Abstract
The paper presents an efficient and highly accurate step-by-step numerical algorithm for computation of stress history from any prescribed strain history in a linear age-dependent viscoelastic material. The method is applicable for any form of the creep function, including the typical case when the slope of the creep curve in the logarithmic time scale is significant over many orders of magnitude of the elapsed time period (i.e retardation spectrum is broad). The time division must be in geometric progression or nearly so. The creep function may be defined by formulas or by a table of values. A FORTRAN program is presented which allows quick and economical computer solution. Numerical examples are given and excellent convergence is demonstrated. For the special case of strains varying linearly with the creep coefficient a useful new theorem is proved.
Résumé
Cet article présente un algorithme numérique pas à pas propre au calcul de la variation des contraintes à partir de la variation de toute déformation déterminée dans un matériau dont les propriétés viscoélastiques sont une fonction linéaire du temps. La base de l'algorithme est une approximation par somme finie de l'intégrale de Stieljes, qui exprime la superposition des déformations en réponse à tous les accroissements de contrainte antérieurs. On présente un programme en FORTRAN IV qui permet d'obtenir par ordinateur une solution rapide et économique. Ce programme peut s'appliquer à quelque forme que ce soit de la fonction fluage, même dans le cas où la pente de la courbe de fluage sur une échelle de temps logarithmique est importante pour plusieurs ordres de grandeur du temps écoulé, c'est-à-dire lorsque le spectre de retardation est étendu. Afin de tenir compte du fait, on divise le temps selon une progression géométrique (ou presque). On peut définir la fonction de fluage par des formules, et aussi bien par une table de valeurs, propre à opérer la conversion directe des déformations mesurées en contraintes. On donne des exemples numériques et une excellente convergence apparaît. On peut entreprendre de même la conversion des données de relaxation des contraintes en données de fluage. Dans le cas particulier des déformations qui varient en fonction linéaire du coefficient de fluage, on démontre un nouveau théorème dont l'utilité est d'établir que la variation de contrainte correspondante est une fonction linéaire de la courbe de relaxation des contraintes. Ce théorème permet une généralisation de la méthode du module effectif, ce qui améliore notablement la précision lorsque les propriétés de fluage dépendent de l'âge.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- E(t)=J(t, t) :
-
Young's modulus (instantaneous)
- E R (t, t′) :
-
relaxation function (Eq. 12)
- E″ :
-
fictitious elastic modulus in Eq. (4) or (14)
- J (t, t′) :
-
creep function (Eq. 1)
- t, t′, t 0 :
-
time, time as integration variable, and time of first stress introduction (all in days)
- ε:
-
normal strain
- ɛ0 :
-
prescribed stress-independent inelastic strain,e.g. thermal dilatation, shrinkage (Eq. 1)
- ɛ″:
-
fictitious inelastic strain in Eq. (5 a) or (15 a)
- ɛ0, ɛ1 :
-
constants in Eq. (16)
- ϕ (t, t′):
-
creep coefficient (Eq. 6)
- σ:
-
normal stress Subscriptsr, s stand for discrete timest r ,t s
References
McMillan F.R.—Discussion of the paper by A.C. Janni: Method of designing reinforced concrete slabs, Trans. Amer. Soc. Civil Engineers, Vol. 80, 1916, p. 1738.
Faber O.—Plastic yield, shrinkage and other problems of concrete, and their effect on design, Minutes of Proceedings of the Institution of Civil Engineers (London), Vol. 225 (1927/28), Nov. 1927, pp. 27–76, with discussions pp. 76–130.
Glanville W.H.—Studies in reinforced concrete III —Creep or flow of concrete under load, Dept of Scientific and Industrial Research, (London), Building Research Technical Paper, No. 12, 1930 (see also: Structural Engineer, Vol. 11, 1933, p. 54, and 1st International Congress on Plain and Reinforced Concrete, Liège, 1930).
McHenry D.—A new aspect, of creep in concrete and its application to design, Proc. American Soc. for Testing Materials, Vol. 43, 1943, 1069–1086.
Arutyunyan N. Kh.—Some problems in the theory of creep (in Russian), Techteorizdat, Moscow 1952 (Engl. transl., Pergamon Press, 1966; Aroutiounian in French transl., Eyrolles, 1957).
Levi F., Pizzetti G.—Fluage, plasticité, précontrainte, Dunod, Paris, 1951.
Hansen T.C.—Estimating stress relaxation from creep data, Materials Research and Standards (ASTM), Vol. 4, 1964, 12–14.
Klug P., Wittmann F.—The correlation between creep deformation and stress relaxation in concrete, Materials and Structures (RILEM, Paris), Vol. 3, 1970, 75–80.
Bažant Z.P.—Creep of concrete and structural analysis (in Czech), SNTL (State Publ. House of Techn. Lit.), Prague, 1966 (186 pp.).
Bažant Z.P.—Phenomenological theories for creep of concrete based on rheological models, Acta Technica ČSAV (Prague), 1966, No. 1, 82–109.
Raphael J.M.—The development of stresses in Shasta Dam, Trans. of American Society of Civil Engineers, Vol. 118 A, 1953, p. 289.
England G.L., Illston J.M.—Methods of computing stress in concrete from a history of measured strain, Civil Engg. & Publ. Works Review (London), 1965, pp. 513–517, 692–694, 846–847.
Bažant Z.P.—Linear creep problems solved by a succession of generalized thermoelasticity problems, Acta Technica ČSAV (Prague), 1967, No. 5, 581–594.
Sackman J.L., Nickell R.E.—Creep of a cracked reinforced beam, J. of the Structural Div., Proc. ASCE Vol. 94, Jan. 1968, 283–308.
Bažant Z.P.—Numerical solution of non-linear creep problems with application to plates, Int. Journal of Solids and Structures, Vol. 7, 1971, 83–97.
Bažant Z.P.—Numerical analysis of creep of reinforced plates, Acta Technica (Ac. Sci. Hung.), Vol. 70, 1971, No. 3–4, 415–428.
Bažant Z.P.—Numerically stable algorithm with increasing time steps for integral-type aging creep, First Intern. Conf. on Struct. Mech. in Reactor Technology (BAM, West Berlin, and Commission of Eur. Communities (Editor T.A. Jaeger), Vol. 3, Paper H2/3, 1971.
Bažant Z.P.—Constitutive equation for concrete creep and shrinkage based on thermodynamics of multiphase systems, Materials and Structures (RILEM, Paris), Vol. 3, 1970, pp. 3–36.
Rektorys K. et al.—Survey of applicable mathematics, Hiffe, London 1969 (USA distribution by M.I.T. Press).
ACI Committee 209/II.—Prediction of creep, shrinkage and temperature in concrete structures, in «Designing for Effects of Creep, Shrinkage and Temperature», Amer. Concrete Inst. Special Publ., 1971, No. 27, 51–94.
Bažant Z.P.—Prediction of concrete creep effects using age-adjusted effective modulus method, Amer. Concrete Institute Journal, Proc. Vol. 69, April 1972, 212–217.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bažant, Z.P. Numerical determination of long-range stress history from strain history in concrete. Mat. Constr. 5, 135–141 (1972). https://doi.org/10.1007/BF02539255
Issue Date:
DOI: https://doi.org/10.1007/BF02539255