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ASCE Workshop DFSAP Presentation ASCE Workshop DFSAP Presentation Presentation Transcript

    • J. P. Singh
    • J. P. Singh & Associates
    • Richmond, California
        • Presented at
        • ASCE Geotechnical Workshop
        • Oakland, California
        • October 21, 2008
    ANALYSIS OF LATERALLY AND AXIALLY LOADED PILES AND SHAFTS USING DFSAP
  • Complexity of the Soil Structure Interaction Problem The Soil-Foundation-Structure Problem involves Kinematic Soil-Foundation Interaction occurring during large (cyclic and permanent) ground deformations as well as Inertial Foundation-Structure Interaction occurring during shaking all of which take place while the soil and possibly structural properties degrade with time.
  • Soil Structure Interaction (SSI) Problem
    • Post Earthquake Damage Recon & Studies
    • Modeling of SSI Effects and their Validation
    • using Full Scale Field Tests
    • Centrifuge Physical Modeling
  • Major Causes of Damage Ground Shaking Site Response Near Fault Effects Ground Deformation Liquefaction Related Soft Soil Related
  • 2001 Bhuj Earthquake
  • Damage to Floating and End Bearing Piles 1964 Niigata Earthquake
  •  
  • Hanshin Expressway Route 5 1995 Kobe Earthquake Permanent Horizontal Displacement of Bridge Piers vs Distance to Waterfront Permanent Horizontal Displacements of Bridge Piers versus Free Field Ground Displacement
  • Important Factors to be considered in Solution of the Complex SSI Problem Thickness and properties (shear strength and passive pressure) of soil strata Geometry and Properties of Foundation Elements Restraining stiffness and strength of Structural Elements Pile Types - Vertical or Batter/End Bearing or Floating
  • Limit Equilibrium Evaluation of Land Road Bridge Foundation 1987 Edgecumbe, New Zealand Earthquake
  • Limit Equilibrium Method for Design of Deep Foundation subjected to Lateral Spreading (Japan Road Association, 1996)
  • NEAR FAULT RESPONSE SPECTRA
  • Port of Oakland - Berth 37 Damage Calibration Study using FLAC Analyses 1989 Loma Prieta Earthquake
  • Berth 37 - Cross Section
  • Berth 37 - Damage Calibration Study Calibration Target Deformations Permanent Horiz. Deck Displacement = 2 - 4 inches Permanent Horiz. Soil Deformation = 6 inches Visible Damage to the Piles Damage to the Piles at Depth ?
  • SUMMARY OF PILE TOP DAMAGE BERTH 37 - LOMA PRIETA EARTHQUAKE Note: Pile Integrity Testing suggests some E-Row piles may be damaged below the liquefiable layer.
  • Orbital Plots of Loma Prieta Records - Port of Oakland (Acceleration) (Velocity) (Displacement) Input Time History to FLAC Model
  • Photo of Pile Top Damage
  • FLAC Model of Berth 37
  • Contours of Horizontal Slope Displacement HG F E D C B A
  • Contours of Vertical Slope Displacement HG F E D C B A
  • Pile Displacement Vector Diagram Permanent Horizontal Deck Displacement = 0.30 feet (feet) Berth 37 (Pre-Loma Prieta Condition) Loma Prieta, S r = 400 PSF
  • 1 2 3 4 Pore Pressure Monitoring Locations Loma Prieta, S r = 400 PSF Berth 37 (Pre-Loma Prieta Condition) B D E F G H A C
  • Pore Pressure Ratios Time (Seconds) Berth 37 Loma Prieta, S r = 400 PSF (Pre-Loma Prieta Condition) Pore Pressure Ratio 4 3 1 2
  • Soil Deformation Time History Near Top of Batter Pile Liquefaction triggered, soil deformation occurs Cyclic motions, no liq.
  • Pile Top Shear Time History - Waterside Batter Pile Inertia Loading Kinematic Loading
  • Berth 37 Loma Prieta, S r = 400 PSF (Pre-Loma Prieta Condition) Time (Seconds) Axial Force at Pile/Deck Connection, Pile Row H -336 kip Axial Force per foot pile spacing (lb) 480 kip
  • Bending Moment at Pile/Deck Connection, Pile Row H 60 ft-kip Mp = 60 ft-kip Berth 37 Loma Prieta, S r = 400 PSF (Pre-Loma Prieta Condition) Time (Seconds) Moment per foot pile spacing (ft/lb)
  • Time (Seconds) Horizontal Deck Displacement Time History Displacement (Feet) Berth 37 Loma Prieta, S r = 400 PSF (Pre-Loma Prieta Condition) 3.6 inches 4.4 inches
  • Moment Diagram, Sr=400 psf
    • SIMPLE ENGINEERING METHODS
    • Traditional p-y Method
    • Strain Wedge Method (SWM)
  • CURRENT PRACTICE
    • The p-y approach represents the most common method
    • in the current practice for lateral load analyses of piles.
    • It is employed in:
        • LPILE
        • GROUP
        • COM624
        • BEAM-COLUMN
        • FLORIDA-PIER
        • ALLPILE
    • p-y curve used in these programs is a function of
    • soil properties and pile width
  • Traditional empirical p-y curves were developed using data from full-scale load tests performed on slender (long) piles as function of soil properties and pile width
      • for sand - Mustang Island Test
      • (2-ft diameter steel pipe pile in medium dense sand)
      • for soft clay - Sabine River Test
      • (10.75-in diameter steel pipe pile in soft clay)
      • for stiff clay - Houston Test
      • (2.5-ft diameter RC pile in med. Stiff clay)
    CURRENT PRACTICE
  • Computer Program DFSAP D eep F oundation S ystem A nalysis P rogram developed using Strain Wedge Method for Washington State Department of Transportation for Analysis of Laterally and Axially Loaded Group of Shafts and Piles
    • 1. Assessment of the lateral response (deflection, shear force
    • and bending moment) for
            • Isolated piles
            • Large Diameter shafts
            • Pile group with/without pile cap
    STRAIN WEDGE METHOD (SWM) AND ITS CAPABILITIES FOR ANALYSIS OF LATERALLY LOADED PILES/SHAFTS
      • 2. Analysis of laterally loaded piles in layered soils
            • Sand
            • Clay
            • C-  soil
            • Weak rock
    • 3. Assessment of the laterally load piles/pile groups considering
            • Soil liquefaction
            • Lateral soil spread
  • THE CAPAPILTIES OF THE SWM PROGRAM FOR LATERALLY LOADED PILES/SHAFTS
      • 4. Consideration of the pile/shaft type (short, intermediate & long)
      • effect on the pile lateral response and resulting p-y curve
      • 5. Evaluation of the bridge foundation stiffnesses,
            • Vertical displacement stiffness
            • Lateral displacement stiffness
            • Rotational stiffness
            • Torsional stiffness
      • 6. Assessment of p-y and t-z curves based on
      • soil and pile properties
      • 7 . Assessment of the piles/shafts behavior under axial loads
            • Pile load - settlement
            • Axial Load distribution along the pile
            • Pile’s skin and tip resistance
  • What are the differences between the SWM approach and the p-y method?
    • p-y curves in SWM Approach accounts for the following:
      • Pile Bending Stiffness (EI)
      • Pile Head Conditions (Free/Fixed)
      • Pile Cross-Section Shape (Square/Circular/H-Shape)
      • Pile-Head Embedment Below Ground
      • Soil Profile Continuity (Winkler Springs)
      • Long/Intermediate/Short Piles
      • Soil Liquefaction and Lateral Soil Spread
      • Pile Group
      • Vertical Side Shear Resistance (Large Diameter Shaft) )
    P-Y CURVES IN STRAIN WEDGE APPROACH
  • Laterally Loaded Pile as a Beam on Elastic Foundation (BEF) y p ( E s ) 1 ( E s ) 3 ( E s ) 4 ( E s ) 2 p p p y y y ( E s ) 5 p y M o P o P v
  • LARGE DIAMETER SHAFT z T y p S o i l - S h a f t H o r i z o n t a l R e s i s t a n c e S o i l - S h a f t S h e a r R e s i s t a n c e T i p R e a c t i o n D u e t o S h a f t R o t a t i o n N e g l e c t e d w i t h L o n g S h a f t s P o M o P v T P o o M o o P v y F P v M t F v F P F P F v F v V t F t
  • The p-y method provides a unique p-y curve for the equal diameter piles in the same soil regardless of the pile’s EI EI & D = 1 ft 0.1 EI & D = 1 ft
  • Variation of soil reaction with the change of the footing stiffness (EI) as presented by Terzaghi (1955) and Vesic (1961) q per unit area B C L q 0.5q K r =  K r = 0 Rigid Footing, K r =  Flexible Footing, K r = 0 Footing H (1-  2 s ) E P H 3 6 (1-  2 P ) E s B 3 K r =
  • The p-y method provides a unique p-y curve for the equal diameter piles in the same soil for piles with free- or fixed-head conditions Load Test by Kim et al. (ASCE J., 2004) to Show the Effect of Pile-Head Fixity on the p-y curve SW Model Analysis
  • Laterally Loaded Pile as a Beam on Elastic Foundation (BEF) Effect of Structural Element Cross-Sectional Shape on Soil Reaction P P K 1 K 2 4 ft 4 ft
  • The SW model is based on, The Basic Strain Wedge Model in Uniform Soil
    • Stress-strain behavior of the soil
    • as assessed in the triaxial test ,
    • Soil effective stress analysis
    • Plane stress problem
    • (Norris 1986 and Ashour et al. 1998)
    • Beam on Elastic Foundation
    SAND CLAY C-  Weak ROCK
  • Horizontal and Vertical Growth in the Soil Passive Wedge Pile Pile Pile head load P o Successive mobilized wedges  m  m Mobilized zones as assessed experimentally
  • Simplified SW Model 6 P o Soil Strain  = y/d , From Triaxial Test Concept , and Stress-Strain Curve,  d =  h , Stress Level= SL & Mobilized friction angle =  m d y x Y o h  m  m  m Pile Real stressed zone F 1 F 1 Triaxial test principle stresses A  Side shear (  ) p = CD *  h + Pile Side Shear (b) Force equilibrium in a slice of the wedge at depth x p Plane taken to simplify analysis (i.e. F 1 ’s cancel) C D A  h d Horizontal Slice (c) Forces at the face of the soil passive wedge (Section elevation A-A) ds dx   h  h * CD * dx =  * CD * ds sin  m  VO  m K  VO Y o h x H i i Sublayer i+1 Sublayer 1  Vertical Slice Beam on Elastic Foundation
  • L = SHAFT LENGTH T = (EI/f ) 0.2 f = Coefficient. of Modulus of Subgrade Reaction Varying Deflection Patterns Based on Shaft Type h = 0.69 X o X o Zero Crossing Deflection Pattern Linearized Deflection Y o  Long Shaft L/T  4 X o > h > 0.69 X o X o Zero Crossing Y o  Linearized Deflection Intermediate Shaft 4 > L/T > 2 Zero Crossing h = X o Y o  Deflection Pattern Short Shaft L/T  2 
  • Different Pile/Shaft Cross-Sections Considered in The SWM Program
  • Stress-Strain Model for Confined Concrete in Compression Pile/Shaft Material Nonlinear Modeling
      • Stress
    Strain f s  s  y Yield Stress (f ) y so E Uniaxial Elastic-Perfectly Plastic Numerical Steel Model E s E s E s f cc E c E cc  cc  cu Compressive Strain,  c Compressive stress, f c
  • SWM Validation Example Single Shaft
  • UCLA/CALTRANS TEST
  • Measured and Predicted Shaft Response of the Las Vegas Test (8-ft Diameter and 32-ft long Shaft) P o COM624P
  • P o 15 ft 4 ft Stiff Clay Su = 5500 psi R/C Shaft Measured and Predicted Shaft Response of the Southern California Test (Pier 1) 0.0095 5500 0 130 22 Clay Layer 1  50 ** S u (psf)  (deg.)  (pcf) Thickness (ft) Soil type Soil layer
  • Pile/Shaft Group
  • PILE GROUP P-multiplier (f m ) concept for pile group y p p single Single pile
    • f m is assumed:
    • to be dependent only on the front pile spacing regardless of the value of the transverse spacing
    • does not consider the soil type or layers
    • to be constant in a given soil layer
    • to be constant regardless of level of loading, and level of deformation
    4 p group = f m p single Pile in a group P o P v S S ? ? P o
    • Current Practice
      • P-multiplier used in the current practice (p-y method) is a reduction
      • factor
    Interaction Among the Piles in a Group (Pile Group Analysis) Different Sets of the P-multiplier from Different Research Sources (Rollins et al. 2006)
  • y p p group = P mult x p single p single Pile in a group Single pile (P mult. ) 1 = (P mult. ) 2 = (P mult. ) 3 =
  • The Overlapping of Passive Soil Wedges and the Interaction among the Piles in a Group at any Step of Lateral Loading 6 Pile Group Analysis in SWM Model No P-multiplier)
  • Horizontal Passive Wedge Interference in Pile Group Response Pile Pile Overlap of stresses based on elastic theory (and nonuniform shaped deflection at pile face) Overlap employed in SW model based on uniform stress and pile face deflection (P o ) g (P o ) g Uniform pile face movement
  • Horizontal (Lateral and Frontal) Interaction for a Particular Pile in a Pile Group at a Given Depth 8
      • No p-multiplier is needed.
      • Interaction among the piles in the Group is Based on
      • Longitudinal and Transverse Pile-Spacing,
      • Level of Loading, and Soil and Pile Properties.
      • The Piles in the Group are Analyzed According to Their
      • Location in the Pile Group.
      • The analysis of the pile cap is part of the pile foundation
      • system and is affected by the pile-head stiffness.
      • Pile response under axial loads
      • (Must be part of the pile group analysis under lateral load)
    Evaluation of Interaction Among Various Piles in a Group
  •  
  • Treasure Island 3 x 3 Pile Group Test (Rollins et al., ASCE J., No. 1, 2005)
  • SWM Validation Example
    • Isolated Shaft and Shaft Group with Cap
    • Effect of Vertical Shear Side Resistance
    • on Large Diameter Shafts
    • Taiwan Test by Brown et. al. 2001
  • Shaft B1 Shaft B2 The Taiwan Test by Brown et al. 2001
  • Traditional p-y curves were modified using LPILE to match the measured p-y data (Brown et al. 2001)
  • 0 40 80 120 160 200 P i l e H e a d D e f l e c t i o n , Y o , m m 0 1000 2000 3000 4000 P i l e H e a d L o a d , P o , k N M e a s u r e d ( B r o w n e t a l . 2 0 0 1 ) P r e d i c t e d ( S W M o d e l ) N o V . S i d e S h e a r W i t h V . S i d e S h e a r S i n g l e 1 . 5 - m - D i a m e t e r Shaft (B1) F r e e - h e a d
  •  
  • Pile Cap Effect and Pile Deflection Patterns
  • y Cap Passive Wedge Pile/Shaft Group with Cap Pile Passive Wedges
  • SWM Example of Pile Group
    • 3 x 3 Pile Group
      • Various Pile Types within Group
      • Pile Cap Contribution
      • Pile-head Effect - Free and Fixed
  • Loading Direction
  • 3 x 3 SHAFT GROUP FREE-HEAD
  • 3 x 3 SHAFT GROUP FIXED-HEAD
  • Effect of Pile-Head Conditions on Cap Resistance at the Same Deflection Value in DFSAP Piles + Cap Piles Cap 320 Piles + Cap Piles Cap 410 Free-Head Fixed-Head
  • Piles/Shafts in Sloping Ground
  • Piles/Shafts in Sloping Ground m tan  m  m D D h  m m C B  x   m (h-x) tan h-x Lateral Load Different Failure Planes Sloping Ground
  • 10 Degree Sloping Ground 0 Degree Sloping Ground
  • Effect of Ground Slope on Pile/Shaft Lateral Response
  • Soil Liquefaction and p-y curves for liquefied soils
  • Current Available Procedures That Assess the Pile/Shaft Behavior in Liquefied Soils (Using the Traditional P-y Curve): 1. Construction of the p-y curve of soft clay based on the residual strength of liquefied sand presented by Seed and Harder (1990) 2. Reduce the unit weight of liquefied sand with the amount of R u (Earthquake effect in the free-field ) and then build the traditional p-y curve of sand based on the new value of the sand unit weight.
  • Pile Deflection, y Soil-Pile Reaction, p Upper Limit of S r using soft clay p-y curve API Procedure Corrected blowcount vs. residual strength, S r (Seed and Harder, 1990) P-Y Curve of Completely Liquefied Soil Lower Limit of S r Treasure Island Test Result (Rollins and Ashford)
            • Post-liquefaction undrained stress-strain behavior of partially or completely liquefied sand
    Post-liquefaction stress-strain behavior of completely liquefied sand (  u c =  3c and R u =1) Axial Strain ,  Deviator Stress,  d Post-liquefaction stress-strain behavior of partially liquefied sand (  u c <  3c and. R u <1) x o  d = 2 S r
  • Effect of Cyclic Loading upon Subsequent Undrained Stress-Strain Relationship for Sacramento River Sand (Dr = 40%) (Seed 1979)
    • SWM Example of Pile in
    • Liquefiable Soil Profile
          • Pile Head Response
          • p-y curves for liquified soil
    • Treasure Island Liquefaction Test (TILT)
  • Peak Ground Acceleration (a max ) = 0.1 g Earthquake Magnitude = 6.5 Induced Porewater Pressure Ratio (r u ) = 0.9 - 1.0 Soil Profile and Properties at the Treasure Island Test S h a f t W i d t h x x L o n g i t u d i n a l S t e e l Steel Shell Soil-Pile Reaction, p Pile Deflection, y Treasure Island Test Result (Rollins and Ashford) Upper Limit of S r using soft clay p-y curve Lower Limit of S r API Procedure
  • 0 100 200 300 400 P i l e - H e a d D e f l e c t i o n , Y o , m m 0 100 200 300 400 500 P i l e - H e a d L o a d , P o , k N C I S S , 0 . 6 1 m E I = 4 4 8 3 2 0 k N - m 2 O b s e r v e d P r e d i c t e d ( S W M ) P r e d i c t e d ( C o m 6 2 4 ) N o - L i q u e f a c t i o n P o s t - L i q u e f a c t i o n ( u x s , f f + u x s , n f )
  • Pile-Head Response (Y o vs. P o ) for 0.61-m Diameter CISS at Treasure Island Test
  • p-y Curve of 0.61-m Diameter CISS in Liquefied Soil ( Treasure Island, After Rollins et al. 2005) 0.2 m Below Ground 1.5 m Below Ground 3.2 m Below Ground
  • p-y Curve Empirical Formula in Liquefied Sand by Rollins et al. 2005 p (d=324 mm) = A(By) C for D r = 50% where: A = 3 x 10 -7 (z+1) 6.05 , B = 2.8 (z+1) 0.11 C = 2.85(z+1) -0.41 z is depth in (m) y is lateral deflection (mm) p multiplier = 3.81 ln d + 5.6 p = p (d=324 mm) x p multiplier
  • p-y Curves for loose and dense sand for M=6.5 and amax=0.35g
  • Loose Sand Profile for Three Levels of Earthquake M=4.5, amax=0.15g; M=5.0, amax=0.25g; M=6.5, amax=0.35g
  • Lateral Soil Spread
  • LATERAL SOIL SPREADING PROBLEM
    • Mobilized Driving Lateral Forces
    • Acting on Piles and Generated
    • by Crust Layer(s)
    • Varying Strength of Liquefied Soil(s)
    • Amount of Soil Lateral Displacement
    Bartlett and Youd, 1995 (Current Practice) Stress-Strain Behavior of Fully Liquefied Sand Axial Strain ,   Deviator Stress,  d x o Soil Lateral Displacement (X o ) in DFSAP Shaft Cross Section Liquefied Soil Soil Flow Around
  • (Ishihara)
  • Clay Shaft Diameter Clay Liquefiable Soil “ Full” Pile-Soil Response Under Lateral Soil Spread Liquefiable Soil “ Partial” P o A x i a l L o a d M o M o M o Phase I y p P-y Curve for Fully Liquefied Soil y p P-y Curve for Partially Liquefied Soil y p P-y Curve for Non- Liquefied Soil y p Lateral Spread Effect P-y Curve for Crust Layer Phase II
  • Comparison of Pile Behavior for - As Is Condition - Liquefaction - Liquefaction with Lateral Spread
  • Pile head load = 100 kN Pile head moment = 316 kN-m No-Liquefaction Liquefaction Liquefaction + Lateral Spread
  • Pile head load = 100 kN Pile head moment = 316 kN-m No-Liquefaction Liquefaction Liquefaction + Lateral Spread
  • UC Davis, Centrifuge Test (Boulanger et al. 2003, and Brandenberg and Boulanger 2004) Dense Sand Loose Sand Clay  = 6 kN/m 3 , Dr = 21-35%  = 30 o ,  50 = 0.01  = 7 kN/m 3 , Dr = 69-83%  = 36 o ,  50 = 0.004 Cu= 44 kPa  = 16 kN/m 3 14.3 9.2 2.2 4.6 0.051 1.17 23.5 Pile Cap Length (m) Pile Cap Width (m) Pile Cap Height (m) Pile Spacing (m) Wall Thick. (m) Diameter (m) Pile Length (m)
  • UC Davis, Centrifuge Test on 2 x 3 Fixed-Head Pile Group (After Brandenberg and Boulanger, 2004) Pile Displacement a max = 0.67 g Magnitude = 6.5 Bending Moment
  • Niigata Court House Bld. 0.35-m-Diam. RC Pile, 1964 Niigata EQ, Yoshida and Hamada, 1991
  • Niigata Court House Bld. 1964 Niigata EQ 0.35-m-Diam. RC Pile (Yoshida and Hamada, 1991)
  • SWM Analysis Based on Shaft Length
  • h = 0.69 X o X o Zero Crossing X o > h > 0.69 X o X o Zero Crossing Zero Crossing h = X o Deflection Pattern Linearized Deflection Y o Y o Y o    Linearized Deflection Deflection Pattern Long Shaft L/T  4 Intermediate Shaft 4 > L/T > 2 Short Shaft L/T  2  L = SHAFT LENGTH T = (EI/f ) 0.2 f = Coefficient. of Modulus of Subgrade Reaction Varying Deflection Patterns Based on Shaft Type
  •  
  • T P o M o P v y 75 ft 6 ft P v = 100 kip P o = 150 kip M o = 800 kip-ft L/T = 3.1 Intermediate Shaft Soil Profile – S5 Short Shaft Analysis Intermediate Analysis Short Shaft Analysis Intermediate Analysis
  • T P o M o P v y 90 ft 6 ft P v = 100 kip P o = 150 kip M o = 800 kip-ft L/T = 4.0 Long Shaft Soil Profile – S5 Short Shaft Analysis Long Shaft Analysis
  • Effect of Soil Liquefaction on Response of Shafts of Different Lengths Effect of Shaft Length and Soil Layers on p-y Curve at Certain Depth
  • T P o M o P v y 65 ft 6 ft P v = 100 kip P o = 800 kip M o = 3000 kip-ft M EQ = 6.0 Soil Profile – S7 Liquefaction
  • T P o M o P v y L 6 ft Soil Profile – S5 Shaft-Length Effect on the p-y Curve P-y Curve at 5 ft depth P-y Curve at 20 ft depth
  • T P o o M o P v y 90 ft 6 ft P v = 100 kip P o = 800 kip M o = 3000 kip-ft M EQ = 6.0 Soil Profile – S7 Liquefaction
  • T P o M o P v y 65 ft 6 ft Soil Profile – S7 Liquefaction Effect of Soil Profile (Liquefaction) on the p-y Curve at the Same Depth
  • Pile and Pile Group Stiffnesses with/without Pile Cap
  • Loads and Axis F1 F2 F3 M1 M2 M3 X Z Y F 1 F 2 F 3 M 1 M 2 M 3 X Z Y
  • Linear Stiffness Matrix K 11 0 0 0 0 -K 16 0 K 22 0 0 0 0 0 0 K 33 K 34 0 0 0 0 K 43 K 44 0 0 0 0 0 0 K 55 0 -K 61 0 0 0 0 K 66 F1 F2 F3 M1 M2 M3
    • Linear Stiffness Matrix is based on
        • Linear p-y curve (Constant E s ), not the real case
        • Linear elastic shaft material (Constant EI), not the actual behavior
        • Therefore,
        •  P, M =  P +  M and  P, M =  P +  M
     1  2  3  1  2  3
  • Shaft Deflection, y Line Load, p y P, M > y P + y M y M y P y P, M y p ( E s ) 1 ( E s ) 3 ( E s ) 4 ( E s ) 2 p p p y y y ( E s ) 5 p y M o P o P v Nonlinear p-y curve As a result, the linear analysis (i.e. the superposition technique ) can not be employed Actual Scenario
  • Nonlinear (Equivalent) Stiffness Matrix K 11 0 0 0 0 0 0 K 22 0 0 0 0 0 0 K 33 0 0 0 0 0 0 K 44 0 0 0 0 0 0 K 55 0 0 0 0 0 0 K 66 F1 F2 F3 M1 M2 M3
        • Nonlinear Stiffness Matrix is based on
        • Nonlinear p-y curve
        • Nonlinear shaft material (Varying EI)
      •  P, M >  P +  M K 11 = P applied /  P, M
        •  P, M >  P +  M K 66 = M applied /  P, M
     1  2  3  1  2  3
  • Pile Load-Stiffness Curve Linear Analysis Pile-Head Stiffness, K11, K33, K44, K66 Pile-Head Load, P o , M, P v P 1, M 1 P 2, M 2 Non-Linear Analysis
  • P L P v M (K22) (K11) (K66) x x K11 0 0 0 0 0 0 K22 0 0 0 0 0 0 K33 0 0 0 0 0 0 K44 0 0 0 0 0 0 K55 0 0 0 0 0 0 K66  1  2  3  1  2  3 (K11) = P L /  1 (K22) = P v /  2 (K33) =  M  3 Group Stiffness Matrix (p v ) M (p v ) M (p v ) Pv (p v ) Pv P L P v (1) M (p L ) PL (Fixed End Moment)
  • THANK YOU!!!