STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS

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SESSION 3 STIFFNESS MATRIX FOR BRIDGE
FOUNDATION AND SIGN CONVETIONS
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Loads and Axis
3
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 )
Single Shaft
Z
Z
Y
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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 / ?

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Linear Stiffness Matrix
K11 PApplied /? K66 MApplied/ ? K61
MInduced /? K16 PInduced/ ?
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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 KF3F3 KF3M1 0 0 0
0 KM1F3 KM1M1 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)

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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?
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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
K43 K44 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

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Actual Scenario
Pv
Mo
p
Po
Nonlinear p-y curve
(
E
)
s
1

Line Load, p
y
p
(
E
)
s
2
y
yM
p
Shaft Deflection, y
yP
(
E
)
s
yP, M
3
y
p
yP, M gt 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
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Nonlinear (Equivalent) Stiffness Matrix
K11 or K33 PApplied /? K66 or K44 MApplied/ ?
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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 gt ?P ?M
• ?P, M gt ?P ?M

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Load Stiffness Curve
Shaft-Head Stiffness, K11, K33, K44, K66
P2, M2
P1, M1
Shaft-Head Load, Po, M, Pv
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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 KF3F3 KF3M1 0 0 0
0 KM1F3 KM1M1 0 0 0 0 0
0 KM2M2 0 -KM3F1 0 0
0 0 KM3M3
Next Slide
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Elements of the Stiffness Matrix
Longitudinal Direction X-X
Next Slide
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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
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MODELING OF INDIVIDUAL SHAFTS AND SHAFT GROUPS
WITH/WITHOUT SHAFT CAP
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Y
Single shaft
K33 F3/?3 K44 M1/?1 K22 F2/ ?2
Z
Z
Y
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Shaft Group with Cap
Pv
Mo
y
Po
Cap Passive Wedge
Shaft Passive Wedge
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Shaft Group (Transverse Loading) (with/without
Cap Resistance)
Ground Surface
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Shaft Group (Longitudinal Loading) (with/without
Cap Resistance)
Ground Surface
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SHAFT GROUP EXAMPLE PROBLEM EXAMPLE PROBLEMS
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Single Shaft with Two Different Diameter
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(No Transcript)
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Example 3, Shaft Group (WSDOT) (Longitudinal
Loading)
Ground Surface
6 ft
20 ft
52 ft
60 ft
8 ft
20 ft
Shaft Group Loads
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Example 3, Shaft Group (WSDOT) Longitudinal
Loading)
Average Shaft (????)
Shaft Group
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Example 3, Shaft Group (WSDOT) (Transverse
Loading)
Ground Surface
8 ft
20 ft
20 ft
Shaft Group Loads
10 ft
60 ft
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Example 3, Shaft Group (WSDOT) (Transverse
Loading)
Average Shaft
Shaft Group
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The moment developed at the column base is a
function of Fv, FH, and ?