ABSTRACT: This paper presents an interlock friction model for single and double U sheet piles which canbe used to predict the structural losses in double U sheet piles due to oblique bending from results of strainmeasurements in single U sheet piles. Some applications of old cases from literature are given.
1 INTRODUCTION
Traditionally, the section modulus and the moment ofinertia of steel sheet piles are related to the centralbending axis of the continuous wall, resulting in thehighest structural resistance. For Z-piles this is neverconsidered as a problem but for U-piles, where theclutches are positioned at the neutral axis of a contin-uous sheet pile wall, lack of shear force transmissionin the interlocks results in a significant decrease ofstructural resistance of sheet pile walls.
U profiles are either installed one by one (singleprofiles) or in pairs where the common interlock iscrimped or welded (double piles); occasionally tripleU profiles are applied. Figure 1 shows these three typesof U profiles where the principal axes of inertia aredrawn in the cross-section.
Lack of shear force transmission capacity in theinterlocks may cause a significant decrease of thestructural resistance. In the extreme case, if the shearfriction capacity of the interlocks is completely absent,single, double and triple U profiles behave as follows:
• Single U-profiles bend about the principal axis ofinertia of an individual pile, which involves aneffective moment of inertia of ca. 30% and an effec-tive section modulus of ca. 50% of the highestcross-sectional resistance
• Double U-profiles have an inclined principal axisof inertia, which causes a deflection both perpen-dicular to the wall and in plane of the wall (oblique
Figure 1. Principal axes of inertia for different U profiles.
bending). The moment of inertia is ca. 50% and theeffective section modulus is ca. 60% of the highestcross-sectional resistance
• Triple U-profiles bend about the principal axiswhich causes in extreme cases a reduction up toca. 90% for the moment of inertia and of ca. 80%for the section modulus
According to Eurocode 3 part 5 reduction factorsshould be applied to the section modulus and to themoment of inertia to account for lack of shear forcetransmission. However, a lot of discussion exists aboutthe magnitude of these factors, especially for doubleU piles because for this type of piles insufficient fieldmeasurements are available where the rotation of thebending axis was measured. On the other hand a lot ofdata from field measurements are available in litera-ture for sheet pile walls composed of single U profiles,e.g., Hebert et al. (1978), Gigan (1979), Gigan (1984).
In this paper it is investigated if field data of singleU piles can be used to predict the effect of interlockfriction to the behaviour of double U profiles.
2 OBLIQUE BENDING
In steel sheet piling oblique bending is a synonym forthe more common term, bending in two directions.Oblique bending is relevant for a double U profile,where the product of inertia Ixy �= 0. When the result-ing load on the sheet pile is perpendicular to the planeof the wall, the neutral axis is rotated with angletan γ ∗ = −Ixy/Iy. The rotated neutral axis involves adisplacement perpendicular to the neutral axis, see
However, more important is that due to the rotatedneutral axis, the effective height of the beam, i.e., thedistance between neutral axis and outermost fibre, isdecreased, which involves a significant loss of stiffnessand strength of the sheet pile.
This loss of structural resistance should be takeninto account in the design when working with planestrain models. The effective moment of inertia Ieff andthe effective section modulusWeff can be derived usingthe reduction factors βI and βW, where
In practice a sheet pile wall is composed of a seriesof U-piles threaded together. Generally guide framesare used to keep the sheet piles in the correct posi-tion but deviations may occur. Due to possible rotationin the clutches, geometrical tolerances of the cross-section, or slightly bent piles, high contact forces canbe generated between the clutches of individual piles,causing high resistance against slipping interlocks. Inaddition, interlock friction may be generated by intru-sion of soil particles in the interlocks voids duringsheet pile installation: intrusion of sand grains mightresult in a different frictional behaviour than that ofclay particles.
The rotation of the neutral axis, γ in Figure 2,decreases when slipping in the interlocks is impededor when shear resistance of the ground against in planebending is generated, and the structural resistance ofthe double U-pile increases.
In general the amount of structural losses due tooblique bending is influenced by (Kort 2002):
• soil-structure interaction• detailing of struts and walings• interlock friction
The next sections focus on the effect of interlockfriction on bending of single and double U profiles.When it can be assumed that structural losses of sin-gle U pile walls due to lack of interlock friction areimputed solely to installation effects, field data fromsingle U pile walls can also be applied to double U-pilewalls to estimate the contribution of interlock frictionto the resistance against oblique bending.
Figure 2. Oblique bending, where the neutral axis is rotated.
3 INTERLOCK FRICTION MODEL
3.1 Lohmeyer
Lohmeyer (1937) developed a general model for inter-lock friction, see Figure 3, which can be applied bothto single and to double U profiles. This model is stillapplied in the engineering practice but unfortunatelyLohmeyer’s solution does not satisfy the equilibriumconditions.
Equilibrium of moment and normal force in thecross-section requires a stress distribution in the cross-section, σ (x∗,y∗), according to
Here is M = the bending moment about the x∗-axis; Ix∗and Iy∗ = moments of inertia about the x∗- and y∗-axis,respectively; As = the area of the single U pile; and Tsand Td = forces modelling interlock friction. Suffixess and d refer to single and double profiles.
Single U profiles are modelled when Td = 0, dou-ble U piles without interlock friction when Ts = 0,and double U piles with interlock friction when0 <Td <Ts, where the neutral axis crosses the com-mon interlock. Based on (3) interlock friction modelsfor single and double U profiles are developed.
3.2 Interlock spring model for single U profiles
Substitution of σ = 0, Td = 0 and y∗ = y∗0 in (3) gives
the friction force in one interlock Ts required to shiftthe neutral axis over distance y∗
0,
The stress distribution in the cross-section and thecurvature of the pile are, respectively
Figure 3. Definitions in the cross-section of a singleU-profile, after Lohmeyer (1937).
where y∗0 = the distance from the principal axis of iner-
tia of the single pile to the neutral axis with the effect ofinterlock friction. Substitutions of y∗
0 = 0 and y∗0 = cs
give the well-known solutions for the single wall andthe continuous wall, respectively.
It follows from the kinematics of a slipping inter-lock (Figure 4) that
where u = the relative slip distance in the interlock; zis an element length along the pile; φ = the rotation ofthe pile along z; and ki = the spring stiffness of theinterlock.
Using φ = κx, equations (8) and (9) are derived:
Equation (9) gives ki = 0 for y∗0 = 0 (no interaction)
and ki → ∞ for y∗0 = cs (full interaction), and these
results may be expected.The element in Figure 4 can be considered as a nodal
rotation spring, defined by
Equation (10) can be implemented in a subgradereaction model; see e.g., Kort (2003).
The reduction factors βI and βW can be derived as afunction of the shift of the neutral axis, using (5) and(6). The interlock friction factor � is introduced
Figure 4. Kinematics of a slipping interlock for single Upiles.
where � = 0 represents perfectly frictionless inter-locks and � = 1 inplies a continuous wall.
The reduction factors βI and βW are expressed by
where EA and EIx refer to the axial and bending stiff-ness of the continuous wall, and where σmax;� is themaximum stress in the cross-section as a function of�, occurring either in the interlock or in the flange ofthe profile.
Figure 5 shows typical interaction relationships fora light PU8 profile and a heavy L607K profile, whereT =Ts: βI versus �, βW versus �, T versus �, and Tversus u. The relationships βI versus � and βW versus� show that the effective stiffness of a row of U-pilesdecreases rapidly when the piles do not fully coop-erate. The normalized interlock force T/T� = 1 mayreach bigger values than unity but this does not have asignificant effect on the reduction factors.
The sudden change of βW near � = 0.2 is caused bythe location of the maximum stress: for smaller valuesof � the maximum stress is in the interlocks, and forlarger values σmax occurs in the flange.
3.3 Interlock spring model for double U profiles
A row of threaded double U piles subjected to a loadperpendicular to the plane of pile row (y-direction)bends about a neutral axis inclined with angle γ ; seeFigures 1 and 2. When the piles are subjected to inter-lock friction, internal forces in the interlock T =Td −Ts are generated; see Figure 3 where for the doubleU profile the neutral axis crosses the left interlock.The interlock force involves a reduction of the rotationangle of the neutral axis γ . It is noted that this angle
may also be reduced by soil-structure interaction orby structural components, e.g., walings or a cappingbeam; these external influences are taken into accountby the bending moment about the y-axis, My. An addi-tional bending moment, My, is generated by theinternal forces of interlock friction:
where bd = the width of the double U profile. Thebending moment about the x-axis, Mx = M, is gener-ated by external loads and is therefore not influencedby interlock friction. Moment equilibrium in the cross-section of the double U profile requires a stressdistribution described by
where
The neutral axis is the axis where the axial stresses σare zero and is the axis about which the cross-sectionbends. Neglecting the effects of axial strain, ε = 0, theneutral axis is described by
For the special case of a double U profile withoutsoil-structure interaction and without interlock frictionwhereT = 0 and My = 0 the neutral axis is rotated withangle γ ∗, for which tan γ ∗ = − EIxy/EIy as is shownin Figure 2.
Substitution of (14), (16) and (17) into (18) givesthe force in an individual interlock as function of therotated neutral axis
Curvature κx as function of the rotated neutral axisfollows from substitution of (14) and (19) into (16):
The reduction factor for the stiffness βI is defined by
and the stress distribution in the cross-section as func-tion of γ follows from substitution of (18) and (20) in(15):
The reduction factor for the section modulus βW canbe expressed similar as for (13) by
where the minimum value following from any (x,y)coordinate on the cross-section is critical.
Figure 6 shows typical relationships of βI, βW andT versus the rotated neutral axis for a light PU8 profileand a heavy L607K profile where My = 0.The rotationof the neutral axis is normalized by 1 − γ /γ ∗ where1 − γ /γ ∗ = 0 implies double U piles without inter-lock friction and 1 − γ /γ ∗ = 1 represents the state ofa continuous wall.
Figure 7 shows a top view of an interlock rotationspring. It can be derived that
Figure 6. Interaction of double U profiles.
∆z
My+ My+ My- My- t
bd
½ ki
T = ½ ki∆z ½ bdφy
�φy
T
T
∆My = ¼ ki∆z bd2φy
t
Figure 7. Nodal interlock spring model for a double Uprofile (top view).
Substitution of (14), (16) and (17) into (24) gives
Equation (25) behaves as may be expected, as substi-tution of γ = 0 (full interaction) gives ki → ∞ andsubstitution of tan γ = − EIxy/EIy and My = 0 (nointeraction) gives ki = 0.
This interlock spring model can be implemented ina subgrade reaction model; see e.g., Kort (2003).
4 RELATIONSHIP BETWEEN SINGLE ANDDOUBLE U PROFILES
4.1 Based on interlock friction models
Figure 8 shows the relationship between reductionfactors for interlock friction of single and double Uprofiles, derived for a simply supported beam with auniformly distributed load. Each diagram shows twocurves for two different profiles: one curve is based onequal ki, modeling interlock friction gradually increas-ing with interlock slip, and the other curve is based onequalT, modeling a certain force to be overcome beforethe interlocks begin to slip.
The curves with equal T in Figure 8 suggest thatoblique bending cannot occur when the reduction fac-tors for the single U profiles are about βI > 0.5 andβW > 0.65, whereas for the same range, the curveswith equal ki imply that oblique bending is of signifi-cance. The question is of course which curve is closerto the “truth”. Some typical physical observations aredescribed:
• Vanden Berghe et al. (2001) measured typically aclose to elastic perfectly plastic behaviour of theinterlock spring in laboratory tests on interlockfriction in dry sand.
• Figure 9 shows the effect of arching that can occurbetween every second pile. Evidence of this mech-anism was measured by DiBiagio (1975) and byGigan (1979) and also in the Rotterdam sheet pile
Figure 8. Relationship between single and double U pro-files. The • and × symbols are indicators, not data points.
wall field test (Kort & Van Tol 2002) a typical posi-tion of the interlocks of all single and double U pileswas observed as indicated in the figure. Schmitt(2002) observed in laboratory tests that the inter-lock friction capacity significantly increases wheninterlocks are prestressed.
In practice resistance against slipping interlocksmay be generated by the following mechanisms:
• The shape or type of an individual interlock (e.g., aLarssen type)
• Sequence of pile driving (e.g., pitch and drive)• High contact stresses in the interlocks due to bad
driving tolerances or slightly bent piles (this maynever be a design parameter)
• Intrusion of soil particles in the interlock voids• Increase of contact stresses due to arching (see
Based on these observations it cannot be concludedthat one of the curves plotted in Figure 8 is better thanthe other; the real behaviour is probably bounded by thetwo curves.
4.2 Based on field measurements
In the Rotterdam sheet pile wall field test field mea-surements were performed on double and single Uprofiles (Kort 2002; Kort & Van Tol 2002). The twoprofiles, however, are not directly comparable, because(1) strain measurements were only performed in thedouble U profile, (2) the single U sheet piles were 2 mlonger than the double U sheet piles and (3) the dou-ble U piles were composed of L607K profiles and thesingle U piles from LX32 profiles; these profiles havemore or less the same geometry and properties. Nev-ertheless Kort & Van Tol (2002) used the followingrelationship between single and double U piles fromthe maximum deflections measured, wmax:
It is noted that the factor βI;double is derivedfrom field measurements and includes the effects
Figure 9. Effect of soil arching on interlock friction.
Figure 10. Field data for single and double U profilesderived from the Rotterdam Field Test and a practicalrecommendation.
of soil-structure interaction and of detailing of wal-ings: βI;double = βI;0 + βI;double;soil+walings + βI;double;
interlock, where βI;0 = the basic reduction factor foundby substituting γ = γ ∗ into (21). When only the contri-bution of interlock friction is considered, then βI;doublewould be smaller, wmax;double would be larger, andwmax;single would be equal, meaning that βI;single mightbe approximately equal.
Figure 10 shows the diagram of βI;double versusβI;single derived from 15 measurements points usingsheet pile data of the L607K profile.
Based on the field measurements in three double Upiles was estimated that βI;double;soil+walings = 0.1 andβI;double;interlock = 0.1 throughout the entire test (Kort2002). When βI;single is equal and βI;double is reducedwith βI;double = 0.1 to correct for the soil structureinteraction and the presence of a waling, Figure 10 sug-gests that the mechanism of equal ki is more likely thanthat of equal T. However, it is noted that this conclu-sion is based on too many statements and assumptionsthat cannot be verified.
4.3 A tentative practical recommendation
A tentative practical recommendation to predict thebehaviour of double piles from that of single piles isindicated by the dashed line in Figure 10.
This recommendation should be verified andupdated using full scale field tests or large scale labo-ratory tests, where the sheet piles should be installedin a realistic manner and the strain distribution in boththe double and the single U profile should be moni-tored using four strain gauges at the level of expectedmaximum deflection.
5 DATA FROM OTHER FIELD TESTS
Strain measurements on quay walls built up of sin-gle U piles and backfilled with sand were performed
Table 1. Contribution of interlock friction to the stiffnessof single U Larssen profiles in backfills and prediction ofβI;double.
Case Profile βI;0;single � βI βI;0;double βI;double
by Hebert et al. (1978) and Gigan (1979, 1984).Based on the strain measurements reported in thesepapers, �, βI;single, βI;single;interlock and βI;double wereestimated using (6), (11), (12), and Figure 10; seeTable 1.
It can be concluded from this table that inter-lock friction is not a promising parameter to con-sider in the design of double U sheet pile walls inbackfills.
6 CONCLUSIONS AND RECOMMENDATIONS
A simple interlock friction model for single and doubleU sheet piles is presented which can be used to predictthe structural losses in double U sheet piles from var-ious field measurements in single U sheet piles. Themodel is based on field measurements and satisfiesthe limits from theoretical derivations.
To obtain more insight in the oblique bendingphenomenon it is recommended to verify the modelusing field measurements of strain distributions ondouble and single profiles loaded under the sameconditions.
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