Skip to main content
SpringerLink
Log in

Rate-type creep law of aging concrete based on maxwell chain

  • Published:
Matériaux et Construction Aims and scope Submit manuscript

Abstract

It is shown that the linear creep law of concrete can be characterized, with any desired accuracy, by a rate-type creep law that can be interpreted by a Maxwell chain model of time-variable viscosities and spring moduli. Identification of these parameters from the test data is accomplished by expanding into Direchlet series the relaxation curves, which in turn are computed from the measured creep curves. The identification has a unique solution if a certain smoothing condition is imposed upon the relaxation spectra. The formulation is useful for the step-by-step time integration of large finite element systems because it makes the storage of stress history unnecessary. For this purpose a new, unconditionally stable numerical algorithm is presented, allowing an arbitrary increase of the time step as the creep rate decays. The rate-type formulation permits establishing a correlation with the rate processes in the microstructure and thus opens the way toward rational generations to variable tempeature and water content. The previously developed Kelvin-type chain also permits such a correlation, but its identification from test data is more complicated.

Résumé

On montre que la loi de fluage linéaire du béton peut être caractérisée, avec toute la précision voulue, par une loi de fluage de type différentiel qu'on peut interpréter par un modèle de Maxwell en chaîne combinant les viscosités en fonction du temps et des modules de ressort. On identifie ces paramètres d'après les résultats d'essai en développant en séries de Dirichlet les courbes de relaxation qui sont elles-mêmes calculées d'après le courbes de fluage expérimentales. L'identification ne comporte qu'une seule solution si une certaine condition de régularisation est imposée aux spectres de relaxation. La formulation est importante lorsqu'on procède à une intégration graduelle dans le temps de systèmes à éléments finis plus grands, car ainsi il n'est pas besoin de stocker les données de l'évolution des contraintes. A cette fin, on propose un nouveau algorithme numérique inconditionellement stable, qui permet un accroissement arbitrare de l'intervalle de temps à mesure que la vitesse de fluage décroît. La formulation de type différential permel d'établir une corrélation avec les processus de mouvement au niveau de la microstructure, et mène ainsi vers des généralisations rationnelles à températures et teneurs en eau variables. La chaîne de type Kelvin précédemment étudiée permet elle aussi une telle corrélation, mais l'identification des résultats d'essai est alors compliquée.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

E(t)=E R (t,t) :

l/J (t,t)=instantaneous Young's modulus of concrete

E″ :

pseudo-instantaneous Young's modulus in Eq. (18)

E R (t,t′) :

relaxation modulus=stress in timet caused by a constant unit strain enforced in timet′ (Eq. (1))

\(\tilde E_R \) :

given data onE R

E μ :

modulus of μth spring in Maxwell chain, fig. 1 Eq. (3)

E ∞ :

ultimate relaxation modulus (fig. 1, Eq. (6))

E μ :

coefficients of smoothing expressions (9), (10), (11)

J(t,t′) :

creep function (or compliance)=stress at timet caused by a constant unit stress acting since timet′ (Eq. 1)

n, m :

number of units in Maxwell chain,m=n—1

t, t′ :

time from casting of concrete (in days)

t′, t 0 :

time of application of constant stress or strain

w 1,w 2,w 3 :

weights in the penalty term in Eq. (8)

ε, σ:

strain and stress

ε0, ε″:

prescribed stress-independent inelastic strain (Eq. (1)) and pseudoinelastic strain in Eq. (18)

σμ :

hidden stresses in Eq. (3)=stress in the μth spring in Maxwell chain (fig. 1, Eq. (3))

ημ :

viscosity of the μth dashpot Maxwell chain (fig. 1, Eq. (3))

λμ :

parameter given by Eq. (16)

τμ=λμ/E μ :

relaxation time of the μth unit in Maxwell chain (fig. 1)

Subscripts:r, s :

for discrete timest r ,t s in step-bystep analysis

α, β:

for selected values oft′ and (t−t′) used in the least square condition

μ:

for the μth unit in Maxwell chain (fig. 1)

\(\dot \varepsilon = d\varepsilon /dt,e.g.\) :

Dot stands for time derivate

References

  1. Bažant Z.P.—Numerical determination of longrange stress history from strain history in concrete. Materials and Structures (RILEM), Vol. 5, 135–141, 1972.

    Google Scholar 

  2. Bažant Z.P.—Prediction of creep effects using the age-adjusted effective modulus method. J. Amer. Concrete Inst., Vol. 69, 212–217, April 1972.

    Google Scholar 

  3. Bažant Z.P.—Theory of creep and shrinkage in concrete structure: A précis of recent development. Mechanics Today, Vol. 2, Pergamon Press (in press).

  4. Bažant Z.P.—Thermodynamics of interacting continua with surfaces and creep analysis of concrete structures. Nuclear Engineering and Design, Vol. 20, 477–505, 1972.

    Article  Google Scholar 

  5. Bažant Z.P., Wu S.T.—Creep and shrinkage law for concrete at variable humidity. J. Eng. Mech. Div. Am. Soc. of Civil Engrs. (under review).

  6. Bažant Z.P., Wu S.T.—Dirichlet series creep function for aging concrete. Proc. ASCE, J. Eng. Mech. Div., Vol. 99, EM2, Apr. 1973, 367–387.

    Google Scholar 

  7. Bažant Z.P., Wu S.T.—Thermoviscoelasticity of aging concrete, sent to Proc. ASCE, J. Eng. Mech. Div.; and Preprint No. 2110, ASCE Annual Meeting, New York, Oct.–Nov. 1973.

  8. Bérès L.—La macrostructure et le comportement du béton sous l'effet de sollicitations de longue durée. Materials and Structures (RILEM), Vol. 2, p. 103–110, 1969.

    Google Scholar 

  9. Browne R.D., Burrow R.E.D.—An example of the utilization of the complex multiphase material behavior in engineering design. Int. Conf. on “Structure, Solid Mechanics and Engineering Design in Civil Engineering”, p. 1343–1378, Editor M. Te'eni, J. Wiley-Interscience, 1971 (Univ. of Southampton, April 1969).

  10. Gamble B.R., Thomas L.H.—The creep of maturing concrete subjected to time varying stress. Proceedings of the 2nd Australasian Conf. on the Mechanics of Structures and Materials, Adelaïde, Australia, Aug., 1969, Paper 24.

  11. Hanson J.A.—A 10-year study of creep properties of concrete. Concrete Laboratory Report No. Sp-38, U.S. Department of the Interior, Bureau of Reclamation, Denver, Colorado, July, 1953.

    Google Scholar 

  12. Hansen T.C.—Estimating stress relaxation from creep data. Materials Research and Standards (ASTM), Vol. 4, 1964, 12–14.

    Google Scholar 

  13. Harboe E.M. et al.—A comparison of the instantaneous and the sustained modulus of elasticity of concrete. Concrete Laboratory Report No. C-854, Division of Engineering Laboratories, U.S. Department of the Interior, Bureau of Reclamation, Denver, Colorado, March, 1958.

    Google Scholar 

  14. Hardy G.M., Riesz M.—The general theory of Dirichlet series, (Cambridge Tracts in Math. & Math. Phys. No. 18), Cambridge Univ. Press. 1915.

  15. Kennedy T.W., Perry E.S.—An experimental approach to the study of the creep behavior of plain concrete subjected to triaxial stresses and elevated temperatures. Res. Report 2864-1, Dept. of Civil Engineering, Univ. of Texas, Austin, 1970.

    Google Scholar 

  16. Klug P., Wittmann F.—The correlation between creep deformation and stress relaxation in concrete. Materials and Structures (RILEM), Vol. 3, 75–80, 1970.

    Google Scholar 

  17. Lanczos C.—Applied Analysis, Prentice Hall, 1964 (p. 272–280).

  18. L'Hermite R., Macmillan M., Lefèvre C.—Nouveaux résultats de recherches sur la déformation et la rupture du béton. Annales de l'Institut Technique du Bâtiment de des Travaux Publics, Vol. 18, No. 207-208, p. 325–360, 1965.

    Google Scholar 

  19. Neville A.M.—Creep of concrete: plain, reinforced, prestressed, North Holland Publ. Co., Amsterdam, 1970.

    Google Scholar 

  20. Neville A.M.—Properties of concrete, J. Wiley, New York, 1963.

    Google Scholar 

  21. Pirtz D.—Creep characteristics of mass concrete for Dworshak Dam. Report No. 65-2. Structural Engineering Laboratory, Univ. of California, Berkeley, California, Oct. 1968.

    Google Scholar 

  22. Roscoe R.—Mechanical models for the representation of viscoelastic properties. British J. of Applied, Physics, Vol. 1, 1950, 171–173.

    Article  Google Scholar 

  23. Ross A.D.—Creep of concrete under variable stress. Amer. Concrete Inst. Journal, Proc. Vol. 54, 739–758, 1958.

    Google Scholar 

  24. Rostásy F.S., Teichen K.T., Engelke H.—Beitrag zur Klärung des Zusammenhanges von Kriechen und Relaxation bei Normalbeton. Bericht, Otto-Graf-Institut, Universität Stuttgart, Strassenbau und Strassenverkehrstechnik, Helf 139, 1972.

  25. Taylor, R.L., Pister K.S., Goudreau G.L. Thermo-mechanical analysis of viscoelastic solids. Intern. J. for Numerical Methods in Engineering, Vol. 2, 45–60, 1970.

    Article  MATH  Google Scholar 

  26. Wittman F. Über den Zusammenhang von Kriechverformung und Spannungsrelaxation des betons. Beton-und Stahlbetonbau, Vol. 66, p. 63–65.

  27. Zienkiewicz O.C., Watson, M., King I.P.—A numerical method of viscoelastic stress analysis. Intern. J. Mech. Sciences, Vol. 10, 807–827, 1968.

    Article  Google Scholar 

Download references

Author information

Author notes

  1. S. T. Wu (Graduate Research Assistant)

    Present address: , Chicago, Ill.

Authors and Affiliations

  1. Northwestern University, 60201, Evanston

    Z. P. Bažant (Professor of Civil Engineering) & S. T. Wu (Graduate Research Assistant)

Authors
  1. Z. P. Bažant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. T. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bažant, Z.P., Wu, S.T. Rate-type creep law of aging concrete based on maxwell chain. Mat. Constr. 7, 45–60 (1974). https://doi.org/10.1007/BF02482679

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02482679

Keywords

Search

Navigation