1.1 grlweap - gray

GRLWEAP - Santa Cruz, 20151
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.
Applications of Stress Wave
Theory to Deep Foundations
with an Emphasis on
“The Wave Equation”
(GRLWEAP)
CONTENTCONTENT
• Introduction
– Dynamic Formula
– Static Formula
• The One‐Dimensional Wave Equation and Wave
Demonstrations
• Wave Equation Models
• Bearing Graph and Driveability
• Example
• Conclusions
WAVE EQUATION OBJECTIVESWAVE 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

Wave Demonstrations
– Slinky
– Pendulum
– Buddies
– Shear Waves
– Compressive Waves
Animation courtesy of Dr. Dan Russell, Kettering Univ.
http://paws.kettering.edu/~drussell/demos.html
WAVES
Example of a Baseball Wave
Example of a Shear Wave

Example of a Compressive Wave
The 1-D Wave Equation
ρ(δ2u/ δt2) = E (δ2u/ δx2)
E … elastic modulus
ρ … mass density
with c2 = E/ ρ ... Wave Speed
Solution: u = f(x‐ct) + g(x+ct)
x … length coordinate
t ... time
u … displacement
f
g
x
x
Time
t
The compression
wave,induced by the
hammer at the pile top,
moves downward a
distance c t during the
time interval t.
Waves in a PileWaves in a Pile

x
Time
t t + t
C t
The compression wave,
induced by the hammer
at the pile top, moves
downward a distance c
t during the time
interval t.
Waves in a PileWaves in a Pile
The compression wave,
arrives at the pile toe where it
is reflected
(on a free pile in tension).
t Time t + t Waves in a Pile
2012 13 Wave Mechanics for Pile Testers
x
u
ρ(δ2u/ δt2) = E (δ2u/ δx2)
E … elastic modulus
ρ … mass density
with c2 = E/ ρ … Wave Speed
x … length coordinate t ... time
u … displacement
THE Wave Equation
Solution: u = f(x-ct) + g(x+ct)

2012 14 Wave Mechanics for Pile Testers
f
g
x
f
g
x
C t
C t
Time
t + t
Time
t
The Solution to the Wave Equation
u = f(x-ct) + g(x+ct)
Force, F – Time to + t
Point A
Point A, like all other
points along the pile, is
at rest at time to (when
contact between ram and
pile top occurs)
Compressed
distance, L
Time to
u
The first instant after impact
∆u is the displacement of a point of pile during time ∆t
F
∆L
Wave travels distance ∆L = c ∆t during time ∆t
Particle Velocity, v = ∆u/ ∆t
but ∆u = ε ∆L and therefore v = ε ∆L / ∆t
and with wave speed c = ∆L / ∆t:
∆u
Force Velocity ProportionalityForce Velocity Proportionality
v = ε c

This is the strain, stress, force-velocity
proportionality
Z = EA/c is the pile impedance (kN/m/s)
This is the strain, stress, force-velocity
proportionality
Z = EA/c is the pile impedance (kN/m/s)
Fd = vd (EA/c)Fd = vd (EA/c)
d = vd(E/c)d = vd(E/c)εd = vd / cεd = vd / c
Strain-Stress-Force Proportionality
Wave travels in one direction only
Strain-Stress-Force Proportionality
Wave travels in one direction only
Express Your ImpedanceExpress Your Impedance
Z = EA/c kN/(m/s)
with c = (E/ρ)1/2 Z = A (E ρ)1/2
with E = c2 ρ Z = A c ρ
with Mp= L A ρ Z = Mp c/ L (Mp ... 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.
A Quick Look at Energy FormulasA Quick Look at Energy Formulas
Energy Dissipated in Soil =
Energy Provided by Hammer
Ru (s + sl) = ηWr h
sl … “lost” set (empirical or measured),
η … efficiency of hammer/driving system
Engineering News: Rallow = Wr h / 6(s + 0.1)

The Gates FormulaThe Gates Formula
Ru = 7 (Wrh)½ log(10Blows/25 mm) ‐ 550
Ru … Nominal Resistance (kN)
Wr… ram weight (kN)
h … actual stroke (m)
log … logarithm to base 10
The Hiley Formula
using Set-Rebound Measurements
The Hiley Formula
using Set-Rebound Measurements
Ru = ηWr h (Wr+ e2 WP)
(s + c/2) (Wr + WP)
Rebound: c
Set = s
Considers combined pile‐soil elasticity effect
Usually with F.S. = 3; η = hammer efficiency.
Bearing Graphs from 2 Energy Formulas
Hammer D 19-42; Er = 59 kJ
Bearing Graphs from 2 Energy Formulas
Hammer D 19-42; Er = 59 kJ
Ru = ηEr /(s + sl)
η = 1/3; sl = 2.5mm
Ru = 1.6 Ep ½ log(10Blows/25mm) – 120 kN
4000
[900]
Ru - kN
[kips]
2000
[450]
0
0 5 10 15 20
Blows/25mm

Shortcomings of FormulasShortcomings 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
Static FormulasStatic Formulas
• Based on Soil Properties
• Always done for any deep foundation type
• Backed up by Static or Dynamic Testing
Static Analysis to Calculate LTSR
Basically for All Soil Types:
Ru = Ru,shaft + Ru,toe
Ru = fsAs + qt At
fs, Ru,shaft, As … Shaft Resistance/Area
qt, Ru,toe, At … End Bearing/Area

The β-Method for Cohesionless Soils
• Ru,shaft = fs As
– fs = ko tan(δ) po
po is the effective overburden pressure
ko is some earth pressure coefficient
– β = ko tan(δ)
• Ru,toe = Nt po At
Nt is a bearing capacity factor
All with Certain Limits
The α-Method for Cohesive Soils
• Ru,shaft = fs As
– fs = α c
c is the undrained shear strength
α is a function of po
• Ru,toe = 9 c At
..... with certain limits
GRLWEAP provides 4 different static analysis methods
ST – based on Soil Type; SA‐ based on SPT‐N; CPT; API
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 Nt Limit (kPa)
degrees kN/m3 Qs Qt
Very loose 2 25 - 30 13.5 0.203 12.1 24 2400
Loose 7 27 - 32 16 0.242 18.1 48 4800
Medium 20 30 - 35 18.5 0.313 33.2 72 7200
Dense 40 35 - 40 19.5 0.483 86.0 96 9600
Very Dense 50+ 38 - 43 22 0.627 147.0 192 19000

ST - INPUTST - INPUT
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 Qs Qt
kPa kN/m3 kPa kPa
Very soft 1 12 17.5 3.5 54
Soft 3 36 17.5 10.5 162
Medium 6 72 18.5 19 324
Stiff 12 144 20.5 38.5 648
Very stiff 24 288 20.5 63.5 1296
hard 32+ 384+ 19 – 22 77 1728
ST - INPUTST - INPUT

The Wave Equation ModelThe 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
The Wave Equation ModelThe 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
GRLWEAP FundamentalsGRLWEAP Fundamentals
• For a pile driving analysis, the “slender,
elastic rod” consists of Hammer+Driving
System+Pile
• The soil is represented by resistance forces
acting on the pile and representing the
forces in the pile‐soil interface
Hammer
D.S.
Pile

Smith’s Numerical Solution of the Wave EquationSmith’s Numerical Solution of the Wave Equation
∆L
ρ(δ2u/ δt2) = E (δ2u/ δx2)
E … elastic modulus ‐ ρ … mass densitywith c2 = E/ ρ ... Wave Speed
Closed Form Solutions to the wave equation are
not practical; we therefore solve the
equation numerically:
(mi/ki)(ui,j+1 ‐2ui,j + ui,j‐1)/Δt2
= (ui+1,j – 2ui,j + ui‐1,j)
This is equivalent to considering mass points
and springs!
i
i+1
i-1
The GRLWEAP Pile ModelThe 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
The Pile ModelThe Pile Model
Relationship between the uniform pile and the
lumped mass model properties:
m k = (ρ A ∆L)(EA/∆L) = A2Eρ = Z2 [kN s/m]2
m/k = (ρ A ∆L)/(EA/∆L) = (ρ/E)∆L2 = (∆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.
∆L

We can model 3 hammer-pile systemsWe can model 3 hammer-pile systems
Ram: A, L for stiffness, mass
Cylinder and upper frame =
assembly top mass
Drop height
External Combustion Hammer Modeling
Ram guides for assembly stiffness
Hammer base =
assembly bottom mass
External Combustion Hammer ModelExternal Combustion Hammer Model
• Ram modeled like rod
• Stroke is an input (Energy/Ram Weight)
• Impact Velocity Calculated from Stroke with Hammer
Efficiency Reduction: vi = (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

External Combustion Hammers
Ram Model
Ram segments
~1m long
Combined Ram‐
H.Cushion
Helmet mass
Assembly model
Assembly model
Assembly segments,
typically 2
Helmet mass
Combined Ram Assembly Model
Combined Ram Assembly Model
Combined Ram-
H.Cushion
Helmet mass
Ram segments
Assembly segments

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
VRAM = (2 g h η)1/2 – 0.001 g
g is the gravitational acceleration
h is the equivalent hammer stroke and η is the hammer efficiency
h = Hammer potential energy/ Ram weight
• Dynamic analysis ends when
– Pile toe has rebounded to 80% of max dtoe
– Pile has penetrated more than 4 inches
– Pile toe has rebounded to 98% of max dtoe and energy
in pile is essentially dissipated
External Combustion Hammer
Analysis Procedure
Diesel HammersDiesel Hammers
• Very popular in the US
• Have their own fuel tank
and combustion “engine”
• Model therefore includes a
thermodynamic analysis
• Stroke is computed

GRLWEAP hammer efficiencies
ηh = Ek/EP
GRLWEAP hammer efficiencies
ηh = Ek/EP
•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
WR
h
ER = WR h
Manufacturer’s Rating
WR
Max ET = ∫F(t) v(t) dt
(EMX, ENTHRU)
ηT = ENTHRU/ ER
(transfer ratio or efficiency) Measure:
Force, F(t)
Velocity, v(t)
Measured Transferred
Energy
Measured Transferred
Energy
Measured Transfer Ratios for Diesels
Steel Piles Concrete Piles

For all impact hammers GRLWEAP
needs impact velocity
WP
mR
h
Er = Wr he = mr g he
he = Er / Wr he – equivalent stroke
he = h for single acting hammers
Epr = η Er Wr he (η = Hammer efficiency )
WR
vi
Ek = Epr = ηh (½ mr vi
2) (kinetic energy)
vi = 2g heηh
GRLWEAP
Diesel hammer efficiencies , ηh
GRLWEAP
Diesel hammer efficiencies , ηh
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
Modern Hydraulic Hammer
Efficiencies, ηh
Modern Hydraulic Hammer
Efficiencies, ηh
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

Vibratory
Hammers
Vibratory
Hammers
Vibratory Force:
FV = me [ω2resin ω t ‐ a2(t)]
FL
FV
m1
m2
• Line Force
• Bias Mass and
• Oscillator mass, m2
• Eccentric masses, me,
radii, re
• Clamp
Vibratory Hammer ModelVibratory Hammer Model
The Driving Systems
Consists of
1. Helmet including inserts to
align hammer and pile
2. Optionally: Hammer Cushion
to protect hammer
3. For Concrete Piles: Softwood
Cushion
Driving System ModelsDriving System Models

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

Segment
i
Segment
i‐1
Segment
i+1
Pile‐Soil Interface
Soil Model ParametersSoil Model Parameters
ki,Rui
Ji
RIGID SOIL
ki+1,Rui+1
Ji+1
ki-1,Rui-1
Ji-1
Fixed
Soil
Smith’s Soil ModelSmith’s Soil Model
Total Soil Resistance
Rtotal = Rsi +Rdi
Total Soil Resistance
Rtotal = Rsi +Rdi
Displacement ui
Velocity vi
Pile
Segment i
The Static Soil ModelThe Static Soil Model
Displacement ui
Velocity vi
Pile
Segment i
Pile Displacement
Rui
Static Resistance
Rui … ult. resistance
qi … quake
ksi = Rui /qi

Recommended Toe Quakes, qtoeRecommended Toe Quakes, qtoe
0.1” or 2.5 mm for
all soil types
0.04” or 1 mm for
hard rock
qtoe
Static Toe Res.
qtoe Ru,toe
Toe Displacement
D/120 for very dense or
hard soils
D/60 for soils which are
not very dense or v. hard
Displacement pilesNon‐displacement piles
D
Toe Quake Effect on Blow CountToe Quake Effect on Blow Count
S200
100m
610x12
95m
Approximatelyy 50% Shaft Resistance
Total No. of Blows: ∞ (qt =D/60); 27,490 (qt=D/120)
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800
DepthofPileToePenetration-m
Blow Count - Blows/m
qt = D/60
qt = D/120
The Dynamic Soil ModelThe Dynamic Soil Model
Displacement ui
Velocity vi
Pile
Segment i
Rd = RuJsv v
Smith‐viscous
damping factor,
Jsv [s/m or s/ft]
For RSA and
Vibratory Analysis
Smith damping
factor,
Js [s/m or s/ft]
Rd = RsJs v
Standard

Recommended Smith damping factors
(Js or Jsv)
Recommended Smith damping factors
(Js or Jsv)
Shaft
Clay: 0.65 s/m or 0.20 s/ft
Sand: 0.16 s/m or 0.05 s/ft
Silts: use an intermediate value
Layered soils: use a weighted average
for bearing graph
Toe
All soils: 0.50 s/m or 0.15 s/ft
Shaft Damping on Blow CountShaft Damping on Blow Count
S200
100m
610x12
95m
Approximatelyy 50% Shaft Resistance
Total No. of Blows: ∞ (Js=0.65 s/m); 27,490 (Js=0.16 s/m)
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800
Js = 0.65 s/m
Js = 0.16 s/m
GRLWEAP’s Static Analysis MethodsGRLWEAP’s Static Analysis Methods
Rs
Rt
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
• 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.

Use of Static Analysis MethodsUse 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
Resistance DistributionResistance 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
IV. CPT Input: Cone Record including Tip
Resistance and Sleeve Friction vs
Depth.
Penetration
All are good for a Bearing Graph
II, III and IV OK for Driveability Analysis
Local experience may provide better values
Mass i
Mass i-1
Mass i+1
Numerical TreatmentNumerical Treatment
• Predict displacements:
uni = uoi + voi ∆t
Fi, ci
uni-1
uni
uni+1
Ri-1
Ri
Ri+1
• Calculate spring compression:
ci = uni - uni-1
• Calculate spring forces:
Fi = ki ci
• Calculate resistance forces:
Ri = Rsi + Rdi

Force balance at a segmentForce balance at a segment
Acceleration: ai = (Fi + Wi – Ri – Fi+1) / mi
Velocity, vi, and Displacement, ui, from Integration
Mass i
Force from upper spring, Fi
Force from lower spring, Fi+1
Resistance force, Ri Weight, Wi
Set or Blow Count Calculation
(a) Simplified: extrapolated toe displacement
Set or Blow Count Calculation
(a) Simplified: extrapolated toe displacement
Static soil Resistance
Pile
Displacement
Final Set
Max. Displacement
Quake
Ru
Extrapolated
Calculated
Blow Count Calculation
(b) Residual Stress Analysis (RSA)
Blow Count Calculation
(b) Residual Stress Analysis (RSA)
Set for 2 Blows
Convergence:
Consecutive Blows
have same
pile compression/sets

RSA Effect on Blow CountRSA Effect on Blow Count
S500
100m
1220x25
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800
Standard
RSA
95m
Total No. of Blows: 8907 (Standard); 6235 (RSA)
Static Equilibrium
Ram velocity
Dynamic analysis
Program Flow – Bearing GraphProgram Flow – Bearing Graph
Model hammer,
driving system
and pile
• Pile stresses
• Energy transfer
• Pile velocitiesChoose first Ru
Calculate Blow
Count
Distribute Ru
Set Soil Constants
Output
Increase
Ru?
Increase Ru
Input
N
Y
Bearing Graph: Variable Capacity, One depth
SI-Units; Clay and Sand Example; D19-42; HP 12x53;
Bearing Graph: Variable Capacity, One depth
SI-Units; Clay and Sand Example; D19-42; HP 12x53;

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

When Should we do the Analysis?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
SummarySummary
• 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).
An example for a Dynamic Test
Preparation
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

Ananlysis of a Pile with Expander BodyAnanlysis of a Pile with Expander Body
Analysis results
Hammers 1 m drop height, 9 inch cushioin
Analysis results
Hammers 1 m drop height, 9 inch cushioin
Thank you for your
attention!
QUESTIONS?
Thank you for your
attention!
QUESTIONS?

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