Vitali Volovoi

On Timoshenko-like modeling of initially curved and twisted composite beams

Abstract A generalized, finite-element-based, cross-sectional analysis for nonhomogenous, initially curved and twisted, anistropic beams is formulated from geometrically nonlinear, three-dimensional (3-D) elasticity. The 3-D strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one-dimensional (1-D) generalized strains and a 3-D warping displacement that is obtained from the formulation, not assumed. The warping is found in terms of the 1-D strains via the variational asymptotic method (VAM). In this paper a Timoshenko-like model is presupposed for a beam with cross-sectional characteristic length h, wavelength of deformation given by l, and the magnitude of the radius of initial curvature and/or twist is taken to be of the order R. First, a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through O(h2/R2) is obtained. Next, the O(h2/l2) correction is computed. It is known that Timoshenko-like theory is not capable of capturing all the O(h2/l2) corrections for generally anisotropic beams. However, if all the O(h2/l2) terms are known, then the corresponding Timoshenko-like theory is uniquely defined. Numerical results are presented to illustrate the trends of the various classical (extension-twist, bending-twist, and extension-bending) and nonclassical couplings (extension-shear, bending-shear, and shear-torsion) as the initial twist and curvatures are varied.

Validation of the Variational Asymptotic Beam sectional Analysis

The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the variational asymptotic method to split a three-dimensional nonlinear elasticity problem into a twodimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with transverse shear and Vlasov refinements, for initially twisted and curved beams with arbitrary geometry and material properties. Several validation cases are presented. First, an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method, and the solution is shown to be identical to that obtained from the theory of elasticity. The shear center locations calculated by VABS for various cross sections agree well with those obtained from common engineering analyses. Comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model. For numerical validation, values of the one-dimensional variables are extracted from an ABAQUS model and compared with results from a one-dimensional beam analysis using cross-sectional constants from VABS. Furthermore, point-wise three-dimensional stress and strain fields are recovered using VABS, and the correlation with the three-dimensional results from ABAQUS is excellent. Finally, classical theory is shown to be insufficient for general-purpose beam modeling. Appropriate refined theories are recommended for some classes of problems.

Modeling of system reliability Petri nets with aging tokens

Abstract The paper addresses the dynamic modeling of degrading and repairable complex systems. Emphasis is placed on the convenience of modeling for the end user, with special attention being paid to the modeling part of a problem, which is considered to be decoupled from the choice of solution algorithms. Depending on the nature of the problem, these solution algorithms can include discrete event simulation or numerical solution of the differential equations that govern underlying stochastic processes. Such modularity allows a focus on the needs of system reliability modeling and tailoring of the modeling formalism accordingly. To this end, several salient features are chosen from the multitude of existing extensions of Petri nets, and a new concept of aging tokens (tokens with memory) is introduced. The resulting framework provides for flexible and transparent graphical modeling with excellent representational power that is particularly suited for system reliability modeling with non-exponentially distributed firing times. The new framework is compared with existing Petri-net approaches and other system reliability modeling techniques such as reliability block diagrams and fault trees. The relative differences are emphasized and illustrated with several examples, including modeling of load sharing, imperfect repair of pooled items, multiphase missions, and damage-tolerant maintenance. Finally, a simple implementation of the framework using discrete event simulation is described.

Theory of Anisotropic Thin-Walled Beams

Asymptotically correct, linear theory is presented for thin-walled prismatic beams made of generally anisotropic materials. Consistent use of small parameters that are intrinsic to the problem permits a natural description of all thin-walled beams within a common framework, regardless of whether cross-sectional geometry is open, closed, or strip-like, Four classical one-dimensional variables associated with extension, twist, and bend-ing in two orthogonal directions are employed. Analytical formulas are obtained for the resulting 4 × 4 cross-sectional stiffness matrix (which, in general, is fully populated and includes all elastic couplings) as well as for the strain field. Prior to this work no analytical theories for beams with closed cross sections were able to consistently include shell bending strain measures. Corrections stemming from those measures are shown to be important for certain cases. Contrary to widespread belief, it is demonstrated that for such classical theories, a cross section is not rigid in its own plane. Vlasovs correction is shown to be unimportant for closed sections, while for open cross sections asymptotically correct formulas for this effect are provided. The latter result is an extension to a general contour of a result for 1-beams previously published by the authors.

Asymptotic construction of Reissner-like composite plate theory with accurate strain recovery

Abstract The focus of this paper is to develop an asymptotically correct theory for composite laminated plates when each lamina exhibits monoclinic material symmetry. The development starts with formulation of the three-dimensional (3-D), anisotropic elasticity problem in which the deformation of the reference surface is expressed in terms of intrinsic two-dimensional (2-D) variables. The variational asymptotic method is then used to rigorously split this 3-D problem into a linear one-dimensional normal-line analysis and a nonlinear 2-D plate analysis accounting for classical as well as transverse shear deformation. The normal-line analysis provides a constitutive law between the generalized, 2-D strains and stress resultants as well as recovering relations to approximately but accurately express the 3-D displacement, strain and stress fields in terms of plate variables calculated in the plate analysis. It is known that more than one theory may exist that is asymptotically correct to a given order. This nonuniqueness is used to cast a strain energy functional that is asymptotically correct through the second order into a simple “Reissner-like” plate theory. Although it is not possible in general to construct an asymptotically correct Reissner-like composite plate theory, an optimization procedure is used to drive the present theory as close to being asymptotically correct as possible while maintaining the beauty of the Reissner-like formulation. Numerical results are presented to compare with the exact solution as well as a previous similar yet very different theory. The present theory has excellent agreement with the previous theory and exact results.

Asymptotic theory for static behavior of elastic anisotropic I-beams

Abstract End effects for prismatic anisotropic beams with thin-walled, open cross-sections are analyzed by the variational-asymptotic method. The decay rates for disturbances at the ends of prismatic beams are evaluated, and the most influential end disturbances are incorporated into a refined beam theory. Thus, the foundations of Vlasovs theory, as well as restrictions on its applicability, are obtained from the variational-asymptotic point of view. Vlasovs theory is proved to be asymptotically correct for isotropic I-beams. The asymptotically correct generalization of Vlasovs theory for static behavior of anisotropic beams is presented. In light of this development, various published generalizations of Vlasovs theory for thin-walled anisotropic beams are discussed. Comparisons with a numerical 3-D analysis are provided, showing that the present approach gives the closest agreement of all published theories. The procedure can be applied to any thin-walled beam with open cross-sections.

Asymptotically accurate 3-D recovery from Reissner-like composite plate finite elements

Abstract An accurate stress/strain recovery procedure for laminated, composite plates that can be implemented in standard finite element programs is developed. The formulation is based on an asymptotic analysis and starts from a three-dimensional, anisotropic elasticity problem that takes all possible deformation into account. After a change of variable, which introduces intrinsic two-dimensional description for the deformation of the reference plane, the variational asymptotic method is then used to rigorously split this three-dimensional problem into two reduced-dimensional problems: a nonlinear, two-dimensional analysis of the reference surface of the deformed plate (an equivalent single-layer plate model), and a linear, one-dimensional analysis of the normal-line element through the thickness. The latter is solved by a one-dimensional finite element method and provides a constitutive law between the generalized, two-dimensional strains and stress resultants for the plate analysis, and a set of recovering relations to approximately express the three-dimensional displacement, strain and stress fields in terms of two-dimensional variables determined from solving the equations of the plate analysis. The strain energy functional that is asymptotically correct through the second-order in the small parameters is then cast into the form of Reissner’s theory. Although it is not in general possible to construct an asymptotically correct Reissner-like composite plate theory, an optimization procedure is used to drive the present theory as close as possible to being asymptotically correct, while maintaining the simplicity and beauty of the Reissner-like formulation. A computer program based on the present procedure, called variational asymptotic plate and shell analysis, has been developed. Its utility is demonstrated by inserting the recovery procedure into the plate element of a general-purpose finite element code. Numerical results obtained for a variety of laminated, composite plates show that three-dimensional field variables recovered from the present theory agree very well with those from exact solutions.

Non-classical effects in non-linear analysis of pretwisted anisotropic strips☆

Abstract The literature on classical analysis of anisotropic beams assumes that all 1D “moment strain” measures (i.e. twist and bending curvatures) are of the same order of magnitude, resulting in a linear cross-sectional analysis. The present paper treats the situation in which one or more of the 1D moment strain measures may be larger than the other(s), resulting in a non-linear cross-sectional analysis. This type of non-classical analysis is needed, for example, in problems where the trapeze effect is important, such as in rotor blades. As a precursor to complicated non-linear sectional analysis of arbitrary cross sections, a non-linear sectional analysis is presented for an anisotropic strip with small pretwist, based on the dimensional reduction of laminated shell theory to a non-linear one-dimensional theory using the variationalasymptotic method. Results obtained from this strip-beam analysis are compared with available theoretical and experimental results for a problem in which the trapeze effect is important. In order to demonstrate the usage of the results in the analysis of structures made of an arbitrary geometrical combination of pretwisted generally anisotropic strips, a closed-form expression is derived for the torsional buckling of a column with a cruciform cross section.

A Fault Diagnosis Method for Industrial Gas Turbines Using Bayesian Data Analysis

This paper presents an offline fault diagnosis method for industrial gas turbines in a steady-state. Fault diagnosis plays an important role in the efforts for gas turbine owners to shift from preventive maintenance to predictive maintenance, and consequently to reduce the maintenance cost. Ever since its birth, numerous techniques have been researched in this field, yet none of them is completely better than the others and perfectly solves the problem. Fault diagnosis is a challenging problem because there are numerous fault situations that can possibly happen to a gas turbine, and multiple faults may occur in multiple components of the gas turbine simultaneously. An algorithm tailored to one fault situation may not perform well in other fault situations. A general algorithm that performs well in overall fault situations tends to compromise its accuracy in the individual fault situation. In addition to the issue of generality versus accuracy, another challenging aspect of fault diagnosis is that, data used in diagnosis contain errors. The data is comprised of measurements obtained from gas turbines. Measurements contain random errors and often systematic errors like sensor biases as well. In this paper, to maintain the generality and the accuracy together, multiple Bayesian models tailored to various fault situations are implemented in one hierarchical model. The fault situations include single faults occurring in a component, and multiple faults occurring in more than one component. In addition to faults occurring in the components of a gas turbine, sensor biases are explicitly included in the multiple models so that the magnitude of a bias, if any, can be estimated as well. Results from these multiple Bayesian models are averaged according to how much each model is supported by data. Gibbs sampling is used for the calculation of the Bayesian models. The presented method is applied to fault diagnosis of a gas turbine that is equipped with a faulty compressor and a biased fuel flow sensor. The presented method successfully diagnoses the magnitudes of the compressor fault and the fuel flow sensor bias with limited amount of data. It is also shown that averaging multiple models gives rise to more accurate and less uncertain results than using a single general model. By averaging multiple models, based on various fault situations, fault diagnosis can be general yet accurate. DOI: 10.1115/1.3204508

Application of Petri nets to reliability prediction of occupant safety systems with partial detection and repair

This paper presents an application of stochastic Petri nets (SPN) to calculate the availability of safety critical on-demand systems. Traditional methods of estimating system reliability include standards—based or field return-based reliability prediction methods. These methods do not take into account the effect of fault-detection capability and penalize the addition of detection circuitry due to the higher parts count. Therefore, calculating system availability, which can be linked to the system’s probability of failure on demand (Pfd), can be a better alternative to reliability prediction. The process of estimating the Pfd of a safety system can be further complicated by the presence of system imperfections such as partial-fault detection by users and untimely or uncompleted repairs. Additionally, most system failures cannot be represented by Poisson process Markov chain methods, which are commonly utilized for the purposes of estimating Pfd, as these methods are not well-suited for the analysis of non-Poisson failures. This paper suggests a methodology and presents a case study of SPN modeling adequately handling most of the above problems. The model will be illustrated with a case study of an automotive electronics airbag controller as an example of a safety critical on-demand system.