Appendix C Estimates of Parameters Used By Model B3 C.1 Equations Used By B3 Model B3 (Bažant and Baweja 1995a, Bažant and Baweja 2000a) covers creep and shrinkage of concrete, including their coupling. The compliance function of concrete is approximated by ( t ) J(t,t )=q 1 + q 2 Q(t,t )+q 3 ln[1+(t t ) n ]+q 4 ln t + J d (t,t ) (C.1) where q 1 = 1/E 0 is the inverse of the asymptotic elastic modulus, the terms containing q 2,q 3 and q 4 represent the aging viscoelastic compliance, non-aging viscoelastic compliance and flow compliance, respectively, and J d (t,t ) is the additional compliance due to drying, which is also influenced by the time t 0 at the beginning of drying. In the definition of the compliance function (C.1), times t and t must be in days, q 1, q 2,q 3 and q 4 are empirical constitutive parameters that depend on the specific type of concrete, and function Q is defined by t Q(t,t ns m )= t (s t )+(s t ds ) 1 n (C.2) in which m = 0.5 and n = 0.1 are empirical parameters whose values can be taken the same for all normal concretes. The values of Q(t,t ) can be calculated by numerical evaluation of the integral in (C.2) or by interpolation from a table given in Bažant and Baweja (1995a), or evaluated (for n = 0.1 and m = 0.5 with an error under 1%) using the approximate explicit formula Q(t,t )=Q f (t ) [ 1+ ( Q f (t ) ] ) r(t ) 1/r(t ) Z(t,t ) (C.3) in which 699
700 C Estimates of Parameters Used By Model B3 r(t )=1.7(t ) 0.12 + 8 Z(t,t )=(t ) m ln[1+(t t ) n ] Q f (t )=[0.086(t ) 2/9 + 1.21(t ) 4/9 ] 1 (C.4) (C.5) (C.6) The average drying shrinkage strain in a cross section, ε sh, and the drying creep compliance, J d, are estimated using formulae ε sh (t)= εsh ( ) 1 h 3 env S(t t0 ) (C.7) J d (t,t )= q 5 e g(t t 0) e g(t t 0 ) (C.8) with auxiliary functions S and g given by ˆt S(ˆt)=tanh, τ sh = k t (k s D) 2 (C.9) τ sh g(ˆt)=8[1 (1 h env )S(ˆt)] (C.10) In the above, h env is the environmental relative humidity, t 0 is the age of concrete at the end of curing, D is the equivalent thickness of the concrete member and k s is a correction factor that takes into account the shape of the member (see Table C.1). Parameters ε sh, k t and q 5 need to be determined by fitting of experimental data or estimated from composition. Table C.1 Values of shape factor k s specimen shape k s infinite slab 1.00 infinite cylinder 1.15 infinite square prism 1.25 sphere 1.30 cube 1.55 C.2 Prediction of Model Parameters Ideally, the material parameters should be determined from long-term creep and shrinkage tests performed on the specific concrete that will be used in the particular structure. In practical applications, it is impossible to wait with the analysis until such tests are finished. The parameters can be crudely estimated from empirical formulae that have been established by fitting a large experimental database for various types of concrete and correlating the obtained model parameters to the concrete strength and composition.
C.2 Prediction of Model Parameters 701 Table C.2 Concrete properties serving as input data Property Symbol in-lb units SI units mean compression strength f c psi MPa water content w lb/ft 3 kg/m 3 cement content c lb/ft 3 kg/m 3 aggregate content a lb/ft 3 kg/m 3 Since the coefficients that appear in the approximate formulae are not dimensionless, their values depend on the choice of the system of units. We present here alternative formulae valid in inch-pound (in-lb) system units and in SI (metric) units. The entire parameter evaluation must be done in one selected system of units. The age of concrete at the onset of drying, t 0, is always given in days. The required input data specifying the strength and composition of concrete are summarized in Table C.2. The compression strength f c should be determined as the statistical average of test results on cylinders of diameter 15 cm (6 inches) and length 30 cm (12 inches) at age 28 days. 1 The content of water, cement and aggregates is the mass of the component per unit volume of concrete mix. If the water and cement contents to be used have not yet been decided, they may be estimated from the empirical correlation of the water/cement ratio to the required design strength (mean compressive cylinder strength after 28 days) of concrete (Neville 1997), ( ) 1 w f c = c + 0.535 (C.11) f ref where the reference strength f ref = 22.8 MPa=3307 psi corresponds to w/c=0.65. The formulae for predicting the creep and shrinkage parameters are presented in Table C.3. 1 Note that design codes deal with a certain safely estimated strength value, which is significantly lower than the mean. The CEB code uses the so-called characteristic strength, f ck, while the ACI code uses the specified design strength, f c; see Appendix E.1 for details.
Table C.3 Prediction formulae for creep parameters according to the B3 model Line Phenomenon Parameter Inch-pound formula Units SI formula Units 1 basic creep q 1 10.526 f c 0.5 10 6 /psi 126.77 f c 0.5 2 q 2 451.1c 0.5 f c 0.9 10 6 /psi 185.4c 0.5 f c 0.9 3 q 3 0.29(w/c) 4 q 2 10 6 /psi 0.29(w/c) 4 q 2 4 q 4 0.14(a/c) 0.7 10 6 /psi 20.3(a/c) 0.7 5 shrinkage k t 190.8t 0.08 0 f c 1/4 days/in 2 0.085t0 0.08 f c 1/4 days/mm 2 6 εs ( α 1 α 2 26w 2.1 f c 0.28 + 270 ) 10 6 ( α 1 α 2 0.019w 2.1 f c 0.28 + 270 ) 10 6 7 ε sh ε s 0.57514 3+14/(t 0 + τ sh ) 10 6 ε s 0.57514 3+14/(t 0 + τ sh ) 10 6 8 drying creep q 5 7.57 10 5 f 1 c ( ) ε 0.6 sh 10 6 /psi 7.57 10 5 f c 1 ( ) ε 0.6 sh 702 C Estimates of Parameters Used By Model B3
C.2 Prediction of Model Parameters 703 The formula in line 7 of Table C.3 originally comes from ε sh = ε s E(7+600) E(t 0 + τ sh ) (C.12) The fraction on the right-hand side is a correction for aging, because the final value of shrinkage depends not only on the material properties, but also on the interplay between hardening and drying. If the drying process starts later and takes more time, concrete becomes stiffer and its shrinkage is reduced. To take that into account, a reference value εs is defined as the final shrinkage strain exhibited by the given concrete if drying starts at 7 days and shrinkage halftime is 600 days. This value is considered as dependent only on material properties and type of curing, and is estimated using the formula in line 6. Under general conditions, it is modified by the above mentioned factor, defined as the ratio between the elastic modulus at 607 days (i.e., somewhere in the middle of the reference drying process) and the elastic modulus at t 0 + τ sh (i.e., somewhere in the middle of the actual drying process). The dependence of elastic modulus on age is then estimated using the adjusted ACI formula (E.29), E(t)=E 28 7t 28+6t (C.13) which leads to E(7+600) 7 607 E(t 0 + τ sh ) = 28+6(t 0 + τ sh ) 14 = 0.57514 + 3 28+6 607 7(t 0 + τ sh ) t 0 + τ sh (C.14) Recall that parameter k t is used in the second part of formula (C.9), which specifies the shrinkage halftime, τ sh. Coefficients α 1 and α 2 that appear in the formulae for the reference shrinkage strain εs in line 6 of Table C.3 are defined as 1.0 for type I cement α 1 = 0.85 for type II cement (C.15) 1.1 for type III cement 0.75 for steam curing 1.2 for sealed or normal curing in air with initial protection α 2 = (C.16) against drying 1.0 for curing in water or at 100% relative humidity If the specific information is not available, the following reasonable default values can be used: type I cement (α 1 = 1.0), curing in air with initial protection (α 2 = 1.2). The types of cement are understood here according to the American classification (ASTM C 150-07: Specification for Portland Cement). Type I is ordinary Portland cement, type II is modified cement, type III is rapid-hardening Portland cement, type IV is low-heat Portland cement, and type V is sulfate-resisting Portland cement.