Academic literature on the topic 'High-Order Shear Deformation Theory'

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.

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.

Response of composite laminates under in-plane compressive or shear loadings is of interest to the analyst and designers. Since they are thin, they are prone to instability under in-plane loads. Transverse shear effects are important even for thin laminates since elastic modulus and shear modulus are independent properties. For very thick laminates neglecting transverse shear effects leads to completely erroneous results. A number of different theories have been suggested by different investigators to account for transverse shear effects. In this investigation, an attempt has been made to take into account transverse shear effects for the stability analysis of moderately thick/very thick composite laminates under in-plane compressive and shear loading using a “SIMPLE HIGHER ORDER SHEAR DEFORMATION THEORY” based on four unknown displacements instead of five which is commonly used for most of the other higher order theories. A C1 continuous shear flexible finite element based on the proposed HSDT is developed using the Hermite cubic rectangular element. The analytical results obtained have been compared with the available results in literature. Effect of various parameters like aspect ratio, thickness to side ratio, fiber orientation and material properties have been studied in detail.

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.

Abstract. In the present study, a static response of functionally graded plate and spherical shells is investigated using higher-order trigonometric shear and normal deformation theory. A need of the shear correction factor is obviated and the effect of actual cross-sectional warping has been considered to get the realistic behaviour of transverse shear stresses across the thickness of the shell. The Navier solution technique has been used to analyse the simply-supported boundary conditions of the shell. To verify the theory, the numerical results obtained using the present theory are compared with other higher-order shear deformation theories available in the literature. The numerical results are obtained with and without considering the effects of transverse normal strain (εz).

In this paper free vibration analysis of skew plates with fully clamped boundary condition made of functionally graded materials is investigated. The study is based on the second order shear deformation plate theory (SSDT) using Generalized Differential Quadrature (GDQ) method. With a proper transformation, partial differential equations of a rectangular plate in Cartesian coordinates into skew coordinates are obtained; the governing differential equations are seven second order partial differential equations. The comparison of simulation results with those presented previously in the literature shows the accuracy of the proposed method. Also, fast rate of convergence is achieved by this method. Finally, the effects of angle of skew plate, power law index and plate geometrical parameters on the natural frequencies of the plate are also obtained and the related results are presented in this paper.

A consistent higher-order shear deformation nonlinear theory is developed for shells of generic shape; taking geometric imperfections into account. The geometrically nonlinear strain-displacement relationships are derived retaining full nonlinear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only nonlinear terms of the von Ka´rma´n type. Results show that inaccurate results are obtained by keeping only nonlinear terms of the von Ka´rma´n type for vibration amplitudes of about two times the shell thickness for the studied case.

West-central Yukon and eastern Alaska are characterized by widespread metamorphic rocks that form part of the allochthonous, composite Yukon-Tanana terrane and parautochthonous North American margin. Structural windows through the Yukon-Tanana terrane expose parautochthonous North American margin in that broad region, particularly as mid-Cretaceous extensional core complexes. Both the Yukon-Tanana terrane and parautochthonous North American margin share the same Late Devonian history, making their discrimination difficult; however, distinct post-Late Devonian magmatic and metamorphic histories assist in discriminating Yukon-Tanana terrane from parautochthonous North American margin rocks. The suture between Yukon-Tanana terrane and parautochthonous North American margin is obscured by many episodes of high-strain deformation. Their main bounding structure is probably a Jurassic to Cretaceous thrust, which has been locally reactivated as a mid-Cretaceous extensional shear zone. Crustal-scale structures within composite Yukon-Tanana terrane (e.g. the Yukon River shear zone) are commonly marked by discontinuous mafic-ultramafic complexes. Some of these complexes represent orogenic peridotites that were structurally exhumed into the Yukon-Tanana terrane in the Middle Permian.

Aeromagnetic and ground gravity data for the Canadian Superior Province, filtered to extract long wavelength components and converted to pseudo-gravity, highlight deep, N-S trending regional-scale, rectilinear faults and margins to discrete, competent mafic or felsic granulite blocks (i.e. at high angles to most regional mapped structures and sub-province boundaries) with little to no surface expression that are spatially associated with lode ('orogenic') Au and Ni-Cu-PGE-Cr occurrences. Statistical and machine learning analysis of the Red Lake-Stormy Lake region in the W Superior Province confirms visual inspection for a greater correlation between Au deposits and these deep N-S structures than with mapped surface to upper crustal, generally E-W trending, faults and shear zones. Porphyry Au, Ni, Mo and U-Th showings are also located above these deep transverse faults. Several well defined concentric circular to elliptical structures identified in the Oxford Stull and Island Lake domains along the S boundary of the N Superior proto-craton, intersected by N- to NNW striking extensional fractures and/or faults that transect the W Superior Province, again with little to no direct surface or upper crustal expression, are spatially associated with magmatic Ni-Cu-PGE-Cr and related mineralization and Au occurrences. The McFaulds Lake greenstone belt, aka. 'Ring of Fire', constitutes only a small, crescent-shaped belt within one of these concentric features above which 2736-2733 Ma mafic-ultramafic intrusions bodies were intruded. The Big Trout Lake igneous complex that hosts Cr-Pt-Pd-Rh mineralization west of the Ring of Fire lies within a smaller concentrically ringed feature at depth and, near the Ontario-Manitoba border, the Lingman Lake Au deposit, numerous Au occurrences and minor Ni showings, are similarly located on concentric structures. Preliminary magnetotelluric (MT) interpretations suggest that these concentric structures appear to also have an expression in the subcontinental lithospheric mantle (SCLM) and that lithospheric mantle resistivity features trend N-S as well as E-W. With diameters between ca. 90 km to 185 km, elliptical structures are similar in size and internal geometry to coronae on Venus which geomorphological, radar, and gravity interpretations suggest formed above mantle upwellings. Emplacement of mafic-ultramafic bodies hosting Ni-Cr-PGE mineralization along these ringlike structures at their intersection with coeval deep transverse, ca. N-S faults (viz. phi structures), along with their location along the margin to the N Superior proto-craton, are consistent with secondary mantle upwellings portrayed in numerical models of a mantle plume beneath a craton with a deep lithospheric keel within a regional N-S compressional regime. Early, regional ca. N-S faults in the W Superior were reactivated as dilatational antithetic (secondary Riedel/R') sinistral shears during dextral transpression and as extensional fractures and/or normal faults during N-S shortening. The Kapuskasing structural zone or uplift likely represents Proterozoic reactivation of a similar deep transverse structure. Preservation of discrete faults in the deep crust beneath zones of distributed Neoarchean dextral transcurrent to transpressional shear zones in the present-day upper crust suggests a 'millefeuille' lithospheric strength profile, with competent SCLM, mid- to deep, and upper crustal layers. Mechanically strong deep crustal felsic and mafic granulite layers are attributed to dehydration and melt extraction. Intra-crustal decoupling along a ductile décollement in the W Superior led to the preservation of early-formed deep structures that acted as conduits for magma transport into the overlying crust and focussed hydrothermal fluid flow during regional deformation. Increase in the thickness of semi-brittle layers in the lower crust during regional metamorphism would result in an increase in fracturing and faulting in the lower crust, facilitating hydrothermal and carbonic fluid flow in pathways linking SCLM to the upper crust, a factor explaining the late timing for most orogenic Au. Results provide an important new dataset for regional prospectively mapping, especially with machine learning, and exploration targeting for Au and Ni-Cr-Cu-PGE mineralization. Results also furnish evidence for parautochthonous development of the S Superior Province during plume-related rifting and cannot be explained by conventional subduction and arc-accretion models.

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.