SESSION # 3 STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS

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

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 P2P2 K 22 K 11 K 66 P1P1 M3M3 Y Y XX P2P2 K 22 K 33 K 44 P3P3 M1M1 Y Y ZZ Loading in the Transverse Direction (Axis 3 or Z Axis )

Steps of Analysis Using SEISAB, calculate the forces at the base of the fixed column (P o, M o, P v ) Use S-SHAFT with special shaft head conditions to calculate the stiffness elements of the required stiffness matrix Longitudinal (X-X) K F1F1 = K 11 = P o /  (fixed-head,  = 0) K M3F1 = K 61 = M Induced /  K M3M3 = K 66 = M o /  (free-head,  = 0) K F1M3 = K 16 = P Induced / 

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

Steps of Analysis K F1F K F1M3 0 K F2F K F3F3 K F3M K M1F3 K M1M K M2M2 0 -K M3F K M3M3 F1 F2 F3 M1 M2 M3 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) 123123123123

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?

Linear Stiffness Matrix K K 16 0 K K 33 K K 43 K K K 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 123123123123

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

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

K K K K K 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  P, M >  P +  M 123123123123

Shaft-Head Stiffness, K11, K33, K44, K66 Load Stiffness Curve Shaft-Head Load, P o, M, P v P 1, M 1 P 2, M 2

Linear Stiffness Matrix and the Signs of the Off-Diagonal Elements K F1F K F1M3 0 K F2F K F3F3 K F3M K M1F3 K M1M K M2M2 0 -K M3F K M3M3 F1 F2 F3 M1 M2 M3 123123123123 Next Slide

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

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

MODELING OF INDIVIDUAL SHAFTS AND SHAFT GROUPS WITH/WITHOUT SHAFT CAP

K 33 = F 3 /  3 K 44 = M 1 /  1 K 22 = F 2 /  2 F2F2 F3F3 M1M1 Y Y ZZ F2F2 F3F3 F2F2 F3F3 K 22 K 33 K 44 Y Y ZZ Single shaft

PvPv PoPo MoMo y Cap Passive Wedge Shaft Passive Wedge Shaft Group with Cap

Ground Surface Shaft Group (Transverse Loading) (with/without Cap Resistance) With Cap P axial = P v / n + P from Mo P o = Pg = P Cap + P h * n P Cap P axial PhPh K axial K Lateral K rot. (free/fixed) n piles Kg axial Kg Lateral Kg rot. No Cap P axial = P v / n P o = Pg = P h * n M shaft = M o /n PvPv PoPo MoMo

Ground Surface Shaft Group (Longitudinal Loading) (with/without Cap Resistance) With Cap (always free) P axial = P v / n P o = Pg = P Cap + P h * n M shaft = M o /n P Cap P axial PhPh K axial K Lateral K rot. (free) n piles Kg axial Kg Lateral Kg rot. No Cap P axial = P v / n P o = Pg = P h * n M shaft = M o /n PvPv PoPo MoMo

SHAFT GROUP EXAMPLE PROBLEM EXAMPLE PROBLEMS

Single Shaft with Two Different Diameter

Example 3, Shaft Group (WSDOT) (Longitudinal Loading) Shaft Group Loads Ground Surface 60 ft 20 ft 8 ft 52 ft 6 ft PvPv PoPo MoMo

Average Shaft (????) Shaft Group Example 3, Shaft Group (WSDOT) Longitudinal Loading)

Example 3, Shaft Group (WSDOT) (Transverse Loading) Shaft Group Loads Ground Surface 60 ft 20 ft 8 ft 52 ft 6 ft 10 ft PvPv PoPo MoMo

Average Shaft Shaft Group Example 3, Shaft Group (WSDOT) (Transverse Loading)

The moment developed at the column base is a function of F v, F H, and 