Back analysis for tunnel engineering as a modern observational method

Tunnelling and Underground Space Technology 18 (2003) 185–196 0886-7798/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved. PII: S0886-7798Ž03.00026-9 Back analysis for tunnel engineering as a modern observational method Shunsuke Sakurai , Shinichi Akutagawa *, Kunifumi Takeuchi , Masato Shinji , Norikazu Shimizua b, c d d Hiroshima Institute of Technology, Hiroshima, Japana Kobe University, Kobe, Japanb Obayashi Corporation, Tokyo, Japanc Yamaguchi University, Ube, Japand Abstract As complexity and unpredictability exist in nature, careful observations and interpretations of what can be measured in the field become prerequisites for geotechnical engineers to conduct safe and economical construction works. Observational methods have evolved from basic visual procedures, conducted on site, to sets of sophisticated procedures using modernized measuring instruments and computer-based back analysis techniques. From the wide range of procedures available for modern tunneling engineering in this field, the present paper tries to address a series of back analysis procedures in which the identification of strain distribution is sought as the primary goal in order to achieve a solid and reliable routine of observations and data interpretations. The discussion starts by identifying structures and flows of forward and back analyses; and it is then expanded to cover several back analysis procedures, including application examples, formulated for linear and non-linear material behaviors. The current status and limitations of the procedures available are also discussed. � 2003 Elsevier Science Ltd. All rights reserved. Keywords: Back analysis; Observational method; Finite element method; Field measurement; Critical strain 1. Introduction General-purpose numerical analysis techniques, such as finite element methods, were developed and prolif- erated since the 1960s to become powerful tools for engineering design procedures. They have, as in other engineering fields, been applied in geotechnical engi- neering problems mostly to perform simulation of con- struction processes in details for design purposes. For the case of construction of underground structures, the numerical simulation would be used to follow the natural sequence of events involved in construction as shown in Fig. 1. However, difficulties in using these methods were soon experienced by geotechnical engineers who tried to simulate or to predict the behaviors of underground structures through limited or incomplete sets of input data. It was natural, therefore, that those working on the simulation of construction of underground structures *Corresponding author. Department of Architecture and Civil Engineering, Kobe University, Rokkodai 1-1, Nada, Kobe 657-8501, Japan. Tel.: q81-78-803-6015; fax: q81-78-803-6069. E-mail address: cadax@kobe-u.ac.jp (S. Akutagawa). shift their focus towards finding ways to identify the missing information from the results of field measure- ments. This trend of utilizing updated information obtained on site effectively has led to the birth and growth of the observational method shown in Fig. 2. The method is characterized by the combination of field measurement, which initially was used only for direct interpretation, and its rational interpretation not only for the evaluation of tunnel stability, but also for the verification or modification of the initial design and the construction method. Numerical procedures uniquely configured in such a way that the results of field measurement could be input to determine some of the controlling parameters to completely describe the analysis model in concern, started to emerge in the late 1970s for geotechnical engineering applications. These methods were termed back analysis and soon became popular to represent family of these analysis techniques, forming core numer- ical tools in the sophistication of observational methods. Interested readers are directed to Gioda and Sakurai (1987) or Sakurai et al. (2001) for a detailed review. 186 S. Sakurai et al. / Tunnelling and Underground Space Technology 18 (2003) 185–196 Fig. 1. Information related to natural simulation of construction. Fig. 2. Cyclic routines in an observational method. 2. Forward analysis vs. back analysis It is now necessary to recognize the versatility of back analysis techniques, which is best explained in comparison with what constitutes a forward analysis and how it may be used in an observational method. As shown in Fig. 3, a forward analysis starts with definition of a mechanical model and parameters required to define its characteristics (see Fig. 1 for more detailed description of items concerned). By following Route F1, a forward analysis is conducted. The results would be compared with measurements for assessment of current state of stress and strain. If necessary, param- eter tuning may be performed for better agreement between computed and measured quantities, which is shown as Route F2. For some cases, parameter tuning only would not lead to a better representation of the reality. It then becomes necessary to adjust the mechan- ical model itself on Route F3. These approaches may be automated with the help of a mathematical optimi- zation tool to form a generalized parameterymodel- tuning tool. These methods, as they are general ones, may be applied to a wide range of problems in linear and non-linear problems, depending on the nature of the projects. However, if a set of assumptions made during a model definition is made such that a forward analysis becomes impossible, a special treatment must be intro- duced to form a core routine to relate the measured data to a list of unknown parameters. The family of this type of analyses, shown as Route B1, is called an inverse analysis. There is a class of problems to which a specially coded inverse analysis could be successfully applied. It is also true that for a given problem, a user has options of selecting a numerical algorithm of an inverse analysis, which may yield different results (see Route B2). One also needs to know that, unlike in a 187S. Sakurai et al. / Tunnelling and Underground Space Technology 18 (2003) 185–196 Fig. 3. Forward analysis versus back analysis. Fig. 4. Behavior of measured quantity with respect to construction stages. forward analysis, each selection of measurement data to be input into a back analysis, leads to different results, depending on quality and quantity of those data, which are shown as Route B3. If one extra parameter is selected as the additional unknown parameter to be determined, however, the inverse analysis might not function just by itself and would probably be best used if it were embedded in a general parameter tuning scheme (Route B4), as was true for the forward analysis case. On top of these, a model tuning phase, Route B5, exists also for back analysis to achieve a better agreement between comput- ed and measured quantities. The history of the application of back analyses in geomechanics (Gioda and Sakurai, 1987; Sakurai et al., 2001) shows that the techniques developed thus far fall onto one of the above-mentioned routes in terms of program structure and fundamental concepts. Differenc- es lie in the selection of the material constitutive laws, the simplification of the geological structure, the treat- ment of multiple excavation stages, the inclusion of measurement errors, and in the way the results of back analysis are utilized. 3. Underground excavation and purpose of back analysis It is an exceedingly demanding task to cover these state-of-the-art developments in full detail. Therefore, this paper intends to pick up several selected back analysis procedures used for interpretation of field meas- urements in tunnels and large underground caverns. Illustrated in Fig. 4 is a typical behavior of measured quantity with respect to progress of construction stages. As underground construction is generally conducted in stages, field measurement and back analysis are often associated with each of those stages. A primary goal of back analysis then is to have the best understanding of what exists and what has happened in that particular stage. For example, if one succeeds to obtain a strain distribution of reasonably accuracy, it may be used for safety assessment for this stage, provided that a strain- based criterion, such as the critical strain (Sakurai and Takeuchi, 1983), is available. It is, however, equally important in underground excavation projects that the information obtained in one stage be best utilized for 188 S. Sakurai et al. / Tunnelling and Underground Space Technology 18 (2003) 185–196 Fig. 5. Multi-staged construction. Fig. 6. Stain distribution measured for a subway tunnel (Hansmire and Cording, 1985). prediction of what will follow in subsequent construction stages. As depicted in Fig. 5, what is found at stage i must or is expected to tell something about what might happen in stages iq1, iq2, and so on. In the following sections, two sets of application examples are introduced for construction of tunnels and large underground caverns, while relating back analysis techniques used to Fig. 3 and significance of conducting back analyses to Figs. 4 and 5. 4. Back analysis for a shallow tunnel in soft ground The first set of back analysis examples deals with a shallow tunnel. It is of primary importance for a tunnel excavated at a shallow depth in a soft ground to complete construction with minimum deformation in a surrounding ground. However, it is not an easy task and deformation of substantial degree may be caused by excavation, as illustrated by an example shown in Fig. 6. It is therefore required that once measured displace- ments become available for interpretation, one needs to be able to say whether deformation around the tunnel is still in an elastic range, whether strain concentration to be noted is occurring, and eventually whether the tunnel structure is stable and if so, for what degree. An example of the actual measurement of displace- ment is shown in Fig. 7 obtained for a shallow tunnel. The first, and the most simplified approach in this case would be to treat this problem as a linear elastic problem. By employing an ad hoc algorithm, called the Direct Back Analysis Program (DBAP), to relate measured displacement to normalized in-situ stress components (Sakurai and Takeuchi, 1983), one would obtain a strain distribution such as that shown in Fig. 8. The back analysis of this type is on Route B1, starting from the assumption of linear elasticity and ending up with strain distribution computed from back analyzed in-situ stresses. Strain distribution is often used for judgment of the state of deformation, however, it depends on the selection of measured displacements (Route B3). Or one may choose to use a different material model (Route B5) from which a new back analysis may be set up. This opens up a whole new selection of alternative back analysis projects, which, however, poses a serious problem of non-uniqueness of back analysis solution. To provide a general forum in which a free interpre- tation is guaranteed for a set of solution, a back analysis technique based on a totally different concept was developed (Sakurai et al., 1995). The non-uniqueness of solution, which in other words the freedom of model set up for a back analysis, comes from a tradition of using an explicitly defined material model to start with. To avoid this, the new strategy suggests the simplest model possible, in this case a linear elastic homogeneous model, to describe the measured deformation at first. If any discrepancy exists between computation and meas- urement, a numerically determined adjustment is given to that simple elastic model to achieve a simulated displacement field almost identical to the measured one. This adjustment is achieved by a set of fictitious forces applied to an elastic medium as shown in Fig. 9, and represents the effects of non-linear material behavior, unexpected inhomogeniety of ground material, failure of rock joint planes, etc. The fictitious forces, which may be directly translated into non-linear strain, obtained in a back analysis may be linked to any of these reasons at the engineer’s will. And even before that reasoning process is finished, one has a simulated 189S. Sakurai et al. / Tunnelling and Underground Space Technology 18 (2003) 185–196 Fig. 7. Displacement measured for a shallow tunnel. strain field at hand from which safety assessment, at least, can be made with no delay. The same example data shown in Fig. 7 were analyzed (Sakurai et al., 1994) by this method with the resulting strain distribution shown in Fig. 10. As is obvious from a comparison with the result shown in Fig. 8, the new results tell at least that the assumption of linear elasticity for this case could have been inappropriate. This method of treating non-linear nature of defor- mation in the form of fictitious forces is in one way an alternative algorithm still on Route B1. This means that even though the method enables modeling a wide range of problems involving non-linearity, the results offer little information on the nature of material model itself. This is not a favored case particularly when the result of a back analysis is expected to offer something for the stages to follow, as was emphasized in Fig. 4. Lastly shown for the interpretation of the measured displacement of a shallow tunnel is an example (Okuda et al., 1999) on Route B4. In this case, a basic material model was selected such that shear stiffness decreases with increasing shear strain. In this model, there is a key parameter a which controls the exponential rela- tionship between shear stiffness and shear strain. Using this basic material model, the non-linear strain (i.e. fictitious forces) was determined for selected zones around the tunnel. Fig. 11 shows a comparison of the maximum shear strain distributions obtained by back analyses for three different values of a. It is observed that as the value of a reaches an optimum value of 2.0, which was judged from displacement comparison, the shear bands devel- oping from the tunnel shoulders upwards grow to be of a typical model of a shallow tunnel in a sandy ground, as shown in Fig. 6. This exemplifies the fact that the appropriate selection of a constitutive law, if available, results in a back analysis solution of higher quality. Although the actual results are not shown here, the constitutive relationship once obtained by the back analysis, could then be used for predictive analyses. 5. Back analysis for a large underground cavern in hard discontinuous rock mass The second set of examples deals with interpretation of measured displacements for a large scale underground cavern for a power house. Fig. 12 shows distribution of Q values obtained for the cavern at the completion of construction. This suggests how complex the rock mass formation is around the cavern. Since the dimension of 190 S. Sakurai et al. / Tunnelling and Underground Space Technology 18 (2003) 185–196 Fig. 8. Maximum shear strain distribution assuming linear elasticity. this opening is much greater than that usually experi- enced for tunnels, the domain in concern contains several major faults and unseen seam planes, etc. All these factors have led to the complex displacement fields shown, for example, in Fig. 13 for one of the monitored sections. Unlike for tunnels, construction of a large scale cavern takes place at a fixed location, though its scale is much greater. There would usually be more displacement measurement points than for tunnels. What dominates a procedure of interpretation of measured displacement should first be judgment of cavern stability, against roof or wall collapse, etc. The subsequence issues to be raised would be identification of the state of deforma- tion, strain, stress for rock masses and for major fault or seam planes. It is also of interest to know, particularly for the case of discontinuous rock mass, behavior of discontinuity planes that affects a global deformation of the cavern. Also to be noted here is that interpretation of measurement and assessment of stability need to be done at each construction stage. In addition, what is found in a stage must be effectively utilized for more reliable prediction of the stages to follow. With these backgrounds and requirements, several approaches would be possible for a back analysis. Figs. 14–16 show strain distributions estimated for the final stage, obtained from three different back analyses. The result from an elastic back analysis (the method on Route B1 using a linear elastic model) is shown in Fig. 14. The strain distribution seems to be of similar quality with those shown in Figs. 15 and 16, however, it took weeks to obtain this result because adjustment of stiff- ness distribution had to be done manually. The pictures in Figs. 15 and 16 are similar and are regarded as having equally important information in themselves, in terms of knowing strain levels and its distribution. The frameworks of theory used to obtain these images are, however, somewhat different. The method used to obtained the results shown in Fig. 15 is called Back Analysis of Non-linear Strain for Jointed rock mass in Incremental form (BANSJI; Hojo et al., 1997). This method is on Route B1 and is based on an equivalent elastic model which is allowed to produce joint slipping, in order to eliminate discrepancy between displacements computed from a reasonably simplified elastic model and those measured in field. By identifying joint slip displacements, which are supposed to happen after shear strength is exceeded, the displace- ments computed match completely with measured val- ues. In addition, one obtains a set of various distribution 191S. Sakurai et al. / Tunnelling and Underground Space Technology 18 (2003) 185–196 Fig. 9. New concept in back analysis. Fig. 10. Strain distribution obtained by a back analysis considering nonlinear strain. of quantities, such as strain distribution caused by elastic deformation of rock core, elastic deformation due to joint planes before failure, non-linear deformation due to joint slipping, etc. Fig. 17 shows, as an example, distribution of locations where rock joint planes have slipped more than 2 mm in the direction indicated. These images certainly help engineers understand what is happening within the rock mass in concern. It must be noticed, however, the method employed here works to identify what is happening now, but tell little of what will happen next, because the results are not associated with a material constitutive relationship. The strain distribution shown in Fig. 16 is obtained from a back analysis method called the Inhomogeneous Non-linear Direct Back Analysis Program (I-N-DBAP; Akutagawa et al., 2000), which is regarded as an alternative method on Route B1. This method also uses fictitious forces to fill gaps between computation and measurement, created by inappropriate assumptions. The algorithm, however, does not stop here. It introduced a 192 S. Sakurai et al. / Tunnelling and Underground Space Technology 18 (2003) 185–196 Fig. 11. Strain distribution with different values of key parameter a. Fig. 12. Distribution of Q values for a cavern. strategy to link non-linear strain caused by the fictitious forces to reduction of stiffness. In other words, it softens materials where non-linear strain is identified by an appropriate amount, so that in the next cycle of com- putation, less amount of non-linear strain would be required to match displacement fields. A comfortable convergence has been found so far leading to the optimized model with unique distribution of rock mass stiffness. The significance of this approach is that the raw information coming from differences between compu- tation and measurement can be c