OHN10109411184
The Inspection of Site Characteristics on Far Field Dynamic Stiffness of Unbounded Media in Coupled Soil-Structure Interaction Analyses Using
the Cone Model
Hossein Rahnema
1, Abdolhossein Baghlani
2, Behtash JavidSharifi
31- Assistant Professor, Shiraz University of Technology, Shiraz, Iran 2- Assistant Professor, Shiraz University of Technology, Shiraz, Iran
3- M.Sc. Student, Shiraz University of Technology, Shiraz, Iran [email protected]
[email protected] [email protected]
Abstract
To anticipate the response of a foundation to seismic excitations including earthquakes, machinery loadings and explosions, it is essential to have at hand the dynamic stiffness of each two media which are attached to each other using the substructure method; otherwise rigorous methods are applied to analyze the whole system at once. In coupled analysis methods, on the other hand, both techniques are made use of each involving its own advantages and the whole putting aside individuals' disadvantages. In this paper, assuming the Cone Model as the method going to be used for the far field impedance calculation, different site characteristics variations have been inspected and their effects on dynamic stiffness of the far field are observed. Having categorized the problem into three major unbounded media classes including uniform infinite sand, continuously non-uniform infinite sand and continuously non-uniform sand resting on bedrock, amplitudes of dynamic stiffness of assumed media are calculated using the mentioned model and it is observed that for the infinite sand, as could be expected, the more the depth gets resulting in a bigger interface between the media, the more the impedance amplitude results; and this is the case for all different site considerations. On the other hand, convergence is observed for non- uniform infinite sand around a frequency of about 10 rad/sec with discrepancies observed for bigger frequencies. This is while for the half space consisting of uniform sand the convergence is observed for a frequency between 20 and 30 rad/sec and no discrepancy is observed afterwards. For the half space restricted to bedrock, the impedance amplitude period against frequency decreases with depth and no convergence is observed except for greater depths.
Keywords: seismic excitation, impedance, far field, Cone Model, half space
1. INTRODUCTION
The importance of soil-structure interaction (SSI) effects has been growing since the effects of incident motion frequency versus the natural frequency of the structure has come into consideration and resonance possibilities have been inspected. However, in order to account well for the interactional effects and interaction forces, one of the most important factors with which calculations and modeling procedures are followed will be the dynamic stiffness coefficient of the free field and of the far field of the site, which is strictly bound to some or all factors including the form of the foundation, geotechnical properties of the site, assumed plastic depth of the half space, geometry of the foundation, physical and geometrical specifications of the strata, etc. considering the method and hypotheses taken to calculate the stiffness coefficients. Cone Model is an approximate method which models the medium and wave propagation with simple hypotheses;
yet the results obtained from this method are of very good accuracy which reveals the method meets engineering requirements for average projects and verifications with rigorous methods all confirm this acceptable accuracy.
The method assumes a one-dimensional model of propagation for the seismic wave whose traveling path is cone-like; as a conical bar conveying a wave through its cross section [2]. Three basic assumptions are considered to use this model to solve problems of interest: First, the Hook's law using the famous elastic- constitutive model with Young's modulus of elasticity; second, dynamic equilibrium; and third, basic equation of motion. Further derivations and expansions of the model have been carried out based on these three assumptions [2, 3]. It should be noted that the Cone Model must be made use of for axisymmetric problems, hence for other types of foundations an equivalent radius must be calculated with an acceptable
cross section of the interface to an equivalent circular one setting the areas equal and carrying out the calculations for the unit of area.
In this study, changes in dynamic stiffness of the far field of a coupled system are observed varying different characteristics of the half space including the compaction of the sand with depth, existence or otherwise of bedrock and uniformity or non-uniformity of the far field material with depth. The method of Cone Model via the program CONAN is employed to carry out the calculations. Mechanical characteristics of the soil layers are considered complying with the proposed elastic behavior section of the UCSD soil model developed by Professor A. Elgamal and his team. To determine the stated coefficients, program CONAN was used which is written in the MATLAB environment. A range of incident frequencies from 0 (i.e. the static case) to 60 rad/sec was inspected for all cases covering commonly observed frequencies in vibratory machinery loadings, earthquakes and explosion incidents. Procedures of dynamic stiffness calculation are carried out only for the translational degree of freedom in this stage and other degrees of freedom will be taken into account in a future work. Results are withdrawn and discussed in details and post- processed, making future developments easier to achieve.
Three major categories of site characteristics are investigated. In the first category, the structure is supposed to be on one infinite layer of uniform sand, which is classified into four separate types: loose, medium, medium dense and dense. The dynamic stiffness is calculated for all four cases each for different depths of plastic soil from zero to 50 meters. The width of the structure is taken tentatively equal to 8 meters with an effective amount of 10 meters. This effective width is added to the horizontal dimension of the near field-far field interface. The second category is a manipulation of the first one involving the sand as compressible with depth, and hence the deeper, the denser. Other mechanical properties vary with depth too.
Calculations for this category too are done for plastic depths varying from zero to 50 meters. Soil shear modulus, Poisson's ratio, density and hysteretic damping ratio are the depth-varying mechanical characteristics used based on Elgamal's suggestion [4]. For this study, the sand is taken pressure-dependent with the p' as suggested in [4]. The third category considered is when the depth-sensitive compressible layer of sand is placed on a bedrock with different depths from the near field-far field interface. For this case of the investigation, the plastic depth is taken as constant and the effect of bedrock depth on elastic far field dynamic stiffness is inspected. The bedrock is assumed in the depths of 50, 100, 150 and 200 meters for each analyses and effects are observed via graphs shown later in this paper. These chosen depths are tried to be in a logical and common range based on the information gathered from the PEER Berkeley site, although many other possibilities and natural cases either exist or have even been observed. For the sake of calculations and comparisons, the magnitude of complex dynamic stiffness is chosen representing the largest value of the force P(a0) for a unit u(a0) [2].
Quite a few works have been done on impedance and dynamic stiffness calculations for different aspects of soil dynamics including soil-structure interaction problems. Aspel RJ and Luco JE [5] used an integral equation approach in 1987 to calculate impedance functions of foundations embedded in a layered medium.
Nick Simos, A. J. Philippacopoulos, Dimitri Papandreou, Norbert J. Krutzik and W. Schutz [2] had a rather comprehensive study in 2001 on impedance calculation of saturated soils. Meek JW and Veletsos AS worked on lateral and rocking movements of foundations in simple models in 1974, when the problem of SSI had not progressed in complication and solution this much. Marco Schauer et al. [7] carried out a very similar job in 2011 large scale simulation of wave propagation in soils interacting with structures using FEM and SBFEM, like the one authors intend to complete in the future proceeding this work, where the SBFEM is used for the far field modeling instead of the Cone Model.
2. MODEL DESCRIPTION
Cone model is an approximate geometric model for wave propagation through soil which is a strength-of- material approach proposed by J. P. Wolf [2]. This model assumes the propagation of wave to be cone-like through the soil and similar to one dimensional wave propagation in a rod whose shape is conical and has a circular cross section [2]. As discussed before, basic assumptions of this model lead to a linear-elastic behavior of soil when seismically induced; hence, linear behavior of the site is assumed, meaning that the soil remains linearly elastic with hysteretic material damping during dynamic excitation.
(1)
Considering the soil as a linear-elastic media may be far from reality for certain cases including seismic loading. But, for the case of far field of a system which manifests reasonable damping through the path of excitement and plasticity in the near field (Figure 1), this assumption may satisfy the reality in the aspect of analytic and numerical solutions when verified with natural and experimental observations. The problem here is to have a model which is logical based on the requirements of the system to be as similar as possible to the nature.
Figure 1 Discretization of the SSI system for parallel solution [1]
Cone model assumes the incident wave to propagate away from the disturbance area spreading its cross section as it travels on, while at each depth the travel path of the wave remains perpendicular to horizon. The derivation of relations is based on the three basic assumptions discussed earlier. For this purpose, an infinitesimal element is inspected as shown in Figure 2.
Figure 2 Wave propagation scheme and equilibrium of infinitesimal element, actual propagation path for a disk [2]
After writing dynamic equilibrium relations, imposing boundary conditions and considering geometrical conditions; and with the constitutive law, Equation 1 will be at hand in time domain the solution of which for the static case will lead to the aspect ratio of the initial cone and the disk (Equation 2).
where z indicates depth of interest from the apex of the cone, u stands for displacement which varies with time (t) and depth (z), cp denotes (compressive) wave velocity, ̈ acceleration, apex height from the foundation, r0 radius of the foundation and v is Poisson's ratio.
Based on equilibrium of the infinitesimal element shown in Figure 2, and with regard to Figure 1, Equations 3 will be concluded in frequency domain:
(4)
(2)
(3)
with the amplitude of the internal force or moment N(z,ω) and density ρ. Simplifying the equations and substituting the force-displacement relationship:
Substituting the force-displacement relationship
Equation 5 leads to the governing differential equation of motion:
The quantity ρc2 is equal to the corresponding elastic modulus (shear modulus G and constrained modulus Ec for the compressible case, respectively). In Equation 7, when n = 2, the equation will be applicable for the translational cone, and when n = 4, for the rotational cone [2].
All degrees of freedom of the disk are illustrated in Figure 3 below [2]. In this study, the horizontal, translational degree of freedom is inspected to make a comparison of effects of site characteristics on coefficients of this degree of freedom considering other factors to be constant in order to be able to make logical judgments regarding the sensitivity of stiffness coefficients to different parameters [5].
Figure 3 Cones for various degrees of freedom with corresponding apex ratio (opening angle), wave velocity and distortion [2]
3. MODEL LIMITATIONS
In order for the program to give accurate enough results, some considerations must be taken into account.
Satisfying limitations of Cone Model will result in the desired accuracy. The first factor to consider is the input layers to the program. Existing layers must be discretized so that the requirement of the model is met.
Equation 8 states the maximum allowable depth of each input layer so that the accuracy of calculations is guaranteed in the tolerance provided:
. 5
d .c (9) Where d is depth of each input layer, π is almost equal to 3.14, c is either the shear or the dilatational wave velocity as stated in Equations 9.a and 9.b and
represents the highest frequency the dynamic model must accurately be able to represent [2].
cl E (10.a) (7) (6)
cs G (10.b) Where cl and cs indicate longitudinal and shear wave velocities respectively, E modulus of elasticity, G shear modulus and ρ indicates the density of soil material.
After incorporating the chosen
along with wave velocities based on the physical properties of the media, the smallest d will be taken the upper limit of input depths. Note that this discretization is merely essential for embedded parts of the foundation, and for layers under the embedded part of the foundation no such limitations are necessary [2, 5]. Since in the present study no real foundation is analyzed and calculations are simply for the sake of modeling the effects of the far field elastic soil on the near field inelastic via the interface, no embedment of the foundation is considered and this limitation does not apply.Other limitations include the equivalent radius for non-axisymmetric foundations and tolerances described earlier.
4. DYNAMIC-STIFFNESS AMPLITUDES
Based on material properties defined for the soil and the half space descriptions, dynamic-stiffness coefficients of interfaces of different dimensions and depths were calculated and changes of stiffness with dimensions and density of the soil were depicted as shown in the following. Figure 4 illustrates changes of dynamic-stiffness coefficients for the whole and per unit areas for the horizontal degree of freedom of interfaces in different depths resulting in different cross section areas, when the structure and its near field are supposed to be placed on a non-uniform infinite sand half space. The sand gets denser as its elevation with depth increases. As observed in this figure, the dynamic stiffness amplitude increases divergently as either the interface area or the incident frequency increases and this is a natural event, yet a convergence point at a frequency around 10 rad/sec is observed for the amplitudes per unit area.
(a)
(b)
Figure 4 Dynamic stiffness amplitude (a) per unit area and (b) for the half space composed of whole interface for the non-uniform infinite sand
In the next stage, the far field of a constant depth resting on bedrock with varying depths was examined and effects of bedrock depths were observed. Apparently, as seen from Figure 5, this depth has significant
Dynamic Stiffness Amplitude (KN/M) Dynamic Stiffness Amplitude (KN/M3)
Excitation Frequency (rad/sec)
Excitation Frequency (rad/sec)
local effects on stiffness amplitudes as the general trend towards the incident frequency remains similar for both cases of the interface whole area and amplitude per unit area, increasing with frequency. As the bedrock gets closer, greater discrepancies are observed in the dynamic stiffness amplitude with frequencies. With the bedrock getting farther, these discrepancies tend to lessen and the curve is smoothed comparatively. This means resonance can be more negotiable for cases in which the bedrock is closer to the coupled system.
(a)
(b)
Figure 5 Changes of dynamic stiffness amplitude (a) per unit area and (b) for the total interface cross with different bedrock depths
The third stage of the job, which is far from reality for the first two cases and close for the second two ones in some special cases in the nature is considering the half space infinitely filled with sand, in four different states, i.e. loose, medium, medium dense and dense. The above calculations are done for the four states of the soil and for each state the infinite layer is supposed uniform to infinity, which is not logical for the loose or medium sand but seems to be rather logical for special cases that are observed frequently in the nature where the bedrock is located at such a far distance from the system that can be neglected and replaced by and infinite half space filled with sand. As the sand gets deeper, obviously its density grows and compared to its density at the infinity it does not play an important role in the beginning of the layer adding that the existence of the structure itself can add to the densifying burden and result in a dense or a medium dense upper part of the far field. Figures 6 illustrate calculation results in summary for all the four states, including effects of different plastic depths of soil with different incident frequencies and for the total area and per-unit area.
Dynamic Stiffness Amplitude (KN/M)
Dynamic Stiffness Amplitude (KN/M3) Excitation Frequency (rad/sec)
Excitation Frequency (rad/sec)
(a)
(b)
(c)
Dynamic Stiffness Amplitude (KN/M) Dynamic Stiffness Amplitude (KN/M) Dynamic Stiffness Amplitude (KN/M) namic Stiffness Amplitude (KN/M) Dynamic Stiffness Amplitude (KN/M3) Dynamic Stiffness Amplitude (KN/M3) namic Stiffness Amplitude (KN/M3) Dynamic Stiffness Amplitude (KN/M3)
Excitation Frequency (rad/sec)
Excitation Frequency (rad/sec) Excitation Frequency (rad/sec)
Excitation Frequency (rad/sec) Excitation Frequency (rad/sec)
Excitation Frequency (rad/sec)
Excitation Frequency (rad/sec) Excitation Frequency (rad/sec)
(d)
Figure 6 Changes of dynamic stiffness amplitude with frequency and density for (a) loos sand, (b) medium density sand, (c) Medium-dense sand, and (d) dense sand
Based on the results of stiffness amplitudes for each depth and pertinent frequencies, logical considerations may be achieved for design purposes in order to decrease vibrational displacements in case any type of seismic excitation occurs. In addition, for the sake of dynamic soil-structure interaction analysis in coupled systems, an approach for the linear elastic far field is developed using the cone model. In the observed cases, the convergence of results, if any, may be a good hint that can govern the criteria considered to control the conditions. Yet, further explorations are still needed to provide comprehensive judgments including all factors affecting the impedance and therefore seismic response of the structure's base. Other degrees of freedom, also, must be checked to reach a comprehensive model for the far field effects on the far field-near field interface for numerical analysis of the interface.
4. CONCLUSIONS
Dynamic stiffness amplitudes of a range of far and near field interfaces subjected to a range of input frequencies covering probable earthquakes have been calculated and plotted. For interfaces on uniform infinite sand four categories are observed each with a class of density for the soil. It is seen that as soil gets denser, its stiffness amplitude increases and for the amplitude per unit area results tend to converge for frequencies more than 10 rad/sec. It is worthwhile mentioning that although the overall impedance amplitude for the whole area of the interface is bigger, for vaster interfaces (as a result of deeper plastic ranges of the system), the impedance amplitude per unit area is smaller for overall bigger interfaces.
For non-uniform infinite sand, in which only one profile instead of four has been inspected, the overall trend is similar to the case before except that the mentioned convergence point for frequencies is achieved.
Seemingly, at this frequency, no matter how deep the interface, the dynamic stiffness amplitude is almost constant. It can be perceived that in case the e.g. earthquake predominant frequency is around this point, the dynamic stiffness amplitude will tend to be constant and not dependent on the plastic depth.
The most complicated case which is presented second in this paper is when a bedrock is assumed under the soil four depths for which are considered in this study, with accordingly assumed densities for the overlying sand, i.e. the deeper the soil, the denser the sand. It is observed that as the bedrock gets deeper, the stiffness amplitude gets more even versus excitation frequencies. Thus, the existence of bedrock is of dominant importance in coupled soil-structure interaction analysis.
Considering the trends discussed and combining other degrees of freedom, the total impedance of interfaces of interest (using the Cone Model in this study) will be determined and coupled SSI models will be efficiently taken advantage of for parallel dynamic analyses.
Summing up the results of further studies may lead to economic designs and more precise dynamic analyses, design and rehabilitation of structures considering soil-structure interaction effects.
5. ACKNOWLEDGMENT
Hereby authors would like to offer their special thanks to Dr. Sassan Mohasseb for providing the code for the analyses and giving precious advice without which no stage of this or any other parallel work would be possible. Reviewers are also to be thanked gratefully. This research was supported by Shiraz University of Technology and the 1st Iranian Congress on Geotechnical Engineering, University of Mohaghegh Ardebili.
6. REFERENCES
1. M. Cemal Genes (2012), "Dynamic analysis of large-scale SSI systems for layered unbounded media via a parallelized coupled finite-element/boundary-element/scaled boundary finite element model", Journal of Engineering Analysis with Boundary Elements.
2. John P Wolf and Andrew J. Deeks, (2004), “Foundation Vibration Analysis: A Strength-of-Materials Approach”, Elsevier, Oxford, Great Bretain; ISBN 0 7506 6164 X.
3. Nick Simos, A. J. Philippacopoulos, Dimitri Papandreou, Norbert J. Krutzik and W. Schutz, (2001),
“Impedance Function Calculations for Saturated Layered Soils,” Paper number 1755 of SMiRT 16 Washington DC Transactions.
4. Zhaohui Yang, Jinchi Lu and Ahmed Elgamal (2008), "OpenSees Soil Models and Solid-Fluid Fully Coupled Elements, User's Manual"
5. Aspel RJ, Luco JE. "Impedance functions for foundations embedded in a layered medium: an integral equation approach." Earthquake Engineering and Structural Dynamics 1987; 15:213–231.
6. Meek JW, Veletsos AS. "Simple models for foundations in lateral and rocking motion." Proceedings of the 5th World Conference on Earthquake Engineering, IAEE, Rome 1974; vol. 2, 2610–2613.
7. Marco Schauer, Sabine Langer, Jose E. Roman, Enrique S. Quintana-Orti (2011), "Large Scale Simulation of Wave Propagation in Soils Interacting with Structures Using FEM and SBFEM", Journal of Cmputational Acoustics, Vol. 19, No. 1 P. 75-93.