Random System Properties Research Articles

Long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos

Polynomial Chaos (PC) is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems. In the case of a structural system under dynamic loading, the PC method is combined with a time integration scheme. The PC method with Galerkin projection converts the stochastic PDE to a set of simultaneous deterministic PDE, which can be solved using any standard time integration scheme. However, the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response. The probability density functions of the responses are not of constant type as the time progresses. However, in the case of generalized PC (gPC), these are approximated using initially considered gPC, thus loses its efficiency. Time-dependent general polynomial chaos (TDgPC) can be considered to overcome these difficulties, where polynomials are generated on the fly to match with the pdf of the responses. The present study considered an investigation of structural mechanics responses with both Gaussian and non-Gaussian random system properties under dynamic loading using TDgPC. An adaptive updation scheme based on the RMS value of the coefficients of PC expansion is investigated for a proper updation of PC expansion. The curse of dimensionality of PC expansion is addressed by considering only the dominant components of the responses evaluated using KL expansion. Further, starting with lower-order PC, the order of PC expansion is increased only after the first updation, thus reducing computational complexities.

Finite Element-Based Stochastic Nonlinear Progressive Failure Response of Piezo-Laminated Composite Plate with Elliptical Cutouts

This paper presents the second-order statistics of hygro-thermo-electrically-induced progressive failure in terms of first-ply failure load (FPFL) and last-ply failure load (LPFL) analysis for laminated composite material plate (LCMP) under out of plane mechanical loading with random system properties. Basic governing equation of nonlinear progressive failure analysis is based on shear deformation theory (higher order) with von-Karman nonlinear kinematics using Newton’s Raphson approach through Tsai–Wu failure criteria. The random input variables are assumed as uncorrelated type and are evaluated using second-order perturbation method (SOPT). Laminated composite plate with elliptical cutouts are subjected to uniformly distributed, point and hydrostatic load. The effect of boundary conditions, temperature variation, moisture content and voltage variations by utilizing piezoelectric layer position and various cutout shapes on the mean and corresponding covariance (COV) of FPFL and LPFL load are evaluated. Convergence of numerical analysis is performed, and results are validated with those available in literatures to check the efficiency of present methodology. It is observed that the presence of elliptical hole always causes an increase in the failure load of plates subjected to bending, even further increase for LPFL due to the reduction of stresses.

Stochastic critical stress intensity factor response of single edge notched laminated composite plate using displacement correlation method

The second-order statistics of critical stress intensity factor (SIF) of single edge notched fiber reinforced composite plates with random system properties and subjected to uniaxial tensile loadings is investigated. This paper is an extension of reference (Lal and Kapania, 2013) by the present authors by considering more number of input random system parameters for higher accuracy. A C0 finite element method based on a higher-order shear deformation plate theory using displacement correlation method via isoparametric quarter point element is proposed for basic formulation. A stochastic finite element method using first-order perturbation technique and Monte Carlo simulation (MCS) is employed to examine the mean, coefficient of variance, and probability density faction of critical first mode SIF. The effect of different fiber orientations, crack length, plate thickness, a number of layers, and the lamination schemes with random system properties on the statistics of SIF of single edge crack laminated composite plate is evaluated. The tensile failure load is predicted using Hashin’s failure criteria. The present approach is validated with results available in literature and by employing independent MCS.

Uncertainty quantification in non-linear dynamic response of functionally graded materials plate

ABSTRACTThis study deals with the stochastic non-linear dynamic response of functionally graded materials (FGMs) plate with uncertain system properties subjected to time-dependent uniformly distributed transverse load in thermal environments. System properties, such as material properties of each constituent's material, volume fraction index, and transverse load, are taken as uncorrelated random input variables. Material properties are assumed as temperature dependent (TD). The formulation is based on higher-order shear deformation theory (HSDT) with von-Karman non-linear strain kinematics using modified C° continuity. A Newton–Raphson-based non-linear finite element method along with a first-order perturbation technique (FOPT) and Monte Carlo sampling (MCS) is outlined to examine the second-order statistics (mean, standard deviation (SD), and probability density function (PDF)) of the non-linear dynamic response of the FGM plate. The governing dynamic equation is solved by Newmark's time integration scheme. The effects of volume fraction index, load parameters, plate thickness ratios, and temperature changes with random system properties are examined through parametric studies. The present outlined approach is validated with the results available in the literature and by MCS.

Stochastic nonlinear bending response of elastically supported nanotube-reinforced composite beam in thermal environment

This paper presents the second-order statistics of the nonlinear bending response of elastically supported single-wall carbon nanotube reinforced composite (CNTRC) beam composed of uniformly distributed (UD) and functionally graded (FG) reinforcements in thermal environments with uncertain system properties. The uncertain system properties such as material properties of matrix and SWCNTs, foundation parameters, thermal expansion coefficients are be modeled as uncorrelated Gaussian random variables. The material properties of FG-CNTRCs are assumed to be graded in the beam thickness direction and are evaluated through a micromechanical model. The higher order shear deformation theories (HSDT) with von-Karman nonlinear strain kinematics are used for the mathematical formulation of CNTRCs beam. The thermal effects are also included in the material properties of CNTRCs which are assumed to be temperature dependent and independent. The second-order perturbation technique (SOPT) and Monte Carlo simulation (MCS) via [Formula: see text] nonlinear finite element method (FEM) through Newton–Raphason method are proposed to examine the mean, COV and probability density function (PDF) of transverse deflection of the beam. Typical numerical results are presented for the different volume fraction of carbon nanotube, slenderness ratios, boundary conditions, foundation parameters, load parameters, CNTRC distribution, temperature dependent and independent material properties with random system properties. The present outlined approach is validated with the results available in the literature and by employing MCS.

Stochastic Fracture Response and Crack Growth Analysis of Laminated Composite Edge Crack Beams Using Extended Finite Element Method

This paper presents a stochastic fracture response and crack growth analysis of mixed-mode stress intensity factors (MSIFs) for edge cracked laminated composite beams subjected to uniaxial, uniform tensile, shear and combined stresses with random system properties. The randomness in material properties of the composite material, lamination angle, laminate thickness, the crack length and the crack angle are modeled as both input uncorrelated and correlated random variables. An extended finite element method (XFEM) through the so-called M-interaction approach combined with the second-order perturbation technique (SOPT) and Monte Carlo simulation (MCS) is used to obtain the statistics in terms of the mean and coefficient of variation (COV) of MSIFs for edge cracked laminated composite beams. The effect of crack propagation on the MSIFs in the presence of tensile, shear and combined stresses using a global tracking algorithm is also investigated. The results using the present approach are compared with the available published results. A good agreement is seen whenever alternative results are available.

Stochastic analysis of thin plates on elastic foundation by combining the generalized polynomial chaos and element free Galerkin method

Uncertainties associated with a physical system are of great importance as they cause the system response to behave randomly. Plated structures exhibit considerable variation in mechanical properties, material density and dimensions due to inadequate control in the manufacturing processes. In this paper, the element free Galerkin method is combined with the generalized polynomial chaos to quantify the uncertainties in the bending analysis of thin plates on a Pasternak elastic foundation with random system properties. The uncertainties in plate and foundation stiffnesses are represented using a truncated Karhunen-Loeve expansion. The applicability of the presented method is demonstrated by solving numerical examples for various combinations of boundary conditions, different types of lateral loading, various values of aspect ratio and different values of coefficient of variation and foundation parameters. Further, by comparing the results of the present method with the results of the Monte Carlo simulations, a very good agreement is obtained.

Stochastic analysis of moderately thick plates using the generalized polynomial chaos and element free Galerkin method

In this paper, the element free Galerkin method is combined with the generalized polynomial chaos to quantify the uncertainties in the bending analysis of shear deformable plates with elastically restrained edges resting on a Pasternak elastic foundation with random system properties. The plate modules of elasticity, stiffnesses of the elastically restrained edges and the foundation stiffnesses are considered as random processes and are represented by using the Karhunen–Loève expansion. It is shown that the results obtained by the presented method are in a very good agreement with the results of Monte Carlo simulations in spite of using low order polynomials in the generalized polynomial chaos expansion. Also the applicability and versatility of the presented method are demonstrated by solving numerical examples for various values of coefficient of variations, aspect ratio, thickness and several combinations of boundary conditions, different types of lateral loading and various values of stiffnesses of restrained edges and elastic foundation.

Nonlinear free vibration analysis of elastically supported carbon nanotube-reinforced composite beam with the thermal environment in non-deterministic framework

AbstractThis paper deals with the investigation of nonlinear free vibration behavior of elastically supported carbon nanotube reinforced composite (CNTRC) beam subjected to thermal loading with random system properties. Material properties of each constituent’s material, volume fraction exponent and foundation parameters are considered as uncorrelated Gaussian random input variables. The beam is supported by a Pasternak foundation with Winkler cubic nonlinearity. The higher order shear deformation theory (HSDT) with von-Karman non-linearity is used to formulate the governing equation using Hamilton principle. Convergence and validation study is carried out through the comparison with the available results in the literature for authenticity and accuracy of the present approach used in the analysis. First order perturbation technique (FOPT),Second order perturbation technique (SOPT) and Monte Carlo simulation (MCS) methods are employed to investigate the effect of geometric configuration, volume fraction exponent, foundation parameters, distribution of reinforcement and thermal loading on nonlinear vibration characteristics CNTRC beam.The present work signifies the accurate analysis of vibrational behaviour influences by different random variables. Results are presented in terms of mean, variance (COV) and probability density function (PDF) for various aforementioned parameters.

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.

Stochastic hygro-thermo-mechanically induced nonlinear static analysis of piezoelectric elastically support sandwich plate using secant function based shear deformation theory (SFSDT)

The second order statistics of transverse nonlinear central deflection of elastically supported piezoelectric laminated composite sandwich plate (ESPLCSP) subjected to hygro-thermo-mechanical loading using micromechanical approach is evaluated in this paper. System randomness as micro-level material properties of fiber and matrix, material properties of piezoelectric, laminate thickness, lamination angle, foundation parameters, and load intensity are taken as independent random variables. The mechanical loading is taken as uniformly distributed and sinusoidal loadings. The secant function based shear deformation theory (SFSDT) with von-Karman nonlinearity is used for basic formulation. The elastic and hygrothermal properties of the composite material are considered to be dependent on temperature and moisture concentration have been evaluated utilized micromechanical modeling. A Newton–Raphson method based on [Formula: see text] nonlinear finite element method combined with mean centered second order perturbation technique (SOPT) proposed by present authors for the composite plate is extended for sandwich composite plate. The effect of random system properties with changing the plate geometry, stacking sequences, support conditions, foundation parameters, piezoelectric layers, fiber volume fraction and temperature, and moisture distribution on ESPLCSP is presented. The performance of proposed approach is validated through comparison with those available in the literature and independent Monte Carlo simulation (MCS).

Stochastic extended finite element implementation for fracture analysis of laminated composite plate with a central crack

A stochastic extended finite element method (SXFEM), developed previously by the first two authors, has been extended for the fracture analysis along with reliability analysis of the central cracked laminated composite plate subjected to uni-axial tension with random system properties. The implemented SXFEM approach is based on the M-integral interaction combined with second-order perturbation technique (SOPT) and independent Monte Carlo simulation (MCS) is performed for the evaluation of statistics of mixed mode stress intensity factors (MMSIFs). The random system properties such as material properties, crack length, crack angle, lamination angle, and uni-axial loading, are assumed as input uncorrelated Gaussian random variables. The effect of the different crack angles, crack lengths, lamination angles, and loading on the statistics of MMSIF in terms of mean, standard deviation (SD), probability density function (PDF) and safety factor of cracked laminated composite plate is examined along with the reliability analysis. The effect of crack propagation and its direction along with its effect on the MMSIFs is carried out through global tracking crack growth algorithm. The results obtained by present approach, are compared with an analytical solution and results available in various literature.

Stochastic dynamic instability response of piezoelectric functionally graded beams supported by elastic foundation

This paper presents the dynamic instability analysis of un-damped elastically supported piezoelectric functionally graded (FG) beams subjected to in-plane static and dynamic periodic thermomechanical loadings with uncertain system properties. The elastic foundation model is assumed as one parameter Pasternak foundation with Winkler cubic nonlinearity. The piezoelectric FG beam is subjected to non-uniform temperature distribution with temperature dependent material properties. The Young's modulus and Poison's ratio of ceramic, metal and piezoelectric, density of respective ceramic and metal, volume fraction exponent and foundation parameters are taken as uncertain system properties. The basic nonlinear formulation of the beam is based on higher order shear deformation theory (HSDT) with von-Karman strain kinematics. The governing deterministic static and dynamic random instability equation and regions is solved by Bolotin's approach with Newmark's time integration method combined with first order perturbation technique (FOPT). Typical numerical results in terms of the mean and standard deviation of dynamic instability analysis are presented to examine the effect of slenderness ratios, volume fraction exponents, foundation parameters, amplitude ratios, temperature increments and position of piezoelectric layers by changing the random system properties. The correctness of the present stochastic model is examined by comparing the results with direct Monte Caro simulation (MCS).

Stochastic progressive failure analysis of laminated composite plates using Puck's failure criteria

ABSTRACTThe present work deals with second-order statistics of the progressive failure response of laminated composite plates subjected to in-plane uniaxial and bi-axial loadings with random system properties. A stochastic finite element method based on higher-order shear deformation theory combined with first- and second-order perturbation technique is used for solution of random progressive failure equation. A Puck failure criterion is used for the evaluation of first ply and last-ply failure load. The results obtained using the present solution approach are validated with the results available in the literature and Monte Carlo simulation.

Probabilistic fracture investigation of symmetric angle ply laminated composite plates using displacement correlation method

AbstractThe second order statistics of mixed mode stress intensity factors (MSIF) of single edge V-notched angle ply laminated composite plate subjected to uniaxial tensile load with uncertinity in the system properties using displacement correlation method (DCM) is evaluated. The random system properties such as material properties, crack opening and crack length are modelled as combined uncorrelated and correlated random system variables. A C0 finite element method (FEM) based on higher order shear deformation plate theory (HSDT) is used for basic formulation. The Taylor series based first order perturbation technique (FOPT), second order perturbation technique (SOPT) are used and direct Monte Carlo simulation (MCS) is performed to evaluate the statistics (mean and coefficient of variance) of the mixed mode SIFs. The present work signifies the accurate analysis of frature behaviour by influence of different random variables and fibre orientations on the fracture behaviour in angle ply laminates.

Stochastic thermal buckling analysis of laminated plates using perturbation technique

Composites are known to display a considerable amount of scatter in their material properties due to large number of parameters involved with the manufacturing and fabrication processes. This paper is concerned with the effect of random system properties on critical thermal buckling temperature of composite laminated plates with temperature dependent properties using micromechanical approach. System parameters are assumed as independent random variables. In this analysis, based on the classical lamination theory in conjunction with the Hamilton’s principle, the basic formulation of random eigenvalue problem has been deduced. A mean-centered first order perturbation technique is used to compute the second-order statistics (mean and standard deviation) of the critical thermal buckling temperature. The performance of outlined stochastic approach has been validated by comparing the present results with those available in the literature and independent Monte Carlo simulation. The effect of random material properties, thermal expansion coefficients, fiber volume fractions, aspect ratios, laying angels and boundary conditions on the critical thermal buckling temperature are presented.

Stochastic Response of Laminated Composite Shell Panel in Hygrothermal Environment

In this article, the effect of random system properties is accounted to estimate the free vibration of multilayered composite shell panel in hygrothermal environment. The majority of previous investigations assumed that the material properties are independent of temperature and moisture. Establishing margin on material properties for design, analysis, and conceptualizing the material is highly difficult and uncertain. To perturb that, a first-order perturbation technique is adopted to obtain the response statistics of the structure by obtaining the mean and variance of random natural frequencies. The higher-order shear deformation theory with 9 degrees of freedom per node with the von-Karman sense of nonlinear kinematics is employed for generating basic formulation. The analysis is carried out by using quadratic C0 eight-noded isoparametric element. The governing equation for free vibration of laminated composite panel is derived using variational principle, which is a generalization of the principle of virtual displacement. The solution methodology is validated with published results. Mean and variance are obtained for different cross-ply of spherical and cylindrical laminates with different stacking sequence and boundary conditions.

Stochastic Thermal Post Buckling Response of Elastically Supported Laminated Piezoelectric Composite Plate Using Micromechanical approach

Abstract In this paper, second order statistics of thermally induced post buckling response of elastically supported piezoelectric laminated composite plate using micromechanical approach is examined. A Co finite element has been used for deriving eigenvalue problem using higher order shear deformation theory (HSDT) with von-Karman nonlinearity. The uncertain system properties such as material properties of fiber and matrix of composite and piezoelectric, fiber volume fraction, plate thickness, lamination angle and foundation are modeled as random variables. The temperature field considered to be uniform temperature distributions through the plate thickness. A direct iterative based nonlinear finite element method combined with mean-centered second order perturbation technique (SOPT) is used to find the mean and coefficient of variance of the post buckling temperature. The effects of volume fraction, fiber orientation, and length to thickness ratio, aspect ratios, foundation parameters, position and number of piezoelectric layers, amplitude and boundary conditions with random system properties on the critical temperature are analysed. It is found that small amount of variations of uncertain system parameters of the composite plate significantly affect the initial and post buckling temperature of laminated composite plate. The results have been validated with independent Monte Carlo simulation (MCS) and those available in literature.

Stochastic Response of Laminated Composite Shell Panel in Hygrothermal Environment

In this article, the effect of random system properties is accounted to estimate the free vibration of multilayered composite shell panel in hygrothermal environment, instead the majority of previous investigations assumed that the material properties are independent of temperature and moisture. Establishing margin on material properties for design, analysis, and conceptualizing the material is highly difficult and uncertain. To perturb that, a first-order perturbation technique is adopted to obtain the response statistics of the structure by obtaining the mean and variance of the random natural frequencies. The higher-order shear deformation theory with nine degrees of freedom (DOFs) per node with von-Karman sense of nonlinear kinematics is employed for generating basic formulation. The analysis is carried out by using quadratic C°eight-noded isoparametric element. The governing equation for free vibration of laminated composite panel is derived using variational principle, which is generalization of the principle of virtual displacement. The solution methodology is validated with published results. Mean and variance is obtained for different cross-ply of spherical and cylindrical laminates with different stacking sequence and boundary conditions.

Thermoelectromechanically induced stochastic post buckling response of piezoelectric functionally graded beam

This paper deals with the stochastic post buckling response the functionally graded material (FGMs) beam with surface bonded piezoelectric layers subjected to thermoelectromechanical loadings. A C0 nonlinear finite element method using higher order shear deformation theory with von-Karman nonlinearity is used for basic formulation. The random system parameter such as material properties of FGM and piezoelectric layers and thermoelectromechanical loadings are modeled as uncorrelated random input variables. The first and second order perturbation method and Monte Carlo sampling (MCS) are proposed to examine the mean, coefficient of variation, probability distribution function and probability of failure of critical post buckling load. Typical numerical results are presented for volume fraction indexes, slenderness ratios, boundary conditions, piezoelectric layers and thermoelectromechanical loadings with random system properties. The present outlined approach is validated with the results available in the literature and by employing MCS.