Algorithm for aging viscoelastic structures under periodic load

A numerical step-by-step algorithm for the analysis of concrete structures exposed to a periodic history of environmental humidity or temperature is presented. The creep law of concrete is assumed to be linear, and the relationship between humidity and shrinkage is also linear. Cracking is assumed to be absent. The effect of concrete age on creep properties is taken into account. The creep law is considered in a rate-type form corresponding to the Maxwell chain model. The well-known exponential algorithm is generalized to complex variables to describe the periodic part of the response. Since this part cannot be separated in advance from the drifting mean response, the standard exponential algorithm in real variables and the new one in complex variables are used simultaneously in each time step to provide the total response. The algorithm allows an arbitrary increase of the time step, and time steps that are orders of magnitude larger than the fluctuation period, as well as the relaxation times, are possible without causing inaccuracies and numerical instability. The algorithm leads to a series of incremental elastic problems in which the stresses, strains, elastic moduli, stiffness matrices, etc., are all complex variables. These spatial problems are solved by finite elements. The proposed algorithm is useful for spectral analysis of the response of concrete structures exposed to random environmental humidity or temperature, and tremendously reduces the computation time when high frequencies are present in the spectral density of environment.