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Prepared by J. P. Singh & Associates in association with Mohamed Ashour, Ph.D., PE

Computer Program DFSAP D eep F oundation S ystem A nalysis P rogram Based on Strain Wedge Method. Prepared by J. P. Singh & Associates in association with Mohamed Ashour, Ph.D., PE West Virginia University Tech and Gary Norris Ph.D., PE University of Nevada, Reno APRIL 3/4, 2006.

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Prepared by J. P. Singh & Associates in association with Mohamed Ashour, Ph.D., PE

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  1. Computer Program DFSAPDeep Foundation System Analysis ProgramBased on Strain Wedge Method • Prepared by • J. P. Singh & Associates • in association with • Mohamed Ashour, Ph.D., PE • West Virginia University Tech • and • Gary Norris Ph.D., PE • University of Nevada, Reno • APRIL 3/4, 2006

  2. Pile and Pile Group Stiffnesses with/without Pile Cap

  3. SESSION I STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVENTIONS • How to Build the Stiffness Matrix of Bridge Pile Foundations (linear and nonlinear stiff. matrix)? • How to Assess the Pile/Shaft Response Based on Soil-Pile-Interaction with/without Soil Liquefaction • (i.e. Displacement & Rotational Stiffnesses)?

  4. Y Z X X Z Column Nodes K11 Foundation Springs in the Longitudinal Direction K22 K66 Y Transverse Longitudinal

  5. Loads and Axis Y F2 Y M2 M1 F1 F2 M3 X M2 F3 M1 X Z M3 F1 F3 Z

  6. x y  zxyz K11 0 0 0 -K15 0 0 K22 0 K24 0 0 0 0 K33 0 0 0 0 K420 K44 0 0 -K51 0 0 0 K55 0 0 0 0 0 0 K66 Force Vector for  x = 1 unit Full Pile Head Stiffness Matrix Lam and Martin (1986) FHWA/RD/86-102

  7. Induced M Applied M   Applied P Applied P = 0  A. Free-Head Conditions B. Fixed-Head Conditions Induced M Applied M  Induced P  = 0 Applied P  = 0  C. Zero Shaft-Head Rotation,  = 0 D. Zero Shaft-Head Deflection,  = 0 Special Conditions for Linear Stiffness Matrix Shaft/Pile-Head Conditions in the DFSAP Program

  8. Y Z X X Y Y Z Column Nodes P2 P2 M3 M1 K11 K11 P1 K33 P3 Foundation Springs in the Longitudinal Direction K22 K66 K66 K44 K22 Y K22 X X Z Z Y Y Loading in the Transverse Direction (Axis 3 or Z Axis ) Loading in the Longitudinal Direction (Axis 1 or X Axis ) Single Shaft

  9. F1 F2 F3 M1 M2 M3 K11 0 0 0 0 -K16 0 K22 0 0 0 0 0 0 K33K34 0 0 0 0 K43K44 0 0 0 0 0 0 K55 0 -K61 0 0 0 0 K66 Steps of Analysis • Using SEISAB (STRUDL), calculate the forces at the base of the fixed column (Po, Mo, Pv) (both directions) • Use DFSAP with special shaft head conditions to calculate the stiffness elements of the required (linear) stiffness matrix. 1 2 3 1 2 3

  10. Applied M Induced M   = 0 Induced P X-Axis Applied P X-Axis  = 0  A. Zero Shaft-Head Rotation,  = 0 B. Zero Shaft-Head Deflection,  = 0 • LINEAR STIFFNESS MATRIX • Longitudinal (X-X) • KF1F1 = K11 = Papplied/1 (fixed-head,  = 0) • KM3F1 = K61 = MInduced / 1 • KM3M3 = K66 = Mapplied / 3(free-head,  = 0) • KF1M3 = K16 = PInduced / 3 K11 = PApplied / K66 = MApplied/  K61 = MInduced / K16 = PInduced/ 

  11. 12312 3 K11 0 0 0 0 -K16 0 K22 0 0 0 0 0 0 K33 K34 0 0 0 0 K43K44 0 0 0 0 0 0 K55 0 -K61 0 0 0 0 K66 Steps of Analysis • Using SEISAB and the above spring stiffnesses at the base of the column, determine the modified reactions (Po, Mo, Pv) at the base of the column (shaft head)

  12. Steps of Analysis • Keep refining the elements of the stiffness matrix used with SEISAB until reaching the identified tolerance for the forces at the base of the column Why KF3M1 KM1F3 ? KF3M1 = K34 =F3 /1 and KM1F3 = K43 = M1 /3 Does the linear stiffness matrix represent the actual behavior of the shaft-soil interaction? 12312 3 F1 F2 F3 M1 M2 M3 KF1F1 0 0 0 0 -KF1M3 0 KF2F2 0 0 0 0 0 0 KF3F3KF3M1 0 0 0 0 KM1F3KM1M1 0 0 0 0 0 0 KM2M2 0 -KM3F1 0 0 0 0 KM3M3

  13. L o n g i t u d i n a l S t e e l Steel Shell x x S h a f t W i d t h Laterally Loaded Pile as a Beam on Elastic Foundation (BEF)

  14. Linear Stiffness Matrix F1 F2 F3 M1 M2 M3 1 2 3 1 2 3 K11 0 0 0 0 -K16 0 K22 0 0 0 0 0 0 K33 K34 0 0 0 0 K43K44 0 0 0 0 0 0 K55 0 -K61 0 0 0 0 K66 • Linear Stiffness Matrix is based on • Linear p-y curve (Constant Es), which is not the case • Linear elastic shaft material (Constant EI), which is not the actual behavior • Therefore, • P, M = P + M and P, M = P + M

  15. Actual Scenario Pv Mo p Po Nonlinear p-y curve ( E ) s 1 y Line Load, p p ( E ) s 2 y yM p Shaft Deflection, y yP ( E ) s yP, M 3 y p yP, M > yP + yM ( E ) s 4 y As a result, the linear analysis (i.e. the superposition technique ) can not be employed p ( E ) s 5 y

  16. Applied M  Applied P  A. Free-Head Conditions Nonlinear (Equivalent) Stiffness Matrix K11 or K33= PApplied / K66 or K44 = MApplied/ 

  17. Nonlinear (Equivalent) Stiffness Matrix F1 F2 F3 M1 M2 M3 1 2 3 1 2 3 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 • Nonlinear Stiffness Matrix is based on • Nonlinear p-y curve • Nonlinear shaft material (Varying EI) • P, M > P + M K11 = Papplied / P, M • P, M > P + M K66 = Mapplied / P, M

  18. Linear Analysis Pile-Head Stiffness, K11, K33, K44, K66 Non-Linear Analysis P2, M2 P1, M1 Pile-Head Load, Po, M, Pv Pile Load-Stiffness Curve

  19. Linear Stiffness Matrix and the Signs of the Off-Diagonal Elements F1 F2 F3 M1 M2 M3 1 2 3 1 2 3 KF1F1 0 0 0 0 -KF1M3 0 KF2F2 0 0 0 0 0 0 KF3F3KF3M1 0 0 0 0 KM1F3KM1M1 0 0 0 0 0 0 KM2M2 0 -KM3F1 0 0 0 0 KM3M3 Next Slide

  20. Y or 2 Y or 2 K11 = F1/1 K61 = -M3/1 K66 = M3/3 K16 = -F1/3 1 X or 1 F1 3 Induced F1 X or 1 Induced M3 Z or 3 M3 Z or 3 Elements of the Stiffness Matrix Longitudinal Direction X-X Next Slide

  21. Y or 2 Y or 2 K44 = M1/1 K34 =F3/1 K33 = F3/3 K43 =M1/3 Induced F3 F3 1 X or 1 X or 1 3 Induced M1 M1 Z or 3 Z or 3 Transverse Direction Z-Z

  22. Linear Stiffness Matrix for Pile group (Lam and Martin, FHWA/RD/86-102)

  23. Linear Analysis Pile-Head Stiffness, K11, K33, K44, K66 Non-Linear Analysis P2, M2 P1, M1 Pile-Head Load, Po, M, Pv Pile Load-Stiffness Curve

  24. Pv M PL (KL)C (KL)2 (KL)1 (KL)3 Rotational angle  (Kv)2 (Kv)1 (Kv)3 Lateral deflection L Axial settlement v (Kv)G (KL)G (KR)G (KL)G =  (KL)i + (KL)C = PL / LL due to lateral/axial loads (Kv)G = Pv / vv due to axial load (Pv) (KR)G = M /   due to moment (M)

  25. Pv Pv M PL M PL (pL)PL (pL)PL (pv)Pv (pv)Pv (pv)Pv (pv)M (pv)M (pv)M (pv)M Pile Cap with Free-Head Piles M PL (Fixed End Moment) (pv)Pv Pile Cap with Fixed-Head Piles Pv z x x z

  26. Pv M Axial Rotational Stiffness of a Pile Group PL Rotational angle  Lateral deflection L Axial settlement v K55 = GJ/L WSDOT MT =  (3.14 D i) D/2 (Li)  = zT / L K55 = MT/ 

  27. Pv Pv (1) M PL M PL (pL)PL (Fixed End Moment) (pv)Pv (pv)Pv (pv)M (pv)M (K22) Group Stiffness Matrix (K11) x x (K66) (K11) = PL / 1 (K22) = Pv / 2 (K33) = M 3 1 2 3 1 2 3 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

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