Rotation Of Cross-section Research Articles

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle.

A class of variational inequalities describing the equilibrium of elastic Timoshenko plates whose boundary is in contact with the side surface of an inclined obstacle is considered. At the plate boundary, mixed conditions of Dirichlet type and a non-penetration condition of inequality type are imposed on displacements in the mid-plane. The novelty consists of modelling oblique interaction with the inclined obstacle which takes into account shear deformation and rotation of transverse cross-sections in the plate. For proposed problems of equilibrium of the plate contacting the inclined obstacle, the unique solvability of the corresponding variational inequality is proved. Under the assumption that the variational solution is smooth enough, optimality conditions are obtained in the form of equilibrium equations and relations revealing the mechanical properties of integrated stresses, moments and generalized displacements on the contact part of the boundary. Accounting for complementarity type conditions owing to the contact of the plate with the inclined obstacle, a primal-dual variational formulation of the obstacle problem is derived. A semi-smooth Newton method based on a generalized gradient is constructed and performed as a primal-dual active-set algorithm. It is advantageous for efficient numerical solution of the problem, provided by a super-linear estimate for the corresponding iterates in function spaces. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Physics-Informed Neural Networks for Solving Forward and Inverse Problems in Complex Beam Systems.

This article proposes a new framework using physics-informed neural networks (PINNs) to simulate complex structural systems that consist of single and double beams based on Euler-Bernoulli and Timoshenko theories, where the double beams are connected with a Winkler foundation. In particular, forward and inverse problems for the Euler-Bernoulli and Timoshenko partial differential equations (PDEs) are solved using nondimensional equations with the physics-informed loss function. Higher order complex beam PDEs are efficiently solved for forward problems to compute the transverse displacements and cross-sectional rotations with less than 1e-3 % error. Furthermore, inverse problems are robustly solved to determine the unknown dimensionless model parameters and applied force in the entire space-time domain, even in the case of noisy data. The results suggest that PINNs are a promising strategy for solving problems in engineering structures and machines involving beam systems.

Analysis of planar arbitrarily curved microbeams with simplified strain gradient theory and Timoshenko–Ehrenfest beam model

As the first endeavor in the context of Mindlin’s strain gradient theory, this study contributes a systematic and rigorous derivation for governing equations and boundary conditions of planar arbitrarily curved microbeams. The Timoshenko–Ehrenfest beam model is incorporated into a simplified version of Mindlin’s strain gradient theory. Kinematic unknowns include displacement components of the beam axis in the local coordinate system and the rotation of cross-section. Since the derived governing equations and boundary conditions are extremely complex, analytical solutions are not achievable for microbeams having non-uniform curvature. To facilitate the numerical analysis, two isogeometric collocation formulations are proposed, that is, displacement-based and mixed formulations. Several tests are designed to evaluate the accuracy and reliability of the proposed isogeometric collocation formulations, especially with respect to the well-known locking pathology. It is found that the mixed formulation is more accurate and robust than the displacement-based one. Therefore, the mixed formulation is then used to numerically investigate the size-dependent behavior and stiffening effect. Furthermore, some informative tests are performed to delineate the significance of the curviness in the prediction of structural responses of planar arbitrarily curved microbeams, which appears to be still an unanswered issue.

Linear composite curvature MITC3+ flat shell elements

This paper introduces a novel linear composite curvature MITC3+ flat shell element, named κMITC3+, that combines the advantages of the MITC3+ theory for flexural (bending and shear) behavior and the Allman-like triangular element (which ensures a true rotation of drilling degrees of freedom and eliminates a spurious energy mode) for membrane behavior. To enhance the bending behavior of the MITC3+ approach, we propose an assumed composite curvature field that is constructed using a polynomial projection technique derived from the orthogonality condition of the compatibility equation in the Hu-Washizu principle. Since two additional internal cross-section rotations are interpolated by cubic bubble functions, a linear polynomial function of the admissible space is chosen that provides a significant enhancement of bending behavior. The κMITC3+ element successfully passes all critical tests and exhibits superior performance compared to the original MITC3+ element and the recent CS-MITC18+ element.

Analytical study of FRP debonding in a real-scale concrete strengthened beam

The exploitation of the mechanical properties of fiber-reinforced polymers (FRP) externally bonded to concrete members is limited by the occurrence of debonding. Single- and double-lap shear tests were performed to study the debonding process. However, the results of these tests fail to describe the debonding process when multiple substrate cracks occur, as in the case of reinforced concrete beams strengthened in flexure. In this paper, an analytical model is proposed to describe the load response of a concrete beam strengthened in flexure with an externally bonded FRP composite. The profiles of the FRP-concrete slip, FRP axial force, and cross-section rotation, the evolution of the cracking process, and the failure mode may be obtained by this model. The results of the model shed light on the debonding process in FRP-strengthened beams and may be used to investigate the relationship between the bond capacity of FRP-concrete interfaces with a single crack and that observed in real-scale strengthened beams.

A Comprehensive numerical study on using lobed cross-sections in spiral heat exchanger: Fluid flow and heat transfer analysis

The present study aims to assess the thermohydraulic characteristics of smooth tri-lobed spiral tube (STST) and twisted tri-lobed spiral tube (TTST) numerically. Water with constant thermophysical properties is considered as the working fluid. The Reynolds number, based on the inlet conditions, varies from 500 to 2500 and a constant temperature of 323.15 K is regarded as the boundary condition of the tube wall. Temperature and flow field inside the tube are simulated utilizing a three-dimensional, stationary finite volume method (FVM) and a computational grid consisting of polyhedral elements. The influence of cross-section rotation for the STST and the effects of twist number, pitch ratio, and different cross-sections for the TTST are evaluated on the thermohydraulic parameters of the heat exchanger. According to the results, the optimized geometric configuration for STST in order to have the best performance occurs when the cross-section of the tube inlet has a rotation angle of −30°, resulting in 11.14% increase in the performance evaluation criteria (PEC) parameter. According to the findings for TTST, the Nusselt number (Nu) and friction factor (f) increase continuously by increasing the number of tube twists. Furthermore, from the PEC point of view, the TTST with a low pitch ratio close to 1 exhibits the best findings. Comparing the optimized results of STST and TTST indicates that the highest value of PEC (1.25) is achieved for TTST with a pitch ratio of 1.25 and a twist of 13 turns, which occurs at the Reynolds number of 1000.

Theoretical thermal damping vibration analysis of functionally graded viscoelastic Timoshenko microbeam with integral nonlocal strain gradient model

In this work, nonlocal strain gradient integral model is applied to investigate the free damping vibration analysis of functionally graded (FG) viscoelastic Timoshenko microbeam with immovable boundary conditions in thermal environment. The microbeam is assumed to be viscoelastic and modeled by Kelvin–Voigt model. The differential governing equations and corresponding boundary conditions are derived with the Hamilton’s principle. Combining nonlocal strain gradient integral model and Kelvin–Voigt viscoelastic model, the integral constitutive equations of nonlocal stress with thermal effect are derived and then converted into differential form plus constitutive constraints. The size-dependent axial force due to thermal expansion is explicitly derived through the immovable boundary conditions. The bending deflection and moment, cross-sectional rotation as well as shear force are explicitly derived with Laplace transformation for linear thermo-elastic vibration. Taking into account the boundary conditions as well as constitutive constraints, one gets a nonlinear equation with complex coefficients, from which one can determine the complex characteristic frequency. A two-step numerical method is proposed to solve the elastic vibration frequency and damping ratio. The influence of length scale parameters, viscos coefficient, FG index, thermal effect, vibration order and ratio between beam length and thickness on the vibration frequencies and damping ratio is investigated numerically for FG viscoelastic Timoshenko microbeams under different immovable boundary conditions.

Indentation of geometrically exact adhesive beams

We investigate the indentation of thin adhesive beams by a rigid frictionless circular punch in two-dimensions through computations and experiments. A non-linear geometrically exact (GE) beam formulation that allows for arbitrarily large displacements and rotations is employed for the thin beam. The Karush–Kuhn–Tucker condition for unilateral contact is adapted to model adhesive interaction in a nonlinear finite element (FE) framework. In this, contact is enforced through a penalty method and adhesion is incorporated through two different cohesive zone models (CZMs), viz. exponential and bilinear. The indentation of the adhesive GE beam is then studied in terms of the variations of the applied force, the pressure distribution, the cross-sectional rotation and the sizes of the contact area and the adhesive zone for different indentation extents, adhesive strengths, punch radii, beam thicknesses and end supports. We find that the system is largely agnostic to the CZM utilized, as well as the type of end supports unless the contact area spans the beam. At the same time, thinner beams tend to initiate contact with a punch more easily, and are harder to detach. We then demonstrate that thin beams may undergo large rotations even if the indentation is small, which necessitates a nonlinear beam theory. Under such conditions approximating a circular punch by a parabolic one used typically when indentations are small introduces errors. Finally, we conduct experiments on clamped adhesive beams made of two different types of PDMS (polydimethylsiloxane) and various thicknesses, and find a good match with our computations.

A Three-Dimensional Vibration Theory for Ultralight Cellular Sandwich Plates Subjected to Linearly Varying In-Plane Distributed Loads.

Thin structural elements such as large-scale covering plates of aerospace protection structures and vertical stabilizers of aircraft are strongly influenced by gravity (and/or acceleration); thus, exploring how the mechanical behaviors of such structures are affected by gravitational field is necessary. Built upon a zigzag displacement model, this study establishes a three-dimensional vibration theory for ultralight cellular-cored sandwich plates subjected to linearly varying in-plane distributed loads (due to, e.g., hyper gravity or acceleration), with the cross-section rotation angle induced by face sheet shearing accounted for. For selected boundary conditions, the theory enables quantifying the influence of core type (e.g., close-celled metal foams, triangular corrugated metal plates, and metal hexagonal honeycombs) on fundamental frequencies of the sandwich plates. For validation, three-dimensional finite element simulations are carried out, with good agreement achieved between theoretical predictions and simulation results. The validated theory is subsequently employed to evaluate how the geometric parameters of metal sandwich core and the mixture of metal cores and composite face sheets influence the fundamental frequencies. Triangular corrugated sandwich plate possesses the highest fundamental frequency, irrespective of boundary conditions. For each type of sandwich plate considered, the presence of in-plane distributed loads significantly affects its fundamental frequencies and modal shapes.

Optimal electrode coverage based on a new criterion for piezoelectric energy harvesters

Piezoelectric energy harvesters (PEHs) are a promising alternative to conventional electrochemical batteries with the advantage of being self-powered and maintenance-free, but their application is severely restricted by the low electric power output. It has been reported that the power output of PEHs can be enhanced by well-designed electrode coverage. A common design criterion for beam-like PEHs is based on the strain node to avoid electrode charge cancellation. This criterion, however, is not feasible for PEHs subject to complex spatio-temporal excitation patterns, where strain nodes change their position. This work proposes a new design criterion for optimal electrode coverage of beam-like PEHs based on the closed-form solution of the circuit equation that expresses voltage as a function of the beam’s dynamic response, specifically the cross-section rotation. The new criterion maximizes the averaged curvature of the beam segment covered by the electrode using data on the instantaneous rotation field. The improved physical significance and reliability of the presented criterion are discussed. The associated electrode optimization procedure is then exemplified for PEHs driven by fluid flow, which helps to realize a complex excitation pattern. Two numerical studies, both including a variety of combinations of fluid densities and inlet velocities, are performed to demonstrate that an optimal electrode configuration can be obtained with the proposed criterion. Comparison of different electrode configurations in above studies finally leads to useful conclusions on the power output and electrode configuration.

Effect of offset between beam outside and neutral surfaces on dynamic soil-structure interactions

In finite element analyses of the dynamic soil-structure interaction (SSI), structures are often modelled as beam or shell elements in structural mechanics, while the soil requires discretization using solid elements in solid mechanics. The beam and shell elements are coupled with the solid elements on their outer surfaces, while the mechanical behaviour of the beam and shell elements are usually represented by variables defined at their neutral surfaces. The offset between the beam outer surface and the neutral surface in several dynamic SSI analyses is often ignored, which can cause large errors in the coupling analysis of solid and structural mechanics. This paper investigates the effect of the offset between the beam outer surface and neutral surface on the dynamic SSIs. First, the coupling conditions between the beam and the solid considering the offset between the beam outer surface and the neutral surface are deduced. In this process, the tangential interaction stress on the beam outer surface is transformed into an axial force and a moment acting on the beam neutral surface, and the tangential displacement on the beam outer surface is related to the axial displacement at the beam neutral surface as well as the cross-section rotation angle. Then, the coupling condition is implemented in the finite element analysis by modifying the mass and stiffness matrices of the beam element. The offset effects between beam outer surfaces and neutral surfaces on dynamic SSIs are finally discussed in detail through the seismic responses of a retaining wall, subway station and tunnel.

A Group-Theoretic Approach to the Bifurcation Analysis of Spatial Cosserat-Rod Frameworks with Symmetry

We present a general approach to the bifurcation analysis of spatial framework structures with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of geometrically exact frameworks is not straightforward. We consider spatial structures comprising assemblages of Cosserat rods; the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a slightly enlarged linear space. The field equations for a framework then comprise the assembly of all such rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the natural symmetry group action, and the implementation of group-theoretic methods is now clear within the enlarged linear space. All potential generic, symmetry-breaking bifurcations are predicted a-priori. We then employ an open-source path-following code, which can detect and compute simple, one-dimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a more complex hexagonal space frame via the same approach. Both structures exhibit simple and double bifurcation points.

ПРО ТОЧНІ РОЗВ’ЯЗКИ РІВНЯНЬ ОСІ Й КУТА ПОВОРОТУ ПЕРЕРІЗУ ПРУЖНОЇ БАЛКИ

Calculation of elastic structures for strength and stiffness is important for their safe operation. The need for calculations is caused by structural deformations under the influence of external forces or temperature.The objects of our research are the equation of the axis of an elastic beam during its flat bending, as well as the equation of the angle of rotation of the beam section. These equations are differential. A certain complexity of their solution led to the simplification of equations in classical sources of information. Approximate solutions of these equations are considered there.But when using approximate calculation methods, you need to be able to assess their accuracy, that is, the degree of approximation to the exact result.The goal of our research was to obtain exact solutions. The exact solution of the beam equations is important to avoid its critical deformations.The article presents the exact analytical solutions we obtained for the exact equations of the bent axis of the beam and the angle of rotation of the beam section. The advantage of the exact solution was revealed, in particular, in the fact that the largest value of the deflection and angle of rotation of the beam section can be obtained directly from the properties of thefunctions that describe the solution.Another advantage of the exact solution was the possibility of obtaining an approximate solution with a predetermined accuracy. In the mentioned classical sources of information, the assessment of accuracy was derived from the range of the maximum deflection of the axis during practical calculations of structures.We have obtained a method of analytical assessment of the accuracy of the obtained solutions. The ability to assess the accuracy of calculation results is an important aspect of their practical application. This is important, in particular, for check-ing and clarifying the safe range of movements of beam points during its operation.An example of calculating the deflection and angle of rotation of the beam cross-section with a given load on the axis of the beam is shown

Slender body theories for rotating filaments

Slender fibres are ubiquitous in biology, physics and engineering, with prominent examples including bacterial flagella and cytoskeletal fibres. In this setting, slender body theories (SBTs), which give the resistance on the fibre asymptotically in its slenderness $\epsilon$ , are useful tools for both analysis and computations. However, a difficulty arises when accounting for twist and cross-sectional rotation: because the angular velocity of a filament can vary depending on the order of magnitude of the applied torque, asymptotic theories must give accurate results for rotational dynamics over a range of angular velocities. In this paper, we first survey the challenges in applying existing SBTs, which are based on either singularity or full boundary integral representations, to rotating filaments, showing in particular that they fail to consistently treat rotation–translation coupling in curved filaments. We then provide an alternative approach which approximates the three-dimensional dynamics via a one-dimensional line integral of Rotne–Prager–Yamakawa regularized singularities. While unable to accurately resolve the flow field near the filament, this approach gives a grand mobility with symmetric rotation–translation and translation–rotation coupling, making it applicable to a broad range of angular velocities. To restore fidelity to the three-dimensional filament geometry, we use our regularized singularity model to inform a simple empirical equation which relates the mean force and torque along the filament centreline to the translational and rotational velocity of the cross-section. The single unknown coefficient in the model is estimated numerically from three-dimensional boundary integral calculations on a rotating, curved filament.

Size-dependent nonlinear post-buckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models

Strain-driven (ɛD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) are applied to study the size-dependent nonlinear post-buckling behaviors of functionally graded porous Timoshenko microbeams. The differential governing equations and boundary conditions of motion are derived on the basis of the principle of minimum potential energy and expressed in nominal form. The integral relation between local and nonlocal stresses is transformed unitedly into equivalent differential form with constitutive constraints for ɛD- and σD-TPNIMs. Bending deflection and cross-sectional rotation are derived explicitly with Laplace transformation for linear buckling. Taking into account boundary conditions and constitutive constraints, a general eigenvalue problem is obtained to determine linear buckling mode shape (LBMS) and nominal buckling load (NBL). Local and nonlocal LBMS based Ritz–Galerkin methods and general differential quadrature method (GDQM) based Newton–Raphson’s method are utilized to obtain the numerical solutions for linear and nonlinear NBLs. Numerical results show that nonlocal LBMS based Ritz–Galerkin method and GDQM would lead to same prediction for linear NBLs as analytical method, and nonlocal LBMSs based Ritz–Galerkin and GDQM based Newton–Raphson’s method would lead to exact same prediction for nonlinear post-buckling loads. However, local LBMS based Ritz–Galerkin method is failed to provide accurate prediction for both linear NBLs and nonlinear post-buckling loads, through the difference between local and nonlocal LBMSs is not significant. The influence of nonlocal parameters, porous distribution patterns and boundary conditions on the linear buckling and nonlinear post-buckling behaviors are investigated numerically.

Lateral-torsional buckling strength of corrugated web girders – Experimental study

Current standards and specifications do not provide specific design methods to determine the lateral-torsional buckling (LTB) strength of girders with trapezoidally corrugated web. In the international literature there are some previous investigations available focusing on the LTB strength of corrugated web girders. Based on numerical calculations analytical solutions have been derived in the past, however, there is a lack of experimental test results to prove or improve these design proposals. The authors performed an extensive experimental research program including eleven large-scale test specimens. All the test girders have different flange sizes, beside, having the same corrugation profile. The test specimens are loaded by pure in-plane bending having simply supported boundary conditions. The tested specimens have limited warping and rotational restraint at the supports. No web distortional type failure is observed. During the tests the longitudinal strains are captured in characteristic cross-sections and the displacements in vertical and lateral directions are measured as well as the rotation of the mid-span cross-section. The measured LTB strength of the test specimens is compared with existing analytical proposals regarding corrugated web girders and the differences are evaluated. It is observed that the imperfections have key role in the LTB behavior. Based on the experimental results buckling curve “b” from Eurocode 3 is preliminary proposed for the determination of the lateral-torsional buckling strength of trapezoidally corrugated web girders. The executed test program gives useful background to further numerical studies.

Locking-free Kriging-based Timoshenko Beam Elements using an Improved Implementation of the Discrete Shear Gap Technique

Kriging-based finite element method (K-FEM) is an enhancement of the conventional finite element method using a Kriging interpolation as the trial solution in place of a polynomial function. In the application of the K-FEM to the Timoshenko beam model, the discrete shear gap (DSG) technique has been employed to overcome the shear locking difficulty. However, the applied DSG was only effective for the Kriging-based beam element with a cubic basis and three element-layer domain of influencing nodes. Therefore, this research examines a modified implementation of the DSG by changing the substitute DSG field from the Kriging-based interpolation to linear interpolation of the shear gaps at the element nodes. The results show that the improved elements of any polynomial degree are free from shear locking. Furthermore, the results of beam deflection, cross-section rotation, and bending moment are very accurate, while the shear force field is piecewise constant.

A Novel Elastomer-Based Inclinometer for Ultrasensitive Bridge Rotation Measurement.

Bridge deformation consists of cross-section rotation and deflection, which are crucial parameters for bridge capacity evaluation and damage detection. The maximum value of deflection usually happens at mid-span while for rotation it happens at two-ends. Therefore, compared with deflection, rotation is more convenient for in-situ measurement since the bridge pier can be the reference point. In this study, a high-precision inclinometer for bridge rotation measurement was conceptualized, designed, and validated. The proposed inclinometer converted the small rotation of bridge section into the deformation of an elastomer. Strain gauges were then utilized to measure the elastomer deformation and thus the bridge rotation can be obtained. The dimensions and modulus of the elastomer were designed and chosen based on the theoretical analysis. Characteristics of the inclinometer were calibrated in lab and in-situ experiments at an in-service bridge were conducted to validate its feasibility and robustness. Test results showed that the proposed inclinometer had excellent performance in resolution and accuracy, which indicate its great potential for future bridge health monitoring.

The static and free vibration analyses of axially functionally graded elliptical beams via mixed FEM

The objective of this study is to investigate the behavior of the static and free vibration analyses of axially functionally graded elliptical planar curved beams using a mixed finite element method (MFEM) based on the Timoshenko beam theory. A two-noded curved mixed finite element has 12 field variables at each node. These variables denote three displacements, three cross-sectional rotations, three forces, two bending moments, and torque, respectively. The functionally graded material is composed of ceramic-particle material and metal-matrix material. The volume fraction of ceramic and metal materials varies along the beam axis. The effective material properties (modulus of elasticity, Poisson's ratio, and density) of the functionally graded material are determined according to the rule of mixture. It is aimed in the benchmark examples to present the influence of ceramic-particle material and non-homogeneity index of material gradation, the minimum radius of the elliptical beam, and boundary condition to the results of static and free vibration analysis in detail.

Analytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory

In the present work, the equations of motion of a thin orthotropic nanoplate were obtained based on the new modified couple stress theory and the third-order shear deformation plate theory. The nanoplate was considered as a size-dependent orthotropic plate. The governing equations were derived using the dynamic version of Hamilton’s principle and natural boundary conditions were formulated. An analytical solution in the form of a double Fourier series was obtained for a simply supported rectangular nanoplate. The eigenvalue problem was set and solved. It was analytically shown that the displacements of the median surface points in the plane of the plate do not depend on the material length scale parameters in the same directions; these in-plane directional displacements depend on the material length scale parameter in the out-of-plane direction only. On the other hand, the out-of-plane directional displacement depends on the length scale parameter in the plane directions only. The cross-section rotation angles depend on all length scale parameters. It was shown that the size-dependent parameters only have a noticeable effect on the deformed state of the plate if their order is not less than the order (plate height)-1.