STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS - PowerPoint PPT Presentation

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STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS

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SESSION # 3. STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS. Loads and Axis ... Direction (Axis 3 or Z Axis ) Steps of Analysis ... – PowerPoint PPT presentation

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Title: STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS


1
SESSION 3 STIFFNESS MATRIX FOR BRIDGE
FOUNDATION AND SIGN CONVETIONS
2
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
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
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 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)

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
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

9
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
10
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 gt ?P ?M
  • ?P, M gt ?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 KF3F3 KF3M1 0 0 0
0 KM1F3 KM1M1 0 0 0 0 0
0 KM2M2 0 -KM3F1 0 0
0 0 KM3M3
Next Slide
14
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
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
Shaft Group (Transverse Loading) (with/without
Cap Resistance)
Ground Surface
20
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
(No Transcript)
24
Example 3, Shaft Group (WSDOT) (Longitudinal
Loading)
Ground Surface
6 ft
20 ft
52 ft
60 ft
8 ft
20 ft
Shaft Group Loads
25
Example 3, Shaft Group (WSDOT) Longitudinal
Loading)
Average Shaft (????)
Shaft Group
26
Example 3, Shaft Group (WSDOT) (Transverse
Loading)
Ground Surface
8 ft
20 ft
20 ft
Shaft Group Loads
10 ft
60 ft
27
Example 3, Shaft Group (WSDOT) (Transverse
Loading)
Average Shaft
Shaft Group
28
The moment developed at the column base is a
function of Fv, FH, and ?
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