etd@IISc Collection:http://hdl.handle.net/2005/332014-12-26T21:18:07Z2014-12-26T21:18:07ZLower Bound Limit Analysis Applications For Solving Planar Stability Problems In GeomechanicsBhattacharya, Paramitahttp://hdl.handle.net/2005/23772014-09-04T10:00:28Z2014-09-03T18:30:00ZTitle: Lower Bound Limit Analysis Applications For Solving Planar Stability Problems In Geomechanics
Authors: Bhattacharya, Paramita
Abstract: Limit analysis based upon the theory of plasticity is one of the very useful numerical techniques to determine the failure loads of different civil and mechanical engineering structures for a material following an associated flow rule. The limiting values of the collapse loads, namely, lower and upper bounds, can be bracketed quite accurately with the application of the lower and upper bound theorems of the limit analysis. With the advancement of the finite elements and different robust optimization techniques, the numerical limit analysis approach in association with finite elements is becoming very popular to assess the stability of various complicated structures. Although two different optimization methods, namely, linear programming and nonlinear programming, have been both successfully implemented by various researchers for solving different stability problems in geomechanics, the linear programming method is employed in the present thesis due to its inherent advantage in implementation and ease in achieving the convergence. The objectives of the present thesis are (i) to improve upon the existing lower bound limit analysis method, in combination with finite elements and linear programming, with an intention of reducing the computational time and the associated memory requirement, and (ii) to apply the existing lower bound finite element limit analysis to various important planar stability problems in geotechnical engineering.
With reference to the first objective of the thesis, two new methods have been introduced in this thesis to improve upon the existing computational procedure while solving the geomechanics stability problem with the usage of the limit analysis, finite elements and linear programming. In the first method, namely, the method-I, the order of the yield polygon within the chosen domain is varied, based on the proximity of the stress state to the yield, such that a higher order polygon needs not to be used everywhere in the problem domain. In the second method, the method-II, it has been intended to use only a few selected sides, but not all, of the higher order yield polygon which are being used to linearize the Mohr-Coulomb yield function. The proposed two methods have been applied to compute the ultimate bearing capacity of smooth as well as rough strip footings for various soil frictional angles. It has been noticed that both the proposed new methods reduce the CPU time and the total number of inequality constraints required as compared to the existing lower bound linear programming method used in literature.
With reference to the second objective, a few important planar stability problems in geomechanics associated with interference of footings and vertical anchors have been solved in the present thesis. Footings are essentially used to transfer the compressive loads of the super structures to underlying soil media. On the other hand, vertical anchors are used for generating passive supports to retaining walls, sheet piles and bulkheads. A large number of research investigations have been reported in literature to compute the collapse load for a single isolated strip footing and a single vertical anchor. It is a common practice to estimate the bearing capacity of footings or pullout capacity of anchors without considering the effect of interference. There are, however, clear evidences from the available literature that (i) the ultimate bearing capacity of footings, and (ii) the ultimate pullout capacity of anchors, are significantly affected by their interference effect. Based on different available methods, the interference of footings, in a group of two footings as well as an infinite number of multiple footings, has been examined by different researchers in order to compute the ultimate bearing capacity considering the group effect. However, there is no research study to find the ultimate bearing capacity of interfering footings with the usage of the lower bound limit analysis. In the present thesis, the ultimate bearing capacity of two and an infinite number of multiple strip footings placed on sandy soil with horizontal ground surface, has been determined. The analysis has been performed for smooth as well as rough footings. The failure loads for interfering footings are found to be always greater than the single isolated footing. The effect of the footings' interference is expressed in terms of an efficiency factor ( ξγ); where, ξγ is defined as the ratio of the magnitude of failure load for a footing of width B in presence of the other footing to the magnitude of failure load of an isolated strip footing having the same width. The effect of the interference on the failure load (i) for rough footings becomes always greater than smooth footings, (ii) increases with an increase in soil frictional angle φ, and (iii) becomes almost negligible beyond the spacing, S > 3B. It is observed that the failure load for a footing in a group of an infinite number of multiple strip footings becomes always greater than that for two interfering footings.
Attempts have been made in this thesis to investigate the group effect of two vertical anchors on their horizontal pullout resistance (PuT). The anchors are considered to be embedded at a certain clear spacing (S) along the same vertical plane. The group effect has been studied separately for anchors embedded in (i) sandy soil, and (ii) undrained clay, respectively. For anchors embedded in clays, an increase of soil cohesion with depth, in a linear fashion, has also been taken into consideration. The magnitude of PuT has been obtained in terms of a group efficiency factor, ηγ for sand and ηc for clay, with respect to the failure load for a single isolated vertical plate with the same H/B. The pullout capacity of a group of two anchors either in sand or in undrained clay becomes quite extensive as compared to a single isolated anchor. The magnitudes of ηγ and ηc become maximum corresponding to a certain critical value of S/B, which has been found to lie generally between 0.5 and 1. The value of ηγ for a given S/B has been found to become larger for greater values of H/B, φ, and δ. For greater values of H/B, the group effect becomes more significant in contributing the pullout resistance.
The horizontal pullout capacity of a single isolated vertical anchor embedded in sand in the presence of pseudo static horizontal earthquake body forces has also been determined by using the lower bound finite element limit analysis. The variation of the pullout factor Fγ with changes in the embedment ratio of the smooth and rough anchor plates for different values of horizontal earthquake acceleration coefficient ( αh) has been investigated. The analysis clearly reveals that the pullout resistance decreases quite significantly with an increase in the magnitude of the earthquake acceleration coefficient.
For the various problems selected in the present thesis, the failure patterns have also been exclusively drawn in order to understand the development of the plastic zones within the chosen domain for solving a given problem. The results obtained from the analysis, for the various problems taken up in this thesis, have been thoroughly compared with those reported in literature.2014-09-03T18:30:00ZStudies On Fatigue Crack Propagation In Cementitious Materials : A Dimensional Analysis ApproachRay, Sonalisahttp://hdl.handle.net/2005/23712014-08-19T06:12:53Z2014-08-18T18:30:00ZTitle: Studies On Fatigue Crack Propagation In Cementitious Materials : A Dimensional Analysis Approach
Authors: Ray, Sonalisa
Abstract: Crack propagation in structures when subjected to fatigue loading, follows three different phases namely - short crack growth, stable crack growth and unstable crack growth. Accurate fatigue life prediction demands the consideration of every crack propagation phase rather than only the stable crack growth stage. Further, the use of existing crack growth laws in structures with small cracks under-predicts the growth rate compared to experimentally observed ones, thereby leading to an unsafe design and keeping the structure in a potentially dangerous state. In the present work, an attempt is made to establish fatigue crack propagation laws for plain concrete, reinforced concrete and concrete-concrete jointed interfaces from first principles using the concepts of dimensional analysis and self-similarity. Different crack growth laws are proposed to understand the behavior in each of the three regimes of the fatigue crack growth curve. Important crack growth characterizing material and geometrical parameters for each zone are included in the proposed analytical models. In real life applications to structures, the amplitude of cyclic loading rarely remains constant and is subjected to a wide spectrum of load amplitudes. Furthermore, the crack growth behavior changes in the presence of high amplitude load spikes within a constant amplitude history and this is incorporated in the model formulation. Using scaling laws, an improved understanding of the scaling behavior on different parameters is achieved. The models describing different regimes of crack propagation are finally unified to obtain the entire crack growth curve and compute the total fatigue life.
In addition, crack growth analysis is performed for a reinforced concrete member by modifying the model derived for plain concrete in the Paris regime. Energy dissipation occurring due to shake-down phenomenon in steel reinforcement is addressed. The bond-slip mechanism which is of serious concern in reinforced concrete members is included in the study and a method is proposed for the prediction of residual moment carrying capacity as a function of relative crack depth.
The application of the proposed analytical model in the computation of fatigue crack growth is demonstrated on three practical problems – beam in flexure, concrete arch bridge and a patch repaired beam. Through a sensitivity study, the influence of different parameters on the crack growth behavior is highlighted.2014-08-18T18:30:00ZNumerical Methods For Solving The Eigenvalue Problem Involved In The Karhunen-Loeve DecompositionChoudhary, Shaluhttp://hdl.handle.net/2005/23082014-05-05T06:00:48Z2014-05-04T18:30:00ZTitle: Numerical Methods For Solving The Eigenvalue Problem Involved In The Karhunen-Loeve Decomposition
Authors: Choudhary, Shalu
Abstract: In structural analysis and design it is important to consider the effects of uncertainties in loading and material properties in a rational way. Uncertainty in material properties such as heterogeneity in elastic and mass properties can be modeled as a random field. For computational purpose, it is essential to discretize and represent the random field. For a field with known second order statistics, such a representation can be achieved by Karhunen-Lo`eve (KL) expansion. Accordingly, the random field is represented in a truncated series expansion using a few eigenvalues and associated eigenfunctions of the covariance function, and corresponding random coefficients.
The eigenvalues and eigenfunctions of the covariance kernel are obtained by solving a second order Fredholm integral equation. A closed-form solution for the integral equation, especially for arbitrary domains, may not always be available. Therefore an approximate solution is sought. While finding an approximate solution, it is important to consider both accuracy of the solution and the cost of computing the solution. This work is focused on exploring a few numerical methods for estimating the solution of this integral equation. Three different methods:(i)using finite element bases(Method1),(ii) mid-point approximation(Method2), and(iii)by the Nystr¨om method(Method3), are implemented and numerically studied. The methods and results are compared in terms of accuracy, computational cost, and difficulty of implementation. In the first method an eigenfunction is first represented in a linear combination of a set of finite element bases. The resulting error in the integral equation is then minimized in the Galerkinsense, which results in a generalized matrix eigenvalue problem. In the second method, the domain is partitioned into a finite number of subdomains. The covariance function is discretized by approximating its value over each subdomain locally, and thereby the integral equation is transformed to a matrix eigenvalue problem. In the third method the Fredholm integral equation is approximated by a quadrature rule, which also results in a matrix eigenvalue problem. The methods and results are compared in terms of accuracy, computational cost, and difficulty of implementation.
The first part of the numerical study involves comparing these three methods. This numerical study is first done in one dimensional domain. Then for study in two dimensions a simple rectangular domain(referred toasDomain1)is taken with an uncertain material property modeled as a Gaussian random field. For the chosen covariance model and domain, the analytical solutions are known, which allows verifying the accuracy of the numerical solutions. There by these three numerical methods are studied and are compared for a chosen target accuracy and different correlation lengths of the random field. It was observed that Method 2 and Method 3 are much faster than the Method 1. On the other hand, for Method 2 and 3, additional cost for discretizing the domain into nodes should be considered whereas for a mechanics-related problem, Method 1 can use the available finite element mesh used for solving the mechanics problem.
The second part of the work focuses on studying on the effect of the geometry of the model on realizations of the random field. The objective of the study is to see the possibility of generating the random field for a complicated domain from the KL expansion for a simpler domain. For this purpose, two KL decompositions are obtained: one on the Domain1, and another on the same rectangular domain modified with a rectangular hole (referredtoasDomain2) inside it. The random process is generated and realizations are compared. It was observed from the studies that probability density functions at the nodes on both the domains, that is, on Domain 1 and Domain 2, are similar. This observation leads to a possibility that a complicated domain can be replaced by a corresponding simpler domain, thereby reducing the computational cost.2014-05-04T18:30:00ZBehaviour Of FRP Strengthened Masonry In Compression And ShearPavan, G Shttp://hdl.handle.net/2005/22922014-04-08T07:26:28Z2014-04-07T18:30:00ZTitle: Behaviour Of FRP Strengthened Masonry In Compression And Shear
Authors: Pavan, G S
Abstract: Masonry structures constitute a significant portion of building stock worldwide. Seismic performance of unreinforced masonry has been far from satisfactory. Masonry is purported to be a major source of hazard during earthquakes by reconnaissance surveys conducted aftermath of an earthquake. Reasons for the poor performance of masonry structures are more than one namely lack of deformational capacity, poor tensile strength & lack of earthquake resistance features coupled with poor quality control and large variation in strength of materials employed. Fibre Reinforced Plastic (FRP) composites have emerged as an efficient strengthening technique for reinforced concrete structures over the past two decades. Present thesis is focused towards analysing the behaviour of Fibre Reinforced Plastic (FRP) strengthened masonry under axial compression and in-plane shear loading. Determination of in-planes hear resistance of large masonry panels requires tremendous effort in terms of cost, labour and time. Masonry assemblages like prisms and triplets that represent the state of stress present in masonry walls and masonry in-fills when under the action of in-planes hear forces present an alternative option for research and analysis purposes. Hence, present research is focused towards analysing the performance of FRP strengthened masonry assemblages and unreinforced masonry assemblages.
Chapter1 provides a brief review on the behaviour of masonry shear walls and masonry in-fills under the action of in-plane shear forces in addition to the performance of masonry structures during past earthquakes. Review of available literature on FRP confinement of masonry prisms with bed joints inclined from 00 to 900 to the loading axis under axial compression, analytical models available for FRP confined concrete, shear strength of masonry triplets attached with FRP is presented.
Chapter 2 primarily focuses on determining the various properties of the materials involved in this research investigation. Test procedure and results of the tests conducted to determine the mechanical and related properties of the materials involved are presented. Elastic properties and stress-strain response of burnt clay brick, mortar and FRP laminates are presented.
Studies conducted on behaviour of GFRP confined masonry prisms under monotonic axial compression are included in Chapter 3. The study comprised of testing masonry prisms, both unconfined and FRP confined masonry prisms under axial compression. Stretcher bond and English bond prisms, with bed joints normal and parallel to loading axis are included in this study. Two grades of GFRP,360g/m2 and 600 g/m2 are employed to confine masonry prisms. The experimental program involved masonry prism types that accounted for variations in masonry bonding pattern, bed joint inclination to the loading axis and grade of GFRP. Review of the available analytical models predicting compressive strength of FRP confined masonry prism is presented. Available models for FRP confinement of masonry are re-calibrated using the present experimental data generating new coefficients for the already existing model to develop new expression for predicting the compressive strength of FRP confined prisms. In addition to the prism types mentioned earlier, behaviour of unconfined and GFRP confined stretcher bond prisms with bed joints inclined at 300, 450 & 600 to the loading axis are further investigated.
Chapter 4 primarily deals with the shear strength and deformational capacity of masonry triplets that represent joint shear failure in masonry. An experimental program involving masonry triplets attached with different types of FRP(GFRP and CFRP), grade of FRP, percentage area covered by FRP and reinforcement pattern is executed. This exercise determined the influence of these parameters over the enhancement achieved in terms of shear strength and ultimate displacement. Results of tests conducted on stretcher bond prisms presented in chapter 3 and results of tests on shear triplets presented in this chapter are combined to study the interaction between shear and normal stresses acting along the masonry bed joint at different angles of inclination.
The thesis culminated with chapter 5 as concluding remarks highlighting the salient
Information pertaining to the behaviour of FRP strengthened masonry under axial compression and in-plane shear loading obtained as an outcome of the research conducted as a part of this thesis.2014-04-07T18:30:00Z