A simple mathematical model for free vibration analysis of combined system consisting of framed tube,shear core,belt truss and outrigger system with geometrical discontinuities

Dynamic analysis of beam-like structure is significantly important in modeling real cases such as tall buildings, aircraft wings, spacecraft antennas and many other applications. This paper tries to determine the first natural frequency of tall buildings including framed tube, shear core, belt truss and outrigger system with multiple jumped discontinuities in the cross section of framed tube and shear core. In this regard, the entire length of the tall building is partitioned into uniform segments between each two successive discontinuity points. The effect of belt truss and outrigger system is modeled as a concentrated rotational spring applied at the belt truss and outrigger system location. Many cantilevered tall structures can be treated as cantilever bars with multiple jumped discontinuities in the cross section for the analysis of their free vibration. In this paper, the continuous approach was accepted and by using the Hamilton’s variational principle, the general form of governing equation for free vibration of tall building can be obtained. By applying the separation of variable method on time and space, the governing Partial Differential Equation (PDE) of motion is reduced to an Ordinary Differential Equation (ODE) with one variable coefficient while the other coefficients are constant based on the assumption that the transverse displacement is a harmonic vibration. To find exact solution of ODE, we must have exact distribution of EI(x), AG(x), N(x) and m(x) in the height of the structure. Some of these parameters such as EI(x), AG(x) and m(x), are constant throughout the height of each segment. These parameters can be expressed exactly by using of multi criteria function, while N(x) is variable in the height of each segment. Therefore, the ODE by using the method of variable separation and partitioned method can be expressed for each segment. We must apply the continuity conditions in conjunction with different segments for obtaining unique mode shape for mentioned system. Tall building characteristics matrix can be derived based on the boundary conditions and the continuity conditions applied at the partitioned points. This matrix is particularly used to find combined system first natural frequency and mode shape. Three numerical examples with different stepped discontinuities in their cross sections are studied to demonstrate the reliability of this method. The results of the proposed mathematical model give a good understanding of the structure’s dynamic characteristics; it is easy to use, yet reasonably accurate and suitable for quick evaluations during the preliminary design stages which require less time.     

A simple analytic method for computing the natural frequencies and mode shapes of tall buildings

A simple, yet accurate analytical approach based on energy principles is developed for quick computation of natural frequencies and mode shapes of multistory buildings constructed using framed tube, shear core and double belt trusses systems. The approach here is based on development of a continuum model that would be equivalent, in major motions, to the actual multistory building. Models studied here are cantilever beams with concentrated moments placed at belt truss locations. Governing equation and boundary conditions of the equivalent beam and moment system were derived using the energy method and Hamilton’s principle. Separation of variables technique is then applied to model’s partial differential equation to obtain the required eigensystem. Robustness and correctness of the proposed method are demonstrated through several numerical examples. Here, 40, 55 and 70-storey tall buildings with combined system of framed tube, shear core and double belt trusses, in which the results obtained from the proposed method, are compared with those obtained from three-dimensional analyses using SAP2000 software. Comparative analyses reveal that the proposed method is simple and efficient; and it provides reasonably accurate results quickly, a feature that is vital during the early stages of building design.     

Cornerning radial tyres

An angular difference between direction of wheel motion and wheel plane produced a lateral force on a rolling tyre. A simple mechanism of shear stress generation in the contact area of tread is used. The coefficient of friction plays a limiting role. The resulting lateral force and self-aligning torque are computed by means of the belt model of a radial tyre. Finally, some numerical results are given to illustrate the theory.     

Multi-scale modeling of beam-like structures: A new boundary condition concept for the RVE

In this work a coupled two-scale beam model using Timoshenko beam elements [1] with finite displacements on the macro scale and fully non-linear 3D brick elements on the micro scale is proposed. The calculation is carried out with the so-called FE² concept. To achieve the coupling between the beam and the brick elements, the algorithm from [2] is adapted. Within the degenerated concept of the Timoshenko beam, the introduction of a pure shear deformation leads to significant problems concerning the equilibrium condition on the micro scale. Applying this deformation mode on the RVE with periodic boundary conditions results in a rigid body rotation. Using linear displacement boundary conditions instead, the wrapping deformation is suppressed on the boundary, leading to a length dependency in the torsional deformation mode. In addition, the shear forces introduce a bending moment, which depends on the length of the RVE and adds spurious normal stresses and a length dependency of the shear stiffness. To overcome these problems, periodic boundary conditions are applied and the displacement assumptions are modified such that the shear deformation is achieved with force pairs on both ends of the RVE. The resulting model leads to length independent results in tension, bending and torsion and a domain which is able to produce a pure shear stress state. Consequently, only this domain of the model should be homogenized which can be accomplished by modifying the variations in the algorithm [2]. The concept is validated by simple linear and non-linear test problems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal sensor placement for spatial lattice structure based on three-dimensional redundancy elimination model

Spatial lattice structures are widely applied in engineering fields attributed to their superiorities in weight and reasonable stress. It is essential to select the best sensor placement layout for structural health monitoring and safety purposes, yet it is neither realistic nor efficient to place sensors in every location possible on the structure. To meet the strong requirements for optimal sensor placement in spatial lattice structure, this paper aims to investigate a combined objective function based on effective independence method and three dimensional redundancy elimination model to balance between optimal sensor placement performance and elimination in redundancy. To eliminate redundant information and resource waste caused by the clustered sensor distribution, the three-dimensional redundancy elimination model is constructed with the consideration of nearer nodes and overall sensor distribution ranges in three-dimensional cases. In addition, the combined function is constructed by giving the two component functions equal significance using weighting factors and normalization, and solved by genetic algorithm. Finally, the proposed method for spatial lattice structure is supported by three numerical examples including a simple lattice structure, a ground spatial truss structure and a space docking modular in space solar power satellite.     

Mathematical model of the pipeline with angular joint of elements

A mathematical model of the pipeline as a Koiter-Vlasov moment shell with kink lines of the surface at the junctions of the pipe segments was constructed and substantiated. The following tasks are solved: The geometric parameters of the mechanical system as a three-dimensional elastic body and as a shell are found; force factors of the shell are expressed in terms of displacements of the middle surface of the wall, taking into account the presence of a kink line; equations of pipe equilibrium are derived as Koiter-Vlasov shells with an edge along the line; forces on oblique sections are expressed as functions of shell movements; the conjugation conditions on the pipe joint line for displacements and the angle of rotation of the normal are imposed and justified; conjugation conditions for bending moments, shear forces, transverse and normal forces are imposed and justified. The presence of the solution singularity at points on the connection line of the pipe segments is theoretically established and illustrated by the numerical example.     

Development of a higher order finite element on a Winkler foundation

Analyzing thick plates as a construction component has been of interest to structural engineering research for several decades. In particular, thick plates resting on elastic foundations are more specific. Mindlin's plate theory for thick plate analysis and the Winkler theory for elastic foundation analyses have wide applications. The current research considers analysis of isotropic plates on a Winkler foundation according to Mindlin's plate theory. The analysis uses a higher order plate element to avoid shear locking phenomena in the plate. The main features of this element are representation of real displacement functions of the plate perfect and shear locking do not occur at the plates modeled with this element. Derivation of the equations for finite element formulation for thick plate theory uses fourth-order displacement shape functions. A computer program using the finite element method, coded in C++, analyzes the plates resting on an elastic foundation. The analysis involves a 17-noded finite element. The study's graphs and tables assist engineers' designs of thick plates resting on elastic foundations. The study concludes with the computer-coded program, which allows effective use for the shear locking-free analysis of thick Mindlin plates resting on elastic foundations.     

A novel dynamics model for railway ballastless track with medium-thick slabs

Motivated by the requirements for elaborated slab ballastless track dynamics analysis in practical engineering application, a novel dynamic model for the railway ballastless tracks with medium-thick slabs is proposed in this work based on the Reissner–Mindlin plate theory, and it is implemented into the coupled dynamics analysis of a vehicle and the ballastless track. First, an efficient and easily programmable computational algorithm is adopted to solve the transverse deflection of the Reissner–Mindlin plate, in which the displacements and shear strains are chosen as the independent variables and subsequently constructed by spline functions, resulting in no shear-locking effect. The involved partial differential equations are transformed into ordinary ones by using the energy variation principle. Further, a mathematical model for the ballastless track dynamics analysis is established, which can consider the effects of the shear deformation and moment of inertia involved in the medium-thick track slab. Experimental verification and comparative analysis with other models demonstrate the accuracy and efficiency of the proposed model. Finally, a spatially coupled dynamics model of a vehicle and the ballastless track is developed, and it is efficiently solved by using the hybrid explicit-implicit time integration method. Compared with the widely used modelling the track slab by elastic thin plate, the reliability and advantages of the proposed vehicle-slab track coupled dynamics model are demonstrated.     

A practical fractional numerical optimization method for designing economically and environmentally friendly super-tall buildings

Due to the complexity of super tall buildings, many well-known optimization algorithms are not well applicable. Using structural lateral system of super tall buildings as engineering background, the paper developed a practical fractional numerical optimization method (FNOM), which applies fractional strategy and quasi-constant assumption, to reduce material cost and embodied carbon cost by searching the optimal structural dimensions. Firstly, two kinds of relationships among optimization variables (structural dimensions), driven design constraints (the interstory drift and the natural period) and optimization objective (cost including material cost and embodied carbon cost) are mathematically modelled. Genetic algorithm (GA) is then introduced to search the optimal structural dimensions based on the quasi-constant assumption of virtual work and internal work of the inactive components. Thirdly, fractional strategy is applied to create assemblies composed of different structural component sets, and the assemblies are then to be optimized in proper sequences. Fourthly, FNOM is implemented as a user-friendly software called C-FNO to practically support the preliminary design of super-tall buildings. Finally, a 700 m high super-tall building is employed to illustrate FNOM by using C-FNO, and the results show that only three design constraints of the interstory drift, the natural period and the stress ratio need to be solved during each optimization step. Belt truss, mega column, outrigger truss and shear wall of the super tall building should be optimized in sequence to save more cost. A great amount of cost can be still saved for the super tall building with the normal traditional design.     

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, a linear size-dependent Timoshenko beam model based on the consistent couple stress theory is developed to capture the size effects. The extended Hamilton's principle is utilized to obtain the governing differential equations and boundary conditions. The general form of boundary conditions and the concentrated loading are employed to determine the exact static/dynamic solution of the beam. Utilizing this solution for the beam's deformation and rotation, the exact shape functions of the consistent couple stress theory (C-CST) is extracted, which leads to the stiffness and mass matrices of a two-node C-CST finite element beam. Due to the complexity and high computational cost of using the exact solution's shape functions, in addition to the Ritz approximate solution, a two primary variable finite element model of C-CST is proposed, and the corresponding general deformation and rotation fields, shape functions, mass and stiffness matrices are calculated. The C-CST is validated by comparing the prediction of different beam models for a benchmark problem. For the fully and partially clamped cantilever, and free-free beams, the size dependency of the formulations is investigated. The static solutions of the classical and consistent couple stress Timoshenko beam models are compared, and a criterion for selecting the proper model is proposed. For a wide range of material properties, the relation between the beam length and length scale parameter is derived. It is shown that the validity domain of the consistent couple stress Timoshenko model barely depends on the beam's constituent material.     

Effect of memory dependent derivative on forced transverse vibrations in transversely isotropic thermoelastic cantilever nano-Beam with two temperature

The present research deals with the study of forced vibrations in transversely isotropic thermoelastic (TIT) nanoscale beam with two temperature (2T). Memory dependent derivative theory of thermoelasticity for clamped-free/cantilever nano-beam has been considered. The mathematical model is prepared for the nanoscale beam in a closed form with the application of Euler Bernoulli beam theory. Laplace transform method is employed to solve the problem. Forced vibrations due to exponential decaying time varying load acting vertically downward along the thickness direction of the nano-beam, Uniform load, Time harmonic load have been considered. Dynamic analysis for these forced vibrations and Static analysis has been carried out in this research. The dimensionless expressions for lateral deflection, thermal moment, temperature change, and axial stress are solved for these three forced vibrations. Response ratio has also been calculated. The analytical results have been numerically analysed using programming in MATLAB. The effect of kernel function has been depicted graphically on the lateral deflection, thermal moment, temperature change, axial stress and response ratio for all the three types of forced vibrations. Some particular cases have also been discussed.     

宾汉流体稠密两相湍流流动中的二阶矩模型

建立的Bingham流体稠密两相流动的二阶矩-颗粒动力论湍流模型（USM-theta模型）既体现了两相的作用,又体现了屈服应力所引起的附加项,并提出了USM-theta模型下考虑浓度修正值影响的两相湍流流动的算法．利用该模型对圆管内Bingham流体的单相湍流流动、稠密液固两相的湍流流动进行了计算,并和五方程湍流模型进行了比较,结果表明该模型的预测效果更好．利用USM-theta模型对含颗粒的Bingham流体的两相湍流流动进行了模拟,随着屈服应力的增加,Bingham流体相与颗粒相在管道中心附近的主流速度减小．液固两相湍流和Bingham流体两相湍流的计算结果表明屈服应力引起的附加项对流动有很重要的影响．     

A Spatial Geometric Nonlinearity Spline Beam Element With Nodal Parameters Containing Strains北大核心CSCD

Many slender rods in engineering can be modeled as Euler-Bernoulli beams. For the analysis of their dynamic behaviors, it is necessary to establish the dynamic models for the flexible multi-body systems. Geometric nonlinear elements with absolute nodal coordinates help solve a large number of dynamic problems of flexible beams, but they still face such problems as shear locking, nodal stress discontinuity and low computation efficiency. Based on the theory of large deformation beams’ virtual power equations, the functional formulas between displacements and rotation angles at the nodes were established, which can satisfy the deformation coupling relationships. The generalized strains to describe geometric nonlinear effects in this case were derived. Some parameters of boundary nodes were replaced by axial strains and sectional curvatures to obtain a more accurate and concise constraint method for applying external forces. To improve the numerical efficiency and stability of the system’s motion equations, a model-smoothing method was used to filter high frequencies out of the model. The numerical examples verify the rationality and effectiveness of the proposed element. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Non-linear oscillations of a thin-walled composite beam with shear deformation

A geometrically non-linear theory is used to study the dynamic behavior of a thin-walled composite beam. The model is based on a small strain and large rotation and displacements theory, which is formulated through the adoption of a higher-order displacement field and takes into account shear flexibility (bending and warping shear). In the analysis of a weakly nonlinear continuous system, the Ritz’s method is employed to express the problem in terms of generalized coordinates. Then, perturbation method of multiple scales is applied to the reduced system in order to obtain the equations of amplitude and modulation. In this paper, the non-linear 3D oscillations of a simply-supported beam are examined, considering a cross-section having one symmetry axis. Composite is assumed to be made of symmetric balanced laminates and especially orthotropic laminates. The model, which contains both quadratic and cubic non-linearities, is assumed to be in internal resonance condition. Steady-state solution and their stability are investigated by means of the eigenvalues of the Jacobian matrix. The equilibrium solution is governed by the modal coupling and experience a complex behavior composed by saddle noddle, Hopf and double period bifurcations.     

Analysis of adiabatic shear bands in thermo-elasto-viscoplastic materials by using piece-wise discontinuous basis functions

An adiabatic shear band (ASB) is a narrow region of intense plastic deformation that forms when some metallic alloys and some polymers are deformed at high strain rates and there is not enough time for the heat generated by plastic deformations to diffuse away. The study of ASBs is important because an ASB is a precursor to shear/ductile fractures. Initial-boundary-value problems simulating the initiation and propagation of an ASB have been analyzed usually using the finite element method (FEM). Because of the large plastic strains involved, the FE mesh needs to be refined several times to delineate the ASB width. Each refinement requires, in turn, interpolation of data from the previous mesh to the new one which causes a smoothening of the sharp gradients of the deformation fields, and affects characteristics of the ASB. In this paper, we propose the application of the finite element method with piecewise discontinuous basis functions for studying the occurrence of ASBs in simple shearing deformations of a body composed of an isotropic and homogeneous thermo-elastoviscoplastic material. The mathematical model of the problem is defined by a system of coupled nonlinear partial differential equations and an inequality constraint associated with the plastic strain rates admissibility.     

Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: Non-transforming spectral element analysis

Engineering systems, such as rolled steel beams, chain and belt drives and high-speed paper, can be modeled as axially translating beams. This article scrutinizes vibration and stability of an axially translating viscoelastic Timoshenko beam constrained by simple supports and subjected to axial pretension. The viscoelastic form of general rheological model is adopted to constitute the material of the beam. The partial differential equations governing transverse motion of the beam are derived from the extended form of Hamilton's principle. The non-transforming spectral element method (NTSEM) is applied to transform the governing equations into a set of ordinary differential equations. The formulation is similar to conventional FFT-based spectral element model except that Daubechies wavelet basis functions are used for temporal discretization. Influences of translating velocities, axial tensile force, viscoelastic parameter, shear deformation, beam model and boundary condition types are investigated on the underlying dynamic response and stability via the NTSEM and demonstrated via numerical simulations.     

A flexible model and efficient solution strategies for discrete location problems

Flexible discrete location problems are a generalization of most classical discrete locations problems like p-median or p-center problems. They can be modeled by using so-called ordered median functions. These functions multiply a weight to the cost of fulfilling the demand of a customer, which depends on the position of that cost relative to the costs of fulfilling the demand of other customers.In this paper a covering type of model for the discrete ordered median problem is presented. For the solution of this model two sets of valid inequalities, which reduces the number of binary variables tremendously, and several variable fixing strategies are identified. Based on these concepts a specialized branch & cut procedure is proposed and extensive computational results are reported.     

Transmuted lower record type inverse rayleigh distribution: estimation,characterizations and applications

This study introduces a new lifetime distribution called the transmuted lower record type inverse Rayleigh which extends the inverse Rayleigh distribution and has the potential to model the recovery times of Covid-19 patients.The new distribution is obtained using the distributions of the first two lower record statistics of the inverse Rayleigh distribution. We discuss some statistical inferences and mathematical properties of the suggested distribution. We examine some characteristics of the proposed distribution such as density shape, hazard function,moments, moment generating function, incomplete moments,Rényi entropy, order statistics, stochastic ordering. We consider five estimation methods such as maximum likelihood, least squares, weighted least squares, Anderson-Darling, Cramér-von Mises for the point estimation of the proposed distribution. Then, a comprehensive Monte Carlo simulation study is carried out to assess the risk behavior of the examined estimators. We provide two real data applications to illustrate the fitting ability of the proposed model, and compare its fit with competitor ones. Unlike many previously proposed distributions, the introduced distribution in this paper has modeled the recovery times of Covid-19 patients.

     

The maximum capture problem with random utilities: Problem formulation and algorithms

A model for the optimal location of new facilities in a competitive market is introduced under the hypothesis that customers' behavior can be modeled by random utility functions. It means that the company, that wished to locate, uses a random utility model to forecast the market share of a location. Therefore the company cannot forecast the behavior of every customer in a deterministic fashion, but is able to embed him by a probability distribution. Three formulations are proposed to compute upper bounds of the objective function and compared in a numerical simulation. A branch and bound method is developed and tested on examples with up to 50 potential locations, and a Variable Neighborhood Search heuristic is proposed to solve larger instances.     

Constitutive equations of non-isothermal polymer melt

Constitutive equations of non-isothermal polymer melt are presented by the analysis of entropic free energy contribution of the macromolecular chains, which are treated as elastic dumbbell models. With describing non-isothermal dumbbell spring, as the function of temperature, the non-linear elastic coefficient expression causes the appearance of temperature gradient in stress constitutive equations. Following the constitutive equation of Hookean dumbbell model, non-isothermal stress constitutive equations of FENE and FENE-P models are derived. In deriving process of constitutive equations, the second moment approximation is used to closure FENE model. Using the non-isothermal constitutive equations, numerical simulations of polymer flow through shear cavity and planar contraction cavity are presented. And the distributions of correlative stress functions and the effects of different temperatures on stress functions are discussed. The present results are shown to explore the non-isothermal constitutive equations of elastic dumbbell models, and to search more accurately describing way of non-isothermal polymer melt.