A quasi-linear programming algorithm for optimising fibre-reinforced structures of fixed stiffness

This paper deals with the problem of optimising multilaminar, fibre-reinforced continua. The main constraint is that of fixed stiffness in the strict sense that deflections are fixed under given loads. It is shown that an algorithm can be proposed, based on the simplex method of linear programming, which solves without linearisation the nonlinear mixed-integer programming problem involved. Numbers of layers, their thicknessesm and their fibre-directions are all optimised simultaneously. Numerical results are presented, and the wider relevance of the restricted problem under consideration is discussed.     

Optimal design of laminated composite beam structures

This work deals with design sensitivity analysis and optimal design of composite structures modelled as thin-walled beams. The structures are treated as a torsion-bending resistant beams. The analysis problem is discretized by a finite element technique. A two-node Hermitean beam element is used. The beam sections are made from an assembly of elements that correspond to flat layered laminated composite panels. Optimal design is performed with respect to the lamina orientations and thickness of the laminates. The structural weight is considered as the objective function. Constraints are imposed on stresses, displacements, critical load and natural frequencies. Two failure criteria are used to limit the structural strength: Tsai-Hill and maximum stress. The Tsai-Hill criterion is also adopted to predict the first-ply-failure loads. The design sensitivity analysis is analytically formulated and implemented. An adjoint variable method is used to derive the response sensitivities with respect to the design. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional structures.     

The application of object-oriented programming to Monte Carlo experiments on beams of phonons in crystals

Recently there has been a lot of interest in experiments with beams of particles and quasi-particles. In particular, beams of phonons (sound quanta) have been used for inventing new types of solid-state spectroscopies. In most real-life experiments, the down-converting long-wavelength acoustic phonons propagate in crystalline media which contain scattering centres. This means that the physical problem considered is quite general and concerns unstable, elastically scattered particles and quasi-particles. The underlying physical problem is so complex that until now the only effective method for the study of such beams has been computer experiments. This paper discuss some aspect of the use of object-oriented programming (OOP) for designing programs of computer experiments on beams of phonons, but the present methods can be applied to quite a wide class of experiments on beams of particles and quasi-particles of the other kinds. The main ideas of OOP are presented and illustrated with examples. The support for OOP concepts that C++ programming language provides is discussed.     

Optimum dynamic design for beam shapes subjected to a moving concentrated load

We are concerned with an optimum dynamic design of beams subjected to a moving concentrated load with constant speed. The influence of the dynamic behaviour of the beams is considered in a proposed optimum design problem. The optimum shapes of beams are determined by the minimization of two kinds of performance indices. The optimization procedure is performed by non-linear programming on the basis of the exterior penalty function and BFGS methods. Optimization is calculated by the modal coordinates transformation and the numerical integration method     

A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint

In this paper, a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity. For various end boundary conditions, we transform the governing equation with varying coefficients to linear algebraic equations; then a characteristic equation in critical buckling loads will be obtained. Several examples of estimating buckling loads under typical end supports are discussed. By comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams, it can be found that our method has fast convergence and the obtained numerical results have high accuracy. Moreover, the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties. The effects of gradient parameters on the critical buckling loads are elucidated. Finally, we give an example to illustrate the enhancement of the load-carrying capacity of tapered beams for admissible shape profiles with constant volume or weight. The proposed method is of benefit to optimum design of beams against buckling in engineering applications.     

An application of geometric programming to structural design

This paper presents an application of generalised geometric programming to an optimal design of a modular floor system, which consists of reinforced solid concrete and voided slab units supported on steel beams. A function representing the cost of the floor system in terms of design variables, length, width and thickness of components, and other engineering and cost parameters is minimized subject to various constraints depending on stresses and deflections. A dual-based algorithm has been used for solving the design problem. Analyses are performed to determine the changes in the optimal values of the design variables with respect to the changes in imposed loads on the floor system.     

Theoretical research on naturally curved and twisted beams under complicated loads

This paper presents an analytical method for the study of naturally curved and twisted beams under complicated loads, with special attention devoted to the solving process of governing equations which take into account the effects of torsion-related warping as well as transverse shear deformations. The solutions derived in this paper can be used for the analysis of the beams, including the calculation of various internal forces, stresses, strains and displacements. These governing equations, in special cases, can be readily solved and yield the solutions to the problem. A generalized warping coordinate for a curved coplanar beam subjected to the action of vertical distributed loads is given for verification.     

The reduced cost branch and bound algorithm for mixed integer programming

A different branch and bound algorithm for mixed integer programming is presented. Unlike standard linear programming based branch and bound algorithms, where a single fractional variable (or Special Ordered Set) is selected for problem separation, the proposed method selects groups of variables for separation on the basis of their reduced cost in an LP relaxation. The proposed method restricts a large portion of the integer variables to zero on one branch. The net effect is that the original integer program is solved by optimizing a series of smaller, more tightly restricted, integer programs. The authors have programmed the algorithm using the Extended Control Language of the IBM MPSX/370-MIP/370 mixed integer programming package. Computational results are presented that demonstrate the efficiency of the method on problems where the $\text{0}\text{1}$ variables are partitioned into multiple choice constraints containing special ordered sets of variables. While the computational results are limited to this class of problems the algorithm can, in theory, be applied to any mixed integer programming problem.     

Elastic stability and buckling loads of multi-span nonuniform beams,shafts and frames on rigid or elastic supports by finite element method using planar uniform line elements

A numerical computer method using planar flexural finite line element for the determination of buckling loads of beams, shafts and frames supported by rigid or elastic bearings is presented. Buckling loads and the corresponding mode vectors are determined by the solution of a linear set of eigenvalue equations of elastic stability. The elastic stability matrix is determined as the product of the bifurcation sidesway flexibility matrix and the second order bifurcation sidesway stiffness matrix which is formed using the element bifurcation sidesway stiffness matrices. The bifurcation sidesway flexibility matrix is determined by partitioning the inverse of the global external stiffness matrix of the system which is formed from the element data using the element stiffness matrices. The method is directly applicable to the determination of the buckling loads of beams and frames partially or fully supported by elastic foundations where the foundation stiffness is approximated by a discrete set of springs. The method of the article provides means to consider complex boundary conditions in buckling problems with ease. Four numerical examples are included to illustrate the industrial applications of the contents of the article.     

Performance evaluation of medium access control for multiple-beam antenna nodes in a wireless LAN

We consider the medium access control (MAC) protocol design for nodes in a wireless LAN that use a wide-azimuth switched beam smart antenna system comprised of a multiple beam antenna array. The one-hop performance of carrier sense multiple access (CSMA) as well as slotted aloha for such a system is presented analytically and through simulation. The problem of synchronization of multiple beams in CSMA is investigated in our analysis. Our results show that, under heavy offered load conditions, CSMA is a good choice with nodes that have multiple-beam smart antennas, despite the performance loss due to the beam synchronization, providing a stable throughput that approaches unity and is invariant to fluctuations in the offered load. Slotted Aloha, on the other hand, is capable of higher peak throughput in a narrow range of offered loads as more switched beams are employed, but performance drastically reduces beyond optimum offered loads. We also introduce a method, expanded receive rule (ERR), whereby the tight synchronization among different beams of a receiver node in CSMA is relaxed, which is observed to provide better throughput. Finally, we also present performance results for a 4-way-handshake-type carrier sense mechanism using multiple beam antennas.     

Inelastic stability of laterally unsupported i-beams

A computer method to study the inelastic stability of laterally unsupported steel I-beams and based on a general non-linear theory is presented.Traditionally, the problem of flexural-torsional stability of beams is treated as a lateral buckling problem. Some of the draw-backs of these earlier studies are given below:The classical theory assumes that the deformations are small. In addition the deformation field is linearized. This theory is therefore valid only when the major axis flexural rigidity is much greater than its minor axis rigidity, so that deformations before the onset of lateral buckling are negligible.The lateral buckling theory is valid for straight beams, with loads applied rigorously in the plane of symmetry. Practical beams have initial imperfections and unavoidable load eccentricities. So the true behavior is better described by the stability phenomenon.For beams of intermediate length for which buckling occurs in the inelastic range, the tangent modulus theory is generally used. For ideally straight beams the tangent modulus theory provides an estimate for the collapse load which is slightly conservative. However, for practical beams with initial deformations, this need not be the case.In the majority of existing studies on inelastic lateral buckling, the differential equations for beams under uniform moment are used without modification for beams under moment gradient. In the later case the shear center line is inclined to the centroidal and geometrical axes. The differential equations for beams under uniform moment should therefore be modified by adding additional terms.The majority of the existing studies are limited to the behavior of isolated beams with simple end-conditions and so the beneficial effect of adjacent members on the beam collapse load cannot be studied accurately.A general non-linear theory to describe the spatial behavior of beams and that doesn't have the deficiencies mentioned above, is developed in the present paper.The paper also presents a computer method of solving these non-linear equations using the method of finite differences. Several numerical examples presented and comparison with the existing theoretical and experimental results show the applicability of the theory to a wide range of problems.     

Buckling, flutter and vibration analyses of beams by integral equation formulations

This paper presents a model for the investigation of buckling, flutter and vibration analyses of beams using the integral equation formulation. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable sections on elastic foundations and subjected to lateral excitation, conservative and non-conservative loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebraic system related to internal and boundary unknowns. Eigenvalue problems related to buckling and vibrations are formulated and numerically solved. Buckling loads, natural frequencies and associated eigenmodes are computed. The corresponding slope, bending and shear forces can be directly obtained. The load-frequency dependence is investigated for various elastic foundations and the divergence critical loads are evidenced. Under non-conservative loads, a dynamic stability analysis is illustrated numerically based on the coalescence of eigenfrequencies. The flutter load and instability regions with respect to the elastic concentrated and distributed foundations are identified. Using the eigenmodes, numerically computed, non-linear vibrations of beams are investigated based on one mode analysis. The presented model is quite general and the obtained numerical results are in agreement with available data.     

Seminumerical solution for buckling of rectangular plates

A semianalytical, seminumerical method of solution is presented for the governing partial differential equation of rectangular plates subjected to in-plane loads. The basic functions in the y-direction are chosen as the eigenfunctions for straight prismatic beams. The classical method of separation of variables is employed to obtain an ordinary differential equation. The resulting equation is solved by a one-dimensional finite difference technique. The problem is then reduced to a typical eigenvalue problem which on solution yields the buckling coefficient of the plate. The method is applied on plates with different edge conditions and under various loading conditions. The results are compared with those of existing solutions. Results for the case when one loaded edge is fixed and the other simply supported were reported in the literature for the first time.     

Optimization of airplane wing structures under taxiing loads

A methodology is presented for the optimum design of aircraft wing structures subjected to taxiing loads. The dynamic stresses induced in the wing as the airplane accelerates or decelerates on the runway during take-off or landing are computed by considering the interaction between the landing gear and the flexible airplane structure. The procedure is capable of taking into account both the effects of discrete runway bumps and the effects of runway unevenness. A numerical step-by-step method is developed for solving the nonlinear differential equations of motion. The optimization methodology is illustrated with two examples. The first example deals with the design of the typical section (symmetric double wedge airfoil). This example is studied by using a graphical procedure mainly to understand qualitatively the behavior of wing structures under taxiing loads and also to obtain a physical insight into the nature of the optimum solution. The second example is concerned with the design of a more realistic wing structure. In this case, the problem is formulated and solved as a constrained nonlinear programming problem based on finite element modeling.     

A NLP approach for evaluating storey-buckling strengths of steel frames under variable loading

The problem of determining elastic buckling strengths for unbraced steel frames under variable loading is investigated in this paper. Whereas the pattern of applied loads is specified prior to stability analysis of a frame under proportional loading, load patterns are not predefined in variable loading. The conventional methods for evaluating the stability strength of unbraced frames under proportional loading are not applicable for variable loading, since the load pattern is unknown. Taking into account the concept of storey-based buckling, the problem of frame stability under variable loading is presented as a pair of minimization and maximization problems subject to stability constraints, which are solved by a nonlinear programming (NLP) method. The proposed variable loading approach takes into account the variability of applied loads during the life span of the structure, and as such, provides accurate evaluation of elastic frame-buckling strengths.     

Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams

The equations of equilibrium which describe the three-dimensional behavior of curved beams are derived. These equations include first order geometric nonlinear influences. The equations are solved by the Galerkin method. The method is very general and allows general variation of the geometric and structural properties along the beam. Any combination of boundary conditions is possible and the most general distribution of loads along the beam can be treated. Numerical results for a few examples are presented and compared to other theoretical and experimental results. The agreement between the results is generally good. It is shown that Galerkin method is an efficient method which enables one to solve the problem using small number of unknowns compared to other methods which are in use.     

Analysis of continuous beams on discrete elastic supports by the five moment equation

The analysis of continuous beams on discrete elastic supports is a complex problem frequently encountered in structural engineering. In structures of this type, the supports displace in direct proportion to the support reactions. These support displacements result in moment redistribution in the beam. The methods in current use for solving this type of problem arc generally complex and in most cases are developed for analysis of more complicated structures.This paper illustrates a technique for computer analysis of continuous beams on discrete elastic supports by the five moment equation. The objective is to present a viable example for electronic computation consistent with the nature of the problem, and which will tend to contribute to the basic understanding of the use of computers in reducing the amount of computation.The five moment equation is an extension of the familiar three moment equation and results in the same simple form of analysis. The solution of the five moment equation for the analysis of continuous beams on elastic supports permits direct determination of the redundant moments at the supports. The result of this simplified form provides practicing engineers with an efficient analysis easily handled on small office computers. For the student of civil engineering, the method illustrated will significantly contribute to development of effective early progress of programming skills.     

A system for the design of packet-switched communication networks with economic tradeoffs

This paper studies the problem of designing a packet-switched communication network with tradeoffs between link costs and response time to users. It consists of assigning capacities to links in the network and determining the routes used by messages for all communicating node pairs in order to minimize total link fixed and variable costs. A tradeoff between link costs and response time to users is achieved by including a constraint that sets an upper limit on the average link queueing delay in the network. The topology of the network and the end-to-end traffic requirements are given. The problem is formulated as a nonlinear integer programming model. Unlike most of previous models, where the best route for a communicating node pair is restricted to a set of prespecified candidate routes, our model considers all possible routes for every communicating node pair. An efficient heuristic based on a Lagrangean relaxation of the problem is developed to generate feasible solutions. The results of extensive computational experiments across a variety of previously used networks are reported. These results indicate that the solution procedure is effective for a wide range of traffic loads and cost structures.     

Allocating power to schedule loads and charge batteries on the space station

The objective of this research is to allocate electric power to schedule loads and charge batteries on the space station. This work is an extension of research performed in the summers of 1986 and 1987 [4]. A simplified model of power flow on the space station is developed. This model forms the basis for a binary integer programming formulation which allocates power to user loads which are scheduled to operate during an orbit. The model is used for developing a procedure for allocating surplus power to charge batteries on the space station. A small example problem is solved by using a commercial integer programming code.