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1OT5206 Offshore Foundations Department of Civil & Environmental Engineering National University of Singapore Pile Driving Analysis Y K Chow 2 3One-Dimensional Wave Propagation in Pile Equilibrium equation (compression as positive) + = dx x PPP t um 2 2 dx x P t uAdx = ρ 2 2 AP σ= x u =ε ρ = density of pile material A = cross-sectional area of pile For a one-dimensional rod Axial strain is given by or where (2) (1) (3) 4Axial stress-strain relationship x uEE =ε =σ Hence from Eqns (1) to (4) 2 2 2 2 x uEA t uA = ρ 2 2 2 2 2 x uc t u = ρ = Ec Eqn (5) is generally known as the one-dimensional wave equation. “c” is the “celerity” or speed of sound in the material, or is simply referred to as the wave speed. For constant E and A, this gives 2 2 2 2 x uE t u = ρ or where (4) (5) 5Typical wave speed: 26 /10207 mkNE ×= 3/83.7 mt=ρ smc /5100≈ Time taken to travel from pile head to pile toe and back to the pile head: c Lt 2= st 067.0 5100 1702 = × = Steel : where L = pile length For example, take L = 170 m (say pile penetration of 50 m + mean water depth of 100 m + nominal 20 m from sea level to top of follower) Steel pile : 6General solution to 1-D wave equation f1(x-ct) = wave propagating in (+)ve x-direction (forward / downward) )()( ‘2 ‘ 1 ctxfctxf x u ++ = ( ) ( )ctxfctxfu ++ = 21 f2(x+ct)= wave propagating in (-)ve x-direction (backward / upward) Proof: )()( ”2”12 2 ctxfctxf x u ++ = )()( ‘2 ‘ 1 ctxcfctxcft u ++ = )()( “2 2” 1 2 2 2 ctxfcctxfc t u ++ = Substitute Eqn (7) into Eqn (5), – (6) – (7) [ ])()()()( “2”12″22″12 ctxfctxfcctxfcctxfc ++ =++ The expressions are identical on both sides of the equation, hence satisfying the wave equation 7( )ctxfu = 1 ( ) ( )[ ]ttcxxfu + += 1 Consider a forward / downward propagating wave at a given time, t At time t+ t , the wave has moved a distance x But x = c t Hence u = f1(x-ct) , i.e. wave shape remains unchanged, the wave has merely advanced a distance x = c t Solutions for velocity and stress : ( ) ( )ctxhctxfE x uE = = =σ 1 ‘ 1)()( 1 ‘ 1 ctxgctxcft uv = = = Obviously, v and σ also propagate with velocity c and do not change in shape in the absence of material damping 8Solution of 1-D Wave Equation 2 2 2 2 2 t u x uc = ( ) ( ) ↑++↓ = )(, 21 ctxfctxftxu ctxyyfctxfLet == );()( 11 Wave equation : General solution : ‘f y f;c t y; x y 1 11 = = = ctxzzfctxfLet +==+ );()( 22 ‘ 2 2;;1 f z fc t z x z = += = ‘ 2 ‘ 1: ffx uStrain = =ε ‘ 2 ‘ 1: cfcft uvvelocityParticle + = = (1) (2) (3) (4) 9No upward propagating wave, i.e. f2(x+ct) = 0 ↓ε= =ε ‘1f ↓= = vcfv ‘1 cvv += ↓ε ↓ = ε No downward propagating wave, i.e. f1(x-ct) = 0 ↑ε= =ε ‘2f ↑=+= vcfv ‘2 cvv = ↑ε ↑ = ε (5) (6) 10 Downward wave : F = EA ε = – EA f1’ Z c EA v F ==↓ : Upward wave : F = EA ε = – EA f2’ Z c EA v F = =↑ : where Z = pile impedance (7) (8) 11 Assuming the pile material remains elastic, the net force and net velocity at any location at a given time can be obtained by superposition of the downward and upward waves: F = F↓ + F↑ (9) v = v↓ + v↑ (10) From Eqns (7) and (8) F↓ = Z v ↓ F↑ = – Zv↑ (11) By combining Eqns (9) – (11), we can separate the downward wave from the upward wave if we know the total (net) force and velocity at a particular point along the pile 22 ZvFFZvFF ↑=+↓= 22 vzFvvzFv ↑=+↓= (12) 12 Boundary Conditions The following boundary conditions are considered: (i) free end (ii) fixed end Stress free boundary condition, i.e. net force at ‘b’, Fb = 0 0↑=+↓= FFFb ↓ ↑= FF A downward propagating compressive wave is reflected at the free end as an upward propagating tensile wave. Implications: Tensile stresses will develop during easy driving conditions. Follower may separate from main pile. Solution: Control drop height of hammer. or Free end (iii) impedance change (iv) external soil resistance 13 Force at ‘b’, ↓↑=+↓= FFFFb 2 A downward propagating compressive wave is reflected at the fixed end as an upward compressive wave. At the fixed end, the compressive stress is doubled. Implications: Potential problems with toe damage when driving piles into very hard stratum (rock), particularly when overburden soil is soft. Boundary condition, vb = 0 0↑=+↓= vvvb ↓ ↑= vv Z F Z F ↓ = ↑ ↓↑=FF or Fixed end 14 rit FFF += c EAZ =Impedance rit vvv += Let subscripts i denote incident wave r denote reflected wave t denote transmitted wave Impedance change At interface “b”, the net force and net velocity is given by the superposition of the incident and reflected wave Fb = Fi + Fr vb = vi + vr (18) This is equal to the transmitted force and velocity: (19) 15 Relationship between transmitted and reflected waves with the incident wave: rit vvv += 112 Z F Z F Z F rit = ( )rit FFZ ZFor = 1 2 Let β = Z2/Z1, then Ft = β ( Fi – Fr) From Eq (19), Fr = Ft – Fi Hence, Ft = β [ Fi – (Ft – Fi) ] or (β + 1)Ft = 2 β Fi it FF 1 2 +β β = (20) 16 +β β = = 1 1 2 iitr FFFF ir FF 1 1 +β β = Then, or (21) Hence, from Eqn (20), it vZvZ 12 1 2 +β β = it vv 1 2 +β =or (22) Similarly from Eqn (21) ir vZvZ 11 1 1 +β β = ir vv β+ β = 1 1 (23) 17 Notes : If an incident wave meets a section with a smaller impedance (β < 1) , the reflected velocity wave is of the same sign as the incident wave. If an incident wave meets a section with a larger impedance (β > 1) , the reflected velocity wave is of the opposite sign as the incident wave. The characteristic of the reflected wave and transmitted wave is entirely a function of the ratio of the impedance of the 2 sections. The analysis for pile with a change of impedance is useful for : (a) interpretation of pile integrity (b) selection of pile follower 1. 4. 3. 2. 18 trib FRFFF +=+= At the interface “b”, the net force and net velocity is given by trib vvvv =+= External Soil Resistance Consider now the effect of an external soil resistance (R) on the wave propagating in the pile. The soil resistance is usually in the form: R = ku + cv (24) (25) From Eqns (25) & (11), Z F Z F Z F tri = tri FFF = or (26) From Eqns (24) & (26) 2 )( RForFFRFF rriri = +=+ (27) 19 From Eqn (26) 2 RFF it = (28) The effect of an external soil resistance (R) on the propagating wave is to create a reflected wave of the same type as R with magnitude R/2 and a transmitted wave (due to soil resistance) of opposite type as R, also with magnitude R/2. From the relationship between force and velocity [Eqn (11)] Z R Z Fv rr 2 = = (29) Note that this reflected velocity has a similar effect compared to when an incident wave meets a section with an increase in impedance ( see Eqn (23) with β > 1 ) 20 One-dimensional wave equation model with soil resistance: Wave Equation Model ( )tP x uEAuk t uc t uA ss = + + ρ 2 2 2 2 pile inertia soil damping soil stiffness pile stiffness Conceptually, the soil is represented as a spring and dashpot. The inclusion of the soil increases the complexity of the problem. Hence, the above equation is generally solved using numerical methods: finite difference method finite element method method of characteristics Modelling of the pile is relatively straight forward. The main difficulty is modelling the soil behaviour. Note: More sophisticated 3-D wave equation model (Chow, 1982) is available that can simulate the pile and soil (especially) in a more rational manner but commercially 1-D wave equation computer program continues to be used 21Schematic in GRLWEAP 22 Typical “quake” value, Qu ( )JvRRD += 1 Soil Models )5.2(1.0 mminQu = Soil resistance during driving ( )mmtointoQu 105.24.01.0= Typical damping coefficient, (a) Smith (1960) Model Shaft : Toe : Soil type Jshaft Jtoe Clay Sand 0.656 0.164 0.033 0.492 ( )msJ / Parameters to define curve: Ru = max static resistance of soil spring Qu = “quake” value – limiting elastic displacement J = damping coefficient R = static soil resistance 23 (b) Lee et al. (1988) Model ss Gk 75.2= sss Grc ρπ= 02 Shaft (per unit length of pile shaft) : Pile toe : s s t v rGk = 1 4 0 s ss t v Gr c ρ = 1 4.3 20 Developed at the National University of Singapore. Theory based on vibrating pile in an elastic continuum. where Gs = soil shear modulus ρs = soil density vs = soil Poisson’s ratio r0 = pile radius The expressions above have physical representations (stiffness and radiation damping) and are characterized by parameters that can be determined in the laboratory. 24 Pile Drivability Analysis Pile drivability analysis is essential for the selection of appropriate hammer for the installation of offshore piles. Example of commercial wave equation program – GRLWEAP. Static Soil Resistance at time of Driving (SRD) The soil resistance at time of driving (SRD) will determine the depth to which a pile can be driven. Unplugged pile Soil column (plug) inside pipe pile moves up the pile during driving – assume that the internal shaft friction is less than end bearing capacity of soil plug. SRD = ∑ fsoAso + ∑ fsiAsi + qbAw (1) where fso = unit shaft friction during driving (outside); Aso = outside shaft area fsi = unit shaft friction during driving (inside); Asi = inside shaft area qb = unit end bearing pressure; Aw = annulus area of pipe pile 25 Plugged pile Soil column (plug) inside pipe pile moves down together with pile during driving – assume that the internal shaft friction is greater than end bearing capacity of soil plug. SRD = ∑ fsoAso + qbAp (2) where fso = unit shaft friction during driving (outside); Aso = outside shaft area qb = unit end bearing pressure; Ap = gross cross sectional area of pile toe 26 Unplugged pile Plugged pile External shaft friction External shaft friction Internal shaft friction End bearing pressure on pile wall End bearing pressure on soil plug & pile wall fsofsi qb fso qb 27 SRD (kN) 28 Notes: 1. Based on statics, it is generally assumed that the pile will behave in a manner which produces least resistance to pile penetration, i.e. at any depth, the SRD will be given by the lesser of Eqn (1) and (2). In reality, pile driving is a dynamic phenomenon and the pile generally behaves as unplugged during driving (especially during continuous driving) (Chow, 1982) 2. The unplugged mode in an all clay soil profile will generally give a higher resistance to pile penetration 3. In commercial wave equation program, the 1-D idealisation of the pile is unable to distinguish between the inside and outside shaft friction and model them as a soil resistance at a particular level of the pile. Although there are specially developed wave equation models that attempt to simulate the behaviour of the soil plug inside the pipe pile, t

hese are essentially 1-D model and are thus approximate. Proper modeling of the soil plug behaviour would require a truly 3-D wave equation model (Chow, 1982; Smith & Chow, 1982) 29 4. It should be noted that in drivability analysis for the selection of hammer, it is conservative to over-estimate the SRD. An underestimate of SRD can lead to premature refusal of piles and has serious cost implications because of the need to remobilize a larger hammer (pile refusal is often taken to be 150 blows per foot (0.3m) or 500 blows/m. 5. On the other hand, over-conservative estimate of SRD may result in piles being driven deeper than the theoretical estimate and may give the mistaken impression that the piles are under-capacity. 6. In offshore pile design, piles are driven to design penetration depth and not to “set” unlike onshore pile design. 30 (a) Unit shaft friction (fs) Clay: fs = cr Remoulded undrained shear strength (cr) – generally estimated from liquidity index based on Skempton & Northey (1952) or using following formula from Wood (1990): cr = 2 x 100(1-LI) kPa where liquidity index , LL = liquid limit, PL is the plastic limit, PI is the plasticity index, and w is the water content. Alternatively, cr = cu/S where S is sensitivity of clay – as a rule of thumb a value of 3 is sometimes used. Sand: K σv’ tan δ (similar to static value) – Limiting value in API applies For pipe piles without internal driving shoe, it is often assumed that: fsi = 0.5 fso (Sam & Cheung, 1993) (b) Unit end bearing pressure (qb) Generally assumed to be similar to static bearing capacity theory: Clay: qb = 9 cu Sand: qb = Nq σv’ where Nq = f(Φ) (API (limiting value applies) or Brinch Hansen) PI PLw PLLL PLwLI = = 31 32 Friction Fatigue Friction fatigue is the phenomenon where shaft resistance decreases as the pile penetrates deeper into the soil during driving. Heerema (1980) introduced this concept that the unit shaft resistance decreases as the piles are driven deeper into the soil. The exponential decrease in soil resistance postulated by Heerema (1980) can be seen in Figure below. Figure: Shaft friction resistance at varying depths (Heerema, 1980) 33 Alm and Hamre (2001) CPT based method incorporating Friction Fatigue Alm and Hamre (2001) proposed a CPT based method incorporating the concept of friction fatigue in sand and clay. General formulation for shaft friction along a pile during driving: s = s,res + (s,i s,res) k(d-p) where s = pile shaft friction (kPa) s,res = residual pile shaft friction (kPa) (function of CPT sleeve friction/tip resistance) s,i = initial pile shaft friction (kPa) (function of CPT sleeve friction/tip resistance) k = shape factor for degradation (function of cone tip resistance and effective overburden pressure) d = depth to actual layer (m) p = pile tip penetration (m) Details given in Alm and Hamre (2001). 34 Hammers Hydraulic or steam hammers used. For deep water depths, underwater hydraulic hammers are used. Southeast Asian waters (typical water depths: 60 – 110 m): Some hammers: (a) IHC90 (b) MHU 500 – 1700 (generally do not use pile cushion) (c) Menck 3000 – 4600 (generally use with pile cushion – bongossi wood) Pile diameter ranges from: 36 – 60 in (0.914 – 1.524 m) Pile wall thickness: 1.25 –2.0 in (32 – 50 mm) In the North Sea, much larger pile sizes are used: Diameter of 84 in (2.13 m) and 96 in (2.43 m) with wall thickness of 2.5 in (63 mm) and 3 in (76 mm) are common 35 Cap Block and Pile Cushion Behaviour Hysteretic behaviour of cap block and pile cushion. Hysteresis (a measure of energy loss): inputenergy outputenergy ABCArea BCDAreae ==2 where e = coefficient of restitution 36 Initial Condition for Computer Program Computer program uses an initial impact velocity assigned to the hammer as the starting condition. Potential energy of hammer is converted to kinetic energy: hgmevm f= 2 2 1 where m = mass of hammer h = hammer stroke or drop height (or equivalent stroke for double acting hammer) ef = efficiency of hammer v = impact velocity of hammer fghev 2= This efficiency, ef, is not to be confused with the measured energy in the pile. Impact velocity of hammer 37 Definition of Pile Penetration per blow (Set) Smith (1960)’s soil model: Pile penetration per blow = δmax – Qu Most computer programs stop computation when the pile toe velocity becomes zero. NUS computer program (and soil model) compute the true set, i.e. gives the final penetration of the pile toe when it comes to rest. 38 Driving Stresses The wave equation program also gives the driving stresses in the pile. The maximum driving stresses should be kept within reasonable limits. Drivability Curves: Blow count versus Depth The blow count versus depth curves should be produced for various hammers to determine suitable hammers to be used for the pile installation Set-up or Relaxation The driving of piles in clay (particularly soft clay) results in the generation of excess pore water pressure. Subsequent consolidation will result in gain in soil strength. Thus if the driving process is interrupted, the soil will exhibit set-up effects, hence driving will be more difficult. Driving in dense sand may give rise to an opposite phenomenon – “relaxation”. A decrease in driving resistance is possible. 39 Pile Refusal Review hammer performance – particularly helpful if pile installation is monitored using a Pile Driving Analyser (PDA) which will allow driving energy to be monitored and compared against rated energy by hammer manufacturer Review SRD used in pile drivability analysis Re-evaluate design penetration – reconsideration of design loads and design soil parameters. Last recourse (very expensive operation) – removal of soil plug inside pile by drilling to reduce driving resistance and continue driving. It may be necessary to subsequently have a grout plug to enable the development of the toe bearing capacity. In more serious cases, it may be necessary to drill past the pile toe to enable driving to continue. 1. Pile refusal is often taken to be 150 blows per foot (per 0.3m) or 500 blows/m 2. When a pile refuses before it reaches design penetration, one or more of the following actions can be taken: 40 41 42 43 44 45 SRD (kN) A typical GRLWEAP output on drivability for another project 46 Methods used to estimate the pile bearing capacity : Dynamic Pile Testing (High Strain Test) (a) Efficiency of piling hammer in driven piles (b) Driving stresses in driven piles (c) Assessment of pile integrity (d) Bearing capacity and load-settlement response of pile (a) Case Method (b) Stress-Wave Matching Technique Objectives – To obtain: Test method: During the impact of the hammer, the stress waves are measured using strain transducers and accelerometers mounted on the pile (at least 1.5 diameter away from the pile head). The force trace is obtained from the strain measurements. From the acceleration trace, the velocity trace is obtained by numerical integration. 47 48 PDA Model PAX Accelerometer and strain transducer mounted on a pile (above water) Underwater strain transducer and accelerometer mounted on pile. 49 Typical force and velocity versus time curves for an offshore pile (Vel Msd above is actually pile impedance Z x Vel Msd) 50 Case Method ( ) ( ) + + ++= c LtvtvZ c LtFtFR 2 2 2 2 1 1111 ( )( )RtFJRR cs = 12 Assuming that all the soil damping is concentrated at the pile toe, the static component or bearing capacity of pile under static load is given by Suggested damping factor, Jc From the force and velocity versus time curves, the total soil resistance (includes both static and dynamic components) is given by Sand : 0.1 – 0.15 ; Silt : 0.25 – 0.4 ; Clay : 0.7 – 1.0 “Correct” Jc value obtain from correlation with static load test or stress wave matching analysis. Silty Sand : 0.15 – 0.25 Silty Clay : 0.4 – 0.7 where t1 is generally taken as the time when F(t1) is maximum and Z is the pile impedance (= EA/c) 51 Available computer programs : Stress-Wave Matching Technique The force-time history or velocity-time history is used as a boundary condition in a wave equation computer program. For instance, if the velocity-time history is used as the input, the wave equation program computes the force-time history and this is compared with the measured values. The soil resistance, soil stiffness and damping values are adjusted iteratively until the computed and measured values agree closely or until no further improvements can be made. When this stage is reached, the soil parameters used in the wave equation model are assumed to be representative of those in the field. The bearing capacity of the pile and the load-settlement response are then determined. CAPWAPC TNOWAVE NUSWAP 52 Notes: 1. The test results are representative of the conditions at the time of testing. For instance in the case of driven piles tested at the end of driving in clay soils, the capacity obtained is generally a lower bound. Pile should be retested a few days after pile installation to allow set- up to occur. 2. If the impact energy used during testing is insufficient to move the pile adequately, the pile capacity obtained may be a lower bound. The capacity obtained is actually the mobilised static resistance. 53 54 55 56 57 58 59 60 61 62 63 64 65 66 References Alm, T. & Hamre L., (2001), Soil model for driveability predictions based on CPT interpretations, Proceedings of the International Conference on Soil Mechanics and Geotechnical Engineering, Balkeema, CONF 15; VOL 2, pp. 1297-1302. Chow, YK (1982) “Dynamic behaviour of piles”, PhD Thesis, University of Manchester, UK Heerema, E. P. (1980). Predicting pile driveability: Heather as an illustration of the friction fatigue theory. Ground Engng 13,15–37. Lee, SL, Chow, YK, Karunaratne, GP and Wong, KY (1988) “Rational wave equation model for pile driving analysis”, Journal of Geotechnical Engineering, ASCE, 114, No 3, pp 306-325. Sam, MT and Cheung LY (1993) “Installation of a 479 feet water depth platform in South China Sea”, Proc 3rd International Offshore and Polar Engineering Conference, Singapore, pp 288-293 Skempton, AW and Northey, RD (1952) “The sensitivity of clay”, Geotechnique, Vol 3, No 1. Smith, EAL (1960) “Pile driving analysis by the wave equation”, Journal for Soil Mechanics and Foundations Division, ASCE, 86, SM4, pp 35-61. Smith, IM and Chow, YK (1982) “Three-dimensional analysis of pile drivability”, Proc 2nd International Conference on Numerical Methods in Offshore Piling, Texas, Austin, pp 1-19. Wong, KY (1988) “A rational wave equation model for pile driving analysis”, PhD Thesis, National University of Singapore. Wood, DM (1990) “Soil behaviour and critical state soil mechanics”, Cambridge University Press.