Title: STIFFNESS MATRIX FOR BRIDGE FOUNDATION AND SIGN CONVETIONS
1SESSION 3 STIFFNESS MATRIX FOR BRIDGE
FOUNDATION AND SIGN CONVETIONS
2Loads and Axis
3Y
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
4Steps 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 / ?
5Linear Stiffness Matrix
K11 PApplied /? K66 MApplied/ ? K61
MInduced /? K16 PInduced/ ?
6Steps 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)
7Steps 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?
8Linear 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
9Actual 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
10Nonlinear (Equivalent) Stiffness Matrix
K11 or K33 PApplied /? K66 or K44 MApplied/ ?
11Nonlinear (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
12Load Stiffness Curve
Shaft-Head Stiffness, K11, K33, K44, K66
P2, M2
P1, M1
Shaft-Head Load, Po, M, Pv
13Linear 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
14Elements of the Stiffness Matrix
Longitudinal Direction X-X
Next Slide
15Y 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
16MODELING OF INDIVIDUAL SHAFTS AND SHAFT GROUPS
WITH/WITHOUT SHAFT CAP
17Y
Single shaft
K33 F3/?3 K44 M1/?1 K22 F2/ ?2
Z
Z
Y
18Shaft Group with Cap
Pv
Mo
y
Po
Cap Passive Wedge
Shaft Passive Wedge
19Shaft Group (Transverse Loading) (with/without
Cap Resistance)
Ground Surface
20Shaft Group (Longitudinal Loading) (with/without
Cap Resistance)
Ground Surface
21SHAFT GROUP EXAMPLE PROBLEM EXAMPLE PROBLEMS
22Single Shaft with Two Different Diameter
23(No Transcript)
24Example 3, Shaft Group (WSDOT) (Longitudinal
Loading)
Ground Surface
6 ft
20 ft
52 ft
60 ft
8 ft
20 ft
Shaft Group Loads
25Example 3, Shaft Group (WSDOT) Longitudinal
Loading)
Average Shaft (????)
Shaft Group
26Example 3, Shaft Group (WSDOT) (Transverse
Loading)
Ground Surface
8 ft
20 ft
20 ft
Shaft Group Loads
10 ft
60 ft
27Example 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 ?