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Sensors 2018, 18(11), 4062; doi:10.3390/s18114062
Validation of Finite Element Model by Smart Aggregate-Based Stress Monitoring
Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China
Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China
School of Civil & Transportation Engineering, South China University of Technology, Guangzhou 510641, China
Author to whom correspondence should be addressed.
Received: 24 September 2018 / Accepted: 15 November 2018 / Published: 21 November 2018
Concrete compressive strength is an important parameter of material properties for assessing seismic performance of reinforced concrete (RC) structures, which has a certain level of uncertainty due to its inherent variability. In this paper, the method of concrete strength validation of finite element model using smart aggregate (SA)-based stress monitoring is proposed. The FE model was established using Open System for Earthquake Engineering Simulation (OpenSEES) platform. The concrete strengths obtained from the material test, peak stress of SA, and estimated concrete strength based on SA stress were employed in FE models. The lateral displacement monitored by Liner variable differential transformer and vertical axial load monitored by load cell in the experiment are applied in the model. By comparing the global response (i.e., lateral reaction force and hysteretic loop), local response (i.e., concrete stress, rebar strain, and cross-section moment) and corresponding root-mean-square error obtained from experiment and numerical analysis, the capabilities of validation of FE model using SA-based stress monitoring method were demonstrated.
Keywords:RC structure; smart aggregate; stress monitoring; compressive strength; OpenSEES; embedded sensor
The assessment of seismic performance of the existing infrastructures is a matter of high priority in earthquake prone areas. Quantifying the material properties of these structures is critical due to the impact that it has on the subsequent application of assessment methods. For reinforced concrete (RC) buildings, concrete compressive strength has emerged as an important material property that requires careful consideration because it directly relates to durability and strength of RC structure.
According to American concrete design code , the average compressive strength of concrete (fcr) obtained from the material test must be higher than the specified compressive strength, fc, to guarantee the safety of structures. A similar requirement also can be found in Chinese concrete design code . Concrete compressive strength has intrinsic variability, which is usually considered as being a random variable with a certain level of uncertainty . In practice, the fcr depends significantly on workmanship including mix, casting and curing operations [4,5,6]. Furthermore, the extreme factors during operation stage of concrete structure, such as degradation of the material induced by freezing and thawing , chemical attack  and fire hazard , may result in the actual fcr being lower than fc. It has been proven that the coefficient of variation (CV) of concrete compressive strength often exceeds 20% . Therefore, it is necessary to estimate the fcr of existing structures to establish the suitable FE model for evaluating the seismic performance of RC structure in advance in case of earthquake.
Non-destructive evaluation (NDT) provides a promising way towards fcr prediction because many NDT parameters are sensitive to concrete strength variations. Two major NDT methods are mostly considered: hardness tests [11,12,13,14] and ultrasonic pulse velocity (UPV) method [15,16,17]. The hardness test can be conducted using rebound hammer based on the principle that the rebound of an elastic mass depends on the hardness of concrete surface against which the mass impinges . However, this method only reflects limited depth close to the concrete surface . Besides, the near surface properties of the concrete such as smoothness, carbonation, size and type of aggregate will significantly affect the test results [13,20]. The expected error of in situ strength prediction with rebound hammer can reach ±30% . In general, UPV measurement are based on the use of a pair of external piezoelectric transducers (i.e., transmitter and receiver) held on the surface of concrete to measure the pulse velocity along the transmitting path . However, this method faces the problem of the unstable coupling between external transducer and concrete surface . Additionally, at least one of the surfaces of the concrete under test should be accessible for transducer deployment. However, most structures need to be inspected in service, which will also have an extra cost for removal of any type of decoration or stop the normal service . To avoid the disadvantages of externally mounted sensors, a promising technique using piezoelectric transducers embedded in concrete structures has been given wide attention [22,23,24,25].
Smart aggregate (SA), as a piezoelectric-based embeddable transducer, is made of a waterproof material packaged PZT patch between concrete or granite blocks to protect it from moisture and fragile broken [25,26]. SA is a multi-functional transducer, which can act as both actuator that generates ultrasonic stress waves and sensor that senses the waves or forces. The result of UPV test using embedded SAs shows good agreement with that using external transducers . Besides velocity, the amplitude of harmonic response was successfully used for evaluating the early-age concrete compressive strength [27,28,29]. The applications of a pair of SAs based on a transmitter–receiver configuration can also be found in structural health monitoring (SHM) of RC structures, such as crack monitoring [24,30,31], damage detection [32,33,34,35] and water seepage monitoring [36,37,38]. Based on sensing configuration, a single SA can also be used for SHM of concrete structures, such as overheight vehicle–bridge collision monitoring , embedded acoustic emission damage detection , and seismic stress monitoring [26,40,41,42,43,44,45]. Since concrete compressive strength is a characteristic value of stress, theoretically, it can be obtained by using the stress-history that SA monitored if the concrete stress reaches peak stress. However, the report of concrete compressive strength evaluation based on sensing configuration has not been found. Compared to the transmitter–receiver configuration, the sensing configuration of SA takes advantage of direct monitoring, reduction of the instruments to generate the high power ultrasonic wave, and low cost.
In this paper, we report on a feasibility study of concrete compressive strength validation of FE model using SA-based stress monitoring method. The FE model was established through the platform of Open System for Earthquake Engineering Simulation (OpenSEES). The global and local response, and corresponding root-mean-square error were compared.
2. SA Sensor
The SA was fabricated by sandwiching a PZT patch between a pair of granite blocks with epoxy, as illustrated in Figure 1. The dimensions of SA and PZT are 25 mm × 25 mm × 25 mm and 15 mm × 15 mm × 0.3 mm, respectively. The details of material properties are reported in the literature . The SA has been proved to have the potential to monitor the exact stress in concrete structure [26,41].
3. Simulation of the RC Column Test
3.1. Description of the Finite Element Model
The two-dimensional FE model was established. The layout of nodes and elements of the FE model is shown in Figure 2a. A zero-length section element located at the bottom of the beam-column element was employed to consider the bond-slip between longitudinal steel bar and concrete , which consists of a pair of nodes (Nodes 1 and 2) with the same vertical and lateral translational degree of freedom. The footing of the column was assumed as rigid due to its relatively large size and stiffness. Since the footing is rigid, it will not influence the deformation of the column and was neglected in the modeling. The other parts of RC column between Nodes 2 and 7 were divided into five elements and modeled using displacement-based nonlinear beam-column element with five Gauss–Legendre integration points. The displacement-based nonlinear beam-column element, implemented in OpenSEES, accounts for linear and nonlinear flexural deformations by assuming plane section at each integration point and capturing the spread of plasticity along the element.
The division of the cross-section at the integration point (140 mm away from Node 2) is presented in Figure 2b. In this cross-section, stresses of the fiber at ten locations that have the same coordinate with SAs were recorded. Since the stresses of two locations with opposite y coordinate and the same x coordinate are equivalent, herein only five locations with passive y coordinate along the loading direction donated as I–V were employed to compare with experimental results. All the divisions of the cross-sections of the column are the same. P-Δ effect of the column was taken into account in the model.
3.2. Section and Material Properties
The material property of the reinforcement fibers in the zero-length section element proposed in the literature  is shown in Figure 3. σu is the ultimate strength of the rebar, which is 1.3 times of yield strength σy. sy and su are loaded-end slips corresponding to the bar stresses of σy and σu, respectively. sy can be calculated using Equation (1) where α is the parameter used in the local bond-slip relation, which is taken as 0.4 in this paper in accordance with provision 6.1 of CEB-FIP Model code 2010 . db is the diameter of the longitudinal steel bar, which is 22 mm in this paper. σy is the yield strength of the rebar, which is tested to be 460 MPa. su = 30–40 sy, which is set to be 35. As shown in Figure 3, the stiffness reduction factor that defines the stiffness degradation after the rebar yielded ranges 0.3–0.5, and is b’ = 0.3 in this study. To employ the model for capturing the pinching characteristic effect of reinforcement in flexural members subjected to reversed cyclic loading, the coefficient Rc, which defines the pinching characteristic, is set to 0.6 according to the literature .
As shown in Figure 4a, the reinforced steel bar follows the nonlinear model of Giuffré–Menegotto–Pinto , as modified by Filippou et al.  to include isotropic-hardening effects. The Young’s modulus of reinforced steel bar, E0, is 210 GPa. The yield strength of the rebar σy is tested to be 460 MPa, corresponding to the yield strain εy of 2190 με. The strain-hardening ratio b is 0.02. The parameter R0, which controls the curvature of the transition from the elastic to plastic-hardening branch, is 20 .
To further improve the efficiency of simulation and adequately capture the major behavior of concrete in compression rather than tension, the constitutive relationship for the concrete is modeled using the Kent–Scott–Park model  with no tension stiffening, as presented in Figure 4b, in which fu is the ultimate residual strength, ε0 is the strain at peak strength, εu is the strain when the residual strength is reached, and E is the Young’s modulus of the concrete.
3.3. Concrete Strength Selection
It should be noted that fc is a statistical parameter, which is a variable for the concrete structures. To identify the average concrete compressive strength, four typical parameters, namely lower limit and average value of concrete compressive strength of concrete cube, the peak stress monitored by SA, and the estimated strength based on concrete stress distribution assumption, were selected as the potential concrete compressive strength in the models and denoted FEA-1–4, respectively.
The average strength of the concrete cube specimen cast with the column was 33.5 MPa with a coefficient of variation of 0.13. The unconfined concrete strength of the RC column is considered to be 0.79 times smaller than that of the concrete cube due to end friction effects , which is 26.5 ± 3.5 MPa. The relationship between confined and unconfined concrete compressive strength is expressed as follows:where fcc is the confined concrete compressive strength induced by stirrups. κ is the ratio between fcc and fc, which can be expressed as :
In this paper, the stirrup is plain bars at 100 mm spacing with a diameter of 10 mm. The volume–stirrup ratio, ρs, is 0.0092. The yield strength of stirrup, fyh, is 300 MPa.
The lower limit and average value of concrete compressive strength of concrete cube are considered as the candidates of the unconfined concrete compressive strength of RC column, which were 23.0 and 26.5 MPa, respectively. The corresponding lower limit and average value of the confined concrete compressive strength can be calculated using Equation (2), which are 26.0 and 29.7 MPa, respectively.
The monitored peak compressive stress of SA in experiment is 39.4 MPa , which is larger than the upper limit of the concrete compressive strength calculated using the concrete cube specimen, and is considered as one of the potential confined concrete compressive strengths. The upper limit of the unconfined concrete compressive strength is 36.5 MPa, which can be obtained from Equation (2) as well.
The stress monitored by SA can be recognized as the average compressive stress of limited zone. Based on the stress distribution assumption, as illustrated in Figure 5, the ratio of concrete compressive strength to average compressive stress is equal to k3/α, where k3 is 0.925 and α is 0.85 for normal strength concrete . The estimated concrete compressive strength can be calculated using peak stress of SA and k3/α, which is 42.9 MPa for confined concrete. Utilizing Equation (2), the estimated unconfined concrete can be obtained, which is 40.0 MPa.
The detailed of the concrete constitutive parameters are shown in Table 1. fu is 0.2 and 0.8 times fc for unconfined and confined concrete, respectively. In accordance with Chinese code for design of concrete structures , the peak strain of unconfined concrete, ε0u and the ratio of the peak strain to ultimate stain of the unconfined concrete, ε0u/εuu, can be obtained. The ratio of peak strain of confined concrete to that of unconfined concrete, ε0c/ε0u, is equal to ε as well, which can be obtained using Equation (3). The confined ultimate strain, εuc, is a constant, 0.02 in this study.
3.4. Loading Scheme
The loading scheme for FE analysis was the same as that of the experiment in . The displacement-controlled lateral displacement with a sine harmonic form and load-controlled vertical load with a variation of amplitude were applied on the top of the FE model, respectively, as shown in Figure 6a,b.
4. Results and Discussion
4.1. Global Response
The global response including lateral load and hysteretic loop were used to evaluate the effectiveness of strength prediction by comparing the results of experiment with that of FE analysis.
4.1.1. Lateral Load
As shown in Figure 7, both the period and variation trend of the lateral load in experiment agree well with that of FE analysis results. With the increase of loading time, the amplitude of each cycle of the curves firstly increased, and then gradually decreased, which reflects the damage process of the specimen until the rebar yield. The curves obtained from FEA-3 that used peak stress of SA and FEA-4 that used estimated strength are almost the same, which is much closer to the experimental results than the FE analysis results that use the concrete cube strength. The experimental peak load was about 250 kN at the eighth cycle, which agrees well with the results of FEA-3 and FEA-4 model.
4.1.2. Hysteretic Loop
The lateral load-displacement hysteretic loops are shown Figure 8a. The results of FEA-3 and FEA-4 are similar, which are much closer to experimental curve than those of FEA-1 and FEA-2. The hysteresis performance of FEA-3 and FEA-4 in terms of reloading stiffness, unloading stiffness, and energy consumption are in good agreement with experimental result.
Figure 8b shows the comparison of envelop curves of RC columns between FE analysis and experiment. It was found that the initial stiffness increased with the increase of fc, which is reasonable based on the relationship between Young’s modulus and fc, as illustrated in Figure 4b. When the peak load and 85% of peak load were reached, the lateral displacement was about 20 mm and 44 mm, respectively, corresponding to a story drift of 1.43% and 3.14%.
4.2. Local Response
Generally, the global response rather than the local response (i.e., concrete stress and rebar strain) was used for evaluating the difference between experimental and FE results, due to the lack of effective local stress and strain monitoring methods, especially in large deformation stage. In this study, utilizing the SAs combined with strain gauge, the local response including concrete stress, rebar strain, and cross-section moment were obtained, providing a new index to evaluate the performance of strength prediction.
4.2.1. Concrete Stress
The stress-histories of five typical locations, as presented in Figure 2b, obtained from experiment and FE models are compared in Figure 9. It is noted that the models FEA-3 and FEA-4 provided a good prediction of the observed response and captured the trend of the stress time-histories. The amplitude of the stress at the locations (Points I and V) away from the center of the cross-section was larger than that of Points II–IV, which are close to the center of the cross-section, due to the larger bending moment. The stress time-history of the SA at the central axial of the cross-section (Point III) is very complicated, which results from the combined effects of the variation of axial load and transformation of neutral axial. As expected, the phase of SAs located at the symmetry positions along the central axial of the cross-section are about 180°. The stress time-histories obtained from FEA-3 and FEA-4 model are similar, which has better agreement with experimental results than FEA-1 and FEA-2 model.
4.2.2. Rebar Strain
The strain time-histories of rebar derived from FE models and experiment are compared in Figure 10. The FE models have a good prediction in the early stage of strain time-history. As shown in Figure 10a,b, the experimental rebar strain mismatches with that of FE results after the second loading cycle, which was probably induced by the uniformity of load. The experimental amplitude of rebar strain clearly increased when the rebar yielded after eighth loading cycle, which might be caused by the uncoordinated deformation between concrete and rebar. Although the bond-slip effect was considered using concentrated plastic constitutive relationship in FE model, distributed plastic behavior still exists between rebar and concrete in plastic hinge zone. Since the part of rebar strain larger than yield strain did not generate any force based on the constitutive relationship of rebar, the actual differences in force borne by rebar between experiment and FE analysis decreased significantly.
4.2.3. Cross-Section Moment
To further validate the feasibility of SA-based concrete strength monitoring method, the typical moment time-history of cross-section A-A obtained from experiment and FE analysis was compared, as illustrated in Figure 11. Both the local concrete stress and rebar strain were adopted for the moment calculation, as reported in the literature . It was found that the analytical moments increased with the increasing of fc. The results of FEA-3 and FEA-4 agree better with experimental moment than those of FEA-1 and FEA-2.
5. Error Analysis
To quantify the difference between the experimental and analytical result, a specific form of root-mean-square (RMS) error was adopted, as expressed in Equation (4)where xexp and xana are the amplitude of each cycle during cyclic loading for experimental and analytical curve, respectively.
5.1. Global Response
Figure 12 shows the variation of the RMS error of lateral load with different concrete strength. The maximum errors of FEA-3 and FEA-4 are very close and less than 10%, which is much lower than those of FEA-1 and FEA-2. The maximum error of global response (i.e., lateral load) of simulations using peak stress of SA and estimated concrete strength are basically same. Thus, the assessment of concrete strength prediction cannot be effectively estimated using global response.
5.2. Local Response
The RMS error of concrete stress at five typical locations (Points I–V) is shown in Figure 13. For Points I and V (Figure 13a,b), the maximum error of FEA-4 is about 5%, which is lower than that of FEA-3. In addition, the errors of FEA-3 and FEA-4 are lower than those of FEA-1 and FEA-2. For Points II and IV (Figure 13c,d), the maximum RMS errors of FEA-3 and FEA-4 are about 15%, which are lower than those of FEA-1 and FEA-2 but larger than at Points I and V. The maximum RMS error of FEA-4 is lower than that of FEA-3, which is similar to those at Points I and V. For Point III (Figure 13c), the error is significantly larger than those obtained at the other points due to the complex stress status. The maximum RMS error of FEA-4 is not larger than that of FEA-3. Therefore, it can be concluded that the concrete stress in FE analysis using the SA stress-based estimated concrete strength agrees better with experimental result than with the strength obtained from the material test.
Figure 14 shows that the RMS error of the rebar strain using four different concrete strengths. The error increases with the increasing of concrete strength. This trend is opposite to lateral load or concrete strength, whereas the error is too large to be used for quantifying the accuracy of simulation.
The RMS error of cross-section moment is shown in Figure 15. The errors of FEA-3 and FEA-4 are smaller than those of FEA-1 and FEA-2. The maximum error of FEA-4 is about 20%, which is basically the same as that of FEA-3. That implies the index of cross-section moment is not suitable for concrete strength prediction.
This paper proposes the concrete strength validation of FE model using smart aggregate-based stress monitoring method. The concrete strengths obtained from material test, peak stress of SA, and estimated concrete strength based on SA stress were employed. The FE models were established, considering the load-end slip of rebar. To verify the proposed method, the global and local responses of the RC column and corresponding root-mean-square (RMS) error obtained from the experimental and numerical analysis were compared. The conclusions are summarized as follows.
The concrete stress of local response is a more sensitive index than global response and other local responses (rebar strain and cross-section moment) to identify the difference between experiment and FE simulation. The concrete compressive strength of the plastic hinge zone of RC structure obtained from standard specimens cast simultaneously provides a conservative estimate. The compressive strength in FE model can be validated using SA-based seismic stress monitoring method.
H.Z. developed the idea, and wrote the paper; S.H. helped write the paper; and J.O. made critical comments on the paper.
This study received joint financial support from China Postdoctoral Science Foundation (grant No. 2017M621281), the Key Lab of Structures Dynamic Behavior and Control (Harbin Institute of Technology), Ministry of Education of the People’s Republic of China (grant No. HITCE201710), and the National Science Foundation of China (grants Nos. U1709207 and 51678250). That support is gratefully acknowledged.
The authors would like to acknowledge Dr. Peizhou Zhang for the special support on the Opensees modeling. We also would like to acknowledge Baoyin Sun and Shuang Zhu for the suggestions on modeling method.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Illustration and photo of the SA.
Figure 2. (a) Finite element model; and (b) division of cross section and SA position of RC column.
Figure 3. Envelope curve for rebar stress versus loaded-end slip relationship.
Figure 4. Material constitutive models: (a) steel; and (b) concrete.
Figure 5. Concrete stress distribution and rectangular stress block.
Figure 6. Loading scheme for: (a) lateral direction; and (b) vertical direction.
Figure 7. The time-history of the lateral load.
Figure 8. Hysteretic responses of the specimens: (a) hysteretic loops; and (b) envelop curves.
Figure 9. Stress time-history for experiment and finite element analysis at: (a) Point I; (b) Point II; (c) Point III; (d) Point IV; and (e) Point V.
Figure 10. Strain time-history for different finite element model at different position along lateral loading direction: (a) −149 mm; and (b) 149 mm.
Figure 11. Time-history of the cross-section moment for experimental and analytical results.
Figure 12. Root-mean-square error of lateral load.
Figure 13. Root-mean-square error of concrete stress at the location of: (a) Point I; (b) Point V; (c) Point II, (d) Point IV; and (e) Point III.
Figure 14. Root-mean-square error of rebar strain at different position along lateral loading direction: (a) −149 mm; and (b) 149 mm.
Figure 15. Root-mean-square error of cross-section moment.
Table 1. Parameters of concrete constitutive relationship.
|Parameter||Lower Limit of Concrete Cube (FEA-1)||Average of Concrete Cube (FEA-2)||Monitored Peak Stress of SA (FEA-3)||Estimated Strength (FEA-4)|
Note: The unit of fcu, fcc, fuu and fuc is MPa. The first subscripts “c“ and “u” stand for concrete strength and ultimate residual strength, respectively. The second subscripts “c” and “u” stand for confined and unconfined of concrete, respectively.
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