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10BUCKLING LOADS OF SLENDERPILESGranholm (1929) showed that for piles of normal dimensions driven through soil,buckling should not take place except in extremely soft soil. However, veryslender and.long piles are increasingly used today (1990). In offshore structures,these piles also extend for a considerable distance mudline. Therefore, thepossibility of buckling of such piles has received considerable atten tion. Researchhas been carried ou t to obtain more accurate estimatesof buckling loads of piles.The majority of analytical methods proposed have employed the subgrade-reaction theory, described in this chapter. Both fully embedded and partiallyembedded piles are considered.10.1 FULLY EMBEDDED PILESEarlier solutions for the elastic buckling loads of embedded piles were based on asubgrade modulus for the soil which was assumed to be constan t over the lengthof the pile. Hetenyi (1946) presented a survey of the work by Forssell(l918, 1926)and Grandholm (1929); the governing differential equation is

d4y d2yd x dxE l - + P T + y = 0where

10.1)

E l = flexural stiffness of the pileP = axial loadk = subgrade modulus

677

Copyright 1990 John Wiley & Sons Retrieved from: www.knovel.com

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678 BUCKLING LOADS OF SLENDER PILES

a) b)

(e) 4 (e)Figure 10.1fixed translating F t ) . (e) fixed F).Pile boundary conditions (a) Coordinate system (b)free (f) , c)pinned ( p ) , d)

All the foregoing quantities were considered to be constants in those solutions.Figure 10.1 shows pile boundary conditions. The solutions of equation 10.1)have been obtained in the nondimensional form, letting

ThenLZ m a x =

whereL = embedded length of the pileR = relative stiffness factorEZ = flexural stiffness of pileZ = nondimensional depth coefficient

(10.2)

(10.3)


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FULLY EM BED DED PILES 679By substituting these definitions into equation (10.1) and rearranging, thefollowing equation is obtained:

d4y P R 2 d2ydz4 E I dz2-+-.- + y = o

Let PR2/EI be the axial load coeficient U; henPcrR2U,, = I

where subscript CY represents the critical values of U, nd P.By substitution, equation (10.6) is obtained:

d4y d2ydz4 d z 2- +u -+y=o

(10.4)

(10.5)

(10.6)The critical values of the axial load coefficient, V,, are obtained by solvingequation (10.6) for U with du e consideration to the pile boundary conditions andthe pile length, Z,,,. The boundary conditions are free f), inned p), fixed-translating ft), and fixed-non-translating, F) see Figure 10.1). An analogcomputer was used to obtain solutions for equation (10.6); the techniques andthe computer program have been presented by Davisson and Gill (1963).Case I: k = COIISCQII~n this solution, the axial load has been assumed to beconstant in the pile, and no load transfer occurs. Th e pile is initially straight. Thesolutions are shown in F igure 10.2 in dimensionless form, as a plot of U, , versusZ for several boundary conditions (e. ft-p, p-p, fr-f f-p,f-f) (Davisson,1963). Figure 10.2 shows that the boundary conditions exert a controllinginfluence on V,,.For pinned ends, the pile deforms into a num ber of sine half-waves, with thenumber of waves depending on the total length of the pile. U,, alues wereobtained for the first three m odes (D avisson, 1963);for all modes, the U,, aluesare above 2, and a t certain values of Z,,, become tangent to the line V,, = 2. Thelowest values of U,, or any given lengthZ are the ones of interest; for practicalpurposes, U,, is considered equal to 2 (Davisson, 1963).Another solution commonly referred to is the one for perfectly free ends f-f),In this case, U,, is zero when Z,,, equals zero and increases with an increasein Z,,, until a maximum value of unity is reached. At this point, a mode changeoccurs and U,, dips below unity, but it returns to unity when the next modechange is about to occur. With increasing pile length, the magnitude of thedeviation from unity becomes negligible. Because in most practical cases Z,,,is greater than 5 U,, an be considered equal to unity (Davisson, 1963). It willthus be seen that the boundary conditions exert a controlling influence on U,,.


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680 BUCKLING LOADS OF SLENDER PILES3.0 1 1 1 1 1 1 1 1 1 1 1

UC,

1

- Legend -f =freep = pinnedf t = fixed

Note: Upper end

- -translating --

condition listed first

0 2 4 6 8 10 122-Figure 10.2 Buckling load vs. length for k, = constant (Davisson, 1963).

For a free head and a pinned tip (f-p) pile (Figure 10.2), U,, increases rapidlywith a n increase inZ,,, up to the limiting value of unity. After first reaching unity,the higher modes indicated U,, values were little different from unity. Thebuckling appears to be controlled by the boundary offering the least restraint. Itmay be reasoned that a pile with a pinned head and a free tip would also have alimiting U alue of unity (Davisson, 1963).A pile with its head fixed against rotation but no t translation ff) represents apile in a group. When com bined with a free tip, the value of U, oscillates slightlyabout 1. When combined with a pinned tip, the value of U,, ecomes tangent to 2.Case 2: k = q * x When a soil profile is considered for which k = n h * x , theboundary condition a t the pile head becomes extremely im portan t compared tothe boundary condition a t the pile tip. Because the pile tends to buckle where thesubgrade modulus is the lowest, instability will tend to occur immediatelyadjacent to the pile head.When k = nhx, equation (10.1) becomes

(10.7)


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FULLY EMBEDDED PILES 681Let

then

where

Xand Z = -T (10.8)

(10.9)

T = relative stiffness factorZ = nondimensional depth coefficientZ,,, = maximum value of the depth coefficientBy substituting the above into equation (10.7) and rearranging , we obtain:

d4y P T 2 d2y-++.- +zy=odz4 E I dz2Let V denote the axial load coefficient, PT2/EI ; hen,

r 2

By substitution, equation 10.10) becomesd 4 y d 2 y-+ v- y=odz4 dz2

(10.10)

(10.11)

(10.12)Equation(10.12) was solved for V,, with the aid of an analog computer

Davisson (1963). V,,versus Z,,, for a pile with a free head and a free tip f-f)sshown in Figure 10.3. V, starts at zero and increases with an increase in Z,,,up to a limiting value of approximately 0.71.Other boundary conditions in Figure 10.3 are a pile with a free head and apinned tip ( f - p ) . Because of the increase in restraint that a pinned tip offers,compared to a free tip, V increases more rapidly with length than for the free-tipcase. The maximum V,, was approximately 0.78, which is only slightly higher thanthat for the free-tip case. Fo r a pinned-head, free-tip pile ( p - f ) a considerablyhigher value of V,, is observed at any given length Z,,,. This i llustrates the effectof the restraint of a pinned-head pile when compared to a free-head pile. Twobuckling modes were observed for this case, but for all practical pile lengthsV,, exceeds 1.44. Generally, a pile will have a length exceeding a Z value of3 to 4 (Davisson, 1963).For a pile with its head fixed against rotat ion but not translation and a free-tip,fig),he minimum value of V, is approximately 0.88; it occurs a t a very shortpile length, nam ely,Z,,, = 2.3. V, increases rapidly for pile lengths greater than


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682 BUCKLING LOADS OF S L E N D E R P I L ES3.0

2.0

vcr

1o

0

I \ I I ILegendf =freep = pinnedp = fixedP-P transiating

Note: Upper end.condition lis ted first

1 2 3 4 5 6zln 3 I

Figure 10.3 Buckling load vs. length for kh= nhx (Davisson, 1963).

Z,,, = 2.3. It can be reasoned that a pile with a fixed-translating head and apinned tip would have higher V , alues, for any given pile length, than the free-tipcase. By similar reasoning, it can be seen that a pile with a fixed-non-translatinghead and either a free tip or a pinned tip would also exhibit higher values. Fo r apile with both ends pinned p-p), the minimum observed Vc, value was 2.30atZ,,, = 2.60.Because most real piles are initially deformed, and because the theoreticalelastic buckling load is an unconservative upper bound to the actual failure load,the computed buckling loads are often only an aid to the judgment of the engineerfaced with the task of predicting the buckling load for a pile. The use of load testsis also unconservative. Most load tests are performed in a relatively sho rt periodof time during which a large patt of the axial load in the pile is dissipated by skinfriction (see Chapter 1). Under service conditions, the skin friction may be muchless than that in short term tests and the tendency to buckle would be greater(Davisson, 1963).Prakash (1987) obtained solutions for buckling loads in dos ed form by energymethods for fully embedded vertical piles for boundary conditions, pinned top-pinned tip (p-p), fixed top-fixed tip F-F),nd a linear variation of soil stiffness


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FULLY EMBEDDED PILES 683

k=O k-constant k= n,, x k = k o + nhFigure 10.4 Soil property variation along the depth of the pile.

with initial value k , >0 (Figure 10.4). The effects of pile leng th, soil stiffness, andboundary conditions on buckling loads and m ode of buckling have been studiedfor pile lengths up to 24m with an E l of 477 tm2,K O rom 0 o 2000 t/m2 and n hfrom 0 o 2000 t/m3.Where k , = k at the top of a fully embedded pile, and k , = k at the tip of afully embedded pile, and n,-constant of subgrade reaction, n, = (k, - o)/L.The variation of coeficient of subgrade reaction with dep th has been shownin Figure 10.4. Four cases are shown:1. Constant with depth k, = k = 0 (Figure 10.4a)2. Constant with depth k, = k , = constant (Figure 10.4b)3. Increasing linearly with depth with zero value at the surface, k = n , x4. Increasing linearly with dep th with nonzero value at the surface, k = k ,(Figure 10.4~)+ nhx in which ko 0, as in Figure 10.4d

The critical load was determined by calculating the smallest eigenvalue of theleading principal submatrix.The buckling loads were determined based on an energy method (Le., theincrement of the strain energy during the beam deflection will be equal to thework done by the external forces). The equations of the deflection curvessatisfying different boundary conditions on the beam have been substituted intothe work energy equation. In order to determine the buckling load P,,, hederivative of the energy equation was set equal to zero and transformed intomatrix notation with a standard eigenvalue form.Efect of Stifness Linearly Increasing with Depth and k = Constant on theBuckring Load Figure 10.5 shows a plot of buckling load P,, and length L of


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4 .4 04.004.20

3.80

-- 0 0 0 n h 0 Case bA A A - nh 100 Case d

the pile for a pile with EZ = 477 m2,(1), = 100 t/m 2 (case b Figure 10.4), nd(2) k, = 100 /m2 nh increasing from zero to 2000 t/m3 (case d). The bucklingmode changes from the first mode to the second and then to the third as thelength of the pile increases. The buckling load in general increases with increasein the value of nh, which is obvious.The minimum buckling load (in case d Figure 10.4)in a higher mode increasesas compared to the corresponding value in the previous mode. This behavioris distinctly different from the situation in which k was constant with depth,that is, in case b (Prakash, 1987).Similar behavior was observed with k , = 500,1000, and 2000 t/m2 (Prakash, 1985).Eflect of Increasing k, Values when n,=Constant on the Buckling LoadsFigure 10.6 shows a plot of buckling load P,, and length of pile L with EZ =477 /m2, ? t h = lo0 /m3 and k, increasing from zero to 2000t/m2. As in theprevious case, the buckling mode changes from the first mode to the secondand then to the third as the length of the pile increases. The buckling load ingeneral increases with the increase in the value of k, nh= constant), which isto be expected.The m inimum buckling load in a higher mode increases as compared to the


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FULLY E MBE DDED PILES 685i t ) X 1000)

4.404.204.003.803.603.403.203.002.802.602.402.202.001.801.601.401.201 oo0.800.60

0.0 0.20 0.40 0.60 0.80 1-00

Legendnun- k, 0 Case cA A A - k, 100 Case d0 0 0 - k, 500 Case d000 k, 1000 Case d000 - K O -2000 ase d

L m) X 10)Figure 10.6 Critical Load of case c and d for a pinned-pinned end Pile when n,, =100 t/m3 (Prakash, 1987).

corresponding value in the previous m ode. This is distinctly different than whenk was constant with depth and n h was zero that is, case b, Figure 10.5. Similarbehavior was observed for nh= 500 t/m3, lo00 t/m3 and 2000 t/m3 (Prakash, 1985).E$ect of Boundary Conditions on the Buckling Load In Figure 10.7,P,, has beenplotted against the length of the p ile for ko = 100 t/m2 and nh= 100 t/m3 (case dFigure 10.4) for two boundary conditions (i.e., pinned top-pinned tip (p-p) andfixed top-fixed tip (F-F)).It will be seen that the buckling load decreases sharplyas the length of the pile increases and attains a minimum value of 724t and1413t for p-p and F-F boundary conditions, respectively. The buckling loadsin the higher modes are larger in both cases. The mode shape in both casesdepends on the length of the pile (i.e., a s the pile length increases, higher bucklingmodes appear). The buckling loads are highest for boundary conditions F-Fand minimum for boundary conditions p-p.The above conclusions are more o r less in the realm of expectation. However,specific numerical values have been determined for the case mentioned above.Similar diagrams for k, = 100 /m2 and n h = 0 case b) and n h = 100 t/m3 andk , = O (case c) have been reported elsewhere (Prakash, 1985). Results asabove will become readily usable by field engineers when these are plotted in


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2.602.402.202.001.801.601.401.201.00

---------

0

4

,,I,

0.80

LegendUDD -F-FA A A - p - p

h a-- * I I I 1 I

L m ) X 10)Figure10.7 Critical load of case d for a pile with different boundary conditions whenk, = I00 and n, = 100t/m3 (Prakash, 1987).non-dimensional forms as in Figures 10.2 and 10.3 for case b and c (Figure 10.4)respectively.

10.2 PARTIALLY EMBEDDED PILESColumn instability is usually a problem in the design of structures su pported bypiles that are partially free standing. Furtherm ore, for structu res such as pierstha t are subjected to both vertical and lateral loads, a flexural analysis of the pilesmay control the design of the foundation. Generally, the analysis is highlyindeterminate and unwieldy unless some simplifying conditions are imposed(Dav isson and Robinson, 1965). In Figure 10.8, L, is the unsupp orted pile lengthabove the g roun d level. The vertical load tends to magnify the deflection causedby Q and M.Solutions for Constant k Davisson and Robinson (1965) have presentedsolutions for buckling loads of partially embedded piles. Th e axial load on the pileis constant and the pile is relatively long. In this analysis, j t has been assumed thatthe actual pile in Figure 10.8a is equivalent to a pile of length Le fixed at the t ip


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PARTIALLY EMBEDDED PILES 687

mnbase

Figure 10.8 Partially embedded pile (a) Actual Pile, (b) equivalent system (Davissonand Robinson, 1965).

(Figure 10.8b).The depth LL may be viewed as one tha t will ma ke the bucklingloads of the actu al system equ al to the equivalent system.By solving equation 10.1) for the freestanding length, the solution has beendeveloped in nondimensional form with the help of the following functions:L:R - R--L UJR = -R

L: = equivalent length of embedded portion of pile (Fig. 10.8Lu unsupported pile length

(10.13)

10.14)

and R is defined in eq uatio n 10.2 with L, = embedded length.


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I

\

IXa)

Figure 10.9 Nondim ensional representation of partially embedded pile (a) Actual pile,(b) equivalent system (Davisson and Robinson, 1965).

1.4Free, free ( f f


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Fixed-translating,

/Free, free ( F f 1

2.0

1.9

S 1.8

1.7

1.6

free ( fr-f

0 2 4 6 8 10 12JT

Figure 10.11 Dimensionless depth of fixity for buckling. Linearly varying k (Davissonand Robinson, 1965).(Reprinted by permissionof UniversityofToronto Press, Canada.)

103.1 Fully Embedded Pilesandwhere

P = Po(l - x / L)P = Po(1 - ( X Z / L 2 )

(10.18a)10.18b)

P o = load at pile headx = depth below surfaceL = pile length= parameter 0< I < 1 )

For I = 0, the pile is an end-bearing pile and for I(/ = 1, the pile is a friction pile.

10.3.2 Partially Embedded Piles

where(10.19)

L, total length of pile (L, +L ),Figure 10.8n = ratio of unsupported length to total length, LJ L, L,)


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r Z m = 4 p Fixed-fixed6 Fixed-pinned5 Fixed-free4321 Freefree

Fixed-freewith sway

0 0.25 0.5 0.75 1.0wfa

Fixed-freeno swayFixed-fixedFixed-freeno sway

Fixed-freewith swayFixed-freewit sway

- ------&.2.._g} Freefree10.25 015 0.;5 110

wfb

Figure 10.12 Effect of skin friction on buckling loads of fully embedded piles for (a)Constant soil modulus, (b) linear soil modulus (Reddy and Valsangkar, 1970).691


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0.50

In this case, the dimensionless length is defined as

Freefree- - 0

z = L, /T 10.20)Also, tb can be greater than one.For long piles Z,,, > 4), the variation of the dimensionless buckling loadsU,,= P, ,R2/EpIpand V,,= P,,T2/E,I, with J is shown in Figure 10.12 for fullyembedded piles. For = 0 and appropriate boundary conditions, the solutions in

2.0

1.5

2 1.0 Fixed-freewith sway0.5 1 Freefree

0.4167 0.8333 1.250 1.6667yl

Figure 10.13 Effect of skin friction on buckling loads forpartially embedded long pilesZ,,, = 4 for k = constant (a) n = 0.2, (b) n = 0.4 (Reddy and Valsangkar, 1970).

2.0Fixed-free

1.52 .0> 1.0

FreefreeO a 5 O a 5

Fixe freewith sway

reefree - 0J0.4167 0.8333 1.250 1.6667

ylb)

Figure 10.14 Effect of skin friction on buckling loads for partially embedded long pilesZ = 4 for k = l h x : (a) n = 0.2, b) = 0.4 (Rcddy and Valsangkar, 1970).


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REFERENCES 693Figures 10.2 and 10.12a agree (k constant with depth). S imilarly for k increasinglinearly with depth, the limiting solutions in Figures 10.3 and 10.12b agree. For$ greater than 0, considerable increase in the buckling load occurs because ofload transfer. For fixed-translating top and free bottom ft-f), the increase inbuckling load is about three times for friction piles ($ = 1) as compared to endbearing piles e=0) for k = constant. Corresponding solutions for partiallyembedded long piles Z,,,= 4) for constant k and n = 0.2 and 0.4 are shownin Figure 10.13(a) and (b). Similar solutions for k = n h * x and n = 0.2 and 0.4have been p lotted in Figure 10.14(a) and (b). The values of $ may be estimatedby a suitable distribution of skin friction.10.4 ROUP ACTIONModel tests by Toakley (1964) with groups of two and three strip piles in soft siltshowed the critical load is reduced by group action. How ever, full scale tests byHoadley et al. (1969) showed little interaction between closely spaced piles.In practice, both vertical and horizontal loads will act on a group of piles. Thechange in the value of k (soil modulus) due to group action was described inChapter 6. It is recommended that the same value of soil modulus be used forcomputing the buckling loads of piles in a group as for computing lateraldeflection. The presence of lateral load is equivalent to introduction ofeccentricity in the vertical load, which reduces the critical buckling load.

REFERENCESDavisson, M. T., Estimating Buckling Loads for Piles, Proceedings Second PanAmerican Conference on Soil Mechanics and Foundation Engineering, Sao Paulo, Vol. 1,Davisson, M. T. and Gill, H. L., Laterally Loaded Piles in a Layered Soil System,J . SoilMech. Found. Diu. Vol. 89, No. SM3, (1963), pp. 63-94.Davisson, M. T. and Robinson, K. E., Bending and Buckling of Partially EmbeddedPiles, Proceedings 6t h International Conference on Soil Mechanics and FoundationEngineering, Montreal, Canada, Vol. 2, (1965), pp. 243-246.

(1963), pp. 351-371.

Forsell, C., Berakning av palar 1918 Stockholm.Forsell, C., Knacksakerhet nos Palar Och Palgrupper Uppsal No. 10,Festskrift kungl.Vag-och Vattenbyggna-dskarem 1926, Stockholm.Grandholm, H.,On Elastic Stability of Piles Surrounded by a Supporting Medium, I n g .Vet. Akad., Hand. 89, (1929), Stockholm.Hetenyi, M., Beams on Elastic Foundations. University of Michigan Press, Ann Arbor

(1946).Hoadley,P. J., Francis, A. J., an d Stevens, L. J.,Load Testing of Slender Steel Piles in SoftClay, Proceedings 7 th International Conference on Soil Mechanics and FoundationEngineering, Mexico, Vol. 2, (1969). pp. 123-130.


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694 BUCKLING LOADS OF SLENDER PILESLee, K. L., Buckling of Partially Embedded Piles in Sand, J. Soil Mech. Found. Diu.,Poulos, H. . and Davis, E. H. ile Foundations. Wiley, New York 1980).Prakash, Sally, Buckling Loads for Fully Embedded Piles, M. S.Thesis University ofPrakash, Sally, Buckling Loads of Fully Embedded Piles, Int. J Computer Geotech.Reddy, A. S . and Valsangkar, A. J., Buckling of Fully and Partially E mbedded Piles, JToakley, A. R., The Behavior of Isolated and Gro up ofSlender Poin t Bearing Piles in Soft

ASCE, Vol. 94, NO. SM1, (1968), pp. 255-270.

Missouri-Rolla (1985).V O ~ ., (1987), pp. 61-83.Soil Mech. Found. Diu., ASCE, Vol. 96, No. SM6, (1970), pp. 1951-1965.Soil, M. . Thesis, University of Melbourne, (1964), Australia.

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