Acta Mech DOI 10.1007/s00707-017-1903-7 ORIGINAL PAPER Alba Sofi Euler–Bernoulli interval finite element with spatially varying uncertain properties Received: 15 January 2017 / Revised: 14 April 2017 © Springer-Verlag GmbH Austria 2017 Abstract The formulation of an Euler–Bernoulli beam finite element with spatially varying uncertain proper- ties is presented. Uncertainty is handled within a non-probabilistic framework resorting to a recently proposed interval field model able to quantify the dependency between adjacent values of an interval quantity that cannot differ as much as values that are further apart. Once the interval element stiffness matrix is defined, the set of linear interval equations governing the interval global displacements of the finite element model is derived by performing a standard assembly procedure. Then, the bounds of the interval displacements and bending moments are determined in approximate explicit form by applying a response surface approach in conjunction with the so-called improved interval analysis via extra unitary interval. For validation purposes, numerical results concerning both statically determinate and indeterminate beams with interval Young’s modulus are presented. 1 Introduction The growing awareness of the influence of inevitable uncertainties on the performance of engineering systems has stimulated an increasing interest toward the development of efficient non-deterministic numerical proce- dures to achieve more robust and reliable designs (see, e.g., [1, 2]). In this context, a major challenge is to incorporate the non-deterministic input parameters into standard finite element (FE) models and then develop efficient propagation strategies. This task is commendably accomplished within the probabilistic framework by the well-established stochastic finite element method (SFEM) (see, e.g., [3]) which may be regarded as the most powerful tool currently available to integrate random input parameters into numerical modeling. Recently, much research effort has been devoted to develop a similar tool within a non-probabilistic framework, giving rise among others to the so-called interval finite element method (IFEM) where the uncertain parameters are modeled as interval variables with given lower bound (LB) and upper bound (UB) [4]. Such a model proves to be very useful when available data are insufficient to build a credible probabilistic distribution of the uncertain parameters, as it happens in early design stages [5]. Furthermore, the propagation of interval uncertainty usu- ally requires less intensive calculations. The main drawback of the interval model is the overestimation of the interval solution range due to the so-called dependency phenomenon [4] which often leads to useless results in the context of engineering design. Several versions of the IFEM have been proposed with the purpose of finding sharp bounds of the interval output [6–11]. For a general overview of the state-of-the-art and recent A. Sofi Department of Architecture and Territory (dArTe), Inter-University Centre of Theoretical and Experimental Dynamics, University “Mediterranea” of Reggio Calabria, 89124 Reggio Calabria, Italy A. Sofi (B ) Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK E-mail: alba.sofi@unirc.it