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 timedependent 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 creepcausing stress. Further, the creep formulation takes advantage of being consistent with the incremental format of agedependent constitutive formulation. The performance of the presented creep equation was investigated with timedependent laboratory experiments on cylindrical specimens and compared with the performances of four existing creep models.
The restraint of creep as well as shrinkage strains causes the mechanical strain and becomes a source of persistent change in the creepcausing 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 timevarying 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 timevarying stress history on the creep model, another aspect concerning the creep model is the formulation format where the creep model is combined with the agedependent 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 timebased or incremental timebased 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 timebased format of the finite element analysis scheme and applied to the timedependent behaviors of concrete structures [
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 [
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 singlecurve representation for depicting creep strains under a timevarying 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, twoway 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 timevarying stress history. The performance of the presented creep model was compared with the effective modulus method, ageing coefficient method, stepbystep method, and RCM.
The RCM, originally proposed by Glanville [
Parallel creep curves: (a) vertical parallel creep curve and (b) horizontal parallel creep curve.
Denoting the vertical parallel creep function as
Concept of the twoway creep model.
Based on the RCM shown in Figure
The increment of creep strain
The integral form of (
A creep curve is introduced under the selfsimilarity assumption of Figure
The incremental expression
Since the actual creep curve shown in Figure
The effective modulus method (EMM), presented by Faber [
An ageing coefficient
The stepbystep method (SSM) superimposes the creep strains due to the loadings at each time increment. In SSM, the continuously varying stress is divided by the specified time intervals, and the creep strain due to a stress increment at a time interval is calculated by the known creep function. Assuming the creep function
Two sets of timedependent laboratory experiments of types A and B were sequentially conducted on cylindrical concrete specimens to determine the value of the ageing factor
All cylindrical concrete specimens were cast with 150 mm diameter and 300 mm height. For each type of test, thirtyfive cylindrical concrete specimens were cast, of which twelve specimens were used for the creep tests in uniaxial compression, three specimens were used to measure the shrinkage, and twenty specimens were used to measure the development of the elastic modulus. Table
Mixture proportions for the two test types of concrete.
Test type  Max. size of aggr. (mm)  W/C ratio (%)  Slump (mm)  Unit weight, (N/m^{3})  

Water  Cement  Fine aggr.  Coarse aggr.  
Type A  20  54  110  1,880  3,480  7,360  10,200 
Type B  20  57  200  2,120  3,720  9,450  7,610 
Stepwise loads applied to (a) type A test and (b) type B test.
Hydraulic pressure was applied by a hydraulic jack. The pressure level was monitored by a pressure meter attached to the jack and a load cell mounted at the top of the specimen throughout the testing period (see Figure
Schematic drawing of creep test setup.
All experiments were conducted in a controlled room with a constant temperature of
When concrete is loaded with a timevarying stress history, the present singlecurve formulation requires a creep curve that describes the creep strain under a constant load. The creep curve was calibrated from the constant load case for both type A and type B tests. The resulting three equations of creep strain to define
Three empirical equations for the two types of tests.
Test type  Parameters  












A  9.5  1.7  51  1.45  4  0.85  10  29,400  32,800  30 
B  11.4  5.1  48.4  0.73  2.6  0.9  7  22,700  25,300  28 



Basic agedependent properties of type A test: (a) total, creep, and shrinkage strains, (b) development of elastic modulus.
The value of the ageing factor
Predicted and measured total strains for the two stepwise load cases of type A test: (a) load case 1 and (b) load case 2.
Predicted and measured creep strains for the two stepwise load cases of type A test: (a) load case 1 and (b) load case 2.
Strain measurements of the two stepwise load cases for type B test in Figure
Predicted and measured total strains for the two stepwise load cases of type B test: (a) load case 1 and (b) load case 2.
Predicted and measured creep strains for the two stepwise load cases of type B test: (a) load case 1 and (b) load case 2.
Strain measurements for the two stepwise load cases of type A and type B tests were predicted and compared using five creep models, including EMM, AEMM, RCM, SSM, and the method presented in this study. A constant value of the ageing coefficient
In the SSM case, the continuous creep function
The types of variables and the model format were compared among the five creep models to identify the model performance depending on those modeling factors. Three modeling factors were verified in the comparisons: whether the creep function accounts for the timevarying stress history, whether the development of the elastic modulus is taken into account in the model formulation, and whether the formulation format is based on incremental time or total time. Regarding the timevarying stress history, three models, AEMM, SSM, and the presented model, account for the timevarying stress history in their creep functions. Regarding the development of elastic modulus, two models, SSM and the presented model, take into account the change of elastic modulus at the age of loadings while the other three models, AEMM, EMM, and RCM, use the elastic modulus evaluated at the age of initial loading. Regarding the formulation format, two models, EMM and AEMM, are based on the total format and three models, RCM, SSM, and the presented model, are based on the incremental format.
Figures
Comparison of the total strain predicted by the five creep models (type A test): (a) load case 1 and (b) load case 2.
Comparison of the total strains predicted by the five creep models (type B test): (a) load case 1 and (b) load case 2.
To investigate the longterm performances of the five models, an axial load of 2 MPa was numerically applied to both of the specimens in type A and type B tests at the ages of 90 days and 103 days, respectively, when the experiments were completed. Strains were computed for 500 days for the five creep models and compared in Figures
Extended predictions and comparison of the total strain by the five creep models (type A test): (a) load case 1 and (b) load case 2.
Extended predictions and comparison of the total strain by the five creep models (type B test): (a) load case 1 and (b) load case 2.
A singlecurve creep model was presented to depict the creep behavior of concrete subjected to timevarying stress history. The formulation was accomplished by introducing a horizontal parallel creep curve assumption and combining with the vertical parallel creep curve concept of RCM. An ageing factor, defining the effect of concrete age on loading, was introduced to combine the two parallel creep curve concepts. Two sets of laboratory experiments on cylindrical specimens cast with different mixture proportions were conducted to determine the value of the ageing factor. The performance of the presented creep formulation was investigated by comparing the measured strains to the predicted strains and then comparing to the existing four creep models. The following conclusions were drawn:
The presented creep model uses a single creep curve to calculate creep strain under a timevarying stress history. This representation of creep simplifies the agedependent formulation of the longterm behavior of concrete structures and requires no memory of stress history.
The presented formulation combines the two creep curve concepts that define the upper and lower bounds of creep strain, thereby providing a reliable level of accuracy for the predicted creep strain.
The presented creep model was formulated in an incremental format that allowed accounting for the timedependent development of the elastic modulus. Furthermore, the incremental form of the presented creep model paves the way for being associated with the incremental form of the timedependent finite element formulation.
A single value of 0.75 was obtained for the ageing factor through the creep tests under stepwise loads. The value of the ageing factor indicated that the creep strain is predominantly predicted by the horizontal parallel creep curve assumption, rather than the vertical creep curve assumption, for concrete loaded at a relatively early age.
The predicted strains in the presented creep model were compared to the measured strains and those of the existing four creep models. This indicated that the predicted strains of ageing coefficient method, stepbystep method, and the presented model agreed well with each other and ranged between those of EMM and RCM.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is financially supported by Korea Ministry of Land, Infrastructure and Transport (MOLIT) as “UCity Master and Doctor Course Grant Program.”