# Variability response functions for statically determinate beams with arbitrary nonlinear constitutive laws

## TL;DR

This paper extends the variability response function (VRF) concept to statically determinate beams with arbitrary nonlinear stress-strain laws, providing a closed-form Green's function that simplifies variance calculations under uncertainty.

## Contribution

It derives closed-form VRFs for nonlinear elastic laws and demonstrates how higher order spectra influence response variance in such beams.

## Key findings

- VRFs are expressed as matrix kernels for nonlinear laws.
- Higher order spectra significantly affect response variance.
- The VRF matrix is unique regardless of the random field's PDF.

## 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.