Transactions, SMiRT-22 GENERAL GUIDELINES FOR APPLICATION OF THE EXTENDED SUBTRACTION METHOD IN SASSI SOIL-STRUCTURE INTERACTION ANALYSIS C. C. Chin 1, Nan Deng 2, and Farhang Ostadan 3 1 Senior Engineer, Bechtel Corporation, San Francisco, CA (ccchin@bechtel.com) 2 Engineering Supervisor, Bechtel Corporation, San Francisco, CA 3 Bechtel Fellow, Chief Engineer, Bechtel Corporation, San Francisco, CA ABSTRACT The computer program SASSI (1999) is widely used for seismic soil-structure interaction (SSI) analysis of nuclear structures. The program offers two main sub-structuring methods: the method and the method. These methods are the basis for computation of the impedance matrix as well as formulation of the equation of motion for typical SASSI SSI models. The difference between the two methods is in the size of impedance problem which is defined by the number of interaction nodes in the free-filed sub-structure model from which the flexibility matrix and subsequently the impedance matrix is computed frequency by frequency. The method, developed in early 1980s as part of the original SASSI program imposes the displacement compatibility between the free-field substructure and the entire excavated soil volume through the common interaction nodes of the two substructures thus the method requires a large group of interaction nodes. However, the strict requirement of the displacement compatibility between the two substructures does make the method capable of producing stable and accurate results within the range of frequencies limited by mesh sizes. In the method, the compatibility of the displacement is maintained for a smaller set of interaction nodes at the common boundary of the free-field substructure and the excavated soil. The limitation of the method associated with maintaining the compatibility of the displacement at limited number of interaction nodes must be recognized and considered in the modeling and analysis. The accuracy of the solution by the method can be improved significantly by extending the method to so called method (ESM). The method is based on the experience of SSI analysis for several verification examples that can in effect reproduce the solution of the method with substantially reduced computational effort. INTRODUCTION The method and the method are provided as two main sub-structuring methods in the computer program SASSI (1999). In the sub-structuring methods, the complete soil-structure interaction (SSI) analysis is subdivided into steps according to each method s own sub-structuring technique. The method is based on a sub-structuring approach enabling simplification of scattering and impedance problems in the SSI analysis considering interaction within a volume of the excavated soil with the free field site. The assumption of a horizontally layered free field site is necessary to make the simplification possible. With the assumption, the scattering problem with a hole in the ground reduces to a free-field site response problem and the impedance problem reduces to dynamic response of a horizontally layered site to a point load. The calculation of the impedance matrix involves performing an inversion of a full flexibility matrix associated with the interaction nodes developed from the axisymmetric point load solution. The method imposes the displacement compatibility between the free-field substructure and the excavated soil volume through the common interaction nodes of the two substructures thus the method requires a large group of interaction nodes. This strict requirement of the
displacement compatibility between the two substructures does make the method capable of producing stable and accurate results within the range of frequencies limited by mesh sizes. In the method, the compatibility of the displacement is maintained for a smaller set of interaction nodes at the common boundary of the free-field substructure and the excavated soil. The method significantly reduces the numerical effort involved in calculation of the impedance matrix for largely embedded structures but suffers from numerical accuracy at high frequencies of the SSI analysis. The accuracy of the solution by the method can be improved significantly by extending the method to so called method (ESM). The Extended method is based on the experience of SSI analysis for several verification examples that can in effect re-produce the solution by the method with substantially reduced computational effort. The method starts by imposing the compatibility of the displacement at the surface of the free-field site as interaction nodes. For deeply embedded structures, additional horizontal layers of interaction nodes may be necessary to ensure stable and accurate results. SCATTERING RESPONSES In this paper, scattering responses of embedded rigid massless cylinders subjected to vertically propagating SV- and P-waves and horizontally propagating SH-wave are computed. Parts of the SV- and SH-wave results of analysis are compared with the solution published by Day (1977). Six cases of SASSI models representing embedded foundations with a dimensionless parameter H of,, and 2.0 are analyzed. The parameter, H, is defined as the ratio of the embedment depth to the radius of the foundation. The SASSI soil layer models consist of 4 (Case 1), 8 (Cases 2 and 4), and 16 (Cases 3, 5, and 6) top layers and a halfspace below the top layers. Each of the top layers has a thickness of 8.125 ft. The halfspace is modeled by an additional 10 layers with varying thicknesses as a function of frequency and a viscous boundary at the base. The shear-wave velocity (Vs) and P-wave velocity (Vp) of the soil layers for the six cases are summarized in Table 1. Other soil properties used are: weight density= 128.68 pcf, Poisson's ratio= 5, and material damping= 1. It is noted that the halfspace in all cases has the same shear-wave (Vs=2000 ft/sec) and P-wave velocity (Vp= 3464.1 ft/sec). The foundations are rigid massless cylinders with a radius (R) of 65 ft and are fully embedded to a depth of 32.5 (Case 1, H=), 65 (Cases 2 and 4), and 130 (Cases 3, 5, and 6, H=2.0) ft. Stiff properties are used to ensure the rigidness of the foundation. MODELING Excavated Soil and Structural Models Half of the cylinder foundations are modeled due to symmetry and discretized using brick elements. Nodes at the bottom of the foundation models are the same for all cases (H=,, and 2.0) as shown in Figure 1(a). The excavated soil models are created similarly to the cylinder foundation models. The finite element models of the excavated soil for the various sub-structure methods in the cases of H=2.0 are also shown Figure 1. It is noted that the symbol, *, is used to indicate interaction nodes in these figures. For example, all nodes of the excavated soil models in the method are interaction nodes as shown in Figure 1(b). SSI Frequencies and Wave Fields The SASSI analysis for this problem is performed for frequencies (f) at 2 Hz and at every 1 Hz from 1 through 35 Hz. The dimensionless frequency ratio, a 0, is computed from
(a) Mesh and Node Numbers at Bottom of Foundation (b) Method (c) Method (d) Method 1 1 additional interaction nodes at surface and mid-depth Note: * s are interaction nodes. Figure 1. Excavated Soil Models for H= 2.0 (e) Method 2 2 additional interaction nodes at surface
a 0 2 f R = π (1) Vs where R is the radius of the cylinder and Vs is the shear wave velocity of the soil material. It is noted that the a 0 is only applicable to Cases 1, 2, and 3 because of the uniform Vs in these cases. Also it is noted that the model mesh constructed for this problem is valid up to the frequency limit of 33.3 Hz corresponding to a 0 = 6.8 for the three cases. The model mesh is valid up to the frequency limit of 16.7 Hz for Cases 4, 5, and 6. The control motion is specified to be vertically propagating SV- or P-wave; or horizontally propagating SH-wave with a control point defined at the ground surface. Table 1: Tables should be centered and preceded by a numbered caption. Case No 1 2 3 4 5 6 Embedment Depth (ft) 32.5 (4 layers) 65.0 (8 layers) 13 (16 layers) 65.0 (8 layers) 13 (16 layers) 13 (16 layers) Vs and Vp (ft/sec) Vs=200 & Vp=3464.1 Vs=200 & Vp=3464.1 Vs=200 & Vp=3464.1 Top 32.5 ft: Vs=200 & Vp=3464.1 Bottom 32.5 ft: Vs=100 & Vp=1732.1 Halfspace: Vs=200 & Vp=3464.1 Top 65.0 ft: Vs=200 & Vp=3464.1 Bottom 65.0 ft: Vs=100 & Vp=1732.1 Halfspace: Vs=200 & Vp=3464.1 Top 65.0 ft: Vs=100 & Vp=1732.1 Bottom 65.0 ft: Vs=200 & Vp=3464.1 Halfspace: Vs=200 & Vp=3464.1 Seismic Loading Vertically Propagating SV- and P-waves; and Horizontally Propagating SH-wave Vertically Propagating SV- and P-waves; and Horizontally Propagating SH-wave Vertically Propagating SV- and P-waves; and Horizontally Propagating SH-wave Vertically Propagating SV- and P-waves Vertically Propagating SV- and P-waves Vertically Propagating SV- and P-waves ANALYSIS RESULTS SV- and SH-Wave Scattering Response of Cases 1, 2, and 3 Two seismic wave loading cases have been analyzed for Cases 1 (H=), 2 (H=), and 3 (H=2.0). It is noted that all three cases have a uniform shear-wave velocity profile (Vs=2000 ft/sec) as shown in Table 1. In the first loading case, the control motion consists of vertically propagating SV-wave with the control point defined at the free surface. The amplitude of the motion, Uo, at the ground surface in the free-field system is for all frequencies with particle displacement in X-direction. In the second loading case, the control motion consists of horizontally propagating SH-wave. The control point is at the ground surface and the particle displacement is in the Y-direction. Similarly, the amplitude of control motion at each frequency is.
It is noted that the method with two additional layers of interaction nodes at the ground surface and at the mid-depth of the foundations are only analyzed for Cases 2 and 3 because the embedment of the foundation in Case 1 is shallow. The amplitude of the translational response motion at Node 1 (X-direction, see Figure 1(a) for locations of nodes) and the amplitude of the rocking motion (Z-direction at Node 66, assuming that the vertical response at Node 1 is negligible) of the foundation subjected to vertically propagating SV-waves for the three cases are normalized to the amplitude of freefield motion and plotted in Figure 2. The results are compared with results reported by Day (1977). As shown in the figure, the results by the method and method with two additional layers of interaction nodes (at the ground surface and at the mid-depth of the embedment) and the results reported by Day are in good agreement in all cases. The accuracy of the results by the method (additional interaction nodes at the ground surface) are in agreement with Day s solutions below a 0 =5 in Case 3 (H=2.0). The method is most efficient and the results are accurate when a 0 is less than 3 in all cases. The horizontal responses of foundation to horizontally propagating SH-wave are shown in terms of the response at Node 1 (Y-direction) and compared with the results reported by Day in Figure 3 for Cases 1, 2 and 3. In addition to the horizontal response motion, SH-waves create some torsional response motion in the foundation. The torsional response is presented by X-direction response at Node 72 (see Figure 1(a) for locations of nodes), assuming that X-direction response at Node 1 is negligible. The results are normalized to the unit amplitude of control motion at the free-field surface and plotted in Figure 3. The results by the method and method with two additional layers of interaction nodes (at the ground surface and at the mid-depth of the embedment) and the results reported by Day are in good agreement in all horizontally propagating SH-wave cases. The accuracy of the results by the method (additional interaction nodes at the ground surface) reduces above a 0 =3 in Case 3 (H=2.0) and the rest of the results are in good agreement with Day s results. As in the vertically propagating SV-wave loading, the method is accurate when a 0 is less than 3 for the horizontally propagating SH-wave loading. SV-Wave Scattering Response of Cases 4, 5, and 6 The loading case of vertically propagating SV-wave with the control point defined at the free surface is analyzed for Cases 4, 5, and 6. The shear-wave velocity (Vs) and P-wave velocity (Vp) of the soil layers in each case are shown in Table 1. In Cases 4 and 5, the shear-wave velocity (Vs) and P-wave velocity (Vp) of the lower half of the soil layers are reduced to 100 and 1732.1 ft/sec, respectively, from those of the uniform soil profiles in Cases 2 and 3. In Cases 6, the Vs and Vp of the upper half of the soil layers are reduced to 100 and 1732.1 ft/sec, respectively, from those of the uniform soil profiles. The amplitude of the motion at the ground surface in the free-field system is for all frequencies with particle displacement in X-direction. The amplitude of the translational response motion at Node 1 (X-direction) and the amplitude of the rocking motion (Z-direction at Node 66, assuming that the vertical response at Node 1 is negligible) of the foundation subjected to vertically propagating SVwaves are normalized to the amplitude of free-field motion and plotted in Figure 4 for the translational and rocking responses of the three cases. As shown in the figures, the results by the method and method with two additional layers of interaction nodes (at the ground surface and at the mid-depth of the embedment) agree with each other in all cases. The accuracy of the results by the method (additional interaction nodes at the ground surface) reduces above a frequency of 9 Hz in Case 5 (H=2.0) and the rest of the results are in good agreement with those by the method. The results by the subtraction method are generally accurate at frequencies of 6 Hz and below.
(a) Translational Response, Case 1, H= (b) Rocking Response, Case 1, H= 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 (c) Translational Response, Case 2, H= (d) Rocking Response, Case 2, H=, 2 Layers, 2 Layers 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 (e) Translational Response, Case 3, H= 2.0 (f) Rocking Response, Case 3, H= 2.0, 2 Layers 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5, 2 Layers 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Figure 2. Translational and Rocking Response of Foundation, Cases 1, 2, and 3, Due to Vertically Propagating SV-Wave
(a) Translational Response, Case 1, H= (b) Torsional Response, Case 1, H= 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Normalized Torsional Response 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 (c) Translational Response, Case 2, H= (d) Torsional Response, Case 2, H=, 2 Layers 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Normalized Torsional Response, 2 Layers 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 (e) Translational Response, Case 3, H= 2.0 (f) Torsional Response, Case 3, H= 2.0, 2 Layers 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Normalized Torsional Response, 2 Layers 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Figure 3. Translational and Torsional Response of Foundation, Cases 1, 2, and 3, Due to Horizontally Propagating SH-Wave
(a) Translational Response, Case 4, H= (b) Rocking Response, Case 4, H= 1.8 1.6 1.4, 2 Layers, 2 Layers 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 (c) Translational Response, Case 5, H= 2.0 (d) Rocking Response, Case 5, H= 2.0 1.8 1.6 1.4, 2 Layers, 2 Layers 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 (e) Translational Response, Case 6, H= 2.0 (f) Rocking Response, Case 6, H= 2.0 1.8 1.6 1.4, 2 Layers, 2 Layers 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 Figure 4. Translational and Rocking Response of Foundation, Cases 4, 5, and 6, Due to Vertically Propagating SV-Wave
P-wave Scattering Response of All Cases P-wave seismic loading cases are analyzed for all cases in Table 1. The control motion consists of vertically propagating P-wave with the control point defined at the free surface. The amplitude of the motion at the ground surface in the free-field system is for all frequencies with particle displacement in Z-direction. The amplitude of the translational response motion at Node 1 (Z-direction) of the foundation subjected to vertically propagating P-waves are normalized to the amplitude of free-field motion and plotted in Figure 5. As shown in the figure, the results by the method and Extended method with two additional layers of interaction nodes (at the ground surface and at the middepth of the embedment) are generally in good agreement up to 14 Hz (Case 5). The accuracy of the results by the method (additional interaction nodes at the ground surface) reduces above 10 Hz in Cases 5 and 6 (H=2.0) and the rest of the results are in good agreement with the results by the method. The method s results are good up to 6 Hz (Case 6).
(a) Translational Response, Case 1, H= (b) Translational Response, Case 2, H=, 2 Layers 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 (c) Translational Response, Case 3, H= 2.0 (d) Translational Response, Case 4, H=, 2 Layers, 2 Layers 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 (e) Translational Response, Case 5, H= 2.0 (f) Translational Response, Case 6, H= 2.0, 2 Layers, 2 Layers 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 2.0 4.0 6.0 8.0 1 12.0 14.0 16.0 Figure 5. Translational Response of Foundation, All Cases, Due to Vertically Propagating P-Wave
CONCLUSION The method is an extension of the method based on the experience of soil-structure interaction (SSI) analysis for several verification examples and nuclear power plant structures that can in effect re-produce the solution by the method with substantially reduced computational effort. For most cases in which the ratio of the embedment depth to the radius of the foundation is less than 2 (H<2), the Extended subtraction method (additional interaction nodes at the ground surface) efficiently produces stable and accurate results. For deeply embedded foundations (H 2), it is recommended that additional layers of interaction nodes should be considered between the ground surface and the bottom of the foundations. REFERENCES University of California at Berkeley (1999). SASSI2000 - A System for Analysis of Soil-Structure Interaction. Day, S. M. (1977). Finite Element Analysis of Seismic Scattering Problems, PhD Thesis, University of California, San Diego, California, USA.