# SOIL-STRUCTURE INTERACTION, WAVE PASSAGE EFFECTS AND ASSYMETRY IN NONLINEAR SOIL RESPONSE

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1 SOIL-STRUCTURE INTERACTION, WAVE PASSAGE EFFECTS AND ASSYMETRY IN NONLINEAR SOIL RESPONSE Mihailo D. Trifunac Civil Eng. Department University of Southern California, Los Angeles, CA URL:

2 INTRODUCTION To design structures for dynamic loads, the engineer begins by creating a model representation of the prototype. This model will form the basis for all subsequent analyses, and it will be used to write the governing equations, to describe the structure by discrete or continuous parameters, and to compute the quantities required for design. Thus, this model must have the properties that will describe the response of the prototype as completely and as accurately as possible. The search for a good model is therefore the first and the most important step preceding the analysis and design processes. After the model has been specified, the engineer can analyze, design, and test only that part of the representation of the real structure that the adopted model is capable of representing.

3 MODELING The forward analyses include only the first three steps (within the dashed box): (1) idealization of real structures, (2) mathematical modeling, and (3) analysis of response. The iterative learning process involves two additional tasks: (4) full-scale experimental verification, and (5) revision of the previously adopted models. In the following, we discuss some of those and illustrate their roles by reviewing examples from our previous work. Other modeling issues will involve consideration of simple versus detailed models, reduction of the degrees of freedom, linear versus nonlinear analysis, various forms of dynamic coupling, dissipation mechanisms, soil-structureinteraction, and completeness in representation of strong ground motion.

4 Simple versus detailed models (a, left) Equivalent SDOF system supported by flexible soil. (b, right) MDOF on independent spread footings on elastic soil. A structure responding to long-period (long-wave) dynamic loads, mainly in its first mode of vibration, could be represented by a single-degree-of-freedom (SDOF) system as shown in Fig. a. The same structure excited by high-frequency (shortwaves) dynamic loads will have to be represented by a detailed (continuous or discrete, 2-D or 3-D) model, which is capable of representing relative deformations and the associated forces within the structure (Fig. b), and which is best formulated in terms of the wave propagation approach.

5 In the example shown at left, the deformation of the structure can be specified by only one coordinate and can be described by a one-dimensional wave equation. For structures with smaller height-to-width ratios and with non-uniform distribution of stiffness (right), the use of 2- or 3-D models and wave propagation analysis will offer advantages over vibrational formulation of response, or may be the only alternative.

6 Soil-structure interaction Analysis of soil-structure interaction requires additional degrees of freedom, and depending on the model it may call for the method of solutions in terms of wave propagation. In general terms, soil-structure interaction will lengthen the apparent period of the system, will increase the relative contribution of rocking excitation of ground motion to the total response, and will usually reduce the maximum base shear. Advantages of including soil-structure interaction in the design result from the scattering of incident waves from the body of the foundation and from additional radiation of structural vibration energy into the soil. When the soil surrounding the foundation experiences small-to-modest levels of nonlinear response, soil-structure interaction will lead to significant energy loss of the input wave energy. Since this energy loss occurs outside the structure, if will be one of the important challenges for the future design of safe structures to quantify this loss and to exploit it.

7 FULL-SCALE EXPERIMENTAL VERIFICATION The laboratory tests can be useful, but are never as complete as full-scale experiments. Even the most carefully and completely planned laboratory work will represent only those aspects of the problem that the experimenter chose to study and that were incorporated into the model. That is, the best and the most complete laboratory tests can be used to verify and quantify mainly those aspects of the problem that the investigator knows. Except when fortunate accidents occur, we do not know how to model what we are not aware of and what we do not understand. The full-scale tests present a completely different set of practical problems, but the as-built environment contains all of the physical properties of reality. We only have to find ingenuous ways to discover, record, and interpret them. Another point is that the physical completeness and reality of the full-scale structures is necessary, but not sufficient, to guarantee the correct end results. The discovery and understanding of the true nature of response tend to be born by the difficult labor involving reconciliation between our imperfect theories, modeling, and analyses, with often incomplete data from the measurements. Experienced experimentalists know that the first test rarely produces results, as we inevitably forget to measure something, or what we measure does not turn out to be useful. Thus, iterations are almost a rule, in both experiments and in the analyses.

8 STRONG GROUND MOTION Strong motion includes translational and rotational propagating excitations which all contribute to the total response. Consideration of only horizontal motion can seriously underestimate the total response.

9 WAVE PASSAGE WILL CHANGE AND AMPLIFY RELATIVE RESPONSE

10 Amplification of relative response of SDOF by wave passage for half transit times equal to 0.005, 0.01, 0.02, 0.05, 0.07, and 0.1 s

11 The transit times in the previous figure are the times it takes the wave with horizontal phase velocity C to propagate across one half of the length of the structure. The phase velocity C depends on the wave type. For body waves it is a function of their incident angle, and for surface waves it is determined by the corresponding dispersion curve of the surface wave mode. Assuming that the seismic wave energy arrives vertically results in (1) Infinite values of C, (2) the transit time equal to zero, and (3) reduction of 2D representation to 1D representation of the physical nature of the problem. Remembering that 70% to 80% of strong motion wave energy arrives at the site as surface waves, and that for small epicentral distances it is not likely that the body waves will arrive vertically towards the site, it should be clear that 1D representation of excitation by vertically incident strong motion waves is not realistic.

12 NONLINAR SSI WILL RESULT IN ASYMETRIC PRMANENT STRAINS IN THE SOIL FOR NONVERTICAL WAVE INCIDENCE In this example we consider elasto-plastic soil and linear foundation and building response. We excite the soil by non-vertically incident SH wave pulse and employ finite differences (FD) to calculate the response.

13 Permanent displacements (left) and strains (right) in the soil for small (C = 1.73), intermediate (C = 1.5), and large nonlinearity (C = 0.8). The angle of incidence is 30 degrees from vertical, the amplitude of the pulse is A = 0.05m, and the dimensionless frequency is 1.5. The properties of the three media (SH wave velocity, density, width, height) are: nonlinear soil (250 m/s, 2000 kg/m³, 95.5 m, m), linear rectangular foundation (500 m/s, 2000 kg/m³, 19.1 m, 9.55 m), and linear building (100 m/s, 270 kg/m³, 19.1 m, m).

14 Principal permanent strain in the soil for h = 0.1, two angles of incidence, three foundation stiffnesses, and intermediate nonlinearity C = 1.5.

15 Reduction of wave energy entering the linear building and linear foundation for different levels of soil nonlinearity (C = 0.8, 0.9, 1.1, 1.3, 1.5, and 1.73) and for different foundation rigidities expressed via shear wave velocity in the foundation equal to 250, 300, 500, and 1000 m/s.

16 Reduction of wave energy entering the linear building and linear foundation by scattering, for different levels of soil nonlinearity (C = 0.8, 0.9, 1.1, 1.3, 1.5, and ) and for different foundation rigidities expressed via shear wave velocity in the foundation 300, 500, and 1000 m/s.

17 Asymmetric permanent strains in the soil will lead to amplification of the effective torsional and rocking excitations, and as precursors of liquefaction can create conditions for asymmetry in effective compliances in SSI. With progression of large amplitudes of excitation a combination of those effects will facilitate faster transition to large nonlinear response and to collapse. In the following I show some observational evidence for asymmetry of response and for nonlinearities in the response of soils during SSI.

18 Hollywood Storage Building

19 Hollywood Storage Bldg. This building was not damaged by any of the earthquakes since 1934, but displays large changes in system frequencies. The observed range of smallest to largest frequencies is 1.9 (for EW) to 2.3 (for NS), so far. Relative to the first recording, system frequencies have increased up to 33% (for NS) and decreased 62% (for EW). Largest drop of system frequency during a single earthquake was 69% (from Trifunac et al. 2001).

20 Van Nuys seven story hotel VN7SH

21 Van Nuys Hotel (VN7SH) VN7SH was damaged by San Fernando 1971, and Northridge 1994 earthquakes. From 1966 to 1994 it displayed large changes in system frequencies. The observed range of smallest to largest frequencies is 3.6 (for NS response). Relative to the first recording, system frequencies have increased up to 68% (for NS) and decreased 47% (for EW). The largest drop of system frequency during a single earthquake was 44% (from Trifunac et al. 2001).

22 Changes in system frequencies Change in system frequencies may be due to change of stiffness of the structure, or of the soil, or of the foundation, or of all of the above. The energy of response is concentrated around the frequencies of the soil-structure system. Therefore, Fourier analyses give the system frequencies, which depend on the properties of the soil, structure and foundation. Majority of studies neglect or ignore the soil structure interaction and thus are based on monitoring changes in the system frequency. This leads to erroneous results except when soil structure interaction effects are small, which is rare for typical buildings.

23 Full-scale observations in buildings have shown that the system frequencies drop during strong shaking, but recover partially or totally. Recoverable changes, not related to damage, can reach and exceed 20% (Udwadia and Trifunac 1974, Trifunac et al. 2001a,b; Todorovska et al. 2006). Most methods for structural identification require separation of the effects of the soil on the measured frequencies of vibration and their changes - i.e. require the structural fixed-base frequency. Other applications may require monitoring of changes of the soilfoundation system i.e. require the rigid-body frequencies.

24 Asymmetry of response of Van Nuys Building

25 Contours of NS motion (normalized amplitudes, shown by heavy lines), and time delays (in seconds) relative to the reference station at B2.

26 The foundation system of VN7SH consists of 38 inch deep pile caps, supported by groups of two to four poured-in-place 24 inch diameter reinforced concrete friction piles. These are centered under the main building columns. All pile caps are connected by a grid of the beams. Each pile is roughly 40 feet long and has design capacity of over 100 kips vertical load and up to 20 kips lateral load. The entire foundation system was designed to be symmetric relative to both longitudinal and transverse axes of this building. We discovered large eccentricity in the response of this building during ambient vibration tests following the Northridge earthquake of Subsequent studies of the recorded earthquake responses in this building have confirmed the presence of this asymmetry during all earthquakes following the San Fernando earthquake of 1971.

27 SUMMARY AND CONCLUSIONS Significant part of strong motion energy arrives at the site horizontally as surface waves, and for small epicentral distances and for shallow faulting, it is not likely that the body waves will arrive vertically. Consequently the 1D engineering representations of site excitation by vertically incident strong motion waves is neither realistic nor conservative. Asymmetric permanent strains in the soil will lead to amplification of the effective torsional and rocking excitations, and as precursors of liquefaction may create conditions for asymmetry in effective compliances in SSI. With progression of large amplitudes of excitation a combination of those effects will speed up the transition towards large nonlinear response and the collapse. Numerical models for engineering analyses of SSI must be capable to take as input realistic representations of strong motion waves, consisting of both body and surface waves, with arbitrary angles of incidence. These models must also be capable to model soil, foundation and the structure in the nonlinear range of response.

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