Geoscience Reference
In-Depth Information
0.25
N
Winkler model without P
effects
FE analysis with P
−δ
effects
−δ
0.2
12m
M
0.15
Q
0.1
0.05
6m
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N: N
u
Fig. 9.14. Finiteelement obtained N-Mfailurecurve for aslender structure on shallow
foundation withP-
δ
effects
isachieved.Moreover,thepassagetotheinelasticsoildomainalsooccursat
χ =
0,when
(
M
elast
)
=
.
=
/
max
0
0833N
u
B
1
6N
u
b
The above
diagram is in excellent accord with the lower-bound solution of
Houlsby and Purzin (1999). It refers however to loads applied at the foundation level,
i.e. without P-
(
M
u
:
N
u
)
effects. When the latter are taken into account, there is a slight drop
in the ultimate capacity. For the aforesaid slender geometry of the bridge pier
δ
(
2h
=
12m
,
B
=
2b
=
6m
)
this drop, asseen inFigure 9.14, isabout 25% sothat now:
maxM
u
≈
0
.
10N
u
B
=
1
/
5N
u
b
More significant however seems to be the reduction in ductility capacity of the P-
δ
affected system, as seen inFigure 9.12.
For completeness, also sketched in Figure 9.14 is the plot on the FE mesh of the
σ
ν
isostress contours, atM
≈
(
2
/
3
)
maxM
u
.
3.3.2. Seismic response
The significance of uplifting and soil inelasticity on the seismic response of the slender
structure of Figure 9.10 is explained in Figures 9.15 and 9.16. Seismic excitation in the
formofalong-periodRickerpulse
isappliedatthebedrock
andispropagatedthroughsoiltoproduceafree-field“inputmotion”ofadominantperiod
T
E
=
(
T
E
=
2
.
2s
,
PGA
=
0
.
20g
)
34g. Such a Ricker-type excitation represents long-period
pulses that areoften attributed tonear-source rupture-directivity effects.
1
.
8s and PGA
=
0
.
The response is highlighted in terms of M-
θ
and M-w hysteresis loops (Figure 9.15), as
wellastime-historiesofrotation
,settlementw,andaccelerationA
cm
atthesuperstruc-
ture (Figure 9.16). Two cases are considered:
θ
•
a “light” weight structure: N
=
(
1
/
4
)
N
u
=
1000kN,corresponding to
χ =
0
.
25
•
a“heavy” weight structure: N
=
(
3
/
4
)
N
u
=
3000kN,corresponding to
χ =
0
.
75
