Rotation Of Cross-section Research Articles

Relevance of soil-pile tangential tractions for the estimation of kinematic seismic forces: Formulation and setting of a Winkler approach

• A BDWF model including the distributed moments produced by the pile rotation and the action of the seismic field is presented. • The performance of the proposed model for computing the pile seismic internal forces is tested against a BE formulation. • The influence of the contact condition between pile and soil on the pile seismic forces is also analysed. • Depending on the contact condition, pile shear forces can be duplicated, while bending moments remain practically unaltered. The effects of considering the tangential tractions that act on the soil-pile interface in the estimation of the pile seismic response are studied through a Beam on Dynamic Winkler Foundation model, which includes the distributed moments produced by the rotation of the pile cross-section and the action of the incident field. The performance of the developed Winkler formulation is evaluated by using a rigorous continuum model based on boundary elements that allows the use of both smooth and welded contact conditions at the soil-pile interface. In order to do the analyses, the seismic response of a fixed-head single pile embedded in different soils subjected to planar shear waves is computed in terms of envelopes of maximum bending moments and shear forces. Two soil profiles are assumed: homogeneous and two-strata halfspaces. The tangential tractions are found to significantly increase the pile maximum shear forces, but to have a minor impact on the pile maximum bending moments. The proposed Winkler model accurately reproduces the results of the continuum formulation for both contact conditions. However, some differences are found in the evaluation of the inter-layer envelopes, for which the simplified model underestimates their values.

Calcite production by the calcifying green alga Phacotus lenticularis

The importance of carbonate precipitation by phytoplankton in fresh water lakes has not been sufficiently considered in global carbon cycles and climate change scenarios. The objective of this study was to determine the influence of the calcifying bivalved phytoflagellate Phacotus lenticularis (Ehrenberg) Deising 1866 on the total calcite precipitation in five European hard-water lakes. For this purpose, an accurate mass determination of single Phacotus lenticularis shells was required. We developed a novel methodological approach to precisely determine the volume and mass of the calcified shells. Focused ion beam (FIB) techniques were employed to investigate internal structural features. Thin layer cross-sections of the shell profiles were reproduced and perforation as well as the crystalline structure of the calcite plates were monitored. 3D-shell models were computed by 360° rotation of the shell cross-sections using a CAD 3D imaging software to calculate precise volumes and estimate realistic masses. In contrast to previous estimates, we determined a 2.8-fold higher shell mass of 0.86 ng CaCO3 (standard deviation SD = 0.18) for the highly massive shells at a mean volume per individual of 334.1 µm³ (SD = 70). An initial shell porosity of less than 5% was derived from thin layer cross-section images, resulting in a presumed mean shell density of 0.0026 ng µm−3. The shell diameter was significantly influenced by the lake’s origin. The shells from each lake displayed substantial variations in diameter and shape. The pores in the shells showed two variations. Wider pore canals penetrated the whole shell wall, whereas small, elongated pores were located along the interspaces between calcite crystals with tabular habit. The approximate average dimensions of these calcite plates were 1.0 × 1.6 × 0.2 µm. The mean lateral wall thickness at the rim and centre of the shell were 1.98 µm (SD = 0.42) and 0.79 µm (SD = 0.17), respectively. The average carbonate precipitation by Phacotus lenticularis in relation to the total epilimnetic suspended calcite precipitation was 6.0% in the oligotrophic lake Großer Ostersee (Bavaria, Germany). During the growing season, Phacotus lenticularis contributed up to 21% of the particulate calcium carbonate in the epilimnion. These findings suggest that Phacotus lenticularis should be considered in the assessment of hard-water lake carbon cycling.

A Series Solution for the Vibration of Mindlin Rectangular Plates with Elastic Point Supports around the Edges

A series solution for the transverse vibration of Mindlin rectangular plates with elastic point supports around the edges is studied. The series solution for the problem is obtained using improved Fourier series method, in which the vibration displacements and the cross‐sectional rotations of the midplane are represented by a double Fourier cosine series and four supplementary functions. The supplementary functions are expressed as the combination of trigonometric functions and a single cosine series expansion and are introduced to remove the potential discontinuities associated with the original admissible functions along the edges when they are viewed as periodic functions defined over the entire x‐y plane. This series solution is approximately accurate in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. The convergence, accuracy, stability, and efficiency of the proposed method have been examined through a series of numerical examples. Some numerical examples about the nondimensional frequency and mode shapes of Mindlin rectangular plates with different point‐supported edge conditions are given.

Nonlinear In-Plane Stability of Deep Parabolic Arches Using Geometrically Exact Beam Theory

In this paper, we investigate the in-plane stability and post-buckling response of deep parabolic arches with high slenderness ratios subjected to a concentrated load on the apex, using the finite element implementation of a geometrically exact rod model and the cylindrical version of the arc-length continuation method enabled with pivot-monitored branch-switching. The rod model used here includes geometrically exact kinematics of the cross-section, exact kinetics, and a linear elastic constitutive law; and advantageously employs quaternion parameters to treat the cross-sectional rotations and to compute the exponential map in the configurational update of rotations. The evolution of the Frenet frame along the centroidal curve is used to determine the initial curvature of the rod. Three sets of boundary conditions, i.e. fixed–fixed (FF), fixed–pinned (FP) and pinned–pinned (PP), are considered, and arches with a wide range of rise-to-span ratios are analyzed for each set. Complete post-buckling response has been derived for all cases. The results reveal that although all the PP arches and all the FF arches (with the exception of FF arches with rise-to-span ratio less than 0.3) considered in this study buckle via bifurcation, the nature of stability of the different solution branches in the post-buckling regime differs from case to case. All FP slender parabolic arches exhibit limit-point buckling, again with several markedly different post-buckling behaviors. Also, some arches in the FF and PP case are shown to exhibit a clear bistable behavior in the post-buckled state.

Universal analytical solution of the steady-state response of an infinite beam on a Pasternak elastic foundation under moving load

In this paper, the steady-state response of a uniform infinite Euler-Bernoulli elastic beam resting on a Pasternak elastic foundation and subjected to a concentrated load moving at a constant velocity along the beam is analytically investigated. A universal closed-form analytical solution is derived through Fourier transform, apt to represent the response for all possible beam-foundation parameters. A rigorous mathematical procedure is formulated for classifying the parametric behavior of the solution, including for viscous damping. Depending on such a classification, different types of bending wave shapes are shown to propagate within the beam, ahead and behind the moving load position, and crucial physical characteristics, such as critical velocity and critical damping, are reinterpreted into a wholly exact and complete mathematical framework. Mechanical features of the solution are revealed for the steady-state response in terms of normalized deflection, cross-section rotation, bending moment and shear force.

Dynamic characteristics for system of bearing on the top of first story column based on geometry nonlinearity

According to the serial system of bearing with column, the natural frequency of serial system is investigated, and the effects of axial load and different rubber bearings are discussed. Taking into account the effects of the cross-section rotation, shear distortion and compression axial force, the mathematical model for free vibration of the system is established. The differential quadrature element method is employed for the discrete processing in governing equations and boundary conditions. The natural frequency of serial system with clampedfree boundary conditions is solved numerically, and the figures of natural frequency with compression axial force are presented. It is shown that the axial force will reduce the lower order natural frequencies significantly, and in certain case of total height, the equivalent bending stiffness of rubber bearings reduces the natural frequencies to different extent.

A layer-wise theory of shallow shells with thin soft core for laminated glass and photovoltaic applications

The aim of this paper is to develop a robust layer-wise theory for structural analysis of curved glass and photovoltaic panels. By the analogy to the existing theories of plates, governing equations for doubly curved layers including kinematic relations, equilibrium conditions and constitutive equations are introduced. Applying assumptions of shear rigidity of skin layers and moments-free core layer as well as approximations of thin shallow shell, a reduced form of governing differential equations is proposed. Compared to the classical theories of shells the derived system includes an additional second order differential equation. As a result, additional boundary conditions should be satisfied for any edge of the shell. The importance of these extensions is demonstrated for long cylindrical panel with for two examples of simple supports: one with free edges, where relative in-plane displacements of skins are allowed, and one with framed edges, where cross-section rotations of all layers are assumed the same. For both cases closed-form analytical solutions related to a shell strip approximation are presented. Displacement bounds in monolithic and layered cases are derived, and the dependence of deformation and stress characteristics on the radius of curvature and types of supports are illustrated.

Static and dynamic analysis of conical helices based on exact geometry via mixed FEM

The first objective of this research is to indicate the steps that are necessary to generate an exact non-circular (Lancert type) helix geometry as an alternative to the widely used approximate geometrical functions obtained by exploiting exact circular helix geometry. A parametric analysis is performed in order to determine the range of the geometric parameters in which the approximate geometry is insufficient to provide the precision required by exact geometry. This investigation is considered important since the validity of approximate helix geometry is satisfied in a limited range of geometric parameters. The exact conical helix geometry is based on associated plane curves, namely either the Archimedean spiral or logarithmic spiral. The second objective of this research is to determine the necessary conditions to use the Archimedean spiral and logarithmic spiral to supply nearly the same geometry of the conical helix. The final aim is to compare the numerical performance of curved mixed type finite elements with straight displacement type elements over exact conical helix geometry. This is based on the Archimedean spiral. For this purpose, static and free vibration analyses are performed. Curved mixed finite element formulation is based on Timoshenko beam theory which considers shear angle and rotary inertias. The degrees of freedom at a node is twelve: three translational displacements, three cross-sectional rotations, three force components, two bending moments and torque. Approaching the exact geometry of the conical helix with approximate geometry assumptions is only possible under some limitations of the taper ratio and the number of active turns of the conical helix. In free vibration analysis, the mixed-curved elements are considerably faster than straight displacement type elements. In the case of static analysis, convergence performance the two different FE formulations differ in terms of boundary conditions.

Quadrature element method for vibration analysis of functionally graded beams

PurposeThe purpose of this paper is to present a general formulation of the quadrature element method (QEM). The method is then used to investigate the free vibration of functionally graded (FG) beams with general boundary conditions and different variations of material properties.Design/methodology/approachThe quadrature elements with arbitrary number of nodes and nodal distributions are established on the basis of two types of FG Timoshenko beam theories. One called TBT-1 takes the cross-sectional rotation as the unknown function and the other called TBT-2 uses the transverse shear strain as the unknown function. Explicit formulas are provided via the help of the differential quadrature (DQ) rule and thus the elements can be implemented adaptively with ease.FindingsThe suitability and computational efficiency of the proposed quadrature elements for the vibration analysis of FG beams are demonstrated. The convergence rate of the proposed method is high. The elements are shear-locking free and can yield accurate solutions with a small number of nodes for both thin and moderately thick beams. The performance of the element based on TBT-1 is better than the one based on TBT-2.Originality/valueThe present QEM is different from the existing one which exclusively uses Gauss–Lobatto–Legendre (GLL) nodes and GLL quadrature and thus is more general. The element nodes can be either the same or different from the integration points, making the selection of element nodes more flexible. Presented data are accurate and may be a reference for other researchers to develop new numerical methods. The QEM may be also useful in multi-scale modeling and in the analysis of civil infrastructures.

A new passive approach to reducing the carbody vertical bending vibration of railway vehicles

ABSTRACTThis paper brings to view a new passive approach regarding the reduction of the flexible carbody vertical bending vibration of railway vehicles via two anti-bending bars fixed onto the longitudinal beams of a carbody underframe. The anti-bending bars operate as elastic elements limiting the rotation of the carbody cross-sections and thus increasing the bending carbody frequency. The anti-bending bars are dimensioned so that the frequency of the first bending mode of the carbody to be convenient in terms of ride comfort. The effectiveness in the use of the anti-bending bars is proven by the lowering of the vibration level and improvement of the vehicle ride comfort.

An arbitrary Lagrangian–Eulerian discretization method for modeling and simulation of reeving systems in multibody dynamics

This paper presents an efficient discretization method for the modeling and simulation of reeving systems using multibody dynamics. Reeving systems are assumed to include rigid bodies connected by a set of sheaves or reels and wire ropes. The method is based on line-parametrized Absolute Nodal Coordinate Formulation (ANCF) defined in the framework of an Arbitrary Lagrangian–Eulerian (ALE) description. In the ALE description the element nodes have variable material coordinates thus allowing the element to change its position within the flexible body. This property is very convenient to model wire ropes rolled in sheaves or reels that have variable-length free spans, as usually occurs in reeving systems. Axial-torsion elastic coupling of the wire ropes is also a very important effect to be considered in reeving systems. A set of locally defined cross-section rotation angles is introduced to describe this effect thus abandoning the ANCF formalism. This paper presents different options for the wire rope modeling depending on the problem dimension (1, 2 or 3D) and the assumed elastic force model. Depending on the application the method offers elements that can show axial, torsion and bending deformation energies or any combination of them. This paper presents a general procedure to model reeving systems and the numerical calculation of the resulting equations of motion. Two examples are presented: the dynamic model of a 2:1-suspended electrically-driven elevator and a quasi-static model of a large crane.

Conforming shear-locking-free four-node rectangular finite element of moderately thick plate

AbstractAn outline of the modified Mindlin plate theory, which deals with bending deflection as a single variable, is presented. Shear deflection and cross-section rotation angles are functions of bending deflection. A new four-node rectangular finite element of moderately thick plate is formulated by utilizing the modified Mindlin theory. Shape functions of total (bending+shear) deflections are defined as a product of the Timshenko beam shape functions in the plate longitudinal and transversal direction. The bending and shear stiffness matrices, and translational and rotary mass matrices are specified. In this way conforming and shear-locking-free finite element is obtained. Numerical examples of plate vibration analysis, performed for various combinations of boundary conditions, show high level of accuracy and monotonic convergence of natural frequencies to analytical values. The new finite element is superior to some sophisticated finite elements incorporated in commercial software.

An Analytical Solution for Two-Layered Composite Beams with Imperfect Shear Interaction

The authors present an analytical solution for two-layered composite beams with interlayer slip. Timoshenko's kinematic assumptions are used for both layers imposing the constraint of the equal cross-sectional rotations. The connection between the beam components is perfect in normal direction, but the axial displacement field may have a jump. The axial force derived from the imperfect connection is proportional to the relative slip occurring between the layers. The determination of the analytical solution of the considered static problem is based on the fundamental solutions. Linear combination of the fundamental solutions, which are fitted to the given loading and boundary conditions, gives the total solution. Three examples illustrate the applications of the presented analytical method.

Slip and stress of tensile armors in unbonded flexible pipes close to end fitting considering an exponentially decaying curvature distribution

An analytical model is given to investigate the end-fitting effect on slip and stress of tensile armors in unbonded flexible pipes under tension, torsion and varying bending in the absence of friction. An exponentially decaying curvature distribution is assumed to represent controlled curvature over the end region. The deviation from the initial helical angle is taken to describe the armor wire path as the pipe is stretched, twisted and bent, which is determined by minimization of the strain energy functional using the Euler equation. The obtained simultaneous differential equations are numerically solved by transforming them into a boundary value problem. An analytical solution is found by neglecting the twisting rotation of the wire cross-section. The developed model is validated with a finite element simulation and geodesic based analytical expressions for constant curvature and with a current numerical model for varying curvature. Reasonable correlations are observed between the model predictions and the results from other methods. The validated model is then applied to typical flexible pipe designs to find the level and location of the greatest increases in stress. The results show that the end restraint could cause a significant stress increase in the armor wire at the end fitting vicinity and the critical location is close to the extreme fiber position on the compressive side of the pipe for typical cases. The effect of end restraint on layer bending stiffness is also evaluated.

The effect of finite strain on the nonlinear free vibration of a unidirectional composite Timoshenko beam using GDQM

In this manuscript, free vibrations of a unidirectional composite orthotropic Timoshenko beam based on finite strain have been studied. Using Green-Lagrange strain tensor and comprising all of the nonlinear terms of the tensor and also applying Hamilton's principle, equations of motion and boundary conditions of the beam are obtained. Using separation method in single-harmonic state, time and locative variables are separated from each other and finally, the equations of motion and boundary conditions are gained according to locative variable. To solve the equations, generalized differential quadrature method (GDQM) is applied and then, deflection and cross-section rotation of the beam in linear and nonlinear states are drawn and compared with each other. Also, frequencies of carbon/epoxy and glass/epoxy composite beams for different boundary conditions on the basis of the finite strain are calculated. The calculated frequencies of the nonlinear free vibration of the beam utilizing finite strain assumption for various geometries have been compared to von Karman one.

Structural theory and finite element modelling of linear elastic sandwich beams subject to severe boundary conditions

Abstract We further develop and improve a structural theory recently proposed by our group, with the aim of determining the simplest kinematics which allows the accurate modelling of any plane sandwich beam in the linear elastic regime. The model builds on Yu–Krajcinovic zig-zag warping, in which each layer, of arbitrary thickness and modulus, is allowed to shear through an independent cross-section rotation. Moreover, the core kinematics is enriched by allowing for a quadratic variation along the core thickness of both the longitudinal and the transverse displacement components. By implementing the proposed theory in a structural finite element, we discuss the contribution to the modelling capability of each independent term entering the chosen core kinematics. Such kinematics, along with a Jourawski-like approach to evaluate the shear stress, leads to a model which can accurately describe the stress state for any relative stiffness between the sandwich layers, even in the case of “severe boundary conditions”, including loading on a specific skin coupled with constraints realised, on certain cross-sections, on the opposite skin only. We demonstrate this claim by considering many benchmarks and by a thorough comparison with the results obtained from continuum plane stress Finite Element (FE) simulations. From such a comparison we also clearly establish the superior computational efficiency of the new structural finite element with respect to the continuum FE analyses.

A geometrically exact Kirchhoff beam model including torsion warping

This paper presents a geometrically exact beam model that includes the Kirchhoff constraint (null shear deformation), torsion-related warping and Wagner effects, and aims at capturing the flexural-torsional behaviour of thin-walled beams undergoing large displacements. The parametrization of the cross-section rotation tensor is based on the composition of a torsional rotation and the so-called smallest rotation to the tangent vector of the beam axis, as proposed in Boyer and Primault (2004). The novel aspects of the proposed model are the following: (i) closed-form expressions for the variations of the strain measures are presented and the equilibrium equations and their linearization are completely written in terms of the independent kinematic parameters, (ii) torsion-warping is allowed, as well as Wagner effects, and (iii) arbitrary cross-sections are considered, namely cross-sections where the shear centre and centroid do not coincide. Moreover, several fundamental aspects related to the cross-section rotation parametrization are discussed. The accuracy and the computational efficiency of the finite element implementation of the proposed model is demonstrated in several numerical examples.

Large deflection and rotation of Timoshenko beams with frictional end supports under three-point bending

Three-point bending of a beam is studied based on the Timoshenko beam theory. Large deflection and large rotation of a beam resting on simple supports with friction are calculated for a concentrated force acting at the midspan. Using the Lagrangian kinematic relations, a system of non-linear differential equations are obtained for a prismatic shear-deformable Timoshenko beam. Exact solutions for the deflection, horizontal displacement, and rotation of cross-section are derived analytically. Two deflections of small and large scale exist under three-point bending. The solutions corresponding to linearized model coincide with the well-known solutions to the classical Timoshenko beams. Numerical calculations are carried out to show the effect of the important parameters such as shear rigidity of the beam and the coefficient of friction at the contact position between the beam and supports on the deflection. The load–deflection curves are graphically presented. A comparison of large deflections and large rotations with their classical counterparts and with experimental data is made. The obtained results are useful in safety design of linear and non-linear beams subject to three-point bending.

Dynamics of a micro scale Timoshenko beam subjected to a moving micro particle based on the modified couple stress theory

In this study, the free and forced vibration analysis of a micro scale Timoshenko beam resting on a Pasternak elastic foundation and subjected to a moving micro particle is presented. Based on the modified couple stress theory and employing Hamilton’s principle, the governing equations along with the boundary conditions are derived. A semi-analytical solution is obtained for the free vibration of the problem by expressing the dynamic lateral displacement and cross-section rotation in terms of the series of Legendre polynomials and extremizing the objective functional of the problem with respect to the unknown displacements and Lagrange multipliers. Correspondingly, the computed eigenvalue information of the system is utilized in the modal expansion technique to obtain the transient dynamic response. For comparison purposes, the free vibration frequencies of the micro beam and the dynamic deflections using the classical Timoshenko beam theory are compared with previously published studies and very good agreements have been observed. Furthermore, more numerical examples for natural frequencies and dynamic deflection of the beam are presented and the effects of some parameters, such as the material length scale parameter, the velocity of micro particle, the Pasternak elastic foundation parameters, shear deformation effects and boundary conditions are examined.

Explicit solutions for depressed masonry arches loaded until collapse—Part I: a one-dimensional nonlinear elastic model

In this paper the equilibrium problem for masonry arches is formulated in terms of a suitable set of nonlinear ordinary differential equations. We show that by making a small number of simple hypotheses it is possible to find the explicit expressions for the displacements and rotations of the cross-sections of an in-plane loaded masonry arch. To this end, the masonry arch is schematised as a curved, one-dimensional nonlinear elastic beam made of a material that is by hypothesis incapable of withstanding significant tensile stresses. In this first part of the two-part paper, the one-dimensional model and the explicit expressions for the displacements and rotations, obtained by integrating the set of differential equations, are presented. In particular, the formal expressions for displacement, stress and strain fields are illustrated in full detail for an explicit, albeit approximate, solution for a statically determinate depressed arch subjected to a uniform vertical load.