Optimal design of rigid-plastic structures under dynamic loading

The optimal design of rigid-plastic beams with piecewise constant thickness under impulsive or dynamic pressure loading is considered. Such dimensions of the structure for which the beam of constant volume attains a minimum of local or mean residual deflection are sought. The equations of motion are integrated (i) exactly and (ii) approximately by making use of the method of mode form solutions. The method of solution is applied also to reinforced beams.The stiffness of the beam can be increased if we provide it with additional supports. The locations of such supports must be chosen so as to minimize the global compliance of the beam. Optimality conditions for this problem are discussed.     

Simultaneous optimization of photostrictive actuator locations,numbers and light intensities for structural shape control using hierarchical genetic algorithm

The present paper introduces an investigation into simultaneous optimization of the PbLaZrTi-based actuator configuration and corresponding applied light intensity for morphing beam structural shapes. A finite element formulation for multiphysics analysis of coupled opto-electro-thermo-mechanical fields in PbLaZrTi ceramics is derived and verified with the theoretical solution and the commercial software ANSYS. This element is then used to simulate beam bending shape control using the orthotropic PbLaZrTi actuators and the simultaneous optimization. In this procedure, the controlling and geometrical variables are simultaneously optimized via a hierarchical genetic algorithm. A bi-coded chromosome is proposed in a hierarchical mode, which consists of some control genes (i.e. actuator location and number) and parametric genes (i.e. applied light intensity). Whether the parametric gene is activated or not is managed by the value of the first-grade control genes. The numerical results demonstrate that the achieved beam bending shapes correlate remarkably well with the expected ones and the simultaneous optimization of photostrictive actuator locations, numbers and light intensities can result in optimal actuator layout with less PbLaZrTi actuators and irradiated light energy. The simulation results also show that the hierarchical genetic algorithm has more superior performance over the conventional real-coded genetic algorithm.     

基于载荷试验的桥梁盖梁承载能力评价

针对某桥梁病害失效问题，对桥梁的外观和材质情况进行检测，对拟更换T梁处的盖梁结构进行安全评估。根据理论分析结果确定盖梁测点布置和加载工况，通过载荷试验分析偏载工况和正载工况下盖梁结构的挠度和应变结果。理论分析和试验结果均表明，该桥跨实际承载能力仍满足正常承载要求，可继续使用。     

Structural optimization under uncertain loads and nodal locations

This paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes. The second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements. We first review the discrete form of the uncertain loads problem, which has been previously solved using a weighted average of multiple load patterns. With minor modifications, we extend this solution to include loads described by continuous joint probability density functions. We then proceed to the main contribution of this paper: structural optimization under uncertainty in the nodal locations. This optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix. It is shown, however, that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads. By expressing these equivalent loads in terms of continuous random variables, we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper. The optimization algorithms are developed in the context of minimum compliance (maximum stiffness) design. Simple examples are presented. The results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design. For structures containing thin substructures under axial loads, it is shown that these uncertainties (a) are of first-order significance, influencing the linear elastic response quantities, and (b) can affect designs by avoiding unrealistically optimistic and potentially unstable structures. The additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem.     

Design of beams,plates and their elastic foundations for uniform foundation pressure

Beams and circular plates on elastic foundations are considered. In some cases, additional elastic supports are present. The stiffness distribution of the foundation is designed so that the pressure on the foundation is uniform. Sometimes the depth of the beam or plate is also varied, with either a piecewise-constant sandwich or solid cross-section, and a global measure of the deflection is minimized. The total stiffness of the foundation and supports is specified, as well as the volume of the structure. In one type of problem, the edges of the structure are displaced downwards; in the other examples, a downward load is applied. Types of loads include a concentrated central load, a uniform load and a parabolic load. The uniform foundation pressure for the resulting design is often substantially lower than the maximum pressure for a corresponding uniform beam or plate on an elastic foundation with uniform stiffness.Virginia Polytechnic Institute and State University, August–December 1989     

On the optimal design of elastic beams

In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of nonlinearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.     

Maximum allowable load of flexible manipulators for given dynamic trajectory

This paper presents the formulation and numerical solution of the dynamic load carrying capacity (DLCC) problem of flexible manipulators. For manipulators under the rigid body assumption, the major limiting factor in determining the maximum allowable load (load mass and load moment of inertia) for a prescribed dynamic trajectory (positions, velocities and accelerations) is the joint actuator capacity. But for a flexible robot, an additional constraint on allowable deformation at the end effector must be imposed because either lighter-weight links or operating at a higher speed could cause unacceptable fluctuations when moving along a trajectory. A Lagrangian assumed mode method was used to model the manipulator and load dynamics, including both joint and deflection motions. The deflection equations are then coupled with robot kinematics to solve for the generalized coordinates. A strategy to determine the DLCC subject to both constraints mentioned above is formulated where the end effector deflection is specified in terms of a series of spherical bounds with a radius equal to the allowable deformation. A general computational procedure for the multiple-link case given arbitrary trajectories is described in detail. Symbolic derivation and simulation by using a PC-based symbolic language MATHEMATICA® was carried out for a two-link planer robot. The results confirmed the necessity of the dual constraints and showed that which constraint is more critical for a given robot and trajectory depends on the required accuracy.     

Optimization of inelastic cylindrical shells with internal supports

A non-linear programming method is developed for optimization of inelastic cylindrical shells with internal ring supports. The shells under consideration are subjected to internal pressure loading and axial tension. The material of shells is a composite which is considered as an anisotropic inelastic material obeying the yield condition suggested by Lance and Robinson. Taking geometrical non/linearity of the structure into account optimal locations of internal ring supports are determined so that the cost function attains its minimum value. A particular problem of minimization of the mean deflection of the shell with weakened singular cross sections is treated in a greater detail.     

Multi-objective optimization of functionally graded materials,thickness and aspect ratio in micro-beams embedded in an elastic medium

Optimal design of micron-scale beams as a general case is an important problem for development of micro-electromechanical devices. For various applications, the mechanical parameters such as mass, maximum deflection and stress, natural frequency and buckling load are considered in strategies of micro-manufacturing technologies. However, all parameters are not of equal importance in each operating condition but multi-objective optimization is able to select optimal states of micro-beams which have desirable performances in various micro-electromechanical devices. This paper provides optimal states of design variables including thickness, distribution parameter of functionally graded materials, and aspect ratio in simply supported FG micro-beams resting on the elastic foundation using analytical solutions. The elastic medium is assumed to be as a two-layered foundation including a shear layer and a linear normal layer. Also, the size effect on the mechanical parameters is considered using the modified strain gradient theory and non-dominated sorting genetic algorithm-II is employed to optimization procedure. The target functions are defined such that the maximum deflection, maximum stress and mass must be minimized while natural frequency and critical buckling load must be maximized. The optimum patterns of FG micro-beams are presented for exponential and power-law FGMs and the effect of theory type and elastic foundation discussed in details. Findings indicate that the elastic foundation coefficients and internal length scale parameters of materials have the significant influences on the distribution of design variables. It is seen that the optimum values of inhomogeneity parameter and aspect ratio for E-FG micro-beams predicted by the modified strain gradient theory are larger than those of the classical continuum theory. Also, the multi-objective optimization is able to improve the normalized values of mass, maximum deflection, buckling load and natural frequency of P-FG micro-beams.     

Optimal shakedown design of beam structures

The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such conditions prove the equivalence of the two types of design problems, provide useful information on the structural behaviour in optimality conditions, and constitute a fifth possible way to determine the optimal design. Whatever approach is used, the strong non-linearity of the corresponding problem does not allow the finding of the analytical solution. Consequently, in the application stage suitable numerical procedures must be employed. Two numerical examples are given.     

Minimum weight design of symmetric angle-ply laminates under multiple uncertain loads

An angle-ply laminated plate is optimized with the objective of minimizing the weight of the plate taking into account uncertainties in the multiple transverse loads. The weight is proportional to the laminate thickness which is minimized subject to deflection and buckling constraints under the least favourable loading with the ply angles taken as design variables. The convex modelling approach is employed to analyse the uncertain loading with the uncertain quantities allowed to vary arbitrarily around their average values subject to the requirements that these variations are bounded inL ₂ norm and represented by a finite number of eigenmodes. The effect of uncertainty on the optimal design is investigated quantitatively. It is shown that the minimum weight increases with increasing level of uncertainty and the optimal ply angles also depend on the level of uncertainty.     

On multimodal optimization of circular arches against plane and spatial creep buckling

The paper deals with the optimal design of circular arches against creep buckling in the Rabotnov-Shesterikov sense. The design vairable, i.e. the cross-sectional area function, is determined so as to minimize the total volume of an arch under given external load (radial pressure) and the critical time. Multimodal formulation of the problem is introduced, i.e. both symmetric and antisymmetric instability modes are considered for in-plane and out-of-plane buckling. The effect of the behaviour of loading in the course of buckling on optimal shapes is analysed. The problem is solved by the use of Pontryagin's maximum principle.     

Design and static modeling of a semiconductor polymericpiezoelectric microactuator

The design of a semicircular polymeric piezoelectric actuator is presented, the models for its deflection and force are derived, and the results of the experiments used to verify the models are reported. The design is a polymeric piezoelectric bimorph formed in a semicircular shape instead of the standard straight cantilever beam bimorph design. The deflection model is adopted from the straight cantilever beam model and relates the free deflection as a function of the applied voltage field. The force model is developed using Castigliano's second theorem. Two experiments were conducted to verify the model: force-voltage and force-deflection. The models show that a bimorph formed in a semicircular shape produces significantly more force than a straight cantilever beam bimorph of the same length without a significant sacrifice in deflection     

Novel Optimal Design Approach for Output‐Feedback H∞ Control of Vehicle Active Seat‐Suspension System

In this paper, the output‐feedback control problem of a vehicle active seat‐suspension system is investigated. A novel optimal design approach for an output‐feedback H_∞ controller is proposed. The main objective of the controller is to minimize the seat vertical acceleration to improve vehicle ride comfort. First, the human body and the seat are considered in the modeling of a vehicle active suspension system, which makes the model more precise. Other constraints, such as tire deflection, suspension deflection and actuator saturation, are also considered. Then the output‐feedback control strategy is adopted since some state variables, such as body acceleration and body deflection, are unavailable. A concise and effective approach for an output‐feedback H_∞ optimal control is presented. The desired controller is obtained by solving the corresponding linear matrix inequalities (LMIs) and by the calculation of equations proposed in this paper. Finally, a numerical example is presented to show the effectiveness and advantages of the proposed controller design approach.     

Optimum integer number and position of several groups of prestressing tendons for given concrete dimensions

The following design problem is solved: Given are the concrete dimensions and the loadings (e.g. dead load plus traffic loading). The engineer can choose the number of groups of tendons he wishes to use. For each group the following data are given: the location of the cross-sections at which the tendons are anchored, the prestressing force of one tendon, which can also be variable along the length of the beam to take friction losses approximately into consideration, the maximum and minimum number of tendons, the lower and upper bounds of the zone permissible for guaranteeing sufficient concrete coverage (these bounds can also vary along the length of the beam), the smallest permissible radius of curvature, as well as a relative price. Several load combinations, under service conditions with specified allowable maximum and minimum concrete stresses are selected by the engineer. A reduction factor for creep and shrinkage is also included in the input.Determined are (a) the overall most favorable integer number of tendons in each group, and (b) the position along the length of the beams, such that the stress margin that results in every section and under the stress conditions formulated under the most unfavorable load positions shall be a maximum.The optimization algorithm employs in part a linear program. Methods to eliminate redundant constraints on the level of the cross-sections and the elements are discussed.The procedure is illustrated with a highway bridge (deck on raking legs), using three groups of tendons.     

Optimization of rigid-plastic shallow curved beams

The minimum weight problem of a shallow circular beam is studied in the case when the beam has a piece-wise constant thickness. The minimum of the weight is sought under the condition that the deflections of the beam of piece-wise constant thickness do not exceed those of the reference beam of constant thickness for given values of the external loading. The beam is subjected to uniformly distributed transverse pressure and to axial dead load. The material of the beam is assumed to be ideally rigid-plastic. Moderately large deflections are taken into account. Necessary optimality conditions are derived and used in order to establish the optimal values of the design parameters.     

Nonlinear optimal tracking control with application to super-tankers for autopilot design

A new method is introduced to design optimal tracking controllers for a general class of nonlinear systems. A recently developed recursive approximation theory is applied to solve the nonlinear optimal tracking control problem explicitly by classical means. This reduces the nonlinear problem to a sequence of linear-quadratic and time-varying approximating problems which, under very mild conditions, globally converge in the limit to the nonlinear systems considered. The converged control input from the approximating sequence is then applied to the nonlinear system. The method is used to design an autopilot for the ESSO 190,000-dwt oil tanker. This multi-input-multi-output nonlinear super-tanker model is well established in the literature and represents a challenging problem for control design, where the design requirement is to follow a commanded maneuver at a desired speed. The performance index is selected so as to minimize: (a) the tracking error for a desired course heading, and (b) the rudder deflection angle to ensure that actuators operate within their operating limits. This will present a trade-off between accurate tracking and reduced actuator usage (fuel consumption) as they are both mutually dependent on each other. Simulations of the nonlinear super-tanker control model are conducted to illustrate the effectiveness of the nonlinear tracking controller.     

VIBRATION CONTROL OF A SMART STRUCTURE USING PERIODIC OUTPUT FEEDBACK TECHNIQUE

Active vibration control is an important problem in structures. One of the ways to tackle this problem is to make the structure smart, adaptive and self‐controlling. The objective of active vibration control is to reduce the vibration of a system by automatic modification of the system's structural response. This work features the modeling and design of a Periodic Output Feedback (POF) control technique for the vibration control of a smart flexible cantilever beam system for a Single Input Single Output case. A POF controller is designed for the beam by bonding patches of piezoelectric layer as sensor/actuator to the master structure at different locations along the length of the beam. The entire structure is modeled in state space form using the Finite Element Method by dividing the structure into 3, 4, 5 elements, thus giving rise to three types of systems, viz., system 1 (beam divided into 3 finite elements), system 2 (4 finite elements), system 3 (5 finite elements). POF controllers are designed for the above three types of systems for different sensor/actuator locations along the length of the beam by retaining the first two vibratory modes. The smart cantilever beam model is developed using the concept of piezoelectric bonding and Euler‐Bernouli theory principles. The effect of placing the sensor/actuator at various locations along the length of the beam for all the three types of systems considered is observed and the conclusions are drawn for the best performance and for the smallest magnitude of the control input required to control the vibrations of the beam. The tip displacements with the controller is obtained. Performance of the system is also observed by retaining the first 3 vibratory modes and the conclusions are drawn.