The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions

Most existing and planned offshore wind turbines (OWTs) are located in shallow water where the possibility of breaking waves impacting the structure may influence design. Breaking waves and their associated impact loads are challenging to model because the breaking process is a strongly non-linear phenomenon with significant statistical scattering. Given the challenges and uncertainty in modeling breaking waves, there is a need for comparing existing models with simultaneous environmental and structural measurements taken from utility-scale OWTs exposed to breaking waves. Overall, such measurements are lacking; however, one exception is the Offshore Wind Turbines at Exposed Sites project, which recorded sea state conditions and associated structural loads for a 2.0 MW OWT supported by a monopile and located at the Blyth wind farm off the coast of England. Measurements were recorded over a 17 month campaign between 2001 and 2003, a period that included a storm that exposed the instrumented OWT to dozens of breaking waves. This paper uses the measurements from this campaign to categorize and identify breaking waves and quantify the variability of their impact loads. For this particular site and turbine, the distribution of measured mudline moments due to breaking waves has a mean of 8.7\ MN-m, a coefficient of variation of 26\% and a maximum of 14.9\ MN-m. The accuracy of several breaking wave limits and impact force models is compared with the measurements, and the impact force models are shown to represent the measurements with varying accuracy and to be sensitive to modeling assumptions.

}, doi = {10.1002/we.1833}, author = {Hallowell, Spencer and Myers, Andrew T. and Arwade, Sanjay R.} } @article {11007, title = {Validity of stationary probabilistic models for wind speed records of varying duration}, journal = {Renewable Energy}, volume = {69}, year = {2014}, month = {09/2014}, pages = {74-81}, chapter = {74}, abstract = {A method for assessing the degree of non-stationarity in annual wind speed records is presented. The method uses quantitative tests on the wind speed records to assess the length of the period over which an assumption of stationarity in the wind record can be considered to provide reasonable engineering accuracy. The tests evaluate stationarity in second moment properties and marginal distribution. Numerical examples are provided for three offshore sites along the Atlantic Coast of the United States\–one off of Virginia, and two off of Maine. The examples illustrate that an assumption of stationarity over a period of one week is largely justified, but that such an assumption over periods of one month is certainly not. Assuming stationarity over a period of a week can lead to errors in model values of the second moment properties of 2\%\–3\% whereas the assumption applied to a monthlong period can lead to error greater than 10\%. Examination of the persistence of marginal distribution reveals that, although true stationarity in marginal distribution persists for a few days at most, there exist two \‘seasons\’, winter and summer, during which the marginal distribution remains relatively consistent, with rapid changes in marginal distribution occurring near the beginning and ends of these seasons. Results are found to be largely consistent across the three sites investigated as numerical examples. The methods and results presented here may be useful to those investigating the potential for offshore wind energy development using stochastic process theory to study wind speed or power production since stationary stochastic models provide simpler and more accessible predictions of quantities such as probabilities of exceedance of threshold values, upcrossing rates, and residence times.

}, keywords = {Mean wind speed, Stationarity, Stochastic process, Wind energy}, doi = {10.1016/j.renene.2014.03.016}, author = {Arwade, Sanjay R. and Gioffre, Massimiliano} } @article {11563, title = {Variability of the mechanical properties of wrought iron from historic American truss bridges}, journal = {Journal of Materials in Civil Engineering, ASCE}, volume = {23}, year = {2011}, month = {05/2011}, pages = {638-647}, chapter = {638}, abstract = {Measurement of the compressive strength of parallel strand lumber (PSL) is conducted on specimens of varying size with nominally identical mesostructure. The mean of the compressive strength is found to vary inversely with the specimen size, and the coefficient of variation of the strength is found to decrease with increasing specimen size, and to be smaller than the coefficient of variation of strength for solid lumber. The correlation length of the compressive strength is approximately 0.5 m, and this correlation length leads to significant specimen-to-specimen variation in mean strength. A computational model is developed that includes the following properties of the PSL mesostructure: the strand length, the grain angle, the elastic constants, and the parameters of the Tsai-Hill failure surface. The computational model predicts the mean strength and coefficient of variation reasonably well, and predicts the correct form of correlation decay, but overpredicts the correlation length for compressive strength, likely because of sensitivity to the distribution of strand length used in the model. The estimates of the statistics of the PSL compressive strength are useful for reliability analysis of PSL structures, and the computational model, although still in need of further development, can be used in evaluating the effect of mesostructural parameters on PSL compressive strength.

}, keywords = {Composite lumber, Measurement, Mechanics, Probability, Random processes, Simulation, Strength, Wood}, issn = {0733-9399}, doi = {10.1061/(ASCE)EM.1943-7889.0000079}, author = {Arwade, Sanjay R. and Winans, Russell and Clouston, Peggi L.} } @article {11223, title = {Variability response functions for effective material properties}, journal = {Probabilistic Engineering Mechanics Probabilistic Engineering Mechanics}, volume = {26}, year = {2010}, month = {04/2011}, pages = {174-181}, chapter = {174}, abstract = {
The variability response function (*VRF*) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a\ *VRF*\ for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a\ *VRF*\ exists for the effective material properties of statically determinate structures. The\ *VRF*\ for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement*VRF*\ to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the*VRF*\ for effective material properties, the method of calculation, and results that can be obtained from it.

This paper applies the Sobol\’ decomposition of a function of many random variables to a problem in structural mechanics, namely the collapse of a two story two bay frame under gravity load. Prior to introduction of this example application, the Sobol\’ decomposition itself is reviewed and extended to cover the case in which the input random variables have Gaussian distribution. Then, an illustrative example is given for a polynomial function of 3 random variables.

In the structural example, the Sobol\’ decomposition is used to decompose the variance of the response, the collapse load, into contributions from the individual input variables. This decomposition reveals the relative importance of the individual member yield stresses in determining the collapse load of the frame. In applying the Sobol\’ decomposition to this structural problem the following issues are addressed: Calculation of the components of the Sobol\’ decomposition by Monte Carlo simulation; the effect of input distribution on the Sobol\’ decomposition; convergence of estimates of the Sobol\’ decomposition with sample size using various sampling schemes; the possibility of model reduction guided by the results of the Sobol\’ decomposition.

}, keywords = {Monte Carlo simulation, Sensitivity analysis, Sobol{\textquoteright} decomposition, Structural collapse analysis, Uncertainty quantification, Variance decomposition}, doi = {10.1016/j.engstruct.2009.08.011}, author = {Arwade, Sanjay R. and Moradi, Mohammadreza and Louhghalam, Arghavan} }