System Reliability Analysis for Truss Structures Based on Automatic Updated Model

Multiple failure modes tend to be identified in the reliability analysis of a redundant truss structure.This identification process involves updating the model for identifying the next potential failure members.Herein we intend to update the finite element model automatically in the identification process of failure modes and further perform the system reliability analysis efficiently.This study presents a framework that is implemented through the joint simulation of MATLAB and APDL and consists of three parts:reliability index of a single member,identification of dominant failure modes,and system-level reliability analysis for system reliability analysis of truss structures.Firstly,RSM(response surface method)combines with a constrained optimization model to calculate the reliability indices ofmembers.Then theβ-unzipping method is adopted to identify the dominant failuremodes,and the system function in MATLAB,as well as the EKILL command in APDL,is used to facilitate the automatic update of the finite element model and realize load-redistribution.Besides,the differential equivalence recursion algorithmis performed to approximate the reliability indices of failuremodes efficiently and accurately.Eventually,the PNET(probabilistic network evaluation technique)is used to calculate the joint failure probability as well as the system reliability index.Two illustrative examples demonstrate the accuracy and efficiency of the proposed system reliability analysis framework through comparison with corresponding references.

Topology Structure Synthesis and Analysis of Spatial Pyramid Deployable Truss Structures for Satellite SAR Antenna

Many attentions for structural synthesis are paid to planar linkages and parallel mechanisms, while design novel pyramid deployable truss structure（PDTS） of satellite SAR mainly depends on experience of designer. To design novel configuration of PDTS, a two-step topology structure synthesis and analysis approach is proposed. Firstly, a conceptual configuration of PDTS is synthesized. Weighted graph and weighted adjacency matrix are established to realize topological description for PDTS. Graph properties are then summarized to distinguish differentia between PDTS and other type structures. According to graph properties, a procedure for synthesis conceptual configuration of PDTS is presented. Secondly, join relationship of components in a PDTS is analyzed. Kinematic chain and corresponding incidence/adjacency matrix are employed to analyze join relationship of PDTS. Properties and simplified rules of kinematic chain are extracted to construct kinematic chain. A procedure for construction kinematic chain of PDTS is then established. Finally, with this two-step approach all 11 rectangular pyramid deployable structures whose folded state is planar are discovered and their kinematic chains are constructed. Based on synthesis results, a novel deployable support structure for satellite SAR is designed. The proposed research can be applied to obtain some novel PDTSs, which is of great importance to design some novel deployable support structures for satellite SAR antenna.

TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURE WITH FUNDAMENTAL FREQUENCY AND FREQUENCY DOMAIN DYNAMIC RESPONSE CONSTRAINTS

In this paper, adaptive genetic algorithm （AGA） is applied to topology optimization of truss structure with frequency domain excitations. The optimization constraints include fundamental frequency, displacement responses under force excitations and acceleration responses under foundation acceleration excitations. The roulette wheel selection operator, adaptive crossover and mutation operators are used as genetic operators. Some heuristic strategies are put forward to direct the deletion of the extra bars and nodes on truss structures. Three examples demonstrate that the proposed method can yield the optimum structure form and the lightest weight of the given ground structure while satisfying dynamic response constraints.

TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURES WITH SYSTEMATIC RELIABILITY CONSTRAINTS UNDER MULTIPLE LOADING CASES

In this paper, a mathematical model for topology optimization oftruss structures with constraints of displacement and systemreliability under multiple loading cases is constructed. In order toavoid the difficulty of computing the structure's system reliability,a solving approach is presented in which the failure probability ofsystem is divided into the sum of a all bars' failures probability bymeans of reliability distribution. In addition, by drawing into thereliability safety factor and the fundamen- tal relationship instructural mechanics, all probability constraints of displacement andstress are equiv- alently displayed as conventional form and linearfunction of the design variables.

DESIGN OPTIMIZATION FOR TRUSS STRUCTURES UNDER ELASTO-PLASTIC LOADING CONDITION

In this paper, a method for the design optimization of elasto-plastic truss structures is proposed based on parametric variational principles （PVPs）. The optimization aims to find the minimum weight/volume solution under the constraints of allowable node displacements. The design optimization is a formulation of mathematical programming with equilibrium constraints （MPECs）. To overcome the numerical difficulties of the complementary constraints in optimization, an iteration process, comprising a quadratic programming （QP） and an updating process, is employed as the optimization method. Furthermore, the elasto-plastic buckling of truss mem- bers is considered as a constraint in design optimization. A combinational optimization strategy is proposed for the displacement constraints and the buckling constraint, which comprises the method mentioned above and an optimal criterion. Three numerical examples are presented to show the validity of the methods proposed.

Multivariate Rational Response Surface Approximation of Nodal Displacements of Truss Structures

Polynomial-basis response surface method has some shortcomings for truss structures in structural optimization,concluding the low fitting accuracy and the great computational effort. Based on the theory of approximation, a response surface method based on Multivariate Rational Function basis(MRRSM) is proposed. In order to further reduce the computational workload of MRRSM, focusing on the law between the cross-sectional area and the nodal displacements of truss structure, a conjecture that the determinant of the stiffness matrix and the corresponding elements of adjoint matrix involved in displacement determination are polynomials with the same order as their respective matrices, each term of which is the product of cross-sectional areas, is proposed. The conjecture is proved theoretically for statically determinate truss structure, and is shown corrected by a large number of statically indeterminate truss structures. The theoretical analysis and a large number of numerical examples show that MRRSM has a high fitting accuracy and less computational effort. Efficiency of the structural optimization of truss structures would be enhanced.

Nonlinear Dynamics Analysis of Joint Dominated Space Deployable Truss Structures

Joints are necessary components in the larger space deployable truss structures which have significant effect on the dynamics behavior of these deployable joint-dominated structures. Four kinds of joints' nonlinear force-displacement relationship are analyzed based on describing function method. The dynamic responses of one-DOF jointed system under different exciting force levels are investigated to understand the influence of joint nonlinearity on dynamic responses. The influences of joint characterizing parameters on joint nonlinearities are analyzed. Dynamic responses of the modular beam-like deployable joint-dominated truss structure are tested under different sinusoidal exciting force levels. The experimental results show obvious nonlinear behaviors contributed by joints that dynamic response shifts to lower resonance frequency and higher amplitude with the increase of exciting force. The nonlinearity of the joints in the tested structure is compared with the theoretical results and identified to meet with the hysteresis nonlinearity.

An improved equivalent beam model of large periodic beam-like space truss structures

Space truss structures are essential components for space-based remote sensing loads with high spatial and temporal resolutions.To achieve high-precision vibration control,an accurate and efficient dynamics model is essential.In addition to the current equivalent beam model(EBM)based on the classical continuum theory,an improved equivalent beam model(IEBM)is proposed that considers the impact of the distinction between trusses and beams on torsional and shear deformations,as well as the impact of shear deformation on flexural rigidity.According to the displacement expressions of spatial beams,torsional,shear,and bending correction coefficients are introduced to derive expressions of strain energy and kinetic energy.The energy equivalence principle is then utilized to calculate the elasticity and inertia matrices,and dynamics equations are established using the finite element method.Subsequently,an IEBM is constructed by employing the particle swarm optimization approach to determine the correction coefficients with the truss natural frequency as the optimization target.The natural vibration characteristics of the structure are estimated for various material properties.Compared with the full-scale finite element model,the EBM reaches a maximum error of 80%for a low modulus of elasticity,while the maximum error of the IEBM is less than 2%for any given parameters,indicating its superior accuracy to the EBM.

Regular truss structure model equivalent to continuum structure

It is difficult to solve the structural problems related to agricultural engineering,due to the wide ranges of the means of related variables and complex structural shapes.For these reasons,discrete models are required that are able to replace or simplify solid structure components used in traditional analysis methods.Therefore,the objective of this study was to develop a regular truss structure model that behaves the same way as a solid structure.It was assumed that if a unit element consists of truss elements with each hinge at the end of the element and the size of the element is infinitesimal,the stress distribution and displacement field will be constant throughout the domain of the unit element.Additionally,the behavior of the truss element was assumed to be in a linear state in a two-dimensional plane.The law of energy conservation,based on the theory of elasticity,was applied to determine the equilibrium conditions between discretized and solid elements.The restrictive condition that we obtained revealed that applications are limited to only ideal elastic materials with a Poisson’s ratio of 1 to 3.The volumetric ratio of the equivalent truss to the continuum structures was 3:1,regardless of the size or number of the mesh.To calculate the internal stress and strain of the unit element,the geometric relationships of each truss member,which has its own role against different stress directions,were used.The calculated von Misses stresses were used to verify this model.Stress concentrations,as explained based on Saint Venant’s principle,were also observed in the equivalent truss structure model.The main stress paths,indicating the areas where reinforcement bars should be placed,were successfully shown without the requirement that each element be transformed in the direction of principal stress;this was done by eliminating elements with only compressive and near-zero stresses.

Crack simulation and probability analysis using irregular truss structure modeling equivalent to a continuum structure

The problems related to agricultural structure engineering for crack simulation and reliability analysis are complicated because its variables contain wide ranges of mean and standard deviation.This paper describes an integrated model to perform crack simulation and reliability analysis of a continuum structure.The structure is assumed to be under a two-dimensional plane stress and the deformation is infinitesimal.A truss structure model that has the same behaviour as a continuum structure was developed using irregular triangle truss components where each element consists of two hinges with an axial degree of freedom at both of their ends.A Monte-Carlo simulation(MCS)was adopted for the reliability analysis.If the length of one side of the irregular triangle mesh is shorter than the thickness of the structure,the slenderness associated with compressive failure needs to be examined only for the short column.For that reason,the failure criterion suitable for the equivalent truss structure model was established by checking only axial stresses acting on truss members.Since nodes of the equivalent truss structure model for the structural analysis in this study consist of hinges,development of plastic hinges that occurred during crack propagation were not considered in this model.To simulate the development of crack,truss members over allowable stresses of tension or compression among truss members with the largest amount of stress at each completed structural analysis time step were sequentially removed.Since irregular triangle meshes have an uncertainty in themselves to compare with regular meshes,the equivalent truss structure model could describe crack propagation more realistically.The failure probabilities of structures under various loads and boundary conditions had good agreement with the analytical solutions directly solved from the limit state equations expressed in the form of moments.

Dynamic Response of Truss Adaptive Structures

It is pointed out that the damping matrix deduced by active members in the finite element vibration equation of a truss adaptive structure generally can not be decoupled, which leads to the difficulty in the process of modal analysis by classical superposition method. This paper focuses on the computational method of the dynamic response for truss adaptive structures. Firstly, a new technique of state vector approach is applied to study the dynamic response of truss adaptive structures. It can make the coeffic lent matrix of first derivative of state vector a symmetric positive definite matrix, and particularly a diagonal matrix provided that mass matrix is derived by lumped method, so the coefficient matrix of the first derivative of state vector can be exactly decomposed by CHOLESKY method. In this case, the proposed technique not only improves the calculation accuracy, but also saves the computing time. Based on the procedure mentioned above, the mathematical formulation for the system response of truss adaptive structures is systematically derived in theory. Thirdly, by using FORTRAN language, a program system for computing dynamic response of truss adaptive structures is developed. Fourthly, a typical 18 bar space truss adaptive structure has been chosen as test numerical examples to show the feasibility and effectiveness of the proposed method. Finally, some good suggestions, such as how to choose complex mode shapes practically in determining the dynamic response are also given. The new approach can be extended to calculate the dynamic response of general adaptive structures.

Optimal Placement of Active Members in Truss Adaptive Structures

The mathematical model of optimal placement of active members in truss adaptive structures is essentially a nonlinear multi-objective optimization problem with mixed variables. It is usually much difficult and costly to be solved. In this paper, the optimal location of active members is treated in terms of （0, 1） discrete variables. Structural member sizes, control gains, and （0, 1） placement variables are treated simultaneously as design variables. Then, a succinct and reasonable compromise scalar model, which is transformed from original multi-objective optimization, is established, in which the （0, 1） discrete variables are converted into an equality constraint. Secondly, by penalty function approach, the subsequent scalar mixed variable compromise model can be formulated equivalently as a sequence of continuous variable problems. Thirdly, for each continuous problem in the sequence, by choosing intermediate design variables and temporary critical constraints, the approximation concept is carried out to generate a sequence of explicit approximate problems which enhance the quality of the approximate design problems. Considering the proposed method, a FORTRAN program OPAMTAS2.0 for optimal placement of active members in truss adaptive structures is developed, which is used by the constrained variable metric method with the watchdog technique （CVMW method）. Finally, a typical 18 bar truss adaptive structure as test numerical examples is presented to illustrate that the design methodology set forth is simple, feasible, efficient and stable. The established scalar mixed variable compromise model that can avoid the ill-conditioned possibility caused by the different orders of magnitude of various objective functions in optimization process, therefore, it enables the optimization algorithm to have a good stability. On the other hand, the proposed novel optimization technique can make both discrete and continuous variables be optimized simultaneously.

An adaptive reanalysis method for genetic algorithm with application to fast truss optimization

Although the genetic algorithm （GA） for structural optimization is very robust, it is very computationally intensive and hence slower than optimality criteria and mathematical programming methods. To speed up the design process, the authors present an adaptive reanalysis method for GA and its applications in the optimal design of trusses. This reanalysis technique is primarily derived from the Kirsch＇s combined approximations method. An iteration scheme is adopted to adaptively determine the number of basis vectors at every generation. In order to illustrate this method, three classical examples of optimal truss design are used to validate the proposed reanalysis-based design procedure. The presented numerical results demonstrate that the adaptive reanalysis technique affects very slightly the accuracy of the optimal solutions and does accelerate the design process, especially for large-scale structures.

Effect of temperature on the compressive behavior of carbon fiber composite pyramidal truss cores sandwich panels with reinforced frames

This paper focuses on the effect of temperature on the out-of-plane compressive properties and failure mechanism of carbon fiber/epoxy composite pyramidal truss cores sandwich panels （CF/CPTSP）. CFJCPTSP with novel reinforced frames are manufactured by the water jet cutting and interlocking assembly method in this paper. The theoretical analysis is presented to predict the out-of-plane compressive stiffness and strength of CFJCPTSP at different ambient temperatures. The tests of composite sandwich panels are per- formed throughout the temperature range from -90℃ to 180℃. Good agreement is found between theo- retical predictions and experimental measurements. Experimental results indicate that the low tempera- ture increases the compressive stiffness and strength of CF/CPTSP. However, the high temperature causes the degradation of the compressive stiffness and strength. Meanwhile, the effects of temperature on the failure mode of composite sandwich panels are also observed.

A NEW METHOD FOR STRUCTURAL TOPOLOGICAL OPTIMIZATION BASED ON THE CONCEPT OF INDEPENDENT CONTINUOUS VARIABLES AND SMOOTH MODEL

A concept of the independent-continuous topological variable is proposed to establish its corresponding smooth model of structural topological optimization. The method can overcome difficulties that are encountered in conventional models and algorithms for the optimization of the structural topology. Its application to truss topological optimization with stress and displacement constraints is satisfactory, with convergence faster than that of sectional optimizations.

Enhanced design of hourglass truss sandwich structures for compressive resistance

In this article,the design of the hourglass truss sandwich structure is improved by optimizing the number of layers to enhance the compressive strength of both the core and the face sheet and then its mechanical performance. The hourglass truss structures characterized by three different numbers of layers are manufactured using an interlocking and vacuum brazing method. The effect of the layer number of the hourglass core panels on their out-of-plane compression and in-plane compression performance is investigated,and the results from calculations and experiments are in reasonable agreement. The results show that as the layer number of the hourglass core increases,the out-of-plane compressive strengths show little change,but their energy absorption properties are effectively increased. The in-plane compressive failure mechanism maps are constructed,and the specimens are designed to examine the local elastic and inelastic buckling failure modes of the face sheets. The results suggest that as the number of layers of the hourglass core increases,its maximum in-plane compressive load increases. The maximum in-plane compressive loads of the two-layer hourglass truss panels are 57%–70% higher than those of the single-layer panels. It can also be concluded that the out-of-plane and in-plane compression mechanical properties of the multilayer hourglass truss outperform those of the pyramidal truss. Furthermore,the number of layers of the hourglass core is optimized in consideration of both mechanical properties and fabrication cost.

Internal force and deformation matrixes and their applications in load path

This paper deals with the internal force and the deformation matrixes, both of which can be used to analyze the topological relationship of a structure. Based on the reciprocal theorem, the relationship between the two matrixes is established, which greatly simplifies the computation of the internal force matrix. According to the characteristics of the internal force matrix, the transfer law of the matrix itself (due to the removal of components) is established based on the principle of linear superposition. With the relation of the two matrixes, the transfer law of the deformation matrix is also obtained. The transfer law illuminates the change regularity of internal force or deformation of the remnant structure when certain members are cut off one after another. The results of numerical examples show that the proposed methods are correct, reliable and effective.