Study of creep test and creep model of hydraulic concrete subjected to cyclic loading

Full text

Journal of Building Engineering 86 (2024) 108891
Available online 26 February 2024
2352-7102/© 2024 Elsevier Ltd. All rights reserved.
Study of creep test and creep model of hydraulic concrete
subjected to cyclic loading
Yaoying Huang
a
,
*
, Yao Zhang
a
, Yiyang He
a
,
b
, Haidong Wei
a
a
College of Hydraulic & Environmental Engineering, China Three Gorges University, Yichang, Hubei, 443002, China
b
Wucheng District Water Bureau of Jinhua, Jinhua, Zhejiang, 321000, China
ARTICLE INFO
Keywords:
Hydraulic concrete
Cyclic loading
Specic creep
Superposition principle
Optimized identication
ABSTRACT
Although the structure of hydraulic concrete depends on the history of cyclic loading and
unloading, most studies focus on a single loading and unloading creep test, rendering dubious the
assumptions of elastic creep theory. To clarify this situation, we designed uniaxial tensile and
compressive cyclic loading and unloading creep tests and implemented them under uniform
conditions. The tests were done with two-grade concrete with a water/binder ratio of 0.5, which
is typical of a concrete dam project. The results elucidate how creep in sealed dam concrete
depends on tensile and compressive cyclic loading and unloading. The Boltzmann superposition
model was then used with the Grey Wolf Optimization algorithm to determine the parameters of
the eight-parameter tensile and compressive creep model. The results show that the tensile spe-
cic creep is about 1.5–2.5 times greater than the compressive specic creep at the same loading
age. When the compressive specic creep is superimposed during unloading for either tensile or
compressive specic creep, the goodness of t between the specic creep curve calculated by
Boltzmann superposition and the measured specic creep curve exceeds 0.85. The results of this
analysis indicate that when the compressive specic creep is superimposed during unloading, the
specic creep calculated based on Boltzmann superposition is consistent with the measured
specic creep.
1. Introduction
Concrete is a creeping material, so the strain increases with time when under continuous load. Concrete creep plays an essential role
in reducing temperature stress and delaying cracking, so it is an important factor that cannot be ignored in the stress analysis of
concrete dams [1,2]. To study the creep characteristics of dam concrete, researchers usually use compression, tension, torsion,
bending, or multi-axial creep tests in indoor laboratories and then use the superposition principle or creep recovery theory to develop
concrete creep models, theories, and methods [3]. The theory of elastic creep is widely used to simulate dam concrete’s temperature
stress eld and convert measured strain to stress for concrete stress monitoring.
The elastic creep theory assumes that creep is linear in stress and obeys the Boltzmann superposition principle, that tensile creep
equals compressive creep, and that loading creep and unloading creep are equal in magnitude but opposite in direction [4–6].
However, these assumptions are controversial [7]. Several viewpoints exist on how tensile specic creep relates to compressive specic
creep: Based on creep tests on concrete with a water/binder ratio of 0.51, Glanville and Thomas [4] reported that, after 30 days of
loading under unit stress, the tensile and compressive creep are essentially the same. However, the creep test results of Davis [4],
* Corresponding author.
E-mail address: huangyaoying@ctgu.edu.cn (Y. Huang).
Contents lists available at ScienceDirect
Journal of Building Engineering
journal homepage: www.elsevier.com/locate/jobe
https://doi.org/10.1016/j.jobe.2024.108891
Received 17 August 2023; Received in revised form 16 February 2024; Accepted 19 February 2024

Journal of Building Engineering 86 (2024) 108891
2
Neville [4], and Huang [8] indicate that the tensile specic creep of concrete is greater than the compressive specic creep under the
same conditions. The studies of the United States Bureau of Reclamation, Huang [3], Wei [9,10], Rossi [11–14], and others show that
the tensile specic creep of concrete is less than the compressive specic creep. In addition, Zhou [15] developed a theoretical method
based on the linear superposition principle to derive the specic creep of concrete from the compressive creep test results where a
certain amount of stress has relaxed. Lye [16] proposed two empirical models to estimate the creep coefcient and shrinkage strain of
concrete covering a wide range of natural, recycled, and secondary aggregates alone or in any combination. Sirtoli [17] measured the
shrinkage and creep in both autogenous and drying conditions from 1 day until 1 year and compared the results with those of empirical
models. The results indicate that standard models originally developed for Portland cement concretes do not allow us to address the
pronounced differences between Portland and CSA-based concretes. In summary, these works used different test conditions (sealing
method, loading level, loading age, measurement method, raw materials, etc.), highlighting the lack of measurement accuracy that
confuses the relationship between tensile specic creep and compressive specic creep and the reasonableness of other assumptions.
Unloading a concrete specimen after a period of loading generates instantaneous elastic recovery strain followed by creep recovery
strain. Numerous reports exist discussing concrete creep and predicting concrete unloading recovery. Creep tests such as Neville [18,
19] and Mei [20] show that the creep recovery strain of concrete is related to the concrete strength. The higher the strength, the higher
the ratio of creep recovery strain to creep strain. In addition, the superposition principle is often used in concrete engineering to predict
creep recovery. The United States Bureau of Reclamation used creep recovery tests of one type of mass concrete to show that when the
stress ratio does not exceed 0.5, the Boltzmann superposition principle is applicable [4]. However, numerous experimental studies
have shown that when tensile and compressive creep are assumed to be equal, the creep recovery calculated by the superposition
principle is greater than the measured value. The experimental studies on tensile and compressive creep by Brooks [6] and Huang [8,
21] show that when the superposition principle is used to analyze the recovery of tensile or compressive creep after unloading, the
predicted creep recovery is more consistent with the measured value when compression creep is superimposed. However, further
research puts this conclusion in doubt [22–24]. Note that the existing reports are mostly predictive analyses based on one-time loading
and unloading creep tests. In contrast, actual hydraulic concrete in the eld undergoes cyclic loading and unloading, which few
predictive studies consider.
To avoid this problem of the confused relationship between tensile and compressive specic creep caused by different test con-
ditions, based on the creep test of hydraulic concrete previously carried out by the project team [8], the present work reports the results
of cyclic tensile and compressive loading and unloading creep tests of hydraulic concrete implemented under uniform test conditions.
Next, based on the tensile and compressive cyclic loading and unloading creep characteristics of sealed hydraulic concrete, we
compare and discuss the creep recovery predicted by the Boltzmann superposition principle with the measured creep recovery.
2. Creep test of hydraulic concrete
2.1. Conditions of creep test
The complexity of concrete materials means that many factors affect concrete creep. All creep tests were carried out under the same
conditions to ensure that the test conditions did not differentiate the results.
2.1.1. Concrete raw materials and mix proportion
All concrete specimens were made from the same raw materials: P⋅O 42.5 Huaxin ordinary Portland cement, secondary y ash
produced in the Yichang area, ne sand from the Yangtze River (Yichang section, neness modulus 2.03), laboratory tap water (in line
with China’s national tap water standard), polycarboxylate superplasticizer produced by Qingdao Hongxia Company, and granite
gravel with particle size 5–40 mm as coarse aggregate. Referring to the concrete mix ratio of the non-foundation constraint area of a
concrete high dam in southwest China and given the laboratory’s raw materials and test conditions, we designed a C30 two-graded
concrete with a water/binder ratio of 0.5. Table 1 details the mix proportion.
2.1.2. Concrete specimen size
The uniaxial tensile, compressive creep specimens and unloaded companion specimens were all prisms measuring 15 cm ×15 cm
×50 cm. In addition, a standard cubic specimen of 15 cm ×15 cm ×15 cm was formed, and the loading of the creep specimen was
obtained by conversion.
2.1.3. Conditions for concrete creep test
The specimens were cured under standard curing conditions (20 ±2 ◦C, relative humidity ≥90%), and the creep test was carried
out in a home-made creep laboratory (20 ±2 ◦C). Given that the concrete in a dam essentially does not exchange humidity with the
environment, the creep specimen and the unloaded companion specimen were immediately wrapped after demolding in an inner layer
consisting of a freshness-maintaining lm and an outer layer consisting of a three-layer tin foil. This procedure prevented humidity
exchange between the test specimen and the environment. Previous research [8] shows that the basic conditions of a creep test can be
ensured by wrapping the specimens in multiple layers of preservative lm and tin foil to prevent humidity exchange.
Table 1
Mix proportion of hydraulic concrete creep test (kg/m
3
).
Water/binder ratio Water reducing agent Cement Water Fly ash Sand Fine aggregate (5–20 mm) Coarse aggregate (20–40 mm)
0.5 1.72 171.60 132.00 92.40 761.26 591.10 886.64
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
3
2.1.4. Load-bearing conditions of concrete creep test
Different specications and standards impose different requirements on the loading stress of creep tests. For example, the stress
ratio of a uniaxial compressive creep test is 0.3 (the ratio of loading stress to concrete compressive strength) in the test code for
hydraulic concrete (SL352-2020) [25], whereas the ASTM standard requires a stress ratio less than 0.4 for uniaxial compressive creep
tests [26]. Studies have shown that when the stress ratio is less than 0.4, the self-shrinkage strain of the creep-loaded specimen and the
unloaded companion specimen can be regarded as equal [27]. The stress level was set to 0.3 for the creep tests reported herein.
2.1.5. Concrete creep test strain-measurement system
For these creep tests, an embedded strain meter was used to measure the strain generated in the specimen during loading and
unloading [23,24]. The strain meter used an S-100 differential resistance strain meter produced by Nanjing Carlson Hydropower
Technology Co., Ltd., which has a strain measurement range of -1500-1000
με
. The strain and temperature were calculated by
measuring the resistance and resistance ratio of the strain meter as follows:
ε
=f(Z−Z0) + qΔT(1)
T=D(R−R0)(2)
where
ε
is the strain (
με
), f is the minimum reading (
με
/0.01%), Z is the resistance ratio (0.01%) Z
0
is the resistance ratio of the full
Wheatstone bridge at the reference time (0.01%), q is the temperature-correction coefcient (
με
/◦C), ΔT is the change of temperature
during the reference time (◦C), T is the temperature (◦C), D is the temperature constant (◦C/Ω), R is the acquired resistance (Ω), and R
0
is the resistance measured at 0 ◦C (Ω).
2.2. Scheme for concrete creep test
For this series of uniaxial tensile and compressive creep tests, the raw materials, test environment, loading stress level, and loading
and unloading times were as consistent as possible. The large tensile and compressive creep specimens and unloaded companion
specimens made them difcult to form, and the creep tests were time-consuming and expensive. Based on the reported uniaxial tensile
and compressive loading and unloading creep test of hydraulic concrete at different loading ages (7d/28d/60d) [8], the uniaxial cyclic
loading and unloading creep test is designed. This designed creep test used one tensile and compressive loading specimen, one
unloaded companion specimen, and three standard cube specimens. The loaded specimen was used to measure the total strain during
creep loading and unloading, the unloaded companion specimen was placed in the same test environment to measure the free volume
strain produced by the whole process, and the standard cube specimen was used to convert the ultimate compressive strength of the
creep loaded specimen. Table 2 details the test scheme.
This series of creep tests used cyclic loading and unloading, which was divided into three stages. The specimen was loaded after
curing 28 days, followed by cyclic loading and unloading creep tests consisting of subsequent loading, unloading, and reloading. The
rst and second stages consisted of loading for 7 days followed by unloading recovery for 7 days. The third stage consisted of loading
for 14 days followed by unloading recovery for 14 days. Fig. 1 shows the loading and unloading diagram of the creep test. The red
curves in the diagram show the loading stage of the specimen in the tensile and compressive creep test, the blue curves show the
unloading stage, and the black lines show the grading loading and unloading stage.
2.3. Concrete creep test steps
The procedure for the concrete creep test included preparing the material before the test, calibrating and embedding the strain
meter, forming and curing the concrete specimen, concrete strength test and conversion, loading and unloading the creep specimen,
and acquiring and analyzing the strain data.
2.3.1. Calibrating and embedding strain meter
To ensure the accuracy of the test data, we adhered to the requirements of GB/T 3408.1-2008 “Instrument for dam monitoring-
Strain meter-Part 1: Unbonded elastic wire resistance strain meter” [28]. The strain meter was calibrated and tested using a small
corrector, a dial indicator, a differential resistance reader, etc., to ensure that the difference between the minimum reading f (sensi-
tivity) of the strain meter and the value provided by the manufacturer was less than 3%. The minimum reading f after calibration was
calculated as follows:
Table 2
Schemes for uniaxial tensile and compressive creep test.
Water/
binder ratio
Creep specimen Cube
specimen
Size Number Load age
(days)
Load holding
time (days)
Unloading recovery
time (days)
stress
ratio
Number
0.5 15 cm ×15
cm ×50 cm
One tensile and compression loading
specimen, one unloaded companion
specimen
28 7 7 0.3 3 per group
42 7 7
56 14 14
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
4
f=ΔL
L(Zmax −Zmin)(3)
where f is the minimum reading (
με
/0.01%), L is the gauge distance of the strain meter (the S-100 unbonded elastic wire resistance
strain meter is 100 mm), ΔL is the deformation of the strain meter in the tension-compression cycle (mm), Z
max
is the resistance ratio
when stretched to the maximum length (0.01%), and Z
min
is the resistance ratio when compressed to the minimum length (0.01%).
In fact, the accuracy of creep test data is closely related to how the strain meter is embedded. Embedding the strain meter manually
in the center of the concrete specimen during the concrete forming process makes it difcult to ensure that the embedding position of
the strain meter remains in the geometric center while the concrete specimen is formed. Therefore, to prevent the deviation of the
embedding position of the strain meter, we drilled holes in the appropriate position in the steel mold before forming the concrete
specimens and used cable ties to x the strain meter in the middle of the mold to ensure accurate positioning of the strain meter. Fig. 2
shows the placement of strain meters.
2.3.2. Concrete specimen strength conversion
After forming the specimen, it was maintained according to the creep test conditions detailed in Sec. 2.1.3. After curing the cube to
the appropriate age, we obtained its compressive strength under the same conditions as the specimens, which allowed us to calculate
the prism compressive strength by reduction. The conversion coefcient (a =0.7–0.8) of the cube compressive strength reduced to the
ultimate compressive strength of the prism is given in SL352-2020 “Test code for hydraulic concrete” [25] and in DL/T 5150-2017
“Test code for hydraulic concrete” [29]. Tang [30] studied how the strength of prism specimens with different heights relates to
the strength of standard cube specimens with the same cross-sectional size and with the same concrete mix ratio as in the present work.
Fig. 1. Loading and unloading diagram for uniaxial creep test.
Fig. 2. Installation of embedded strain meters.
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
5
The results show that the conversion coefcient for the compressive strength of standard cube specimens and prism specimens with
heights of 45 and 55 cm is 0.65–0.68.
When the conversion coefcient changes slightly, the strain in the concrete specimen is elastic creep because the stress ratio is 0.3.
Therefore, we used a conversion coefcient of 0.7 for our series of uniaxial compression creep tests. The compressive strength of the
prism specimens was obtained by converting the compressive strength of the cube specimens with the same loading age.
To obtain the ultimate tensile strength f
t
of the prism, the cube compressive strength f
cu
was converted by using the following
empirical formulas [31–33]:
ft=0.395f0.55
cu (4)
ft=0.26f2/3
cu (5)
ft=0.332f0.6
cu (6)
After conversion, the values given by the three empirical formulas are relatively similar. For this reason, the axial tensile strength of
this series of tensile creep tests was assigned the average of the results obtained by Eqs. (4)–(6).
2.3.3. Methods for loading and unloading concrete specimens
Before the creep specimen is loaded on the frame, a horizontal check of the uniaxial compressive creep apparatus is done with a
horizontal ruler to ensure that the specimen is in a state of axial compression. Given the cube compressive strength in the loading age,
the load (kN) was calculated using the conversion coefcient of 0.7 and a stress ratio of 0.3. Before the formal loading, a hydraulic jack
was used for two preloadings. The preloaded load was 50% of the loading load, and the preloading process took 10 min. The formal
loading was done immediately after the preloading by hierarchical loading, which is generally divided into ve stages. Considering
that long-term loading can lead to stress relaxation, the load sensor must be monitored and supplemented as required during the load-
holding stage. When the load holding time reaches the design age, unloading is done by step unloading, with the whole unloading
process completed within 10–15 min. Given the design loading and unloading age shown in Fig. 1, loading and unloading were done
sequentially at the corresponding times until the end of the test.
Fig. 3 shows the uniaxial tensile and compressive creep test apparatus. For the tensile creep specimen, the threaded pull rod of 115
mm in length and 36 mm in cross-sectional diameter is used to apply the tensile stress. One end of the rod is embedded in the concrete
creep specimen with a length of 80 mm, and the other end is connected to the bottom of the device and the load cell. Table 3 lists the
compressive strength of the cube specimen and the required load for the creep test.
3. Analysis of hydraulic concrete creep test data
3.1. Analysis of hydraulic concrete creep strain
This experiment used a differential resistance strain meter to measure the strain data. The strain mainly includes the creep strain
caused by the load and the free volume strain of the specimen (including the temperature strain and the autogenous volume strain).
Fig. 3. Uniaxial tension, compression creep test device diagram.
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
6
The free volume strain of the specimen can be measured by the unloaded companion specimen, which is not loaded in the same test
environment. Because the specimen is loaded as a sealed package, only the basic creep of the specimen is considered. At this time, the
measured strain of the loading specimen is
ε
total =
ε
creep +
ε
a+
ε
T(7)
where
ε
total
is the total strain of the tensile (compressive) creep specimen (10
−6
),
ε
creep
is the measured strain of the tensile
(compressive) creep specimens (10
−6
), and
ε
a
and
ε
T
are the autogenous volume strain and temperature strain of the specimen,
respectively (10
−6
).
Sec. 2.1.4 explains that because the stress ratio of this creep test is the same and less than 0.4, the autogenous volume deformation
Table 3
Cubic compressive strength and creep test load.
Average cube compressive strength (MPa) Axial compressive strength (MPa) Stress ratio Uniaxial compression and uniaxial tension (kN)
26.45 18.52 0.3 125.0/16.0
Fig. 4. Strain curves for uniaxial creep test.
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
7
of the loaded specimen is the same as that of the unloaded companion specimen. In addition, throughout the creep test process, the
loaded specimen and the unloaded companion specimen are sealed, wrapped, and maintained in the same test environment, so they
have the same temperature strain. The basic creep of the loaded specimen is obtained by subtracting the free volume strain of the
unloaded companion specimen from the measured total strain of the loaded specimen. Fig. 4 shows the resulting strain curve of the
loaded specimen.
3.2. Analysis of concrete creep
Specic creep refers to the creep strain under unit stress, which is usually denoted C (t,
τ
), that is, the creep strain at time t for
loading time
τ
, which can be calculated as follows:
C(t,
τ
) =
ε
creep(t,
τ
)
σ
(
τ
)(8)
where C (t,
τ
) is the specic creep of concrete at time t for loading time
τ
(10
−6
MPa
−1
),
ε
creep
(t,
τ
) is the creep strain of concrete at time t
for loading time
τ
(10
−6
),
σ
(
τ
) is the loading stress (MPa) of the creep specimen for loading time
τ
.
Fig. 5 shows the creep produced by the uniaxial test calculated by applying Eq. (8) to the measured creep of Fig. 4. The specic
creep in tension and compression of typical loading times is shown in Table 4.
Fig. 5 and Table 4 show that (1) the specic creep of concrete decreases gradually with increasing loading age. The specic creep of
Fig. 5. Longitudinal specic creep of tensile and compressive specimens.
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
8
concrete increases gradually with increasing loading time, but the specic creep rate decreases gradually with increasing loading time,
which conforms to the general law of concrete creep. (2) The tensile specic creep is about 1.5–2.5 times greater than the compressive
specic creep at the same age.
3.3. Model for predicting tensile and compressive creep of concrete
3.3.1. Superposition principle for tensile and compressive creep of concrete
Assuming a linear relationship between concrete creep and stress under sustained load, the Boltzmann superposition principle may
be used to express the total strain under variable stress as the sum of the strain caused by each increment in stress:
ε
(t) =
σ
(t0)J(t,t0) + t
t0
J(t,t0)d
σ
dt dt (9)
For one-dimensional stress, the stress increment method transforms Eq. (9) to
ε
(t) = 
n
i=0
Δ
σ
i1
E(
τ
i)+C(t,
τ
i)(10)
where
σ
(t
0
) is the stress increment at t
0
, J (t,t
0
) is the creep function, Δ
σ
i
is the stress increment at
τ
i
, E (
τ
i
) is the elastic modulus of
concrete at
τ
i
, C (t,
τ
i
) is the creep at time t for loading age
τ
i
, and the unit stress continues at time t.
Based on the stress diagram for uniaxial tensile and compressive creep loading and unloading (Fig. 6), the strain formula is
ε
(t) = Δ
σ
01
E(
τ
0)+C(t,t0)+
5
i=1
Δ
σ
i1
E(
τ
i)+C(t,ti)(11)
The absolute value of the change in stress at each stage of the creep test is the same (Δ
σ
0
= − Δ
σ
1
=Δ
σ
2
= − Δ
σ
3
=Δ
σ
4
= − Δ
σ
5
. For
the convenience of calculation, only the creep strain is considered:
ε
creep(t) = Δ
σ
0C(t,
τ
0) + Δ
σ
1C(t,
τ
1) + ⋯+Δ
σ
5C(t,
τ
5)(12)
The creep test data given in Sec. 3.2 show that the tensile specic creep of concrete differs from the compressive specic creep of
concrete. Therefore, when only the creep under unit stress is considered, the different values of C (t,
τ
) in Eq. (12) can be divided into
the following four cases :
(1) When C (t,
τ
i−1
) is the compressive specic creep, it is unloaded at
τ
i
. Assuming that C (t,
τ
i
) is the compressive specic creep for
the loading age
τ
i
, the predicted values for this time is C
R1
and is calculated as follows:
CR1=CC(t,
τ
0) − CC(t,
τ
1) + CC(t,
τ
2) − CC(t,
τ
3) + CC(t,
τ
4) − CC(t,
τ
5)(13)
where the subscript “C” stands for compression.
(2) When C (t,
τ
i−1
) is compressive specic creep, it is unloaded at
τ
i
. Assuming that C (t,
τ
i
) is the tensile specic creep for the loading
age
τ
i
, the predicted values for this time is C
R2
and is calculated as follows:
CR2=CC(t,
τ
0) − CT(t,
τ
1) + CC(t,
τ
2) − CT(t,
τ
3) + CC(t,
τ
4) − CT(t,
τ
5)(14)
where the subscript “T” stands for tension.
(3) When C (t,
τ
i−1
) is tensile specic creep, it is unloaded at
τ
i
. Assuming that C (t,
τ
i
) is the compressive specic creep for the loading
age
τ
i
, the predicted values for this time is C
R3
and is calculated as follows:
CR3=CT(t,
τ
0) − CC(t,
τ
1) + CT(t,
τ
2) − CC(t,
τ
3) + CT(t,
τ
4) − CC(t,
τ
5)(15)
(4) When C (t,
τ
i−1
) is tensile specic creep, it is unloaded at
τ
i
. Assuming that C (t,
τ
i
) is the tensile specic creep for the loading age
τ
i
, the predicted values for this time is C
R4
and is calculated as follows:
CR4=CT(t,
τ
0) − CT(t,
τ
1) + CT(t,
τ
2) − CT(t,
τ
3) + CT(t,
τ
4) − CT(t,
τ
5)(16)
Table 4
Specic creep in tension and compression with typical load holding time (10
−6
/MPa).
Stress state Loading age (days) Load holding time (days) Specic creep (10
−6
/MPa)
Tension 28 7 15.299
42 7 12.875
56 14 10.667
Compression 28 7 −10.737
42 7 −7.538
56 14 −6.190
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
9
3.3.2. Optimization of superposition based on concrete creep test data
We now calculate the tensile and compressive specic creep for cyclic loading and unloading, and then compare the results with the
measured tensile and compressive specic creep for cyclic loading and unloading. Given the time-consuming calculation required to
treat the cyclic loading and unloading creep test data and optimize the various combinations of Boltzmann superposition in Sec. 3.3.1,
we combine (i) the reported eight-parameter creep model optimized from the creep test data for hydraulic concrete with the same
water/binder ratio over 30 days of aging by the project team [8] with (ii) the various superposition combinations from Sec. 3.3.1. The
eight-parameter specic creep model is proposed on the basis of the superposition principle and was improved by Zhu [2]. The model
is as follows:
C(t,
τ
) = 
2
i=1
(fi+gi
τ
−pi)1−e−ri(t−
τ
)= (f1+g1
τ
−p1)1−e−r1(t−
τ
)+ (f2+g2
τ
−p2)1−e−r2(t−
τ
)(17)
where C (t,
τ
) is usually expressed in 10
−6
MPa
−1
; f
i
, g
i
, p
i
, r
i
are constant parameters to be determined in the same series of creep tests.
The rst part in Eq. (17) represents the recoverable creep in the early load holding time, and the second part represents the
recoverable creep in the late load holding time. Table 5 shows the eight parameters for this specic creep model [8]. Fig. 7 compares
the creep curves for cyclic loading and unloading produced by the various superposition combinations of Sec. 3.3.1 with the measured
creep curves for cyclic loading and unloading.
We use the mean absolute percentage error (MAPE) and root mean square error (RMSE) to evaluate the error produced by using the
superposition principle to predict the elastic aftereffect of tensile and compressive specimens after unloading. These are calculated as
follows:
MAPE =100%
N
N
i=1
yi−yi
yi(18)
RMSE =1
N
N
i=1
(yi−yi)21
2
(19)
where y
i
is the measured value, 
yi is the calculated value, and yi is the average of the measured value.
The results given in Fig. 7 and Table 6 show that, if C (t,
τ
i−1
) is the compressive specic creep, it is unloaded at
τ
i
. Assuming that C
(t,
τ
i
) is the compressive specic creep for the loading age
τ
i
, the calculated cyclic loading and unloading compressive specic creep
obtained by Boltzmann superposition is close to the corresponding measured value. If C (t,
τ
i−1
) is the tensile specic creep, it is also
unloaded at
τ
i
. Assuming that C (t,
τ
i
) is the compressive specic creep for loading age
τ
i
, the calculated cyclic loading and unloading
tensile specic creep obtained by Boltzmann superposition is close to the corresponding measured value. However, comparing the
Fig. 6. Stress diagram for uniaxial creep loading and unloading.
Table 5
Parameters for eight-parameter specic creep model of hydraulic concrete with the same water/binder ratio.
Loading f
1
g
1
p
1
r
1
f
2
g
2
p
2
r
2
Tension 4.00 89.65 0.832 1.512 11.40 8.19 0.51 0.121
Compression 2.18 52.37 0.895 0.864 6.79 24.17 0.623 0.065
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
10
Fig. 7. Calculated specic creep compared with measured specic creep for various superposition combinations.
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
11
curves reveals differences between the cyclic loading and unloading calculated specic creep and the measured specic creep. In other
words, the parameters for the specic creep require further optimization, and new parameters may need to be introduced.
3.3.3. Optimization of model of hydraulic concrete creep based on superposition
Eq. (17) shows that the eight-parameter specic creep expression C (t,
τ
) has the eight parameters f
i
, g
i
, p
i
, r
i
(i =1, 2) to solve for.
The measured specic creep curve obtained by the concrete creep test is denoted C
M
(t,
τ
). The specic creep curve obtained by using the
eight-parameter specic creep model based on Boltzmann superposition is denoted C
J
(t,
τ
). The residual sum of squares between C
M
and C
J
is the objective function, and the non-negativity of the model parameters is the constraint. The eight parameters of the specic
creep model are optimized as follows:
F=CM(t,
τ
) − CJ(t,
τ
)2→min,
xi≥0.(20)
We use the Grey Wolf Optimizer (GWO) to optimize the specic creep model because it converges rapidly [34]. The GWO algorithm
is an intelligent optimization algorithm based on imitating the leadership class and hunting mechanism of a grey wolf group. In the
grey wolf population, the social hierarchy is divided into four categories:
α
, β, δ, and
ω
from high to low. In the GWO algorithm, each
grey wolf represents a candidate solution in the population. According to the grey wolf population hierarchy, the wolf
α
is at the
highest level and is the optimal solution. The wolf β is located in the second level, which is a sub-optimal solution; the wolf δ is located
in the third level, which is the third optimal solution; the wolf
ω
has the lowest level, obeys the leadership of the rst three wolves and
belongs to the remaining solution.
In grey wolf hunting, wolf
α
, wolf β, and wolf δ are the best groups to perceive prey information. That is to say,
α
, β, and δ are the
three optimal solutions obtained so far in parameter optimization. The three positions are used to judge the target position, and the
remaining grey wolf individuals are forced to update their positions according to the optimal grey wolf individual position. The
following formula updates the individual grey wolf position:



D
α
= |C1·X
α
(t) − X(t)|
Dβ=C2·Xβ(t) − X(t)
Dδ= |C3·Xδ(t) − X(t)|
(21)



X1(t) = X
α
(t) − A1·D
α
X2(t) = Xβ(t) − A2·Dβ
X3(t) = Xδ(t) − A3·Dδ
(22)
X(t+1) = X1(t) + X2(t) + X3(t)
3(23)
where D
α
, Dβ and Dδ represent the distance vectors between individual grey wolves and wolf
α
, wolf β, and wolf δ, respectively. The
notation |⋅| represents the vector obtained by taking the absolute value of the difference between the corresponding components of two
position vectors. X
α
(t), Xβ(t), and Xδ(t)represent the current positions of wolf
α
, wolf β, and wolf δ, respectively. X1(t), X2(t), and X3(t)
respectively represent the positions that
ω
wolves need to adjust after being affected by the positions of wolf
α
, wolf β, and wolf δ.
X(t+1)is the updated position of grey wolf i at the next moment after being adjusted according to the positions of wolf
α
, wolf β, and
wolf δ, i ∈N, where N is the population number. A1, A2, A3, C1, C2, and C3 are coefcient vectors, and the calculation formula is as
follows:
A=2a·r1−aI(24)
C=2r2(25)
a=2−2t
tmax
(26)
where a is the convergence factor, which decreases linearly from 2 to 0 with the number of iterations, I is a unit vector, and r1 and r2 are
random variables between [0,1], t is the current number of iterations, and t
max
is the maximum number of iterations.
The GWO will start from a set of random solutions, nd the optimal parameter group in the continuous update iteration, and
automatically stop the output of the optimal solution when the accuracy is satised. The ow chart of the GWO algorithm is generally
Table 6
Evaluation of the prediction of C
R1
–C
R4
elastic aftereffect.
Evaluation method C
R1
C
R2
C
R3
C
R4
RMSE 7.50 23.32 7.23 8.91
MAPE 49.87% 146.68% 33.30% 41.45%
max|△| 11.34 35.63 10.22 15.13
min|△| 2.89 2.89 0.52 0.52
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
12
shown in Fig. 8.
The sum F in Eq. (20) is used as the tness value for the GWO, that is, the objective function value, to search for the optimal
parameters. Given that multiparameter optimization is time-consuming, a wide range of parameter optimization results were collected
from measured creep of hydraulic concrete and the corresponding specic creep expressions [2]. Next, the range and magnitude of
each parameter in the specic creep model was estimated. In the GWO, the solution vector for f
i
to r
i
(i =1, 2) is set to [0, 0, 0, 0, 0, 0, 0,
0] to [10, 200, 1, 1, 10, 200, 1, 1], and the parameters are set as follows: the problem dimension D is 16, the population number N is
100, the iteration threshold of the termination algorithm is 3000, and the accuracy of the termination algorithm is 1 ×10
−3
. The
results of parameter optimization are given in Table 7, and Fig. 9 compares the calculated specic creep with the measured specic
creep.
In addition to the RMSE and the MAPE, the goodness of t R
2
is introduced to evaluate how well the calculated results reproduce the
measured results. The formulas for R
2
are
R2=1−SSE
SST (27)
SSE =
n
i=1
(yi−yi)2(28)
SST =
n
i=1
(yi−yi)2(29)
where y
i
is the measured value, 
yi is the calculated value, and yi is the average of the measured values.
Table 8 gives the R
2
, RMSE, and MAPE results for the specic creep calculated by the superposition method and the measured
specic creep.
Although Figs. 7 and 9 show that the calculated curve for the rst loading stage agrees well with the measured curve, Tables 5 and 7
show that the parameter g
i
optimized based on the cyclic loading and unloading creep test data different from the parameter g
i
optimized based on the creep test data for hydraulic concrete with the same water/binder ratio over 30 days of aging. This result is
attributed to the non-uniqueness of the multi-parameter optimization inversion. Because the cyclic loading and unloading fully reect
the creep characteristics of concrete, the parameters inverted based on the cyclic loading and unloading creep test data are more
Fig. 8. Flow chart of GWO algorithm.
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
13
reasonable.
When using the GWO to obtain the parameters for the specic creep expression based on the superposition determined in Sec. 3.3.2,
the goodness of t R
2
between the specic creep curve calculated by Boltzmann superposition and the measured specic creep curve
exceeds 0.85, the maximum RMSE is not more than 1.5
με
/MPa, and the maximum MAPE is not more than 10%. Thus, the t is good.
4. Conclusion
Most previous research focuses on one-time loading and unloading creep tests, which ignores the fact that actual hydraulic concrete
structures are subjected to cyclic loading and unloading. Thus, the assumptions used in the conventional elastic creep theories are
dubious. The present work thus uses uniaxial tensile and compressive creep tests under cyclic loading and unloading to investigate how
these conditions affect sealed hydraulic concrete. By comparing the measured data from creep tests with calculations based on
Table 7
Optimized parameters for concrete specic creep model.
Parameter f
1
g
1
p
1
r
1
f
2
g
2
p
2
r
2
Tension 2.386 99.702 0.325 0.031 1.420 92.555 0.739 0.498
Compression 1.010 98.668 0.272 0.026 0.391 54.425 0.621 0.499
Fig. 9. Measured curve compared with calculated curve using optimized parameters for specic creep.
Table 8
Fitting of specic creep calculated with optimized parameters to measured curve.
Specic creep superposition curve R
2
RMSE MAPE
C
R1
0.897 1.392 9.08%
C
R3
0.980 0.839 4.00%
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
14
Boltzmann superposition of the specic creep produced by loading and unloading, we arrive at the following conclusions:
(1) The comparative analysis of tensile and compressive specic creep curves under cyclic loading and unloading at loading ages
28, 42, and 56 days shows that when the load holding time is less than 14 days, the tensile specic creep is about 1.5–2.5 times
greater than the compressive specic creep at the same age.
(2) When using the Boltzmann superposition principle and the Grey Wolf Optimization algorithm to obtain the parameters of the
improved eight-parameter creep model, the results of this analysis indicate that, when in the state of either tensile or
compressive specic creep, the compressive specic creep is superimposed during unloading, the goodness of t R
2
between the
specic creep curve calculated by Boltzmann superposition and the measured specic creep curve exceeds 0.85, the maximum
RMSE does not exceed 1.5
με
/MPa, and the maximum MAPE is no greater than 10%. Namely, when the compressive specic
creep is superimposed during unloading, the specic creep calculated based on Boltzmann superposition is consistent with the
measured specic creep.
(3) Due to the complexity of concrete materials, more different water-binder ratios and more specimen sizes should be investigated
to enhance the applicability and practicability of the research results.
CRediT authorship contribution statement
Yaoying Huang: Conceptualization, Investigation, Methodology, Writing – original draft, Writing – review & editing. Yao Zhang:
Writing – review & editing, Writing – original draft, Investigation, Data curation. Yiyang He: Writing – original draft, Investigation,
Data curation. Haidong Wei: Supervision, Investigation.
Declaration of competing interest
The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to
inuence the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgements
This study was supported by the National Natural Science Foundation of China under Grant Nos. 52179135 and 51779130.
References
[1] C.S. Gu, H.Z. Su, S.W. Wang, Advances in calculation models and monitoring methods for long-term deformation behavior of concrete dams, J. Hydroelectr. Eng.
35 (5) (2016) 1–14.
[2] B.F. Zhu, Temperature Stress and Temperature Control of Mass Concrete, second ed., China Water & Power Press, Beijing, 2012.
[3] G.X. Huang, R.Y. Hui, X.J. Wang, Creep and Shrinkage of Concrete, China Electric Power Press, Beijing, 2012.
[4] A.M. Neville, W.H. Dilger, J.J. Brooks, Creep of Plain and Structural concrete[M], London and New York: Construction Press, London and New York, 1983.
[5] S.G. Kim, Y.S. Park, Lee, Y.H. Lee, Comparison of concrete creep in compression, tension, and bending under drying condition, Mater 12 (20) (2019) 3357.
[6] J.J. Brooks, A.M. Neville, Comparison of creep, elasticity and strength of concrete in tension and in compression, Mag. Concr. Res. 29 (100) (1977) 131–139.
[7] A. Hilairea, F. Benboudjema, A. Darquennes, et al., Modeling basic creep in concrete at early-age under compressive and tensile loading, Nucl. Eng. Des. 269
(2014) 222–230.
[8] Y.Y. Huang, T. Xie, Y. Ding, et al., Comparison of tensile and compressive creep of hydraulic concrete considering loading/unloading under unied test
conditions, Construct. Build. Mater. 286 (2021) 122763.
[9] Y. Wei, J.S. Huang, S.M. Liang, Measurement and modeling concrete creep considering relative humidity effect, Mech. Time-Dependent Mater. 24 (2) (2020)
161–177.
[10] Y. Wei, Z.H. Wu, J.S. Huang, et al., Comparison of compressive, tensile, and exural creep of early-age concretes under sealed and drying conditions, J. Mater.
Civ. Eng. 30 (11) (2018) 04018289.
[11] P. Rossi, J.L. Tailhan, F.L. Maou, Creep strain versus residual strain of a concrete loaded under various levels of compressive stress, Cement Concr. Res. 51
(2013) 32–37.
[12] P. Rossi, J.L. Tailhan, F.L. Maou, et al., Basic creep behavior of concretes investigation of the physical mechanisms by using acoustic emission, Cement Concr.
Res. 42 (1) (2012) 61–73.
[13] P. Rossi, J.P. Charron, M.M. Bastien, et al., Tensile basic creep versus compressive basic creep at early ages: comparison between normal strength concrete and a
very high strength bre reinforced concrete, Mater. Struct. 47 (10) (2014) 1773–1785.
[14] P. Rossi, J.L. Tailhan, F.L. Maou, Comparison of concrete creep in tension and in compression: inuence of concrete age at loading and drying conditions,
Cement Concr. Res. 51 (1) (2013) 78–84.
[15] Y.Q. Zhou, W.Y. Chen, P.Y. Yan, Measurement and modeling of creep property of high-strength concrete considering stress relaxation effect, J. Build. Eng. 56
(2022) 104726.
[16] C.Q. Lye, R.K. Dhir, G.S. Ghataora, Estimation models for creep and shrinkage of concrete made with natural, recycled and secondary aggregates, Mag. Concr.
Res. 74 (8) (2021) 392–418.
[17] S. Davide, M. Wyrzykowski, R. Paolo, et al., Shrinkage and creep of high-performance concrete based on calcium sulfoaluminate cement, Cem. Concr. Compos.
98 (2019) 61–73.
[18] A.M. Neville, Recovery of creep and observations on the mechanism of creep of concrete, Appl. Sci. Res. 9 (1) (1960) 71–84.
[19] J.M. Torrenti, Basic creep of concrete-coupling between high stresses and elevated temperatures, Eur. J. Environ. Civ. Eng. 22 (12) (2017) 1419–1428.
[20] S.Q. Mei, J.C. Zhang, Y.F. Wang, et al., Creep-recovery of normal strength and high strength concrete, Construct. Build. Mater. 156 (2017) 175–183.
[21] Y.Y. Huang, L. Xiao, J. Gao, et al., Tensile creep and unloading creep recovery testing of dam concrete with y ash, J. Mater. Civ. Eng. 31 (5) (2019) 05019001.
[22] J.P. Forth, Predicting the tensile creep of concrete, Cem. Concr. Compos. 55 (2015) 70–80.
[23] N. Ranaivomanana, S. Multon, A. Turatsinze, Basic creep of concrete under compression, tension and bending, Construct. Build. Mater. 38 (1) (2013) 173–180.
Y. Huang et al.

Journal of Building Engineering 86 (2024) 108891
15
[24] A.E. Klausen, T. Kanstad, O. Bjontegaard, et al., Comparison of tensile and compressive creep of y ash concretes in the hardening phase, Cement Concr. Res. 95
(2017) 188–194.
[25] SL352-2020, Test Code for Hydraulic Concrete, Standards for Water Conservancy Industry in People’s Republic of China, 2020.
[26] ASTM C512/C512M-15, Standard Test Method for Creep of Concrete in Compression. ASTM International.
[27] M. Wyrzykowski, P. Lura, The effect of external load on internal relative humidity in concrete, Cement Concr. Res. 65 (12) (2014) 58–63.
[28] GB/T 3408.1-2008, Instrument for Dam Monitoring-Strain Meter-Part 1: Unbonded Elastic Wier Resistance Strain Meter, National Standards of People’s
Republic of China, 2008.
[29] DL/T 5150-2017, Test Code for Hydraulic Concrete, Standards for Electric Power Industry in People’s Republic of China, 2017.
[30] T.F. Tang, Research on Creep Experiment and Prediction Model of High Stress Ratio Concrete, Yichang: China Three Gorges University, Yichang, 2014.
[31] GB 50010-2010, Code for Design of Concrete Structures (2015 Edition), National Standards of People’s Republic of China., 2015.
[32] DL/T 5057-2009, Design Specication for Hydraulic Concrete Structures, Standards for Electric Power Industry in People’s Republic of China, 2009.
[33] B.F. Zhu, Relation between extensibility, age, tensile and compressive strength of concrete, China Civ. Eng. J. 29 (5) (1996) 72–76.
[34] S. Mirjalili, S.M. Mirjalili, A. Lewis, Grey wolf optimizer, Adv. Eng. Software 69 (3) (2014) 46–61.
Y. Huang et al.