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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 Computer Program DFSAP Deep Foundation System Analysis Program Based on Strain Wedge Method

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Pile and Pile Group Stiffnesses with/without Pile Cap

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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)?

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Y XX Z Z Y Foundation Springs in the Longitudinal Direction K 11 K 22 K 66 Column Nodes Longitudinal Transverse

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Loads and Axis F1 F2 F3 M1 M2 M3 X Z Y F1F1 F2F2 F3F3 M1M1 M2M2 M3M3 X Z Y

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K11 0 0 0 -K15 0 0 K22 0 K24 0 0 0 0 K33 0 0 0 0 K42 0 K44 0 0 -K51 0 0 0 K55 0 0 0 0 0 0 K66 x y z x y z Force Vector for x = 1 unit Full Pile Head Stiffness Matrix Lam and Martin (1986) FHWA/RD/86-102

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

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Y XX Z Z Y Foundation Springs in the Longitudinal Direction K 11 K 22 K 66 Column Nodes Loading in the Longitudinal Direction (Axis 1 or X Axis ) Single Shaft K 22 Y P2P2 K 11 K 66 P1P1 M3M3 Y XX P2P2 K 22 K 33 K 44 P3P3 M1M1 Y Y ZZ Loading in the Transverse Direction (Axis 3 or Z Axis )

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Steps of Analysis Using SEISAB (STRUDL), calculate the forces at the base of the fixed column (P o, M o, P v ) (both directions) Use DFSAP with special shaft head conditions to calculate the stiffness elements of the required (linear) stiffness matrix. K11 0 0 0 0 -K16 0 K22 0 0 0 0 0 0 K33 K34 0 0 0 0 K43 K44 0 0 0 0 0 0 K55 0 -K61 0 0 0 0 K66 F1 F2 F3 M1 M2 M3 123123123123

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K11 = P Applied / K66 = M Applied / K61 = M Induced / K16 = P Induced / B. Zero Shaft-Head Deflection, = 0 = 0 Applied P = 0 Induced P Applied M Induced M A. Zero Shaft-Head Rotation, = 0 X-Axis LINEAR STIFFNESS MATRIX Longitudinal (X-X) KF1F1 = K11 = P applied / 1 (fixed-head, = 0) KM3F1 = K61 = M Induced / 1 KM3M3 = K66 = M applied / 3 (free-head, = 0) KF1M3 = K16 = P Induced / 3

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Steps of Analysis Using SEISAB and the above spring stiffnesses at the base of the column, determine the modified reactions (P o, M o, P v ) at the base of the column (shaft head) 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 1 2 3 1 2 3

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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 K F3M1 K M1F3 ? K F3M1 = K 34 = F 3 / 1 and K M1F3 = K 43 = M 1 / 3 Does the linear stiffness matrix represent the actual behavior of the shaft-soil interaction? KF1F1 0 0 0 0 -KF1M3 0 KF2F2 0 0 0 0 0 0 KF3F3 KF3M1 0 0 0 0 KM1F3 KM1M1 0 0 0 0 0 0 KM2M2 0 -KM3F1 0 0 0 0 KM3M3 F1F2F3M1M2M3F1F2F3M1M2M3 1 2 3 1 2 3

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Laterally Loaded Pile as a Beam on Elastic Foundation (BEF)

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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 ), 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 123123123123

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Shaft Deflection, y Line Load, p y P, M > y P + y M yMyM yPyP 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 MoMo PoPo PvPv Nonlinear p-y curve As a result, the linear analysis (i.e. the superposition technique ) can not be employed Actual Scenario

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Applied P Applied M A. Free-Head Conditions K11 or K33 = P Applied / K66 or K44 = M Applied / Nonlinear (Equivalent) Stiffness Matrix

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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 123123123123

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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

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Linear Stiffness Matrix and the Signs of the Off-Diagonal Elements K F1F1 0 0 0 0 -K F1M3 0 K F2F2 0 0 0 0 0 0 K F3F3 K F3M1 0 0 0 0 K M1F3 K M1M1 0 0 0 0 0 0 K M2M2 0 -K M3F1 0 0 0 0 K M3M3 F1 F2 F3 M1 M2 M3 123123123123 Next Slide

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F1F1 X or 1 Z or 3 Y or 2 Induced M 3 11 K 11 = F 1 / 1 K 61 = -M 3 / 1 X or 1 Z or 3 Y or 2 M3M3 33 K 66 = M 3 / 3 K 16 = -F 1 / 3 Induced F 1 Elements of the Stiffness Matrix Next Slide Longitudinal Direction X-X

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F3F3 X or 1 Z or 3 Y or 2 Induced M 1 33 K 33 = F 3 / 3 K 43 = M 1 / 3 X or 1 Z or 3 Y or 2 11 K 44 = M 1 / 1 K 34 = F 3 / 1 M1M1 Induced F 3 Transverse Direction Z-Z

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

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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

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(K L ) 1 (K L ) 2 (K L ) 3 (K v ) 2 (K v ) 1 (K v ) 3 (K L ) C (K v ) G (K L ) G (K R ) G (K L ) G = (K L ) i + (KL) C = P L / L L d ue to lateral/axial loads (K v ) G = P v / v v d ue to axial load (P v ) (K R ) G = M / d ue to moment (M) PLPL PvPv M Rotational angle Lateral deflection L Axial settlement v

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PLPL PvPv M PvPv (p v ) M (p v ) Pv (p L ) PL PLPL PvPv M Pile Cap with Free-Head Piles x x z z (p v ) M (p v ) Pv PLPL M (p L ) PL Pile Cap with Fixed-Head Piles (Fixed End Moment)

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PLPL PvPv M Rotational angle Lateral deflection L Axial settlement v Axial Rotational Stiffness of a Pile Group K 55 = GJ/LWSDOT M T = (3.14 D i ) D/2 (L i ) = z T / L K 55 = M T /

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PLPL PvPv 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 123123123123 (K11) = P L / 1 (K22) = P v / 2 (K33) = M 3 Group Stiffness Matrix (p v ) M (p v ) Pv PLPL P v (1) M (p L ) PL (Fixed End Moment)

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