Some shortcomings in Michell's truss theory

Michell (1904) derived his well-known optimality criteria for trusses in the context of unequal permissible stresses in tension and compression. For the same design constraints, Hemp (1973) and the author (e.g. 1989) have obtained optimality conditions which are different from those of Michell, and also lead, in general, to different, and lighter trusses. The reasons for this contradiction are examined and it is found that for unequal permissible stresses Michell's optimality conditions are only valid for a highly restricted class of support conditions.     

Optimum design of steel frames with stability constraints

Optimum design algorithms based on the optimality criteria approach are proven to be efficient and general. They have the flexibility of accomodating variety of design constraints such as displacement, stress, stability and frequency in the design problem. The design methods developed recently, although considering one or more of these constraints, lack the necessity of referring to any relevant design code. The algorithm presented for the optimum design of street frames implements the displacement and combined stress limitations according to AISC. The recursive relationship for design variables in the case of dominant displacement constraints is obtained by the optimality criteria approach. The combined stress inequalities which include in-plane and lateral buckling of members are reduced into nonlinear equations of design variables. The solution of these equations gives the values of bounds for the variables in the case where the stress constraints are dominant in the design problem. The use of effective length in the combined stress constraints makes it possible to study the effect of the end rigidities on the final designs. The design procedure is simple and easy to program which makes it particularly suitable for microcomputers. A number of design examples are considered to demonstrate the practical applicability of the method. It is also shown that the design procedure can be employed in selecting the optimum topology of steel frames.     

Plastic behaviour and stability constraints in the shakedown analysis and optimal design of trusses

A shakedown analysis and optimum shakedown design of elasto-plastic trusses under multi-parameter static loading are presented. To control the plastic behaviour of the truss, bounds on the complementary strain energy of the residual forces and on the residual displacements are applied and for the bars under compression critical stresses updated during the iteration are taken into consideration. The formulation of problems is suitable for nonlinear mathematical programming which is solved by the use of an iterative procedure. The application of the method is illustrated by three test examples.     

Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization

In this work, we explore simultaneous geometry design and material selection for statically determinate trusses by posing it as a continuous optimization problem. The underlying principles of our approach are structural optimization and Ashby’s procedure for material selection from a database. For simplicity and ease of initial implementation, only static loads are considered in this work with the intent of maximum stiffness, minimum weight/cost, and safety against failure. Safety of tensile and compression members in the truss is treated differently to prevent yield and buckling failures, respectively. Geometry variables such as lengths and orientations of members are taken to be the design variables in an assumed layout. Areas of cross-section of the members are determined to satisfy the failure constraints in each member. Along the lines of Ashby’s material indices, a new design index is derived for trusses. The design index helps in choosing the most suitable material for any geometry of the truss. Using the design index, both the design space and the material database are searched simultaneously using gradient-based optimization algorithms. The important feature of our approach is that the formulated optimization problem is continuous, although the material selection from a database is an inherently discrete problem. A few illustrative examples are included. It is observed that the method is capable of determining the optimal topology in addition to optimal geometry when the assumed layout contains more links than are necessary for optimality.     

Optimum least-cost design of a truss roof system

An algorithm is presented encompassing the application of optimization methods to the least-cost elastic design of roof systems composed of rigid steel trusses, web joists and steel roof deck. The method is capable of designing rigid trusses that can be fabricated from various grades of steel and several types of standard sections. The selection of open web joist is presently limited to standard H-series, and decking material is standard 22 gage. The design is based upon AISC allowable values where combined stresses resulting from axial forces and secondary bending moments are considered. The effective column lengths are computed using the characteristic buckling equation for a member whose ends are elastically restrained against rotation. The procedure developed considers changes in the mechanical properties of the members, geometric variations in the truss configuration and changes in topology. Selected sets of members may be chosen to be identical, and chord members may be defined as continuous over several panels. Also investigated is the problem of finding the design containing the optimum number of trusses. A number of examples are presented which demonstrate the flexibility and generality of the design approach developed.     

Optimum geometry design of nonlinear braced domes using genetic algorithm

Domes are lightweight and cost effective structures that are used to cover large areas. They are mainly comprised of a complex network of triangles made out of slender members. The behaviour of flexible dome is nonlinear under the external loads which makes it necessary to consider the geometrical non-linearity in their analysis to obtain realistic response of these structures. Furthermore, instability check during the nonlinear analysis is of prime importance. In this paper, an algorithm is presented for the optimum geometry design of nonlinear braced domes. The height of crown is taken as design variable in addition to the cross-sectional properties of members. A procedure is developed that calculates the joint coordinates automatically for a given height of the crown. The optimum design algorithm takes into account the nonlinear response of the dome due to the effect of axial forces on the flexural stiffnesses of members. It considers serviceability requirements as well as combined strength limitations set by BS 5950. The solution of the design problem is obtained by genetic algorithm. The elastic instability analysis is then carried out for each individual in the initial population until the ultimate load factor is reached. During this analysis, checks on the overall stability of the dome is conducted. If the loss of stability takes place, this individual is taken out of the population and replaced by a new randomly generated individual. This replacement policy is repeated until an individual is found which does not have instability problem. Once the initial population is established where all the individuals are free of instability problem, the regular genetic operations are applied to generate a new population. Number of design examples are included to demonstrate the application of the algorithm.     

Exact analytical theory of topology optimization with some pre-existing members or elements

This note deals with topological optimization of structures in which some members or elements of given cross-section exist prior to design and new members are to be added to the system. Existing members are costless, but new members and additions to the cross-section of existing members have a non-zero cost. The added weight is minimized for given behavioural constraints. The proposed analytical theory is illustrated with examples of least-weight (Michell) trusses having (a) stress or compliance constraints, (b) one loading condition and (c) some pre-existing members. Different permissible stresses in tension and compression are also considered. The proposed theory is also confirmed by finite element (FE)-based numerical solutions.     

Truss topology optimization including bar properties different for tension and compression

The problem of optimum truss topology design based on the ground structure approach is considered. It is known that any minimum weight truss design (computed subject to equilibrium of forces and stress constraints with the same yield stresses for tension and compression) is—up to a scaling—the same as a minimum compliance truss design (subject to static equilibrium and a weight constraint). This relation is generalized to the case when different properties of the bars for tension and for compression additionally are taken into account. This situation particularly covers the case when a structure is optimized which consists of rigid (heavy) elements for bars under compression, and of (light) elements which are hardly/not able to carry compression (e.g. ropes). Analogously to the case when tension and compression is handled equally, an equivalence is established and proved which relates minimum weight trusses to minimum compliance structures. It is shown how properties different for tension and compression pop up in a modified global stiffness matrix now depending on tension and compression. A numerical example is included which shows optimal truss designs for different scenarios, and which proves (once more) the big influence of bar properties (different for tension and for compression) on the optimal design.     

Discrete optimization of geometrically nonlinear truss structures under stability constraints

The paper deals with discrete optimization of elastic trusses with geometrical nonlinear behaviour and constraints on stability. The problem consists of minimizing the weight and determining the optimal member distribution so that the external load does not cause a loss of stability of the structure. Member cross-sections are selected from a catalogue of available sections. Element stresses, elment stability and global structural stability constraints are considered. A controlled enumeration method according to the increasing value of the objective function is applied. Shallow space trusses are numerically analysed.     

Theoretically optimal bracing for pre-existing building frames

Bracing is commonly used to provide resistance to lateral forces in building structures. However, traditional bracing design approaches appear not to be underpinned by clear fundamental principles. Here, theoretically optimal arrangements of bracing members are sought for pre-existing building frames, already designed to carry gravity loads. For sake of simplicity existing frame elements are assumed to be capable of carrying additional loads and three types of bracing are considered: tension only bracing, bracing intersecting only at the corners of the existing frame, and unconstrained optimal bracing, where bracing elements can intersect at any location. Layout optimization techniques are used to identify initial design solutions; these are then related to Michell trusses to obtain exact reference volumes, against which the efficiency of other bracing layouts can be judged. It is shown that from a theoretical standpoint tension only bracing is inefficient and that the optimal angle of intersection between a pre-existing frame member and intersecting tension/compression bracing member pairs is 45^°, something that can potentially be adopted as a basic principle when designing bracing for a pre-existing frame.     

Optimum design of a reinforced plastic bridge girder

This paper describes the results of an analytical and experimental investigation concerned with the optimum design of a fiberglass-reinforced plastic flexural number. In the initial phase of the study, seven members were designed, fabricated, load-tested and subsequently analyzed. Design modifications were made in each subsequent specimen based on results from the experimental and analytical study. The initial configurations were selected on the basis of intuitive arguments and empirical optimum design concepts. In order to achieve a more efficient and economical structural system, an iterative algorithm for optimum geometric configuration was developed and used in conjunction with an optimality criterion for optimum member sizing. This procedure was applied to the design of five separate flexural members and final design data for these members are presented and discussed.     

Optimum shape design of cold-formed thin-walled steel sections

The algorithm presented in this study obtains the optimum cross-sectional dimensions of cold-formed thin-walled steel beams subjected to general loading. It has the flexibility of considering different cross-sectional shapes such as symmetrical or unsymmetrical channel, lipped channel or Z-sections. The algorithm treats the cross-sectional dimensions such as width, depth and wall thickness as design variables and considers the displacement as well as stress limitations. The presence of torsional moments causes warping of thin-walled sections. The effect of warping in the calculation of normal stresses is included using Vlasov theorems. These theorems require the computation of sectorial properties of cross-sections. A general numerical procedure is presented for obtaining these properties. The optimum design problem of thin-walled open sections subjected to combined loading turns out to be a highly nonlinear problem. It is shown that optimality criteria method can effectively be used to obtain its solution. A number of design examples are presented to demonstrate the application of the algorithm.     

On the validity of Prager's example of nonunique Michell structures

Prager (1974) demonstrated through an example that the optimal layout of least-weight trusses for a stress constraint — termed also Michell trusses—can be nonunique. The strain field for the above example, however, seems to violate Michell's optimality criteria. It is shown in this note that the above example of nonuniqueness is completely correct if we restrict the truss members to a smaller subset of the plane.     

Design of structures using parametric quadratic programming

This paper presents a method for the interactive analysis design of planar steel trusses. Given an initial feasible design, that is, a design that satisfies stress and displacement performance conditions, and fabrication conditions, a parametric analysis is used to identify the extent to which one or two members can be redesigned while still satisfying design feasibility. The parametric analysis is obtained from the solution of a parametric Hessian quadratic programming problem (PHQP). The analysisdesign methodology is demonstrated with a three member truss.     

Force density method for simultaneous optimization of geometry and topology of trusses

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.     

Optimal design of planar frames based on structural performance criteria

The aim of this study is to present a new approach, nearer to engineer's point of view, for optimizing the structures, in particular for the planar frames. As a means of improving the overall stability of the structure the objective function chosen to be maximized is the eigenvalues of the buckling modes. The optimization process involves two stages: a preliminary design when a prescribed value for the natural fundamental vibration period is the main constraint and a second stage in which the usual stress, displacement, and side constraints are taken into account. The algorithm used for optimization is based on a classical optimality criteria approach. The steps of the algorithm are illustrated by some examples which demonstrate the effectiveness of the proposed procedure in improving design.     

Benchmark case studies in optimization of geometrically nonlinear structures

A structural optimization algorithm is developed for truss and beam structures undergoing large deflections against instability. The method combines the nonlinear buckling analysis using the displacement control technique, with the optimality criteria approaches. Several benchmark case studies illustrate the procedure and the results are compared with examples reported in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass or overestimates the buckling load for these types of structures, so a design based on the linear buckling analysis may result in catastrophic failure. The effect of geometrical nonlinearities and element imperfections has also been studied.     

Nonlinear analysis of reinforced concrete structures

For the prediction of yield and failure of concrete under combined stress, a generalization of the Mohr-Coulomb behavior is made in terms of the principal stress invariants. The generalized yield and failure criteria are developed to account for the two major sources of nonlinearity: the progressive cracking of concrete in tension, and the nonlinear response of concrete under multiaxial compression. Using these criteria, incremental stress-strain relationships are established in suitable form for the nonlinear finite element analysis.For the analysis of reinforced concrete members by finite elements, a method is introduced by which the effect of reinforcement is directly included. With this approach, the stress-strain laws for the constituent materials of reinforced concrete are uncoupled permitting efficient and convenient implementation of a finite element program. The applicability of the method is shown on sample reinforced concrete analysis problems.     

Nonlinear influence of contraflexure migration on near-curtailment stresses in hyperstatic FRP-laminated steel members

The flexural characteristics of hyperstatic steel members with adhesively bonded Fibre reinforced polymer (FRP) laminates are considered. It is shown that, during elasto-plastic activity, the migration of the points of contraflexure along such members has two nonlinear effects on the stresses in both the laminate and the adhesive. First, the lengths of laminate traversed by these migrating points undergo reversal of axial stress from tension to compression, which exposes the laminate to potential buckling. Second, the bond stresses in the adhesive near laminate curtailment increase dramatically, and so may initiate brittle failure near curtailment by separation of the laminate from the steel member. Nonlinear finite element (FE) analyses are used to illustrate both these effects. The FE model is verified by comparison with published data, and is then used to analyse an isostatic and a hyperstatic FRP-laminated steel member. The results show that the above nonlinear influences are pronounced in the hyperstatic member, but are absent from the isostatic member. Tapering of both the laminate and adhesive profiles near curtailment of the laminate is investigated as a potential means of reducing these effects in the hyperstatic member. Finally, the implications of the above effects for the performance of hyperstatic FRP-laminated steel members in practice, are discussed.     

Nonparametric gradient-less shape optimization for real-world applications

A nonparametric gradient-less shape optimization approach for finite element stress minimization problems is presented. The shape optimization algorithm is based on optimality criteria, which leads to a robust and fast convergence independent of the number of design variables. Sensitivity information of the objective function and constraints are not required, which results in superior performance and offers the possibility to solve the structural analysis task using fast and reliable industry standard finite element solvers such as ABAQUS, ANSYS, I-DEAS, MARC, NASTRAN or PERMAS. The approach has been successfully extended to complex nonlinear problems including material, boundary and geometric nonlinear behavior. The nonparametric geometry representation creates a complete design space for the optimization problem, which includes all possible solutions for the finite element discretization. The approach is available within the optimization system TOSCA and has been used successfully for real-world optimization problems in industry for several years. The approach is compared to other approaches and the benefits and restrictions are highlighted. Several academic and real-world examples are presented.