Congreso Internacional de Fundaciones Profundas de Bolivia Santa Cruz, Bolivia, 12 al 15 de Mayo de 2015 Day 1: Software Demonstrations

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1 Applications of Stress Wave Theory to Deep Foundations with an Emphasis on The Wave Equation (GRLWEAP) Congreso Internacional de Fundaciones Profundas de Bolivia Santa Cruz, Bolivia, 12 al 15 de Mayo de 2015 Day 1: Software Demonstrations Frank Rausche, Ph.D., P.E., D.GE - Pile Dynamics, Inc. 1 CONTENT Introduction Dynamic Formula Static Formula The One Dimensional Wave Equation and Wave Demonstrations Wave Equation Models Bearing Graph and Driveability Example Conclusions 2

2 WAVE EQUATION OBJECTIVES Smith s Basic Interest: Allow for realistic stress calculations Replace Unreliable Energy Formulas Use improved models elastic pile elasto plastic static resistance viscous dynamic (damping) resistance detailed driving system representation 4 Wave Demonstrations Slinky Pendulum Buddies Shear Waves Compressive Waves 5

3 WAVES Example of a Baseball Wave Animation courtesy of Dr. Dan Russell, Kettering Univ. 6 Example of a Shear Wave Animation courtesy of Dr. Dan Russell, Kettering Univ. 7

4 Example of a Compressive Wave Animation courtesy of Dr. Dan Russell, Kettering Univ. 8 The 1-D Wave Equation f ρ(δ 2 u/ δt 2 ) = E (δ 2 u/ δx 2 ) E elastic modulus ρ mass density g x with c 2 = E/ ρ... Wave Speed Solution: u = f(x ct) + g(x+ct) x length coordinate t... time u displacement 9

5 t Time Waves in a Pile The compression wave,induced by the hammer at the pile top, moves downward a distance c t during the time interval t. x 10 t Time t + t Waves in a Pile The compression wave, induced by the hammer at the pile top, moves downward a distance c t during the time interval t. C t x 11

6 t Time t + t Waves in a Pile The compression wave, arrives at the pile toe where it is reflected (on a free pile in tension) Wave Mechanics for Pile Testers THE Wave Equation ρ(δ 2 u/ δt 2 ) = E (δ 2 u/ δx 2 ) E elastic modulus ρ mass density with c 2 = E/ ρ Wave Speed u x Solution: u = f(x-ct) + g(x+ct) x length coordinate t... time u displacement 13

7 Wave Mechanics for Pile Testers The Solution to the Wave Equation u = f(x-ct) + g(x+ct) Time t Time t + t C t f f g g C t x x 14 The first instant after impact Time t o Force, F Time t o + t Point A u Point A, like all other points along the pile, is at rest at time t o (when contact between ram and pile top occurs) Compressed distance, L 15

8 Force Velocity Proportionality Wave travels distance L = c t during time t F u is the displacement of a point of pile during time t L u Particle Velocity, v = u/ t but u = ε L and therefore v = ε L / t and with wave speed c = L / t: v = ε c 16 Strain-Stress-Force Proportionality Wave travels in one direction only ε d = v d / c F d = v d (EA/c) d = v d (E/c) This is the strain, stress, force-velocity proportionality Z = EA/c is the pile impedance (kn/m/s) 17

9 Express Your Impedance Z = EA/c kn/(m/s) with c = (E/ρ) 1/2 Z = A (E ρ) 1/2 with E = c 2 ρ with M p = L A ρ Z = A c ρ Z = M p c/ L (M p... pile mass) The Pile Impedance is a force which changes the pile velocity suddenly by 1 m/s. Reversely, if the velocity changes by 1 m/s then pile will develop a force equal to Z. 18 A Quick Look at Energy Formulas Energy Dissipated in Soil = Energy Provided by Hammer R u (s + s l ) = ηw r h s l lost set (empirical or measured), η efficiency of hammer/driving system Engineering News: R allow = W r h / 6(s + 0.1) 19

10 The Gates Formula R u = 7 (W r h) ½ log(10blows/25 mm) 550 R u Nominal Resistance (kn) W r ram weight (kn) h actual stroke (m) log logarithm to base The Hiley Formula using Set-Rebound Measurements R u = ηw r h (W r + e 2 W P ) (s + c/2) (W r + W P ) Considers combined pile soil elasticity effect Usually with F.S. = 3; η = hammer efficiency. Rebound: c Set = s 21

11 Bearing Graphs from 2 Energy Formulas Hammer D 19-42; E r = 59 kj [900] 3500 R u -kn 3000 [kips] 2500 Capacity in kn [450] R u = ηe r /(s + s l ) η = 1/3; s l = 2.5mm R u = 1.6 E p ½ log(10blows/25mm) 120 kn Blows/0.25 m Blows/25mm Gates - w/ calculated Stroke ENR - Ru = Rd x 2 22 Shortcomings of Formulas Rigid pile model Poor hammer representation Inherently inaccurate for both capacity and blow count predictions No stress results Unknown hammer energy Relies on EOD Blow Counts 23

12 Static Formulas Based on Soil Properties Always done for any deep foundation type Backed up by Static or Dynamic Testing 24 Static Analysis to Calculate LTSR Basically for All Soil Types: R u = R u,shaft + R u,toe R u = f s A s + q t A t f s, R u,shaft, A s Shaft Resistance/Area q t, R u,toe, A t End Bearing/Area 25

13 The β-method for Cohesionless Soils R u,shaft = f s A s f s = k o tan(δ) p o p o is the effective overburden pressure k o is some earth pressure coefficient β = k o tan(δ) R u,toe = N t p o A t N t is a bearing capacity factor All with Certain Limits 26 The α-method for Cohesive Soils R u,shaft = f s A s f s = α c c is the undrained shear strength α is a function of p o R u,toe = 9 c A t... with certain limits GRLWEAP provides 4 different static analysis methods ST based on Soil Type; SA based on SPT N; CPT; API 27

14 GRLWEAP: ST Method Non-Cohesive Soils (after Bowles) Soil Parameters in ST Analysis for Granular Soil Types Soil Type SPT N Friction Angle Unit Weight, γ β N t Limit (kpa) degrees kn/m 3 Q s Q t Very loose Loose Medium Dense Very Dense ST - INPUT 29

15 GRLWEAP: ST Method Cohesive Soils (after Bowles) Soil Parameters in ST Analysis for Cohesive Soil Types Soil Type SPT N Unconfined Compr. Strength Unit Weight γ Q s Q t kpa kn/m 3 kpa kpa Very soft Soft Medium Stiff Very stiff hard ST - INPUT 31

16 The Wave Equation Model The Wave Equation Analysis calculates The displacement of any point along a slender, elastic rod at any time durting and after impact From the displacements forces, stresses, velocities The calculation is based on rod properties: Length Cross Sectional Area Elastic Modulus Mass density 32 The Wave Equation Model The Wave Equation Analysis calculates The displacement of any point along a slender, elastic rod at any time durting and after impact From the displacements forces, stresses, velocities The calculation is based on rod properties: Length Cross Sectional Area Elastic Modulus Mass density 33

17 GRLWEAP Fundamentals For a pile driving analysis, the slender, elastic rod consists of Hammer+Driving System+Pile Hammer D.S. The soil is represented by resistance forces acting on the pile and representing the forces in the pile soil interface Pile 34 Smith s Numerical Solution of the Wave Equation ρ(δ 2 u/ δt 2 ) = E (δ 2 u/ δx 2 ) E elastic modulus ρ mass densitywith c 2 = E/ ρ... Wave Speed Closed Form Solutions to the wave equation are not practical; we therefore solve the equation numerically: (m i /k i )(u i,j+1 2u i,j + u i,j 1 )/Δt 2 = (u i+1,j 2u i,j + u i 1,j ) This is equivalent to considering mass points and springs! L i-1 i i+1 35

18 The GRLWEAP Pile Model Each segment has a mass and spring stiffness of length L 1 m (3.3 ft) with mass m = ρ A L and stiffness k = E A / L there are N = L / L pile segments which allow us to solve the wave equation numerically. L 36 The Pile Model Relationship between the uniform pile and the lumped mass model properties: m k = (ρ A L)(EA/ L) = A 2 Eρ = Z 2 [kn s/m] 2 m/k = (ρ A L)/(EA/ L) = (ρ/e) L 2 = ( L/c) 2 [s] 2 Or Z = (mk) 1/2 (pile impedance) and t= (m/k) 1/2 (wave travel time) Note: the smaller L, the smaller L and that means the higher the frequencies that can be represented. 37 L

19 We can model 3 hammer-pile systems 40 External Combustion Hammer Modeling Cylinder and upper frame = assembly top mass Ram guides for assembly stiffness Drop height Ram: A, L for stiffness, mass Hammer base = assembly bottom mass 41

20 External Combustion Hammer Model Ram modeled like rod Stroke is an input (Energy/Ram Weight) Impact Velocity Calculated from Stroke with Hammer Efficiency Reduction: v i = (2 g h η) ½ Assembly also modeled because it may impact during pile rebound Note approximation in data file: Assembly mass = Total hammer mass Ram mass 42 External Combustion Hammers Ram Model Ram segments ~1m long Combined Ram H.Cushion Helmet mass 43

21 External Combustion Hammers Assembly model Assembly segments, typically 2 Helmet mass 44 External Combustion Hammers Combined Ram Assembly Model Ram segments Assembly segments Combined Ram- H.Cushion Helmet mass 45

22 External Combustion Hammer Analysis Procedure Static equilibrium analysis Dynamic analysis starts when ram is within 1 ms of impact. All ram segments then have velocity V RAM = (2 g h η) 1/ g g is the gravitational acceleration h is the equivalent hammer stroke and η is the hammer efficiency h = Hammer potential energy/ Ram weight 46 External Combustion Hammer Analysis Procedure Dynamic analysis ends when Pile toe has rebounded to 80% of max d toe Pile has penetrated more than 4 inches Pile toe has rebounded to 98% of max d toe and energy in pile is essentially dissipated 47

23 Diesel Hammers Very popular in the US Have their own fuel tank and combustion engine Model therefore includes a thermodynamic analysis Stroke is computed 48 GRLWEAP hammer efficiencies η h = E k /E P The hammer efficiency reduces the impact velocity of the ram; it is based on experience Hammer efficiencies cover all losses which cannot be calculated Diesel hammer energy loss due to pre compression or cushioning can be calculated and, therefore, is not covered by hammer efficiency 58

24 Measured Transferred Energy Max E T = F(t) v(t) dt (EMX, ENTHRU) W R W R h η T = ENTHRU/ E R (transfer ratio or efficiency) E R = W R h Manufacturer s Rating Measure: Force, F(t) Velocity, v(t) 60 Measured Transfer Ratios for Diesels Steel Piles Concrete Piles 62

25 For all impact hammers GRLWEAP needs impact velocity m R E r = W r h e = m r g h e v i W R h h e = E r / W r h e equivalent stroke h e = h for single acting hammers E pr = η E r W r h e (η = Hammer efficiency ) E k = E pr = η h (½ m r v i2 ) (kinetic energy) W P v i = 2g h e η h 63 GRLWEAP Diesel hammer efficiencies, η h Open end diesel hammers: 0.80 uncertainty of fall height, friction, alignment Closed end diesel hammers: 0.80 uncertainty of fall height, friction, power assist, alignment 64

26 Modern Hydraulic Hammer Efficiencies, η h Hammers with internal monitor: 0.95 uncertainty of hammer alignment Hydraulic drop hammers: 0.80 uncertainty of fall height, alignment, friction Power assisted hydraulic hammers: 0.80 uncertainty of fall height, alignment, friction, power assist 65 Vibratory Hammers 68

27 Vibratory Hammer Model Line Force F L Bias Mass and m 1 Oscillator mass, m 2 Eccentric masses, m e, radii, r e m 2 F V Clamp Vibratory Force: F V = m e [ω 2 r e sin ω t a 2 (t)] 69 The Driving Systems Consists of Driving System Models 1. Helmet including inserts to align hammer and pile 2. Optionally: Hammer Cushion to protect hammer 3. For Concrete Piles: Softwood Cushion 71

28 Driving System Model Example of a diesel hammer on a concrete piles Hammer Cushion: Spring plus Dashpot Helmet + Inserts Pile Cushion Pile Top: Spring + Dashpot 72 The Soil Model After Smith Soil outside of interface: Rigid Interface Soil: Elasto Plastic Springs and Viscous Dashpots 75

29 Soil Resistance Soil resistance slows pile movement and causes pile rebound A very slowly moving pile only encounters static resistance A rapidly moving pile also encounters dynamic resistance The static resistance to driving (SRD) differs from the soil resistance under static loads 76 Soil Model Parameters RIGID SOIL Pile Soil Interface Segment i 1 Segment i Segment i+1 k i-1,r ui-1 J i-1 k i,r ui J i k i+1,r ui+1 J i+1 77

30 Smith s Soil Model Total Soil Resistance R total = R si +R di Pile Segment i Fixed Soil Displacement u i Velocity v i 78 The Static Soil Model Static Resistance R ui Pile Segment i Displacement u i Velocity v i Pile Displacement k si = R ui /q i R ui ult. resistance qi quake 79

31 Recommended Toe Quakes, q toe Non displacement piles 0.1 or 2.5 mm for all soil types 0.04 or 1 mm for hard rock Displacement piles D/120 for very dense or hard soils D/60 for soils which are not very dense or v. hard Static Toe Res. q toe R u,toe D q toe Toe Displacement 82 Toe Quake Effect on Blow Count S200 Blow Count - Blows/m x m 95 m Depth of Pile Toe Penetration - m qt = D/60 qt = D/ Approximatelyy 50% Shaft Resistance Total No. of Blows: (q t =D/60); 27,490 (q t =D/120) 83

32 The Dynamic Soil Model Standard R d = R s J s v Smith damping factor, J s [s/m or s/ft] Pile Segment i Displacement u i Velocity v i For RSA and Vibratory Analysis R d = R u J sv v Smith viscous damping factor, J sv [s/m or s/ft] 84 Recommended Smith damping factors Shaft Clay: Sand: Silts: Layered soils: Toe All soils: (J s or J sv ) 0.65 s/m or 0.20 s/ft 0.16 s/m or 0.05 s/ft use an intermediate value use a weighted average for bearing graph 0.50 s/m or 0.15 s/ft 85

33 S200 Shaft Damping on Blow Count Blow Count - Blows/m x m 95 m Depth of Pile Toe Penetration - m Js = 0.65 s/m Js = 0.16 s/m Approximatelyy 50% Shaft Resistance Total No. of Blows: (J s =0.65 s/m); 27,490 (J s =0.16 s/m) 86 GRLWEAP s Static Analysis Methods Q Icon Input Basic Analysis ST Soil Type Effective Stress, Total Stress SA SPT N-value Effective Stress CPT R at cone tip and sleeve Schmertmann API φ, Su Effective Stress, Total Stress R s GRLWEAP s static analysis methods may be used for dynamic analysis preparation (resistance distribution, estimate of capacity for driveability). For design, be sure to use a method based on local experience. R t 88

34 Use of Static Analysis Methods Should always be done for finding reasonable pile type and length For driven piles static analysis is only a starting point, since pile length is determined in the field (exceptions are piles driven to depth, for example, because of high soil setup) For LRFD when finding pile length by static analysis method use resistance factor for selected capacity verification method 89 Resistance Distribution 3. More Involved: I. ST Input: Soil Type II. SA Input: SPT Blow Count, Friction Angle or Unconfined Compressive Strength III. API (offshore wave version) Input: Friction Angle or Undrained Shear Strength Penetration IV. CPT Input: Cone Record including Tip Resistance and Sleeve Friction vs Depth. All are good for a Bearing Graph II, III and IV OK for Driveability Analysis Local experience may provide better values 92

35 Numerical Treatment Predict displacements: u ni = u oi + v oi t Calculate spring compression: u ni-1 Mass i-1 R i-1 c i = u ni -u ni-1 F i, c i Calculate spring forces: u ni Mass i R i F i = k i c i Calculate resistance forces: R i = R si + R di u ni+1 Mass i+1 R i+1 94 Force balance at a segment Force from upper spring, F i Resistance force, R i Mass i Weight, W i Force from lower spring, F i+1 Acceleration: a i = (F i + W i R i F i+1 ) / m i Velocity, v i, and Displacement, u i, from Integration 95

36 Set or Blow Count Calculation (a) Simplified: extrapolated toe displacement Static soil Resistance Max. Displacement Calculated R u Extrapolated Pile Displacement Final Set Quake 97 Blow Count Calculation (b) Residual Stress Analysis (RSA) Set for 2 Blows Convergence: Consecutive Blows have same pile compression/sets 100

37 RSA Effect on Blow Count S x m 95 m Blow Count Blow (bl/m) Count - Blows/m Depth of Pile Toe Penetration - m 10 No RSA RSA 20 Standard 30 RSA Total No. of Blows: 8907 (Standard); 6235 (RSA) 101 Program Flow Bearing Graph Input Distribute Ru Set Soil Constants Increase Ru Model hammer, driving system and pile Choose first Ru Static Equilibrium Ram velocity Dynamic analysis Pile stresses Energy transfer Pile velocities Increase R u? N Y Calculate Blow Count Output 103

38 Bearing Graph: Variable Capacity, One depth SI-Units; Clay and Sand Example; D19-42; HP 12x53; 104 Driveability Analysis Analyze a series of Bearing Graphs for different depths for SRD and/or LTSR Put the results in sequence so that we get predicted blow count and stresses vs pile toe penetration Note that, in many or most cases, shaft resistance is lower during driving (soil setup) and end bearing is about the same as long term In the few cases of relaxation, the toe resistance is actually higher during driving than long term 107

39 Program Flow Driveability Input Model Hammer & Driving System Calculate Ru for first gain/loss Analysis Next G/L Choose first Depth to analyze Pile Length and Model Increase Depth Y Increase G/L? N Increase Depth? N Y Output 108 Driveability Result During a driving interruption soil setup occurs 109

40 When Should we do the Analysis? Before pile driving begins Equipment selection for safe and efficient installation Preliminary driving criterion After initial pile tests have been done Refined Wave Equation analysis for improved driving criterion For different driving systems In preparation of dynamic testing 110 Summary The wave equation analysis simulates what happens in the pile when it is struck by a heavy hammer input. It calculates a relationship between capacity and blow count, or blow count vs. depth. The analysis model represents hammer (3 types), driving system (cushions, helmet), pile (concrete, steel, timber) and soil (at the pile soil interface) GRLWEAP provides a variety of input help features (hammer and driving system data, static formulas among others). 111

41 An example for a Dynamic Test Preparation Prepare dynamic test on a 400 mm dia. pile with Expander Body of 600 mm diameter and 2000 mm length. Sand and Gravel Drop Weights 5 and 8 tons Drop Height 1.2 m Cushion 100 mm 112 Ananlysis of a Pile with Expander Body 113

42 Analysis results Hammers 1 m drop height, 9 inch cushioin GRL Engineers, Inc. Expander Body: 8ton ram; 6" cushion Expander Body: 5ton ram; 6" cushion 30-Apr-2015 GRLWEAP Version GRL 8 ton GRL 5 ton Compressive Stress (MPa) Tension Stress (MPa) Stroke m Ram Weight kn Efficiency Helmet Weight kn Pile Cushion kn/mm COR of P.C Skin Quake mm mm Toe Quake mm mm Skin Damping sec/m sec/m Toe Damping sec/m sec/m Pile Length m Pile Penetration m Pile Top Area cm2 0 0 Pile Model Skin Friction Distribution Pile Model Skin Friction Distribution 4000 Ultimate Capacity (kn) Blow Count (blows/.10m) Res. Shaft = 20 % (Proportional) Res. Shaft = 20 % (Proportional) 114 Thank you for your attention! QUESTIONS? 115