In this paper the rocking response of slender/rigid structures stepping on a viscoelastic foundation is revisited. The study examines in depth the motion of the system with a non-linear analysis that complements the linear analysis presented in the past by other investigators. The non-linear formulation combines the fully non-linear equations of motion together with the impulse-momentum equations during impacts. The study shows that the response of the rocking block depends on the size, shape and slenderness of the block, the stiffness and damping of the foundation and the energy loss during impact. The effect of the stiffness and damping of the foundation system along with the influence of the coefficient of restitution during impact is presented in rocking spectra in which the peak values of the response are compared with those of the rigid block rocking on a monolithic base. Various trends of the response are identified. For instance, less slender and smaller blocks have a tendency to separate easier, whereas the smaller the angle of slenderness, the less sensitive the response to the flexibility, damping and coefficient of restitution of the foundation.

In structural mechanics there are several occasions where a linearized formulation of the original nonlinear problem reduces considerably the computational effort for the response analysis. In a broader sense, a linearized formulation can be viewed as a first-order expansion of the dynamic equilibrium of the system about a `static¿ configuration; yet caution should be exercised when identifying the `correct¿ static configuration. This paper uses as a case study the rocking response of a rigid block stepping on viscoelastic supports, whose non-linear dynamics is the subject of the companion paper, and elaborates on the challenge of identifying the most appropriate static configuration around which a first-order expansion will produce the most dependable results in each regime of motion. For the regime when the heel of the block separates, a revised set of linearized equations is presented, which is an improvement to the unconservative equations published previously in the literature. The associated eigenvalues demonstrate that the characteristics of the foundation do not affect the rocking motion of the block once the heel separates.

京都大学 / 0048 / 新制・論文博士 / 博士(地球環境学) / 乙第12827号 / 論地環博第7号 / 新制||地環||24(附属図書館) / 31314 / (主査)教授 勝見 武, 教授 三村 衛, 准教授 乾 徹 / 学位規則第4条第2項該当 / Doctor of Global Environmental Studies / Kyoto University / DFAM

[pt] Esta dissertação apresenta uma análise numérica do fluxo não saturado em uma pilha de lixiviação de minério de cobre. Adicionalmente, foi feito também um estudo da estabilidade estática e sísmica da pilha de lixiviação, por sua construção estar planejada em região de alta sismicidade no sul do Peru. A pilha será construída com minério de cobre não tratado sobre base impermeável para evitar a contaminação do meio ambiente por fluxo da solução ácida através da fundação. Os resultados obtidos nas análises numéricas indicam que a pilha de lixiviação apresenta fatores de segurança satisfatórios considerando sismos de magnitude até 7,5 com aceleração máxima de até 0,30g. Da mesma forma, constatou-se que a posição da linha freática formada pela solução de lixiviação não atinge as camadas previstas para proteção das tubulações de drenagem. / [en] This dissertation presents a numerical analysis for the non saturated flow throughout a copper ore heap leach. Additionally, static and dynamic stability analyses were also made because the heap leach is planned to be built in a highly seismic region in the south of Peru. The heap will be constructed with non-treated copper ore (run of mine) on impermeable pad specially devised to avoid any possibility of ground contamination by the flow through the foundation of the acid solution used for the lixiviation process. The numerical results obtained in the stability analyses indicate that the heap leach presents satisfactory safety factors even when considering earthquakes of magnitude 7.5 with maximum acceleration of 0.30g. It has been also observed that the phreatic line formed by the lixiviation fluid does not reach the layers devised to protect the drainage pipes of the lixiviation system.

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第21357号 / 工博第4516号 / 新制||工||1703(附属図書館) / 京都大学大学院工学研究科都市社会工学専攻 / (主査)教授 大津 宏康, 准教授 PIPATPONGSA Thirapong, 教授 三村 衛 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DGAM

[pt] Uma das atividades relacionadas à recuperação de barragens envolve o alteamento de barragens existentes, normalmente com o objetivo de aumentar a capacidade de armazenamento dos reservatórios, melhorar o fator de segurança dos taludes ou a proteção da estrutura contra possíveis cheias. O alteamento pode estar previsto no projeto original da barragem, mas na maioria dos casos trata-se de um novo estudo, com a barragem em operação, devendo-se verificar as novas condições de fluxo, efeitos na estabilidade de taludes e na resposta da barragem a carregamentos estáticos e sísmicos, estes principalmente em regiões de alta sismicidade, como no sul do Peru, onde se enontram a barragem de terra de Viña Blanca, aqui considerada. Nesta dissertação o método dos elementos finitos e o método de equilíbrio limite, isolada ou conjuntamente, são empregados para análise estática e dinâmica destas barragens de terra considerando diversas opções de alteamento, como a construção de muros parapeito, muros de gabião, solo compactado, solo reforçado com geotêxteis e solo reforçado com revestimento de concreto. De estudos de perigo sísmico efetuados no local das barragens, selecionou-se o valor de aceleração horizontal máximo para ser utilizado nos registros de aceleração ocorridos nos terremotos de Lima (1974) e de Moquegua (2001). As análises numéricas efetuadas mostram que as opções de alteamento consideradas não alteram significativamente as condições de segurança das barragens existentes, tanto do ponto de vista hidráulico como da estabilidade de taludes e resposta dinâmica durante a incidência de terremotos. / [en] One of the activities related to dam constructions involves the raising of the existent structure, normally done with the objective of increasing the water storage capacity of the reservoirs, improving the safety factor of the embankment slopes or to ensure a better protection against possible water flooding. The raising of an earth dam can be predicted in the original dam plan, but in mostly situations consists of a new design, with the dam fully operational, where the effects of a new dam height and reservoir level should be assessed with respect to flow conditions, stability of the embankment soil slopes and the response of the revised structure under static and seismic loads, mainly in highly seismic regions, as in the South of Peru where the earth dam of Viña Blanca, herein studied, was constructed. In this dissertation, the finite element method and the limit equilibrium method were used for the static and dynamic analyses of these earth dams, in their original geometry as well after dam raising with reinforced soil, compacted soil and concrete or gabion structures, among other options. From studies of seismic risk analyses carried out at the dam sites, the value of maximum horizontal acceleration equal to 0.4g was chosen to be used as the peak acceleration in the Lima (1974) and Moquegua (2001) acceleration time histories. The numerical results indicate that all dam raising options investigated in this work do not affect the safety conditions of the dams significantly, either under the point of view of the hydraulic behavior as well as soil slope stabilities or the dynamic response of the earth dams to seismic loads.

This thesis presents the implementation of the upper bound limit analysis in combination with finite elements and linear optimization for solving different stability problems in geomechanics under plane strain conditions. Although the nonlinear optimization techniques are becoming quite popular, the linear optimization has been adopted due to its simplicity in implementation and ease in attaining the convergence while performing the analysis. The objectives of the present research work are (i) to reduce the computational effort while using an upper bound finite element limit analysis with linear programming in dealing with geotechnical stability problems, and (ii) to obtain solutions for a few important geotechnical stability problems associated with reinforced earth, unsupported tunnels and a group of anchors. It is also intended to examine the developments of the failure patterns in all the cases. For carrying out the analysis for different stability problems, three noded triangular elements have been used throughout the thesis. The nodal velocities are treated as basic unknown variables and the velocity discontinuities are employed along the interfaces of all the elements. The soil mass is assumed to obey the Mohr-Coulomb’s failure criterion and an associated flow rule. The Mohr-Coulomb yield surface is linearized by means of an exterior regular polygon circumscribing the actual yield circle so that the finite element formulation leads to a linear programming problem. A simple technique has been proposed for reducing the computational effort while solving any geotechnical stability problem by using the upper bound finite element limit analysis and linear optimization. In the proposed method, the problem domain has been discretized into a number of different regions in which a particular order (number of sides) of the polygon has been specified to linearize the Mohr-Coulomb yield criterion. A greater order of the polygon needs to be chosen only in that part of the domain wherein the rate of the plastic strains becomes higher. The computational effort required to solve the problem with this implementation reduces considerably. By using the proposed method, the bearing capacity has been computed for smooth as well as rough strip footings and the results obtained are found to be quite satisfactory. The ultimate bearing capacity of a rigid strip footing placed over granular, cohesive-frictional and purely cohesive soils, reinforced with single and a group of two horizontal layers of reinforcements has been determined. The necessary formulation has been introduced to incorporate the inclusion of reinforcement in the analysis. The efficiency factors, and , to be multiplied with Nc and Nγ for finding the bearing capacity of reinforced foundations, have been established. The results have been obtained (i) for different values of soil friction angles in case of granular and cohesive-frictional soils, and (ii) for different rates at which the cohesion increases with depth for purely cohesive soil under undrained condition. The optimum positions of the reinforcements' layers corresponding to which and becomes maximum, have been established. The effect of the length of the reinforcements on the results has also been analyzed. As compared to cohesive soil, the granular soils, especially with greater values of frictional angle, cause much more predominant increase in the bearing capacity. The stability of a long open vertical trench laid in a fully cohesive and cohesive-frictional soil has been determined with an inclusion of single and a group of two layers of horizontal reinforcements. For different positions of the reinforcement layers, the efficiency factor (ηs), has been determined for several combinations of H/B, m and where H and B refer to height and width of the trench, respectively, and m accounts for the rate at which the cohesion increases linearly with depth for a fully cohesive soil with = 0. The effect of height to width of the long vertical trench on the stability number has been examined for both unreinforced and reinforced soils. The optimal positions of the reinforcements layers, corresponding to which becomes maximum, have been established. The required length of reinforcements to achieve maximum efficiency factor corresponding to optimum depth of reinforcement has also been determined. The magnitude of the maximum efficiency factor increases continuously with an increase in both m and . The effect of pseudo-static horizontal earthquake body forces on the stability of a long unsupported circular tunnel (opening) formed in a cohesive frictional soil has been determined. The stability numbers have been obtained for various values of H/D (H = tunnel cover, D = diameter of the tunnel), internal friction angle of soil, and the horizontal earthquake acceleration coefficient The computations revealed that the values of the stability numbers (i) decreases quite significantly with an increase in , and (ii) become continuously higher for greater values of H/D and . The failure patterns have also been drawn for different combinations of H/D, and . The geometry of the failure zone around the periphery of the tunnel becomes always asymmetrical with an inclusion of horizontal seismic body forces. The interference effect on the stability of two closely spaced parallel (twin) long unsupported circular tunnels formed in fully cohesive and cohesive-frictional soils has been evaluated. The variation of the stability number with S/D has been established for different combinations of H/D, m and ; where D refers to the diameter of each tunnel, S is the clear spacing between the tunnels, and is the internal friction angle of soil and m accounts for the rate at which the cohesion increases linearly with depth for a soil with = 0. On account of the interference of two tunnels, the stability number reduces continuously with a decrease in the spacing between the tunnels. The minimum spacing between the two tunnels required to eliminate the interference effect increases with (i) an increase in H/D and (ii) a decrease in the values of both m and . The failure patterns have also been generated for a few cases with different values of S/D. The size of the failure zone is found to become smaller for greater values of m and . The horizontal pullout capacity of a group of two vertical strip anchors embedded, along the same vertical plane in sand, at shallow depths has been determined. At collapse, it is assumed that the anchor plates are subjected to the same uniform horizontal velocity without any bending or tilt. The pullout resistance increases invariably with increases in the values of embedment ratio, friction angle of the sand mass and anchor-soil interface friction angle. The effect of spacing (S) between the anchors on their group collapse load is examined in detail. For a given embedment ratio, the total group failure load becomes maximum corresponding to a certain optimal spacing (Sopt). The values of Sopt increases with an increase in the value of , but the changes in the value of H/B and do not have any significant effect on Sopt. The vertical uplift capacity of a group of two horizontal strip plate anchors with the common vertical axis buried in purely cohesive as well as in cohesive frictional soil has been computed. The variation of the uplift factors Fc, Fq and F , due to the contributions of soil cohesion, surcharge pressure and unit weight, respectively, has been evaluated for different combinations of S/B and H/B. As compared to a single isolated anchor, the group of two anchors generates significantly greater magnitude of Fc. On the other hand, the factors Fq and F , for a group of two anchors are found to become almost equal to that of a single isolated anchor as long as the levels of the lower plate in the group and the single isolated anchor are kept the same. For the group of two horizontal strip plate anchors in purely cohesive soil, an increase of cohesion of soil mass with depth and the effect of self weight of the soil have been incorporated. The uplift factor Fcy both due to cohesion and unit weight of the soil has also been computed for the anchors embedded in clay under undrained condition. For given embedment ratios, the factor Fcy increases linearly with an increase in the normalized unit weight of soil mass upto a certain value before attaining a certain maximum magnitude. The computational results obtained for different research problems would be useful for design.

Design of retaining structures depends upon the load which is transferred from backfill soil as well as external loads and also the resisting capacity of the structure. The traditional safety factor approach of the design of retaining structures does not address the variability of soils and loads. The properties of backfill soil are inherently variable and influence the design decisions considerably. A rational procedure for the design of retaining structures needs to explicitly consider variability, as they may cause significant changes in the performance and stability assessment. Reliability based design enables identification and separation of different variabilities in loading and resistance and recommends reliability indices to ensure the margin of safety based on probability theory. Detailed studies in this area are limited and the work presented in the dissertation on the Optimum design of retaining structures under static and seismic conditions: A reliability based approach is an attempt in this direction. This thesis contains ten chapters including Chapter 1 which provides a general introduction regarding the contents of the thesis and Chapter 2 presents a detailed review of literature regarding static and seismic design of retaining structures and highlights the importance of consideration of variability in the optimum design and leads to scope of the investigation. Targeted stability is formulated as optimization problem in the framework of target reliability based design optimization (TRBDO) and presented in Chapter 3. In Chapter 4, TRBDO approach for cantilever sheet pile walls and anchored cantilever sheet pile walls penetrating sandy and clayey soils is developed. Design penetration depth and section modulus for the various anchor pulls are obtained considering the failure criteria (rotational, sliding, and flexural failure modes) as well as variability in the back fill soil properties, soil-steel pile interface friction angle, depth of the water table, total depth of embedment, yield strength of steel, section modulus of sheet pile and anchor pull. The stability of reinforced concrete gravity, cantilever and L-shaped retaining walls in static conditions is examined in the context of reliability based design optimization and results are presented in Chapter 5 considering failure modes viz. overturning, sliding, eccentricity, bearing, shear and moment failures in the base slab and stem of wall. Optimum wall proportions are proposed for different coefficients of variation of friction angle of the backfill soil and cohesion of the foundation soil corresponding to different values of component as well as lower bounds of system reliability indices. Chapter 6 presents an approach to obtain seismic passive resistance behind gravity walls using composite curved rupture surface considering limit equilibrium method of analysis with the pseudo-dynamic approach. The study is extended to obtain the rotational and sliding displacements of gravity retaining walls under passive condition when subjected to sinusoidal nature of earthquake loading. Chapter 7 focuses on the reliability based design of gravity retaining wall when subjected to passive condition during earthquakes. Reliability analysis is performed for two modes of failure namely rotation of the wall about its heel and sliding of the wall on its base are considering variabilities associated with characteristics of earthquake ground motions, geometric proportions of wall, backfill soil and foundation soil properties. The studies reported in Chapter 8 and Chapter 9 present a method to evaluate reliability for external as well as internal stability of reinforced soil structures (RSS) using reliability based design optimization in the framework of pseudo static and pseudo dynamic methods respectively. The optimum length of reinforcement needed to maintain the stability against four modes of failure (sliding, overturning, eccentricity and bearing) by taking into account the variabilities associated with the properties of reinforced backfill, retained backfill, foundation soil, tensile strength and length of the geosynthetic reinforcement by targeting various component and system reliability indices is computed. Finally, Chapter 10 contains the important conclusions, along with scope for further work in the area. It is hoped that the methodology and conclusions presented in this study will be beneficial to the geotechnical engineering community in particular and society as a whole.