Academic literature on the topic 'First order shear deformation'

This book, based on laboratory, teaching and field experience, has a strong focus towards hard rocks and magmatic rocks, from both the continental crust worldwide, where quartz and granites are dominant, and the mantle dominated by olivine in peridotites. With more than 250 figures, most of them original, the book develops, in addition to classical structural geology objects, the fundamentals of brittle fracturing of materials, plastic deformation of ice, quartz and olivine, and fabric acquisition in rocks and magmas. Measurement and orientation of stress axes, bases of neotectonics and geophysics, and practical tools such as magnetic fabrics not commonly treated in geological books, are also provided. Emblematic tectonic and geodynamic sites are presented, both from the oceanic and continental crust, for instance the Oman ophiolites, and the India-Eurasia collision and its associated shear zones. Since the targeted readers are present-day young students, a few structural geology exercises are also included in order to improve their abilities.

The aim of the paper was to determine the influence of root systems of chosen tree species found in the Polish Flysch Carpathians on the increase of soil shear strength (root cohesion) in terms of slope stability. The paper's goal was achieved through comprehensive tests on root systems of eight relatively common in the Polish Flysch Carpathians tree species. The tests that were carried out included field work, laboratory work and analytical calculations. As part of the field work, the root area ratio (A IA) of the roots was determined using the method of profiling the walls of the trench at a distance of about 1.0 m from the tree trunk. The width of the. trenches was about 1.0 m, and their depth depended on the ground conditions and ranged from 0.6 to 1.0 m below the ground level. After preparing the walls of the trench, the profile was divided into vertical layers with a height of 0.1 m, within which root diameters were measured. Roots with diameters from 1 to 10 mm were taken into consideration in root area ratio calculations in accordance with the generally accepted methodology for this type of tests. These measurements were made in Biegnik (silver fir), Ropica Polska (silver birch, black locust) and Szymbark (silver birch, European beech, European hornbeam, silver fir, sycamore maple, Scots pine, European spruce) located near Gorlice (The Low Beskids) in areas with unplanned forest management. In case of each tested tree species the samples of roots were taken, transported to the laboratory and then saturated with water for at least one day. Before testing the samples were obtained from the water and stretched in a. tensile testing machine in order to determine their tensile strength and flexibility. In general, over 2200 root samples were tested. The results of tests on root area ratio of root systems and their tensile strength were used to determine the value of increase in shear strength of the soils, called root cohesion. To this purpose a classic Wu-Waldron calculation model was used as well as two types of bundle models, the so called static model (Fiber Bundle Model — FIRM, FBM2, FBM3) and the deformation model (Root Bundle Model— RBM1, RBM2, mRBM1) that differ in terms of the assumptions concerning the way the tensile force is distributed to the roots as well as the range of parameters taken into account during calculations. The stability analysis of 8 landslides in forest areas of Cicikowicleie and Wignickie Foothills was a form of verification of relevance of the obtained calculation results. The results of tests on root area ratio in the profile showed that, as expected, the number of roots in the soil profile and their ApIA values are very variable. It was shown that the values of the root area ratio of the tested tree species with a diameter 1-10 ram are a maximum of 0.8% close to the surface of the ground and they decrease along with the depth reaching the values at least one order of magnitude lower than close to the surface at the depth 0.5-1.0 m below the ground level. Average values of the root area ratio within the soil profile were from 0.05 to 0.13% adequately for Scots pine and European beech. The measured values of the root area ratio are relatively low in relation to the values of this parameter given in literature, which is probably connected with great cohesiveness of the soils and the fact that there were a lot of rock fragments in the soil, where the tests were carried out. Calculation results of the Gale-Grigal function indicate that a distribution of roots in the soil profile is similar for the tested species, apart from the silver fir from Bie§nik and European hornbeam. Considering the number of roots, their distribution in the soil profile and the root area ratio it appears that — considering slope stability — the root systems of European beech and black locust are the most optimal, which coincides with tests results given in literature. The results of tensile strength tests showed that the roots of the tested tree species have different tensile strength. The roots of European beech and European hornbeam had high tensile strength, whereas the roots of conifers and silver birch in deciduous trees — low. The analysis of test results also showed that the roots of the studied tree species are characterized by high variability of mechanical properties. The values Of shear strength increase are mainly related to the number and size (diameter) of the roots in the soil profile as well as their tensile strength and pullout resistance, although they can also result from the used calculation method (calculation model). The tests showed that the distribution of roots in the soil and their tensile strength are characterized by large variability, which allows the conclusion that using typical geotechnical calculations, which take into consideration the role of root systems is exposed to a high risk of overestimating their influence on the soil reinforcement. hence, while determining or assuming the increase in shear strength of soil reinforced with roots (root cohesion) for design calculations, a conservative (careful) approach that includes the most unfavourable values of this parameter should be used. Tests showed that the values of shear strength increase of the soil reinforced with roots calculated using Wu-Waldron model in extreme cases are three times higher than the values calculated using bundle models. In general, the most conservative calculation results of the shear strength increase were obtained using deformation bundle models: RBM2 (RBMw) or mRBM1. RBM2 model considers the variability of strength characteristics of soils described by Weibull survival function and in most cases gives the lowest values of the shear strength increase, which usually constitute 50% of the values of shear strength increase determined using classic Wu-Waldron model. Whereas the second model (mRBM1.) considers averaged values of roots strength parameters as well as the possibility that two main mechanism of destruction of a root bundle - rupture and pulling out - can occur at the same. time. The values of shear strength increase calculated using this model were the lowest in case of beech and hornbeam roots, which had high tensile strength. It indicates that in the surface part of the profile (down to 0.2 m below the ground level), primarily in case of deciduous trees, the main mechanism of failure of the root bundle will be pulling out. However, this model requires the knowledge of a much greater number of geometrical parameters of roots and geotechnical parameters of soil, and additionally it is very sensitive to input data. Therefore, it seems practical to use the RBM2 model to assess the influence of roots on the soil shear strength increase, and in order to obtain safe results of calculations in the surface part of the profile, the Weibull shape coefficient equal to 1.0 can be assumed. On the other hand, the Wu-Waldron model can be used for the initial assessment of the shear strength increase of soil reinforced with roots in the situation, where the deformation properties of the root system and its interaction with the soil are not considered, although the values of the shear strength increase calculated using this model should be corrected and reduced by half. Test results indicate that in terms of slope stability the root systems of beech and hornbeam have the most favourable properties - their maximum effect of soil reinforcement in the profile to the depth of 0.5 m does not usually exceed 30 kPa, and to the depth of 1 m - 20 kPa. The root systems of conifers have the least impact on the slope reinforcement, usually increasing the soil shear strength by less than 5 kPa. These values coincide to a large extent with the range of shear strength increase obtained from the direct shear test as well as results of stability analysis given in literature and carried out as part of this work. The analysis of the literature indicates that the methods of measuring tree's root systems as well as their interpretation are very different, which often limits the possibilities of comparing test results. This indicates the need to systematize this type of tests and for this purpose a root distribution model (RDM) can be used, which can be integrated with any deformation bundle model (RBM). A combination of these two calculation models allows the range of soil reinforcement around trees to be determined and this information might be used in practice, while planning bioengineering procedures in areas exposed to surface mass movements. The functionality of this solution can be increased by considering the dynamics of plant develop¬ment in the calculations. This, however, requires conducting this type of research in order to obtain more data.

Constant and variable stiffness strategies have been developed to design a composite laminate. With the former, each layer is designed with straight fibers that have the highest stiffness and strength in the fiber direction. With the latter, on the other hand, the stiffness can change within each layer by placing the fibers along a curvilinear fiber path. A variable stiffness design results in improved structural performance, as well as opens up opportunities to search for trade-off among structural properties. During the manufacture of a variable stiffness design with Automated Fiber Placement, certain defects in the form of gaps and overlaps could appear within the laminate and affect the laminate performance. In this study, we use the first-order shear deformation theory to assess the effect of transverse shear stresses on the critical buckling load, free and forced vibration of a variable stiffness laminate with embedded defects, an issue so far rarely examined in literature. The governing differential equations for the static analysis are first derived. A semi-analytic solution is then obtained using the hybrid Fourier-Galerkin method and the numeric time integration technique. The eigenvalue analysis is also conducted to determine the fundamental frequency and critical buckling load of the plate. It is found that the behavior of a variable stiffness plate is much more affected by the shear stresses than a constant stiffness plate. Ignoring the effect of transverse shear stresses results in 34% error in the predicted buckling load of a variable stiffness laminate with overlaps and a length-to-thickness ratio of 10.

Buckling analysis of simply supported functionally graded cylindrical shells under mechanical loads is presented in this paper. The Young’s modulus of the shell is assumed to vary as a power form of the thickness coordinate variable. The shell is assumed to be under three types of mechanical loadings, namely, axial compression, uniform external lateral pressure, and hydrostatic pressure loading. The equilibrium and stability equations are derived based on the first order shear deformation theory. Resulting equations are employed to obtain the closed-form solution for the critical buckling load. The influences of dimension ratio, relative thickness and the functionally graded index on the critical buckling load are studied. The results are compared with the known data in the literature.

Abstract The dynamic stiffness method for free vibration of beams and frameworks is developed using a higher order shear deformation theory. Starting with the displacement field, the potential and kinetic energies of the beam in flexural vibration, are first formulated. Then, Hamilton’s principle is applied to derive the governing differential equations and associated natural boundary conditions. Next, the differential equations are solved to obtain the expressions for flexural displacement, bending rotation and the first derivative of the flexural displacement. The expressions for the shear force, bending moment and the higher-order moment are obtained from the natural boundary conditions resulting from the Hamiltonian formulation. Finally, the force vector comprising the amplitudes of the shear force, bending moment and the higher-order moment is related to the amplitudes of the displacement vector comprising the flexural displacement, bending rotation and the first derivative of the flexural displacement through the frequency-dependent dynamic stiffness matrix. The dynamic stiffness matrix for axial motion which is uncoupled from the flexural motion is now implemented to the dynamic stiffness matrix in flexural motion to analyse individual beams and frameworks for their free vibration characteristics by applying the Wittrick-Willaims algorithm. Illustrative examples are given, and significant conclusions are drawn.

Abstract The influence of transverse shear strain in the lamination theory modeling of Printed Wiring Board (PWB) deflection due to support conditions was examined. The in-plane mechanical properties of the core materials of a commercial PWB were measured as a function of temperature. Classical laminated plate theory and first-order shear deformation theory solutions for the out-of-plane deflection of a bare board configuration with two opposite edges simply supported and the remaining edges free were obtained. The weight of the board was approximated as a distributed transverse load. The effect of material property decrease with temperature and FR-4 layer thickness were examined to compare first-order shear and plane strain assumptions for the predicted warpage.

Abstract Higher order Shear Defamation Plate Theories (HSDPT) are improved theories over Mindlin plate theory because their assumptions are closer to reality. However, they are seldom used in solving ordinary engineering works. This is due to the fact that mathematical formulations and computations are so lengthy that time and efforts required are close to solving a exact 3-D model. For problems involving sharp stress variation, higher order theories are anticipated to give better results. The combination of HSDPT and Finite Element Modelings are especially attractive because a finite element modeling is much simpler. The current research develops a finite element model on the higher order shear deformation theory. A plate vibration problem was solved. The plates contain square interior cutout. Stress distributions are much complicated than whole plates. Results of HSDPT are compared with FSDPT (First order Shear Deformation Plate Theories) and CPT (Classical Plate Theory). Better accuracies are obtained by using the HSDPT.

This report describes several advances in the cyclic failure assessment of silt soils with immediate and practical benefit to the geotechnical earthquake engineering profession. First, a database of cyclic loading test data is assembled, evaluated, and used to assess trends in the curvature of the CRR-N (cyclic resistance ratio - the number of equivalent cycles) relationship. This effort culminated in a plasticity index-dependent function which can be used to estimate the exponent b in the power law describing cyclic resistance, and may be used to estimate the cyclic resistance of silt soils as well as the number of equivalent loading cycles anticipated for subduction zone earthquakes. Statistical models for the cyclic resistance ratio and cyclic strength ratio are presented in this report. The SHANSEP (Stress History and Normalized Soil Engineering Properties)-inspired functional form of these models have been trained and tested against independent datasets and finalized using a combined dataset to provide reasonable estimates of resistance based on the available data. These models can be used to provide provisional estimates of the CRR-N and cyclic strength ratio power laws for cyclic shear strain failure criteria ranging from 1 to 10%, within certain stated limitations. The ground motion records within the NGA Subduction Project which have been released to the public to-date are implemented to examine the role of subduction zone earthquake characteristics on the number of equivalent loading cycles for a wide range of soils with exponents b ranging from 0.05 (moderate plasticity silt and clay) to 0.35 (dense sand). This analysis shows that the number of loading cycles for a given magnitude subduction zone earthquake is larger than those previously computed, whereas the corresponding magnitude scaling factors for use with the Simplified Method span a smaller range as a result of the ground motion characteristics. Owing to the large variability in the computed equivalent number of loading cycles, consideration of the uncertainty is emphasized in forward analyses. The work described herein may be used to estimate cyclic resistance of intact non-plastic and plastic silt soils and corresponding factor of safety against cyclic failure for a range in cyclic shear strain failure criteria, to plan cyclic laboratory testing programs, and to calibrate models for use in site response and nonlinear deformation analyses in the absence of site-specific cyclic test data. As with any empirical approach, the models presented herein should be revised when additional, high-quality cyclic testing data become available.

The geology of the Montney Formation and the geochemistry of its produced fluids, including nonhydrocarbon gases such as hydrogen sulfide were investigated for both Alberta and BC play areas. Key parameters for understanding a complex petroleum system like the Montney play include changes in thickness, depth of burial, mass balance calculations, timing and magnitudes of paleotemperature exposure, as well as kerogen concentration and types to determine the distribution of hydrocarbon composition, H2S concentrations and CO2 concentrations. Results show that there is first-, second- and third- order variations in the maturation patterns that impact the hydrocarbon composition. Isomer ratio calculations for butane and propane, in combination with excess methane estimation from produced fluids, are powerful tools to highlight effects of migration in the hydrocarbon distribution. The present-day distribution of hydrocarbons is a result of fluid mixing between hydrocarbons generated in-situ with shorter-chained hydrocarbons (i.e., methane) migrated from deeper, more mature areas proximal to the deformation front, along structural elements like the Fort St. John Graben, as well as through areas of lithology with higher permeability. The BC Montney play appears to have hydrocarbon composition that reflects a larger contribution from in-situ generation, while the Montney play in Alberta has a higher proportion of its hydrocarbon volumes from migrated hydrocarbons. Hydrogen sulphide is observed to be laterally discontinuous and found in discrete zones or pockets. The locations of higher concentrations of hydrogen sulphide do not align with the sulphate-rich facies of the Charlie Lake Formation but can be seen to underlie areas of higher sulphate ion concentrations in the formation water. There is some alignment between CO2 and H2S, particularly south of Dawson Creek; however, the cross-plot of CO2 and H2S illustrates some deviation away from any correlation and there must be other processes at play (i.e., decomposition of kerogen or carbonate dissolution). The sources of sulphur in the produced H2S were investigated through isotopic analyses coupled with scanning electron microscopy, energy dispersive spectroscopy, and mineralogy by X-ray diffraction. The Montney Formation in BC can contain small discrete amounts of sulphur in the form of anhydrite as shown by XRD and SEM-EDX results. Sulphur isotopic analyses indicate that the most likely source of sulphur is from Triassic rocks, in particular, the Charlie Lake Formation, due to its close proximity, its high concentration of anhydrite (18-42%), and the evidence that dissolved sulphate ions migrated within the groundwater in fractures and transported anhydrite into the Halfway Formation and into the Montney Formation. The isotopic signature shows the sulphur isotopic ratio of the anhydrite in the Montney Formation is in the same range as the sulphur within the H2S gas and is a lighter ratio than what is found in Devonian anhydrite and H2S gas. This integrated study contributes to a better understanding of the hydrocarbon system for enhancing the efficiency of and optimizing the planning of drilling and production operations. Operators in BC should include mapping of the Charlie Lake evaporites and structural elements, three-dimensional seismic and sulphate ion concentrations in the connate water, when planning wells, in order to reduce the risk of encountering unexpected souring.

The application of stainless-clad (SC) bimetallic steel in various conditions such as offshore and marine environment requires members designed in different cross-sectional shapes, which consist of both internal and outstand elements. To form a comprehensive understanding of buckling behaviour of the SC bimetallic steel members, the behaviour of outstand compression plates needs to be investigated. In this study, the theoretical elastic buckling stress of outstand SC bimetallic steel plates subjected to uniformly distributed uniaxial compression is derived. Considering the position of neutral surface, the energy method and Ritz formulation are used to solve the buckling stress. Adaptation of the first-order shear deformation plate theory (FSDT) is used to modify the solution, which is further compared with finite element analyses. The influence of different parameters such as cladding configuration, clad ratio, elastic modulus ratio, aspect ratio and width-to-thickness ratio on the elastic buckling behaviour of SC bimetallic plates is analysed. The simplified design formulae and design requirements are summarized to form a comprehensive design method.

The application of stainless-clad (SC) bimetallic steel in various conditions such as offshore and marine environment requires members designed in different cross-sectional shapes, which consist of both internal and outstand elements. To form a comprehensive understanding of buckling behaviour of the SC bimetallic steel members, the behaviour of outstand compression plates needs to be investigated. In this study, the theoretical elastic buckling stress of outstand SC bimetallic steel plates subjected to uniformly distributed uniaxial compression is derived. Considering the position of neutral surface, the energy method and Ritz formulation are used to solve the buckling stress. Adaptation of the first-order shear deformation plate theory (FSDT) is used to modify the solution, which is further compared with finite element analyses. The influence of different parameters such as cladding configuration, clad ratio, elastic modulus ratio, aspect ratio and width-to-thickness ratio on the elastic buckling behaviour of the SC bimetallic plates is analysed. The simplified design formulae and design requirements are summarized to form a comprehensive design method.

Due to the complex connection of components and significant initial defects in the formwork structure, the utilization of a direct analysis method to design the formwork structure can effectively address the shortcomings of traditional first-order linear design methods in formwork design standards. Taking a practical engineering project as an example, a direct analysis method for the construction of a SCP structure was proposed. Based on its design results, a comparative analysis with the traditional first-order linear design method was conducted to clarify the influence of considering initial defects and second-order effects on support reaction force, deformation, and stress ratio. This provided a reference basis for the design of SCP structures. Based on the results, cross-sectional optimization was carried out on the key components of the SCP structure. Following optimization, the stress ratio distribution of the structure was found to be more reasonable, and a 25.63% reduction in steel consumption was achieved, thus demonstrating the feasibility of the direct analysis method in the design and optimization of the SCP structure.

To study the mechanism of load transfer in double-side-plate connections between I-beams and wall-type concrete-filled steel tubular columns, a pseudo-static experiment and finite element analysis were conducted for two full-scaled specimens. The results revealed that the primary load was transmitted along an S-shaped path in the side plate, and the primary strain occurred in an X-shaped region between the left and right steel beam flanges. The shear force in the steel beam web was transmitted first to the side plate centre and then to the joint area, where the side plate, steel tube web, and concrete all resisted the internal force. Based on principal component methods, a calculation formula was established for initial rotational stiffness that comprehensively considers the influence of the tensions, compression, and shear deformation of the cover plate, side plate, and web. Comparing this formula with an existing model showed that the proposed formula is suitable for new types of side plate joints. Moreover, it can accurately calculate the initial rotational stiffness of the joint, thus providing a reliable basis for future engineering design.