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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

    Address: 901 12th Ave., Seattle, WA 98122-1090, USA. Tel.: +1 206 296 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


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