BACKGROUND AND INTRODUCTION

Transcription

1 WAVE MECHANICS As applied to pile testing 1 BACKGROUND AND INTRODUCTION Wave Mechanics 2 1

2 HISTORIC TOOLS AND MATERIALS Timber piles Drop Hammers 10 Days for the Romans to build the bridge over the Rhine River (Lasting till the next flood) TODAY WE HAVE: High strength materials: concrete, steel Steam, air, diesel, hydraulic hammers Powerful drilling equipment High capacity lifting equipment Routine dynamic measurements and analyses High capacity piles 2

3 PILE TESTING METHODS Load testing Top loaded static test Two-way static (O-cell) Dynamic and Rapid (DLT and RLT) Integrity Pulse Echo (PIT) Cross Hole (CHA) Thermal Integrity Profiling (TIP) Gamma-Gamma (GGL) Length of Existing Foundations Parallel seismic (PST) Inductive (LITE) ONE STATIC LOAD TEST FAILURE CRITERION Failure Load or Nominal Resistance Load Δ Δ = 0.15" (3.8 mm) + D/120 (Davisson Offset) EA/L Set 3

4 LRFD: LOADS AND CAPACITY Load R ult Factored Load: Q i f Li φ R ult Un-Factored Loads, Q i φ R ult > Q i f Li Set DYNAMIC LOAD TESTING METHODS Capacity assessment by dynamic formula Capacity assessment by wave equation analysis Dynamic Load Testing Rapid Load Testing 4

5 INTRODUCTION Hammer causes a downward travelling stress-wave to enter the pile Soil resistance and section changes cause upward stress-wave reflections Stress in pile can be represented by 1-dimensional Wave Theory These stress-waves can be analyzed by measured pile top force and velocity 10 F v Gages mounted near the top of the pile WAVE PROPAGATION 1 Stress wave arrives at the gage location 2 Stress wave reaches its initial peak velocity 3 Stress wave arrives at the pile toe and starts to reflect up the pile 4 initial response from the pile toe arrives at the gage location 5 Peak response from the pile toe arrives at the gage location 15 5

6 F v WAVE PROPAGATION The response from the pile toe will not be observed at the gage location until the initial stress wave has travelled down the pile and back again L Initial rise Initial peak Initial rise + 2L/c Initial peak + 2L/c If the length of the pile below the gage location and wavespeed, c, of the pile material are know then we can calculate the point in time where that toe response will occur: 2L/c 2L/c 16 THE WAVE EQUATION 17 6

7 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 Solution: u = f(x-ct) + g(x+ct) x x length coordinate t... time u displacement No.18 The Solution to the Wave Equation Time t Time t + t u = f(x-ct) + g(x+ct) C t f f g g C t x x No.19 7

8 PROPORTIONALITY Wave Mechanics 20 PROPORTIONALITY Viewing the rod to the right we see a small particle at some point along the pile. Upon impact from the hammer a compression wave is generated travelling down the pile at wavespeed, c. As the stress wave encounters the particle; the particle is deformed and accelerated down the pile 21 8

9 PROPORTIONALITY The Pile has a known section area,, and modulus of Elasticity, Δ The Compression will cause a strain, The Compression also results in a particle velocity,, and wavespeed, 22 PROPORTIONALITY A deformation,, is created by strain,, over a distance, and by substitution Δ A particle travelling a distance,, over a time,, has a velocity, Finally by solving for and substituting into the previous equation we can establish: 23 9

10 PROPORTIONALITY We can now put this in terms of stress or And then in terms of Force where The proportionality constant,, is referred to as the impedance of the pile units of kn/m/s (kip/ft/sec) 24 PILE TOP FORCE AND VELOCITY FROM PDA Force ½ 1 2 ½ Velocity -200 Scales are Proportional by Pile Impedance Z

11 WAVE SPEED Wave Mechanics 26 WAVESPEED The Force,, acting on the cross section will cause a downward acceleration According to Newton s Second Law where Substitution yields: From previous slides we have defined and 27 11

12 WAVESPEED or the wave speed, c, is the square root of the product of mass density and elastic modulus depends only on the pile material properties and not, the frequency or amplitude of the applied force this is only true for our simplifying assumptions of a very slender, elastic rod. 28 UPWARD AND DOWNWARD TRAVELLING WAVES Wave Mechanics 29 12

13 SIGN CONVENTIONS Force: Compression positive (+) Tension negative (-) Velocity: Downward positive (+) Upward negative (-) 30 Compression Wave Compression wave begins travelling down the pile DOWNWARD TRAVELLING Small particle located somewhere along the length of the pile Once the compression wave encounters the particle; the particle is instantaneously accelerated down the pile WAVES Note that with our sign convention the force will be positive (compression) and our velocity will be positive(downward particle velocity) Therefore: i.e. downward travelling waves will have similar sign conventions 31 13

14 Once the compression wave encounters the particle; the particle is instantaneously accelerated up the pile Small particle located somewhere along the length of the pile UPWARD TRAVELLING WAVES Note that with our sign convention the force will be positive (compression) but the velocity will be negative(upward particle velocity) Therefore: + Compression Wave Compression wave begins travelling UP the pile 32 Once the tension wave encounters the particle; the particle is instantaneously accelerated down the pile Small particle located somewhere along the length of the pile UPWARD TRAVELLING WAVES Note that with our sign convention the force will be negative (tension) but the velocity will be positive (downward particle velocity) Therefore: Tension Wave Tension wave begins travelling UP the pile 33 14

15 UPWARD/DOWNWARD TRAVELLING WAVES From the previous illustrations and based on our Proportionality Law we can summarize: Downward travelling force and velocity waves will have similar sign conventions proportional by the pile impedance Upward travelling force and velocity waves will have opposite sign conventions proportional by the impedance 34 Downward Travelling Wave F v Up to this point we have assumed small pulse inputs as modeled here Upward Travelling Wave DOWNWARD & UPWARD TRAVELLING WAVES Our theoretical model 35 15

16 DOWNWARD & UPWARD TRAVELLING WAVES Velocity*Z Force 36 Downward Travelling Wave F v Upward Travelling Wave DOWNWARD & UPWARD TRAVELLING WAVES Reality stress waves are much longer duration At gage location we see the superposition of the upward and downward waves 37 16

17 Velocity Force DOWNWARD & UPWARD TRAVELLING WAVES 38 UPWARD/DOWNWARD TRAVELLING WAVES Measured force and velocity are comprised of both Upward and Downward travelling waves We can rewrite the second equation in terms of Force using the impedance: 39 17

18 UPWARD/DOWNWARD TRAVELLING WAVES With two equations and two unknowns we can solve for the Downward travelling Force Wave ( ) and the Upward Travelling Force Wave ( ): Therefore the Downward Travelling Force Wave is the average of the measured Force and the measured velocity times Impedance. 2 And the Upward Travelling Force Wave is half the difference of the measured Force and the measured velocity times Impedance Force Velocity*Z UPWARD/ DOWNWARD TRAVELLING WAVES Wave Up Wave Down 41 18

19 END EFFECTS Wave Mechanics 42 FREE-END FREE BODY DIAGRAM Pile Forces Pile Velocities +F C T -F +v +v Equilibrium Continuity F=0 +2v 43 19

20 time C T C FREE PILE DISPLACEMENT 1. Compression wave travels down the pile 2. Wave arrives at time, L/c 3. Free End reflects compression wave as tension 4. Tensile waves arrives at the top of the pile at time 2L/C length 1 2 C T Free End reflects tension wave as Compression 44 time FREE PILE VELOCITY 1. Downward velocity taken as positive 2. Velocity at Free End doubles at time, L/c 3. Wave returns to top at time, 2L/c, where again, velocity doubles length 45 20

21 time FREE PILE STRESS 1. Compression taken as positive 2. Stress at Free End must be zero 3. Wave changes from compression to tension at L/c and tension to compression at 2L/c length Stress a toe must be zero 46 Force Velocity FREE PILE EXAMPLE easy driving Wave Up Wave Down 47 21

22 FREE END For a free-end, the upward travelling force wave will be in Tension With little shaft resistance early in driving a majority of the initial impact force will be delivered to the pile toe APPLICATION If easy driving; the tensile stresses are greatest Tensile stresses become a concern for concrete piles whose tensile strength is limited To reduce tensile stresses, increase pile cushion or decrease impact force 48 FIXED-END FREE BODY DIAGRAM Pile Velocities Pile Forces +v +F C C -v +F Continuity Equilibrium v=0 C +2F 49 22

23 time C C T FIXED PILE DISPLACEMENT 1. Compression wave travels down the pile 2. Wave arrives at time, L/c 3. Fixed End reflects compression wave as compression 4. Compression waves arrives at the top of the pile at time 2L/C length C C 5. Free End at 2L/c reflects compression wave as tension time FIXED PILE VELOCITY 1. Downward velocity taken as positive 2. Velocity at Fixed End must be zero by definition 3. Wave returns to top at time, 2L/c, where, velocity doubles length Velocity a toe must be zero 51 23

24 time FIXED PILE STRESS 1. Compression taken as positive 2. Because of Fived end condition, stress at toe doubles 3. Wave stays in from compression to tension at L/c and switches to tension at 2L/c length 52 Force Velocity*Z FIXED PILE EXAMPLE Hard driving Wave Up Wave Down 53 24

25 FIXED END If the pile toe is a fixed end (refusal driving) the resistance at the toe can be twice the magnitude as the impact force Practically speaking the resistance at the toe is generally reduced due to the damping effect of the soil as well as the soil stiffness APPLICATION If driving at refusal; the critical stresses will be at the pile toe These stresses become even more critical if there is Little Soil Shaft Resistance Short Pile Length To continue moving the pile; the impact force needs to be increased 54 SUMMARIZING THUS FAR Proportionality Where Wavespeed Sign Convention: Force Velocity Downward Waves will have the same sign (+F/+v) (-F/-v) Upward wave will have opposite sign (+F/-v) (-F/+v) Upward/Downward Travelling Waves 2 Free 2 Fixed

26 SOIL RESISTANCE Wave Mechanics 56 Compression wave, F, begins travelling down the pile At some distance down the pile the compression wave encounters a Resistance, R SOIL RESISTANCE The activated resistance creates an upward travelling compression wave with magnitude of R/2 and a downward travelling tension wave with magnitude of - R/2 Therefore: the initial downward compression wave will decrease by R/

27 Velocity Force F v SOIL RESISTANCE The friction effect at the gage location corresponds to the time it requires the compression wave to travel down to that resistance and back again 58 Velocity Force F v SOIL RESISTANCE In reality, instead of one point of resistance along the shaft of the pile there will be resistance all along the length of the pile having a cumulative effect as the Force and Velocity progressively diverge 59 27

28 Force Velocity PILE WITH SHAFT RESISTANCE EXAMPLE Wave Up Wave Down 60 SOIL RESISTANCE Theoretically: The Stress Wave will continue down the pile as long as the shaft resistance is less than twice the impact force: 2 Practically: limit is reduced due to soil damping effects Application: If driving is refusal, increase impact force 61 28

29 CASE-GOBLE CAPACITY Wave Mechanics 62 A pile is struck at time 1,, generating a downward travelling force 1 CASE-GOBLE CAPACITY The resistance wave returns at together with all resistance waves 63 29

30 Wave Up F v CASE-GOBLE CAPACITY Considering the Stress wave as it travels down and encounters resistance we see the sum effect of all stress waves returning to the top of the pile at time 2 / 64 CASE-GOBLE CAPACITY From the previous Slide we see that a time 2L/c we see the Upward Travelling Force Wave can be

31 THE CASE DAMPING FACTOR Wave Mechanics 66 CASE DAMPING FACTOR The Total Resistance calculated is made up of a dynamic resistance and a static resistance. Obviously the static resistance is of concern, therefore: Dynamic resistance is velocity dependent; therefore we can estimate it from the pile toe velocity 67 31

32 Dynamic resistance is velocity dependent the greater the velocity the higher the dynamic resistance CASE DAMPING FACTOR Someone about to encounter some dynamic resistance 68 CASE DAMPING FACTOR To calculate static from total resistance, a viscous damping parameter ( ) is introduced for multiplication of computed toe velocity, or Non-dimensionalisation leads to the Case Damping Factor, : 69 32

33 CASE-GOBLE ) 1 1 / /2 70 CASE DAMPING FACTOR A larger damping factor produces a lower static Case Method capacity With the RMX method, the Case Damping factor varies typically between 0.4 and 1.0 for non-cohesive and cohesive soils, respectively When driving is hard, the pile velocities are low. Then there is little damping and the Case Method capacity becomes insensitive to the choice of damping factor 71 33

34 CASE DAMPING FACTOR VALUES FOR RMX Gravel Sand Reducing Grain Size Increasing Damping factor Silt Clay IMPACT STRESSES 80 34

35 In absence of helmet and cushion: At the first instant of Ram-Pile contact, the pile top particles assume the velocity of the ram Ram impact velocity: v i = 2ghη) 1/2 Pile top force: F i = v i EA/c v i W R h IMPACT STRESS Pile Impedance: Z = EA/c = A (E ρ) 1/2 F i Z, W P RAM MASS EFFECT Pile Top Force, F Reduction of pile top force with time W R F Light Ram Heavy Ram Z, W P Time 35

36 WAVE ARRIVAL AT TOE Force at fixed pile bottom Time L/c Heavy Ram, toe Heavy Ram, top Light Ram, toe Light Ram, top W R Z, W P F i The force at the fixed bottom is twice the force at the top: R = 2F i R Theoretical pile top force over time Time 2L/c Heavy Ram Light Ram W R F i Z, W P REFLECTED WAVE RETURNS TO TOP The greater the ram weight, the higher the force at wave return 36

37 ENERGY Wave Mechanics 92 HAMMER PERFORMANCE Why is it important? Ensure proper hammer operation To install pile to design depth (capacity) Quality control tool (blow count criteria) 93 37

38 Ram Weight, W Potential Energy, E P Drop Height, h Potential Energy ENERGY Kinetic Energy, E K Kinetic Energy Transferred Energy, EMX F v 1 2 Where 2 (ideal) 2 Where ; 1 94 E( t) t EMX = maximum of E ( t ) ETR = EMX / (max hammer rated E) STK = ½*g*( T/2 ) 2-0.1(m); STK is open end Diesel hammer stroke; T is time between blows ENERGY ETH = EMX (Wr x STK) diesels only 0 F( t) V ( t) dt 95 38

39 Velocity becomes negative FORCE VELOCITY ENERGY Determining the shaft resistance along any point on the pile Energy being returned from Pile to Hammer ENERGY Energy being transferred from hammer to Pile Pile comes to rest at final displacement Pile reached peak displacement and begins unloading DISPLACEMENT 96 HAMMER EFFICIENCY Hammer efficiencies are specific to hammer type 97 39

40 Comparison of Hammer Efficiency vs Pile Type and Hammer Type 100% SAD - Steel SAD - Concrete SAH - Steel SAH - Concrete Percentile 90% 80% 70% 60% 50% 40% HAMMER EFFICIENCY Note that reduced efficiency base on pile type is largely due to the use of a hammer cushion on concrete piles 30% 20% 10% 0% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Energy Transfer Ratio (EMX/Rated Energy) 98 NET MEASURED EFFICIENCIES (STEEL PILES) 0% 20% 40% 60% 80% 100% Hydraulic Hammers Air/Steam Hammers Diesel Hammers Drop Hammers 60% 100% 40% 70% 30% 60% 20% 80% Efficiencies on concrete piles are lower ~ 10% 40

41 OTHER HAMMER PERFORMANCE MEASUREMENTS RADAR (HPA) measures v i OTHER HAMMER PERFORMANCE MEASUREMENTS Saximeter measures stroke of open end diesels hammers E-Saximeter measures impact velocity 41

42 SUMMARY Wave propagation theory helps to interpret measurements. Classical solutions for fixed and free pile help understanding of stresses occurring during impact. Complex situations and bearing capacity determination has to resort to numerical solutions (GRLWEAP, CAPWAP ). SUMMARY Impact stress is function of impact velocity Tension stresses can be equal to compressive stresses in free and fixed pile. Compressive stresses at the pile toe can be twice those at impact. High ram mass causes higher top stresses than at impact when wave returns (maybe even twice). Stresses at a point along pile are the sum of stresses in upward and downward wave. 42

43 SUMMARY Pile toe force can reach twice force at top. Forces in pile are a function of pile impedance (Z = EA/c). Pile forces (stresses, strains) are proportional to positive particle velocity in a downward travelling stress wave. Pile forces (stresses, strains) are proportional to positive particle velocity in an upward travelling stress wave. SUMMARY Energy concepts are important to judge hammer potential and performance. It is important to distinguish between Potential energy (energy before any losses) Rated energy (manufacturer s assessment of potential energy) Kinetic energy (just before impact) Transferred energy (what gets into the pile) 43

44 The End Questions: