Rate‐Type Concrete Creep Law with Reduced Time

Abstract

For creep analysis of large structural systems, the linear aging integral-type creep law needs to be converted to a rate-type form which consists of a system of first-order linear differential equations with age-dependent coefficients. The system may be visualized by the Kelvin chain model with agedependent elastic moduli and viscosities. In the existing formulation, the independent variable is actual time. It is shown that, by using as the independent variable a certain reduced time which increases with time at a gradually declining rate, one reduces the number of differential equations needed to describe creep within the given time range, thereby making numerical structural analysis more efficient. An algorithm for identifying the material parameters from given creep data, based on minimization of a sum of squared deviations, is also presented. Thermodynamic restrictions on the material coefficients are analyzed. Finally, the capability of closely approximating available creep data is demonstrated. INTRODUCTIONFor numerical creep analysis of large structural systems, particularly finite element analysis, the rate-type form of a linear aging creep law is far more efficient than the integral-type form based directly on the principle of superposition. Instead of storing during step-by-step computation the values of stresses or strains in all previous steps, it suffices to store only the current values of several internal variables. The number of arithmetic operations also gets significantly reduced (3,5,6,10,11,19).The existing rate-type formulation of the aging creep law of concrete consists of a system of first or second-order linear differential equations with the actual time as an independent variable. This paper examines the question whether the rate-type formulation can be made more efficient by means of introducing as the independent variable a certain monotonically increasing function of time, called the reduced time.Linear Aging Creep Law Based on Principle of Superposition.-For stresses less than about one-half of the strength, the creep law of concrete may be approximately considered as linear, i.e., obeying the principle of superposition. The linearity assumption greatly simplifies structural analysis, and is generally used in practice. According to the principle of superposition, the uniaxial creep law of concrete may be written in the form Vol. 110, No.3, March, 1984. ©ASCE, ISSN 0733-9399/84/0003-0329/$01.00. Paper No. 18621. 329 in which CT, E = uniaxial stress and strain; EO = prescribed stress-independent strain representing shrinkage and thermal dilatation; and J (t, It may now be checked by substitution into Eqs: ~ and 1 that the ~ol lowing linear differential equations are always satisfied by the precedmg expression for ElL:dy~ dYIL CIL(t) dYIL Expressing the derivative d'YlL/dYIL from Eq. 5, we may verify that functions "I IL always satisfy the differential equationVariables EILand "I IL are related to E by the differential equations . (8 )It can be further shown that int...

Strain Softening with Creep and Exponential Algorithm

“…At the same time, studies of fracture test data of concrete showed that strain-softening stress-strain relations are inevitable for de-SCribing the observed deviations from linear elastic fracture mechanics or from strength criteria, and for obtaining the correct structural size effect (12,31). Also, it has been found that the same strain-softening stressstrain relations yield correct deflections of reinforced concrete beams in the cracking stage (10), and explain the existing test data on the shear transmission across cracks with the associated dilatancy (8). Thus, the tensile strain softening-a property well-established experimentally (19,(21)(22)(23)25,26,31)-emerges as a fundamental property of concrete.…”

Nonlinear long-term behaviour of spherical shallow thin-walled concrete shells of revolution

“…This procedure is difficult, consuming in computational effort, and in essence only provides an approximation to the compliance function. Moreover, in many cases the use of differential type constitutive relations based on Kelvin or Maxwell chains violates basic thermodynamics laws, where the spring moduli and dashpot viscosities become negative (see Bazant and Chern, 1984). Hence, the integral-type constitutive relation, which is also applicable to any compliance function, is preferred and used here.…”

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“…The rate of creep of the concrete material approaches zero as time increases. Hence, when this method is applied for the buckling analysis of concrete structures, where creep causes a significant variation of the internal stresses in time or when the structure is gradually loaded, it may lead to erroneous results because of its inability to accurately asses the creep caused by loads that are applied at older ages [Bažant 1988;Gilbert 1988]. Therefore, a model that is able to account for these effects needs to be implemented in the creep buckling analysis of concrete domes.…”

“…The effects of creep and shrinkage on the behaviour of concrete structures have been widely studied and reported [Bažant 1988;Gilbert 1988]. However, very little advanced research appears to have been reported on these effects in concrete shell structures, and in shallow concrete domes in particular.…”

“…The variational principle of virtual work is used to derive the field equations and the boundary conditions for the problem. Due to the use of an implicit integral-type constitutive relation, a time incremental superposition approach is invoked to formulate the time-dependent governing equations, where the creep and shrinkage strains are treated as initial strains and kept constant with each time step interval [Bažant 1988]. The time integration scheme utilises a slightly variable time step, to account for the decrease of the creep rate, and to account for the rapid change of the structural response prior to buckling.…”