DRYING AND CRACKING EFFECTS IN BOX-GIRDER BRIDGE ...
DRYING AND CRACKING EFFECTS IN BOX-GIRDER BOX-GIRDER BRIDGE SEGMENT SEGMENT By Zdenek Bazant, Fellow, Fellow, ASCE, ASCE,'1 Vladimir Vladimir Kfistek, Kfistek,22 and Jan L. Vitek3 Zdenek P. Kazant, paper deals with the effect effect of of drying as a moisture diffusion This paper diffusion process in a segment of a typical box-girder box-girder bridge. Pore relative humidity humidity distrisegment of butions throughout of the girder are calculated calculated from throughout the cross sections of the plates of from the diffusion equation, taking its nonlinearity nonlinearity and aging into account. The effect diffusion equation, effect of creep with aging on the stresses produced produced by drying shrinkage shrinkage is taken into account assuming a creep law based on the principle principle of of superposition superposition and described described effects between between creep and drying, by an aging spring-dashpot spring-dashpot chain model. The cross effects expressed as stress-induced stress-induced shrinkage shrinkage (equivalent considered. expressed (equivalent to drying creep), are considered. softening (smeared account in the form of an Tensile strain softening (smeared cracking) is taken into account energetic fracture-mechanics fracture-mechanics additive cracking strain. Localization Localization of of cracking and energetic algorithm for step-by-step step-by-step integration aspects are neglected. A general algorithm integration in time is presented and verified presented verified by a convergence convergence study. An example of of analysis of the cross feasibility of the analysis and section of one recently recently built bridge confirms confirms the feasibility distributions indicates that the effect effect of the diffusion diffusion process of of drying on the stress distributions is very large, in fact, so large that the stress values obtained obtained by customary customary methods fictitious. are merely fictitious. ABSTRACT: ABSTRACT:
INTRODUCTION
The long-time drying process, as well as temperature fluctuations, in concrete structures cause either tolerable or intolerable cracking. To ensure such damage would be tolerable, a more realistic method of analysis than those used in the current practice is needed. The current methods, based on codes and standard recommendations, generally do not yield adequate information on the stress and strain distributions produced by changes of information of furnishing only apmoisture content or temperature. They are capable of furnishing proximate, and sometimes quite inaccurate, values of normal forces and bending moments in the cross sections of beams and plates. These limitations are due mainly to the omission of moisture diffusion diffusion analysis and inadequacy simplified shrinkage and creep laws in use at present. These laws of the simplified characterize only the overall behavior of the cross section, and especially diffusion aspects of the drying process. Properly, a material ignore the diffusion constitutive equation equation that describes the of aa small small material element constitutive that describes the behavior behavior of material element rather than than the the average average response of the cross section section should should be used. rather response of the cross be used. Free, i.e., unrestrained, shrinkage is an abstraction, nonexistent in reality. During the process of drying, as well as heating or cooling, the drop of of moisture content or temperature occurs first in the surface layers and only diffusion equation much later in the core, as described by the solution of the diffusion for moisture transport. The material shrinkage and thermal expansion first develop in the surface layers. This produces tension in the surface layers, which must be balanced by compression in the core of the cross section. As
P. Murphy Prof. Civ. Civ. Engrg., Northwestern Univ., Evanston, IL 60208. 'Walter P. 2 2Prof. Prof. Civ. Civ. Engrg., Engrg., Czech Czech Tech. Tech. Univ., Univ., Prague, Prague, Czechoslovakia; Czechoslovakia; formerly, formerly, Visiting Visiting Scholar, Northwestern Univ., Evanston, IL. 3 3Res. Res. Engr., Czech Tech. Tech. Univ., Prague, Czechoslovakia. Note. Note. Discussion open until June 1, 1992. 1992. To extend the closing date one month, Journals. The manuscript a written request must be filed with the ASCE Manager of Journals. 19, 1990. 1990. for this paper was submitted for review and possible publication on March 19, ofStructural Structural Engineering, No.January, 1, January, Journal of Engineering, Vol.Vol. 118,118, No.1, This paper is part of the Journal 1992. ©ASCE, ISSN 0733-9445/92/0001-0305/$1.00 + $.15 per page. Paper No. 1992. 26538. 26538. 305
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the the drying drying process process advances, advances, the the shrinkageshrinkage- and and temperature-induced temperature-induced stresses stresses vary throughout throughout the the cross cross section. section. Because Because of of irreversible irreversible strains strains due due to to vary distributed micro microcracking, as well well as as additional additional stress-induced stress-induced shrinkage, shrinkage, distributed cracking , as the process process does does not not end end up up with with aauniform uniform state state of ofstress stress and andstrain. strain. Rather, Rather, the remain. residual stresses remain. Intense research has has now now finally crystallized into aa material model that that ought to to give give aa realistic picture of stresses and and strains in in concrete structures structures exposed to to drying and and temperature changes. changes. After aa period of development development of realistic constitutive relations for for shrinkage and and basic creep of concrete concrete 1988; Bazant and and Panula, 1978), 1978), two two advances that seem to to comcom("State" 1988; the requisite material model have recently been made: made: (1) (1) The The conplete the for creep has has been extended to to also cover the the cross effects effects stitutive relation for as either stress-induced shrinkage shrinkage with shrinkage and thermal strains, known as and as as stress-induced thermal strain [e.g., Bazant (1972), or drying creep, and Thelandersson (1983), (1983), Bazant and and Chern (1985a, 1985b)]; 1985b)]; and and (2) (2) the the law the strain softening due due to to distributed tensile cracking has been that governs the formulated and and verified experimentally [e.g., ACI ACI Committee 446, (1991), Bazant (1986)]. The objective of this paper is is to to apply the the aforementioned aforementioned advances in material modeling to to the the problem of drying effects in in the the cross section of a (Fig. 1). 1). A A numerical segment of aa prestressed concrete box-girder bridge (Fig. the cross section of one one recently completed bridge will be be used example of the to examine the the effects of drying from the the designer's viewpoint and and to to demthe feasibility of this type of analysis in practice. onstrate the MATERIAL MODEL AND AND GOVERNING EQUATIONS
Diffusion and and Aging Diffusion Prior to the solution of stresses, it is necessary to calculate the distributions of pore relative humidity, 17, h, as well as temperature through the entire We will assume the pore humidity and temperstructure at various times. We ature to be governed by uncoupled diffusion diffusion equations. The uncoupling appears to be an acceptable approximation, unless the cracks open widely (Bazant et al. 1988) or the moisture and temperature gradients are high al. 1988) (Bazant and Thonguthai (1979), which is not normally the case. We will further further assume that the moisture movement movement is one-dimensional, occurring only in the directions normal to the plates of the cross section, i.e., wle will will neglect moisture migrations along the plates and in the longitudinal direction of of the girder. The pore relative humidity as a function function of time t and of the the
FIG. 1. FIG.
Box Girder Girder Bridge Bridge Segment Segment Analyzed Box
306 306
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transverse cross section coordinate c~ordinate z of the walls is governed by the following following nonlinear parabolic partial differential differential equation: 8/? bh _8_ C(t,h) ~~J ~ = 8~ [C(t,h) Sz 8z ht
.........................................
(1)
The boundary conditions consist of prescribed values of h representing the environmental conditions. The diffusivity, diffusivity, C(t,h), C(t,h), is variable, due to aging of concrete. It depends not only on time, but also on the local moisture function of h. The dependence of Con C on h is very strong content, which is a function and may be approximately described by the following empirical (but physically explicable) equation (Bazant and Najjar Najjar 1972; "State" 1988):
1
C(te,H) = C,{te)
1 +
(1 - h)»
........................ (2) (2)
(1 - K)'\
in which C,(te) Ct{te) == reference diffusivity diffusivity of moisture in concrete at full saturation (h == 1), which depends on the equivalent hydration period, te; te; and a0, n, and he hc == empirical constants, a 0o =~ 0.05, he hc =~ 0.75, n == 6. 6. Furthermore Furthermore ao, 1975; Bazant and Chern 1985a): (Bazant 1975;
C,(te) 10- 6 C,(Q == Co(0.3 + 3.6t;1/2) 3.6t-m) 10-
(m2/day) (m2/day) ..................... (3) (3)
in which te te must be given in days; and Co Cu == an empirical constant, roughly Co0 = representing the diffusivity diffusivity at saturation at 28 days of age, typically C ~ 0.1 cm 22/day. The evolution of the equivalent hydration period reflects the effect of pore humidity and temperature on the local rate of cement hyeffect dration. It may be approximately characterized by the equation
1:
tete == Jo Aw,,iwi are the preceding iteration, respectively. If R > given small tolerance (e.g., 0.005), go go and start the the next iteration of this step. Otherwise go go to to 11 and and start to step 2 and the next time step. Note that the the Newton iterations for S £ could be eliminated from the the foregoing algorithm if (l3) (13) could be be explicitly inverted. (13) yields a horizontal tangent at the the peak stress point may The fact that (13) may difficulties for states at or very near the the peak point. Their cause numerical difficulties accurate handling would necessitate a more sophisticated algorithm (e.g., the arc-length method known from nonlinear finite element analysis). However, for practical purposes where high accuracy is is not necessary, numerical difficulties difficulties be avoided just by by taking a larger time step that straddles the the peak point can be the a( cr(0 diagram. of the s) diagram. In the the numerical example that follows, only the the effect of drying will be be anfor the the sake of simplicity. However, the the effect of temperature changes alyzed, for be analyzed in in an an analogous manner. could be EXAMPLE: Box BOX GIRDER BRIDGE BRIDGE
an example, we we now now present the the analysis of the the behavior of the the cross As an the Kishwaukee River Bridge in in Illinois, shown in Fig. 3(a). 3(a). The The section of the reinforcement and and prestress of the the girder is is assumed to to have longitudinal reinforcement effect in the the present analysis. analysis. The The cross section is is reinforced reinforced by by mild no effect steel [Fig. [Fig. 3(a)]. 3(a)]. The The girder is is assembled from precast segments, and and we the behavior of one segment in in the the transverse plane, assuming twoanalyze the dimensional response and and plane strain conditions. The The segment is is cured in and at at the the age age of seven days, days, the the formwork is is stripped. At the formwork and that moment, the the drying of concrete begins and and the the own own weight begins to act. act. temperature The effects of the fluctuations of environmental humidity and temperature in this example. Their realistic analysis would necessitate a are neglected in 312 312
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105
KH
701
17n
@ -
6 bars dlo. 20 mm *r m
-
5 bars die, 16 mm p~r m
i
i
i i
371
Dimensions in em
Element Thick""," Element Thdness type In em tVDe In em 1 25.4 10 25 2 28 11 22.9 3 32 12 60 42 4 36 13 41 35.5 5 14 6 46 15 46 16 28 7 40 8 35 17 203 30 9
FIG. FIG. 3. 3. (a) Reinforcement of Box Girder in Cross-Section Plane; Plane; (b) Subdivision in Finite Elements; and (c) Geometry of Neutral Axis
probabilistic approach that takes into account the randomness of the process (Bazant and Wang 1984a, 1984b; Bazant and Xi 1989). After stripping, the surfaces of concrete are exposed to the environmental After humidity, which is considered to be h == 0.5 on the outside and 0.6 on the inside of the box. The reason for assuming the average humidity at the inner surface to be higher than at the outer surface is that the interior space is not as well ventilated as the exterior space and is not illuminated by sun. So the bridge segment loses moisture faster to the outside than to the inside. As the boundary condition for the diffusion diffusion equation of drying, the pore relative humidities at the exterior and interior surfaces of the cross section plates are assumed to approach the environmental values gradually, as shown in Fig. 4. This assumption does not describe the ideal situation of a structure suddenly exposed to an environment of constant controlled relative humidity, as in the laboratory. Rather, this assumption is intended to approximately reflect reflect field conditions. After After the stripping of the form, a box-girder subjected for some time to periodic, although perhaps sposegment is still subjected radic, moist treatment. This causes the average daily relative humidity at the surface of concrete to approach the average environmental humidity If the ideal conditions with a considerable delay rather than immediately. If of a sudden drop of surface humidity to its constant environmental value were considered, the effects of drying would be more severe than those calculated in this paper. For the purpose of analysis, the plates of the cross section are subdivided 3(b). b ). Each finite element is subdivided into finite elements as shown in Fig. 3( into 14 equally thick layers. The geometry of the neutral axis of the beam elements is shown in Fig. 3(c). We analyze the deformations deformations of the cross section and the stresses caused by drying during a period of 55 days after after stripping. The period from strip313
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J
.c
1.0 -
.t-
I
I
'0
I I
E
~
interior surface
1) 0.6 -t----;,de'riOr~iia,;;
---
&0.5~-----------~
~
I I
.~
I
I
I
2
w
I
I
I
0
I
I
+-+.c------------=-=!_=_85.5 7 Time, tt (days) (days) 85.5 Time,
FIG. 4. Assumed Variation of Relative Humidity at Interior and Exterior Surfaces FIG. 4. of Plates of Cross Section
f -ih-
N
i
40
30 20
10
@
I
'k}
i
i
i i
50 40 30 20
10
----.~==;e:;==::j
FIG. FIG. 5. 5. (a) (a) Calculated Deformation of of Cross Section; Section; (b) (b) Distribution of of Bending Moments; Moments; (c) Distribution of of Normal Forces
ping to to joining of the segment with its neighbors may sometimes be be as long as this, this. Fig, Fig. 5(a) shows exaggerated deformation of the cross section at at the final final time (55 (55 days after stripping), stripping). The cross section is is assumed in in the the analysis be simply supported at at the the bottom corners of the the box. Obviously, the to be the is caused by own weight, weight. The drying shrinkage of major part of deflections is of is manifested principally by shortening of of the distance between the concrete is is, however, relatively small compared to to transverse transverse opposite plates, which is, of the plates. plates. deflections of Fig. Fig. 5(a) shows the distribution of the bending moments at at the final time, time. at the the upper corners at at which the top plates As we see, their maxima arise at the web join, join. The moments caused by by nonuniformity of of shrinkage are and the 314
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manifested in Fig. Fig .. 5( b). The final distribution of the normal not particularly manifested 5(b). forces in the plates of the cross section is shown in Fig. 5(c). Again the effect of own weight dominates, and even the small tensile force in the effect bottom plate seems to be caused by the transmission of the own weight component from the web to the bottom plate. The final distributions of the normal stresses in the axial directions are shown for several typical cross sections in Figs. 6(a-d). 6(a-d). In contrast to the bending and axial internal force resultants in the cross sections, the stress influenced by nonuniform nonuniform drying as well as strain distribution is strongly influenced softening. Even in regions where the usual analysis without drying would indicate compression, significant significant tensile stresses arise due to shrinkage. The stresses in steel in the compressed zones reach values around 20 MPa. In these distributions, one can clearly see the effect softening due to effect of strain softening microcracking, which causes the drop of stress near the surfaces. The largest compression stresses are reached in the bottom plate, which is the thinnest (see the distribution for element 33); the reason is that the tensile stresses produced by drying in the surface layers are counteracted counteracted in compression by concrete of a smaller thickness than in the other plates. This is also the reason why the axial shrinkage in the bottom plate is the largest. For comparison purposes, the alternative case, in which the segment is reinforcement, is free from own weight, and is affected affected only by without reinforcement, drying has also been solved. Fig. 7(a) shows the final deformation deformation of the cross section due to shrinkage. The bottom plate shrinks more than the upper plate because it is thinner. This difference difference in shrinkage causes the distribution of axial forces shown in Fig. 7(b), and the distribution of bending moments, shown in Fig. 7(c). The bending moments are produced by the
Elerrent 33
interior
~po
g{= -16.4 MPa
~po
6s = 0.2 MPa
exterior 1
I
1
-1 + $ -• T
2 (MPa]
~ ~2 I~Po) (MPa)
1
Element 23 exterior
11.2 MPo MPa r--1l';:= -11.2
Element 3
exterior
~MPO
~MPO
-= -6.4 MPa
_S;=-10.8MPa
!nterlor
exterior
. -~ $ 1~ ~2 lWo) (MPa)
2 (MPa)
FIG. 6. Calculated Stress Distributions Over Cross Sections of of Four Finite Elements in Box 315
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-I "T
mm
j
1-
\ \ \ \
Elo.3t 0.3 ~
E'
@
-gf 0.2 0.2+ ciS 0.1
99
6
112 12
No. No. of of beam beam elements elements
hh = 0.626 0.626 0
©
E o
n;::; 0.532
TenSion ---+.I---+----+----+~
Compressio Compression
-+-1-'----+-1 ·2 ·1
0
3
M .:=: 0,233 0.233 kNm kNm
FIG. 10. Convergence when Element Subdivision of Beam is Refined: (a) Convergence of Bending Moments at Two Locations; (b) Corresponding Stress Distribution at Midspan for Finest Element Subdivision
incorporation of of strain-softening effects into any algorithm for concrete allows incorporation strain-softening effects creep, not just the exponential algorithm used before. ACKNOWLEDGMENT
under NSF Grant Grant number number MSM-8815166 to Partial financial support support under Northwestern Northwestern University University is gratefully gratefully acknowledged. APPENDIX. ApPENDIX.
REFERENCES
"Thermodynamics of of interacting continua with surfaces surfaces and Bazant, Z. P. (1972). "Thermodynamics Nud. Engrg. Des., 20, 20,477-505. of concrete structures." Nucl. creep analysis of 477-505. "Theory of of creep and shrinkage in concrete structures: a precis Bazant, Z. P. (1975). "Theory Pergamon Press, New York, of recent recent developments." Mechanics today, Vol. 2, Pergamon of N.Y.,1-93. N.Y., 1-93. Bazant, Z. P. (1982a). "Input "Input of of creep and shrinkage characteristics characteristics for a structural Materials and analysis program." Materials and Structures, Structures, Research Researchand andTesting Testing(RILEM, (RILEM,Paris), Paris), 15(88), 283-90. Bazant, Bazant, Z. P. (1982b). "Mathematical "Mathematical models for for creep creep and shrinkage shrinkage of of concrete." and shrinkage shrinkage in concrete structures, Z. P. Bazant Bazant and and F. H. Wittmann, Creep and eds., John Iohn Wiley Wiley and and Sons, London, London, England, England, 163-256. eds., Bazant, Appl. Mech. Mech. Reviews, Bazant, Z. P. (1986). "Mechanics "Mechanics of of distributed distributed cracking." Appl. 39, 39, 675-705. Bazant, I.-C. (1985a). "Concrete "Concrete creep creep at at variable variable humidity: Bazant, Z. P., and and Chern, Chern, J.-C. Materials and and Structures (RILEM, (RILEM, Paris), Paris), 18, 18, Constitutive law and and mechanism." mechanism." Materials Constitutive 1-20. 1-20. 319 319
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J .-C. (1985b). "Strain softening softening with creep and exponential Bazant, Z. P., and Chern, J.-C. J. Engrg. Mech., ASCE 111(3), 391-415. algorithm." J. J.-c. (1987). "Stress-induced "Stress-induced thermal and shrinkage strains Bazant, Z. P., and Chern, J.-C. 1493-1511. J. Engrg. Mech., ASCE, 113(10), 1493-1511. in concrete." J. Najjar, L. J. (1972). "Nonlinear "Nonlinear water water diffusion diffusion in non saturated Bazant, Z. P., and Najjar, nonsaturated Materials and Structures (RILEM, (RILEM, Paris), 5, 3-20. concrete." Materials 3-20. Oh, B. B. H. (1984). "Deformation "Deformation of progressively progressively cracking reinBazant, Z. P., and Oh, ACI J. 81, 81,226-278. forced concrete beams." ACI 226-278. prediction of time-dependent time-dependent dedeBazant, Z. P., and Panula, L. (1978). "Practical prediction (RILEM, Paris), 11(65), 11(65),307formations of concrete." Materials and Structures {RILEM, 30728, (66), 415-34; and 12(69), (1979), 169-83. P., and and Prasannan, S. Bazant, Z. P., S. (1989). "Solidification "Solidification theory for concrete creep: I. Formulation, and and II. II. Verification Verification and and application." /. J. Engrg. Mech., ASCE, 115(8),1691-1725. 115(8), 1691-1725. Z. P., P., Sener, S., and and Kim, Kim, J. K. "Effect of cracking on drying Bazant, Z. K. (1987). "Effect and diffusivity diffusivity of concrete." ACI ACI Mater. Mater. J., 84(Sept), 351-357. permeability and P., and and Wang, T.-S. T.-S. (1984a). "Spectral Bazant, Z. P., "Spectral analysis of random shrinkage J. Engrg. Mech., ASCE, 110, . stresses in concrete." /. 110, 173-186. element analysis of random Z. P., P., and Wang, T. T. S. Bazant, Z. S. (1984b). "Spectral finite element J. Struct. Engrg., ASCE, 110, 110,2196-211. shrinkage in concrete." J. 2196-211. P., and and Xi, Xi, Y. Y. (1989). "Probabilistic prediction prediction of creep and shrinkage Bazant, Z. P., Proc., 5th 5th Int. and spectral approach." Proc, in concrete structures: combined sampling and Int. Conf. Vol. I, A. H.-S. Ang Ang et al., al., Conf. on Struct. Safety SaJety and and Reliability (ICOSSAR), (ICOSSAR), Vol. eds., held in San San Francisco, Published by ASCE, New York, N.Y. N.Y. 803-808. 803-808. New York, Brooks, J. and strength J. J., Neville, A. A. M. M. (1977). "A comparison of creep, elasticity and and in compression." Mag. Con cr. Res. Res. 29(100), 131-141. of concrete in tension and Concr. ACI Committee 446 (1991), "Fracture mechanics of concrete: Concepts, models and and determination of material properties." ACI 446.IR-XX properties ."ACI 446. IR-XX (state-oj-art (state-of-artreport), report),AmerAmerican Concrete Institute (ACI), Detroit, Mich. Glucklich, J., and J. Am. and Ishai, O. O. (1962). "Creep mechanism in cement mortar." /. Am. Concr. Concr. Inst., 59, 59, 923-948. Gopalaratnam, V. and Shah, S. P. (1985). "Softening V. S., S., and S. P. "Softening response of concrete in ACI J., 81, direct tension." ACI 81, May-Jun., 310-323. Hanson, J. J. A. A. (1953). A ten ten year study of creep properties of concrete." Coner. Concr. Lab. Report No. Sp-38, U.S. Dept. of the Interior, Bureau of Reclamation, Denver, Colo. reL'Hermite, R. R. G., G., Mamillan, M., M., Lefevre, C. C. (1965). "Nouveaux resultats de reAnnales de cherches sur sur la deformation et la rupture du du beton." Annales de l'Institut I'Institut Techn. Techn. et des des Travaux Publics, Publics, 18(207-208) 325; 325; see see also Int. Int. Conf. Conf. on on the the du Batiment et Structure oj of Concrete, Concrete, Cement and and Concrete Concrete Association, London, 1968, 423. Structure McDonald, J. J. E. E. (1975). (1975). "Time dependent deformation deformation of concrete under multiaxial stress conditions." Tech. Tech. Report C-75-4, C-75-4,U.S. U.S.Army ArmyEngrg. Engrg.Waterways WaterwaysExperiment Experiment Sta., Vicksburg, Miss. Miss. Petersson, P. P. E. E. (1981). (1981). "Crack growth and and development of fracture zones in plain 1006, Lund Inst. concrete and and similar materials." Report Report TVBM TVBM1006, Inst, of Tech., Lund, Sweden. Pickett, G. G. (1942). (1942). "The effect of change in moisture content on the creep of concrete under a sustained load." J. J. Am. Concr. Concr. Inst., 47,165-204,361-397. 47, 165-204, 361-397. Reinhardt, H. H. W., W., and and Cornelissen, H. H. A. A. W. W. (1984). "Post-peak cyclic behavior of concrete in in uniaxial tensile and and alternating tensile and and compressive loading." Cem. Concrete Concrete Res., Res., 14, 14, 263-270. 263-270. Cem. Ruetz, W. W. (1966). (1966). "A hypothesis for for the the creep of hardened cement paste and and the Int. Conf. Conf. Struct. Struct, oj of Concrete, Concrete, held in in Loninfluence of simultaneous shrinkage." Int. 1965, Cement and and Concrete Association, London, United Kingdom, 1968, don, 1965, 1968, see also Deutscher Ausschuss fUr fur Stahlbeton, Heft 183, 183, 1966. 1966. pp. 365-387, see "State of the art art in in mathematical modeling of creep and and shrinkage of concrete" (by (1988). Mathematical modeling of creep and and shrinkage RILEM Committee TC69) (1988). of concrete, Z. Z. P. P. Bazant, ed., ed., J. J. Wiley and and Sons, New New York, York, N.Y., N.Y., 57-392. 57-392.
320
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Thelandersson, S. (1983). "'On "On the multiaxial behavior of concrete exposed to high temperature." Nucl. Engrg. Engrg. Des., 75(2),271-82. 75(2), 271-82. Troxell, G. E., Raphael, J. M., Davis, R. W. (1958). "Long-time "Eong-time creep and shrinkage tests of plain and reinforced ASTM, 58, 151-158. reinforced concrete." Proc. Proc. ASTM, Tsubaki, T. et al. (1988). "Probabilistic models." Mathematical modeling of creep of creep and shrinkage of of concrete. concrete. Z. P. Bazant, ed., J. Wiley and Sons, New York, N.Y., 311-383. Ward, M. A., Cook, D. J. (1969). "The mechanism of tensile creep in concrete." Mag. Mag. Caner. Concr. Res., 21(68), 151-158. Wittmann, F. H., Roelfstra, P. E. (1980). (1980). "Total deformation deformation of loaded drying concrete." Cern. Cem. Caner. Concr. Res., 10,601-610. 10, 601-610.
321 321
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INTRODUCTION
The long-time drying process, as well as temperature fluctuations, in concrete structures cause either tolerable or intolerable cracking. To ensure such damage would be tolerable, a more realistic method of analysis than those used in the current practice is needed. The current methods, based on codes and standard recommendations, generally do not yield adequate information on the stress and strain distributions produced by changes of information of furnishing only apmoisture content or temperature. They are capable of furnishing proximate, and sometimes quite inaccurate, values of normal forces and bending moments in the cross sections of beams and plates. These limitations are due mainly to the omission of moisture diffusion diffusion analysis and inadequacy simplified shrinkage and creep laws in use at present. These laws of the simplified characterize only the overall behavior of the cross section, and especially diffusion aspects of the drying process. Properly, a material ignore the diffusion constitutive equation equation that describes the of aa small small material element constitutive that describes the behavior behavior of material element rather than than the the average average response of the cross section section should should be used. rather response of the cross be used. Free, i.e., unrestrained, shrinkage is an abstraction, nonexistent in reality. During the process of drying, as well as heating or cooling, the drop of of moisture content or temperature occurs first in the surface layers and only diffusion equation much later in the core, as described by the solution of the diffusion for moisture transport. The material shrinkage and thermal expansion first develop in the surface layers. This produces tension in the surface layers, which must be balanced by compression in the core of the cross section. As
P. Murphy Prof. Civ. Civ. Engrg., Northwestern Univ., Evanston, IL 60208. 'Walter P. 2 2Prof. Prof. Civ. Civ. Engrg., Engrg., Czech Czech Tech. Tech. Univ., Univ., Prague, Prague, Czechoslovakia; Czechoslovakia; formerly, formerly, Visiting Visiting Scholar, Northwestern Univ., Evanston, IL. 3 3Res. Res. Engr., Czech Tech. Tech. Univ., Prague, Czechoslovakia. Note. Note. Discussion open until June 1, 1992. 1992. To extend the closing date one month, Journals. The manuscript a written request must be filed with the ASCE Manager of Journals. 19, 1990. 1990. for this paper was submitted for review and possible publication on March 19, ofStructural Structural Engineering, No.January, 1, January, Journal of Engineering, Vol.Vol. 118,118, No.1, This paper is part of the Journal 1992. ©ASCE, ISSN 0733-9445/92/0001-0305/$1.00 + $.15 per page. Paper No. 1992. 26538. 26538. 305
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the the drying drying process process advances, advances, the the shrinkageshrinkage- and and temperature-induced temperature-induced stresses stresses vary throughout throughout the the cross cross section. section. Because Because of of irreversible irreversible strains strains due due to to vary distributed micro microcracking, as well well as as additional additional stress-induced stress-induced shrinkage, shrinkage, distributed cracking , as the process process does does not not end end up up with with aauniform uniform state state of ofstress stress and andstrain. strain. Rather, Rather, the remain. residual stresses remain. Intense research has has now now finally crystallized into aa material model that that ought to to give give aa realistic picture of stresses and and strains in in concrete structures structures exposed to to drying and and temperature changes. changes. After aa period of development development of realistic constitutive relations for for shrinkage and and basic creep of concrete concrete 1988; Bazant and and Panula, 1978), 1978), two two advances that seem to to comcom("State" 1988; the requisite material model have recently been made: made: (1) (1) The The conplete the for creep has has been extended to to also cover the the cross effects effects stitutive relation for as either stress-induced shrinkage shrinkage with shrinkage and thermal strains, known as and as as stress-induced thermal strain [e.g., Bazant (1972), or drying creep, and Thelandersson (1983), (1983), Bazant and and Chern (1985a, 1985b)]; 1985b)]; and and (2) (2) the the law the strain softening due due to to distributed tensile cracking has been that governs the formulated and and verified experimentally [e.g., ACI ACI Committee 446, (1991), Bazant (1986)]. The objective of this paper is is to to apply the the aforementioned aforementioned advances in material modeling to to the the problem of drying effects in in the the cross section of a (Fig. 1). 1). A A numerical segment of aa prestressed concrete box-girder bridge (Fig. the cross section of one one recently completed bridge will be be used example of the to examine the the effects of drying from the the designer's viewpoint and and to to demthe feasibility of this type of analysis in practice. onstrate the MATERIAL MODEL AND AND GOVERNING EQUATIONS
Diffusion and and Aging Diffusion Prior to the solution of stresses, it is necessary to calculate the distributions of pore relative humidity, 17, h, as well as temperature through the entire We will assume the pore humidity and temperstructure at various times. We ature to be governed by uncoupled diffusion diffusion equations. The uncoupling appears to be an acceptable approximation, unless the cracks open widely (Bazant et al. 1988) or the moisture and temperature gradients are high al. 1988) (Bazant and Thonguthai (1979), which is not normally the case. We will further further assume that the moisture movement movement is one-dimensional, occurring only in the directions normal to the plates of the cross section, i.e., wle will will neglect moisture migrations along the plates and in the longitudinal direction of of the girder. The pore relative humidity as a function function of time t and of the the
FIG. 1. FIG.
Box Girder Girder Bridge Bridge Segment Segment Analyzed Box
306 306
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transverse cross section coordinate c~ordinate z of the walls is governed by the following following nonlinear parabolic partial differential differential equation: 8/? bh _8_ C(t,h) ~~J ~ = 8~ [C(t,h) Sz 8z ht
.........................................
(1)
The boundary conditions consist of prescribed values of h representing the environmental conditions. The diffusivity, diffusivity, C(t,h), C(t,h), is variable, due to aging of concrete. It depends not only on time, but also on the local moisture function of h. The dependence of Con C on h is very strong content, which is a function and may be approximately described by the following empirical (but physically explicable) equation (Bazant and Najjar Najjar 1972; "State" 1988):
1
C(te,H) = C,{te)
1 +
(1 - h)»
........................ (2) (2)
(1 - K)'\
in which C,(te) Ct{te) == reference diffusivity diffusivity of moisture in concrete at full saturation (h == 1), which depends on the equivalent hydration period, te; te; and a0, n, and he hc == empirical constants, a 0o =~ 0.05, he hc =~ 0.75, n == 6. 6. Furthermore Furthermore ao, 1975; Bazant and Chern 1985a): (Bazant 1975;
C,(te) 10- 6 C,(Q == Co(0.3 + 3.6t;1/2) 3.6t-m) 10-
(m2/day) (m2/day) ..................... (3) (3)
in which te te must be given in days; and Co Cu == an empirical constant, roughly Co0 = representing the diffusivity diffusivity at saturation at 28 days of age, typically C ~ 0.1 cm 22/day. The evolution of the equivalent hydration period reflects the effect of pore humidity and temperature on the local rate of cement hyeffect dration. It may be approximately characterized by the equation
1:
tete == Jo Aw,,iwi are the preceding iteration, respectively. If R > given small tolerance (e.g., 0.005), go go and start the the next iteration of this step. Otherwise go go to to 11 and and start to step 2 and the next time step. Note that the the Newton iterations for S £ could be eliminated from the the foregoing algorithm if (l3) (13) could be be explicitly inverted. (13) yields a horizontal tangent at the the peak stress point may The fact that (13) may difficulties for states at or very near the the peak point. Their cause numerical difficulties accurate handling would necessitate a more sophisticated algorithm (e.g., the arc-length method known from nonlinear finite element analysis). However, for practical purposes where high accuracy is is not necessary, numerical difficulties difficulties be avoided just by by taking a larger time step that straddles the the peak point can be the a( cr(0 diagram. of the s) diagram. In the the numerical example that follows, only the the effect of drying will be be anfor the the sake of simplicity. However, the the effect of temperature changes alyzed, for be analyzed in in an an analogous manner. could be EXAMPLE: Box BOX GIRDER BRIDGE BRIDGE
an example, we we now now present the the analysis of the the behavior of the the cross As an the Kishwaukee River Bridge in in Illinois, shown in Fig. 3(a). 3(a). The The section of the reinforcement and and prestress of the the girder is is assumed to to have longitudinal reinforcement effect in the the present analysis. analysis. The The cross section is is reinforced reinforced by by mild no effect steel [Fig. [Fig. 3(a)]. 3(a)]. The The girder is is assembled from precast segments, and and we the behavior of one segment in in the the transverse plane, assuming twoanalyze the dimensional response and and plane strain conditions. The The segment is is cured in and at at the the age age of seven days, days, the the formwork is is stripped. At the formwork and that moment, the the drying of concrete begins and and the the own own weight begins to act. act. temperature The effects of the fluctuations of environmental humidity and temperature in this example. Their realistic analysis would necessitate a are neglected in 312 312
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105
KH
701
17n
@ -
6 bars dlo. 20 mm *r m
-
5 bars die, 16 mm p~r m
i
i
i i
371
Dimensions in em
Element Thick""," Element Thdness type In em tVDe In em 1 25.4 10 25 2 28 11 22.9 3 32 12 60 42 4 36 13 41 35.5 5 14 6 46 15 46 16 28 7 40 8 35 17 203 30 9
FIG. FIG. 3. 3. (a) Reinforcement of Box Girder in Cross-Section Plane; Plane; (b) Subdivision in Finite Elements; and (c) Geometry of Neutral Axis
probabilistic approach that takes into account the randomness of the process (Bazant and Wang 1984a, 1984b; Bazant and Xi 1989). After stripping, the surfaces of concrete are exposed to the environmental After humidity, which is considered to be h == 0.5 on the outside and 0.6 on the inside of the box. The reason for assuming the average humidity at the inner surface to be higher than at the outer surface is that the interior space is not as well ventilated as the exterior space and is not illuminated by sun. So the bridge segment loses moisture faster to the outside than to the inside. As the boundary condition for the diffusion diffusion equation of drying, the pore relative humidities at the exterior and interior surfaces of the cross section plates are assumed to approach the environmental values gradually, as shown in Fig. 4. This assumption does not describe the ideal situation of a structure suddenly exposed to an environment of constant controlled relative humidity, as in the laboratory. Rather, this assumption is intended to approximately reflect reflect field conditions. After After the stripping of the form, a box-girder subjected for some time to periodic, although perhaps sposegment is still subjected radic, moist treatment. This causes the average daily relative humidity at the surface of concrete to approach the average environmental humidity If the ideal conditions with a considerable delay rather than immediately. If of a sudden drop of surface humidity to its constant environmental value were considered, the effects of drying would be more severe than those calculated in this paper. For the purpose of analysis, the plates of the cross section are subdivided 3(b). b ). Each finite element is subdivided into finite elements as shown in Fig. 3( into 14 equally thick layers. The geometry of the neutral axis of the beam elements is shown in Fig. 3(c). We analyze the deformations deformations of the cross section and the stresses caused by drying during a period of 55 days after after stripping. The period from strip313
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J
.c
1.0 -
.t-
I
I
'0
I I
E
~
interior surface
1) 0.6 -t----;,de'riOr~iia,;;
---
&0.5~-----------~
~
I I
.~
I
I
I
2
w
I
I
I
0
I
I
+-+.c------------=-=!_=_85.5 7 Time, tt (days) (days) 85.5 Time,
FIG. 4. Assumed Variation of Relative Humidity at Interior and Exterior Surfaces FIG. 4. of Plates of Cross Section
f -ih-
N
i
40
30 20
10
@
I
'k}
i
i
i i
50 40 30 20
10
----.~==;e:;==::j
FIG. FIG. 5. 5. (a) (a) Calculated Deformation of of Cross Section; Section; (b) (b) Distribution of of Bending Moments; Moments; (c) Distribution of of Normal Forces
ping to to joining of the segment with its neighbors may sometimes be be as long as this, this. Fig, Fig. 5(a) shows exaggerated deformation of the cross section at at the final final time (55 (55 days after stripping), stripping). The cross section is is assumed in in the the analysis be simply supported at at the the bottom corners of the the box. Obviously, the to be the is caused by own weight, weight. The drying shrinkage of major part of deflections is of is manifested principally by shortening of of the distance between the concrete is is, however, relatively small compared to to transverse transverse opposite plates, which is, of the plates. plates. deflections of Fig. Fig. 5(a) shows the distribution of the bending moments at at the final time, time. at the the upper corners at at which the top plates As we see, their maxima arise at the web join, join. The moments caused by by nonuniformity of of shrinkage are and the 314
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manifested in Fig. Fig .. 5( b). The final distribution of the normal not particularly manifested 5(b). forces in the plates of the cross section is shown in Fig. 5(c). Again the effect of own weight dominates, and even the small tensile force in the effect bottom plate seems to be caused by the transmission of the own weight component from the web to the bottom plate. The final distributions of the normal stresses in the axial directions are shown for several typical cross sections in Figs. 6(a-d). 6(a-d). In contrast to the bending and axial internal force resultants in the cross sections, the stress influenced by nonuniform nonuniform drying as well as strain distribution is strongly influenced softening. Even in regions where the usual analysis without drying would indicate compression, significant significant tensile stresses arise due to shrinkage. The stresses in steel in the compressed zones reach values around 20 MPa. In these distributions, one can clearly see the effect softening due to effect of strain softening microcracking, which causes the drop of stress near the surfaces. The largest compression stresses are reached in the bottom plate, which is the thinnest (see the distribution for element 33); the reason is that the tensile stresses produced by drying in the surface layers are counteracted counteracted in compression by concrete of a smaller thickness than in the other plates. This is also the reason why the axial shrinkage in the bottom plate is the largest. For comparison purposes, the alternative case, in which the segment is reinforcement, is free from own weight, and is affected affected only by without reinforcement, drying has also been solved. Fig. 7(a) shows the final deformation deformation of the cross section due to shrinkage. The bottom plate shrinks more than the upper plate because it is thinner. This difference difference in shrinkage causes the distribution of axial forces shown in Fig. 7(b), and the distribution of bending moments, shown in Fig. 7(c). The bending moments are produced by the
Elerrent 33
interior
~po
g{= -16.4 MPa
~po
6s = 0.2 MPa
exterior 1
I
1
-1 + $ -• T
2 (MPa]
~ ~2 I~Po) (MPa)
1
Element 23 exterior
11.2 MPo MPa r--1l';:= -11.2
Element 3
exterior
~MPO
~MPO
-= -6.4 MPa
_S;=-10.8MPa
!nterlor
exterior
. -~ $ 1~ ~2 lWo) (MPa)
2 (MPa)
FIG. 6. Calculated Stress Distributions Over Cross Sections of of Four Finite Elements in Box 315
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-I "T
mm
j
1-
\ \ \ \
Elo.3t 0.3 ~
E'
@
-gf 0.2 0.2+ ciS 0.1
99
6
112 12
No. No. of of beam beam elements elements
hh = 0.626 0.626 0
©
E o
n;::; 0.532
TenSion ---+.I---+----+----+~
Compressio Compression
-+-1-'----+-1 ·2 ·1
0
3
M .:=: 0,233 0.233 kNm kNm
FIG. 10. Convergence when Element Subdivision of Beam is Refined: (a) Convergence of Bending Moments at Two Locations; (b) Corresponding Stress Distribution at Midspan for Finest Element Subdivision
incorporation of of strain-softening effects into any algorithm for concrete allows incorporation strain-softening effects creep, not just the exponential algorithm used before. ACKNOWLEDGMENT
under NSF Grant Grant number number MSM-8815166 to Partial financial support support under Northwestern Northwestern University University is gratefully gratefully acknowledged. APPENDIX. ApPENDIX.
REFERENCES
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Proc, in concrete structures: combined sampling and Int. Conf. Vol. I, A. H.-S. Ang Ang et al., al., Conf. on Struct. Safety SaJety and and Reliability (ICOSSAR), (ICOSSAR), Vol. eds., held in San San Francisco, Published by ASCE, New York, N.Y. N.Y. 803-808. 803-808. New York, Brooks, J. and strength J. J., Neville, A. A. M. M. (1977). "A comparison of creep, elasticity and and in compression." Mag. Con cr. Res. Res. 29(100), 131-141. of concrete in tension and Concr. ACI Committee 446 (1991), "Fracture mechanics of concrete: Concepts, models and and determination of material properties." ACI 446.IR-XX properties ."ACI 446. IR-XX (state-oj-art (state-of-artreport), report),AmerAmerican Concrete Institute (ACI), Detroit, Mich. Glucklich, J., and J. Am. and Ishai, O. O. (1962). "Creep mechanism in cement mortar." /. Am. Concr. Concr. Inst., 59, 59, 923-948. Gopalaratnam, V. and Shah, S. P. (1985). "Softening V. S., S., and S. P. "Softening response of concrete in ACI J., 81, direct tension." ACI 81, May-Jun., 310-323. Hanson, J. J. A. A. (1953). A ten ten year study of creep properties of concrete." Coner. Concr. Lab. Report No. Sp-38, U.S. Dept. of the Interior, Bureau of Reclamation, Denver, Colo. reL'Hermite, R. R. G., G., Mamillan, M., M., Lefevre, C. C. (1965). "Nouveaux resultats de reAnnales de cherches sur sur la deformation et la rupture du du beton." Annales de l'Institut I'Institut Techn. Techn. et des des Travaux Publics, Publics, 18(207-208) 325; 325; see see also Int. Int. Conf. Conf. on on the the du Batiment et Structure oj of Concrete, Concrete, Cement and and Concrete Concrete Association, London, 1968, 423. Structure McDonald, J. J. E. E. (1975). (1975). "Time dependent deformation deformation of concrete under multiaxial stress conditions." Tech. Tech. Report C-75-4, C-75-4,U.S. U.S.Army ArmyEngrg. Engrg.Waterways WaterwaysExperiment Experiment Sta., Vicksburg, Miss. Miss. Petersson, P. P. E. E. (1981). (1981). "Crack growth and and development of fracture zones in plain 1006, Lund Inst. concrete and and similar materials." Report Report TVBM TVBM1006, Inst, of Tech., Lund, Sweden. Pickett, G. G. (1942). (1942). "The effect of change in moisture content on the creep of concrete under a sustained load." J. J. Am. Concr. Concr. Inst., 47,165-204,361-397. 47, 165-204, 361-397. Reinhardt, H. H. W., W., and and Cornelissen, H. H. A. A. W. W. (1984). "Post-peak cyclic behavior of concrete in in uniaxial tensile and and alternating tensile and and compressive loading." Cem. Concrete Concrete Res., Res., 14, 14, 263-270. 263-270. Cem. Ruetz, W. W. (1966). (1966). "A hypothesis for for the the creep of hardened cement paste and and the Int. Conf. Conf. Struct. Struct, oj of Concrete, Concrete, held in in Loninfluence of simultaneous shrinkage." Int. 1965, Cement and and Concrete Association, London, United Kingdom, 1968, don, 1965, 1968, see also Deutscher Ausschuss fUr fur Stahlbeton, Heft 183, 183, 1966. 1966. pp. 365-387, see "State of the art art in in mathematical modeling of creep and and shrinkage of concrete" (by (1988). Mathematical modeling of creep and and shrinkage RILEM Committee TC69) (1988). of concrete, Z. Z. P. P. Bazant, ed., ed., J. J. Wiley and and Sons, New New York, York, N.Y., N.Y., 57-392. 57-392.
320
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Thelandersson, S. (1983). "'On "On the multiaxial behavior of concrete exposed to high temperature." Nucl. Engrg. Engrg. Des., 75(2),271-82. 75(2), 271-82. Troxell, G. E., Raphael, J. M., Davis, R. W. (1958). "Long-time "Eong-time creep and shrinkage tests of plain and reinforced ASTM, 58, 151-158. reinforced concrete." Proc. Proc. ASTM, Tsubaki, T. et al. (1988). "Probabilistic models." Mathematical modeling of creep of creep and shrinkage of of concrete. concrete. Z. P. Bazant, ed., J. Wiley and Sons, New York, N.Y., 311-383. Ward, M. A., Cook, D. J. (1969). "The mechanism of tensile creep in concrete." Mag. Mag. Caner. Concr. Res., 21(68), 151-158. Wittmann, F. H., Roelfstra, P. E. (1980). (1980). "Total deformation deformation of loaded drying concrete." Cern. Cem. Caner. Concr. Res., 10,601-610. 10, 601-610.
321 321
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