Analysis of beam-like lattice trusses

A simple procedure is presented for predicting the thermoelastic and free vibration responses of large repetitive beam-like trusses. The procedure is based on replacing the original lattice structure by an equivalent continuum beam model and obtaining closed-form (exact) solutions for the beam model. The equivalent beam model accounts for warping and shear deformation in the plane of the cross-section and is characterized by its thermoelastic strain and kinetic energies, from which the equations of motion and constitutive relations can be derived. The high accuracy of the results obtained by the proposed approach is demonstrated by means of numerical examples.     

Analysis of spatial beamlike lattices with rigid joints

Micropolar beam models are developed for the static, free vibration and buckling analysis of repetitive spatial beamlike lattices with rigid joints. The micropolar beam models have independent microrotation and displacement fields and are characterized by their strain energy, potential energy due to initial stresses and kinetic energy from which the governing differential equations and boundary conditions can be derived. The procedure for developing the expression for the strain energy of the micropolar beam involves introducing basic assumptions regarding the variation of the displacement and microrotation components in the plane of the cross-section, and obtaining effective elastic coefficients of the continuum in terms of the material properties and geometry of the original lattice structure. The high accuracy of the solutions obtained by the micropolar beam models is demonstrated by means of numerical examples for vierendeel and double-laced lattice girders with triangular cross-sections.     

Stability of dynamic models obtained by ARMAX lattice predictor

A new ARMAX lattice algorithm could be used for identification and prediction of dynamic processes. Stability analysis shows that the effect of the error propagation for the lattice form decays exponentially. The proposed algorithm is tested on hydrological records from the Nile.     

Pareto optima of isostatic trusses

Multicriterion optimization of elastic stress limited isostatic trusses is considered and a numerical method for determining the Pareto optimal set of the problem is developed. The weight of the structure and some chosen nodal displacements are taken as design criteria, and member areas are used as design variables. The corresponding bicriterion problem with weight and one displacement in the objective function is solved exactly, and this result is used as a basis of the proposed method. By introducing certain parameters, each joined with one displacement criterion, the Pareto optimal solutions of the general problem may be obtained with any accuracy. Two examples are given to illustrate the method.     

On automated optimum design of trusses

In the course of formulating the automated design of trusses for a project, some hitherto unexplored ideas have been encountered. These ideas are developed in this article.To explore these ideas systematically, truss design for single and multiple load conditions are considered separately. In the first part the applicability of linear programming is illustrated for single load conditions to safeguard against failures by yielding, buckling, and excessive deformations. Postoptimal behaviour of the structure is also studied.In the second part nonlinear programming is used as the design tool for multiple loading conditions. In this procedure, the members are grouped, a fully utilized design is taken as the starting point, and the condensed matrix displacement method of reanalysis is adopted during optimization iterations.Practical design examples of trussed towers are given to illustrate the capabilities of the computer program developed.     

On optimal physically nonlinear trusses

Optimization problems for physically nonlinear hyperelastic trusses statically loaded by a single system of loads are investigated. Several possible formulations of the problems for a prescribed truss layout are considered. It is demonstrated that the problems lead to the design of equal absolute stress values in some rods and of active technology constraints in other rods. Some general properties of the trusses are examined. Optimal physically nonlinear hyperelastic truss layout problems under static loading are also considered. It is shown that the well-known results of Maxwell, Michell and Prager concerning linear-elastic structures are generalized for the considered problems. It is demonstrated that the results of the paper are extended to optimization problems for structures in stationary creep with an arbitrary concave stress-strain velocity relation.     

On optimal geometrically non-linear trusses

The paper is devoted to the investigation of regularities inherent to optimal geometrically non-linear trusses. The single static loading case is considered, a single structural material is used (except specially indicated cases) and buckling effects are neglected. The so-called small strains and large rotations case is investigated. Some regularities inherent to the kinematic and static variational principles for geometrically non-linear trusses are considered. Then the strain compatibility conditions resulting from the static variational principle are obtained and explored. It is shown that 1) the conditions are linear with respect to subcomponents of rod Green strains such as rotations and geometrically linear strains, 2) strains (in particular, rotations and geometrically linear strains) within rods which are not members of the so-called basic structure are fully determined by geometrically linear strains in rods of the basic structure.Extensions of some theorems (Maxwells theorem, Michells theorem, theorems on the stiffness properties of equally-stressed structures, etc.) known for geometrically linear structures are proved.Conditions assuring better or worse quality of equally-stressed geometrically non-linear truss as compared to geometrically linear ones are obtained.It is shown that in numerical optimization of geometrically non-linear trusses in the case of negligible rotations of compressed rods some updated analytical optimization algorithms (derived earlier for geometrically linear case) are monotonic. A simple numerical example confirming the features is presented.     

Nonlinear dynamic analysis of space trusses

A computational procedure is presented for predicting the dynamic response of space trusses with both geometric and material nonlinearities. A mixed formulation is used with the fundamental unknowns consisting of member forces, nodal velocities and nodal displacements. The governing equations consist of a mixed system of algebraic and differential equations. The temporal integration of the differential equations is performed by using an explicit half-station leap-frog method. The advantages of the proposed computational procedure over explicit methods used with the displacement formulation are discussed. The high accuracy of the procedure is demonstrated by means of numerical examples of plane and space trusses. The constitutive relations in these examples are assumed, for convenience, to be represented by the Ramberg-Osgood polynomials. Comparison is also made with solutions obtained by using implicit multistep temporal integration schemes.     

Computation of non-unique solutions of elastic-plastic trusses

It has recently been shown that the solution to certain well-defined boundary value problems for elastic/perfectly-plastic structures is non-unique. The present paper is concerned with the ability of finite-element computer programs to handle such solutions. Some problems arising from non-uniqueness are documented and techniques for circumventing those problems are evaluated by exploring the response of the program NonSAP to a variety of trusses with non-unique solutions.     

A language for defining spatial trusses

This paper describes a computer language enabling its users to define three-dimensional truss structures. The language can be used either as a design tool in specifying the geometrical characteristics of these structures or as an input language to a program that actually performs their structural analysis. The main characteristic of the language is that it is procedural; programs defining a nucleus of a structure can be made into procedures (subroutines) which can be called within programs and other procedures in order to define more complex structures composed of that nucleus or its variants. This characteristic is particularly useful in defining repetitive geometrical patterns which themselves appear within repetitive patterns. The paper also describes a syntax-directed compiler whose input is a sentence in the language and whose output is interpretable code which, when executed, produces drawings and provides listings of the coordinates of the points and the linkages defined by the user in his source program. Several examples of structures defined using this language are included in the paper.     

Hencky-Prandtl nets and constrained Michell trusses

We study minimum weight trusses (or truss-like continua) subject to technological constraints which limit the member forces. In the optimal design the principal strains are constant over part of the domain, and the principal stresses over another part—leading to a combination of a Michell truss and a slip line net. We begin with a variational treatment of the unconstrained Michell problem, indicating one possible numerical approach and also suggesting spaces of stress and displacement fields which will allow a proof of the existence of optimal trusses for very general boundary conditions.     

Multiobjective optimization of trusses using genetic algorithms

In this paper we propose the use of the genetic algorithm (GA) as a tool to solve multiobjective optimization problems in structures. Using the concept of min–max optimum, a new GA-based multiobjective optimization technique is proposed and two truss design problems are solved using it. The results produced by this new approach are compared to those produced by other mathematical programming techniques and GA-based approaches, proving that this technique generates better trade-offs and that the genetic algorithm can be used as a reliable numerical optimization tool.     

Optimum design of geometrically nonlinear space trusses

A structural optimization algorithm is developed for geometrically nonlinear three-dimensional trusses subject to displacement, stress and cross-sectional area constraints. The method is obtained by coupling the nonlinear analysis technique with the optimality criteria approach. The nonlinear behaviour of the space truss which was required for the steps of optimality criteria method was obtained by using iterative linear analysis. In each iteration the geometric stiffness matrix is constructed for the deformed structure and compensating load vector is applied to the system in order to adjust the joint displacements. During nonlinear analysis, tension members are loaded up to yield stress and compression members are stressed until their critical limits. The overall loss of elastic stability is checked throughout the steps of algorithm. The member forces resulted at the end of nonlinear analysis are used to obtain the new values of design variables for the next cycle. Number of design examples are presented to demonstrate the application of the algorithm. It is shown that the consideration of nonlinear behaviour of the space trusses in their optimum design makes it possible to achieve further reduction in the overall weight. The other advantage of the algorithm is that it takes into account the realistic behaviour of the structure, without which an optimum design might lead to erroneous result. This is noticed in one of the design example where a tension member changed into a compression one at the end of nonlinear analysis.     

On the optimal layout of multi-purpose trusses

This paper provides a solution to the problem of minimum mass design of multi-purpose trusses for which the design variables are not only the areas of the bars but also the positions of the joints. Displacement constraints and non-constant stress constraints (stability) are taken into account.

With multiple loading systems, the optimal structure is normally statically indeterminate and generally not even ·fully stressed”. The solution is obtained by successive iterations, using a gradient method with move-limits. For each iteration only the critical forces and displacements are considered and trusses with up to 40 joints have been optimized.

Analytical expressions are derived for the necessary gradients, i.e. for the partial derivatives of the displacements and forces with respect to the bar areas and joint coordinates.     

Optimal design of trusses by approximate compatibility

Optimal design of elastic trusses is formulated as an approximate linear programming problem. Using the force method of analysis, the redundant forces are expressed in linearized terms of the design variables. The solution of the resulting linear programming problem can be viewed as an exact optimum for a truss with different displacements corresponding to the unknown redundants. The latter displacements, computed directly from the linear programming solution, indicate the degree of not satisfying the compatibility conditions. This information can be used to introduce imaginary displacements in subsequent iteration cycles.An iterative procedure of solution is proposed in which both the design and the imaginary displacements are modified until the compatible optimal solution is reached. Each iteration cycle requires the solution of a linear programming problem. The proposed procedure provides more flexibility in the solution process than the usual algorithms based on a sequence of linear programs and may improve the convergence. Numerical examples illustrate the application of this procedure in optimal design of simple trusses.     

Analytic solutions of elastically supported Michell trusses

Structural and Multidisciplinary Optimization - In this short note, analytic solutions of elastically supported Michell truss with varying support stiffness and support positions are presented....     

Dynamic response analyses of trusses under harmonic loading

The stiffness method of analysis is used to compute displacements and end-bar forces for pinconnected trusses under the action of harmonic forces. Three different formulations are presented to account for the inertia forces: (1) the ‘exact’ analysis based on the Bernoulli-Euler equation, (2) the series expansion approximation, and (3) the lumped mass method. A computer program is provided with options to perform the analysis by any of these three methods.     

Exact geometry considerations in buckling analysis of trusses

This paper is concerned with the exact geometrical formulation for truss buckling. Based on the assumptions of large member rotations but small strains, a generalized formulation is presented for the nonlinear analysis of trusses in their deformed state. Conditions for instability are included and the snap-through load is accurately predicted for various examples by using the Newton-Raphson approach. The formulation is used to study the response of cable networks to external loads. Under uniformly distributed loads the networks behaved linearly with very small nodal displacements. In other loading cases, where the load distribution is unsymmetrical and the total load applied is smaller than the uniform loading, very large displacements occurred with significant nonlinearities.