ROSE SCHOOL SEISMIC VULNERABILILTY OF MASONRY ARCH BRIDGE WALLS

Transcription

1 I.U.S.S. Istituto Universitario di Studi Superiori Università degli Studi di Pavia EUROPEAN SCHOOL OF ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK ROSE SCHOOL SEISMIC VULNERABILILTY OF MASONRY ARCH BRIDGE WALLS A Dissertation Submitted in Partial Fulfilment of the Requirements for the Master Degree in EARTHQUAKE ENGINEERING By MARIA ROTA Supervisors: Prof. ALAIN PECKER Dr. RUI PINHO May, 24

2 The dissertation entitled Seismic vulnerability of masonry arch bridge walls, by Maria Rota, has been approved in partial fulfilment of the requirements for the Master Degree in Earthquake Engineering. Rui Pinho Alain Pecker

3 ACKNOWLEDGEMENTS The author would like to thank Dr. Rui Pinho for his guidance and support during the work and for his useful critique during the final stages of the dissertation. The author is also grateful to Prof. Alain Pecker for his valuable help and encouragement during the period spent in Paris, and for his careful review of the manuscript. Last but not least, the important contribution of Mr. Davide Bolognini is acknowledged, for his insightful comments and suggestions. 3

4 Table of contents TABLE OF CONTENTS 1 INTRODUCTION ARCH MASONRY BRIDGES General characteristics of masonry bridges Characteristics of the filling material Seismic damage to masonry bridges SEISMIC OUT-OF-PLANE CAPACITY OF WALLS Introduction Out-of-plane capacity of isolated walls Out-of-plane capacity of typical masonry arch bridge walls STATIC SOIL PRESSURES ON A RETAINING WALL Introduction Static soil pressure Check for sliding failure of the wall Check for overturning failure of the wall Check for shear failure of the wall DYNAMIC SOIL PRESSURES ON A RETAINING WALL State of the art Method by Mononobe and Okabe Description of the model Shortcomings of the model Method by Scott (1973) General derivation of the model Discussion of some details of the model a) Boundary effects...44 b) Effect of the wall flexibility Shortcomings of the model Method by Veletsos and Younan [1994b] Description of the model Shortcomings of the model Further developments of the method [Veletsos and Younan, 1997 and 2]

5 Table of contents 5.5 Comparison of the results obtained with the three methods implemented COMBINED MODEL COMPUTATION OF CAPACITY Introduction Parametric study on the reciprocal influence of the two walls Overview of capacity model Introduction Determination of the acceleration capacity of the wall Calibration of the parameters of Veletsos and Younan [1994b] method Stiffness of the rotational spring Reduction of shear modulus Frequency of excitation Discussion on the influence of acceleration on the results Results obtained COMPUTATION OF DEMAND Introduction Amplification of the seismic input to the base of the wall Amplification of the seismic input through the wall Results obtained COMPARISON OF CAPACITY AND DEMAND CONCLUSIONS AND FUTURE DEVELOPMENTS REFERENCES

6 List of figures LIST OF FIGURES Fig. 2.1: identification of the different parts constituting a masonry arch bridge [Galasco et al., 24]...13 Fig. 2.2: typical thickness of the filling material...15 Fig. 2.3: overturning of the walls for a railway masonry bridge, after the Fig. 2.4: clear evidence of overturning of a bridge wall, after Koyna earthquake, India, 1967 [ASC]...17 Fig. 2.5: overturning of the wall of a masonry bridge, after the Umbria-Marche earthquake, Fig. 3.1: constitutive law used for masonry...2 Fig. 3.2: forces acting on the wall...21 Fig. 3.3: stress diagrams on the masonry section at the different significant points of the λ curve. The tensile stresses are not represented, since the masonry does not resist in tension. Note that this is the same procedure used for a reinforced concrete section Fig. 3.4: example of λ - curve Fig. 3.5: maximum acceleration capacity obtained for the different walls considered Fig. 3.6: values of maximum acceleration obtained for walls of different slenderness h/t Fig. 3.7: values of maximum acceleration obtained for walls with different axial load P Fig. 4.1: factors of safety obtained against sliding (C S ), for the different wall cases considered Fig. 4.2: factors of safety obtained against overturning (C O ), for the different wall cases considered...33 Fig. 4.3: factors of safety obtained against shear (C V ), for the different wall cases considered Fig. 5.1: forces acting on an active wedge in M-O analysis, for a dry cohesionless backfill (from Wood [1973])...39 Fig. 5.2: system considered by Scott [1973]...42 Fig. 5.3: System considered by Veletsos and Younan [1994b] Fig. 5.4: Model of the soil stratum: elastically constrained bar Fig. 5.5: comparison of base shear, obtained with the two methods, for different values of R θ...55 Fig. 5.6: comparison of the points of application of the thrust resultant, obtained with the two methods, for different values of R θ...56 Fig. 5.7: comparison of base moment, obtained with the two methods, for different values of R θ

7 List of figures Fig. 6.1: graphical representation of a complex quantity, c...59 Fig. 6.2: system considered by Wood [1973]...6 Fig. 6.3: dimensionless thrust (left) and moment (right) factors... 6 Fig. 6.4: base shears and base moments obtained for different values of L/H...61 Fig. 6.5: forces acting on the wall in the combined model Fig. 6.6: pressure distribution on the wall for different values of rotational stiffness R θ...66 Fig. 6.7: effect of wall flexibility on the base shear, for a statically excited system Fig. 6.8: effect of wall flexibility on the base moment, for a statically excited system Fig. 6.9: stiffness of the rotational spring for the different cases considered Fig. 6.1: shear modulus reduction curve [Seed et al., 1986]...7 Fig. 6.11: reduction of the shear modulus for the different cases considered, for ω = Fig. 6.12: variation of the soil pressure distribution for different values of ω, for a fixed-base wall (left) and for a rotationally constraint wall, with R θ = 1. E7 (right) Fig. 6.13: variation of G/G max, h Vel and λ max, with the level of acceleration A, for case 7 and ω = Fig. 6.14: variation of G/G max, h Vel and λ max, with the level of acceleration A, for case 4 and ω = Fig. 6.15: capacity of the walls considered, for 5 different bridge typologies Fig. 6.16: variation of the base shear with the ratio frequency of excitation - first frequency of the wall, for different values of wall flexibility d θ [Veletsos and Younan, 1994b] Fig. 6.17: variation of the first frequency of the soil layer, with the frequency of excitation... 8 Fig. 6.18: variation of the ratio ω/ω 1 with the height of the soil layer... 8 Fig. 6.19: capacity of the first 7 cases of wall considered...81 Fig. 6.2: variation of the acceleration capacity with the height of the wall, h...82 Fig. 6.21: acceleration capacity of the different wall cases obtained with and without considering the effect of the infill material...82 Fig. 7.1: example of elastic horizontal response spectra, for seismic zone 4 (PGA =.5 g) Fig. 7.2: acceleration response factor for a damping ratio ξ = Fig. 7.3: acceleration response factors for the case of ω = Fig. 7.4: variation of the acceleration demand with the soil category, for the different bridge typologies and the different walls considered

8 List of figures Fig. 7.5: variation of the acceleration demand with the seismic zone, for the different bridge typologies and the different walls considered Fig. 7.6: variation of the acceleration demand with the seismic zone and the different soil categories, for the wall of case Fig. 7.7: variation of the acceleration demand with the bridge typology, for the wall of case Fig. 8.1: comparison of acceleration capacity and demand, for all the bridges and walls cases considered, assumed to be on soil of type B, C, and E and in seismic zone Fig. 8.2: ratio between demand and capacity obtained for the different bridge typologies Fig. 8.3: ratio between demand and capacity obtained for the different bridge typologies and for the different soil types...99 Fig. 8.4: ratio between demand and capacity, for the different bridge typologies and seismic zones...11 Fig. 8.5: variation of acceleration demand and capacity with bridge typology, for the wall of case 9 and for each seismic zone...13 Fig. 8.6: variation of acceleration demand and capacity with bridge typology, for the wall of case 9 and for each soil type

9 List of tables LIST OF TABLES Table 2.1: values of specific weight for some types...15 Table 3.1: characteristics of the walls considered in the parametric study Table 5.1: characteristics of the system considered...54 Table 5.2: results obtained from the comparison of Veletsos and Younan and M-O methods, for the case of fixed massless wall and static excitation Table 5.3: results obtained with the three method for a fixed-base wall, excited with a frequency of 4 rad/s Table 6.1: percentage variation of base shear and base moment Table 6.2: characteristics of the system considered

10 Chapter 1 Introduction 1 INTRODUCTION Many railway and road bridges in Italy (and, more generally, in Europe) are arch masonry bridges. It is a very old typology of bridges, which employment can be traced back to the ancient Romans period, or even earlier. The durability of this kind of bridges is remarkable, as demonstrated by the many roman masonry arch bridges, as well as some aqueducts, which are still perfectly working. This construction technique has been transmitted from one generation to the other, with only minor variations. Recently, new bridges are more often steel or concrete structures, but many masonry bridges still exist and their seismic performance is a major concern. The potentially high seismic vulnerability of masonry railway bridges has not yet been fully perceived, maybe because there have not been many significant damage evidences after recent earthquakes. While this can be simply due to chance, it is a matter of fact that it is hard to find any specific research study on this subject in the literature. In particular, the problem of the interaction between the infill material and the side walls of the bridges under seismic excitation has not been studied yet, but appears to be a potentially relevant damage mode. Above all, a problem to be considered is the interaction between infill material and parapet walls, in relation with a possible out-of-plane collapse. This failure mode is expected to occur at the very beginning of the seismic excitation, before other types of mechanisms may have damaged the structure. Also, it is believed that this kind of out-of-plane collapse may occur even for low levels of acceleration. This mechanism of failure has been studied in the current work, through a parametric study on a set of bridges and walls typologies. For each case, acceleration demand and capacity are compared and some conclusions are drawn. The present work is organised in 9 Chapters, which will be briefly described here. Following the introduction, Chapter 2 is an overview of the main characteristics of arch masonry bridges, particularly railway bridges, that will be studied in this work. The different parts composing such a structure are identified and special attention is given to the characteristics of the filling material, introduced between the two walls of the bridge, in order to create a horizontal plane for the rails. Also, an overview of typical seismic damages, occurred in the past to this typology of bridges, is presented. It is shown that local failures, such as overturning of the bridge walls, probably due to the interaction with the infill material, are very common for masonry arch bridges. 1

11 Chapter 1 Introduction In order to study whether this failure mechanism occurs or not for a given bridge typology, a comparison between capacity and demand is carried on, for a wide range of cases. Attention is given particularly to the determination of the capacity of the walls, including the effect of the infill material. For this scope, a capacity model is presented, obtained from the combination of different sub-models, each one describing a single aspect of the problem. In a first moment, the different sub-models are introduced separately and, only in a second time, they are assembled, to obtain the combined model, as explained more in detail in what follows. In Chapter 3, a method developed by Priestley [1985] and Paulay and Priestley [1992] for the out-of-plane seismic capacity of an isolated wall is introduced. Using this force-based model, it is possible to determine the acceleration-displacement curve of an unreinforced masonry wall and therefore to evaluate the maximum acceleration that the wall can resist before collapse. The described method is applied to a set of typical arch masonry walls and the results obtained are presented. After the maximum acceleration that a bridge wall can resist is determined, the thrust of the infill material on the walls needs to be taken into account, since it influences the capacity of the wall. This thrust can be subdivided into a static and a dynamic part. The static thrust, which has been calculated using Coulomb s theory, is described in Chapter 4. Then, the typical bridge walls previously introduced are checked with respect to three static mechanisms of failure, i.e. sliding, overturning and shear. The calculation of the dynamic component of the infill thrust is discussed in detail in Chapter 5. Three different methods for the evaluation of this dynamic thrust are implemented in the current work, respectively proposed by Mononobe-Okabe, Scott and Veletsos and Younan. The results obtained with these three methods are reported and compared. At the end, it is shown that the method of Veletsos and Younan is preferable in the current framework. The model for the isolated wall (described in Chapter 3) and the two models for the static and dynamic components of the thrust (described in Chapters 4 and 5), are combined, in order to form a complete capacity model for the system consisting in the walls and the infill material. This combined model is described in detail in Chapter 6. In order to apply this model, it is necessary to calibrate some parameters used in Veletsos and Younan method, to determine the dynamic thrust. These are essentially the stiffness of the rotational spring at the base of the wall, the frequency of excitation and the reduction of the shear modulus of the soil, depending on the level of deformation. Moreover, some other details of Veletsos method, concerning the maximum acceleration at the base of the wall, need to be analysed. Each one of these aspects is 11

12 Chapter 1 Introduction described in detail in Chapter 6. The acceleration capacity, obtained for each one of the wall cases considered, after application of this combined model, is also reported and discussed. Chapter 7 is devoted to the computation of the level of demand on the walls considered. This demand is calculated in two different steps: in a first moment, the acceleration demand at the base of the bridge wall is determined, using an elastic response spectrum. This spectrum is constructed according to the new Italian Seismic Code [O.P.C.M. n. 3274, 23] and the acceleration corresponding to the dominant frequency of oscillation of the bridge in the transverse direction is read from the spectrum. In particular, the acceleration at the base of the wall for all the wall cases considered is determined, assuming that the bridge belongs to each one of the four seismic zones described in the Code (each one corresponding to a different level of peak ground acceleration on rock) and assuming it is located on each one of the three soil categories. Then, the acceleration demand at the barycentre of the wall is determined, after application of an acceleration response factor. The results obtained are also reported and discussed in the same Chapter. Chapter 8 summarises the most relevant results obtained from the comparison of the acceleration capacity and demand for the different cases. In particular, the ratio between demand and capacity is analysed, since it is believed to be of some importance not only to check if a wall fails or survives, but also to quantify how it is far from the limit condition, corresponding to demand equal to capacity. Finally, Chapter 9 summarises the most relevant conclusions derived from this work and outlines suggestions for future research, with particular emphasis for those issues where further investigation is required. 12

13 Chapter 2 Arch masonry bridges 2 ARCH MASONRY BRIDGES 2.1 General characteristics of masonry bridges As stated by Resemini [23], the modern Italian masonry bridges, especially railway bridges, have been built in a time period of 1 years, approximately between 183 and 193. This very short period determines the homogeneity of some of the construction techniques and of the geometry of such structures. Nevertheless, they show some different details, depending on the year of construction, on the geographical area and, most likely, also on the designer. For a more detailed overview of the history of masonry arch bridges construction, the reader is referred to the work of Resemini [23]. Generally, the parts constituting a masonry bridge (shown in Fig. 2.1) are: the arch, which is the structural part of the bridge; the elements supporting the arch, i.e. abutments and piers; the foundations and the non-structural parts, located above the arch, to create a horizontal plane, such as the filling material. The filling material is laterally constrained by two walls, which are located above the exterior part of the arch. Some more detail on the characteristics of this infill material will be discussed in the next section. Fig. 2.1: identification of the different parts constituting a masonry arch bridge [Galasco et al., 24] 13

14 Chapter 2 Arch masonry bridges For existing masonry bridges, one of the major difficulties consists in the determination of the characteristics of the materials used in the construction. This is mainly due to the intrinsic characteristics of masonry, which is a very anisotropic material, whose mechanical properties are strongly dependent on the properties of its constituents. In particular, for existing structures it is far from trivial to determine the consistency of the grout, when present, or the quality of the structural elements (bricks or stones) used. Often, different types of materials may be used for different parts of the bridge, due to structural reasons (need for higher resistance of the more heavily stressed parts and for lower weight of the non-structural parts), as well as to economical reasons. [Resemini, 23] The geometry of the bridge is strongly influenced by the orography of the valley to be crossed. Wide and deep valleys are often crossed by bridges called viaducts, with more spans on high piers, whilst wide but shallow valleys are crossed by bridges with more spans on short piers. Narrow valleys and minor streams are usually crossed by single-span bridges. [Resemini, 23] 2.2 Characteristics of the filling material The space between the two walls of a masonry arch bridge is filled up with some material, in order to create a horizontal plane. This infill material must be light and able to drain the water; moreover it should contribute to the repartition to the arch of the concentrated loads applied on the horizontal plane on top of it. The filling material is usually incoherent material (such as soil or mucking resulting from mines excavation) or, in order to reduce the thrust on the walls, it may be constituted by dry stones, coarse aggregate, gravel, ballast or, more recently, low resistance concrete. In case of viaducts, especially when they have piers of significant height, it is not uncommon to find some brick vaults instead of the filling material. The reasoning behind this solution is not clear, but it may have been adopted to reduce the weight acting on the arch. For the sake of simplicity, this infill material will be always referred to as soil, in what follows. Some typical values of the specific weight of the infill material, suggested by Gambarotta et al. [21b], are summarised in Table 2.1. In that work, it is also stated that, when precise information on the type of material used is lacking, a value between 17 and 19 KN/m 3 may be reasonably assumed for the specific weight. In the present work a value of 17 KN/m 3 has been selected. 14

15 Chapter 2 Arch masonry bridges Table 2.1: values of specific weight for some types of filling material Material Specific weight [kn/m 3 ] Incoherent material Dry stones Aggregate or gravel Low resistance concrete 21 According to Albenga [1953], for railway bridges, the height of the soil between the horizontal plane and the top of the arch must not be less than 4 cm. However, for lower-height bridges, this limit may be brought down to 3 cm, but never less than 15 cm. Generally, the thickness of the stratum is equal to the arch thickness at the apex stone, as illustrated in Fig Fig. 2.2: typical thickness of the filling material The filling material produces, as obvious, some pressures on the side walls of the bridges. This pressure can be subdivided into static (always present) and dynamic (developed only when the structure is subjected to dynamic loads) parts. The calculation of the static part is described in detail in the chapter 4, whilst the dynamic part will be described in chapter Seismic damage to masonry bridges As stated above, most of the existing Italian masonry bridges have been built between 183 and 193. Therefore their seismic history is quite short, with a consequent lack of information about seismic damages to these structures. It seems likely that, for earthquakes of moderate intensity, a masonry bridge, without particular structural deficiencies, will survive without being heavily damaged; nevertheless, the response of these bridges to major earthquakes requires more investigation [Resemini, 23]. 15

16 Chapter 2 Arch masonry bridges Information about seismic damage to masonry bridges throughout the world is very limited. In fact, in the less industrial areas of the world, masonry structures are either of limited extension, or information is difficult to be obtained, whilst in the more industrial zones, modern infrastructures are not realized in masonry, as anticipated in the introduction. Also, the seismic risk of this kind of structures is not associated with human lives loss, but with the functionality of some parts of the system of infrastructures. This is another reason of the scarce information available, especially for those areas in which an earthquake causes many victims and therefore this kind of infrastructural damage is not considered of significance. Therefore, the seismic vulnerability of masonry bridges seems to be an interesting issue only in Europe and, in particular, considering the seismicity of the area, in Italy [Resemini, 23]. Even if the construction techniques may vary from area to area, due to local knowledge and in situ availability of materials, a study of the evidences of seismic damage to masonry bridges throughout the world may help in the knowledge of the damage mechanisms of these structures. The Bhuj earthquake (magnitude 7.7), which occurred in January 21 in the district of Gujarat, India, caused damage to several masonry bridges. Most of them were short span bridges, with short piers. The damages which have been observed consist in cracking and skews of the elements of the arches, damages to the piers and overturning of the bridge walls. An example of a railway masonry bridge, which has been damaged during the Bhuj earthquake, is shown in Fig This 88 years old bridge is already under repair, as indicated in the picture. It can be seen that the earthquake caused the overturning of the walls and the falling-off of the infill material, exposing the train rails. Fig. 2.3: overturning of the walls for a railway masonry bridge, after the Bhuj earthquake, 21, India [Gisdevelopment, 21] 16

17 Chapter 2 Arch masonry bridges Another example of seismic damage to a masonry bridge is due to the earthquake of Koyna, which occurred in India (1967) and is shown in Fig The damage clearly consists in the overturning of one of the bridge walls. The infill material between the two walls, which is visible in the picture, consists in some kind of soil. Fig. 2.4: clear evidence of overturning of a bridge wall, after Koyna earthquake, India, 1967 [ASC] Another example of damage due to overturning of the bridge walls is related to the earthquake occurred in the Italian regions of Umbria and Marche, in The damaged arch masonry bridge is shown in Fig Also in this case, the overturning of the wall exposes the filling material, even if, from this picture, it is not easy to identify the type of material constituting the filling for this particular bridge. 17

18 Chapter 2 Arch masonry bridges Fig. 2.5: overturning of the wall of a masonry bridge, after the Umbria-Marche earthquake, 1997 [Resemini and Lagomarsino, 24] The above puts in evidence that one of the most frequent types of failure consists in the overturning of the bridge walls. For these bridges, local failures, such as overturning of the bridge walls, probably due to the interaction with the infill material, are very common. The present work will focus on the study of this particular failure mechanism. 18

19 Chapter 3 Seismic out-of-plane capacity of walls 3 SEISMIC OUT-OF-PLANE CAPACITY OF WALLS 3.1 Introduction The determination of the capacity of unreinforced masonry walls to out-of-plane seismic excitation is one of the most complex and ill-understood areas of seismic analysis [Griffith et al, 23]. One of the reasons for this is the fact that all modern codes for new design of masonry structures give dimensioning and detailing rules that make out-of-plane collapse of walls extremely unlikely. Nevertheless, out-of-plane collapse is a crucial issue for old existing masonry structures [D Ayala and Speranza, 22]. There exist different types of out-of-plane collapse mechanisms that can be observed, being related to the overall geometry of the wall, its boundary conditions and the quality of the materials [D Ayala and Speranza, 22; Lagomarsino, 1999]. In this work, the attention is restricted to the simplest mechanism, the one way bending condition, which occurs in cantilever walls, since this is the typology of restraint typical of masonry arch bridge walls. In this case, the formation of cracks does not constitute wall failure, even in unreinforced walls. Failure can occur only when the resultant force in the compression zone of the central crack, R, is displaced outside the line of action of the applied gravity loads, as will be discussed later in further details. A simplified model is used to determine the acceleration-displacement curve and therefore to obtain the maximum level of acceleration that a given wall can resist prior to collapse. This model has been applied to a set of walls, which are considered typical for masonry arch bridges, and the maximum values of acceleration that each wall can resist, without considering the effect of the infill material, have been calculated. The results obtained are summarised in section Out-of-plane capacity of isolated walls In this work, the approach developed by Priestley [1985] and Paulay and Priestley [1992] for a masonry wall is followed, in order to analytically evaluate the capacity of a wall; in particular, the acceleration-displacement curve is determined. The wall is modelled as a single-degree-offreedom structure and is subjected to a seismic action in the direction perpendicular to the wall. The following assumptions are involved in the method: The response acceleration is constant over the height of the wall. Therefore the lateral inertia force per unit area will be w = ma, where m is the mass per unit area of wall 19

20 Chapter 3 Seismic out-of-plane capacity of walls surface. This force can also be written as w = λ p, where p is the weight per unit area of wall surface and λ is the seismic coefficient, given by λ = a/g. The model considers a wall of constant, rectangular section and unitary width. P - effects are included in the model. The constitutive relationship for the material is nonlinear, for which reason the elastoperfectly-plastic behaviour shown in Fig. 3.1 has been assumed for the masonry. σ f max Fig. 3.1: constitutive law used for masonry ε Failure of the wall will occur for out-of-plane deflection, after the formation of a plastic hinge at the base. In particular, referring to Fig. 3.2, the wall will fail when the point of application of the resultant of the vertical loads (P and W) is aligned with the point of application of the reaction R. The model can consider the presence of an axial load P, acting on top of the wall (in case of arch masonry bridge walls there could be a parapet on top of each wall). The method conservatively assumes that the displacement at the top of the wall increases proportionally to the curvature. A more accurate estimate of the displacement could be obtained by integrating the curvature distribution, but this is probably not warranted, since the errors deriving from this approximation are typically not large, until very large displacements are reached [Paulay and Priestley, 1992]. The model was originally developed by Paulay and Priestley for two different cases of boundary conditions, i.e. for a wall fixed at the base and free at the top and for a wall simply supported at both the ends. In this work, for obvious reasons, only the first condition has been considered. 2

21 Chapter 3 Seismic out-of-plane capacity of walls P λp λp /2 W h h/2 o x R Fig. 3.2: forces acting on the wall The seismic coefficient λ = a/g is considered in this model as a load factor and is used to construct the λ - curve. This curve will be referred in what follows as the accelerationdisplacement curve, since λ is effectively a dimensionless acceleration, normalised by a constant parameter (g). The acceleration-displacement curve is obtained by sequentially evaluating its points, increasing progressively the curvature at the top section of the wall and imposing equilibrium of external and internal forces. The forces acting on the wall are represented in Fig The force indicated as λp is the inertia force produced by the axial load P. This force is proportional to P through the seismic coefficient λ. For a generic value of eccentricity x of the reaction R, the seismic coefficient λ can be determined through the equation of equilibrium of moments around the point O in Fig. 3.2: λp h R x = λp h + P + W 2 (3.1) from which: λ = W Rx P + 2 ph h + P 2 (3.2) 21

22 Chapter 3 Seismic out-of-plane capacity of walls As stated above, the λ - curve needs to be defined by a series of discrete points. This is because the relationship between acceleration and displacement is influenced by the nonlinear constitutive law of the material and cracking effects. The first point of the λ - curve to be determined is the point of first cracking. Up to this point, the section is intact and therefore the λ - curve is linear elastic. At this point, the stress diagram is triangular, as can be seen from Fig. 3.3, and therefore the maximum eccentricity x is given by t/6, if t is the thickness of the wall. The cracking point is particularly important for this method, since the determination of the displacements is based on the values of displacement and curvature at cracking, as will be shown later in this section. origin f=r/t cracking f=2r/t after cracking 2R/t<f<fmax yielding b1 f=fmax after yielding b f=fmax ultimate t f=fmax Fig. 3.3: stress diagrams on the masonry section at the different significant points of the λ curve. The tensile stresses are not represented, since the masonry does not resist in tension. Note that this is the same procedure used for a reinforced concrete section The seismic coefficient at cracking, indicated as λ *, can be obtained by calculating the overturning moment from equilibrium considerations and by setting it equal to Rt/6 (the area of the triangular stress distribution in Fig. 3.3). It should be pointed out that, at this stage, the 22

23 Chapter 3 Seismic out-of-plane capacity of walls second order moment due to P and W is negligible and therefore it is not included in the expression for λ * : * λ = t R 6 ph h + P 2 (3.3) Before cracking, the wall behaviour is linear elastic and therefore the displacement can be calculated applying the superposition of the effects due to the inertia force uniformly distributed along the wall (λ p) and to the inertia force concentrated at the top (λ P). From classic structural mechanics formulae, the displacement at cracking * is thus given by: 4 3 * 1 * ph 1 * Ph = λ + λ (3.4) 8 EI 3 EI i.e.: where E is the elastic modulus of masonry and I is the moment of inertia of the intact section. The corresponding curvature φ * can be obtained dividing the moment at cracking (M crack ) by EI, R t φ M * = crack = EI 6 EI (3.5) After first cracking has occurred, the process becomes nonlinear. The crack propagates through the section and the stress diagram remains triangular until the maximum resistance, f max, of the masonry is reached. This situation corresponds to the point of first yielding, which is the next significant point of the λ - curve. In this range of the curve, λ is calculated according to Eq. (3.2). The stress f in the masonry is given by: f 2R = t 3 x 2 (3.6) 23

24 Chapter 3 Seismic out-of-plane capacity of walls The curvature φ can be obtained from: φ f 2R = t 3 E x t x = 2 E (3.7) With the assumption of displacement at the top proportional to the curvature, can be calculated as: * = φ * φ (3.8) After yielding, the stress diagram becomes trapezoidal, as shown in Fig For a given value of b, b 1 can be calculated as: b 1 = 2R f max b (3.9) and the eccentricity x can be obtained from: x = 1 2 ( b b1 ) b + b1 + b t b + b (3.1) The curvature is given by: φ = f max E ( b b1) (3.11) The seismic coefficient and the displacement are obtained respectively according to Eq. (3.2) and Eq. (3.8). Finally, the last point of the curve to be determined is the ultimate point, corresponding to the failure of the wall. The ultimate displacement u occurs when the resisting moment becomes 24

25 Chapter 3 Seismic out-of-plane capacity of walls zero, i.e. when the reaction R is aligned to the resultant of the vertical loads. Thus, from moment equilibrium: P + W u = xmax where: (3.12) P + W / 2 t a 1 R x = = max t (3.13) f max In the previous equation, a represents the width of the rectangular stress block at ultimate conditions. As expected, to the ultimate displacement corresponds a seismic coefficient λ =. An example of λ - curve, obtained with the method described above, is reported in Fig. 3.4, just to give an idea of the shape of this kind of curves λmax λ = a/g u Displacement [m] Fig. 3.4: example of λ - curve A couple of comments to the described model must be made: The assumption that the displacement at the top of the wall is proportional to the curvature is not really accurate. Hence, the values of displacement derived with this method are not accurate. Nevertheless, in this work, attention is paid more to the value of maximum acceleration that a wall can withstand than to the corresponding displacement. Therefore, it is believed that the model is accurate enough for the purposes of this study. 25

26 Chapter 3 Seismic out-of-plane capacity of walls According to Paulay and Priestley [1992], in calculating the response using the methodology outlined above, some account of vertical acceleration should be taken, since this reduces the equivalent acceleration necessary to induce failure in the wall. Conservatively, a value equal to two-thirds of the peak lateral ground acceleration is suggested by Paulay and Priestley. However, the vertical acceleration contribution has been neglected in this work. 3.3 Out-of-plane capacity of typical masonry arch bridge walls A parametric study has been performed, considering a set of walls of different dimensions and characteristics, which are summarised in Table 3.1. The values of the geometrical parameters have been chosen based on realistic considerations on the dimensions of existing bridges and based on limitations due to the selected method for the calculation of the dynamic thrust of the filling material on the wall, which will be discussed later. These issues will be analysed in some detail in the next chapters. Table 3.1: characteristics of the walls considered in the parametric study. Parameters common to all cases: f max = 1MPa, E = 8 f max Case h [m] t [m] P/W Case h [m] t [m] P/W Values of slenderness (h/t) of the walls approximately equal to 2 and 2.5, typical for this kind of walls, have been used. In particular, the height is assumed to vary between 1 and 2.5 m, while the 26

27 Chapter 3 Seismic out-of-plane capacity of walls thickness ranges from.4 to 1.2 m. These significant values of thickness are also confirmed by the observations of Torre [23], who states that a thickness value of 1 m is quite typical for the walls of a masonry arch bridge and the range of variation of the thickness may be between.8 and 1.5 m, according to the type of infill material and therefore to the thrust acting on the walls. It should be noted that the actual height of the walls of arch bridges varies along the arch. In this study, a constant height has been assumed, ideally equal to the weighted mean height of the wall. The eventual presence of a parapet on top of the walls is considered in some cases, throughout the application of an axial load P. This parapet would not in any case be very heavy and therefore an axial load equal to half of the weight of the wall has been assumed. Concerning the resistance of the masonry, a maximum value of 1 MPa has been considered. This value comes from the considerations reported by Gambarotta et al. [21a] about realistic values of resistance of the different types of masonry used in the past for the construction of arch bridges. In particular, Gambarotta and co-researchers suggest mean values of resistance ranging from 1 to 3 MPa for brick masonry and from 5 to 3 MPa for stone masonry. From these ranges, the assumption of a resistance of 1 MPa seems to be a conservative value and was thus adopted in this work. The Young modulus E has been assumed to be 8 times the masonry resistance, with 8 being a reasonable value based on previous studies on arch masonry bridges [Picchi, 21]. A masonry unit weight of 18 kn/m3 has been used to compute the wall weight. The model described in the previous section for the determination of the accelerationdisplacement curve of an isolated wall has been applied to the selected cases (refer to Table 3.1). For each case, the maximum acceleration λ max that the wall can withstand before failing has been determined. The values obtained for the different cases are summarised in Fig It can be observed that the maximum acceleration the walls can survive ranges from a minimum value of.285 g, to a maximum value of.488 g. 27

28 Chapter 3 Seismic out-of-plane capacity of walls.6 Maximum acceleration capacity w/o soil.5.4 λ max cases considered Fig. 3.5: maximum acceleration capacity obtained for the different walls considered As expected, the more slender the wall is (i.e. the higher is the ratio h/t), the lower the value of λ max it can resist, as shown in Fig In this plot, walls of the same height and with the same axial load P =, but with different slenderness values are compared. In particular, the maximum accelerations obtained for walls with slenderness values approximately equal to 2 and 2.5 are compared..6.5 h/t = 2 h/t = λ max Height of the walls [m] Fig. 3.6: values of maximum acceleration obtained for walls of different slenderness h/t Also, the walls with an axial load on top can resist lower levels of acceleration than the walls with the same geometry, but without axial load, as shown in Fig This effect is due to the 28

29 Chapter 3 Seismic out-of-plane capacity of walls destabilizing effect of P and of the horizontal inertia force λp induced by the axial load. In Fig. 3.7, only the walls with a slenderness value approximately equal to 2.5 are reported. If we look for example at the first couple of cases in the plot, corresponding to walls of height h = 1 m, the decrease of maximum acceleration due to the axial load is in the order of 26%..5.4 P = P =.5.3 λ max Height of the walls [m] Fig. 3.7: values of maximum acceleration obtained for walls with different axial load P 29

30 Chapter 4 Static soil pressures on a retaining wall 4 STATIC SOIL PRESSURES ON A RETAINING WALL 4.1 Introduction In the previous chapter, a simplified model for the behaviour of an isolated wall has been set up. It is now necessary to consider the pressure that the infill material exerts on the wall. It has already been anticipated that this effect can be subdivided into a static component and a dynamic component. The static thrust of the soil on the walls, which is more straightforward, will be discussed in this chapter, whilst the dynamic thrust will be analysed in greater detail in chapter 5. Moreover, before calculating the effect of the dynamic pressure induced by the presence of the soil, the out-of-plane stability of the walls under the static thrust has been checked, in order to make sure that the selected parameters (geometry and characteristics of the walls) are realistic. In particular, the static resistance of the walls has been checked with respect to three possible failure mechanisms: sliding, overturning and shear. It should be underlined that the first failure mechanism, sliding, concerns the overall stability of the wall, whilst the two others, overturning and shear, are checked at the section level. 4.2 Static soil pressure The static thrust exerted on a wall by the infill soil has been calculated using Coulomb s theory [e.g. Kramer, 1996]. According to this theory, the soil thrust is determined from force equilibrium, for both active and passive conditions. The basic assumption is that the force acting on the back of a retaining wall results from the weight of a wedge of soil, limited by a planar failure surface. A significant number of failure surfaces should be analysed, in order to identify the critical failure surface, i.e. the surface that produces the greatest active thrust or the smallest passive thrust. It is evident that in the present work, the situation of interest is the active case, corresponding to the soil exerting a force on the wall. The active soil thrust on a wall retaining a cohesionless soil can be determined using the following formulae: R st 1 = K A γ h 2 2 (4.1) where: 3

31 Chapter 4 Static soil pressures on a retaining wall K A = cos 2 β cos( δ + β ) cos ( φ β ) sin( δ + φ)sin( φ i) cos( δ + β )cos( i β ) 2 (4.2) In the above formulae: γ is the unit weight of the soil h is the height of the soil i is the angle of inclination from horizontal of the backfill β is the angle of inclination from vertical of the wall φ is the friction angle of the soil δ is the angle of interface friction between the wall and the soil Coulomb theory does not give explicitly the distribution of active pressure, but it can be assumed that, for linear backfill surfaces without surface loads, this distribution is triangular. In such cases, the static pressure R st acts at a height of h/3, with h being the height of the wall and this is what has been assumed in the current work. 4.3 Check for sliding failure of the wall A sliding type of failure occurs when horizontal force equilibrium is not maintained, i.e. when the lateral pressures on the back of the wall produce a thrust that exceeds the available sliding resistance at the base of the wall. The sliding resistance at the base of the wall is given by the product of the vertical forces times the tangent of the angle of interface friction. Therefore, the sliding check is satisfied when C S > 1, with C S defined by Eq. (4.3): C S ( W + P + R ) st v = (4.3) R st, h, tanδ where R st,v and R st,h are the vertical and horizontal components of the thrust, W is the weight of the wall and P is the axial load applied at the top of the wall. The results obtained for C S (see Table 3.1 for the identification of the cases considered), are summarised in Fig It can be observed that, for all the cases considered, the factor of safety 31

32 Chapter 4 Static soil pressures on a retaining wall against sliding is well above the limit condition, corresponding to C S = 1. For the soil density, a value of ρ = 17 Kg/m 3 has been used throughout the entire work. factor of safety cases considered Fig. 4.1: factors of safety obtained against sliding (C S ), for the different wall cases considered 4.4 Check for overturning failure of the wall Overturning moment failures occur when moment equilibrium is not satisfied, i.e. when the overturning moment, caused by the thrust, exceeds the resisting moment, due to the vertical forces acting on the wall. It is important to point out that all the calculations are made per linear meter of wall. In particular, the resisting moment is given by: M res t a = R 2 2 (4.4) where t is the thickness of the wall and R is given by the sum of the vertical forces acting, i.e.: R = W + P + R st, v (4.5) with W being the weight of the wall and P being the eventual axial force acting on top of the wall (both for an out-of-plane length of 1 m). Finally the term a in Eq. (4.4) is the width of the compressed zone at the ultimate condition (when the maximum resistance of the wall is reached), given by: 32

33 Chapter 4 Static soil pressures on a retaining wall a = R f max (4.6) where f max is the maximum resistance of the masonry. The overturning check is then satisfied when C O > 1, with C O defined by: C O = t a R 2 2 h Rst, h 3 (4.7) The results obtained for C O for the different cases considered (see Table 3.1), are summarised in Fig It can be observed that, for all the cases considered, also the factor of safety against overturning is well above the limit condition, corresponding to C O = 1. factor of safety cases considered Fig. 4.2: factors of safety obtained against overturning (C O ), for the different wall cases considered 4.5 Check for shear failure of the wall Shear failure occurs when the shear resistance V t of the section is exceeded by the shear force acting on the wall, which is the horizontal component of the static soil thrust, R st,h. Again, all the calculations are made per linear meter of wall. The shear check is satisfied when C V > 1, with C V defined as: C V = Vt R st, h (4.8) 33

34 Chapter 4 Static soil pressures on a retaining wall In the equation above, V t is calculated according to: V t t f = ' γ M vk (4.9) where t is the thickness of the compressed zone of the wall, defined as : t' = β t with (4.1) 6 M 1 if 1 t N 3 3 M 6 M β = if > 1 (4.11) 2 t N t N 2 M if 1 t N where - M is the bending moment around the axis perpendicular to the plane of the figure, induced by the earth thrust: M = R st,h *h/3 + R st,v *t/2 - t is the thickness of the wall - N is the axial force acting, given by the weight of the wall, W, plus the axial force P, due to the eventual parapet on top of the wall γ M is a safety factor, used only when designing new structures and therefore assumed equal to one in this work, consisting in an assessment of the vulnerability of existing bridges. f vk is the shear resistance of masonry, defined as: f = f σ 1. 4 f (4.12) vk vk +.4 m 1.5MPa bk where - f vk is the characteristic value of the shear resistance, when no compression is applied to the section 34

35 Chapter 4 Static soil pressures on a retaining wall - σ m is the average normal tension, calculated in this case as σ m = N/t - f bk is the characteristic value of the vertical compression resistance of masonry blocks The results obtained for C V for the different cases considered (see Table 3.1), are summarised in Fig It can be observed that, for all the cases considered, also the factor of safety against shear is well above the limit condition, corresponding to C V = 1. factor of safety cases considered Fig. 4.3: factors of safety obtained against shear (C V ), for the different wall cases considered 35

36 Chapter 5 Dynamic soil pressures on a retaining wall 5 DYNAMIC SOIL PRESSURES ON A RETAINING WALL 5.1 State of the art Despite the multitude of studies that have been carried out over the years, the dynamic response of vertical retaining walls is still not well understood. The methods that have been used until now can be classified into three categories: (1) limit state analyses, in which the relative motions of the wall and backfill material are sufficiently large to induce in the soil a limit or failure state; (2) methods in which the wall and soil movements are limited, so that it can be assumed that the backfill material deforms in its linear elastic range. This approach is often referred to as the elastic approach, since the soil is modelled as a linear elastic material. (3) intermediate cases, where the soil is neither failed nor elastic, but the actual non-linear hysteretic properties of the soil are modelled. A typical example of the first category is the Mononobe-Okabe method [Mononobe and Matsuo, 1929], [Okabe, 1924], with its variants [Nadim and Whitman, 1983], [Richards and Elms, 1979], [Richards et al., 1999], in which a wedge of soil bounded by the wall is assumed to move as a rigid block, with the same acceleration as the ground. In the vein of the second group are the methods proposed by Wood [1973], by Scott [1973], by Veletsos and Younan [1994a, 1994b, 1997, 2], by Li [1999] and by Ortigosa and Musante [1991]. In particular, Wood [1973] analysed the dynamic response of a homogeneous linear elastic soil, trapped between two rigid walls connected to a rigid base, providing an analytically exact solution. The approximate model proposed by Scott [1973] represents the restraining action of the backfill by a set of massless, linear horizontal springs. The stiffness of the springs is defined as the subgrade modulus. Veletsos and Younan [1994a, 1994b, 1997, 2] improved Scott s model, by using a set of semi-infinite, elastically supported, horizontal bars with distributed mass, to include the radiational damping of the soil and by using horizontal linear springs of constant stiffness, to model the horizontal shearing action of the stratum. Li [1999], following Veletsos and Younan s analyses, introduced the effects of foundation flexibility and damping into the model. In this study, the rigid wall and the viscoelastic backfill are considered to rest on a viscoelastic half-space foundation. Ortigosa and Musante [1991] proposed a simplified kinematic method, in which the wall is supported in several locations and the possible wall movement is the 36

37 Chapter 5 Dynamic soil pressures on a retaining wall flexural deformation. The free-field shear modulus is used directly by the authors to calculate the subgrade modulus, without considering the variation of the stress field near the wall. The elastic approach is not widely accepted in current design procedures, because early elastic approaches assumed the wall to be rigid and fixed against both deflection and rotation, such that the soil pressures and forces so computed are significantly larger (2.5 to 3 times, according to Veletsos and Younan [1997] and 1 to 2 times according to Li and Aguilar [2]) than those calculated using limit-state methods. However, some studies carried out by Veletsos and Younan [1994a, 1994b, 1997, 2] have shown that both the magnitude and distribution of the dynamic wall pressures are very sensitive to base constraints and wall flexibility. Therefore, if realistic values are used both for wall and base constraints, the computed pressures are significantly lower than the values obtained with fixed-base rigid walls and potentially of the same order of magnitude as those computed by the Mononobe-Okabe method. Moreover, Li [1999] showed that, if foundation effects are taken into account, the computed base shear may be of the same order of that estimated with Mononobe-Okabe, even for a rigid gravity wall. Therefore, after these studies, the initial limitations to the elastic approach seem to be overcome and this method can be considered a valuable tool for the seismic design of non-yielding walls. Finally, an example of the third group of methods is the work by Richards et al. [1999], in which consideration is given to the plastic response characteristics of the soil in the free-field, where horizontal stress increases nonlinearly beyond yield. The soil is modelled by a series of springs, the stiffness of which is given by the subgrade modulus, determined by knowing the value of elastic or secant shear modulus of the soil in the so-called free-field. Again to the third group belongs the work of Siller et al. [1991], on the behaviour of gravity and anchored walls. Three of the methods cited above (one limit state and two elastic approaches) have been implemented in the current work. They will be described in more details in the following sections. 5.2 Method by Mononobe and Okabe Description of the model The Mononobe-Okabe (M-O) method [Mononobe and Matsuo, 1929], [Okabe, 1924] is a direct extension of the static Coulomb s theory to pseudo-static conditions: pseudo-static vertical and horizontal accelerations are applied to a Coulomb wedge and the pseudo-static soil thrust is then obtained through equilibrium of forces acting on the wedge (see Fig. 5.1). The original method 37

38 Chapter 5 Dynamic soil pressures on a retaining wall has been developed only for the case of active earth pressures; later, it has been extended (without any experimental validation) to the case of passive pressures. The main assumptions involved in the method are: The wall is assumed to displace sufficiently at the base to mobilise the maximum shearing resistance of the backfill, so that the classical Coulomb s wedge theory can be applied. The soil is assumed to satisfy the Mohr-Coulomb failure criterion. Failure of the soil is assumed to occur along a plane surface, passing through the toe of the slope and inclined at some angle from the horizontal. The wedge of soil behind the wall behaves as a rigid body and its acceleration is uniform. The wedge of soil bounded by the wall and the failure plane is assumed to be in equilibrium under the forces acting on it: gravity, earthquake and boundary forces along the wall and the failure surface (see Fig. 5.1). The earthquake loading is accounted for by equivalent static horizontal and vertical forces k h W and k v W (with k h and k v horizontal and vertical earthquake coefficients and W the weight of the wedge of soil), applied at the centre of gravity of the wedge. The use of constant coefficients implies that dynamic amplification due to site effects (propagation of seismic waves through a soft layer of soil, resonance effects, topographic effects, etc.) is neglected. The selection of the appropriate earthquake coefficients is not a simple matter. Since the peak ground acceleration occurs only for a very short period during an earthquake, it is clear that k h should correspond to a certain fraction of the peak ground acceleration; k v will be a fraction of k h. Different percentages of the maximum acceleration have been suggested in the literature [D Ayala and Speranza, 22], but consensus has not yet been reached. It can be noted that EC8 [23] has proposed a relation between allowable wall displacement and percentage of maximum acceleration to be used in the M-O method. 38

39 Chapter 5 Dynamic soil pressures on a retaining wall Fig. 5.1: forces acting on an active wedge in M-O analysis, for a dry cohesionless backfill (from Wood [1973]) A positive horizontal seismic coefficient causes the total active thrust to exceed the static active thrust whilst it causes the total passive thrust to be less than the static passive thrust. Therefore, the active case has been considered, since it is the more critical and the formulae reported from now on will refer to this case. It is recalled that the active pressure condition corresponds to a situation in which the soil is exerting a force on the retaining system, whilst the passive pressure condition corresponds to the opposite case of the retaining system exerting a force on the soil. The total active thrust can be obtained as: 1 2 P AE = K AE γ h (1 ± kv ) (5.1) 2 where γ is the unit weight of the soil, H is the height of the wall, k v is the vertical earthquake coefficient, defined as k v = a v / g, with a v the vertical pseudo-static acceleration. Finally, K AE is given by: K AE = cosθ cos 2 2 cos ( φ θ β ) β cos( δ + β + θ ) 1 + sin( δ + φ)sin( φ θ i) cos( δ + β + θ )cos( i β ) 2 (5.2) 39

40 Chapter 5 Dynamic soil pressures on a retaining wall in which, for a dry soil, the angle θ is calculated as: k h θ = tan 1 (5.3) 1± kv The meaning of the different angles appearing in the Eq. (5.2) is explained graphically in Fig The method, as described above, gives the magnitude of the total force acting on the wall, but does not state anything about its point of application or the pressure distribution. Although the M-O method seems to imply that the total thrust acts at a height of h/3 above the base of the wall, experimental results [Seed and Whitman, 197] suggest that, under dynamic conditions, it actually acts at a higher point. The total active thrust P AE, determined from Eq. (5.1) can be subdivided into a static component R st, acting at h/3 from the base and a dynamic component P AE, acting at approximately.6h [Seed and Whitman, 197]. The static component can be obtained using the classic Coulomb s theory (see Eqs. (4.1) and (4.2)), while the dynamic component can be easily obtained from the difference between the total and the static parts. On this basis, the total active thrust will act at a height H R ( h / 3) + P P st AE = (5.4) AE (.6h) The formulae reported above refer to the case of a dry cohesionless backfill. The method can easily be extended to include soil cohesion, by considering the equilibrium of the wedge with the addition of cohesive forces acting along the wall boundary and the failure surface. The M-O method, very simple and straightforward, has been widely used by designers, because experimental and theoretical studies have shown that it gives satisfactory results in cases where the backfill deforms plastically and the wall movement is large and irreversible [Whitman, 199]. However, there are many practical cases, such as massive gravity walls or basement walls braced at top and bottom, where the wall movement is not sufficient to induce a limit state in the soil, and the dynamic soil pressures should be computed using elastic or viscoelastic methods, such as those explained in more detail in the following sections. 4

41 Chapter 5 Dynamic soil pressures on a retaining wall Shortcomings of the model 1. Representation of the complex, transient dynamic effects of earthquake shaking by a single constant unidirectional pseudo-static acceleration is a crude approximation. The vertical and horizontal pseudo-accelerations are chosen to be significantly less than the peak ground accelerations expected to occur during the design earthquake. This reduction seems to be based on the acceptance of a certain permanent outward movement of the wall. The amount of reduction to be applied is not clearly defined, even if EC8 [23] suggests some criteria for its evaluation. 2. No accounting for the flexibility of the wall is possible in the analysis. 3. The method is not applicable to soils experiencing a significant strength degradation during earthquakes, such as potentially liquefiable soils. 4. Resonance effects and amplification of earthquake motion as a result of propagation of seismic waves through a relatively soft soil layer behind the wall are not taken into account. 5.3 Method by Scott (1973) General derivation of the model Probably the simplest available approximate model for evaluating the dynamic soil pressures induced by ground shaking on walls retaining an elastic stratum is the one proposed by Scott [1973]. The soil-wall system considered in this work is a semi-infinite uniform layer of viscoelastic soil, free at its upper surface, bonded to a non-deformable rigid base and retained along either one or both of its vertical boundaries by rigid walls. The walls are assumed to be of the same height as the stratum and their mass is neglected in the model. The walls may be either fixed or restrained at the base by a rotational spring. Both the walls and the shear beam are presumed to be excited by the same horizontal ground motion, of acceleration A(t) at any time t. The system considered is shown in Fig

42 Chapter 5 Dynamic soil pressures on a retaining wall Rigid wall k Elastic Shear Beam k ρ, G h x, u Rθ y Rigid foundation b L A(t) Rθ Fig. 5.2: system considered by Scott [1973] The response of the medium in the absence of the wall (the so-called free-or far-field response) is evaluated considering it to respond as a base-excited, one-dimensional, vertical cantilever shearbeam. This beam is attached to the wall or walls by springs, representing the soil-wall interaction. At each level, y, of the shear beam, a resisting dynamic pressure is developed. This pressure is proportional to the instantaneous lateral displacement, u, at that level, which is the relative motion between the shear-beam and the wall at that height. The constant of proportionality is the spring stiffness K. It is implicit therefore that only normal pressures can develop between the wall and the soil, that these pressures will be both tensile and compressive and that shearing vertical stresses are neglected in the model. In general, the soil properties (density ρ, Young s modulus E and shear modulus G) vary with depth, and therefore the density and shearing stiffness of the beam, as well as the spring constant K, also vary with depth. This variation can be easily accounted for in the model, but for simplicity the equations reported here refer to the case of soil properties constant with depth. From the equations of motion of the problem, the eigenfrequencies of the different modes can be obtained as: ( 2n 1) 2 2 π G 2K ωn = + (5.5) 2 4h ρ bρ 42

43 Chapter 5 Dynamic soil pressures on a retaining wall where n is the number of the mode being considered, h is the height of the walls and b is the width of the shear beam. The first term on the right hand side of Eq. (5.5) is the frequency of the unrestrained shear beam, while the second term accounts for the effect of the restraining springs. For the case of b = L, with L being the distance between the two walls, and assuming the walls to be rigid, the spring constant K is given by: ( 1 ν ) ( 1 2ν ) 8G K = (5.6) L The participation factors for each mode, in the case of constant soil properties, can be calculated as: α = n h h cosλ x n 2 cos λ x n dx 4sin λnh = 2λnh + sin 2λnh dx (5.7) where the terms λ n are obtained from λ = n ( n 1) 2 π 2h For a given design earthquake, the maximum deflection at a certain height y for the nth mode is given by: u mn S α cos( λ y) ω vn n n = (5.8) n where S vn is the pseudo relative velocity response at the frequency ω n, obtained from the design spectrum, with a reasonable value of damping. The combination of the contributions of the different modes, performed using the square-rootof-the-sum-of-the-squares method, gives the deflection at elevation y: u = 2 Svn n cos( n y) ( umn ) = α λ (5.9) ωn 43

44 Chapter 5 Dynamic soil pressures on a retaining wall Therefore, the pressure at this elevation y is given by: p = K u (5.1) Analyses performed by Scott [1973] have shown that the pressure distribution is dominated by the first mode contribution and that higher modes are negligible. Considering only the first mode, the maximum pressure distribution on the wall is given by: p m1 = 4KSv πω 1 1 πy cos = 2h p πy cos 2h (5.11) and the resultant of the dynamic pressure acting on the wall is 2 p m1 = p h (5.12) π From geometrical considerations on the cosine pressure distribution, the point of application of the above resultant is located at a distance 2h/π from the base of the wall. Therefore, the maximum moment per linear meter at the base can be calculated as: M 2h π 4 π 2 m1 = pm 1 = p 2 h (5.13) Discussion of some details of the model a) Boundary effects The model has been developed for the case of two walls, located at a distance L. The boundary effects exerted by one wall on the other vary, as obvious, with the distance L. When L becomes large with respect to the height of the walls, h, the restraining effect of the second wall becomes insignificant, and the pressures exerted by the soil on the considered wall become equivalent to the pressures exerted by a backfill of unlimited extent. A limit value of L/h = 1 has been 44

45 Chapter 5 Dynamic soil pressures on a retaining wall arbitrarily proposed by Scott, to model an infinite backfill. In this case, the same formulae can be used, but with a value of L/h = 1. On the other hand, for small values of L/h (1. may be assumed as a limit) the model is not applicable, due to its intrinsic one-dimensional nature. For L/h < 1, indeed, the two-dimensional nature of the real motion cannot be neglected. b) Effect of the wall flexibility Up until now, the walls have been considered to be rigid. However, in reality, a wall deflects both by bending and by rotation. The wall flexibility is accounted for in the model by considering a rigid wall hinged at its base, with a rotational spring of stiffness R θ (refer to Fig. 5.2). With this spring at the base, if a moment M is applied to the wall, the angular deflection of the wall is thus given by M = R θ θ. Bending deflections of the walls are neglected, since their incorporation in the model would be difficult. The effect of wall flexibility can be easily included in the model only in case of a linear first mode of vibration. This is the case when soil properties increase with depth. For a linear first mode, the pressure, given by K times the difference between the soil and wall displacements, increases linearly with height. This is due to the fact that the lateral displacement of the wall is linear with height (because the wall is rigid) and the dynamic lateral soil displacement is also linear (because the first mode is linear). The effect of the base rotation of the wall can therefore be simply considered as a reduction of the effective soil stiffness K. From moments considerations, the new soil stiffness K can be calculated as: 1 K = Kh 1+ 3R ' 3 θ K (5.14) Shortcomings of the model 1. The only damping included in the model is that involved in the shear beam itself. For the case of an infinite backfill, the model indeed does not consider the radiational damping capacity of the soil. As a result, when the exciting frequencies are close to the natural frequencies of the stratum, the model may significantly overestimate the response of the 45

46 Chapter 5 Dynamic soil pressures on a retaining wall system. This limitation however does not apply when there are two walls, since in this case the waves are trapped between the walls and no radiation damping can take place. 2. This soil spring stiffness K (or K ) is a constant value, which does not depend on the characteristics of the ground motion. The uncertainty in the model lies in the choice of this value K, since forces and pressures are proportional to it. Therefore, any error in the determination of K is directly reflected in the other response quantities. Moreover, as the Poisson s ratio tends to.5, K tends to infinity, as well as forces and pressures. This result is clearly unrealistic. 3. The model is based on the assumption that the demand due to the ground motion is resisted by shearing action of the medium in the far-field and by column-like extensional behaviour in the medium between the far field and the medium. The capacity of the medium adjacent to the wall to transfer forces vertically by horizontal shearing resistance is not considered. 4. It s a mono-dimensional method and therefore cannot be applied when the soil has values of width and height comparable. In these cases, indeed, the bi-dimensionality of the problem cannot be neglected. 5.4 Method by Veletsos and Younan [1994b] The following description of the method makes reference essentially to the work proposed by the authors in the paper indicated as 1994b, since this is the model that has been implemented in this study. Nevertheless, some references are made to the successive works of the authors and to the modifications introduced to the original model Description of the model The authors investigated a system consisting in a uniform layer of linear viscoelastic material that is free at its upper surface, bonded to a non-deformable rigid base and retained along one of its vertical boundaries by a rigid wall. The wall may be either fixed or elastically constrained against rotation at its base, by a spring of stiffness R θ. The wall and the soil stratum have the same height h. Both the base of the layer and the wall are subjected to a space-invariant, harmonic, uniform horizontal motion, a(t), at any time t., characterised by a frequency ω and a maximum amplitude A. The system considered is shown in Fig

47 Chapter 5 Dynamic soil pressures on a retaining wall y Rigid wall h Rθ a(t) x Fig. 5.3: System considered by Veletsos and Younan [1994b] The main hypotheses involved in the method are: Constant hysteretic, frequency independent material damping, characterised by a coefficient δ, which is the same for both shearing and axial deformations. Soil movement is only induced by horizontal earthquake excitation in the layer bottom. Therefore it has been assumed in the model that the vertical normal stress does not change during vibration. The model has been developed for vertically propagating shear waves, with the assumption that horizontal variation of vertical displacements in the soil medium is negligible. Only normal pressures develop along the wall-soil interface. The wall can resist both compressive and tensile stresses, which is equivalent to assume that there is complete bonding between the wall and the retained medium. According to Veletsos and Younan [1994b], one of the main problems of Scott method [1973] is its inability to account for the vertical stresses due to horizontal shearing. Therefore, they proposed to include this variation of horizontal shearing stresses, which can be expressed as: τ τ = y xy 1 τ xy = h η (5.15) with η being the dimensionless distance given by η = y/h. Neglecting the horizontal variation of the vertical displacements, 47

48 Chapter 5 Dynamic soil pressures on a retaining wall 2 G u G u τ xy = and τ = (5.16) 2 2 h η h η where u is the relative horizontal displacement of the medium with respect to the moving base. If u is expressed by the method of separation of variables as a linear combination of modal terms, through mathematical manipulation [Veletsos and Younan, 1994b], the nth component of τ, denoted as ( τ) n, may be recognized to be: ( τ ) = ρω u (5.17) n 2 n n where u n is the nth component of the displacement u and ω n is the circular natural frequency of the stratum, considered to respond as a cantilever shear-beam, given by ( 2n 1) πvs ωn = (5.18) 2h where n refers to the mode being considered and V s is the shear wave velocity of the stratum. Eq. (5.17) represents a force per unit of length, that is identical to the force induced by a massless linear spring of stiffness k n given by: k n 2 2 (2n 1) π G = ρω n = (5.19) 2 2 h This corresponds to modelling the shearing action of the medium, for each modal component, with a set of horizontal linear springs of constant stiffness k n, connected at their lower ends to the common base, subjected to the prescribed ground acceleration A(t). On the other end, the medium is modelled by a series of semi-infinitely long, elastically supported horizontal bars with distributed mass, as shown in Fig In reality, it is sufficient to determine the response of a single bar for each modal component, since the responses of the other bars are proportional to sin[(2n-1)πη/2]. 48

49 Chapter 5 Dynamic soil pressures on a retaining wall x Elastic bar with distributed mass kn kn kn Fig. 5.4: Model of the soil stratum: elastically constrained bar For a viscoelastic bar, with frequency independent damping δ, subjected to a harmonic motion of circular frequency ω, the impedance or dynamic stiffness K n can be calculated according to Eq. (5.2). It should be pointed out that δ is the loss coefficient, equal to twice the damping ratio β, commonly used in soil dynamics. K 2 = ( K ) (1 + iδ )(1 φ iδ ) (5.2) n st n n + where φ n is the dimensionless frequency ratio φ n = ω/ω n and (K st ) n is the static frequency, defined as ( K ) (2n 1) π ψ G = with 2 h st n 2 ψ = (5.21) 1 ν The real part of the impedance K n is related to the force component which is in phase with the excitation, while the imaginary part represents the component 9 out of phase. The instantaneous value of the far field, horizontal, steady-state displacement, relative to the moving base, of a point of the stratum situated at a distance η from the base, can be expressed as the superposition of modal components as [Veletsos and Younan, 1994a]: u f ( η, t) = U n sin n= 1 2 (2n 1) π η e iωt where: (5.22) U n 16 ρah 3 π G 2 1 (2n 1) 1 1 φ + iδ = 3 2 n (5.23) 49

50 Chapter 5 Dynamic soil pressures on a retaining wall 5 If we consider the wall to be elastically restrained against rotation at the base by a spring of stiffness R θ, the instantaneous value of steady-state horizontal displacement of the wall at a distance η from the base, is given by: t i e h t w ω η η = Θ ), ( (5.24) where Θ is the complex-valued amplitude of the wall rotation, which can be determined as explained later. The above equation, expanded in terms of the natural modes of vibration of the stratum, becomes: t i n n e n W t w ω η π η = = 2 1) (2 sin ), ( 1 where: (5.25) h n W n n Θ = ) (2 1) ( 8 π (5.26) The instantaneous pressure exerted from the backfill on the wall can then be obtained as the product of each displacement component and the corresponding bar impedance: t i n n n n n n t i n n n e n i i n G i i n Ah t e i e n W U K t ω ω η π δ φ δ π ψ δ φ δ ρ π ψ η σ η π η σ + + Θ + + = = = = = 2 1) (2 sin ) )(1 (1 1) (2 1) ( ) (2 1 8 ), (.. 2 1) (2 )sin ( ), ( (5.27) It should pointed out that, in the expression above, the terms involving the displacements U n represent the pressure induced on a fixed-base wall, while the remaining terms (involving W n ) represent the contribution of the wall rotation.

51 Chapter 5 Dynamic soil pressures on a retaining wall The shear and bending moment at the base of the wall can then be easily obtained by integration of the pressure. It should be pointed out that the last two terms of Eqs. (5.28) and (5.31) represent the effect of the wall inertia, which may be easily included in this model, with µ being the mass per unit of plan area of the wall. The base shear is thus given by: Q iωt b ( t) = Qb ΘQb µ Ah + µ h ω Θ e where (5.28) Q b 16ψ = ρah 2 3 π n= 1 1 (2n 1) 3 1+ iδ 2 1 φ + iδ n (5.29) 8 2 n 1 1 ψ ( 1) Qb = Gh (1 iδ )(1 φ iδ ) n + π = (2n 1) n 1 (5.3) While the base moment is given by: M iωt b ( t) = M b ΘM b µ h A + µ h ω Θ e where (5.31) M b 32ψ = ρah 3 4 π = n 1 n+ 1 ( 1) (2n 1) 4 1+ iδ 2 1 φ + iδ n (5.32) M 16ψ = Gh (1 δ )(1 φ 3 + i 3 n π = (2n 1) 1 b + n 1 iδ ) (5.33) The rotation amplitude Θ can be obtained from the equilibrium of moments around the wall base: M b iωt ( t) = M b ΘM b µ h A + µ h ω Θ e = Rθ Θ (5.34) 2 3 It should be pointed out that the response is evaluated here for a harmonic excitation. The response to an arbitrary ground motion can be determined using Fourier transform techniques. 51

52 Chapter 5 Dynamic soil pressures on a retaining wall Once the geometry and the properties of the system have been fixed, the value and distribution of the pressure acting on the wall are still not determined, because some parameters, such as the stiffness of the rotational spring and the characteristics of the excitation (frequency and maximum acceleration for a harmonic motion), still need to be evaluated. The calibration of such parameters is not trivial and the procedure that has been followed in the present work, in order to determine these parameters, will be discussed later Shortcomings of the model The rotational stiffness of the spring should be assigned as an input data. In practice, this stiffness value is difficult to determine a priori and can be known only from results obtained from a previous dynamic interaction study. In fact, for the elastic behaviour of a rigid wall, the wall rotation is directly related to the bedrock flexibility. It may be therefore preferable to express the rotational stiffness by other frequency-dependent dynamic stiffnesses that are related, in turn, to the soil shear modulus and damping, as well as to the base geometry of the wall. On the other hand, if the bedrock flexibility is considered for the wall, it must also be accounted for in the backfill analysis. This can be done by including the damping capacity of the bedrock along its large contact area with the backfill, as suggested by Li [1999]. The rotational stiffness is a real-valued quantity. The fact that it does not have a complex part overamplifies the wall dynamic responses. To overcome this drawback, Li [1999] replaced this rotational stiffness by the complex-valued dynamic stiffness of a rigid strip foundation. In this model, the damping capacity of the wall is neglected. Therefore, all waves impinging on it cannot be dissipated and the rigid movement of the wall increases the wave amplitude. As a result, the dynamic wall shear is more amplified for a wall with a larger rotation, which is not what would be expected. The assumption of complete bonding between the wall and the retained soil, which makes possible the development of tensile pressures on the wall, is clearly unrealistic. In a real case, when the tensile pressures exceed the gravity-induced compressive pressures, the backfill would tend to separate from the wall and this separation would increase the wall shears and bending moments with respect to the case of complete bonding. In fact, with the assumption of complete bonding, the portions of the wall subjected to tensile pressure will produce contributions to shear and moment of 52

53 Chapter 5 Dynamic soil pressures on a retaining wall opposite sign with respect to the compressed zones, thus reducing the total shear and moment at the base. The assumption of uniform properties for the soil is also unrealistic, since the shear modulus of the backfill is likely to increase with depth and this variation will affect both the magnitude and distribution of the pressure. For a rigid wall, it has been shown [Veletsos and Younan, 1994b] however that the variation of the soil properties results in smaller wall forces than in the considered case of constant properties. Therefore it could be argued that this effect compensates for the effect of the assumption of complete bonding, which is of opposite sign. However, these two aspects require further studies Further developments of the method [Veletsos and Younan, 1997 and 2] As already stated, the method described above is the one proposed by Veletsos and Younan in More recently, the authors have added some other features to the model. In particular, in the versions proposed in 1997 and 2, the flexibility of the wall can also be included in the model. The effect of wall flexibility is to reduce the forces acting on the wall itself. Also, different boundary conditions are introduced in the model, with the wall being either free or hinged at the top. In particular, the force reduction due to wall flexibility is more significant for cantilever walls than for top-constrained walls, due to the greater effective stiffness of topsupported walls. However, it was felt that such refinements were not needed within the scope of the current work. 5.5 Comparison of the results obtained with the three methods implemented Some comparisons have been made of the results given by the three methods implemented. First of all, the Veletsos and Younan (V-Y) and Mononobe-Okabe (M-O) methods have been compared for the case of a massless and fixed-base wall, retaining a backfill of the same height. The system is subjected to an excitation, characterised by a value of frequency very small with respect to the fundamental natural frequency of the stratum. This is the so-called static case [Veletsos and Younan, 1994b], [Li and Aguilar, 2], which is equivalent to applying the loading slowly enough not to induce dynamic amplification. Therefore, it is meaningful to compare the solution obtained with V-Y for the static case to that of M-O, since this latter method neglects the dynamic amplification due to the backfill or to the wall responses. 53

54 Chapter 5 Dynamic soil pressures on a retaining wall The characteristics of the system considered in this example are summarised in Table 5.1. It should be pointed out that the values of the soil parameters, described in the table, will be kept constant throughout all this work. In V-Y method, 2 modes have been considered, even if, for the case of a fixed-base wall, the higher modes are often negligible. In M-O, a horizontal seismic coefficient equal to the PGA has been used, while the vertical seismic coefficient is equal to half of the PGA. It has to be noted that, in the M-O approach, only the dynamic contribution is shown, after the static contribution has been subtracted. The results obtained are shown in Table 5.2. Table 5.1: characteristics of the system considered WALL SOIL EXCITATION Height [m] Thickness [m] Specific weight [N/m 3 ] Mass density [Kg/m 3 ] Poisson ratio [-] Damping coeff Max acceleration [g] Frequency [rad/s].25 1 Table 5.2: results obtained from the comparison of Veletsos and Younan and M-O methods, for the case of fixed massless wall and static excitation Veletsos et al. M-O Dynamic base shear [N/m] Point of application [m].6.6 Dynamic base moment [Nm/m] In can be observed that: the base shear calculated by the elastic approach (3919 N/m) is approximately equal to ρah 2 (=4169 N/m), and is approximately 3 times the base shear obtained with M-O (1268 N/m). This result agrees very well with what reported by Li and Aguilar [2]. the point of application of the resultant of the pressure obtained with the two methods is practically identical. This height is close to the 2/π value, corresponding to a pressure distribution that increases as a quarter-sine from base to top. The latter distribution is obtained assuming that the soil responds is its fundamental mode of vibration [Veletsos and Younan, 1997]. 54

55 Chapter 5 Dynamic soil pressures on a retaining wall From the two previous points, it follows that the base moment obtained with V-Y is again approximately 3 times that obtained with M-O The same system has been considered again, subjected to the same static excitation (A =.25 g and ω = 1), but with the addition of a rotational constraint at the base of the wall in Veletsos method. The results obtained have been compared to those of the M-O method. It can be observed that: as the flexibility increases, the base shear calculated by Veletsos method decreases and hence approaches the value obtained with M-O; for values of R θ between 1E6 and 2E7, it is shown in Fig. 5.5 that the two models give quite close results. This trend was expected since, as explained in section 5.2, one of the assumptions of M-O method is that the wall displaces sufficiently at the base to mobilise the maximum shearing resistance of the backfill. As R θ decreases and the base of the wall becomes more flexible, the wall displacements become larger and therefore the results given by V-Y approach those obtained with M-O. The variation of the point of application of the resultant of the pressure with R θ is shown in Fig. 5.6, for the two models. It can be observed that, as R θ increases, the result obtained with V-Y tends to that obtained with M-O, showing a trend which is opposite to that of the base shear. On the other hand, the base moment shows a dependence on R θ which is very similar to what has been observed regarding the base shear, as shown in Fig Base shear [N/m] V-Y M-O 5 1.E+6 2.E+7 4.E+7 6.E+7 8.E+7 1.E+8 R θ [N/rad] Fig. 5.5: comparison of base shear, obtained with the two methods, for different values of R θ 55

56 Chapter 5 Dynamic soil pressures on a retaining wall.7 Point of application of the thrust resultant [m] V-Y.1 M-O 1.E+6 1.E+8 2.E+8 3.E+8 4.E+8 5.E+8 R θ [N/rad] Fig. 5.6: comparison of the points of application of the thrust resultant, obtained with the two methods, for different values of R θ 25 Base moment [Nm/m] V-Y M-O 1.E+6 1.E+8 2.E+8 3.E+8 4.E+8 5.E+8 R θ [N/rad] Fig. 5.7: comparison of base moment, obtained with the two methods, for different values of R θ A third comparison between the different methods has been made for the case of the same system, with a fixed-base wall, excited by a harmonic motion of circular frequency ω = 4 rad/s and maximum acceleration A =.3 g. In this case, also Scott s method has been included in the comparison, but only the fundamental mode has been considered, since, according to the author, higher modes effects are negligible. Concerning M-O method, k h = PGA and k v = k h /2 have been used. In Veletsos model, the mass of the wall has been neglected and 2 modes have been considered. The results obtained are summarised in Table 5.3:. 56

57 Chapter 5 Dynamic soil pressures on a retaining wall Table 5.3: results obtained with the three method for a fixed-base wall, excited with a frequency of 4 rad/s Veletsos et al. Scott M-O Dynamic base shear [N/m] Point of application [m] Dynamic base moment [Nm/m] As expected, the M-O method results in the lowest base shear. This is because this method relies on the wall movement to relieve the pressure behind the wall [Ostadan and White, 1998]. Also, V-Y method gives the highest base shear. It can be observed, on the other hand, that the three methods give very close values for the point of application of the resultant of the thrust. After this quick comparison of the three methods, it has been decided to continue the work using the model proposed by Veletsos and Younan [1994b] since, as seen above, this method features the following advantages: dynamic amplification (induced by the backfill or by the wall) is included in the solution the capacity of the medium adjacent to the wall to transfer forces vertically by horizontal shearing resistance is not neglected the radiational damping capacity of the soil is considered the wall inertia effect can be included in the model the flexibility of the wall can be accounted for Therefore, all the results reported in the subsequent chapters will refer to this method for the calculation of the dynamic thrust of the soil on the wall. 57

58 Chapter 6 Combined model computation of capacity 6 COMBINED MODEL COMPUTATION OF CAPACITY 6.1 Introduction In order to determine the capacity of the considered masonry bridges walls, including also the effect of the infill material, it has been necessary to combine the model proposed by Veletsos and Younan [1994b], for the dynamic thrust of the soil on the walls, and the model developed by Priestley [1985] and Paulay and Priestley [1992], for the seismic response of an isolated wall. Also the static thrust, calculated according to Coulomb s theory, will be considered in the combined model. Before going into detail about how this combined model has been realised, one of the limitations of Veletsos method - the fact that it considers only one wall, thus neglecting the effect of the presence of a second wall - will be discussed. In particular, it will be shown that, for typical values of the ratio width of the bridge to height of the walls, this effect is actually negligible. Then, an overview of the combined model is presented. However, in order to apply this combined model, some parameters need to be discussed and calibrated. These are essentially the stiffness of the rotational spring at the base of the wall, the frequency of excitation and the reduction of the shear modulus of the soil, depending on the level of deformation. Each one of these aspects will be described in detail in the following sections. Moreover, some other details of Veletsos method, concerning the maximum acceleration at the base of the wall, will be discussed. Before going any further, it is important to note that the Veletsos method gives, as results, complex-valued, time dependent quantities, in which the real part corresponds to the component which is in phase with the excitation applied to the system, and the imaginary part represents the component which is 9 out of phase. These quantities vary with time, since the excitation, which is assumed to be harmonic, is not constant with time. It is evident that, in order to incorporate this results into the model for the isolated wall, real-valued quantities, constant in time, are required. Therefore, the modulus of the complex quantities obtained with Veletsos method has been used. The reason for this choice is that what has actually a physical meaning is the real part of the response and what is required in the model is the maximum over time of this real part. In this particular case of a harmonic excitation, characterised by a single frequency and with a given amplitude, which is constant in time, the maximum of the real part coincides with the modulus of the complex quantity. This can be easily understood by observing the representation of a given 58

59 Chapter 6 Combined model computation of capacity complex quantity, c, as a vector rotating in the imaginary-real plane, with a given modulus, as shown in Fig It is clear then from the figure that: max t [ Re() c ] c (6.1) Im Im(c) c o Re(c) Re max[re(c)] c Fig. 6.1: graphical representation of a complex quantity, c 6.2 Parametric study on the reciprocal influence of the two walls As explained in the previous chapter, it has been decided to calculate the dynamic thrust of the soil on the wall using the method proposed by Veletsos and Younan [1994b]. Nevertheless, this method has been developed for a single wall, retaining a uniform layer of viscoelastic soil, whilst in arch masonry bridges, there are actually two parallel walls, retaining the infill material. It is easy to understand that, if the two walls are spaced far apart, the pressures on one wall will not be strongly influenced by the presence of the other. In order to estimate the minimum distance at which the interaction between the two walls can be neglected and therefore the approximation introduced by using Veletsos method for the dynamic thrust is acceptable, a parametric study has been performed, using the model proposed by Wood [1973]. This model considers a homogeneous linear elastic soil, trapped between two rigid walls, connected to a rigid base, as shown in Fig The system is subjected to a horizontal, harmonic base acceleration, of amplitude a h. 59

60 Chapter 6 Combined model computation of capacity Fig. 6.2: system considered by Wood [1973] The resultant of the dynamic thrust (which is the base shear) and the dynamic overturning moment (about the base of the wall) can be expressed as: Q = a g 2 h γ H F (6.2) p M = a g 3 h γ H F (6.3) m where γ is the unit weight of the soil, H is the height of the soil, F m and F p are the dimensionless dynamic thrust and moment factors, shown in Fig Fig. 6.3: dimensionless thrust (left) and moment (right) factors for various geometries and Poisson s ratios [Wood, 1973] 6

61 Chapter 6 Combined model computation of capacity The values of dynamic base shear and base moment have been calculated for different values of the ratio L/H and for the following parameters: height: H = 2 m PGA: a h =.1 g Poisson s ratio: ν =.33 The results obtained are summarised in Fig It can be noticed that, for L/H > 5 the variation of base shear and moment becomes approximately nil. This can be even better appreciated from the observation of the percentage difference between the values obtained with increasing values of L/H, and the values corresponding to the case of an infinite backfill, taken as L/H = 8, summarised in Table 6.1. It can be noted that, for L/H = 5, the percentage difference is less than 1% for both the quantities. Therefore, the ratio L/H = 5 can be viewed as the limit beyond which the influence of the second wall becomes insignificant and the backfill can be considered unlimited. Nevertheless, even for L/H = 4, the variation in the base shear is negligible (1%) and the variation in the base moment (5%) is not very large. Thus, even L/H > 4 can be set as a limit for the unlimited backfill, with an acceptable level of approximation. Assuming L/H = 4 as the threshold value and considering that typical values of bridge width range approximately from 4 to 6 m for one rail bridges and from 8 to 12 m for two rails bridges, it results obvious that Veletsos model is acceptable for values of wall height between 1 and 3 m. Since, in this work, walls of height ranging from 1 to 2.5 m have been considered, the approximation introduced by Veletsos method (a single wall with an unlimited backfill) is applicable. Influence of the distance between the two walls 71 base shear [N/m] and base moment [Nm/m] base shear base moment L / H Fig. 6.4: base shears and base moments obtained for different values of L/H 61

62 Chapter 6 Combined model computation of capacity Table 6.1: percentage variation of base shear and base moment with respect to an infinite backfill, taken as L/H = 8 L/H % variation base shear % variation base moment Overview of capacity model Introduction The combined model consists essentially in the model proposed by Priestley [1985] and Paulay and Priestley [1992], for the seismic response of an isolated wall, with the addition of the effect of the infill material. As explained in the previous chapters, the effect of the soil is a thrust, acting on the wall. This thrust has actually two components: a dynamic component, calculated using Veletsos and Younan model and a static component, calculated using Coulomb s method Determination of the acceleration capacity of the wall The model of the isolated wall, proposed by Priestley [1985] and Paulay and Priestley [1992], has been described in detail in chapter 3. This model allows to compute the maximum acceleration that a wall can resist before collapse, through moment equilibrium considerations. The forces acting on the wall, in absence of infill material, have been shown in Fig The static soil thrust is included in the model decomposed into a horizontal and a vertical components, and is applied at h/3. On the other hand, the effect of the dynamic component of the thrust, calculated according to Veletsos method, is included in the model through a force λr Vel, applied at a certain height h Vel, as shown in Fig The exact meaning of this force λr Vel will be explained in what follows. 62

63 Chapter 6 Combined model computation of capacity P λp λrvel hvel /2 Rst,v W h Rst,h h/3 o R h/2 Fig. 6.5: forces acting on the wall in the combined model It is important to underline that, in the combined model, the wall inertia effects are not represented anymore explicitly by the distributed force λp, as it was in the model for the isolated wall. This is because, as explained in chapter 5, the wall inertia effect can be easily included in Veletsos model; in this case, the output given by Veletsos considers both the pressure exerted by the soil on the wall and the wall inertia. In particular, the resultant of these effects, λr Vel, is simply the shear at the base of the wall (multiplied by the gravity acceleration and divided by the acceleration A), whilst its point of application h Vel is a weighted average of the points of application of the resultants of the two effects, and is calculated from the ratio between base moment and base shear. It is important to note that the force λr Vel can be written in a form proportional to λ = a/g, since both the soil pressure and the wall inertia are proportional to the acceleration. This aspect will be discussed in more detail in a following section. The acceleration-displacement curve of this combined model (wall plus soil) can be calculated according to a procedure which is very similar to what has been described in chapter 3 for the isolated wall. In particular, the seismic coefficient λ is now given by: 63

64 Chapter 6 Combined model computation of capacity λ = W h R x P + Rst 2 3 R h + P h Vel Vel, h t + R 2 st, v (6.4) where the reaction R is obtained as: R P + W + R st, v = (6.5) As already explained in section 3.2, the cracking point determination is very important in this method, for the evaluation of the displacements. The seismic coefficient at cracking, λ, is calculated as: t h R Rst, h * λ = 6 3 R h + P h Vel Vel (6.6) The corresponding displacement at cracking,, is given by: * = * λ P h EI R Vel h 2 2 Vel h h 3 Vel R EI st, h h 3 (6.7) Eqs. (3.5) through (3.11) described in chapter 3 still hold. The maximum acceleration that the wall can withstand can thus be obtained as: a max = λ max g (6.8) In order to evaluate the amplification of the seismic input through the wall, which will be discussed in section 7.3, it is necessary to estimate the first circular frequency of the wall, ω w.. This frequency can be calculated as: 64

65 Chapter 6 Combined model computation of capacity ω K n m λ g max = = (6.9) w λmax where: K n is the secant stiffness corresponding to λ max on the λ curve, which can be evaluated according to: λ g m λ max K n = (6.1) max λ max is the displacement corresponding to the attainment of λ max It is clear that the value of ω w just calculated depends on the value of the displacement λ max. The method proposed by Priestley [1985] and Paulay and Priestley [1992] is a force-based procedure, in which the evaluation of the displacement is approximated. Therefore also the value obtained for ω w is subjected to this approximation. In any case, it will be shown that the influence of ω w on the results is negligible. 6.4 Calibration of the parameters of Veletsos and Younan [1994b] method Stiffness of the rotational spring From a physical point of view, the wall does not have a rotational capability at its base, because it is more or less clamped in the arch (see Fig. 2.1). Nevertheless, a fictitious rotational stiffness can be used to account for the flexibility of the wall, due to its own deformation. That flexibility indeed reduces the earth thrust, as the rotational stiffness does [Veletsos and Younan, 1997]. Since the impact of the wall flexibility on the dynamic thrust is not taken into account in the model developed in this work, the approach of a fictitious rotational stiffness is considered to be a valuable alternative. The stiffness of the rotational spring R θ is a very important parameter in V-Y method. It is to be noted, indeed, that a variation of the value of R θ does not only determine a different value of the resultant of the pressure on the wall, but it also determines a different shape of the pressure distribution. This effect is very clear for the case of a statically excited system (in the meaning 65

66 Chapter 6 Combined model computation of capacity explained in section 5.5), as can be noticed from Fig. 6.6, where the normalized pressure σ/ρah is plotted. The results in Fig. 6.6 refer to the massless wall of case 1 (see Table 3.1), subjected to an excitation of frequency ω = 1 and maximum acceleration A =.2 g. It can be observed that, as the flexibility at the base of the wall increases, the wall pressures decrease and change their shape. 1.2 Adimensional height η = y / h R θ = 1E2 1. R θ = 1E R θ = 1E8 R θ = 1E Adimensional pressure σ / (ρah) Fig. 6.6: pressure distribution on the wall for different values of rotational stiffness R θ The change in shape arises from the higher modes contributions, which become more significant as the flexibility increases. These results are consistent with what has been observed by Wood [1973] and by Veletsos and Younan [1994b]. Therefore, a significant number of modes of vibration is necessary to accurately represent the pressure distribution on walls which are elastically constrained against rotation at the base. Nevertheless, the base shear is well approximated using only the contributions of the first two modes of vibration and the base moment is well approximated using the fundamental mode only. This is shown in Fig. 6.7 and Fig. 6.8, where the normalised values of these quantities are plotted, for the same system considered in Fig. 6.6, subjected to an excitation of frequency ω = and maximum acceleration A =.1 g. 66

67 Chapter 6 Combined model computation of capacity Qb/(ρAh 2 ) modes only 2 modes 1st mode only d θ = Gh 2 /R θ Fig. 6.7: effect of wall flexibility on the base shear, for a statically excited system.6 M b/(ρah 3 ) modes 1st mode only d θ = Gh 2 /R θ Fig. 6.8: effect of wall flexibility on the base moment, for a statically excited system For each one of the walls considered in this study, the stiffness value to be assigned to the rotational spring at its base has been determined by an iterative process, using a fibre element software, called Seismostruct [Seismosoft, 24], developed for nonlinear analyses of structures. The following steps have been performed: the dynamic pressure of the soil on the wall is evaluated using Veletsos and Younan method, with a trial value of R θ. A simple finite element model of the wall is constructed. The wall is modelled as a fixed-base cantilever, made of 3D inelastic beam-column elements. The material 67

68 Chapter 6 Combined model computation of capacity constitutive model is elasto-perfectly plastic. The wall is loaded by the pressure distribution obtained from the previous step. A static push-over analysis is performed on the model and the moment-curvature plot at the base of the wall is obtained. From this plot, it is possible to read the effective rotational stiffness corresponding to the moment given by Veletsos method. This stiffness is compared to the initially assumed value and the procedure is repeated until convergence of the two values is reached. The values of R θ obtained for the different walls considered, for the case of ω = 3, after application of the procedure described above, are summarised in Fig The case numbers refer to the different wall typologies and are described in Table E+12 Rotational stiffness 1.5E E+12 R θ 9E+11 6E+11 3E cases considered Fig. 6.9: stiffness of the rotational spring for the different cases considered It can be observed that, as expected, the stiffness of the rotational spring increases as the dimensions of the walls augment (with increasing case number). Also, it should be pointed out that, despite some differences in the stiffness values for the various cases, all the walls can be considered to be fixed-base, since the values of the rotational stiffness are very high. This means that, in the current work, the flexibility of the wall is neglected and the approach of using a fictitious rotational spring to account for it would have been a valuable alternative if finite equivalent values of stiffness were found, which is not the case. 68

69 Chapter 6 Combined model computation of capacity Reduction of shear modulus It is well known that the deformation characteristics of soil are highly nonlinear and this is manifested in the shear modulus and damping ratio, which vary significantly with the amplitude of shear strain under cycling loading. In particular, for this work, what is of interest is the reduction of the shear modulus, for increasing levels of deformation. This effect has been accounted for in the model, as described in the following. The low-amplitude shear modulus, or maximum shear modulus, G max, which depends essentially on the characteristics of the soil and on the height of the wall, has been first evaluated according to the following equation: G max ' σ m = k pa (6.11) p a where: ' σ m is the mean effective stress p a is the atmospheric pressure k is a coefficient, which depends on the relative density. For low values of relative density, typical of the problems under study, a value of k = 1 is commonly used [Pecker, personal communication]. The next step in the evaluation of the reduction of the shear modulus is the selection of a degradation curve, which is a plot of G/G max versus the shear strain γ. There are many such curves in the literature, developed for different types of soil. Since the type of soil used as filling material in masonry arch bridges may vary significantly from bridge to bridge and this work does not focus on a particular bridge, but is a parametric study of different possible bridge typologies, it is not an easy task to select a curve. Within this framework, a curve developed for gravelly soils has been adopted, since it is believed that gravel has properties which may be considered intermediate among the range of properties of the different types of soil possibly used as infill material in masonry bridges. In particular, the curve proposed by Seed et al. [1986] and represented in Fig. 6.1 has been selected. 69

70 Chapter 6 Combined model computation of capacity Fig. 6.1: shear modulus reduction curve [Seed et al., 1986] The last element necessary to evaluate the shear modulus reduction is the level of strain developed in the soil due to the earthquake, in order to be able to enter the curve and obtain a value. This shear strain can be easily calculated in Veletsos method, remembering that: γ = 1 h u η (6.12) where u is the displacement, calculated according to Eq. (5.22), and η is the dimensionless height, η = y/h. Since u / η is a complex quantity, in order to calculate a single value of strain, the modulus has been used. The quantity expressed by Eq. (6.12), evaluated at mid-height (for η = ½), is the maximum shear strain in the soil, γ max. The reduction of shear modulus has been evaluated for a level of shear deformation corresponding to 2/3 γ max. This arbitrary reduction factor of 2/3 has been introduced in order to be more conservative, since a higher level of deformation corresponds to a lower value of G/G max (as can be seen from Fig. 6.1) and therefore to a smaller soil thrust. It is important to outline that, for each case considered, the procedure used to evaluate this shear modulus reduction is actually iterative, since the shear strain depends on the shear modulus of the soil which, in turns, depends on the shear strain. Therefore, for each case, it has been necessary to perform few iterations, until the pair of values of shear strain and modulus belongs to the selected curve. 7

71 Chapter 6 Combined model computation of capacity The values of G/G max obtained for the different walls considered, for the case of ω = 3, after application of the procedure described above, are summarised in Fig It can be observed that the amount of reduction of the shear modulus depends on the height of the wall, but not on its thickness, since both G max and γ do not depend on the thickness of the wall. Reduction of shear modulus G/G max cases considered Fig. 6.11: reduction of the shear modulus for the different cases considered, for ω = Frequency of excitation The results given by Veletsos method are sensitive to the characteristics of the excitation applied at the base of the wall and, especially, to the circular frequency ω, which has to be given as an input. In particular, the height of application of the resultant of the soil thrust does not depend on the ground motion characteristics, as stated by Veletsos and Younan [1997], but the value of the resultant is very sensitive to them. Since Veletsos method considers the effect of dynamic amplification, it is obvious that, if the frequency of excitation approaches the frequency of the soil layer, the response is amplified, due to the well known resonance effect. This can be easily demonstrated, by looking at an example. If one considers a system with the characteristics reported in Table 6.2, subjected to excitations of different frequencies and constant PGA A =.1g, the first frequency of the soil layer can be calculated as: V ω s 1 = π (6.13) 2h 71

72 Chapter 6 Combined model computation of capacity In the case-study analysed herein, ω 1 results to be 235 rad/s. It is evident, from the results obtained and reported in Fig. 6.12, that the response (in this case the soil pressure) is significantly amplified for ω = 2 rad/s, with respect to the other values of frequency considered. Table 6.2: characteristics of the system considered WALL SOIL EXCITATION Height [m] Thickness [m] Specific weight [N/m 3 ] Mass density [Kg/m 3 ] Poisson s ratio [-] Damping coeff Max acceleration [g] Adimensional height η = y/h ω = 1 ω = 1 ω = 2 ω = 3 ω = 4 ω = 5 Adimensional height η = y/h ω = 1 ω = 1 ω = 2 ω = 3 ω = 4 ω = soil pressure [N/m 3 ] soil pressure [N/m 3 ] Fig. 6.12: variation of the soil pressure distribution for different values of ω, for a fixed-base wall (left) and for a rotationally constraint wall, with R θ = 1. E7 (right) The circular frequency ω that is applied at the base of the wall in V-Y method has been chosen as the dominant frequency of oscillation in the transverse direction of the bridge, which is below 72

73 Chapter 6 Combined model computation of capacity the walls, because that structure acts as a narrow band filter in the broad band frequency content of the earthquake. Therefore, the choice of a particular configuration of bridge is equivalent to the selection of a value of frequency. Few values of frequency have been used in this work, i.e. ω = 5, 4, 3, 2 and 1 rad/s. These frequencies correspond respectively to values of the dominant period of vibration of the bridge in the transverse direction T 1 =.13,.16,.2,.31 and.63. These values have been selected based on the results of a number of 3D finite element modal analyses. The upper limit of the range of frequencies corresponds to a bridge characterised by very short and stiff piers. This value of frequency has been successively reduced to account also for bridges which are less stiff. When analysing a specific bridge, the value of the dominant frequency of excitation ω can be evaluated more accurately. This can be obtained for example running a modal analysis of the structure, in order to get the periods of vibration. From observation of the response spectrum corresponding to a given ground motion, in correspondence to the periods of vibration determined from the modal analysis, it should be possible to identify the dominant frequency of the bridge, for the chosen input motion. Alternatively, the frequency can also be determined running a time-history analysis of the bridge, subjected to a given accelerogram, and then constructing the Fourier spectrum of the response. The dominant frequency is identified by the peak of the Fourier spectrum Discussion on the influence of acceleration on the results In principle, the capacity of a structure should be independent of the level of seismic excitation applied. This is true, as expected, also for the described model, but some aspects of it deserve a more detailed discussion, since they can create some problems. First of all, in Veletsos method, the maximum acceleration at the base of the wall, A, has to be given as an input, and its value influences the soil pressure. However, the value of A does not modify the shape of the soil pressure and hence the height of application of the resultant of the thrust does not depend on the level of acceleration applied. Moreover, the base shear, which includes also the effect of the wall inertia, is directly proportional to A. Since the results of Veletsos method used in the combined model are indeed the point of application and the base shear normalised with respect to the acceleration (and multiplied by a constant value, g), it is obvious that the results obtained from this combined model, i.e. the capacity of the different systems, do not depend on the level of excitation. 73

74 Chapter 6 Combined model computation of capacity There are still a couple of aspects to be clarified. As described in the previous sections, the combined model requires calibration of some parameters, i.e. the stiffness of the rotational spring at the base of the wall, the reduction of the shear modulus and the frequency of excitation. As explained previously, the frequency of excitation has been related only to the chosen bridge typology. This corresponds to the assumption that the range of frequencies considered includes all the possible frequencies excited for the selected population of bridges, for any given acceleration. Therefore it can be assumed that the selected values of frequency are independent of the level of acceleration. On the other hand, the determination of the other two parameters requires some iterations, which involve other quantities calculated using Veletsos method and hence apparently dependent on the acceleration. In order to be able to state that the capacity does not depend on the demand of acceleration, it is necessary to show that also the values of these two parameters are independent of it. It can be easily demonstrated that the stiffness of the rotational spring, R θ, is not influenced by the value of acceleration. We know from Eq. (5.34) that: M b R = (6.14) θ Θ where M b is given by Eq. (5.31) and Θ can be written, from Eq. (5.34) as: 1 2 M b µ h A Θ = M b µ h ω + R 3 θ (6.15) Since both the terms at the numerator of Eq. (6.15) are directly proportional to A (see Eq. (5.32) for the term M b), it can be concluded that also Θ is proportional to A. Similarly, by looking at Eq. (5.31), it can be easily noted that all the terms defining M b are directly proportional to A and hence M b is proportional to it as well. From this considerations, it is evident, from Eq. (6.14), that since both the numerator and the denominator are proportional to A, R θ does not depend on it. When using the iterative procedure described in the section 6.4.1, the moment-curvature plot, obtained from the finite element model, depends only on the shape of the distribution of the soil 74

75 Chapter 6 Combined model computation of capacity pressure (which is not influenced by the acceleration) and therefore is independent of A. On the other hand, the value of dynamic base moment, given by Veletsos method and used to enter the moment-curvature plot and calculate the corresponding R θ is proportional to A. However it has been observed that, since the walls considered are all very stiff, it is correct to use for R θ the value of initial stiffness, which is independent of A. For what concerns the reduction of shear modulus, it is more complicated to demonstrate that its value does not depend on the level of acceleration. It has been shown that the value of G/G max depends on the value of G max and on the level of shear strain, necessary to enter the reduction curve. G max depends only on the geometrical characteristics of the system and on the properties of the soil. On the other hand, as obvious, the shear strain depends also on the level of acceleration applied. A parametric study has been performed, in order to investigate the effect of this dependence of G/G max on the acceleration, on the values of capacity, obtained with the combined model. Only two cases of this parametric study are reported here. The first example concerns the wall of case 7, subjected to an excitation of frequency ω = 4 and different levels of acceleration. The results obtained are shown in Fig Fig. 6.13: variation of G/G max, h Vel and λ max, with the level of acceleration A, for case 7 and ω = 4 The second example concerns the wall of case 4, subjected to an excitation of frequency ω = 1. The results obtained for this case are shown in Fig

76 Chapter 6 Combined model computation of capacity Fig. 6.14: variation of G/G max, h Vel and λ max, with the level of acceleration A, for case 4 and ω = 1 From these two examples it can be observed that the level of acceleration influences the values of G/G max, and this, in turns, influences the point of application of the dynamic soil thrust, h Vel (which on itself does not depend on the acceleration), and the maximum acceleration, λ max. How significant is this influence varies with the frequency of excitation. In particular, for the first example, it can be observed that, even if the values of G/G max and of h Vel vary significantly, the results obtained for the capacity (λ max ) are almost constant, with a maximum variation, among the different cases, of.77%. In the second example, the difference in the results for the various cases is more significant, with a maximum variation of λ max of 22%. From the results of this parametric study it can be concluded that: As A increases, G/G max and hence the soil thrust decreases and the capacity of the wall increases. For high values of the frequency of excitation ω, the influence of the level of acceleration on the reduction of shear modulus and hence on the capacity is negligible. For lower values of the frequency of excitation ω, this influence may become significant. In what follows, all the results for capacity refer to the case of A = 1 m/s 2. For low values of frequency, in which case the dependence of G/G max on the acceleration may be significant, the results obtained for A = 1 m/s 2 represent lower bounds, with respect to the results that would be obtained with higher levels of acceleration. 76