Abstract
The dynamic behavior of reinforced concrete beams are inevitably affected by loading rate. This article reports the effect of loading rate on the dynamic behavior of reinforced concrete beams under cyclic loading. Dynamic tests on reinforced concrete beams at various loading rates were carried out using the MTS electro-hydraulic servo system. The cyclic loading was controlled by displacement with a loading rate in the range from 0.1 to 10 mm/s. Based on the test results, the effect of the loading rate on the failure shape and the loading–displacement curve of reinforced concrete beams was investigated. The research focuses on the effect of loading rate on strength and deformation behavior, including cracking, yielding, ultimate, and failure strengths as well as displacements. Finally, the effect of loading rate on the ductility, stiffness degradation, and dissipated energy capabilities of reinforced concrete beams was also critically examined.
Introduction
The dynamic behavior of reinforced concrete (RC) members, such as RC beams and RC columns, are inevitably affected by loading rate. On one hand, concrete and steel are typical rate-dependent materials: their strengths, stiffness, and brittleness (or ductility) are affected by loading rate. The general observation is that the dynamic tensile and compressive strengths and the elastic modulus of concrete increase with increasing loading rate. The yield strength and the corresponding strain of steel reinforcement increase with increasing the loading rate, but the elastic modulus is rate-independent. On the other hand, the failure shape of RC members due to dynamic load was different from that of static load. With the exception of the familiar flexural failure, the brittle shear failure may have occurred on some circumstances, even though RC members were designed according to the flexural failure; in other words, the shear failure occurred before the flexural failure. This phenomenon has been confirmed through the dynamic tests of RC members (Ghabossi et al., 1984; Krauthammer, 1984; Krauthammer et al., 1986).
Recently, more and more researchers have studied the effect of loading rate on the dynamic behavior of RC members. Krauthammer (1984) presented a method for the analysis of RC box-type structures under the effect of severe dynamic loading conditions, demonstrated by employing it for the analysis of seven different events, and then evaluating its accuracy by comparing numerical and experimental results (Krauthammer et al., 1986). Afterward, with the rate-dependent model compared with the rate-independent model, the dynamic responses of RC beams under the dynamic loading condition were further studied (Al-Haddad, 1995; Beshara and Virdi, 1992; Farag and Leach, 1996; Kulkarni and Shah, 1998). Kunnath and Reinhorn (1990) presented an efficient model for inelastic biaxial bending interaction of RC sections, and then demonstrated the validity of the proposed scheme through the analytical simulation of available biaxial experiments on RC columns in comparison with other analytical models. Fu et al. (1991) presented a survey on the behavior of RC subjected to dynamic loading, reviewing, and then discussing the response of RC materials and elements to various strain rates. If significant, the increase in the flexural capacity of individual members as a result of high strain rates might transform a structure’s failure shape from a preferred ductile manner to a less-desirable brittle mode. A rate- and history-dependent constitutive model of concrete was developed to nonlinearly analyze both the plane and axisymmetric RC structures subjected to transient impulsive loading (Beshara and Virdi, 1992). Considering the actual properties of the reinforcing steel under both low and high strain rates of loading, Al-Haddad (1995) undertook a parametric study of the curvature ductility capacity of RC sections. Farag and Leach (1996) proposed an improved material model for concrete, which included the effect of high strain rate upon both the stiffness of the material and upon the crushing strength, and the expressions for the yield and failure surfaces of concrete, which account for the effects of high strain rate. Seven pairs of singly reinforced beams (without shear reinforcement) were tested under displacement control with a closed-loop servo-hydraulic testing machine to investigate the dynamic behavior of RC beams at high rate of loading (Kulkarni and Shah, 1998). In order to improve the computational efficiency in the transient dynamic nonlinear analysis of RC plates subjected to blast or seismic loading, a parallel scheme for the time-marching procedure using the explicit Newmark’s algorithm was presented (Sziveri et al., 1999).
Kwak and Kim (2001) proposed a hysteretic moment–curvature relationship to simulate the behavior of a RC beams under cyclic loading and analyze the nonlinear behavior of RC beams subjected to flexural cyclic loading. Nürnbergerová et al. (2001) presented a theoretical model for the determination of the moment versus curvature, and shear force versus shear deformation relationships based on the stress–strain curves of the materials. In order to study the relationship between the nonlinear dynamic behavior of RC beams and their damage levels, the time–frequency response analyses from the impact excitation vibration tests were carried out on RC beams with various damage levels (Wang et al., 2006). Cotsovos et al. (2008) described the numerical investigation into the dynamic response of RC beams subjected to high rates of transverse loading. Different from other studies, Cotsovos attributed the effect of the applied loading rate on the exhibited structural response to the inertia forces that developed within the beam instead of the loading rate sensitivity of the materials. Valipour et al. (2009) took into account the effect of strain rate at the fiber level using the dynamic increase factor (DIF) concept for steel and concrete to improve upon an one-dimensional flexibility of fiber element that ignored the shear effect at the material level. Fujikake et al. (2009) examined the impact responses of RC beams through an experimental study, and presenting an analytical model, they developed to predict the maximum mid-span deflection and maximum impact load. Abbas et al. (2010) investigated key aspects of structural response, such as the loading-deformation behavior, crack patterns, and the strength and failure shapes of RC-wide beams under low-rate (static) and high-rate (impact) concentrated loading applied at their mid-span. An iterative approach for the computation of a curvature ductility factor for doubly RC sections was proposed by taking into account the strain rate sensitive properties of concrete and steel, the confinement of core concrete, and the degradation of cover concrete during load reversal under earthquake loading (Pandey, 2011). Through an experimental study, Fukuda et al. (2011) investigated the dynamic shear failure behavior of RC beams under rapid loading. The behavior of RC beams under varying rates of concentrated loading was studied to investigate the effect of loading rate on RC beams and also to propose empirical equations in terms of various parameters to predict the DIF of maximum resistance of RC beams under various loading rates (Adhikary et al., 2012). Pandey (2013) has taken into account the strain rate sensitive constitutive behavior of concrete and steel to compute the flexural ductility of RC beam sections. The parametric studies indicated that the flexural ductility factor decreased with increasing strain rates. Li and Li (2013) studied the effect of loading rate on RC beams experimentally and numerically at a strain rates in the range between 10−5 and 0.1/s that could potentially be experienced during earthquakes. Afterward, the dynamic behavior of RC beams was given through experiments, and the effect of loading rate on the bearing capacity, ductility, stiffness, failure shape, and energy dissipation capacity of beams were analyzed.
The behavior of RC members on the cyclic loading has often been used to measure the capacity of energy dissipation and the aseismic capacity. The character of RC members under cyclic loading resulted in researchers’ interest. The development and application of an energy dissipation index to characterize RC beams under cyclic loading were described and used to evaluate the results of five major experimental investigations (Darwin and Nmai, 1986). Reversed cyclic loading tests for RC beams were performed to standardize the dissipated energy; consequently, a new method of ductility evaluation based on the standardized dissipated energy of RC beams was proposed (Miyauchi, 1996). Also, a new method of damage evaluation for RC beams based on the dissipated energy of a perfectly elasto-plastic body was likewise proposed (Miyauchi, 1997). Maruo and Tanabe (1998) used a microplane model in which the micro behavior of concrete was focused to analyze an RC beam subjected to cyclic load. Kwak and Kim (2001) simulated the hysteretic moment–curvature relationship of an RC beam under cyclic loading and analyzed the nonlinear behavior of RC beams subjected to flexural cyclic loading using the proposed model. Through experimentation, Khan et al. (2010, 2013) investigated the effect of cyclic thermal loading on the shear strength and the bond strength of a RC beam. Vaz et al. (2014) studied the behavior under the cyclic loading of RC beams strengthened for bending through the addition of concrete and steel on their tension side using expansion bolts as shear connectors, denominated as partial jacketing. Through an experimental campaign, Gião et al. (2014) studied the effect of the gravity load on the RC beam connection to the column subjected to cyclic loading.
The above-mentioned studies indicate that the effect of loading rate on the dynamic behavior of RC members is significant. However, the research on the dynamic behavior of RC members under the cyclic load was very limited; In fact, only a few design codes take into account the effect of loading rate on RC structures because there are insufficient studies on the effect of loading rate. In this article, the dynamic tests on RC beams under cyclic loads at various loading rates were carried out to study the effect of loading rate on the structural behavior of RC beams. According to the test results, the effect of loading rate on the failure shape and loading–displacement curve of RC beams was critically examined. The cracking, yielding, ultimate, and failure strengths as well as deformation, ductility, and dissipated energy capability of RC beams are reported.
Experimental program
Beam design and material properties
The test RC beams were designed based on the consideration of the fundamental period of the beams consistent with that of general concrete buildings. Each of the RC beams with dimensions 1.4 m in length, 150 mm in width, and 200 mm in depth was simply supported within a 1.2-m span, as shown in Figure 1. Portland cement was used in all the mixtures; its concrete grade was designed as 25 MPa. The mix proportion of constituents by weight was 1.00: 0.40: 1.18: 2.36, which corresponded to cement, water, sand, and gravel, respectively. The average 28-day concrete compressive strength for all the beams was 25 MPa. The tensile strength was adopted as one-tenth of the compressive strength. The longitudinal steel reinforcement consisted of four rebars, 10 mm in diameter, horizontally placed in the top and bottom of the beam. The transverse stirrups were 6 mm in diameter and 150 mm in distributed distance. The design strength of reinforcing bars of the RC beams was 320 MPa.
Details of the RC beam.
Effect of loading rates on strength of materials
In order to study the effect of loading rate on the strength of materials, the uniaxial compressive test of concrete cubic specimen with 100 mm in length was carried out. The dynamic strength of concrete at various loading rates was illustrated in Table 1. In this article, the loading rate 0.1 mm/s was regarded as the quasi-static loading rate. The strength of concrete at this loading rate was taken as the quasi-static strength. Compared to the quasi-static strength of concrete, the uniaxial compressive strengths of concrete at loading rates 0.5, 1.0, 5.0, and 10.0 mm/s increase 6.42%, 11.35%, 25.21%, and 30.06%, respectively.
Dynamic uniaxial compressive strengths of concrete.
DIF: dynamic increase factor.
Similarly, the uniaxial tensile test of steel bar with 10 mm in diameter and 50 mm in length was carried out. The dynamic yield and ultimate strengths of steel bar at various loading rates were illustrated in Table 2. Compared to the quasi-static strength of steel bar, the uniaxial tensile yield strengths of steel bar at loading rates 0.5, 1.0, 5.0, and 10.0 mm/s increase 4.67%, 5.24%, 9.59%, and 26.42% and the ultimate strengths increase 1.81%, 4.89%, 6.87%, and 12.89%, respectively.
Dynamic yield and ultimate strengths of steel bar.
DIF: dynamic increase factor.
Loading setup
The tests were carried out in the structural laboratory at Shenyang Jianzhu University, China. The multi-channel MTS servo-hydraulic loading system was employed for loading. The maximal thrust loading was up to 350 kN, the maximal push loading was up to 240 kN, and the maximal displacement was up to 500 mm. It was needed that the stiffness of the test frame for RC beams was sufficiently large to reduce the measurement error. In order to avoid a local failure of the boundary concrete, two 20-mm-width armor plates were imbedded into the bottom and the top of the RC beams to support the beam and the loading steel beam. The setup and the loading equipment are shown in Figure 2.
Experiment setup: (a) load system and (b) setup.
Measurement setup
Because the maximum loading rate was 10 mm/s, a high-speed data acquisition system was needed to prevent the test data from loss. The dynamic data acquisition system DH5937 was therefore utilized in this experiment. The system collected various electrical signals with a maximum acquisition frequency of 20 kHz. All eight channels worked at the same time, and the simultaneous collection frequency of every channel could achieve 2.5 kHz. In order to collect sufficient, but not overabundant data, different collection frequencies were adapted at various loading rates. The collection frequencies of five test beams from BC-1 to BC-5 were 20, 50, 200, 1000, and 2000 Hz, respectively. During the tests, the strain of reinforcements, the mid-span displacement of the beam, and the MTS loading actuator were acquired.
Loading program and test procedure
In order to study the effect of loading rate on the dynamic behavior of RC beams, the cyclic displacement-controlled load with various loading rates (i.e. 0.1, 0.5, 1, 5, and 10 mm/s) were imposed on five beams from BC-1 to BC-5, respectively. Because the reversed loading could not be imposed on the beam, the loading was unloaded according to the same loading rate until the force was equal to zero. The typical load cycle in the proposed test procedure is shown in Figure 3.
Loading procedure.
Experimental results
Failure shape of RC beams
The failure shape of RC beam was affected by the shear span ratio. Generally, when the shear span ratio is greater than 3.0, the RC beam would fail with the diagonal tension failure shape. When the shear span ratio is less than 1.0, the RC beam would fail with the diagonal compression failure shape. When the shear span ratio is greater than 1 and less than 3, the failure mode of RC beam would fail with the shear compression failure shape. In order to avoid the diagonal tension and compression failure shape, the shear span ratio of RC beam is selected between 1.0 and 3.0. In this article, the experiment was designed to study the effect of loading rate on the dynamic behavior of the pure bending segment, and the shear span ratio was selected as 1.5, which is equal to 300 divided by 200.
The crack pattern and failure shape of RC beams at various loading rates are shown in Figure 4. The crack pattern and failure shape were the same at various loading rates; tension caused the vertical cracks of concrete located at the bottom of the beam. Through careful observation and comparison, the crack width decreases slightly with increasing the loading rate, and the cracks also distribute more uniformly when the displacements for all the RC beams are the same. The reason for this might be that the rapid loading rate delayed the propagation of any internal micro-cracks and enlarged the crack field, which caused the developments of the external cracks. Consequently, when the displacements for all the RC beams are the same, more visible cracks appeared, and the crack width decreased with increasing the loading rate. Because the ultimate displacement of RC beams increases with increasing the loading rate, the final crack width of concrete may be same or increase slightly with increasing the loading rate. In addition to the vertical cracks, few diagonal cracks appeared on the beam surface under the loading plates, which was caused by the combined action of shear and bending moment.
The failure configurations of RC beams.
Effect of loading rate on the loading–displacement curves
Loading–displacement curves of RC beams at various loading rates
The mid-span loading–displacement curve was an important factor in evaluating the mechanical behavior of the simply supported RC beams. Figure 5(a) to (e) illustrates the five mid-span loading–displacement hysteretic curves of the RC beams at various loading rates. From these curves, three main parts of the behavior were identified. In the first cycle, a progressive decrease in the stiffness occurred due to cracking that appeared. For the second cycle, a plastic plateau appeared that corresponded to the yielding of the reinforcing bars. For the remaining six cycles, the negative stiffness occurred because RC beams reached their ultimate strength.
Mid-span loading–displacement curve: (a) loading rate: 0.1 mm/s, (b) loading rate: 0.5 mm/s, (c) loading rate: 1.0 mm/s, (d) loading rate: 5.0 mm/s, and (e) loading rate: 10.0 mm/s.
Comparison with the existing models of RC members
The hysteretic moment–curvature relationships to simulate the behavior of RC beams under cyclic loading were proposed by many researchers. Early in 1987, Roufaiel and Meyer (1987) proposed a theory for an enhanced mathematical model of R/C frame members and verified its accuracy by simulating various laboratory tests for which data were available in the literature. Taking the bond–slip effect into account by defining the initial loading branch on the basis of the monotonic moment–curvature relationship, Kwak and Kim (2001) presented a curved hysteretic moment–curvature relationship of RC members, in which the fixed-end rotation at the beam–column joint interface and the pinching effect caused by the applied shear force were also taken into consideration. Finally, the validity of the proposed model was established by comparing the analytical predictions with those from the experimental and previous analytical studies. These two models were accepted by many researchers and adopted to compare with their models or experimental results. In this article, the experimental results of RC beams at various loading rates were compared with these two models, shown in Figure 6(a) to (e). It can be observed from these figures that in the loading branches, the two model predictions agree well with the test outcomes, especially with the Kwak’s model because this model takes into account the bond–slip effect of concrete and steel. But in the unloading branches, the two model predictions do not descend obviously, and the two models overestimate the stiffness degradation of RC beams.
Comparison of test results with the current model of RC beams: (a) loading rate: 0.1 mm/s, (b) loading rate: 0.5 mm/s, (c) loading rate: 1.0 mm/s, (d) loading rate: 5.0 mm/s, and (e) loading rate: 10.0 mm/s.
Effect of loading rate on the envelope curves of RC beams
Similar to the loading–displacement curve of a RC beam under the monotonic load, the envelope curve of RC beams under the cyclic load was used to investigate the structural behavior of RC beams. Figure 7 shows a sketch map of the loading–displacement envelope curve for RC beams. Generally, the plot consists of four segments. The first straight line shows the linear behavior before the first crack (point A). The second slope corresponding to the cracked section followed until point B, where the flexural reinforcement yielded. At point B, the displacement of the beam began to increase at a higher rate as greater load was applied. After the point B, the nonlinearity of RC beams appeared obviously because the cracked concrete and the yielding reinforcement led to the degradation of the RC beam’s stiffness. The beam’s stiffness would be descended to zero at point C where the ultimate loading corresponded to the nominal flexural capacity of the cross section. After the point C, the beam’s ability to distribute the load throughout the cross section began to decrease with the increasing deformation until the failure point D, where the carrying ability was defined to the 85% of the ultimate loading.
Typical loading–displacement curve of RC beams.
To study the effect of loading rate on the dynamic behavior of RC beam, Figure 8 illustrates the five mid-span loading–displacement envelope curves of the RC beams subjected to various loading rates. When the five curves are compared, it can be found that at the beginning of the load, the RC beams are within their elastic stages. The loading–displacement curves appeared as a straight line, and the effect of the loading rate was negligible. With increasing the loading rate, the effect of strain rate on the RC beams is more obvious and the loading–displacement curves appeared to be nonlinear. The yielding load and the corresponding displacement as well as the ultimate load and the corresponding displacements increase with increasing the loading rate.
Envelope curves of the RC beams at various loading rates.
Effect of loading rate on the crack, yield, and ultimate strengths
The crack, yield, and ultimate strengths were defined as the carrying capacity of points A, B, and C of the RC beams in Figure 7. Table 3 lists the crack, yield, and ultimate strengths of RC beams at specimens BC-1 to BC-5. When compared with specimen BC-1, the crack strengths of specimens BC-2, BC-3, BC-4, and BC-5 increase 1.92%, 2.15%, 7.51%, and 10.40%, respectively. Similarly, the yield strengths of specimens BC-2, BC-3, BC-4, and BC-5 increase 0.11%, 6.32%, 8.64%, and 11.87% and the ultimate strengths increase 3.40%, 4.85%, 8.52%, and 11.21%, respectively.
The crack, yield, and ultimate strengths of RC beams.
RC: reinforced concrete.
According to reference (Bischoff and Perry, 1991), the increase in concrete strength follows a linear logarithmic relationship when the loading rate increases. Similarly, Figure 9 illustrates the relationship between the crack, yield, and ultimate strengths of RC beams and the logarithm of the loading rate. It can be seen that the crack, yield, and ultimate strengths increase notably. Compared the crack strength with the yield and ultimate strengths, the slopes of the yield and ultimate strengths are somewhat greater than those of the crack strength. The reason for this is that the effect of the loading rate on concrete and steel reinforcement was different at various stages. Most of the loads were carried by the concrete before it cracked, so the effect of the loading rate on the crack loading of the RC beam was closer to the effect on the concrete. On the yield and ultimate states, most of the tensile force at the bottom of the beam was carried by the tensile steel reinforcement; however, most of the compressive loadings at the top of the beam were carried by the compressive concrete, so the effect of the loading rate on the yield and ultimate strengths of the RC beams was determined by the combined effects of the loading rate on the concrete and steel reinforcement. Obviously, the combined effect was greater than that of concrete at the cracking state.
Effect of loading rate on the strengths of RC beams.
Effect of loading rate on the deformation
The mid-span crack, yield, ultimate, and failure displacements at various loading rates are listed in Table 4. It can be seen that the mid-span crack, yield, ultimate, and failure displacements of the RC beams increased significantly with increasing the loading rate. In comparison with specimen BC-1, the mid-span crack displacement of specimens BC-2, BC-3, BC-4, and BC-5 increased 3.39%, 15.25%, 22.03%, and 33.90%, respectively. Similarly, the mid-span yield displacement of specimens BC-2, BC-3, BC-4, and BC-5 increased 6.09%, 11.54%, 16.99%, and 21.47%, the mid-span ultimate displacement increased 7.53%, 7.00%, 5.38%, and 16.00% and the failure displacement increased 4.50%, 10.03%, 11.55%, and 14.31%, respectively. These test results agree with the previous tests of RC members (Li and Li, 2013).
The crack, yield, ultimate, and failure displacement of RC beams.
RC: reinforced concrete.
Effect of loading rate on the stiffness degradation
Nearly all RC members would exhibit some level of stiffness degradation when subjected to several large cyclic load reversals, which is caused by the results of cracking, loss of bond, or interaction with high shear or axial stresses. The level of stiffness degradation depends on the characteristics of the RC members as well as on the loading history.
In this article, the stiffness degradation is defined as the ratio of the unloading stiffness to the initial stiffness. Usually, the stiffness degrades as displacement increases. Figure 10 illustrates the stiffness degradation of the RC beams with the displacement at the unloading point at different loading rates. From this figure, the effect of loading rate on the stiffness degradation is not obvious, and a fit exponential curve of stiffness degradation with the displacement for all the RC beams is also given in Figure 10.
Effect of loading rate on the stiffness degradation of RC beams.
Effect of loading rate on the displacement ductility factor
Ductility characterizes the deformation capacity of members (structures) after yielding, or their ability to dissipate energy. In this article, the displacement ductility factor is defined by the ratio of the deformation on the failure state to that on the yield state as follows
where
Figure 11 lists the displacement ductility factor curve that is a function of the loading rate. It is clear that the displacement ductility factor decreased obviously as the loading rate increased. The reason was that the yield displacement of RC beam increased observably but the failure displacement increased slightly with increasing the loading.
Effect of loading rate on the displacement ductility factor.
Effect of loading rate on the energy dissipation capacity
One of the most important aspects of structural performance under seismic loading is the ability of a structure to adequately dissipate energy. The RC beams’ capacity for energy dissipation is defined by its ability to absorb the energy dissipated by concrete and reinforcing steel during one loading cycle. Generally, the energy dissipation of RC member under the cyclic load could be calculated by accumulating the energy dissipation of each hysteresis loop, which is equal to the area of loading–displacement curve with the horizontal axial. But the numbers of the hysteresis loop at the various loading rates are different; it is difficult to study the effect of loading rate on the energy dissipation by comparing them. So, in this article, the energy dissipation at cyclic loading is calculated as same as that at the monotonic loading. Simply, the energy dissipation capacity was obtained by calculating the area under the load versus the mid-span displacement curve OABCD, shown in Figure 7.
The energy dissipation capacity depends on various parameters, such as reinforcement ratio, arrangement of reinforcing bars, and the shape and size of the member’s cross section. The energy dissipation of RC beams at different loading rates is shown in Figure 12. It is clear that the energy dissipation capacity increased dramatically with increasing the loading rate. Compared with specimen BC-1, the energy dissipation capacity of specimens BC-2, BC-3, BC-4, and BC-5 increased 16.77%, 6.96%, 26.63%, and 28.54%, respectively. The energy dissipation capacity was enhanced because increasing the loading rate not only increased the crack, yield, and ultimate strengths of the RC beams but also improved the RC beams’ deformation ability.
Effect of loading rate on the energy dissipation capacity.
Conclusion
Based on the experimental and analytical results reported in this article, the following conclusions can be drawn:
Subjected to various rate loadings, the failure shape of five RC beams were almost same; but the crack width decreased slightly with increasing the loading rate, and the cracks distributed more uniformly.
The loading–displacement curves and the envelope curves of RC beams at various loading rates were similar but the strength and displacement increased with the increasing loading rate.
The crack, yield, and ultimate strengths of RC beams increased with increasing the loading rate; the relationship between the strength-increasing factor and the logarithm of the loading rate is approximately linear. The crack, yield, and ultimate displacements of RC beams increased with increasing the loading rate.
The stiffness degraded exponentially as the displacement increased, but it is not dependent of the loading rate.
The displacement ductility factor decreased with increasing the loading rate because the yield displacement of RC beam increased observably but the failure displacement increased slightly with the increasing the loading.
The energy dissipation capacity of RC beams increased obviously with increasing the loading rate because increasing the loading rate not only increased the crack, yield, and ultimate strengths of the RC beams but also improved the RC beams’ deformation ability.
Footnotes
Acknowledgements
The authors thank the staff of the department of Civil Engineering at the Shenyang Jianzhu University for supporting the experimental studies.
Declaration of Conflicting Interests
The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: The first author spent 1 year sabbatical in concrete structure research at the University of Houston, supported by the China Scholarship Council.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by the National Science Foundation of China under grant nos 51178082 and 51421064.