Title: Variability response functions for statically determinate beams with arbitrary nonlinear constitutive laws,
Abstract: The variability response function (VRF) is generalized to statically
determinate Euler Bernoulli beams with arbitrary stress-strain laws following
Cauchy elastic behavior. The VRF is a Green's function that maps the spectral
density function (SDF) of a statistically homogeneous random field describing
the correlation structure of input uncertainty to the variance of a response
quantity. The appeal of such Green's functions is that the variance can be
determined for any correlation structure by a trivial computation of a
convolution integral. The method introduced in this work derives VRFs in closed
form for arbitrary nonlinear Cauchy-elastic constitutive laws and is
demonstrated through three examples. It is shown why and how higher order
spectra of the random field affect the response variance for nonlinear
constitutive laws. In the general sense, the VRF for a statically determinate
beam is found to be a matrix kernel whose inner product by a matrix of higher
order SDFs and statistical moments is integrated to give the response variance.
The resulting VRF matrix is unique regardless of the random field's marginal
probability density function (PDF) and SDFs.