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1. Introduction
Seismic rehabilitation standards [1] for buildings located
inmoderate and high seismic risk zones rethe interaction between
the structure and
Studies on soilstructure interaction profar back as the work
conducted by Lam
tion ofon thperhaal and(rockinte on
General formulas and charts for the calculation of
impedances(dynamic stiffness and damping) of surface and embedded
founda-tions of any shape were presented by Gazetas [8] and
incorporatedinto the pre-standard FEMA 273 [9]. The formulas were
derived onthe basis of Finite Element and Boundary Element analyses
and
included eight modes of vibration: (i) lateral sway (2
directions),(ii) rocking (2 axis), (iii) torsion, (iv) vertical
displacement, (v) rock-ing coupled with lateral sway.
of this manuscript. The static stiffnesses, on the other hand,
consistof simplied expressions obtained by Pais and Kausel [17]
usingelastic half space theory; in the case of rocking about the
short axisof a footing, the static rotational stiffness for a
shallow foundationis given by:
Kh GB3
1 m 0:4LB
0:1
1
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206 296 5901;fax: +1 206 296 2173.
Engineering Structures 34 (2012) 572580
Contents lists available at
g
lseE-mail address: [email protected]
presentation of decades the of theoretical andexperimental research
on vibration of foundations is presented inthe classic textbook by
Richart et al. [4].
Some of the rst studies on soilstructure interaction that
usedthe analytical and experimental results from elastic half-space
the-ory were carried out by Hall [5], Parmelee [6] and Parmelee et
al.[7]. Although the main limitation was the dependency of
theimpedance on the exciting frequency, these became the
rstattempts to establish a bridge between the elastic half space
theoryand the mass-spring-dashpot system.
programs which range from monotonic to cyclic loading and
cen-trifuge tests [1315].
The macro models and beam-on-nonlinear-Winkler foundationmodels
can be very precise but often require test results to cali-brate
the multiple parameters involved.
Standard ASCE 41-06 for the seismic rehabilitation of
structuresallows modeling the foundation by means of a set of
uncoupledelasto-plastic springs at the base of columns and walls.
Themoment capacity, in particular, is calculated using
Meyerhofsequivalent width concept [16] as further described in
Section 4hundred years ago, on the propagaby the application of a
point loadsemi-innite media. Bycroft [3] wascomplete set of
solutions for verticand rotation about a horizontal axisaxis
(torsion) of a rigid circular pla0141-0296/$ - see front matter
2011 Elsevier Ltd. Adoi:10.1016/j.engstruct.2011.10.005quire
consideration ofthe supporting soil.blems can be traced asb [2],
more than oneelastic waves inducede surface of an elasticps the rst
to present ahorizontal translation,g) and about a verticalan
elastic half-space. A
In recent studies, the dynamic response of footings has
beendescribed by using macro models that capture the coupled
nonlin-ear material and uplift response at the soil-foundation
interface[1012]. Macro elements consist of joint elements in global
coordi-nates and variables (forces and displacements) located at
the baseof columns and walls, which can be directly incorporated in
non-linear nite element models of the entire
soil-foundation-structuresystems [11]. In addition to macro models,
detailed soil-foundationmodels using
beam-on-nonlinear-Winkler-foundations have alsobeen developed and
calibrated through extensive experimentalShort communication
Design aids for simplied nonlinear soil
J. Paul Smith-Pardo Department of Civil and Environmental
Engineering, Seattle University, Seattle, WA, USA
a r t i c l e i n f o
Article history:Received 17 February 2011Revised 25 September
2011Accepted 3 October 2011Available online 18 November 2011
a b s t r a c t
This manuscript presents tquickly quantify the restrations. The
formulation is dtion that uses two basic pstiffness parameter and
ththe intrinsic non-linear behand axial loads. Using thecombined
axial load and m
Engineerin
journal homepage: www.ell rights reserved.ructure interaction
analyses
development of charts that structural engineering practitioners
can use tog effect at the bases of columns and walls supported on
shallow founda-ed based on compatibility, equilibrium, and a
simplied constitutive rela-meters to characterize the
soil-foundation: the subgrade modulus theltimate bearing capacity
the strength parameter. The model capturesor of the soil with
increasing loading, and the coupling between momentsposed charts,
calculated rocking responses for rigid footing models underent are
found to compare reasonably well with experimental results.
2011 Elsevier Ltd. All rights reserved.
Structures
vier .com/locate /engstructSciVerse ScienceDirect
where, B and L are the footings width (perpendicular to the axis
ofrotation) and length, G and m are the effective shear modulus
andPoissons ratio of the soil.
ASCE 41-06 establishes four different levels of analysis for
theseismic rehabilitation of structures which, in order of
complexityare: linear static procedure (LSP), linear dynamic
procedure (LSP),nonlinear static procedure (NSP), and nonlinear
dynamic proce-dure (NDP). Nonlinear procedures are permitted for
any of therehabilitation strategies dened in this Standard. Upper
and lowerbound models of the soil-foundation need to be considered
byusing half and twice of the best estimate of the elastic
stiffnessand corresponding plateau. It is apparent, therefore, that
practi-tioner engineering standards advocate the use of simple
methodsgiven the inherent uncertain nature of the input parameters
forsoil-foundation interaction models.
This manuscript presents the development of
non-dimensionalcharts for the estimation of the rocking response of
footings undercombined axial load and moment. The formulation is
useful to
along the width of the footing as given by Eq. (4) but modied
to
J.P. Smith-Pardo / Engineering Strperform nonlinear static
lateral load (pushover) analyses ofsoil-foundation-structure
systems. It is not intended to serve fornonlinear dynamic analyses
as the inuence of inertial effects isnot addressed and because the
formulation assumes monotonicloading only. The charts account for
nonlinearities such as soften-ing of the soil and foundation
partial loss of contact and also for thecoupling between axial load
and moment. The formulation is basedon the foundation subgrade
modulus as the stiffness parameterand the bearing capacity as the
strength parameter. Issues relatedto scale effects for these input
parameters are beyond the scope ofthis paper.
2. Soil-foundation model
Consider a soil for which the average normal stress (r
versusnormalized settlement (dn = dB) response is given
schematicallyby Fig. 1. The use of normalized settlement is
advantageous be-cause it alleviates foundation size effects [1820].
Adjusting the re-sponse to a hyperbolic function, the relation can
be written interms of the initial slope-Ks0B, and the asymptote,
r0:
r Ks0Bdn1 ndn 2
where, n is the stiffness to strength ratio:
n Ks0Br0
3
/B
= P/(BL)
Ks0B
10
Conditions for (/B):
Hyperbolic fit: (/B)
Test results
P
B
Concentric load test
0B/0s )B/(
)B/(BK=
=
)B/(limB/0
=Fig. 1. Concentric load test in terms of subgrade modulus and
bearing capacitystress.account for foundation partial loss of
contact:
K5a 1
1 ndna for dnaP 00 for dna 6 0
8
