Benefits of Collaboration between Centrifuge Modeling and Numerical Modeling. Xiangwu Zeng Case Western Reserve University, Cleveland, Ohio

Transcription

1 Benefits of Collaboration between Centrifuge Modeling and Numerical Modeling Xiangwu Zeng Case Western Reserve University, Cleveland, Ohio ABSTRACT There is little doubt that collaboration between centrifuge modeling and numerical modeling has mutual benefit. A quantitative analysis of the experimental data from most earthquake centrifuge tests needs numerical procedures. Development of numerical procedures needs experimental data for verification purpose and data from centrifuge tests is probably the most frequently used one. In addition to the abovementioned benefits, collaboration between centrifuge modeling and numerical modeling can help us to understand the techniques used in modeling and in some cases even improve the techniques. This paper uses two examples to demonstrate how carefully planned collaboration can benefit each other. The first example shows that numerical simulation can help us to quantify the influence of variation of centrifugal acceleration and model container size on accuracy of centrifuge test. The second example shows how development in centrifuge modeling technique can determine some of the previous unknown soil parameters that have significant influence on the results of numerical prediction. For centrifuge modeling, the variation of centrifugal acceleration in a model and the boundary conditions imposed by a model container have significant influence on the accuracy of test results. It is difficult to determine the effect quantitatively by doing tests because it is impossible to prepare identical models. On the other hand, in numerical simulation, identical soil parameters and initial conditions are easy to achieve. Thus, the results of numerical simulation can be used as a yardstick to determine the effect, provided that the numerical code has been verified beforehand. For earthquake centrifuge tests, the effect of model containers is particularly important and more research is needed. For numerical simulation of earthquake problems, soil parameters (such as G max and K 0 ) that determine initial stress and strain conditions are important. However, in the past, these parameters are not measured in centrifuge tests and in particular not measured during the flight of a centrifuge model. Numerical modelers have to guess these parameters in their prediction and, in the example presented here, a wide range of values was used. It was not surprising that the results were not satisfactory. In addition, it is impossible to determine how much the difference in these soil parameters contributed to the difference in the results of numerical prediction. The recent development of new measuring devices that can be operated during the flight of a centrifuge can make valuable contribution to the improvement of numerical simulation by making accurate measurement of these parameters during the flight of a model. INTRODUCTION There has been considerable progress made in both centrifuge modeling and numerical modeling. For centrifuge modeling, larger and more powerful shakers have been developed. New types of transducers and better data acquisition systems allow us to get more and better quality data in model tests. For numerical simulation, a few effectivestress based fully-coupled numerical codes with new constitutive models have been 1

2 developed. Together with the enormous increase in computational power provided by new generation of computers, numerical simulation of complex problems in geotechnical earthquake engineering is now achievable at a reasonable cost. Both centrifuge modeling and numerical modeling have become powerful research tools in geotechnical earthquake engineering. However, each has its own limitations. As an experimental technique, centrifuge modeling has inherited inaccuracies arising from factors such as boundary conditions imposed by model container, inability to satisfy all the scaling laws in certain situations, and system limitations of equipment, transducers, and data acquisition systems. The application of the centrifuge modeling technique and the accuracy of testing data depend critically on how well the effects of these problems are understood and addressed. In recent years, a significant number of research projects have been conducted to study problems of transducer response (Kutter et al, 1990, and Lee, 1990), boundary effects in earthquake centrifuge tests (Hushmand, et al, 1988, and Zeng and Schofield, 1996), and the use of viscous pore fluids (Zeng et al, 1998, Ko and Dewoolkar, 1999). For numerical simulation, verification of a code is a complex process. There are many factors, such as soil parameters, constitutive model, and numerical methods used, which can influence the final results. It would be very useful for the verification process if the influence of or uncertainties associated with some of the factors can be determined or eliminated. NUMERICAL SIMULATION HELPS THE UNDERSTANDING OF CENTRIFUGE MODELING One of the major differences between a centrifuge model and the corresponding prototype is the variation of centrifugal acceleration in both the magnitude and direction. For a soil layer in the prototype and its corresponding centrifuge model shown in Fig.1, there are obvious differences in the stress field. For a typical geotechnical problem in the field, the gravitational acceleration everywhere can be considered heading vertically down and has a constant magnitude. On the other hand, in a centrifuge model, the magnitude of centrifugal acceleration increases with radius and the direction is always in the radial direction. Therefore, centrifugal acceleration varies from point to point. In addition, artificial boundaries (usually rigid end walls) are imposed by model containers, which would further affect the stress distribution in the model. The end walls can be smooth or rough and in each case, it will cause stress condition different from that in the field. The effect of the variation of centrifugal acceleration on vertical stress at the center of a model was analyzed by Schofield (1980) using a one-dimensional approach. It was found that the stress in the upper part of a centrifuge model would typically be lower than that in the corresponding prototype while at the bottom of a model, the stress would be higher than its counterpart in the field. It was shown that if the depth of a model is onetenth of the radius of a centrifuge, the magnitude of vertical stress error is under ± 2% (+2% at bottom, -2% near the top). However, in reality the problem is at least twodimensional and the boundary conditions imposed by the model container are a further source for inaccuracy. Apart from that, accuracy in the simulation of horizontal stress and shear stress is also very important. Thus a more comprehensive analysis which can take into account these differences is needed. Here the results of a numerical simulation of this problem using a finite element program is presented. 2

3 soil σ v ' σ h ' gravitational acceleration g a) Gravitational acceleration and the resulting stresses in a prototype centrifugal acceleration model container soil b) Centrifugal acceleration in a model Fig. 1 A soil layer in prototype and its corresponding centrifuge model Methodology Ideally, the best way to investigate the influence of the non-uniform centrifugal acceleration on the accuracy of a centrifuge test is to compare directly the stresses and other properties measured in the field to those measured in the corresponding centrifuge model. However, in reality, a number of conditions make such a comparison impossible. First, soils in the field are most likely to be anisotropic and inhomogeneous. These field conditions are very difficult, if not impossible, to be replicated in a small-scale model. Second, there will be some discrepancies between the properties of soil in the model and those in the field, which would contribute to some of the differences in the results. Last but not least, there will be experimental inaccuracies arising from instrumentation and operation of devices used both in the field and centrifuge tests. Therefore, it is impossible to identify how much of the difference is due to each of the contributing factors. An alternative approach is to apply an analytical method to this problem assuming that all the soil properties are identical in the prototype and the model. Thus it can exclude the influence of other factors so as to investigate the inaccuracy caused exclusively by the variation of centrifugal acceleration and the artificial boundary conditions imposed by model containers. To simplify the analysis, soils in both the prototype and the model are assumed to be isotropic and homogeneous. For the stress field in the prototype, assuming that the uniform soil layer is of infinite lateral extent, vertical and horizontal effective stresses can be determined by hand calculation. On the other hand, the stress field in the model is a complex two-dimensional problem, which can be solved by using a finite element code. Of course, the finite element code used has to be one that is well established. For this project, the finite element code used is the 3

4 SIGMA/W developed by Geo-Slope International (1995). Details of this study are reported by Zeng and Lim (2002) and a summary is presented here. The accuracy of a finite element simulation is affected by the number of elements and the type of elements used. In this study, 8 noded quadrilateral elements with integration order of 9 were used. A simple problem of a soil layer of 10 m depth with an infinite lateral extent was simulated using a varying number of elements. It was found that when 50 (10 5) elements were used, the difference in stresses between the analytical solution and the numerical simulation was less than 1%. Therefore, in the following study, a finite element mesh with 50 elements was adopted. Simulation of Stresses in a Soil Layer Induced by Self-Weight In the first attempt to investigate the influence of the variation of centrifugal acceleration on the accuracy of centrifuge modeling, the stresses induced by self-weight of the soil in a horizontal soil layer of infinite lateral extent is calculated and compared to the stresses in the corresponding centrifuge model. The soil layer in the prototype is 10 meters thick and has a saturated and buoyant unit weight of kn/m 3 and 10 kn/m 3, respectively. The water table is at the surface of the layer. The test is assumed to be conducted at a centrifugal acceleration of 50g at the center of the model and hence the thickness of the soil layer in the model is 20 cm. The study will be concentrated on the influence of the radius of the centrifuge R (defined as the distance from the rotating center of the centrifuge to the center of the model) and the width of the model container B. To illustrate the variation of centrifugal acceleration in a centrifuge model, the models of the 10 m soil layer when tested on centrifuges with radii of 1m and 4m, respectively, and a model container 0.6m wide are shown in Fig. 2. Clearly, the variation in centrifugal acceleration in the model is quite significant for the case of a small centrifuge. Two types of constitutive models were used in the finite element simulation: linearly elastic and the original Cam-Clay models. Two parameters (Young s modulus E and Poisson s ratio ν) are required in a linearly elastic model and the values used in this study are E = 30 MPa and ν = (which would result in a K 0 = 0.5). For the original Cam-Clay model, soil parameters used are: λ = 0.193, κ = 0.047, Μ = 1.2, G = 2.4 MPa, and ν = When the linearly elastic model is used, the stress calculation is achieved in one load increment. On the other hand, for a non-linear model such as the Cam-Clay, a small load increment needs to be used in order to achieve accurate results. Also the first initial stress state has to be given or calculated by other procedures. In this study, stress calculation when using the Cam-Clay model was finished in 50 load steps. For the stress calculation in the field, the first step uses a linearly elastic model and a unit weight of 10/50 = 0.2 kn/m 3. Then the next 49 steps use the Cam-Clay model and each step has the same unit weight increase of 0.2 kn/m 3 with the stresses from the previous step used as the initial stresses. For the stress calculation in the centrifuge model, the initial stresses due to 1g gravity are calculated by the linearly elastic model. Using this as the initial stress state, the stresses at 2g are calculated using Cam-Clay model with the body forces increased by an amount corresponding to an increase of centrifuge acceleration of 1g at the center of the model. This procedure is repeated until the final body force is increased to that corresponding to a centrifugal acceleration of 50g at the center of the model. 4

5 1.0 m 18.4 o 0.3m 0.3m 45g 47.4g 50g 0.2m 15.2 o 55g 57g a) Centrifuge model with R = 1m, B = 0.6m 4.0 m 4.4 o 48.9g 4.2 o 51.4g 0.3m 0.3m 48.8g 50g 51.3g 0.2m b) Centrifuge model with R = 4m, B = 0.6m Fig. 2 Variation of centrifuge acceleration in models of a 10 m soil layer Simulation of Centrifugal Forces The effect of the centrifugal acceleration is simulated by applying body forces on to each element. The body forces have both the X and Y components as shown in Fig. 3. Supposing the center of element i has a coordinate of X i and Y i, the centrifugal acceleration at this point would be a i = ω 2 (Y i 2 + X i 2 ) where ω = angular velocity of the centrifuge. The resulting body forces per unit volume of this element in x and y directions are f xi = ρω 2 X i f yi = ρω 2 Y i 5

6 where ρ = submerged mass density of soil. Obviously, variation in body forces can be quite significant between elements near the center and elements at the four corners. X Rotation Center Y Yi R element i f xi f yi Xi B Fig. 3 Body forces induced by centrifugal acceleration Results from Linearly Elastic Model For a soil layer that is 10 meter thick in the prototype and has a saturated and buoyant unit weight of kn/m 3 and 10 kn/m 3, respectively, the vertical and horizontal normal stress distributions are linearly increase with depth for the linearly elastic soil. There is no shear stress on the horizontal plane or vertical plane. When the same soil layer is simulated in a model container 0.6m wide with rigid and smooth end walls on a centrifuge with a radius of 1m, the stress distributions are shown in Fig. 4. The vertical stress is well simulated throughout the model but there are some differences in the horizontal stress in the model in comparison to the stress in the prototype, especially near the end walls. In addition, there is a small shear stress in the horizontal and vertical planes in the model. As the radius of the centrifuge increases, the difference in the horizontal stress drops quickly and the difference in the vertical stress become even less. For example, if the radius of the centrifuge is increased to 4m with the size of the model container remaining at 0.6m, the difference between the horizontal stress in the model and prototype becomes much smaller and may be neglected for most purposes, as shown in Fig. 5. Results from Original Cam-Clay Model When the same layer is simulated on a centrifuge with a radius of 1m and a model container 0.6m wide using the Cam-clay model, the stress distributions are shown in Fig. 6. Again, the vertical stress is well simulated but there is noticeable difference in horizontal stress even near the centerline. 6

7 Fig. 4. Effective stresses for a soil layer in a centrifuge model (R = 1m, B = 0.6m) Fig. 5 Horizontal effective stress in the centrifuge model (R = 4m, B = 0.6m) 7

8 Fig. 6 Effective stresses in the centrifuge model (R = 1m, B = 0.6m) The difference in horizontal stress distribution still exists even when the radius of the centrifuge is increased to 4m. On the other hand, if the width of the model container is increased to 0.8m, the difference in horizontal stress distribution is significantly reduced, suggesting that boundary effect of the end walls plays a more important role for this type of situation. An Example Problem In order to evaluate the effect of difference in stress field between the model and the prototype on test results, the settlement of a 6 m wide flexible strip footing subject to 100 kpa surface loading is analyzed. The footing is founded on a 10 m thick normally consolidated (by self weight) and soft clay layer and the original Cam-Clay model is used in the finite element calculation. The load is applied in 50 steps. Assuming the same structure is simulated in centrifuge tests with different centrifuge radii and model 8

9 container width, the settlement and stress conditions in each model are also calculated using the approaches mentioned earlier in this paper. For the problem in the prototype, the settlement of the footing and the foundation soil is shown in Fig. 7. The maximum settlement of the footing is m. If the model test is conducted on a centrifuge with a radius of 1m and a model container 0.6m wide, the maximum settlement is increased to m, resulting in a 12.4% difference. The magnitude of the difference may not be significant from an engineering standpoint but it is a concern for a numerical simulation. The stress distribution in the foundation looks similar but there are some differences in the magnitude of stresses. However, when the radius of the centrifuge is increased to 4m, the difference between the results of the prototype and the centrifuge model becomes negligible. The maximum settlement of the footing is now 1.256m, only a 2.5% difference compared to the prototype. The stress distributions and their magnitudes in the foundation in the model are similar to that in the prototype. Several other simulations were conducted and the results are shown in Table 1. For this particular problem, it seems that a centrifuge radius of 2m will produce satisfactory accuracy. Fig. 7 Settlement of a foundation in prototype (maximum settlement 1.288m) It is important to point out that the influence of the centrifugal acceleration, the radius of the centrifuge, and the size of model container vary from problem to problem and from parameter to parameter measured. For the example footing problem, if a centrifuge has a radius of 2m, the settlement of the footing would be simulated with satisfactory accuracy. For other types of problems, the requirement may be different. Also for the examples here, it is assumed that the tests are conducted at 50g. For tests conducted at other centrifugal accelerations, the results may be slightly different for analyses using the Cam-Clay model. Therefore, it is highly recommended that analyses similar to what are reported in this paper be conducted to determine a suitable size of the centrifuge and model container or at least determine the range of accuracy of the experimental data due to the imperfect simulation of gravitational acceleration on a centrifuge and the boundary conditions imposed by a model container in a centrifuge test. 9

10 Table 1 Maximum Settlement of a Footing by Different Tests Test Case Settlement (m) Diffenernce (%) Prototype R = 1m, B = 0.6m R = 1m, B = 0.8m R = 2m, B = 0.6m R = 4m, B = 0.6m DEVELOPMENT OF CENTRIFUGE MODELING TECHNIQUES ENHANCES NUMERICAL SIMULATION A numerical simulation is critically dependent on the input parameters used. If the parameters used in a constitutive model can be accurately measured during the flight of a centrifuge model, it can enhance a numerical simulation through the elimination of uncertainties associated with input parameters. Some latest development in instrumentation for earthquake centrifuge modeling can provide valuable information about soil properties during the flight of a centrifuge and thus, improve the usefulness of centrifuge test data, which in return, benefit the development of numerical simulation as data from centrifuge tests are the ones that most frequently used for the verification of numerical codes. For example, the initial stress and strain state of soil is very important in the analysis of soil liquefaction. Therefore, accurate measurement of parameters such as G max and K 0 can play a critical role in the numerical simulation. In the past, these parameters are not measured in a centrifuge model during the flight of a centrifuge. Numerical predictors have to assume the values of these parameters based on empirical formula or experience, creating uncertainties in the numerical prediction because it is unknown how much these estimations will contribute to the difference between numerical simulation and experimental data. For example, during the VELACS project, numerical predictors used a wide range of values for G max and K 0 for Nevada sand as shown in Table 2. It means that the initial stress state was quite different between different predictors. For instance, the corresponding initial stress state when the vertical effective stress is 100 kpa while K 0 are 0.70 or 0.36 (the maximum and minimum value used by the predictors) is shown in Fig. 8. Using an analogy to triaxial tests, if we had two soil samples made of the same soil but with such different initial stress states, we would expect quite different response of these samples to external loading applied. Therefore, it was not surprising that there were large discrepancies between the results of numerical predictions. Moreover, it is not possible to identify the cause of these discrepancies because of the differences in the initial stress and strain state. This particular problem was not caused by either numerical predictors (since they did not have these values provided by centrifuge modelers) or centrifuge modelers (since they were not asked to provide such parameters, or even if asked, could have not been able to do so because technology of measuring these parameters during the flight of a centrifuge model was not available at that time). It was the consequence of insufficient communication between numerical modelers and centrifuge modelers in a joint research project. 10

11 A new technique that can carry out the measurement of G max and K 0 of soils in a centrifuge model during the flight of a centrifuge is being developed at the Case Western Reserve University using bender element technique. The project is sponsored by the National Science Foundation and is expected to be finished in about a year. With accurate measurement of G max and K 0, numerical predictors can get the initial stress and strain state correct before carrying out dynamic simulation, thus excluding one of the uncertainties in the prediction. Table 2 Parameters of Nevada Sand used by predictors in VELACS Project Predictor n in G max = Kσ n K 0 Anandarajah Aubry 0.5 Bardet Been Chan 0.4 Iai Ishihara 0.52 Kimura 0.5 Lacy 0.5 Li Prevost 0.7 Roth 0.5 Shiomi 0.5 Siddharthan σ v = 100 kpa σ v = 100 kpa σ h = 70 kpa σ h = 36 kpa K 0 = 0.7 K 0 = 0.36 Fig. 8 Difference in initial stress state with different K 0 value CONCLUSIONS In conclusion, it is obvious that collaboration between numerical modeling and centrifuge modeling can bring mutual benefits. For earthquake centrifuge modeling, there are a few problems related to modeling technique that can benefit from numerical simulation such as the influence of boundary conditions of different types of model 11

12 container, the effect of using a viscous fluid, and the effect of smooth or frictional end walls. For numerical modeling, development of in-flight measuring device and better instrumentation can provide more and better data for verification purpose. In order to maximize the benefits, the collaboration needs to start from the planning stage. REFERENCES Kutter, B. L., Sathialingan, N., and Herrmann, L. R., 1990, Effects of arching on response time of miniature pore pressure transducers in clay, Geotechnical Testing Journal, ASTM, Vol. 13, No. 3, pp Lee, F.-H., 1990, Frequency response of diaphragm pore pressure transducers in dynamic centrifuge model tests, Geotechnical Testing Journal, ASTM, Vol. 13, No. 3, pp Hushmand, B., Scott, R.F. and Crouse, C.B., 1988, Centrifuge liquefaction tests in a laminar box, Geotechnique, London. U.K., Vol. 38, No.2, pp Zeng, X. and Schofield, A. N., 1996, Design and Performance of An Equivalent-Shear- Beam Container for Earthquake Centrifuge Modeling, Geotechnique, London, U.K., Vol. 46, No. 1, pp Zeng, X., J. Wu, and Young, B.A., 1998, Influence of viscous fluids on properties of sand, Geotechnical Testing Journal, ASTM, Vol.21, No.1, pp Zeng, X. and Lim, S.L., 2002, The influence of centrifugal acceleration and model container size on accuracy of centrifuge test, Geotechnical Testing Journal, ASTM, March. Ko, H.Y. and Dewoolkar, 1999, Modeling liquefaction in centrifuges, Proceedings of the International Workshop on the Physics and Mechanics of Soil Liquefaction, edited by P.V. Lade and J.A. Yamamuro, A.A. Balkema, Rotterdam, pp Schofield, A.N., 1980, Cambridge geotechnical centrifuge operation, Geotechnique, London, U.K., Vol. 30, No.3, pp Geo-slope International, 1995, Sigma/W user s guide, version 3, Calgary, Alberta, Canada. 12