Short communication Design aids for simplified nonlinear soil–structure interaction analyses 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 2011 Revised 25 September 2011 Accepted 3 October 2011 Available online 18 November 2011 a b s t r a c t This manuscript presents the development of charts that structural engineering practitioners can use to quickly quantify the restraining effect at the bases of columns and walls supported on shallow founda- tions. The formulation is derived based on compatibility, equilibrium, and a simplified constitutive rela- tion that uses two basic parameters to characterize the soil-foundation: the subgrade modulus – the stiffness parameter and the ultimate bearing capacity – the strength parameter. The model captures the intrinsic non-linear behavior of the soil with increasing loading, and the coupling between moments and axial loads. Using the proposed charts, calculat ed rocking responses for rigid footing models under combined axial load and moment are found to compare reasonably well with experimental results. 2011 Elsevier Ltd. All rights reserved. 1. Introduction Seismic rehabilitation standards [1] for building s loca ted in moder ate and high seismic risk zones requir e consi dera tion of the interaction between the structure and the supporting soil. Studies on soil–structure interaction problems can be traced as far back as the work conduct ed by Lamb [2], more th an one hundred years ago, on the propagation of elastic waves induced by the app lic ati on of a poi nt load on the surf ace of an ela sti c semi-infinite media. Bycroft [3] was perhaps the first to present a complete set of solutions for vertical and horizontal translation, and rotation about a horizontal axis (rocking) and about a vertical axis (torsion) of a rigid circular plate on an elastic half-space. A compr ehen sive presenta tion of deca des the of theo reti cal and experimental research on vibration of foundations is presented in the classic textbook by Richart et al. [4] . Some of the first studies on soil–structure interaction that used the analytical and experimental results from elastic half-space the- ory were carried out by Hall [5] , Parmelee [6] and Parmelee et al. [7]. Althou gh the main limita tion wa s the dep enden cy of the imp eda nce on the exc iti ng fre quenc y, the se be came the fir st attempts to establish a bridge betwe en the elastic half space th eory and the mass-spring-dashpot system. 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 incorporated into the pre-standard FEMA 273 [9]. The formulas were derived on the 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) vert ical displace ment, (v) rock- ing coupled with lateral sway. In recent studies, the dynamic response of footings has been described by using macro models that capture the coupled nonlin- ear material and uplift response at the soil-foundation interface [10–12] . Macro elements consist of joint elements in global coordi- nates and variables (forces and displacements) located at the base of columns and walls, which can be directly incorporated in non- linear finite element models of the entire soil-foundation- structure systems [11]. In addition to macro models, detailed soil-foundation models using beam-on-nonlinear-Winkler-foundations have also been deve lope d and calibrated thro ugh ext ensive expe rime ntal programs which range from monotonic to cyclic loading and cen- trifuge tests [13–15]. The macro models and beam-on-nonlinear-Winkler foundation models 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 structure s allows modeling the foundation by means of a set of uncoupled ela sto -pl as tic spr ing s at the ba se of col umns and wa lls . The mome nt capacity, in part icul ar, is calc ulat ed usin g Mey erho f’s equivalent width concept [16] as further described in Section 4 of this manuscript. The static stiffnesses, on the other hand, consist of simplified expressions obtained by Pais and Kausel [17] using elastic half space theory; in the case of rocking about the short axis of a footing, the static rotational stiffness for a shallow foundation is given by: K h ¼ GB 3 1 m 0:4 L B þ 0:1 ð1Þ 0141-0296/$ - see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.10.005 ⇑ Address: 901 12th Ave., Seattle, WA 98122-1090, USA. Tel.: +1 206 296 5901; fax: +1 206 296 2173. E-mail address: [email protected] Engineering Structures 34 (2012) 572–580 Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct