Large deflections of a cantilever beam subjected to a rotational distributed loading

Large deflection problems of a uniform cantilever beam under a rotational distributed loading are formulated by means of a second order nonlinear integro-differential equation. The problem is numerically solved by considering a uniform rotational distributed load and a linearly varying rotational distributed load along the span of the beam. The details of load deflection curves are presented. Assuming Dirac delta function as a load distribution function along the span of the beam, the present general formulation yields the solution for the problem of a uniform cantilever beam with end rotational concentrated load. The numerical results for this case are found to be in good agreement with existing closed form solutions. As the formulation is general, the problem with nonuniform rotational distributed load of any complexity can be solved following the present numerical procedure which is quite simple, accurate and involves less computational time.     

Multi-Disciplinary Optimization for Multi-Objective Uncertainty Design of Thin Walled Beams

The focus of this paper is concentrated on multi-disciplinary and multi-objective optimization for thin walled beam systems considering safety, normal mode, static loading-bearing and weight, in which the uncertainties of the parameters are described via intervals. The size and shape of the cross-section are treated as design parameters during optimization. Considering the lightweight and safety, the design problem is formulated with two individual objectives to measure structural weight and maximum energy absorption, respectively, constrained by the average force, normal mode and maximum stress. The optimization problem with uncertainties is further transformed into a deterministic optimization based on interval number programming. The approximation models, coupled with the design of experiment (DOE) technique, are employed to construct objective functions and constraints. The uncertain optimization problem characterized with these approximation models is performed and applied to a practical thin walled beam design problems.     

An uncertain multidisciplinary design optimization method using interval convex models

This article proposes an uncertain multi-objective multidisciplinary design optimization methodology, which employs the interval model to represent the uncertainties of uncertain-but-bounded parameters. The interval number programming method is applied to transform each uncertain objective function into two deterministic objective functions, and a satisfaction degree of intervals is used to convert both the uncertain inequality and equality constraints to deterministic inequality constraints. In doing so, an unconstrained deterministic optimization problem will be constructed in association with the penalty function method. The design will be finally formulated as a nested three-loop optimization, a class of highly challenging problems in the area of engineering design optimization. An advanced hierarchical optimization scheme is developed to solve the proposed optimization problem based on the multidisciplinary feasible strategy, which is a well-studied method able to reduce the dimensions of multidisciplinary design optimization problems by using the design variables as independent optimization variables. In the hierarchical optimization system, the non-dominated sorting genetic algorithm II, sequential quadratic programming method and Gauss–Seidel iterative approach are applied to the outer, middle and inner loops of the optimization problem, respectively. Typical numerical examples are used to demonstrate the effectiveness of the proposed methodology.     

Shear deformation effect in flexural–torsional vibrations of beams by BEM

In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.     

SPACE–TIME SPECTRAL ELEMENT METHOD FOR OPTIMAL SLEWING OF A FLEXIBLE BEAM

A new computational approach to modelling and control of a flexible beam is proposed. The structural modelling and the control design problems are formulated in a unified mathematical framework that allows simultaneous structural and control design iterations that result in an optimal overall system performance. The method employs the space–time spectral elements for simultaneous space and time discretizations of a Timoshenko beam model. Dimensionless equations of motion are derived using Hamilton's principle of variable action and an integral formulation in the framework of space–time spectral elements is introduced. An optimal control problem formulated for the continuum model is transformed by the space–time spectral element formulation into an optimization problem in a finite-dimensional parameter space. Dynamic programming is then used to obtain both open and closed loop control laws. A simulation study shows good performance of the control law applied to the nominal model. It is also demonstrated that proper discretization yields performance robustness of the system with respect to modal truncation.     

A firefly algorithm for optimum design of new-generation beams

This research addresses the minimum weight design of new-generation steel beams with sinusoidal openings using a metaheuristic search technique, namely the firefly method. The proposed algorithm is also used to compare the optimum design results of sinusoidal web-expanded beams with steel castellated and cellular beams. Optimum design problems of all beams are formulated according to the design limitations stipulated by the Steel Construction Institute. The design methods adopted in these publications are consistent with BS 5950 specifications. The formulation of the design problem considering the above-mentioned limitations turns out to be a discrete programming problem. The design algorithms based on the technique select the optimum universal beam sections, dimensional properties of sinusoidal, hexagonal and circular holes, and the total number of openings along the beam as design variables. Furthermore, this selection is also carried out such that the behavioural limitations are satisfied. Numerical examples are presented, where the suggested algorithm is implemented to achieve the minimum weight design of these beams subjected to loading combinations.     

基于灰色关联分析法的线切割机床模块化设计评价研究

目的 为了客观地评价线切割机床外观设计的设计重点,提升实际操作的满意度、操作效率而展开研究。方法 采用灰色关联分析法（GRA）,通过一系列的设计调研、研究分析线切割机床的设计特点和操作存在的问题,得出机床功能模块区域的评价指标,并对各指标进行灰色关联度的定量计算和排序,从而找出机床外观设计的重点功能区域为人机交互的操控区域,在此基础上对操控区域进行重新规划和设计,同时也对机床的其他功能模块区域进行优化设计。最后利用灰色关联系数求得各评价指标的权重,通过模糊综合评价法（FCE）对最终的设计效果进行满意度评价。结论 采用模块化设计评价方法对机床进行设计,优化和重新设计的功能模块区域,可以提高用户体验的整体满意度,能有效解决在实际操作过程中的问题,为线切割机床的外观设计提供客观的数据支撑与参考。     

A superconducting secondary beam line in the 12 GeV proton synchrotron at KEK

Design and performance of a superconducting high energy secondary beam line, π1, are reported. The π1 beam line was constructed to transport an 8 GeV/c unseparated secondary beam to the experimental area of the 12 GeV proton synchrotron at KEK. The beam line has been operated for five years without any serious problems and was shut down in the summer of 1986. We describe its design, construction and operational results.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

A Displacement Solution to Transverse Shear Loading of Composite Beams by BEM

In this paper the boundary element method is employed to develop a displacement solution for the general transverse shear loading problem of composite beams of arbitrary constant cross section. The composite beam (thin or thick walled) consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross section, while its axes are not necessarily the principal bending ones. The transverse shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of a warping function. The shear deformation coefficients are obtained from the solution of two boundary value problems with respect to warping functions appropriately arising from the aforementioned one using only boundary integration, while the coordinates of the shear center are obtained from these functions using again only boundary integration. Three boundary value problems are formulated with respect to corresponding warping functions and solved employing a pure BEM approach. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the obtained values of the resultant transverse shear stresses compared with those obtained from an exact solution is remarkable.     

Analysis of plates stiffened by parallel beams

In this paper a general solution for the analysis of plates stiffened by parallel beams subjected to an arbitrary loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. The aforementioned integrated tractions result in the loading of the beams as well as the additional loading of the plate. Their distribution is established by applying continuity conditions in all directions at the interfaces. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second‐order effects. Six boundary value problems with respect to the plate transverse deflection, to the plate inplane displacement components, to the beam transverse deflections, to the beam axial deformation and to the beam non‐uniform angle of twist are formulated and solved using the analog equation method (AEM), a boundary element method (BEM)‐based method employing a boundary integral equation approach. The solution of the aforementioned plate and beam problems, which are non‐linearly coupled, is achieved using iterative numerical methods. The adopted model describes better the actual response of the plate beams system and permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The evaluated lateral deflections of the plate–beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. Copyright © 2006 John Wiley & Sons, Ltd.     

Fully coupled non‐linear analysis of piezoelectric solids involving domain switching

Domain switching is the cause of significant non‐linearity in the response of piezoelectric materials to mechanical and electrical effects. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The constitutive equations are non‐linear. Moreover, due to the domain switching phenomenon, the resulting governing equations become highly non‐linear. The corresponding non‐linear finite element equations are derived and solved by using an incremental technique. The developed formulation is first verified against a number of benchmark problems for which a closed‐form solution exists. Next, a cantilever beam made of PZT‐4 is studied to evaluate the effect of domain switching on the overall force–displacement response of the beam. A number of interesting observations are made with respect to the extent of non‐linearity and its progressive spread as the load on the beam increases. Copyright © 2002 John Wiley & Sons, Ltd.     

Parametric Collaborative Design of Network-based Mechanical Products

Aiming at the problems of low-level information sharing,slow transmission and repetitive work in the designing process of product series,the internet-oriented parametric collaborative design method is proposed,in which the problems of sharing conflict and network heterogeneous in the distributed collaborative design are analyzed,and the construction method of collaborative design platforms based on PDMWorks Workgroup is put forward. Through studying the mechanism of roles distribution and function allocation and data concurrency control,the communication mechanism of internet-oriented collaborative design is formulated. On the basis of structure features of overhead travelling crane,through combining parametric variant design with collaborative design,internet-oriented parametric collaborative design system of overhead travelling crane is developed and verified through main girder design. In the paper,the internet-oriented parametric collaborative design method is proposed,aiming to solve the problems of low-level information sharing,slow transmission and repetitive work in the designing process of product series. The problems of sharing conflict and network heterogeneous in the distributed collaborative design are analyzed. The construction method of collaborative design platforms based on PDMWorks Workgroup is put forward. The communication mechanism of internet-oriented collaborative design is formulated,through studying the mechanism of roles distribution and function allocation and data concurrency control. On the basis of structure features of overheadtravelling crane,through combining parametric variant design with collaborative design,internet-oriented parametric collaborative design system of overhead travelling crane is developed and verified through main girder design.     

Generalized Timoshenko modelling of composite beam structures: sensitivity analysis and optimal design

This article describes a new approach to design the cross-section layer orientations of composite laminated beam structures. The beams are modelled with realistic cross-sectional geometry and material properties instead of a simplified model. The VABS (the variational asymptotic beam section analysis) methodology is used to compute the cross-sectional model for a generalized Timoshenko model, which was embedded in the finite element solver FEAP. Optimal design is performed with respect to the layers’ orientation. The design sensitivity analysis is analytically formulated and implemented. The direct differentiation method is used to evaluate the response sensitivities with respect to the design variables. Thus, the design sensitivities of the Timoshenko stiffness computed by VABS methodology are imbedded into the modified VABS program and linked to the beam finite element solver. The modified method of feasible directions and sequential quadratic programming algorithms are used to seek the optimal continuous solution of a set of numerical examples. The buckling load associated with the twist–bend instability of cantilever composite beams, which may have several cross-section geometries, is improved in the optimization procedure.     

Hybrid neural network model for the design of beam subjected to bending and shear

There is no direct method for design of beams. In general the dimensions of the beam and reinforcement are initially assumed and then the interaction formula is used to verify the suitability of chosen dimensions. This approach necessitates few trials for coming up with an economical and safe design. This paper demonstrates the applicability of Artificial Neural Networks (ANN) and Genetic Algorithms (GA) for the design of beams subjected to moment and shear. A hybrid neural network model which combines the features of feed forward neural networks and genetic algorithms has been developed for the design of beam subjected to moment and shear. The network has been trained with design data obtained from design experts in the field. The hybrid neural network model learned the design of beam in just 1000 training cycles. After successful learning, the model predicted the depth of the beam, area of steel, spacing of stirrups required for new problems with accuracy satisfying all design constraints. The various stages involved in the development of a genetic algorithm based neural network model are addressed at length in this paper.     

Closed form equations for FRP flexural strengthening design of RC beams

FRP external strengthening has proven its success in structural rehabilitation and upgrade. Researchers and practicing engineers are working towards introducing its design procedures to standard codes of practice. The current state of the art flexural design process suggests an iterative approach, which may lead to tedious calculations. This paper provides a direct approach furnishing calculation simplicity and design efficiency. It also proposes equations to design doubly strengthened sections for the first time. The expressions derived for doubly reinforced rectangular and Tee sections are written in a compact form based on those formulated for singly reinforced rectangular sections. Verifications against experimental results are performed. The solution, using the closed form equations, is compared to that of other procedures available in the literature through design examples. Tee section design is also presented and illustrated through a comparison with an analysis example. A doubly strengthened beam design example is also solved. The prevention of premature FRP plate separation using U-wraps to develop the full flexural capacity is also discussed.     

Homotopy methods for constraint relaxation in unilevel reliability based design optimization

In reliability based design optimization, a methodology for finding optimized designs characterized with a low probability of failure the main objective is to minimize a merit function while satisfying the reliability constraints. Traditionally, these have been formulated as a double-loop (nested) optimization problem, which is computationally intensive. A new efficient unilevel formulation for reliability based design optimization was developed by the authors in earlier studies, where the lower-level optimization was replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper-level optimization and imposed as equality constraints. But as most commercial optimizers are usually numerically unreliable when applied to problems accompanied by many equality constraints, an optimization framework for reliability based design using the unilevel formulation is developed here. Homotopy methods are used for constraint relaxation and to obtain a relaxed feasible design and heuristic scheme is employed to update the homotopy parameter.     

Foundations for low-loss fiber gradient-index lens pair coupling with the self-imaging mechanism

A fiber-optic collimator that emits a Gaussian beam with its beam waist at a certain distance after the exit face of the lens is labeled a self-imaging collimator. For such a collimator, the waist of the emitted Gaussian beam and its location are partly dependent on the properties of the gradient-index (GRIN) lens. Parameters for the self-imaging collimator are formulated in terms of the parameters of a GRIN lens (e.g., pitch, core refractive index, gradient index, length) and the optical wavelength. Next, by use of the Gaussian beam approximation, a general expression for the coupling power loss between two self-imaging-type single-mode fiber (SMF) collimators is, for the first time to our knowledge, derived as a function of three types of misalignment, namely, separation, lateral offset, and angular tilt misalignment. A coupling experiment between two self-imaging collimators with changing separation distance is successfully performed and matches the proposed self-imaging mechanism coupling loss theory. In addition, using a prism, lateral offset, as well as angular tilt, misalignments are experimentally simulated for a two self-imaging collimator coupling condition by a single collimator reflective test geometry. Experimental results agree well with the proposed loss formulas for self-imaging GRIN lenses. Hence, for the first time to our knowledge, the mathematical foundations are laid for employing self-imaging-type fiber collimators in SMF-based free-space systems allowing optimal design for ultra-low-loss coupling.     

Structural optimization of non-linear systems under stochastic excitation

This work concentrates on the structural optimization of a class of non-linear systems with deterministic structural parameters subject to stochastic excitation. The optimization problem is formulated as the minimization of an objective function subject to constraints on the response level. The stochastic response is characterized by its first two statistical moments, which are computed by a statistical equivalent linearization technique. The implicit structural optimization problem is replaced by a sequence of explicit sub-optimization problems. The sub-problems are constructed by using a conservative first-order approximation of the objective and constraint functions. The applicability of the proposed design process is demonstrated in three numerical examples where the methodology is applied to systems with nonlinearity of hardening and hysteretic type. The effects of the nonlinearity on the general performance of the final designs are discussed. At the same time, some engineering implications of the results obtained in this work are addressed.     

Advanced 3D beam element of arbitrary composite cross section including generalized warping effects

In this paper, an advanced 20 × 20 stiffness matrix and the corresponding nodal load vector of a member of arbitrary composite cross section is developed taking into account shear lag effects due to both flexure and torsion (the case of the three‐dimensional beam element of arbitrary homogeneous cross section is treated as a special one). The composite member consists of materials in contact each of which can surround a finite number of inclusions. Nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions. Ten boundary value problems with respect to the kinematical components are formulated and solved employing the analog equation method, a BEM‐based technique. The aforementioned boundary value problems are formulated employing either an improved stress field arising from the correction of shear stress components or the stress field arising directly from displacement considerations. The warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure BEM approach, that is, only boundary discretization of the cross section is used. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The deviations arising from the use of the advanced 20 × 20 stiffness matrix and the classical 12 × 12 or 14 × 14 ones employed in commercial software packages are illustrated through examples of great practical interest. Moreover, the influence of nonuniform warping effects necessitating the use of the aforementioned ‘advanced’ stiffness matrix is also demonstrated. Copyright © 2015 John Wiley & Sons, Ltd.