Minimization Solutions for Vibrations Induced by Underground Train Circulation Carlos Dinis da Gama 1, Gustavo Paneiro 2 1 Professor and Head, Geotechnical Center of IST, Technical University of Lisbon, Portugal 2 Researcher, Geotechnical Center of IST, Technical University of Lisbon, Portugal ABSTRACT Both structural damage and human discomfort may result when underground trains circulate in urban areas. Many factors are involved in this phenomenon, either related with train maintenance, tunnel and track characteristics or ground dynamic properties (mainly attenuation).those factors are referred in ISO 14837 (2005), which provides a general guidance on ground-borne vibration generated by operations of rail systems, and the resultant ground-borne noise in buildings. The problem is aggravated when those effects are felt in metropolitan lines, which cannot be submitted to comprehensive revamping workings. The article describes the research conducted in the Lisbon Metro, in old and new tracks, with and without anti-vibration mantles, in order to identify the most influent variables in these situations. An innovative solution for existing tracks is suggested and it is analyzed through finite element simulations. 1. INTRODUCTION The existing computation modeling of ground behavior allows the simulation of rock and soil excavations, considering all their constitutive elements, both under static or dynamic conditions (Brady & Brown, 1985). The usage of these calculation methods has provided the opportunity to devise innovative solutions for design, construction and maintenance problems related to underground openings at decreasing computer costs. The stress and strain analysis with computational methods can be divided in two main types: differential methods and integration methods. In the first ones, the problem domain may be divided in several sub-domains or elements. Therefore, the procedure to obtain a solution can be achieved by numerical approaches of equilibrium differential equations involving the relations between stress, strain and displacement, and from the stress-strain equations, like the classical methods of finite differentiation. The main feature of the integration methods for stress analysis is related to the specification and problem solution terms of the stress and displacement field variables. Since the problem boundary conditions are defined and discretized, the so-called boundary element method gives a unit reduction in the dimensional order of the problem. This fact constitutes an advantage in computational efficiency, in comparison with the differential methods. The existing differences in terms of problem definition between differential and integration methods leads to the detection of several operational features. For the finite element method (FEM), for example, the non linear and heterogeneous material properties can be accommodated; however, the problem domain beyond the boundary conditions is arbitrarily defined, conducting to discretization errors in the domain. On the other hand, boundary element models create models with more realistic 571
boundary conditions, reducing discretization errors and assuring the stress and strain continuum variation. However, these methods usually conduct to better results in materials with linear behavior. Regarding underground excavations, the finite element method defines the problem domain that encircles an opening and divides it in an interacting and discrete element structure. This is the methodology used in the analyses presented in this paper, with special reference to dynamic conditions. 2. DYNAMIC ANALYSIS USING THE FINITE ELEMENT METHOD Usually the finite element method considers a continuum solid as an element association in which the boundaries are defined by nodal points or nodes, and assumes that its behavior submitted to static or dynamic loadings, can be defined by the movements of those nodes. To obtain a good computational efficiency, the minimization of the element number in a finite element analysis is desirable. The maximum element dimensions are generally controlled by the wave propagation velocity and frequency, so the minimization of the element number is usually related with the minimization of the discretized region dimension. As the size of this region reduces, the boundary conditions become more important (Kramer, 1996). For most problems of dynamic behavior involving soil-structure interactions, a boundary rigid material is established at considerable distances to the study zone, particularly in the horizontal direction. As a result, the incident wave runs all the area beyond the study region and can be permanently removed from that region, so for a dynamic analysis using the finite element code that is a significant advantage (Plaxis, 2004). 3. CASE STUDY OF VIBRATIONS GENERATED BY UNDERGROUND RAIL TRAFFIC The finite element model for the cross-section of a urban underground railway track, was developed as shown in Figure 1. Ground dynamic properties were needed for the input and so two types were considered, as Table 1 summarizes. Fig. 1. Geometric model lay-out used in the FE analysis. Table 1. Properties of the ground materials considered in the FE model. PARAMETERS UNITS SAND CLAY TUNNEL FLOOR FILL Behavior type - Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Weight (γ) kn/m 3 17.0 16.0 20.0 Young s Module (E) kpa 2.0 x 10 4 1.5 x 10 4 1.0 x 10 5 Poisson s Ratio (ν) Adim. 0.3 0.3 0.2 Cohesion (c) kpa 1.0 2.0 10.0 Friction angle (φ ) º 35.0 24.0 40.0 572
For the rest of the model elements, like the tunnel walls and superficial buildings, the considered input properties are shown in Table 2. Table 2. Element properties of tunnel lining and nearby buildings, used in the FE model. PARAMETERS UNITS TUNNEL WALLS BUILDINGS Behavior Type - Elastic Elastic Normal Stiffness (EA) kn/m 7.6 X 10 6 5.0 X 10 6 Flexural Rigidity (EI) kn.m 2 /m 2.4 X 10 4 9.0 X 10 3 Weight (W) kn/m/m 5.0 5.0 Poisson s Ratio (ν ) Adim. 0.0 0.0 Figure 2 represents a graphical output of the model with all elements considered in the dynamic analyses. Fig. 2. Graphical output with the FE model elements. A Fig. 3. Detail of FE mesh to study the vibration amplitude generated by traffic train in the tunnel, measured at point A in the lower floor of the building. 573
For the output results, it was chosen a nodal point of the model, where the values of vibration velocities are generated by a train traveling in the tunnel (Figure3). In this model, as it can be seen, the origin of the dynamic events is located in the vertical position above node A. In this way, the main component of the vibration amplitude to be considered is the vertical one. Making the calculations, it can be shown that in sandy soils, the vibration amplitude in point A is lower than in clayey soils, as it can be seen in Figures 4, and in the graph of Figure 5. a) b) Fig. 4. Initial situation of vibration velocities in sandy (a) and clayey (b) soils. Sandy soil Clayey soil Fig. 5. Time vs. vibration velocity variation for clayey and sandy soils. The maximum amplitude obtained for clayey soils was 0.088 mm/s and for sandy soils 0.068 mm/s. 4. VIBRATION MINIMIZATION DUE THE OPENING OF LATERAL CAVITIES Using the developed model shown before, and some knowledge from mining engineering, two side cavities were created near the dynamic forces origin (Figure 6). The cavities geometry was based in the conical vibration propagation theory, with a 50º inclination angle and their length is related to the base perimeter of the tunnel. Upon several FE simulations the main obtained results are shown in Figure 7. Comparing these graphical results with the previous ones, it can be inferred that vertical vibration velocity is now much greater near the origin. This may be cause by the complex mechanisms of reflection and refraction caused by the constructed holes. 574
Fig. 6. Lay-out of the input model, with two lateral holes placed near the rail track. a) b) Fig. 7. Vibration velocity variation in sandy (a) and clayey (b) soils, considering two cavities in the track, near the origin of the dynamic forces caused by train traffic. In Figure 8a, a 48.5% reduction of vibration velocity at node A is observed upon the excavation of cavities, and in Figure 8b a 55.6% reduction in clayey soils is reached. 0.068 mm/s 0.088 mm/s 0.035 mm/s 0.039 mm/s No cavities With cavities a) b) Fig. 8. Graphical comparison of vibration velocity in node A before and after the opening of two cavities in sandy (a) and clayey (b) soils. 5. CONCLUSIONS The applied methodology involved the outlining of a geometric model to model the real case of a railway tunnel under a building structure located at the surface. Using the vibrations generated by train circulation within the tunnel as the dynamic source, it was found that: 575
Outputs of FE programs provide valuable information on the dynamic behavior of the composite structure formed by tunnel, soil and building. For two different soil types that were analyzed, it was observed that clayey soils provided less attenuation to vibrations than the sandy soils. Upon creating two lateral cavities at tunnel floor the same model supplied the following conclusions: a) The induced vibration attenuation reached 48.5% in the clayey soils and 55.6% in the sandy soils, so in the average about 50% of reducing vibration levels at the building. b) A superposing effect of attenuation phenomena was also verified, for greater attenuation occur for the soil type which granted higher vibration absorption with the lateral cavities. c) Therefore, the proposed constructive methodology may be useful for existing situations where no interruptions to train traffic are allowed. d) The formulation of alternatives related to constructive methodologies, insuring the efficiency of the proposed methodology is aimed to optimize solutions at any location. REFERENCES Bernardo, P. A. M., 2004. Environmental impacts of using explosives in excavations, with emphasis on vibrations. PhD thesis in Mining Engineering, IST Technical University of Lisbon (in Portuguese). Brady, B.H.G, Brown, E.T., 1985. Rock mechanics for underground mining. George Allen & Unwin (Publishers) Ltd., London, UK. Hoek, E., Brown, E.T., 1980. Underground excavations in rock. Institution of Mining and Metallurgy, London. ISO 14837-1, 2005. Mechanical vibration Ground-borne noise and vibration arising from rail systems Part 1: General guidance. International Standard Organization. Switzerland. Kramer, S.L., 1996. Geotechnical Earthquake Engineering. Prentice-Hall, Inc. Paneiro, G., 2006. Allowable vibrations in human beings and their repercussions in the design of antivibration tracks. MSc Thesis, IST Technical University of Lisbon (in Portuguese). Plaxis User's Manual, 2004. Dynamic Module. Delft, Netherlands. 576