On Optimality Conditions for Trusses with Nonuniform Stress Constraints*

ABSTRACT

The optimal design of a truss subjected to a single loading system and stress constraints, which are not necessarily the same in each bar, is considered. Sufficient conditions for global optimality are derived by variational methods. While these optimality criteria lead to a linear programming formulation of the problem, they show in a clear physical way how the optimal design is found, and that advantages accrue from incorporating the optimality criteria in a numerical scheme.     

A Dual Approach to Optimum Truss Design

Abstract

This paper considers the problem of finding member sizes which minimize the weight of a pin-jointed truss of fixed geometry while satisfying constraints upon joint displacements, member stresses, and minimum sizes. Aspects of both mathematical programming methods and optimality criteria methods for designing large trusses are discussed. The optimality criteria approach is further extended and the whole truss design problem is recast into a new dual formulation in which constraint activity levels are used as variables in a mathematical programming solution method. This new dual formulation unifies both the optimality criteria and mathematical programming approaches to the problem of truss design. The paper is theoretical in nature, being largely devoted to a proof of the dual method. A discussion of the likely implications and usefulness of the dual approach to truss design is given with comments upon its possible modes of practical use.     

Optimal Shape Design of Multicomposite Structures

Abstract

The problem of the optimal shape design of an elastic composite structure with unspecified interfaces is discussed for the case of mean compliance constraint. The optimality conditions are derived, and the optimization of a circular plate with segmentwise constant thickness is considered in order to illustrate these conditions. The case of optimal plastic design is obtained as a limiting case of mean compliance design.     

The solution of viscous incompressible jet flows using non‐staggered boundary fitted co‐ordinate methods

A new approach for the solution of the steady incompressible Navier–Stokes equations in a domain bounded in part by a free surface is presented. The procedure is based on the finite difference technique, with the non‐staggered grid fractional step method used to solve the flow equations written in terms of primitive variables. The physical domain is transformed to a rectangle by means of a numerical mapping technique. In order to design an effective free solution scheme, we distinguish between flows dominated by surface tension and those dominated by inertia and viscosity. When the surface tension effect is insignificant we used the kinematic condition to update the surface; whereas, in the opposite case, we used the normal stress condition to obtain the free surface boundary. Results obtained with the improved boundary conditions for a plane Newtonian jet are found to compare well with the available two‐dimensional numerical solutions for Reynolds numbers, up to Re=100, and Capillary numbers in the range of 0≤Ca<1000. Copyright © 2001 John Wiley & Sons, Ltd.     

Optimal Design of Thin-Walled Cylinders of Variable Wall-Thickness Subject to Flexural and Torsional Stiffness Constraints

ABSTRACT

This paper considers the problem of determining center-line shape and wall-thickness distribution of a thin-walled cylinder of given center-line length that uses the minimum possible amount of material to achieve prescribed minimum stiffnesses in torsion and bending in a given plane. Necessary optimality conditions are derived and the solution is found partly in closed form and partly by numerical methods. Optimal solutions are presented for various stiffness ratios and compared with other admissible designs.     

Generalization of an Energetic Optimality Condition for Nonconservative Systems

ABSTRACT

An optimality condition based on strain energy densities, derived by earlier investigators, has been generalized. The generalized condition for an optimum structure is in bilinear form. The concept of the adjoint system is utilized. The resulting condition is applied to the minimization of the mass of a column loaded by a tangential follower force. Some results are presented.     

Application of Laser Deposition to Mechanical Characterization of Advanced High Strength Steels Subject to Non-Proportional Loading

Background

Characterization of hardening and fracture limits of advanced high strength steels (AHSSs) undergoing strain path changes (SPCs) are particularly challenging for plane strain condition, which commonly occurs in sheet metal forming. There is a need for a simple, engineering-friendly method to characterize materials subjected to complex loading paths that mimic stress conditions in actual forming processes.

Objective

Experimental additive manufacturing techniques have been applied to reinforce AHSS specimens subjected to SPCs in order to broaden capabilities for characterizing hardening behavior and fracture limits.

Methods

Hardening curves subject to SPCs (e.g. uniaxial tension or equi-biaxial tension followed by plane strain) have been obtained with a programmable biaxial tensile testing system using cruciform-shaped specimens with load-bearing arms reinforced by laser deposition. A notched specimen selectively reinforced by laser deposition was newly designed to characterize fracture limits subjected to SPCs ending with plane strain condition.

Results

Complex loading histories were successfully enabled by applying laser deposition technology. Results show that both hardening behavior and fracture limits of a TRIP-assisted steel and a dual-phase steel are dependent on loading history.

Conclusions

It appears that the laser deposition technique can be used for material characterization under specific SPCs. Hardening behavior of AHSSs under SPCs ending with plane strain is quite different from traditional uniaxial tension-uniaxial compression tests. For materials sensitive to SPCs, multi-step forming can be a great option to reach the targeted forming shape.

     

Simulation of the stochastic damage evolution process under low cycle fatigue by the finite element method

A numerical simulation of stochastic damage evolution process in the condition of low cycle fatigue loading is discussed. The relations between damage variables and micro-cracks are obtained by means of the micro-mechanics model of the representative volume element proposed by Lemaitre and Dufailly^[10]. The stochastic initial damage values are introduced in consideration of the inherent micro-defects in materials. The model combined with a finite element method is applied to simulate the damage evolution process under low cycle fatigue loading. The micro-cracks on the sur face of a specimen of 19Mn6 alloy steel are measured with a replica technique. The numerical results show that the nonhomogeneity of damage and the localization of the fatigue failure are well shown by the proposed simulations, and the fatigue lives are reasonably predicted.     

Basic Geometrical Properties of Optimal Flexural Force Transmission Fields

Abstract

On the basis of theories of optimal design by Masur, Prager and Shield, kinematic optimality conditions for elastic and elastic-plastic plane flexural systems of maximum strength and maximum stiffness were previously derived. In the present paper, general geometrical properties of moment and displacement fields in various types of optimal regions are outlined with a view to constructing optimal solutions for any arbitrary set of boundary conditions.     

Analytical Treatment of Some Extended Problems in Structural Optimization,Part I

ABSTRACT

Optimization problems involving supports of unspecified location as well as variable external actions and reactions of nonzero cost are discussed in the context of both plastic and elastic beams and frames, of non-preassigned, partially preassigned, and preassigned cost distribution, and of strength or deflection constraints.

The static-kinematic optimality criteria for various classes of problems are illustrated with simple examples and the results are checked by an independent method, i.e., by differentiation of the total cost with respect to the unknown variables. It is shown that for certain classes of problems the conditions introduced reduce to existing criteria by Prager and Masur.     

Optimal Design of Supports of Elastic Structures Subjected to Loads and Initial Distortions

ABSTRACT

A problem of optimal location, stiffness, and prestress of supports in an elastic frame structure that is subjected to external loads, displacements, and initial distortions is formulated. Sensitivity analysis and optimality conditions are discussed for the assumed objective functional, accounting for conflicting design requirements that correspond to stiffness and stress constraints. A simple example is presented that illustrates the general theory.     

基于拓展多尺度有限元的点阵材料结构最小柔顺性设计

本文应用拓展的多尺度有限元法（Extended Multiscale Finite Element Method),以微观构件的截面积为设计变量,研究了体积约束下点阵材料构成结构的最小柔顺性设计问题。建立了适应具有复杂几何形状和载荷边界的点阵材料结构的优化模型,应用序列二次规划算法对悬臂梁和L形梁算例在线性边界条件和周期性边界条件下进行了优化设计,讨论了点阵材料微结构尺寸效应对优化结果的影响,验证了优化模型和求解算法的可靠性,为点阵材料应用于复杂实际工程结构的优化设计提供了新的技术手段。     

Optimal Design of Viscoelastic Structures under Forced Steady-State Vibration

ABSTRACT

ABSTRACT With minimum dynamic response as the design criterion, we derive by means of variational analysis a general set of equations governing optimal design of one-dimensional, viscoelastic structures acted on by harmonically varying external loading. The equations are specialized to problems of minimizing transverse vibrational response of beams by attaching optimal, nonuniform cover layers made of a solid, viscoelastic material on the beams. Several numerical solutions to such problems are presented.     

Analysis of Damage and Failure in Anisotropic Ductile Metals Based on Biaxial Experiments with the H-Specimen

Background

Dependence of strength and failure behavior of anisotropic ductile metals on loading direction and on stress state has been indicated by many experiments. To realistically predict safety and lifetime of structures these effects must be taken into account in material models and numerical analysis.

Objective

The influence of stress state and loading direction on damage and failure behavior of the anisotropic aluminum alloy EN AW-2017A is investigated.

Methods

New biaxial experiments and numerical simulations have been performed with the H-specimen under different load ratios. Digital image correlation shows evolution of strain fields and scanning electron microscopy is used to visualize failure modes on fracture surfaces. Corresponding numerical studies predict stress states to explain damage and fracture processes on the micro-scale.

Results

The stress state, the load ratio and the loading direction with respect to the principal axes of anisotropy affect the width and orientation of localized strain fields and the formation of damage mechanisms and fracture modes at the micro-level.

Conclusions

The enhanced experimental program with biaxial tests considering different loading directions and load ratios is suggested for characterization of anisotropic metals.

     

Minimal loading conditions for higher-order numerical homogenisation schemes

The present study deals with the formulation of minimal loading conditions for microscale applications in numerical two-scale modelling (FE²) approaches. From the homogenisation concept, a set of volume average rules constrains the microscale PDE to be solved. They are considered to be the minimal set of loading conditions and can be specified by additional polynomial or periodic assumptions, for example, on the microscale displacement field. Whereas the resulting volume integrals can be transformed into surface integrals for so-called first-order homogenisation schemes, this is not possible for a second-order homogenisation of second gradient or micromorphic effective media substituting a heterogeneous microcontinuum represented by a volume element on the microscale. Several numerical examples compare the minimal loading condition concept with standard techniques discussed in literature.     

Limit Design of Frames with Axial Force-Bending Moment Interaction

ABSTRACT

A method for optimum plastic design of plane frames is studied, taking into account the axial force-bending moment interaction plastic behavior. The frame is regarded as a discrete model, formed by rigid elements which are separated by (generalized) plastic hinges. Alternative loading conditions are considered, in addition to the action of the design-dependent self-weight. The optimization problem is solved as a problem of linear programming into which some technological and construction requirements are also introduced. Optimality conditions are also discussed.     

Displacement Bounding Principles in the Dynamics of Elastoplastic Continua

Abstract

General bounds on the total displacements of structures subjected to any dynamic loading process are developed for an elastic perfectly plastic material. These bounds are subsequently extended to linear kinematic hardening materials. Previously developed bounds for the dynamics of impulsively loaded structures are recognized to be particular cases of the present formulation. A simple example indicates that the proposed technique is relatively easy to apply for numerical computations     

Finite Element Implementation of the Microplane Theory for Simulating a Rigid Concrete Pavement-Vehicle Interaction∗

Abstract

This paper presents an approach for modeling concrete pavement, based on the constitutive implementation of Bazant's microplane theory, for the purpose of predicting pavement response due to complex loading by vehicles. This includes implementation of the microplane theory in a three dimensional finite element code and verification of its numerical accuracy. The analytical method is then verified. The program's accuracy under simple static loading is verified by comparison with two of the most widely used pavement design codes. Experimental data from the literature are used to verify the approach developed for both cyclic response and prediction of material softening, a critical feature of the Portland Cement Concrete (PCC) concrete material used in pavement. The analysis is also verified against experimental influence function data for a single axle, Finally, the analytically predicted pavement response is verified for dynamic multi-axle truck loading. Based on agreement with experimental data, the model developed captures the essential characteristics of concrete pavement subjected to complex     

Closed-form Solutions in Optimal Design of Structures with Nonlinear Behavior

ABSTRACT

Optimal design problems for flexural systems with a nonlinear constitutive law are considered, in the presence of constraints on displacements. A general nonlinear holonomic moment-curvature relationship is assumed and a direct variational method is applied in order to obtain optimality criteria. Accordingly, a general method of solution is proposed and some examples are solved.