

axial load on the torsional pile response. The torsional pile capacity can vary

from zero to 6900 k2{.m depending on the applied axial load.

303

8

T=O

6 I - T : 4000 kN.m

Q

i 0.0 0.04 0.0B 0.12 0.16

PILE-HEAD SETTLEMENT (m)

Predicted offshore pile response FIGURE 7.

304

CONCLUSIONS

The experimental work presented in this paper has proved that the interac-

tion between axial and torsional pile responses has an important influence on

global pile behavior.

A torque applied to the pile head results, in addition to the rotation of

the pile, in reducing the ultimate axial bearing capacity and increasing the

settlement. The same conclusion was also derived from a numerical analysis

which considers the soil as two series of interacting non-linear axial and tor-

sional springs.

The simplicity of the analysis, coupled with the good agreement achieved

between the predicted and measured pile responses to combined axial / torsional

loads, suggests that it may prove to be of considerable value. The conventional

approach of separate examination of the axial and torsional pile responses may

in some cases overestimate seriously the pile capacity and underestimate the

pile movements.

I.

2.

3.

5.

6.

7.

REFERENCES

Butterfield, R. and Banerjee, P.K., The elastic analysis of compressible piles and pile groups. Geoteehnique 21 (1971) 43-60.

Poulos, H.G. and Davis, E.H., Pile Foundation Analysis and Design, J.Wiley and Sons, New York (1980).

Randolph, M.F. and Wroth, C.P., Analysis of deformation of vertically loaded piles. J.Geot.En~.Div.A.S.C.E. I04, GT12 (1978) 1465-1488.

Armaleh, S. and Desai, C.S., Load-deformation response of axially loaded piles. J.Geot.En~.A.S.C.E. 113, GT12 (1987) 1483-1500.

Coyle, H.M. and Reese, L.C., Load transfer for axially loaded piles in clay. J.Soil Mecb.Fdns.A.S.C.E. 92, SM2 (1966) 1-26.

Georgiadis, M. and Butterfield, R., Laterally loaded pile behavior. J.Geot.Ens.Div.A.S.C. E. I0__88, GTI (1982) 155-165.

Kraft, L.M., Ray, R.P. and Kagawa, T., Theoretical t-z curves. J.Geot. Eng.Div.A.S.C.E. 107, GT11 (1981) 1543-1561.

305

8. Reese, L.C., Cox, W.R. and Koop, F.D., Field testing and analysis of laterally loaded piles in clay. Proc.Tth Offshore Technolo~D" Conference, Houston Texas, OTC 2312 (1975) 671-690.

9. Vijayvergiya, V.N., Load-movement characteristics of piles. Proc. PORTS'77, Long Beach California (1977) 269-28h.

10. Poulos, H.G., Torsional response of piles. J.Geot.En~.Div.A.S.C.E. 101, GTI0 (1975) 1019-1035.

11. Randolph, M.F., Piles subjected to torsion. J.Geot.En~.Div.A.S.C.E. 107, GT8 (1981) 1095-1111.

12. Chow, Y.K., Torsional response of piles in nonhomogeneous soil. J.Geot.Ens.A.S.C.E. 111, GT7 (1985) 9h2-9h7.

13. Georgiadis, M., Interaction between torsional and axial pile responses. Int.J.Numer.Anal.Methods Geomech. 11, (1987) 645-650.

lb. American Petroleum Institute, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms. API RP 2A (1985).

15. Timoshenko, S.P. and Goodier, J.N., Theory of Elasticity, 3rd ed., McGraw-Hill Book Co.Inc., New York (1970).

16. Georgiadis, M., Flexible Landing Mats on Clay, Ph.D.Thesis, Dept.of Civil Engineering, Univ.of Southampton, U.K. (1979).

17. Pestel, E.C. and Leckie, F.A., Matrix Methods in Elastomechanics, McGraw Hill, London (1963).

Received 20 June 1990; revised version received 12 September 1990; accepted 15 September 1990