Effect of concrete carbonation on natural frequency of reinforced concrete beams

Abstract

We carried out an experimental investigation to study the influence of concrete carbonation on the natural frequency of simply supported reinforced concrete beams. A total of 10 reinforced concrete test beams and 12 concrete-carbonation test specimens were subjected to different accelerated carbonation stages for 0, 7, 14, 21, and 28 days. Modal tests were performed on reinforced concrete test beams after the accelerated carbonation stages. In order to reduce the effect of other factors on the modal tests, constant temperature, relative humidity, and boundary conditions of the test beams were maintained in the experimental process. The experimental results show a trend of the natural frequencies of reinforced concrete test beams to decrease with the increase in concrete-carbonation depths. With statistical analyses of experimental data, this study established the relationship between concrete-carbonation depths and natural frequencies. Fitting lines for the drop in natural frequencies and carbonation depths are obtained for the first four modal frequencies. Based on the analysis of the physicochemical processes of concrete carbonation, the main reason behind the drop in natural frequencies is the increase in mass after concrete carbonation. The percentage composition of increase in mass after complete carbonation is obtained based on the analysis of the physicochemical process. This analysis demonstrates part of the reason for the drop in natural frequencies and proves that the experimental results are reliable and credible. This study provides further insight into the use of modal parameters to assess damage in concrete structures in structural health monitoring.

Keywords

concrete carbonation modal test natural frequency reinforced concrete structures

Introduction

Reinforced concrete (RC) structures are used widely in a large number of bridges and other civil engineering structures. Because RC structures are exposed to many different types of environment, it is considerably important to understand the durability of RC. In these respects, the performance of RC structures has generally been given more attention (Maalej et al., 2010; Page and Treadaway, 1982). In the early development of RC structures, interest was motivated in the use of many nondestructive system identification (SI) techniques to monitor and assess the performance of structures (Chang et al., 2001). A better approach to assess bridge working conditions is one that uses global indices. The vibration test and modal analysis techniques enable global dynamic properties to be measured for the purpose of assessing performance integrity (Esfandiari et al., 2013; Salawu and Williams, 1995). The main purpose of structural health monitoring (SHM) is to detect damage in its early stage and minimize or prevent premature failure. Most popular SI methods are based on changes in measured vibration responses. These measured vibration responses include natural frequencies, mode shapes, and modal damping. The principle of SI methods is that measured system parameters, including natural frequencies, mode shapes, and modal damping, are functions of the physical properties of structures. These physical properties of structures include stiffness, mass, and damping, among other properties. In fact, any change in the physical properties due to damage results in measurable change in the system dynamic response. Natural frequencies give a good overview of the global structural condition (Siringoringo and Fujino, 2012). In some applications, these parameters have proven to be a reliable and sensitive indicator of structural performance (Bayraktar et al., 2015; Esfandiari et al., 2013; Vestroni and Capecchi, 2000). It is well known that natural frequencies are easy to monitor in many structures.

Concrete in bridge structures is constantly subjected to $C O_{2}$ ingression as well as deterioration processes caused by repeated vehicle loads (Chen et al., 2011; Xia et al., 2013), environmental temperature, and relative humidity (RH) among other factors. Concrete carbonation is a common issue that inevitably affects RC structures in an atmospheric environment. Bernal et al. (2015) addressed the effects of RH and carbon dioxide ( $C O_{2}$ ) concentration on the rate and effects of accelerated carbonation in alkali-activated slag–metakaolin (MK) concretes. Papadakis et al. (1991) described the physicochemical processes in concrete carbonation and modeled them mathematically. As is well known, the steel reinforcement embedded in concrete is easily corroded because of concrete carbonation. The effect of corrosion on natural frequency and damping ratio had been studied by Razak and Choi (2001). They used two corroded beams with slight spalling and moderate cracking and showed a decrease in natural frequencies ranging from 4% to about 11% for spalling and from 1% to 4% for cracking compared to an intact control beam. Changes in modal damping ratio were inconsistent with the changes in frequencies. According to the results from previous research (Duran-Herrera et al., 2015; Silva et al., 2014), carbonation itself does not degrade the concrete strength. However, it decreases the alkalinity of the concrete and, thus, depassivates the passive layer of steel reinforcement there. Once the protective layer is destroyed, steel reinforcement will be corroded easily, giving rise to performance deterioration of RC structures. Therefore, concrete carbonation draws significant attention.

The use of natural frequencies as parameters for SHM is popular and effective but needs to be further developed. Most numerical or experimental studies focus on a particular type of damage (Li et al., 2014; Ni et al., 2006), for example, the deterioration of stiffness due to severe damage or cracking. For RC structures exposed in an atmospheric environment, the first step of deterioration is concrete carbonation. The time for carbonation to reach the surface of the steel reinforcement is likely to be several decades, and prior to concrete carbonation, the degree of steel reinforcement corrosion is limited. In this process, the main effect of the degree of concrete carbonation is on the natural frequencies of RC structures. This article is concerned with the changes in natural frequencies caused by the concrete-carbonation process with the view of making the SHM technique more reliable and effective. The primary objective of the experimental study is to establish relationships between the degree of concrete carbonation and the changes in natural frequencies. Comparing the experimental data with the analysis of the physicochemical processes of concrete carbonation, we have determined the effect of concrete carbonation on natural frequencies. The main reason for this phenomenon is the increase in mass due to concrete carbonation. Using the experimental test data, we have established the fitting equations that describe the relationship between the natural frequencies and concrete-carbonation depth. This is the main new result of this study. The results may provide further insight into the use of modal parameters to assess damage in concrete structures in SHM.

Experiment setup

The experimental program is designed to investigate the effect of concrete carbonation on the natural frequencies of RC structures. In order to accelerate the concrete-carbonation process, a carbonation chamber was employed. The length and width of the carbonation chamber in our laboratory are 4.0 and 3.0 m, respectively. To ensure good pneumatic transport, the distance from the edge of the test beam to the carbonation chamber wall should be not <0.4 m. Therefore, the length of the RC test beam should be not >3.2 m. The tested beams were simply supported using a fixed-pin pedestal and hinged support. The boundary conditions were unchanged during the experimental process. The clear span of the RC test beam was 3.1 m. The modal test was performed in the carbonation chamber, and the carbonation depths were measured from concrete-carbonation test specimens at the same time.

The time spans of accelerated carbonation were 0, 7, 14, 21, and 28 days, defined as stages 0, 1, 2, 3, and 4, respectively. At the end of each stage, every RC test beam was subjected to a modal test to estimate its natural frequencies. Then, a dynamic test response was obtained using a fast Fourier transform (FFT). In order to measure the carbonation depths at the end of each stage, a total of six prism-shaped concrete-carbonation test specimens, sized at $150 \times 120 \times 400 m m^{3}$ , were poured for this experiment. The carbonation depths were obtained from these concrete-carbonation test specimens at the end of each stage using an indicator test.

Specimens

Two kinds of specimens were poured for this experiment at the same time. The first are the RC test beams, and the second are concrete-carbonation test specimens. Each RC test beam had a length of 3.2 m, a width of 0.15 m, and a height of 0.12 m. The main reinforcement comprised $6 Φ 10$ ( $A_{s} = A'_{s} = 235.6 m m^{2}$ ) with reinforcement ratios of 2.62%. The dimension of stirrups is $Φ 6$ . The spacings of stirrups are 0.2 m except at the supports where they are 0.1 and 0.07 m to improve shear capacity. Figure 1 shows the detailed information of the RC test beam configuration. In order to investigate the effect of concrete carbonation on modal parameters, two groups of five RC test beams measuring $0.15 \times 0.12 \times 3.2 m^{3}$ were poured for this research. One group of five was poured using concrete of strength grade 30, and the other group of five was poured using concrete of strength grade 40. Tables 1 and 2 present the mix proportions of each concrete.

Figure 1.

Detailed configuration of RC test beam (mm).

Table 1.

Mix proportions of concrete of strength grade 30.

Material	Type	Ratio	Content (kg/m³)
Cement	#425 PO	1	382
Water	Tap water	0.487	186
Coarse aggregate	Gravel 5–16 mm	2.636	1007
Fine aggregate	Natural river sand	2.071	791

Table 2.

Mix proportions of concrete of strength grade 40.

Material	Type	Ratio	Content (kg/m³)
Cement	#425 PO	1	414
Water	Tap water	0.425	176
Coarse aggregate	Gravel 5–25 mm	2.447	1013
Fine aggregate	Natural river sand	1.845	764

Table 3 shows the serial numbers of RC test beams, serial numbers of concrete-carbonation test specimens, and the type of test undergone. A total of 10 RC test beams (C30N01–C30N05 and C40N01–C40N05) were subjected to modal tests. A total of 12 concrete-carbonation test specimens (C30-1–C30-6 and C40-1–C40-6) were used for carbonation depth tests. The cross section of each concrete-carbonation test specimen is the same as the cross section of the RC test beams, namely, a width of 150 mm and a height of 120 mm. The length of the concrete-carbonation test specimens is 400 mm. The concrete-carbonation test specimens were split off four times to test carbonation depths after accelerated carbonation of 7, 14, 21, and 28 days. The locations of splitting are shown in Figure 2.

Table 3.

Test arrangement.

Specimens	Serial number	Modal test	Carbonation depth test
RC test beams	C30N01–C30N05	Yes	No
	C40N01–C40N05	Yes	No
Concrete-carbonation test specimens	C30-1–C30-6	No	Yes
	C40-1–C40-6	No	Yes

RC: reinforced concrete.

Figure 2.

Details of concrete-carbonation test specimen.

Boundary conditions of RC test beams

Modal parameters, such as natural frequencies, mode shapes, and modal damping, are affected by mass, damping, stiffness, and boundary conditions. In order to remove or reduce the effect of boundary conditions on modal parameters, invariable boundary conditions are necessary. In this experimental study, four concrete support abutments were poured, and bolts ( $Φ 20$ ) were embedded to fasten the RC test beams. The height of the concrete support abutments was 795 mm. The length of the bolts was 500 mm, and the embedded length in the concrete support abutments was 400 mm. The pitch of bolts was 200 mm. Therefore, the distance between each RC test beam was 50 mm. Five RC test beams were installed on each pair of concrete support abutments. Figure 1 shows the details of the boundary conditions of RC test beams. The nut, bolt, steel plate, hinged support, and fixed-pin pedestal were processed with austenitic steel (304 18 Cr-8Ni). At the beginning of this experiment, the RC test beams were installed on the concrete support abutments as shown in Figure 1. The hinged support and fixed-pin pedestal below the RC test beams are monolithic and above the RC test beams are segregated. The nut was tightened and retained unchanged at each stage in the whole experimental process. The whole experiment included five stages. The first stage was intact stage including tests before accelerated carbonation in the carbonation chamber. The other four stages (stages 1–4) were exposure to the CO₂-rich atmosphere in the carbonation chamber for 7, 14, 21, and 28 days, respectively. The concrete support abutments and RC test beams were invariable in this experimental process. Under these conditions, we can guarantee that the boundary conditions of the RC test beams were unchanged, which means the effect of boundary conditions on the modal test is minimal. Therefore, the test data of this experiment are more reliable because the effect of boundary conditions can be ignored.

Accelerated carbonation

There are several deterioration processes that affect RC structures at an early stage leading to loss of performance. The most common reason for deterioration is corrosion of the steel reinforcement. In most cases, the concrete cover should protect the steel reinforcement from corrosion because it provides an alkaline environment. If the pH value of concrete decreases below 9, the steel reinforcement embedded in concrete should be corroded easily. Generally, the acidic substances of concern are atmospheric carbon dioxide ( $C O_{2}$ ) and, in polluted environments, other gases such as sulfur dioxide ( $S O_{2}$ ). These react with the alkaline constituents of the cement paste to form a carbonated zone. As we know, in an alkaline environment, the metal may remain passive over a wide range. The primary mechanism of corrosion protection in concrete involves passivation of embedded steel reinforcement resulting from the chemically inhibitive influence of hydroxyl ions. The kinetics of carbonation for various types of concrete in different environments have been widely researched. It is well known that the RC structures are not only protected steel reinforcement from corrosion but also protected concrete from carbonation. Carbonation should not penetrate to an extent that endangers the passivity of steel reinforcement during the design life of a structure.

Portland cement is mainly composed of the following four minerals: tricalcium silicate ( $3 CaOSi O_{2}$ ), dicalcium silicate ( $2 CaOSi O_{2}$ ), tricalcium aluminate ( $3 CaOA l_{2} O_{3}$ ), tetracalcium alumino-ferrite ( $4 CaOA l_{2} O_{3} F e_{2} O_{3}$ ); these minerals are conventionally referred to as $C_{3} S$ , $C_{2} S$ , $A_{3} S$ , and $C_{4} AF$ . When Portland cement is mixed with water, the constituent minerals undergo a complex sequence of hydration reactions that transform the paste to a hardened matrix of hydrated products. The most significant feature of cement hydration is that the aqueous phase rapidly acquires a high pH value. In fact, after a few weeks of hydration, pH values exceed 13. Therefore, the concrete cover provides an important physical barrier.

For the purpose of measuring the carbonation depth, two groups, each with six concrete-carbonation test specimens (C30-1–C30-6, C40-1–C40-6) were poured. These two groups of concrete-carbonation test specimens were subjected to the same curing conditions (curing time, curing temperature, and curing RH) as the RC test beams at the same time. Five lateral surfaces of all concrete-carbonation test specimens in the carbonation test were sealed by paraffin wax, leaving open the top surface, guaranteeing that $C O_{2}$ diffuses into the concrete in one dimension. Figure 2 shows the top surface is divided into five parts by the lines A-A, B-B, C-C, and D-D. These sections are split off at the end of different accelerated carbonation stages. That is, the carbonation test specimens were split off through the A-A section at the end of stage 1 (accelerated carbonation after 7 days), B-B section at the end of stage 2 (accelerated carbonation after 14 days), C-C section at the end of stage 3 (accelerated carbonation after 21 days), and D-D section at the end of stage 4 (accelerated carbonation after 28 days).

In this experiment, 12 partially sealed carbonation test specimens (C30-1–C30-6 and C40-1–C40-6) and 10 simply supported RC test beams (C30N01–C30N05 and C40N01–C40N05) were subjected accelerated carbonation in the carbonation chamber with the following experimental conditions: $20^{\circ} C$ , $70 %$ RH, and a $C O_{2}$ concentration of $20 %$ . The process of accelerated carbonation was interrupted four times. They were the RC test beams and concrete-carbonation test specimens had been exposed to the CO₂-rich atmosphere in the carbonation chamber after 7, 14, 21, and 28 days. At the end of stage 1 (accelerated carbonation after 7 days), stage 2 (accelerated carbonation after 14 days), stage 3 (accelerated carbonation after 21 days) and stage 4 (accelerated carbonation after 28 days), the concrete-carbonation test specimens were split off along, respectively, the A-A, B-B, C-C, and D-D sections. A phenolphthalein pH-indicator was sprayed over the surfaces cut from concrete-carbonation test specimens. After 3 min, the indicator is colorless if the area is carbonated and red if it is noncarbonated. Figure 2 shows that the distance of carbonation depth measurement points is 10 mm. Because the width of the exposed surface is 150 mm, there are 16 measurement points at each split section. The first measurement point and the last measurement point were removed because they are located at edges. Then, the average carbonation depth was obtained from the remaining 14 measurement points.

Modal test

A total of 10 RC test beams (C30N01–C30N05 and C40N01–C40N05) were subjected to the modal test at the end of each accelerated carbonation stage. The boundary conditions are unchanged in the whole experimental process. The experimental stages were accelerated carbonation after 0, 7, 14, 21, and 28 days. Figure 3 illustrates seven vertical accelerometers mounted on the top surface of each RC test beam.

Figure 3.

Sensor locations for modal test.

Piezoelectric accelerometers were selected for vertical measurements because of their high performance and shockproof qualities. The initial axial sensitivity of each piezoelectric accelerometer is $150 mv / (m / s^{2})$ . The transverse sensitivity is <5%. The frequency response is 0.2–400 Hz. The operating temperature range is $- 40^{\circ} C to {+ 150}^{\circ} C$ . The measurement range is 600 g. The weight is 38 g. Before actual measurements, the accelerometers were calibrated using a shaking-table test. Calibration instruments consisted of a signal generator, shaking table, charge optimal modulator, laser displacement meter, and computer. The results of calibration for axial sensitivity are shown in Table 4. We can see that different piezoelectric accelerometers have different axial sensitivities. In order to maintain consistency in every stage, the locations of the piezoelectric accelerometers are unchanged in each testing stage. The measurement points and the serial numbers of the piezoelectric accelerometers are shown in Figure 1 and Table 4.

Table 4.

Results of calibration for piezoelectric accelerometers.

Accelerometer serial number	Axial sensitivity ( $mv / (m / s^{2})$ )	Measurement points
44835	141.628	1
44755	145.424	2
66533	134.485	3
44845	145.051	4
44816	145.052	5
44807	144.876	6
66563	129.426	7

As shown in Figure 3, to ensure stability, each accelerometer was fixed with a bolt ( $Φ 5$ ) that was placed on the top surface of RC test beams. The bolt is permanently fixed at each measurement point. In every test, the accelerometer is installed at each measurement point by bolt. In order to decrease the effect of temperature and RH on natural frequencies (Ding and Li, 2011), after opening the carbonation chamber door, the modal test was completed within 3 h.

Figure 4 is a photograph of the modal test configuration in the carbonation chamber. The five RC test beams (C30N01–C30N05) at the left-hand side were poured using concrete of strength grade 30. The five RC test beams (C40N01–C40N05) at the right-hand side were poured using concrete of strength grade 40. The instruments we used include a computer, 16-channel oscilloscope, charge optimal modulator, piezoelectric accelerometer, impulse hammer, and signal cables. The piezoelectric accelerometers were connected to the 16-channel oscilloscope to capture acceleration signals in real time. The output from the piezoelectric accelerometers and the load input were amplified and filtered using a charge optimal modulator. The impulse hammer was used to excite the RC test beam. The inputs to the RC test beams were tested with an accelerometer installed on the head of the hammer. The impacts imposed by the impulse hammer were controlled to have a peak force between 100 and 1000 N. Measurement point 5, which can be seen in Figure 1 and which is located 3/8 of the way along the length of the RC test beam, was selected as the excitation location because this excitation location can excite first four modal parameters for a simply supported beam. Theoretical analysis demonstrated that the fourth-order frequency of the RC test beam is 326.647 Hz. Based on the sampling theorem, the sample frequency was selected as 1000 Hz. The measurements were made using a block size of 250,000 lines, thus giving a resolution of 0.002 Hz per spectrum line. With modal analysis software, the dynamic load test response was obtained using an FFT to extract natural frequencies.

Figure 4.

Modal test for RC test beams.

Results and discussions

Carbonation test results

Figure 5 shows a phenolphthalein pH-indicator sprayed over the surfaces split from the concrete-carbonation test specimen (C30-2) after the specimen had been exposed to a CO₂-rich atmosphere in the carbonation chamber for 21 days. The carbonated area, which is colorless, and the noncarbonated area, which is red, can be clearly seen.

Figure 5.

Split surface sprayed with phenolphthalein.

According to the measurement data from 12 concrete-carbonation test specimens (C30-1–C30-6 and C40-1–C40-6), the mean values of concrete-carbonation depths were determined and are shown in Tables 5 and 6. The standard deviation is used to quantify the amount of variation in the data.

Table 5.

Result of carbonation depths of concrete of strength grade 30.

Specimens	Carbonation depth (mm)
	Exposure: 7 days	Exposure: 14 days	Exposure: 21 days	Exposure: 28 days
C30-1	15.2	20.4	25.5	26.4
C30-2	13.1	19.6	24.2	26.6
C30-3	12.8	18.6	23.5	25.1
C30-4	12.8	17.9	21.7	24.5
C30-5	14.4	20.5	24.0	29.0
C30-6	13.4	20.5	23.4	28.6
Mean value	13.6	19.6	23.7	26.7
Standard deviation	0.97	1.09	1.25	1.81

Table 6.

Result of carbonation depths of concrete of strength grade 40.

Specimens	Carbonation depth (mm)
	Exposure: 7 days	Exposure: 14 days	Exposure: 21 days	Exposure: 28 days
C40-1	0.8	4.5	10.2	14.1
C40-2	0.6	6.2	6.0	10.3
C40-3	4.1	6.8	9.0	10.4
C40-4	2.4	6.6	10.2	10.8
C40-5	2.4	5.4	12.3	13.9
C40-6	1.9	5.9	9.4	11.2
Mean value	2.0	5.9	9.5	11.8
Standard deviation	1.29	0.82	2.05	1.76

From the experimental results of concrete-carbonation depths, we can see that the degree of concrete carbonation is different between the concrete of strength grade 30 and the concrete of strength grade 40. In the same exposure environment, the carbonation depth of the concrete of strength grade 30 is more severe. Some of the factors that affect the degree of concrete carbonation include cement species and dosage, water–cement ratio (W/C), pore structure, and curing condition (Jiang et al., 2015). Because the concrete-carbonation test specimens and the RC test beams are exposed to the same environment, we take the carbonation depths in RC test beams to be the same.

Modal test results

When a structure is subjected to a certain degree of deterioration, the stiffness or mass is changed. The dynamic parameters, such as natural frequencies, mode shapes, and modal damping, will be changed consequently (Fertis, 1973). In this experimental study, it is apparent that the changes in natural frequencies at different accelerate carbonation stages are different. These stages are accelerated carbonation after 7, 14, 21, and 28 days. The main cause of this phenomenon is the difference in the carbonation depth at each stage. After the modal test at each stage, a total of 350 FFTs (7 measurement points, impulse excitation five times on every beam, and a total of 10 RC test beams) were computed. Figure 1 shows the locations of the measurement points on the RC test beam. There are seven vertical accelerometers mounted on the top surface of every beam. The distance between these vertical accelerometers is the same, which means the RC test beam is divided into eight equal parts by the seven vertical accelerometers. According to the dynamic response analysis of a simply supported beam, in the measurement of the second-order modal frequency, the measurement data coming from the accelerometer located at measurement point 4 should be removed because measurement point 4 is located at the mid-span of the test beam. Similarly, in the measurement of the fourth-order modal frequency, the measurement data coming from accelerometers located at measurement points 2, 4, and 6 should be removed. Therefore, the typical time history of acceleration and the power spectral density (PSD) at every measurement point are obtained. Figure 6 shows the time history of acceleration and PSD of C30N01 RC test beam at the end of stage 3, which is accelerated carbonation for 21 days. C30N01 is the serial number of RC test beam; T04A denotes the modal test at the end of stage 3; S5 denotes these test data obtained from accelerometer located at measurement point 5; and P1, P2, P3, P4, and P5 denote the times of impulse excitation. For example, C30N01T04A-S5-P1 means the serial number of the RC test beam is C30N01, testing was at the end of stage 3, the test data come from accelerometer located at measurement point 5, and it was the first of the five impulse excitations. In this study, the modal test was performed at the end of each of the five different stages. Therefore, the changes in natural frequencies should be obtained in this experimental process. Previous research (Ding and Li, 2011; Zhou et al., 2015) has shown that ambient temperature and RH are principal factors affecting structural modal parameters in the long term. In order to reduce the effect of temperature and RH on the modal tests, the tests should be completed as soon as possible after opening the carbonation chamber door (<3 h). Using a hammer-impacting method and the modal analysis program, the natural frequencies are extracted. Figures 7 and 8 show the typical relations between natural frequencies and carbonation depths at the end of each stage for concrete of strength grade 30 (C30N01 RC test beam) and strength grade 40 (C40N01 RC test beam), respectively. Because the carbonation depths of the concrete of strength grade 40 are small at the end of stages 1 and 2, the effect of concrete carbonation on natural frequencies is limited. The modal test results at the end of stages 0, 3, and 4 are shown in Figure 8. The modal test data at the end of stages 1 and 2 are not included because the changes from stage 0 are small.

Figure 6.

Typical time history of acceleration and PSD plots.

Figure 7.

Typical results of relations between natural frequencies and carbonation depths (C30N01) $μ_{0}$ , $μ_{1}$ , $μ_{2}$ , $μ_{3}$ and $μ_{4}$ denote mean value of frequency at the end of stage 0, stage 1, stage 2, stage 3 and stage 4, respectively; $σ_{0}$ , $σ_{1}$ , $σ_{2}$ , $σ_{3}$ and $σ_{4}$ denote standard deviation of frequency at the end of stage 0, stage 1, stage 2, stage 3 and stage 4, respectively; PDF denotes probability density function.

Figure 8.

Typical results of relations between natural frequencies and carbonation depths (C40N01) μ₀, μ₃ and μ₄ denote mean value of frequency at the end of stage 0, stage 3 and stage 4, respectively; σ₀, σ₃ and σ₄ denote standard deviation of frequency at the end of stage 0, stage 3 and stage 4, respectively; PDF denotes probability density function.

From Figures 7 and 8, we see a tendency of natural frequencies of the RC test beams to decrease with the increase in carbonation depth. Figure 7(a) shows that the mean value of the first-order frequency decreases from 22.239 to 21.075 Hz, a drop of 5.23%, when the carbonation depth increases from 0 to 26.7 mm. In the same manner, Figure 7(b) to (d) shows that the mean values of the second-order, third-order, and fourth-order frequencies decrease from 81.352 to 75.558 Hz, from 181.243 to 171.645 Hz, and from 306.763 to 287.035 Hz, reductions of 7.12%, 5.30%, and 6.43%, respectively, when the carbonation depth increases from 0 to 26.7 mm. Figure 8(a) shows that the mean value of the first-order frequency decreases from 22.189 to 21.540 Hz, a drop of 2.92%, when the carbonation depth increases from 0 to 11.8 mm. In the same manner, Figure 8(b) to (d) shows that the mean values of the second-order, third-order, and fourth-order frequencies decrease from 80.535 to 78.615 Hz, from 187.909 to 183.818 Hz, and from 318.130 to 309.694 Hz, reductions of 2.38%, 2.18%, and 2.65%, respectively, when the carbonation depth increases from 0 to 11.8 mm. Statistical analysis shows that at stage 4, which is accelerated carbonation for 28 days, the reductions in first four natural frequencies of the RC test beams poured with concrete of strength grade 30 are in the ranges 3.99%–6.67%, 7.12%–8.34%, 4.48%–7.95%, and 5.17%–8.69%, respectively. In the same way, the reductions in the first four natural frequencies of the RC test beams poured with concrete of strength grade 40 are in the ranges of 2.46%–2.98%, 2.38%–3.35%, 0.87%–2.49%, and 0.91%–2.65%, respectively. These experimental test results obviously show the tendency of the natural frequencies to decrease with the increase in the degree of concrete carbonation.

Because of the many factors influencing the experimental process, there are two groups of data that are inconsistent with the tendency of the frequencies to decrease with increased carbonation. The first dataset is the fourth modal test from stage 3 to stage 4 for test beam C30N01. The second is the third modal test from stage 3 to stage 4 for test beam C40N02. The reason that these two datasets do not agree with the general trend is that the change in carbonation depths is so small that the modal test is not sensitive to any difference. Table 5 shows that the carbonation depths in the concrete of strength grade 30 at the end of stage 3 (exposure: 21 days) and at the end of stage 4 (exposure: 28 days) are 23.7 and 26.7 mm, respectively. The change in carbonation depths in the concrete of strength grade 30 is only 3.0 mm. Table 6 shows that the carbonation depths in the concrete of strength grade 40 at the end of stage 3 (exposure: 21 days) and at the end of stage 4 (exposure: 28 days) are 9.5 and 11.8 mm, respectively. The change in carbonation depths in the concrete of strength grade 40 is only 2.3 mm.

The two datasets described account for only 2.5% of the total modal test data. The tendency of natural frequencies to decrease with increasing carbonation depths is a real effect.

It is important to note that there are some errors in the survey of natural frequencies due to the influences of environmental noise and other factors. The data exhibit quite a significant variation of natural frequencies for a particular level of carbonation depth. In order to illustrate that the frequency change in this experiment is caused by the difference in carbonation depths, we use statistical methods to compare the variation of identified frequency with the statistical significance of the frequency change due to the difference in carbonation depths.

In statistics, the standard deviation ( $σ$ ) is a measure that quantifies the amount of variation in a set of data values (Bland and Altman, 1996). A standard deviation close to zero indicates that the data points tend to be very close to the mean or expected value of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values. In the case where $X$ takes random values from a finite dataset $X_{1}, X_{2}, \dots, X_{N}$ with each value having the same probability, the standard deviation can be expressed as

σ = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (x_{i} - μ)}

(1)

where $μ = (1 / N) \sum_{i = 1}^{N} x_{i}$ .

In this experiment, there are 35 measurements for every RC test beam at each stage because there are seven measurement points and five impulse excitations. Thus, we select the mean value of measured frequencies as the relevant frequency. Figure 7 shows the probability density function (PDF) and standard deviation ( $σ$ ) of the frequency test data at different stages for test beam C30N01. The results of statistical analysis of test beams C30N01, C30N02, C30N03, C30N04, and C30N05 show that the maximum values of the standard deviation ( $σ$ ) are 0.075 (C30N01, the first-order frequency), 0.08 (C30N04, the second-order frequency), 0.431 (C30N04, the third-order frequency), and 1.029 Hz (C30N05, the fourth-order frequency). The corresponding changes in the natural frequencies are 1.02, 5.85, 8.46, and 20.47 Hz. Similar to Figure 7, Figure 8 shows the PDF and standard deviation ( $σ$ ) of the frequency test data at different stages for test beam C40N01. The statistical analysis results of test beams C40N01, C40N02, C40N03, C40N04, and C40N05 show that the maximum values of standard deviation ( $σ$ ) are 0.032 (C40N01, the first-order frequency), 0.088 (C40N03, the second-order frequency), 0.574 (C40N02, the third-order frequency), and 1.052 Hz (C40N01, the fourth-order frequency). The corresponding changes in natural frequencies are 0.649, 1.920, 1.893, and 8.435 Hz. Comparing the standard deviation ( $σ$ ) of the frequency test data with the corresponding changes in natural frequencies, we can ascertain that the variation of natural frequencies observed in the modal tests is caused by variation of carbonation depths.

The PDF was used to assess statistical properties for the test data. Stochastic analysis results of measured natural frequencies for test beam C30N01 at each test stage are shown in Figure 7. The statistical results of mean value $μ$ and lower and upper fractiles $μ \pm 2 σ$ of natural frequencies can be obtained from measured data. Figures 7 and 8 demonstrate that the experimentally measured frequencies data are mostly within 2σ of the mean, µ, and illustrate the discreteness of test data is very small.

In order to obtain the quantitative relationship between concrete-carbonation depths and natural frequencies, statistical methods and curve fitting were used to analyze the experimental data. The data are categorized into two groups according to the strength grade of the concrete. Figure 9 shows the relationships between concrete-carbonation depths and first four modal frequencies from the five test beams C30N01–C30N05. Discrete points represented by different colors and markers denote the mean values of modal frequencies. From the experimental test results of concrete-carbonation depths, the carbonation depths at the end of stages 0, 1, 2, 3, and 4 are 0, 13.6, 19.6, 23.7, and 26.7 mm, respectively. Straight lines are fit to the data for the first four modal frequencies. The fitting equations are shown in Figure 9. In the same way, Figure 10 shows the relationship between concrete-carbonation depths and the first four modal frequencies from the five test beams C40N01–C40N05. The carbonation depths at the end of stages 0, 3, and 4 are 0, 9.5, and 11.8 mm, respectively. The fitting equations for the first four modal frequencies are shown in Figure 10. The results of the experiment should provide a reference and innovative insight for SHM using modal frequencies.

Figure 9.

Fitting results for the drop in natural frequencies as a function of carbonation depth (concrete of strength grade 30).

Figure 10.

Fitting results for the drop in natural frequencies as a function of carbonation depth (concrete of strength grade 40).

Analysis of the main reason for drop in natural frequencies

Some reasons give rise to the above experimental results. The primary explanation of the experimental results is that the carbonation reaction increases the mass of RC test beam.

For a simply supported beam, the modal frequency is given by

f_{n} = \frac{ω_{n}}{2 π} = \frac{n^{2} \cdot π}{2} \cdot \sqrt{\frac{EI}{\bar{m} L^{4}}}

(2)

where $f_{n}$ denotes the natural frequency, $\bar{m}$ denotes the mass per unit length, $n$ is the mode number, $L$ is the span length, and $E$ is elastic modulus. The variable factors that affect the natural frequency are $EI$ and $\bar{m}$ . According to previous studies, the elastic modulus is almost invariant in the concrete-carbonation process. Therefore, the primary factor causing the drop in natural frequency is the change in mass per unit length $\bar{m}$ .

If the RC test beam is simply supported and the span length is unchanged in the whole experimental process, then

f_{n}^{2} = {(\frac{n^{2} \cdot π}{2})}^{2} \cdot \frac{EI}{L^{4}} \cdot \frac{1}{\bar{m}}

(3)

Differentiation of the above equation yields

2 f_{n} d (f_{n}) = {(\frac{n^{2} \cdot π}{2})}^{2} \cdot \frac{EI}{L^{4}} \cdot \frac{1}{{\bar{m}}^{2}} d (- \bar{m})

(4)

According to equation (3)

2 f_{n} d (f_{n}) = f_{n}^{2} \cdot \frac{1}{\bar{m}} d (- \bar{m})

(5)

That is

2 (\frac{1}{f_{n}}) d (f_{n}) = (\frac{1}{\bar{m}}) d (- \bar{m})

(6)

From equation (6), it is apparent that the fractional change in natural frequency of the beam is proportional to the fractional change in mass per unit length. Because of the minus sign, an increase in mass per unit length gives rise to a reduction in the natural frequencies. According to equation (6), a 2% change in mass per unit length would result in a 1% change in natural frequency. It is worth noting that the above analysis assumed ideal conditions. Namely, material and geometric properties, including mass, stiffness, damping, changes to the boundary conditions, and environmental conditions, are consistent with the assumed conditions of the theoretical analysis. In fact, there must be difference between theoretical analysis and engineering practice. The theoretical analysis should illustrate the phenomena in this article, but it is not the complete explanation.

The mass of concrete will increase after carbonation because the carbonatable substances react with gaseous $C O_{2}$ . Thus far, research into carbonation and engineering approaches to the problem have ignored the mass increase. Research on concrete carbonation has focused primarily on predicting the degree and depth of carbonation (Bernal et al., 2015; Han et al., 2013; Silva et al., 2014; Suzuki et al., 1985).

Concrete carbonation is a complex physicochemical process in which $C O_{2}$ penetrates the intricate porous system of concrete and simultaneously reacts with carbonatable constituents. By thermogravimetric analysis (TGA) and quantitative x-ray diffraction analysis (QXDA; Suzuki et al., 1985), the present and previous research has shown that $C O_{2}$ in the gaseous phase that diffuses through the concrete pores reacts not only with $Ca {(OH)}_{2}$ of the hardened cement paste but also with calcium silicate hydrate $3 CaO \cdot 2 Si O_{2} \cdot 3 H_{2} O$ (denoted by $CSH$ ), and with the yet unhydrated silicates, that is, tricalcium silicate $3 CaO \cdot Si O_{2}$ (denoted by $C_{3} S$ ) and dicalcium silicate $2 CaO \cdot Si O_{2}$ (denoted by $C_{2} S$ ). $Ca {(OH)}_{2}$ (denoted by CH) and $CSH$ are the products of hydration of $C_{3} S$ and $C_{2} S$ . This process can be described by the following reactions (Papadakis et al., 1991)

2 (3 CaO \cdot Si O_{2}) + 6 H_{2} O \overset{r_{H, C_{3} S}}{\to} (3 CaO \cdot 2 Si O_{2} \cdot 3 H_{2} O) + 3 Ca {(OH)}_{2}

(7)

2 (2 CaO \cdot Si O_{2}) + 4 H_{2} O \overset{r_{H, C_{2} S}}{\to} (3 CaO \cdot 2 Si O_{2} \cdot 3 H_{2} O) + Ca {(OH)}_{2}

(8)

where $Ca {(OH)}_{2}$ is also consumed during the hydration of the other constituents of cement. These chemical reactions proceed at constituent molar rates $r_{H, i}$ in moles of constituent $i$ reacting per unit volume of concrete per second.

The carbonation of $CSH$ and the yet unhydrated $C_{3} S$ and $C_{2} S$ of hardened cement paste takes place according to the reactions

(3 CaO \cdot 2 Si O_{2} \cdot 3 H_{2} O) + 3 {CO}_{2} \to (3 CaC O_{3} \cdot 2 Si O_{2} \cdot 3 H_{2} O)

(9)

3 CaO \cdot Si O_{2} + 3 C O_{2} + γ H_{2} O \to Si O_{2} \cdot γ H_{2} O + 3 CaC O_{3}

(10)

2 CaO \cdot Si O_{2} + 2 C O_{2} + γ H_{2} O \to Si O_{2} \cdot γ H_{2} O + 2 CaC O_{3}

(11)

According to a study by Papadakis et al. (1991), the mathematical model of the carbonation process is based on the differential mass balances of gaseous $C O_{2}$ , which account for the production, diffusion, and consumption of $C O_{2}$ .

The dissolved $C O_{2}$ per unit volume of completely carbonated concrete, denoted as $m_{0}$ , should be obtained according to the constituent materials of the cement and their proportions (Jiang et al., 2015). The carbonatable substances of cement are calcium oxide $CaO$ , iron oxide $F e_{2} O_{3}$ , and aluminum oxide $A l_{2} O_{3}$ . Therefore

m_{0} = \frac{c \cdot [CaO] %}{M_{CaO}} - \frac{c \cdot 4 [{Fe}_{2} O_{3}] %}{M_{{Fe}_{2} O_{3}}} - \frac{c \cdot 4 [{Al}_{2} O_{3}] %}{M_{{Al}_{2} O_{3}}}

(12)

where $[CaO] %$ denotes the percentage composition of calcium oxide $CaO$ in the cement, $[{Fe}_{2} O_{3}] %$ denotes the percentage composition of iron oxide $F e_{2} O_{3}$ , $[{Al}_{2} O_{3}] %$ denotes the percentage composition of aluminum oxide $A l_{2} O_{3}$ , $M_{CaO} = 56 g / mol$ denotes the molar mass of calcium oxide $CaO$ , $M_{F e_{2} O_{3}} = 160 g / mol$ denotes the molar mass of iron oxide $F e_{2} O_{3}$ , $M_{A l_{2} O_{3}} = 102 g / mol$ denotes the molar mass of aluminum oxide $A l_{2} O_{3}$ , and $c$ ( $kg / m^{3}$ ) denotes the cement content.

Equation (12) yields

m_{0} = {\frac{c \cdot [CaO] %}{56} - \frac{c \cdot 4 [{Fe}_{2} O_{3}] %}{160} - \frac{c \cdot 4 [{Al}_{2} O_{3}] %}{102}} \times 10^{3}

(13)

The dissolved $C O_{2}$ per unit volume of completely carbonated concrete, $m_{0}$ , is calculated from equation (13). $m_{0}$ is mainly affected by the percentage composition of calcium oxide $CaO$ , iron oxide $F e_{2} O_{3}$ , and aluminum oxide $A l_{2} O_{3}$ . In order to ascertain $m_{0}$ , the percentage composition of the three oxides ( $CaO$ , $F e_{2} O_{3}$ , and $A l_{2} O_{3}$ ) should be known first. Common Portland cement is widely used in China. The percentage composition of oxides of typical Portland cement is shown in Table 7 (Mindess and Young, 1981).

Table 7.

Oxides percentage composition of common Portland cement.

Chemical name	Abbreviation	Percentage composition (%)
Calcium oxide	$CaO$	63
Silicon oxide	SiO	22
Aluminum oxide	$A l_{2} O_{3}$	6
Iron oxide	$F e_{2} O_{3}$	2.5

Therefore, equation (13) can be written as follows

m_{0} = {\frac{c \cdot 63 %}{56} - 4 \times \frac{c \cdot 2.5 %}{160} - 4 \times \frac{c \cdot 6 %}{102}} \times 10^{3} = 8.27 \cdot c

(14)

In this experimental study, the cement content of concrete of strength grade 30 is $382 kg / m^{3}$ . Because the molar mass of $C O_{2}$ is 44 g/mol, the dissolved $C O_{2}$ per unit volume of completely carbonated concrete is $m_{0} = 139 kg / m^{3}$ . The mass per unit volume of concrete is estimated to be about $2500 kg / m^{3}$ . Therefore, the increased mass is 5.56% after complete carbonation. In the same way, the cement content of concrete of strength grade 40 is $414 kg / m^{3}$ . The increased mass is 6.03% after complete carbonation. According to equation (6), the decrease in natural frequency is 2.88% after complete carbonation for concrete of strength grade 30; the decrease in natural frequency is 3.02% after complete carbonation of concrete of strength grade 40.

Comparison of theoretical analysis and experiment results

In this experiment (see Figure 1), each RC test beam has a length of 3.2 m, a width of 0.15 m, and a height of 0.12 m. According to the test results of carbonation depths, the carbonation depth of the concrete of strength grade 30 is 26.7 mm at the end of stage 4. Table 8 shows that the percentage of concrete carbonated is $1 - (0.15 - 0.0267 \times 2) \times (0.12 - 0.0267 \times 2) / (0.15 \times 0.12) = 64.26 %$ . From equation (14), the increased mass is 5.56% after complete carbonation. Therefore, the percentage increase in mass of the RC test beam is 3.58% at the end of stage 4. According to the conclusion in equation (6), the drop in natural frequency of a simply supported RC test beam is 1.79% at the end of stage 4 for concrete of strength grade 30. In the same way, we can come to the conclusion that the drop in natural frequency of a simply supported RC test beam is 0.97% at the end of stage 4 for concrete of strength grade 40.

Table 8.

Theoretical analysis results of drop in natural frequencies.

Concrete mark	C30	C40
Cement content (kg/m³)	382	414
Carbonation depth at stage 4 (mm)	26.7	11.8
Percentage of concrete carbonated	64.26	32.31
Raise percentage in mass	3.58	1.95
Drop in natural frequency (%)	1.79	0.97

The drops in natural frequencies observed in the experiments are more than those predicted by theoretical analysis. Many factors give rise to the above results. First, the theoretical analysis depends on the percentage composition of chemical materials. The standard percentage composition of chemical materials differs from the composition of the experimental specimens. Second, the theoretical analysis is based on an ideal simply supported beam, and there are some differences between experimental boundary conditions and simply supported conditions. Third, the theoretical analysis considers only the completely carbonated volume. There must be a partially carbonated volume that gives rise to an increase in mass per unit length in the experiment that is higher than in the theoretical analysis. Therefore, we can understand that the drop in natural frequencies in the experiment is higher than in the theoretical analysis. Last but not least, the theoretical analysis considers only the increase in mass per unit length. In fact, many factors, such as stiffness, boundary conditions, temperature, and RH, slightly affect the natural frequencies in the experiment. To sum up, the experimental test and theoretical analysis demonstrate a tendency of the natural frequencies of the concrete test beams to drop as the carbonation depth increases.

In order to verify the accuracy of the mass increase estimated from physicochemical reactions, eight concrete block specimens of size $120 \times 75 \times 80 m m^{3}$ were subjected to accelerated carbonation and were weighed. Changes in mass should be obtained after accelerated carbonation. The time span of accelerated carbonation is 14 days. The mean values of carbonation depth for concrete of strength grade 30 and strength grade 40 are 12.4 and 10.1 mm, respectively. The comparison of experimental weighing results with physicochemical analysis results is shown in Figure 11. The experimental results are very close to physicochemical analysis results. Therefore, the mass increase estimated from physicochemical reaction analysis can be reliably used when test conditions do not allow for weighing specimens.

Figure 11.

Comparison of increase in mass determined from weight with that from physicochemical analysis.

Conclusion

The experimental results presented show that concrete carbonation in RC test beams causes significant changes in natural frequencies. These results can apply SHM more effectively for RC structures based on natural frequencies. The following conclusions are drawn:

A tendency of natural frequencies of RC test beams to decrease with increasing carbonation depths is observed. At each successive stage in the accelerated carbonation process, there is a higher change in natural frequencies for every RC test beam. The natural frequencies decrease as the carbonation depths increase. The fitting lines provide quantitative relationships between carbonation depths and natural frequencies.

The main reason for the reduction in natural frequencies is the increase in concrete mass due to concrete carbonation. Based on an analysis of the physicochemical process of concrete carbonation, we obtain the percentage increase in mass after complete carbonation. For concrete of strength grade 30 and strength grade 40 at stage 4, the increases are 3.58% and 1.95%, respectively. The comparison of experimental weighing results with physicochemical analysis results demonstrates that experimental weight increases compare well with predicted weight increases based on physicochemical analysis.Therefore, the physicochemical analysis method is reliable when test conditions do not allow for weighing specimens.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by grants from Subject of Project of National Key Fundamental Research Development Program (973 Program; grant no. 2013CB036305), Ministry of Transport of the People’s Republic of China (grant no. 2015318J38230), and National Science and Technology Support Plan (grant no. 2012BAJ11B01).

References

Bayraktar

Turker

Altunisik

(2015) Experimental frequencies and damping ratios for historical masonry arch bridges. Construction and Building Materials 75(1): 234–241.

Bernal

Provis

de Gutierrez

. (2015) Accelerated carbonation testing of alkali-activated slag/metakaolin blended concretes: effect of exposure conditions. Materials and Structures 48(3): 653–669.

Bland

Altman

(1996) Statistics notes: measurement error. British Medical Journal 312(7047): 1654.

Chang

TYP

Zhang

(2001) Ambient vibration of long-span cable-stayed bridge. Journal of Bridge Engineering 6(1): 46–53.

Chen

Zheng

. (2011) Experimental research on the behaviour of high frequency fatigue in concrete. Engineering Failure Analysis 18(7): 1848–1857.

Fertis

(1973) Free vibration of two-degree spring-mass systems. In: Dynamics and Vibration of Structures. London: Wiley, pp. 31–35.

Ding

(2011) Temperature-induced variations of measured modal frequencies of steel box girder for a long-span suspension bridge. International Journal of Steel Structures 11(2): 145–155.

Duran-Herrera

Mendoza-Rangel

De-Los-Santos

. (2015) Accelerated and natural carbonation of concretes with internal curing and shrinkage/viscosity modifiers. Materials and Structures 48(4): 1207–1214.

Esfandiari

Bakhtiari-Nejad

Rahai

(2013) Theoretical and experimental structural damage diagnosis method using natural frequencies through an improved sensitivity equation. International Journal of Mechanical Sciences 70(1): 79–89.

10.

Han

Park

Yang

(2013) Evaluation of concrete durability due to carbonation in harbor concrete structures. Construction and Building Materials 48(1): 1045–1049.

11.

Jiang

Zhang

. (2015) Modeling of carbonation in tensile zone of plain concrete beams damaged by cyclic loading. Construction and Building Materials 77(1): 479–488.

12.

Hao

Xia

. (2014) Damage detection of shear connectors in bridge structures with transmissibility in frequency domain. International Journal of Structural Stability and Dynamics 14(2): 1–33.

13.

Maalej

Chhoa

Quek

(2010) Effect of cracking, corrosion and repair on the frequency response of RC beams. Construction and Building Materials 24(5): 719–731.

14.

Mindess

Young

(1981) Concrete. Englewood Cliffs, NJ: Prentice Hall, Inc.

15.

Zhou

(2006) Experimental investigation of seismic damage identification using PCA-compressed frequency response functions and neural networks. Journal of Sound and Vibration 290(1–2): 242–263.

16.

Page

Treadaway

KWJ

(1982) Aspects of the electrochemistry of steel in concrete. Nature 297(5862): 109–115.

17.

Papadakis

Vayenas

Fardis

(1991) Fundamental modeling and experimental investigation of concrete carbonation. ACI Materials Journal 88(4): 363–373.

18.

Razak

Choi

(2001) The effect of corrosion on the natural frequency and modal damping of reinforced concrete beams. Engineering Structures 23(9): 1126–1133.

19.

Salawu

Williams

(1995) Bridge assessment using forced-vibration testing. Journal of Structural Engineering: ASCE 121(2): 161–173.

20.

Silva

Neves

de Brito

(2014) Statistical modelling of carbonation in reinforced concrete. Cement and Concrete Composites 50(1): 73–81.

21.

Siringoringo

Fujino

(2012) Estimating bridge fundamental frequency from vibration response of instrumented passing vehicle: analytical and experimental study. Advances in Structural Engineering 15(3): 417–433.

22.

Suzuki

Nishikawa

Ito

(1985) Formation and carbonation of C-S-H in water. Cement and Concrete Research 15(2): 213–224.

23.

Vestroni

Capecchi

(2000) Damage detection in beam structures based on frequency measurements. Journal of Engineering Mechanics: ASCE 126(7): 761–768.

24.

Xia

Nassif

Hwang

. (2013) Optimization of design details in orthotropic steel decks subjected to static and fatigue loads. Transportation Research Record: Journal of the Transportation Research Board 2331(1): 14–23.

25.

Zhou

Sun

Peng

(2015) Mechanisms of thermally induced deflection of a long-span cable-stayed bridge. Smart Structures and Systems 15(3): 505–522.