Journal of the Engineering Mechanics Division Stochastic Process for Extrapolating Concrete Creep
Creep ?f :on~rete IS modeled as a process with independent increments : of locally gamma dlstnb~b<?n. The proce~s is transformed to a stationary gamma : proce~s. The ~ean predlcbon agrees With the deterministic double power law I est~bh~he~ p~evIOusly. Infinite divisibility of the increment distribution is assumed t Th18. 18 J.ustified by ~dditivit~ of, def~rmations and of stresses: and also by I c~nsld~ratlons ~f the mlcroscoP.lC mecJ:tan18m of creep, assuming creep to be due to : mlg~a~o:ts o~ w~dely spaced sohd particles along micropore passages whose length is I statl~tlC y d!Stnbute~. The treatment of creep as a stochastic procC$S allows extracting : considerable Inf?rmatl<?n from measurements even on one specimen, although a greater : numbe~ of specimens .IS prefera~le. The main use of the model is in extrapolation of : dhort tl.~e creep data mto long times, and calculation of confidence limits. Methods of : .eterm~nmg process parameters from creep test data are given. Monte Carlo I Simulations demonstrate reasonable agreement with test data. : REFERENCE:Cinlar, .Erhan, Bazant, Zdenek P., and Osman, ElMamoun, "Stochastic : ~?c.e~ for Extrapolatmg Concrete Creep," Journal of the Engineedng Mechanics : 1~;:lOn, ASCE, Vol. 103, No. EM6, Proc. Paper 13447, December, 1977, pp. 1069· 1 I I I I I I EM3 DISCUSSION STOCHASTIC PROCESS FOR EXTRAPOLATING CONCRETE CREEpa Discussion by Ian J. Jordaan 4 485 The authors have presented an interesting and complex paper on a problem that is of practical importance. Creep of concrete is in certain respects a complicated phenomenon but the writer feels that the problem of stochastic variation of creep may be looked at in a much simpler light than that proposed by the authors. Very often seemingly complicated problems have simple solutions. The writer's hypothesis is merely the following. For a specimen of macroscopic dimensions, creep is not a stochastic process, in the sense that the variation of an individual creep curve about its mean is negligible, given stable environmental conditions of constant temperature and humidity. However, there are variations in scale, which the writer ascribes (30) principally to gage length effects (e.g., Ref. 29). The variations in scale persist, i.e., one creep curve remains in effect a constant multiple of another. Creep curves obtained in experiments often exhibit scatter about their means but the writer attributes these to: (I) Gage variations; and (2) other experimental factors, such as small differences in temperature between. the creep specimen and companion (shrinkage) specimens. Thus the writer would question the principal result of the authors' investigation, i.e., the simulation as shown in Fig. 1. The variation along each curve, including curves crossing over each other, is simply not found in experiments in which gage and other variation have been eliminated. As noted, the writer considers the scatter of data along a particular curve to be the result of gage and experimental variation, and considers that it should not be modeled as a consequence of the micromechanism of creep. In order to support these observations some of the writer's data are included (Table 2). The values presented are three realizations from an extensive set which show similar features to those illustrated. Deviatoric strains are given as these, are independent of volumetric strains, and therefore problems induced in correcting for shrinkage are absent. The writer's hypothesis, as noted herein, stems from the simple observation that each increment in creep, measurable on a macroscopic specimen, is the sum of a very large number of molecular events. One has merely to consider the size of particles as noted by the authors and the fact that a single event occurring somewhere in the cross section of a macroscopic specimen could only contribute a very small part of the local deformation to the deformation of the specimen as a whole. Therefore, the scatter observed on a creep curve (e.g., Fig. 1) is unlikely to be associated with the molecular behavior unless all the events happen at once! The writer is particularly uneasy about the authors' "December, 1977, by Erhan ~in1ar, Zdenek P. BaZant, and EIMamoun Osman (Proc. Paper 13447). 4 Prof. of Civ. Engrg., The Univ. of Calgary, Calgary, Alberta, Canada. 486 JUNE 1979 EM: assumption of sparseness, both in time and in space. This highlights one aspect of the a~thors' approac~ to analyzing experimental data, i.e., that of ignoring those POInts corresponding to decreaSing values of creep. The writer agrees, of course, that the decreases stem from experimental variation, but how can the authors assume that other points do not contain such variations? The admission of variation due to factors other than those related to the micromechanisms of creep must then logically apply to all data points. The writer emphasizes that he considers practically all of the variation modeled by the authors to be du~ to extraneous ~actors unrelated to the physical mechanism of creep. B~SlC~y, th~ ~~estlO~ o~ sp~rseness results from the modeling of creep as ~ ~~tely dIVISIble distnbutton of the kind stated in the paper, and the distnbutlOns selected must therefore be questioned. The authors have started out with the assumption that an increment of creep of a specimen of macroscopic : ABLE 2.-Dev~atoric Strains x 10 -8 from Uniaxially Loaded Specimens and NormalIzed Creep Strains for Each Specimen (Equal to Creep Strain at Given Time Divided by Creep Strain at 63 days) I Elastic plus creep strains Normalized creep strains Days after from specimen number from specimen number loading 1 2 3 1 2 3 0 205 209 219 I 250 254 267 0.29 0.28 0.28 2 261 268 279 0.36 0.37 0.35 3 273 277 289 0.43 0.43 0.41 5 284 289 303 0.51 0.50 0.49 7 295 300 314 0.57 0.57 0.55 14 316 321 337 0.70 0.70 0.69 21 327 335 351 0.77 0.78 0.77 35 344 351 369 0.88 0.89 0.88 49 355 361 381 0.95 0.95 0.95 63 363 369 391 1.00 1.00 1.00 Stress, in megapascals 6.58 6.37 7.08 , , I I = size in a specifie~ time interval is a random variable, which can be represented as the ~um of n mdependent random variables (with a common distribution). The wnte~'s analysis. proceeds in the opposite direction. Since creep is not a stochastIc process (m the sense described earlier), there are sufficiently many events so that the central limit theorem effectively ensures that the increments of creep in companion specimens are the same, except for the scale factor noted previously. The result is simpler, accords with one's common sense and does not lead to increments of macroscopic creep being a stochastic proces~ as depicted in Fig. I of the author's paper. Aging effects have been eliminated in the authors' analysis by means of a transformation (that is reminiscent of the use of "pseudo-time" by the writer and others); and in addition the particular transformation is based on the representation of creep as a product of age and duration functions. Thus the extrapolation of mean values using these functions appears to have whatever EM3 DISCUSSION 487 advantages or disadvantages are characteristic of these functions as mean values of creep; in particular, it is to be noted that they model adequately only reversible creep defonnation (30). In conclusion, this discussion represents a critique-frank, but hopefully courteous-of the spirit rather than the detail of the authors' paper. In essence, I feel that the authors have modeled experimental "noise" as a micromechanical phenomenon, and that the basic problem they are addressing is not, in any practical sense, a problem at all. The writer would particularly value the authors' interpretation of the results in Table 2 in tenns of their model. ApPENDIX.-REFERENCES 29. Binns, R. D., and Mygind, H. S., "The Use of Electrical Resistance Strain Gauges and the Effect of Aggregate Size on Gauge in Connection with the Testing on Concrete," Magazine of Concrete Research, Vol. 1, Jan., 1949, pp. 3~-39.. ... 30. Jordaan, I. J., "Models for Creep of Concrete, with SpeCial EmphaSIS on Probabilistic Aspects," presented at the April, 1978, Reunion Internationale des Lab~ratoires d'Essais et de Recherches sur les Materiaux et les Constructions ColloqUIUm on Creep of Concrete, held at Leeds, England. Closure by Erhan {:inlar,~ Zdenek P. Bazant,6 M. ASCE, and ElMamoun Osman 7 Thanks are due to Professor Jordaan for his courteous discussion. His fundamental position is that basic creep (i.e., creep under constant temperature and humidity and other environmental factors) is detenninistic and that any observed variations from the deterministic actual creep value are due to experimental errors (gage effects. control of environmental conditions. load, etc.). We think that his belief is experimentally unjustified, and that the arguments he presents in favor of his position are faulty. The following is a step by step analysis of his assertions. Firstly, suppose that the basic creep curve is some deterministic function, c(t). as Jordaan claims. If the observations are taken at times tp t l , ... , tn' then the values read are c(t,) + Ep c(t 2 ) + E l .... , c(t n ) + En in which E" E 2 , ... , En are the error values introduced by gage effects and other experimental factors. Since these errors are external, Ep .... En must be nonsystematic. and are therefore independent and identically distributed random variables having the nonnal distribution with mean 0 and variance (T2 (unknown). Thus. the observed scatter around the mean curve c(t) must have the same variability for all t. It is well known that the rate of variability is greater for small t 5 Prof. of Applied Mathematics, Industrial Engrg., and Mathematics, Northwestern Univ., Evanston, Ill. • Prof. of Civ. Engrg., Northwestern Univ., Evanston, Ill. 'Instr., Univ. of Petroleum and Minerals, Dhahram, Saudi Arabia.
Models for creep of concrete, with special emphasis on probabilistic aspects
Six simple requirements are presented for constitutive equations for creep of concrete stressed within the working range. The resulting theory is not complete, but a unique representation of irreversible creep is obtained. The use of products of age and duration functions is shown to be unsuitable for representation of total creep, but is useful for reversible creep. There are several adequate representations of reversible creep and some of these are reviewed. The use of the methods of viscoelastic theory in stress analysis and some aspects of non-linear behaviour are briefly discussed. The question of whether creep of a macroscopic specimen at constant temperature and humidity is a stochastic process is examined on the basis of a set of experimental results and by taking into account the fact that macroscopic creep is the sum (both spatially and temporarily) of a very large number of molecular events. It is concluded that creep curves are different in scale but are not stochastic, in the sense that they exhibit negligible scatter about the mean of the particular curve being measured. However, gauge, temperature and humidity variation can cause stochastic variation. Lastly, a Bayesian approach to the engineering problem of dealing with uncertainty regarding the magnitude of creep is suggested.RésuméDans cet article on décrit les six conditions à remplir par les équations constitutives pour le fluage du béton soumis à des contraintes dans les conditions de service. Il en résulte une théorie qui n'est pas complète, mais on obtient une représentation unique du fluage irréversible. On montre que l'utilisation des produits des fonctions de l'âge au moment du chargement et dans l'intervalle qui suit, ne convient pas à la représentation du fluage total, mais que celle-ci est utile dans le cas du fluage réversible. Il existe plusieurs représentations adéquates du fluage réversible, dont quelques-unes sont présentées. L'utilisation des méthodes de la théorie visco-élastique dans l'analyse de la contrainte des structures en béton est brièvement résumée; ainsi que quelques aspects des effects non linéaires. On examine la question suivante: le fluage d'un spécimen macroscopique à une température et une humidité constantes est-il un processus stochastique? Cette étude s'appuie sur un ensemble de résultats expérimentaux, et tient compte du fait que le fluage macroscopique est la somme, sur un plan spatial ainsi que temporel, d'un très grand nombre d'évènements moléculaires. On en conclut que les courbes de fluage sont à des échelles différentes, mais ne sont pas stochastiques, dans le sens qu'elles présentent une dispersion négligeable autour de la moyenne de la courbe particulière que l'on mesure. Néanmoins, les variations de température et d'humidité peuvent causer une variation stochastique. Enfin, on propose d'appliquer le théorème de Bayes au problème technique que pose le manque de certitude à propos du degré de fluage.
Same-Sex Marriage in the United States
This Chapter gives a brief analysis of the status of same-sex marriage in the United States prior to the US Supreme Court decisions of 2013 and the status of litigation and political reforms triggered in part by these court decisions. It shows that marriage is a central institution in the country’s rationale of family law in ways that separate it from other western countries that have allowed same-sex marriage.