# SESSION # 3 STIFFNESS MATRIX FOR BRIDGE FOUNDATION

SESSION # 3 STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS Loads and Axis Y F2 M2 M3 F3 Y M1 F1 F2 X M2 M1 Z M3 Z F1 F3

X Y Z X X Z Column Nodes K11 Y Y P2 M1 P3 Single Shaft K11 K66 K22

Z Y M3 Y Loading in the Transverse Direction (Axis 3 or Z Axis ) Z P2 P1 K33 K44 K22 K22 K66 Foundation Springs in the Longitudinal Direction Loading in the Longitudinal Direction (Axis 1 or X Axis ) X

X Y 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 / KF1M3 = K16 = PInduced / (free-head, = 0) Linear Stiffness Matrix

Applied M Induced M Applied P X-Axis Induced P =0 =0 A. Zero Shaft-Head Rotation, = 0 B. Zero Shaft-Head Deflection, = 0 K11 = PApplied / K66 = MApplied/ K61 = MInduced / K16 = PInduced/ X-Axis Steps of Analysis F1 1 2

3 1 2 KF1F1 0 0 0 0 -KM3F1 F2 0 KF2F2 0 0 0 0 F3 0 0 KF3F3 KM1F3 0 0 M1

0 0 KF3M1 KM1M1 0 0 M2 0 0 0 0 KM2M2 0 M3 -KF1M3 0 0 0 0 KM3M3 3 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) 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? Linear Stiffness Matrix F1 1 2 3 1 2 3 K11 0 0 0 0 -K61 F2

F3 M1 M2 0 K22 0 0 0 0 0 0 K33 K43 0 0 0 0 K34 K44 0 0 0 0 0 0 K55 0

M3 -K16 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 Actual Scenario Po Nonlinear p-y curve Line Load, p Pv Mo p (Es)1

y p (Es)2 y yM yP Shaft Deflection, y p (Es)3 yP, M yP, M > yP + yM As a result, the linear analysis (i.e. the superposition technique ) can not be employed y p (Es)4 y p (Es)5 y Nonlinear (Equivalent) Stiffness Matrix Applied M

Applied P K11 or K33 = PApplied / K66 or K44 = MApplied/ A. Free-Head Conditions Nonlinear (Equivalent) Stiffness Matrix F1 K11 0 0 0 0 0 1 2 3 1 2 3

F2 F3 M1 M2 0 K22 0 0 0 0 0 0 K33 0 0 0 0 0 0 K44 0 0 0 0 0 0

K55 0 Nonlinear Stiffness Matrix is based on Nonlinear p-y curve Nonlinear shaft material (Varying EI) P, M > P + M P, M > P + M M3 0 0 0 0 0 K66 P2, M2 P1, M1 Shaft-Head Stiffness, K11, K33, K44, K66 Load Stiffness Curve Shaft-Head Load, Po, M, Pv Linear Stiffness Matrix and the Signs of the Off-Diagonal Elements F1 1

2 3 1 2 KF1F1 0 0 0 0 -KM3F1 F2 0 KF2F2 0 0 0 0 F3 0 0 KF3F3 KM1F3 0 0 M1 0

0 KF3M1 KM1M1 0 0 M2 0 0 0 0 KM2M2 0 M3 -KF1M3 0 0 0 0 KM3M3 3 Next Slide Elements of the Stiffness Matrix Y or 2 Y or 2

3 K66 = M3/3 K11 = F1/1 K16 = -F1/3 K61 = -M3/1 Induced F1 X or 1 F1 M3 Z or 3 1 Induced M3 Z or 3 Longitudinal Direction X-X Next Slide X or 1 Y or 2 1

Y or 2 Ind K44 = M1/1 K33 = F3/3 K34 = F3/1 K43 = M1/3 F d e uc 3 F3 X or 1 X or 1 Z or 3 Z or 3 In du

M 1 ce d M 1 3 Transverse Direction Z-Z MODELING OF INDIVIDUAL SHAFTS AND SHAFT GROUPS WITH/WITHOUT SHAFT CAP Y Single shaft F2 K33 = F3/3 F2 F3 F3 K44 = M1/1 K22 = F2/ 2

Y F2 M1 F3 K33 K22 K44 Z Z Z Z Y Y Shaft Group with Cap Pv Po Mo y Cap Passive Wedge

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

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

Po = Pg = PCap + Ph * n Mshaft = Mo/n KLateral Krot. (free) SHAFT GROUP EXAMPLE PROBLEM EXAMPLE PROBLEMS A x ia l L o a d M o o G ro u n d S u rfa c e 5 0 = = 4 2 O , = 7 4 p c f , 0 .0 0 2 5 SAND Layer # 3 Rock q u= 3 0 0 0 0 p s f = 7 5 p c f

5 0 = 0 . 0 0 0 4 D ia m e te r o f S h a ft S e g m e n t # 2 = 6 .0 ft L o n g itu d in a l S te e l x x 3 ft Segm ent # 2 = 1 7 ft D ia m e t e r o f S h a f t S e g m e n t # 1 = 1 0 ft = 6 4 p c f 5 0 = 0 . 0 2 3 7 ft S u= 4 0 0 p s f S e g m e n t # 1 = 4 3 ft Layer # 1 C la y 2 5 ft

P W a te r T a b le L o n g it u d in a l S te e l x x S h a f t W id t h S h a ft W id th S h a ft S e c tio n # 1 N o n lin e a r a n a ly s is S e g m e n t le n g t h = 4 3 f t S h a ft d ia m e te r = 1 0 ft fc o f c o n c re te = 5 K s i fy o f th e s te e l b a rs = 6 0 K s i R a tio o f S te e l b a r s ( A s/A c) = 1 .1 % R a t io o f T r a n s v e r s e s t e e l ( A 's / A c ) = 0 . 5 % C o n c r e te c o v e r = 6 .0 in S h a ft S e c tio n # 2 N o n lin e a r a n a ly s is S e g m e n t le n g th = 1 7 f t S h a f t d ia m e t e r = 6 . 0 f t fc o f c o n c re te = 5 K s i fy o f th e s te e l b a rs = 6 0 K s i R a t io o f S t e e l b a r s ( A s / A c ) = 3 % R a t io o f T r a n s v e r s e s t e e l ( A 's / A c ) = 0 . 5 % C o n c r e te C o v e r = 4 .0 in

Single Shaft with Two Different Diameter Example 3, Shaft Group (WSDOT) (Longitudinal Loading) Ground Surface Pv Mo Po 20 ft 52 ft 8 ft 20 ft 60 ft 6 ft Shaft Group Loads L o n g itu d in a l S te e l S h a ft P r o p e r tie s S h a ft le n g th = 5 2 ft S h a ft d ia m e te r = 8 .0 ft fc o f c o n c r e te = 5 k ip s

R a t io o f s te e l r e b a r s ( A s / A c ) = 1 .5 % x x S h a ft W id t h Example 3, Shaft Group (WSDOT) Longitudinal Loading) Average Shaft (????) Shaft Group Example 3, Shaft Group (WSDOT) (Transverse Loading) Ground Surface Pv Mo Po 52 ft 6 ft 8 ft 20 ft Shaft Group Loads 20 ft

10 ft 60 ft Average Shaft Shaft Group Example 3, Shaft Group (WSDOT) (Transverse Loading) FV FH P- EFFECT KH The moment developed at the column base is a function of F K1v, FH, and K2

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