wsdot 04 06(v stiffness) apresentacao
TRANSCRIPT
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Prepared by
J. P. Singh & Associates
in association withMohamed Ashour, Ph.D., PE
West Virginia University Techand
Gary Norris Ph.D., PE
University of Nevada, Reno
APRIL 3/4, 2006
Computer Program DFSAP
Deep Foundation System Analysis ProgramBased on Strain Wedge Method
Washington StateDepartment of Transportation
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Pile and Pile Group Stiffnesseswith/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
X X
Z
Z
Y
Foundation Springs in
the Longitudinal Direction
K11
K22K66
Column Nodes
Longitudinal
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Loads and Axis
F1
F2
F3
M1M2
M3 X
Z
Y
F1
F2
F3
M1
M2
M3X
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
= 0Induced PApplied 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 forLinear Stiffness Matrix
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Y
X X
Z
Z
Y
Foundation Springs inthe 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)
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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
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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 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
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 forthe forces at the base of the column
Why KF3M1KM1F3 ?
KF3M1 = K34 =F3 /1 and KM1F3 = K43= M1/3Does 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
M1M2
M3
1 2 3 1 2 3
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y
p
(Es)1
Po
(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)
ShaftWidth
x x
Longitudinal
Steel
Steel Shell
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Linear Stiffness Matrix
K11 0 0 0 0 -K160 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
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
1
2
3
1
2
3
A t l S i
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Shaft Deflection, y
LineLoad,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
MoPo
Pv
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= PApplied/
K66 or K44 = MApplied/
Nonlinear (Equivalent) Stiffness Matrix
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Nonlinear (Equivalent) Stiffness Matrix
K11 0 0 0 0 00 K22 0 0 0 0
0 0 K33 0 0 0
0 0 0 K44 0 0
0 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, MP, M> P+ M K66 = Mapplied/ P, M
1
2
3
1
2
3
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Pile Load-Stiffness Curve
Linear Analysis
Pile-He
adStiffness,K
11,
K33,
K44,K
66
Pile-Head Load, Po, M, Pv
P1,
M1
P2,
M2
Non-Linear Analysis
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Linear Stiffness Matrixand
the Signs of the Off-Diagonal Elements
KF1F1 0 0 0 0 -KF1M3
0 KF2F2 0 0 0 00 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
Next Slide
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F1 X or 1
Z or 3
Y or 2
Induced M3
1
K11= F1/1K61 =-M3/1
X or 1
Z or 3
Y or 2
M3
3
K66= M3/3K16 =-F1/3
Induced F1
Elements of the Stiffness Matrix
Next SlideLongitudinal Direction X-X
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F3X or 1
Z or 3
Y or 2
K33 = F3/3K43 =M1/3
X or 1
Z or 3
Y or 2
1
K44= M1/1K34 =F3/1
Transverse Direction Z-Z
Linear Stiffness Matrix for Pile group
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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-HeadStiffness,K
11,
K33,
K44,K
66
Pile-Head Load, Po, M, Pv
P1,
M1
P2,
M2
Non-Linear Analysis
P
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(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 Ldue to lateral/axial loads
(Kv)G= Pv / v vdue to axial load (Pv)
(KR)G= M / due to moment (M)
PL
Pv
M
Rotational angle
Lateral deflection L
Axial settlement v
P P
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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)
P
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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
/ L
K55= MT/
P
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PL
Pv
M
(K22)(K11)
(K66) xx
K11 0 0 0 0 00 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
1
2
3
1
2
3
(K11) = PL/ 1
(K22) = Pv/ 2
(K33) = M3
Group Stiffness Matrix
(pv)M(pv)M
(pv)Pv(pv)Pv
PL
Pv (1)
M
(pL)PL
(Fixed End Moment)
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