The spectral method, previously generalized for aging linear systems, is applied in conjunction with the finite element method to an~lyze shrinkage stresses in aging viscoelastic structures exposed to random enVIronmental humidity. The age-dependence of both drying diffusivity and creep properties are taken into account. The solution of pore humidity is obtained from a matrix differential equation in time, with complex-valued matrices. Elastic shrinkage stresses are then obtained from the matrix equations of the finite element method, in which the matrices are also complex-valued. The stresses in presence of aging creep are detennined by a sllperposition integral in time based on the relaxation function. Numerical examples concerning a long cylindrical vessel exposed at the outer surface are given. The standard deviations of pore humidity and of stresses significantly vary with time, and their standard deviation exhibits fluctuations about a drifting mean. The solution is practically meaningful only if concrete does not crack, e.g., when a prestress sufficient to prevent cracking is introduced. For environmental fluctuations of long periods, such as one year, the computation is quite efficient; however, if shorttime fluctuations are considered, the computing time becomes very large.

A numerical step-by-step algorithm for the analysis of concrete strud';1res exposed to a periodic history of environmental humidity or temperatun; IS p~sented. The ~~ law of concrete is assumed to be linear, and the relationship between hwrudity and shrinkage is also linear. Cracking is assumed to be absent. The effect of concrete age on creep properties is taken into account. Th~ creep law is considered in a rate-type form corresponding to the Maxwell ~ model. The well-known exponential algorithm is genera1ized to complex Variables to describe the periodic part of the response. Since this part cannot ~ separat~d in .advance from the drifting mean response, the standard exponentia! algonthm m !eal va~bles and the new one in complex variables are u.sed slIDultaneously m each time step to provide the total response. The algonthm allo,,:s an arbitrary increase of the time step, and time steps that are ~rders of magrutude larger than the fluctuation period, as well as the relaxation times: are possible without causing inaccuracies and numerical instability. The alg~nthm le~ds to a s~rie~ of increme?tal elastic problems in which the stresses, str~s, 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 en~o~ental humi~ity or temperature, and tremendously reduces the co~putation time when high frequencies are present in the spectral density of enVironment. IProf. o~ Civ. Engrg. and Dir., Center for Concrete and Geomaterials, The Tet~?!ogJcal Inst., Northwestern Univ., Evanston, lli. 60201.. VlSlting Scholar, Northwestern UniV.i on leave from the Huai River CommisSion , Bangbu, Anhui, China. Note.-Discus~ion open until November 1, 1984. To extend the closing date one month,.a wntten ~eq1;1est must be filed with the ASCE Manager of Technical an~