1 / 28

SESSION # 3 STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS

SESSION # 3 STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS. Loads and Axis. Y. F2. Y. M2. M1. F1. F 2. M3. X. M 2. F3. M 1. X. Z. M 3. F 1. F 3. Z. Y. Z. X. X. Z. Y. Column Nodes. P 2. K 11. M 3. K 11. P 1. Foundation Springs in

claudeg
Download Presentation

SESSION # 3 STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SESSION # 3 STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS

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

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

  4. Steps of Analysis • Using SEISAB, calculate the forces at the base of the fixed column (Po, Mo, Pv) • Use S-SHAFT with special shaft head conditions to calculate the stiffness elements of the required stiffness matrix • Longitudinal (X-X) • KF1F1 = K11 = Po / (fixed-head,  = 0) • KM3F1 = K61 = MInduced /  • KM3M3 = K66 = Mo /  (free-head,  = 0) • KF1M3 = K16 = PInduced / 

  5. 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 K11 = PApplied / K66 = MApplied/  K61 = MInduced / K16 = PInduced/ 

  6. Steps of Analysis 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 • 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)

  7. 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?

  8. 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

  9. 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

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

  11. 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 • P, M > P + M

  12. Load Stiffness Curve Shaft-Head Stiffness, K11, K33, K44, K66 P2, M2 P1, M1 Shaft-Head Load, Po, M, Pv

  13. 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

  14. 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

  15. 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

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

  17. Y Y F2 M1 F2 F3 F3 F2 F3 Z Z K33 Y K44 K22 Single shaft K33 = F3/3 K44 = M1/1 K22 = F2/ 2 Z Z Y

  18. Shaft Group with Cap Pv Mo y Po Cap Passive Wedge Shaft Passive Wedge

  19. Pv Mo Po Paxial Kgrot. PCap Ph KLateral KgLateral Krot. (free/fixed) Kaxial Kgaxial With Cap Paxial = Pv/ n + Pfrom Mo Po = Pg = PCap + Ph * n n piles No Cap Paxial = Pv/ n Po = Pg = Ph * n Mshaft = Mo/n Shaft Group(Transverse Loading) (with/without Cap Resistance) Ground Surface

  20. Pv Mo Po Paxial Kgrot. PCap Ph KLateral KgLateral Krot. (free) Kaxial Kgaxial With Cap (always free) Paxial = Pv/ n Po = Pg = PCap + Ph * n Mshaft = Mo/n n piles No Cap Paxial = Pv/ n Po = Pg = Ph * n Mshaft = Mo/n Shaft Group(Longitudinal Loading) (with/without Cap Resistance) Ground Surface

  21. SHAFT GROUP EXAMPLE PROBLEM EXAMPLE PROBLEMS

  22. Single Shaft with Two Different Diameter

  23. Example 3, Shaft Group (WSDOT) (Longitudinal Loading) Pv Mo Po Ground Surface 6 ft 20 ft 52 ft 60 ft 8 ft 20 ft Shaft Group Loads

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

  25. Example 3, Shaft Group (WSDOT) (Transverse Loading) Pv Mo Po 6 ft 52 ft Ground Surface 8 ft 20 ft 20 ft Shaft Group Loads 10 ft 60 ft

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

  27. F V F H P- EFFECT K H K K 1 2 The moment developed at the column base is a function of Fv, FH, and 

More Related