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Prepared byJ. P. Singh & Associates
in association with
Mohamed Ashour, Ph.D., PE West Virginia University Tech
andGary Norris Ph.D., PE
University of Nevada, Reno
APRIL 3/4, 2006
Computer Program DFSAPDeep Foundation System Analysis Program
Based on Strain Wedge Method
Washington StateDepartment of Transportation
Pile and Pile Group Stiffnesses with/without Pile Cap
SESSION ISTIFFNESS MATRIX FOR BRIDGE STIFFNESS MATRIX FOR BRIDGE
FOUNDATION AND SIGN CONVENTIONSFOUNDATION 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 onSoil-Pile-Interaction with/without Soil Liquefaction(i.e. Displacement & Rotational Stiffnesses)?
Y
X X
Z
Z
Y
Foundation Springs in the Longitudinal Direction
K11
K22K66
Column Nodes
Longitudinal
Transvers
e
Loads and Axis
F1
F2
F3
M1M2
M3 X
Z
Y
F1
F2
F3
M1
M2
M3
X
Z
Y
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
= 0
Applied P
Applied M
Applied P
Induced M
A. Free-Head Conditions B. Fixed-Head Conditions
= 0
Applied P
= 0Induced P
Applied MInduced 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
Y
X X
Z
Z
Y
Foundation Springs in the Longitudinal Direction
K11
K22K66
Column Nodes
Loading in the Longitudinal Direction (Axis 1 or X Axis )
Single Shaft
K22
Y
P2
K11
K66
P1
M3
Y
X X
P2
K22
K33
K44
P3
M1
Y
Y
Z Z
Loading in the Transverse Direction (Axis 3 or Z Axis )
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.
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
1
2
3
1
2
3
B. Zero Shaft-Head Deflection, = 0
= 0
Applied P
= 0Induced P
Applied MInduced M
A. Zero Shaft-Head Rotation, = 0
X-Axis X-Axis
LINEAR STIFFNESS MATRIXLongitudinal (X-X)
KF1F1 = K11 = Papplied /1 (fixed-head, = 0) KM3F1 = K61 = MInduced / 1
KM3M3 = K66 = Mapplied / 3 (free-head, = 0)
KF1M3 = K16 = PInduced / 3
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)
K11 0 0 0 0 -K16
0 K22 0 0 0 00 0 K33 K34 0 00 0 K43 K44 0 00 0 0 0 K55 0-K61 0 0 0 0 K66
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 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?
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
F1
F2
F3
M1
M2
M3
1 2 3 1 2 3
y
p(Es)1
P o
(Es)3
(Es)4
(Es)2p
p
p
y
y
y
(Es)5
p
y
Laterally Loaded Pile as a Beam on Elastic Foundation (BEF)
Shaft Width
x x
Longitudinal Steel
Steel Shell
Linear Stiffness Matrix
K11 0 0 0 0 -K16
0 K22 0 0 0 00 0 K33 K34 0 00 0 K43 K44 0 00 0 0 0 K55 0-K61 0 0 0 0 K66
F1 F2 F3 M1 M2 M3
• 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 behaviorTherefore,
P, M = P + M and P, M = P + M
1
2
3
1
2
3
Shaft Deflection, y
Lin
e L
oad
, p
yP, M > yP + yM
yM
yPyP, M
y
p(Es)1
(Es)3
(Es)4
(Es)2p
p
p
y
y
y
(Es)5
p
y
Mo
Po
Pv
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 = PApplied /
K66 or K44 = MApplied/
Nonlinear (Equivalent) Stiffness Matrix
Nonlinear (Equivalent) Stiffness Matrix
K11 0 0 0 0 00 K22 0 0 0 00 0 K33 0 0 00 0 0 K44 0 00 0 0 0 K55 00 0 0 0 0 K66
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 K11 = Papplied / P, M
P, M > P + M K66 = Mapplied / P, M
1
2
3
1
2
3
Pile Load-Stiffness Curve
Linear Analysis
Pile
-Hea
d S
tiff
nes
s, K
11, K
33, K
44, K
66
Pile-Head Load, Po, M, Pv
P 1, M
1
P 2, M
2
Non-Linear Analysis
Linear Stiffness Matrix and
the Signs of the Off-Diagonal Elements
KF1F1 0 0 0 0 -KF1M3
0 KF2F2 0 0 0 00 0 KF3F3 KF3M1 0 00 0 KM1F3 KM1M1 0 00 0 0 0 KM2M2 0-KM3F1 0 0 0 0 KM3M3
F1 F2 F3 M1 M2 M3
1
2
3
1
2
3
Next Slide
F1X or 1
Z or 3
Y or 2
Induced M3
1
K11 = F1/1
K61 = -M3/1
X or 1
Z or 3
Y or 2
M3
3
K66 = M3/3
K16 = -F1/3
Induced F1
Elements of the Stiffness Matrix
Next SlideLongitudinal Direction X-X
F3X or 1
Z or 3
Y or 2
Indu
ced M
1
3
K33 = F3/3
K43 = M1/3
X or 1
Z or 3
Y or 2
1
K44 = M1/1
K34 = F3/1
M 1
Induced F 3
Transverse Direction Z-Z
(Lam and Martin, FHWA/RD/86-102)
Linear Stiffness Matrix for Pile group
Pile Load-Stiffness Curve
Linear Analysis
Pile
-Hea
d S
tiff
nes
s, K
11, K
33, K
44, K
66
Pile-Head Load, Po, M, Pv
P 1, M
1
P 2, M
2
Non-Linear Analysis
(KL)1(KL)2(KL)3
(Kv)2 (Kv)1(Kv)3
(KL)C
(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)
PL
Pv
M
Rotational angle
Lateral deflection L Axial settlement v
PL
Pv
M
Pv
(pv)M(pv)M
(pv)Pv(pv)Pv
(pL)PL
PL
Pv
M
Pile Cap with Free-Head Piles
xx
z
z
(pv)M(pv)M
(pv)Pv(pv)Pv
PLM
(pL)PL
Pile Cap with Fixed-Head Piles
(Fixed End Moment)
PL
Pv
M
Rotational angle
Lateral deflection L Axial settlement v
Axial Rotational Stiffness of a Pile Group
K55 = GJ/L WSDOT
MT = (3.14 D i) D/2 (Li)
= zT / LK55 = MT/
PL
Pv
M
(K22)(K11)
(K66) xx
K11 0 0 0 0 00 K22 0 0 0 00 0 K33 0 0 00 0 0 K44 0 00 0 0 0 K55 00 0 0 0 0 K66
1
2
3
1
2
3
(K11) = PL / 1
(K22) = Pv / 2
(K33) = M 3
Group Stiffness Matrix
(pv)M(pv)M
(pv)Pv(pv)Pv
PL
Pv (1)
M
(pL)PL
(Fixed End Moment)
