Advances in Materials Science and Engineering

Volume 2016, Article ID 9370514, 13 pages

http://dx.doi.org/10.1155/2016/9370514

## Description of Concrete Creep under Time-Varying Stress Using Parallel Creep Curve

^{1}Department of Civil Engineering, Konkuk University, 120 Neungdong-ro, Gwangjin-gu, Seoul 143-701, Republic of Korea^{2}Department of MOT, Konkuk University, 120 Neungdong-ro, Gwangjin-gu, Seoul 143-701, Republic of Korea

Received 2 September 2015; Accepted 17 January 2016

Academic Editor: Jun Liu

Copyright © 2016 Yeong-Seong Park et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An incremental format of creep model was presented to take account of the development of concrete creep due to loading at different ages. The formulation was attained by introducing a horizontal parallel assumption of creep curves and combining it with the vertical parallel creep curve of the rate of creep method to remedy the disadvantage of the rate of creep method that significantly underestimates the amount of creep strain, regardless of its simple format. Two creep curves were combined by introducing an ageing parameter whose value was obtained from two sets of time-dependent laboratory experiments on cylindrical specimens. The presented creep description takes the advantage that a single creep curve due to the initial loading describes the entire development of creep under the persistent change of creep-causing stress. Further, the creep formulation takes advantage of being consistent with the incremental format of age-dependent constitutive formulation. The performance of the presented creep equation was investigated with time-dependent laboratory experiments on cylindrical specimens and compared with the performances of four existing creep models.

#### 1. Introduction

The restraint of creep as well as shrinkage strains causes the mechanical strain and becomes a source of persistent change in the creep-causing mechanism. This type of creep development forms a circulating loop with a mechanical significance that is the same as the creep strain developed under a time-varying stress history. Mathematical modeling of the creep mechanism is formulated in terms of the age and concrete properties at loading. Aside from the effect of time-varying stress history on the creep model, another aspect concerning the creep model is the formulation format where the creep model is combined with the age-dependent constitutive model that relates the change of stress with the mechanical strain developed due to the restraint of creep and shrinkage. When the creep model is combined with the constitutive model, a consistent formulation framework is required between the two models, such as the total time-based or incremental time-based formulation frameworks. The consistency requirement in the formulation framework is further extended to the global equilibrium equations to compute the nodal displacements whose formulation strongly depends on the type of constitutive descriptions. A number of studies regarding the formulation of global equilibrium equations have been presented in the total time-based format of the finite element analysis scheme and applied to the time-dependent behaviors of concrete structures [1–3]. In this paper, a creep model is formulated within the incremental framework of the formulation to be consistent with the incremental format of the constitutive model.

Most studies regarding creep models have been performed to model the creep phenomena under various conditions of mix proportions, curing environments, ages of loading, and geometrical shape and dimensions [4–11]. However, focusing on the circulating loop phenomenon occurring in the time-dependent analysis of concrete structures, the creep model concentrates on depicting the creep strain developed under the time history loadings applied at different ages. A simple and robust method to model creep strain under time-varying stress history is a step-by-step method based on the principle of superposition. The formulation subdivides the entire stress history with stepwise variations of stress, which are called incremental stresses, applied at the end of a small time interval [11]. However, it is a difficult task to define the number of creep curves corresponding to the number of time intervals with stress increments. A great deal of effort has been made to model creep curves. The related studies are well introduced in the books of the ACI Committee 209 [5], Bažant and Wittmann [12], CEB [9], Ghali et al. [13], Gilbert [14], Gilbert and Ranzi [11], Jirasek and Bažant [15], and Rusch et al. [16]. Among the number of studies presented in the literature, four creep models draw attention to depict the circulating loop phenomenon: the rate of creep method (RCM) [17], step-by-step method [11], effective modulus method [18], and ageing coefficient method [19]. The performance of the models had been extensively studied by Gilbert and Ranzi [11] through a comparison of the mathematical formulations and numerical applications. Each model shows a different performance depending on the assumption made regarding the ageing effects of concrete loaded at different ages. Among these four models presented in the literature, special attention is given to RCM [17] because of its simple single-curve representation of creep behavior under a time-varying stress history. However, this method significantly underestimates the creep strain when it is applied to the creep problem subjected to persistent change in creep-causing stress. This paper presents a model to improve the disadvantage observed in RCM by introducing a horizontal parallel creep curve.

RCM is often referred to as the parallel creep curve method because the creep curves due to the loadings at different ages are assumed to be parallel. The term “parallel” in this method means that the tangents of the creep curves are identical along a vertical line. However, the method suffers from a significant underestimation of creep strain because of the parallel assumption when the sustained loads are applied at different ages. To remedy this inherent limitation, a horizontal parallel creep curve assumption is introduced and combined with the vertical parallel creep assumption of RCM. This type of formulation provides a creep strain bounded between upper and lower limits, where the lower limit is defined by RCM and the upper limit is defined by the horizontal parallel creep curve assumption. As a result, the presented creep formulation has the advantage of a single-curve representation for depicting creep strains under a time-varying stress history.

Two sets of laboratory experiments were sequentially conducted on cylindrical specimens to obtain the model parameters and to investigate the performance of the presented, two-way parallel creep curve formulation. Both sets of experiments included two cases of axial loads, including both constant and stepwise loads, where the stepwise loads were designed to depict the time-varying stress history. The performance of the presented creep model was compared with the effective modulus method, ageing coefficient method, step-by-step method, and RCM.

#### 2. Formulation of Creep

The RCM, originally proposed by Glanville [17], assumes that the change in the rate of creep with time is independent of the age of loading. This means that the creep curves for concrete loaded at different times are assumed to be parallel. The meaning of “parallel” is illustrated in Figure 1(a), which shows creep curves corresponding to loads applied at different times . RCM underestimates creep strain because of the inherent limitation imposed by the parallel assumption, as observed from Figure 1(a). Therefore, a parallel creep curve along the horizontal direction is introduced, as shown in Figure 1(b), and combined with the vertical parallel creep curve of RCM. The relationship between the creep function , creep coefficient , and creep compliance function under a constant stress applied at time follows the convention of