Peer-reviewed Stress-strain Curves for High Strength Concrete at Elevated Temperatures
High-strength concrete (HSC) has several well-known technical, aesthetic, and economical advantages over normal-strength concrete (NSC), which explains the increasing popularity of the former material in the structure domain. As in the case of NSC, however, high temperature adversely affects HSC mechanical properties even more than than in NSC, as indicated by the many studies performed so far on HSC at loftier temperature (hot backdrop) or past a thermal cycle at high temperature (residual properties). Since many code provisions concerning concrete backdrop versus high temperature were adult for ordinary concrete and the available models (in terms of stress-strain relationship) come up mostly from the tests on NSC—as the tests on HSC are less numerous—developing predictive relationships for HSC exposed to high temperature is yet an open upshot, peculiarly with reference to many parameters affecting concrete compressive strength, like temperature as such, heating rate, water-to-binder ratio, and strength in compression, to cite the most relevant parameters. To this purpose, a large database (more than than 600 tests) is examined in this newspaper, which is focused on HSC residuum properties and on the variables affecting its residual force. Available design models from diverse guidelines, standards, codes, and technical reports are tested against the database, and new regression-based models and blueprint formulae are proposed for HSC strength in pinch, after the exposure to high temperature.
1. Introduction
High-strength concrete (HSC) is gaining popularity over normal-strength concrete (NSC) in the blueprint of various structural components because of many advantages of such concrete [1]. As per ACI 363R-ten [2], HSC is divers as concrete having compressive forcefulness greater than 41.four MPa (6000 psi). This criterion is followed in the current written report to identify the two concrete types. The advantages of HSC over NSC include better mechanical backdrop such as improved durability, higher stiffness, and strength. The economic advantages are obtained as a result of reductions in section dimensions of structural members thereby increasing the usable space [3].
High-temperature exposure is known to adversely affect the mechanical properties of physical [4–xv]. There be three steady-country temperature tests available in the literature to determine the mechanical properties of physical at elevated temperature, as shown in Figure 1 [16]. In the first blazon, i.e., the stressed tests, the specimen is preloaded before heating and the preload is maintained during the heating menses. The preload may vary from 20 to twoscore% of the compressive strength of concrete at ambient temperature. The rate of increase in heat is commonly kept constant till the desired temperature is achieved, to guarantee quasi-steady thermal conditions during the heating process. After a rest period at the desired (i.e., nominal) temperature to make the thermal field as uniform as possible, the load is increased until the failure of the specimen in compression (Figure ane(a)). In the second blazon exam (i.e., unstressed test), at that place is no preload and the specimen is directly heated to the desired temperature usually at a constant rate. In this test too, the top temperature is sustained for prescribed period for achieving the thermal steady state. Subsequently, the specimen is loaded up to failure (Figure 1(b)). In the third type of test (i.e., unstressed residue test), no preload is practical during the heating process and the specimen is cooled down to room temperature (possibly under controlled conditions, i.due east., at a constant cooling rate) earlier being loaded in compression until concrete crushing (Figure 1(c)).
Because of enquiry on the impact of high-temperature exposure on concrete properties [17–26], it has been reported that the operation of NSC at elevated temperature is different from HSC at the same temperature. The review of behavior of HSC in fire [27, 28] revealed two dominant distinctions between NSC and HSC at high-temperature exposure: (i) at temperatures ranging from 100°C to 400°C, there is a deviation in relative forcefulness loss, and (2) at temperatures ranging from 200°C to 400°C, the explosive spalling was noticed in HSC test specimens. The explosive spalling in HSC, one of the major concerns, is mainly due to its denser microstructure causing depression permeability. This makes information technology difficult for the h2o vapor to escape to the exterior environs and consequently increases the pore pressure in cement paste which is mainly responsible for the explosive spalling in HSC. Thus, in that location is a demand for resolving the questions almost the functioning of HSC exposed to elevated temperature (or fire). Moreover, the available fire-design guidelines—developed typically based on the test data of NSC—cannot be simply extended to HSC.
There are several existing models for assessing the residual compressive strength of physical later exposure to elevated temperature [four–6, 29–34]. Nevertheless, these bachelor models were either developed for NSC or derived from inadequate test data on HSC, which was non sufficient to encompass all affecting parameters. The goal of this study is to come with predictive relationships taking care of the major parameters affecting concrete strength at high temperature and the strength loss. It should be noted that deterministic prediction of physical behavior may exist hard due to its heterogeneous nature that leads to substantial inconsistency in its backdrop. Theoretical developments are non, therefore, the objective of this research, and the models volition be developed on the ground of statistical regression analysis of the bachelor experimental database.
Aslani and Bastami [31] developed predictive models for both NSC and HSC subjected to high temperature. These relationships were obtained for tensile and compressive strengths, modulus of elasticity, and stress-strain (tensile and compressive) beliefs of unconfined physical at elevated temperatures. Nevertheless, the database employed to generate the models for HSC is inadequate (less than 100 exam specimens were utilized), and information technology is also limited to siliceous-aggregate physical. A recommendation was given to add together more test results to the database in order to cover wider range of affecting parameters.
As a effect of the research carried out since the 1950s on HSC exposed to loftier temperature, there exist extensive database on the functioning of HSC under loftier-temperature exposure. An inclusive assessment of these information is missing, withal. The residual compressive forcefulness models adult in this enquiry intend to bridge this gap through employing a comprehensive database for increasing the statistical sturdiness. A sufficiently big experimental database (617 data points)—obtained from the accessible literature pertaining to the tests on HSC exposed to elevated temperature environments—was thoroughly examined to assess the impact of various parameters on the residual compressive strength of HSC with strength exceeding 100 MPa. The collected database was employed to assess the applicability of the available models from guidelines, standards, codes, and researchers. New regression-based models were as well developed for the prediction of residuum compressive strength of HSC later on beingness exposed to elevated temperature. The exam results were subsequently compared with the predicted results of the regression-based models. Finally, new residuum concrete forcefulness pattern equations for HSC were developed in order to be utilized in fire pattern of physical structures.
2. Existing Models in Codes and Inquiry Publications
Tabular array 1 shows the models for the assessment of the balance concrete forcefulness postexposure to elevated temperature that are available in codes and standards such as Eurocode 2: EN 1992-1-2 [4] and ACI 216.one-07 [5] in addition to ASCE Manual of practice [6]. These are 1-variable models, which depend only on the elevated temperature, T. The event of all other variables is ignored except the model of ACI 216.one-07 [5], which employs different curves for siliceous and calcareous aggregates. Table i besides lists the models developed by researchers [29–34] for predicting residuum physical force. The following distinguishing features are noted on these models: (i) Some of the collected information points demonstrate enhancement in physical strength when exposed to moderately lower temperatures; however, this trend is not shown in any of the available models. A reduction in concrete strength for temperature higher than 20°C is demonstrated in virtually of the existing models [33]. However, the model of the ASCE Manual of exercise [half-dozen] reveals no change of compressive strength for temperatures up to 450°C. (ii) The model of Eurocode 2 [four] is not valid for HSC with forcefulness exceeding 90 MPa. Although the effect of blazon of aggregate (calcareous or siliceous) is considered for physical with < 55.0 MPa, information technology is not taken into account for HSC having ≥ 55.0 MPa. (iii) The curves given in ACI 216.ane-07 [5] clarify that the unstressed residual concrete forcefulness is the lowest amidst the three test procedures presented in Figure ane. Information technology should be noted that the height temperature reached in these curves is 871°C (i.e., 1600°F). The trend of curves indicates improve high temperature resistance (unstressed balance) for siliceous-aggregate physical in comparison with calcareous-aggregate physical. The effect of blazon of aggregate on the balance forcefulness of HSC is unidentified as these curves are generated only for NSC; however, in that location is no limit in the lawmaking on the concrete strength for the use of these curves. Moreover, no other code or researchers' models consider difference between the aggregate types (calcareous or siliceous) for HSC except Aslani and Bastami [31]. Although ASCE Manual [half dozen] besides uses the aforementioned curves as given in ACI 216.one-07 [five], a unproblematic linear model is suggested for assessment of the residuum concrete strength regardless of the aggregate type. Even though the two codes do not differentiate betwixt NSC and HSC, information technology is apparent that these models [5, 6] are only intended for NSC. (iv) The models given by some researchers [29, 32, 33] are piecewise linear for dissimilar ranges of T. Yet, Choe et al. [32] considered 7 ranges of T thereby providing five different formulae for assessing the residual physical strength. (five) The models of both Nielsen et al. [30] and Hertz [34] are the only ones that are applicable for whole temperature range. The model developed by Nielsen et al. [30] is a single quadratic equation in T. However, the equation proposed past Hertz [34] has a numerator of 1 and a 64-degree polynomial of T every bit the denominator. (vi) Aslani and Bastami [31] proposed several formulae for diverse ranges of T and . The adult formulae are linear and cubic in T.
| | |
| Code/researcher | Compressive strength afterward high temperature exposure ( ) |
| | |
| Eurocode 2: EN 1992-1-2 [4] | (i) If 41.4 < < 55.0 MPa, use Tabular array 3.1 of the code. (two) If ≥ 55.0 MPa, employ Table 6.1N of the code. |
| ACI 216.i-07 [5] | (i) For siliceous aggregate, use curve from Effigy two.12(a) of the code. (ii) For calcareous (carbonate) aggregate, utilize bend from Figure 2.12(b) of the code. |
| ASCE Manual [six] | |
| Kodur et al. [29] | |
| Nielsen et al. [thirty] | |
| Aslani and Bastami [31] | For siliceous aggregate, if 41.four < < 55.2 MPa, If 55.2 ≤ ≤ 80 MPa, If > 80 MPa, For calcareous amass, |
| Choe et al. [32] | |
| Phan and Carino [33] | |
| Hertz [34] | |
| | |
three. Experimental Data Fix
A significantly large data set of available experimental results on the residual compressive strength of HSC subsequently being exposed to high temperature was compiled. The experimental information were considered sufficient for deep investigation of different affecting parameters. These information were compiled from published sources including briefing proceedings, journal papers, student theses, and technical reports. The data taken from the literature were for those studies wherein nigh of the geometric dimensions of specimens and material backdrop were available. The database contains results of 617 specimens (including those at room temperature), out of which 485 specimens were exposed to high temperature. Summary of the collected experimental database is given in Table two. The data were obtained from 54 studies reporting experiments performed during the flow from 1965 to 2017 [vii, 10, eleven, 24, 35–84]. The following criteria were used for achieving consistency in the information set: (i) Data are but for the HSC specimens with > 41.4 MPa (following the limits of ACI 363R-x [2]). (two) Data are for unreinforced physical specimens such as cylinders and cubes. (iii) Concrete mix contains no fibers. (iv) The concrete mix contains mineral additives (such as fly ash and silica fume) not more than 15% of the weight of cement. (5) Ordinary Portland cement is utilized in the concrete mix. (6) The specimens are subjected to heating regimes at rates between 0.5 and xx°C/min, and the elevated temperature is sustained to guarantee quasi-steady thermal land. Then, natural air cooling is used to bring the temperature of specimens down to ambient temperature. Side by side, the specimens are concentrically loaded upward to failure to assess their residual strength. (vii) Enough details for different geometric and material characteristics are available for achieving better confidence in the derivation of results.
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ↑NA = not available. Values outside brackets are the total number of test specimens including those at room temperature, whereas values within brackets are the number of specimens exposed to high temperature. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
It should be noted that the database shown in Tabular array 2 is for different sizes of the cylinder and cube specimens. The formulae proposed by Yi et al. [85] were employed to convert the compressive strength measured for cubes and cylinders of dissimilar dimensions to that of the standard 150 × 300 mm cylinder as follows: where is the compressive force of the standard 150 × 300 mm concrete cylinder, is the compressive force of the general cylinder of diameter d (in mm) and height h (mm), and is the compressive strength of the general cube with size d (mm).
Table 3 provides the statistics of the information employed in the analysis. The statistics include data range (maximum and minimum values), mean value, standard deviation (SD), coefficient of variation (CV), skewness, and kurtosis. In social club to check unevenness in the distribution of the data, the skewness was estimated. For highly skewed data, skewness is greater than i. For determination of the shape of information distribution, the kurtosis of the information, listed in Table 3, was also estimated. The kurtosis of data should be 3 in society for the data to conform with the normal distribution curve. Form Table 3, it is clear that information are spread over a wide range due to the large values of coefficient of variation for all independent variables (varying from 28.ii% to 99.4%). The aggregate-to-binder ratio varies from 0.82 to 4.59 thus covering a wide range. The same was besides observed for the water-to-binder ratio which varies from 0.xiii to 0.half dozen due to concrete mixes that were designed with or without superplasticizers. It is explicable from Tabular array three that the superlative heating rate of 20°C/min is less than that reached in standard fire tests [86–88]. Yet, the maximum temperature reached in many experiments is 1200°C that agrees with the peak burn down temperature. Information technology should be noted from Table three that the skewness of heating charge per unit (H r) is 1.507, which indicates highly skewed information. It is worth mentioning here that combining the test information for specimens with wide range of heating rates varying from 0.5 to 20°C/min in developing the remainder force models of HSC was like to other published piece of work in the literature [31, 33, 89]. HSC specimens of the compiled database used by Phan and Carino [33] were exposed to high temperatures at wide range of heating rates varying from 0.2 to 32°C/min. Aslani and Bastami [31] developed models for NSC and HSC using test data with heating rates varying from very low rates up to standard fire curve as per the ASTM E119 [87]. Also, Knaack et al. [89] developed compressive forcefulness relationships for both NSC and HSC nether elevated temperatures using published experimental data with heating rates ranging from ii to 93.33°C/min. The concrete compressive force at room temperature varies from the start of the range of HSC and includes fifty-fifty the ultrahigh-strength physical.
| | |||||
| Statistical parameter | Aggregate/binder ratio, a/b | H2o/binder ratio, w/b | Heating rate, H r (°C/min) | Elevated temperature, T (°C) | Physical strength at room temperature, (MPa) |
| | |||||
| Min. value | 0.82 | 0.13 | 0.50 | 50 | 41.5 |
| Max. value | iv.59 | 0.60 | 20.0 | 1200 | 140.six |
| Hateful | 2.29 | 0.36 | v.78 | 479 | 67.3 |
| Standard deviation (SD) | 0.65 | 0.11 | 5.74 | 261 | 19.vii |
| Coefficient of variation (CV) (%) | 28.2 | 31.2 | 99.4 | 54.5 | 29.2 |
| Skewness | 0.232 | 0.564 | 1.507 | 0.529 | 0.646 |
| Kurtosis | −0.042 | −0.569 | i.235 | −0.255 | 0.263 |
| | |||||
Figure 2 shows the breakup of the data with respect to the type of the test specimen, the type of coarse amass, and the concrete strength at room temperature ( ). It should be noted from Figure 2(c) that the breakup of the information for is based on the limits given in Eurocode ii: EN 1992-ane-2 [four] (for the 55 MPa limit), and the model of Aslani and Bastami [31] (for both 55 and eighty MPa limits). The limit of 100 MPa was selected as an onset for ultrahigh-force concrete. The information cover cubes of 70, 100, and 150 mm and cylinders of 75 × 150, 100 × 200, and 150 × 300 mm. It is observed from Effigy 2(a) that data for cubes (57.five%) are slightly more than than that of cylinders. Maximum data are for cubes of 100 mm size (41.5%). It should be noted that even though the force loss at high temperatures measured on cylinders may be greater than that measured on cubes as reported by Bamonte and Gambarova [xc] (based on the data obtained from di Prisco et al. [91]) for temperatures across 400°C, equations (1) and (2) are assumed to exist valid for all range of elevated temperatures. This is due to the big variance in the sizes of cylinders and cubes used in this study which were not all covered by the data of di Prisco et al. [91]. Therefore, information of cubes and cylinders of different sizes were mixed in this study while developing residue force models for HSC exposed to elevated temperatures. This was similar to other work published in the literature [31, 33, 89]. For case, Phan and Carino [33] compiled test information for HSC cylinders of height/diameter ratio less than ii (28 × 52 mm and 57 × 100 mm) with HSC cylinders of the aspect ratio at least 2 (51 × 102 mm, sixty × 180 mm, 75 × 150 mm, 80 × 300 mm, 100 × 200 mm, and 150 × 300 mm) along with HSC cubes of 100 mm size. The study of Aslani and Bastami [31] fifty-fifty combined the data of HSC cubes and cylinders of different sizes and aspect ratios with standard burn down tests on big-scale reinforced HSC columns of dimensions 305 × 305 × 3810 mm. Likewise, in the work of Knaack et al. [89], data of physical cylinders and prisms with unlike sizes and aspect ratios were mixed to come up with compressive strength relationships for both NSC and HSC under elevated temperatures. Equally seen in Figure two(b), the data of concrete having siliceous aggregates (52%) more than that of calcareous aggregates (36%). Even though a wide range of concrete compressive strength is included in the information (from 41.five to 140.6 MPa), the data for the physical strength greater than 100 MPa are fairly low (5.7%). As seen in Figure ii(c), about 40.0% of the data of concrete strength ranges from 55.1 to lxxx.0 MPa with 32.12% data for lower strength ranging from 41.5 to 55 MPa, which is likewise articulate from the positive skewness of 0.646 as seen in Table iii.
Figure three shows the surface plot of normalized physical strength (i.e., the ratio of the remainder strength of physical subsequently high-temperature exposure to the concrete force at room temperature) against two variables. One of the variables is the elevated temperature, T, and the remaining variables are taken in turn for the 2nd. The use of siliceous or calcareous aggregates does not indicate any definite trend, every bit seen in Figure 3(a). The plots prove the occurrence of random waves in the surface profiles especially in the low-temperature range for most of the variable combinations (Figures 3(b) to 3(e)). These waves are maximum in water-to-binder ratio, which may exist owing to the hostile effects of superplasticizers. It is thus explicable that at that place is no definite trend of variation for dissimilar variables other than the exposure temperature, T.
Although there are some other variables such as moisture content, type of superplasticizer, and cooling regime that may affect the concrete strength, they were not taken into account due to the nonavailability of their values. It is worth noting that these variables are as well omitted in the models available in the literature.
four. Sensitivity Analysis
The models based on the artificial neural network (ANN) accept been detailed in several studies [92–99]. The main idea of ANN models originates from forming a mathematical relation between the dependent and independent variables by training the models using available data [94–96]. Sensitivity tests were performed using artificial neural networks (ANN) to appraise the relative importance of the variables in the interpretation of balance forcefulness of HSC after beingness exposed to elevated temperature. In the ANN modeling, the input independent variables were aggregate-to-binder ratio, a/b; water-to-binder ratio, w/b; heating rate in °C/min, H r ; exposure temperature in °C, T; and concrete strength at room temperature in MPa, . The output dependent variable is the ratio of residue forcefulness of HSC after high temperature exposure, , to . The network architecture of ANN contains twelve neurons and one subconscious layer. The tansig transfer function was employed forth with the back propagation (BP) formulation [92–94].
Tan-sigmoid transfer office: where x and y are the independent and dependent variables, respectively, whereas W and b are the weight and biases. Subscripts i and j stand for number of independent variables and the number of neurons, respectively. The values of West and b, of the higher up equation, were estimated in such a fashion that the energy office is minimized. The architecture of the neural network employed in the study is given in Effigy 4. The data were candy through three phases of ANN modeling, namely, grooming, validation, and testing, and a discrete data ready was used for each phase. It is worth mentioning hither that ideal distribution of data between testing/validation and training does non be. However, a review of literature shows that the pct of data used for grooming varies from 67% to 90%, and the remaining data are used for validation and testing [89–95]. In the nowadays study, the grooming was done using 2/3rd of the data, whereas the validation and testing were performed using the remaining 33% information [92, 93]. The data for the iii phases were selected randomly.
The sensitivity assay was carried out via removing each input at a time from the model. The outcome of elimination of a variable from the model on the calculation of residual physical forcefulness after high-temperature exposure was adamant in terms of the error estimate parameters listed in Table 4. These parameters include mean per centum error (MPE), mean absolute pct fault (MAPE), root mean foursquare mistake (RMSE), and correlation coefficient (CC). They are defined in the post-obit formulae: where north is the number of test specimens; R exp is the measured normalized concrete strength, which is the ratio of the measured remainder concrete strength after high temperature exposure in MPa ( ) exp to the measured concrete strength at room temperature in MPa ( ) exp; and R th is the predicted normalized concrete strength. The positive value of MPE means overestimation, whereas its negative value shows underestimation. However, the desired value of MPE is cypher. For the best predictive model, the error parameters MPE, MAPE, and RMSE should be zero, while CC should be unity. The value of epochs was input as 100.
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| MPE, hateful percentage error; MAPE, hateful absolute percent error; RMSE, root mean foursquare error; CC, correlation coefficient. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
The output results reported in Tabular array 4 were reached afterwards 100 successful runs of ANN for all cases. Although T is clearly the most sensitive variable, the remaining variables show low level of sensitivity which is nigh the aforementioned for all. The removal of temperature, T, beingness the nigh sensitive parameter, brings the value of CC downwards from 0.95 to 0.51. As observed from Table ane, most of the models of codes and researchers for the assessment of residual concrete strength only incorporate T except Eurocode two [iv] and Aslani and Bastami [31] where dissever equations are suggested for different ranges of concrete strength, . All the same, the effect of its removal has virtually no issue on fault estimated with RMSE increasing from 0.10 to 0.11 and CC reducing from 0.95 to 0.94. Keeping only T depicts small-scale reduction in CC from 0.95 to 0.90. Moreover, the effect of individual elimination of balance of the variables on error estimates is virtually insignificant. Therefore, the institution of the predictive model using T lonely may be acceptable.
five. Regression Models
As regression-based models are ordinarily in the form of equations relating the independent variable with one or more than of dependent variables, they are most favored by codes, standards, and blueprint guidelines. These models are also elementary to be used past practicing engineers. The goal of this research is to devise simple regression-based models to be included in burn down-design codes and guidelines for the assessment of remainder compressive strength of HSC after its exposure to high temperature or burn. Accordingly, the ANN models discussed in the previous department were not assessed further and thus were not compared with existing models available in the literature, as the focus will be on regression-based models.
Equally reported in the models available in the literature, for NSC, the siliceous-aggregate concrete has meliorate resistance to high temperature (unstressed remainder) as compared to the calcareous-aggregate concrete [5]. Nevertheless, for HSC, but the model proposed by Aslani and Bastami [31] differentiates between the amass types. Even so, the effect of type of aggregate on HSC is not obviously identified in other models. In fact, as mentioned previously, the upshot of blazon of amass (calcareous or siliceous) is considered in the model of Eurocode 2 [4] for concrete having < 55.0 MPa; however, it is not taken into business relationship for HSC having ≥ 55.0 MPa. As seen from Figure two(b), 52% and 36% of the compiled data were for HSC with siliceous and calcareous aggregates, respectively; however, the remaining data were for either unknown type of aggregate or HSC with silico-calcareous aggregate. Therefore, in the development of regression-based models in this study, the compiled information were not separated based on the type of aggregate every bit the employ of siliceous or calcareous aggregates did not betoken any definite tendency, equally seen from the surface plot in Effigy three(a) and also for the models to be applicative for the data for which type of amass is unidentified.
A regression model (Model i) of the following form was used by employing all variables for assessing the residual compressive force of HSC: where R is the normalized concrete strength, which is the ratio of the residual concrete strength afterward existence exposed to the elevated temperature in MPa ( ) to the concrete strength at room temperature in MPa ( ); T is the elevated temperature in °C; a/b is the amass-to-binder ratio; west/b is the water-to-binder ratio; and H r is the heating rate in °C/min.
The regression analysis was used to obtain the model parameters C1 to C7, which were found every bit 0.018, −0.027, 0.002, −0.036, −0.168, −0.006, and −0.092, respectively (Table 5). The scatter in the prediction by this model is shown in Effigy 5(a). It is worth mentioning here that the values of all parameters were not bachelor for all data points, thus the to a higher place model was adult based on 460 data points for which the values of all parameters were available.
| | ||||||
| Constant | Model ane | Model 2 | Model three | Model 4 | Model v | Model 6 |
| | ||||||
| C1 | 0.018 | 0.041 | 0.021 | 0.053 | −0.2074 | 0.234 |
| C2 | −0.027 | one.331 | −0.030 | 1.254 | 0.0150 | 0.644 |
| Ciii | 0.002 | −0.120 | 0.002 | — | −0.0005 | — |
| C4 | −0.036 | −0.267 | — | — | — | — |
| C5 | −0.168 | 0.021 | — | — | — | — |
| Cvi | −0.006 | −0.239 | — | — | — | — |
| C7 | −0.092 | — | — | — | — | — |
| | ||||||
The above model was simplified by replacing the cubic polynomial of elevated temperature, T, by the power function. The revised model (Model two) is given past
The values of the model parameters C1 to Chalf dozen are given in Table 5, and the scatter in the prediction using this model is plotted in Figure v(b).
A comparison of the predictive models, in terms of the mistake estimates, is given in Table half-dozen. The error estimates used for the evaluation of different models are MPE, MAPE, RMSE, pct information for error inside 15%, and per centum fault enveloping 80% of the data. It is observed from the table that there is marginal deterioration in the error estimates when polynomial office of T used in Model one is replaced by power function in Model two with the MAPE, RMSE, and percentage error enveloping 80% data increasing from 20.04 to 26.68, from 0.14 to 0.15, and from 31.8 to 34.iv, respectively. On the contrary, there is small reduction in the MPE and per centum information for error within xv% from half-dozen.70 to 5.88 and from 54 to 45, respectively. Thus the overall outcome of this replacement is well-nigh negligible.
| | ||||
| Parameter for error estimate | Normalized concrete forcefulness (R = ) | |||
| Regression Model 1 | Regression Model 2 | Regression Model three | Regression Model 4 | |
| | ||||
| Mean pct error (MPE) | 6.lxx | v.88 | 6.71 | 5.66 |
| Mean accented percent fault (MAPE) | 20.04 | 26.68 | 19.95 | 28.twenty |
| Root mean square error (RMSE) | 0.xiv | 0.15 | 0.14 | 0.15 |
| Percent information for error inside 15% | 54 | 45 | 55 | 46 |
| Percentage error enveloping 80% information | 31.8 | 34.4 | thirty.eight | 38.4 |
| | ||||
Every bit the sensitivity analysis highlighted the significance of models containing the elevated temperature, T, lonely, another regression model (Model three) was developed as a role of T, thus eliminating all other variables:
The in a higher place model is a cubic polynomial for whole range of the exposure temperature with the model parameters given in Table five. Figure 5(c) shows the besprinkle in the prediction of residuum physical compressive strength using this model.
Another simplified course of the higher up model was tried (Model 4) in which the cubic polynomial of T was replaced by the power function:
The model parameters C1 and C2 obtained by regression assay are given in Tabular array 5. The scatter in the assessment of balance compressive strength of concrete using this model is plotted in Figure v(d). The result of model simplification involving the replacement of cubic polynomial of T by the power part on error estimates is marginal with some parameters such equally MAPE, RMSE, and percent error enveloping 80% data showing small deterioration, whereas the remaining two mistake estimates (i.e., MPE and pct data for error within xv%) prove nominal comeback.
Figure half dozen shows the T versus R plots along with the prediction models given past using equations (vii) and (viii) (i.eastward., Model iii and Model 4). Information technology is discernable from the figure that in that location is no definite trend for dissimilar pct ranges of temperature and thus the development of above models for the whole range of temperature is justified.
(a)
(b)
Figure 7 shows a comparing between the normalized forcefulness of HSC assessed by proposed predictive models with experiments. The histogram of mistake for the four predictive models is shown in Figure viii. It is indicated that the errors in prediction of the normalized force of HSC past Model 1 is almost the same every bit those predicted using Model 3 and errors estimated using predictive Model 2 are near identical to those predicted using Model 4. The values of the statistical indicators such every bit mean, SD, CV, CC, and R 2 for the proposed predictive models are listed in Table vii, which shows good correlation for all the iv models with marginal difference. Information technology is observed from the mistake estimates given in Table half dozen and the statistical indicators listed in Table 7 that the models developed based on the temperature T alone (i.east., Model 3 and Model four) are equally proficient as the remaining two models. Thus the incorporation of the variables other than temperature, T, has almost negligible effect on the prediction of R, which was also evident in the sensitivity analysis. It is due to this reason that Model three and Model 4 will exist used for the development of the proposed pattern models.
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| SD, standard divergence; CV, coefficient of variation; CC, coefficient of correlation; R 2, coefficient of conclusion. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
half dozen. New Proposed Blueprint Models
For making bourgeois estimates of the balance compressive force of physical after high-temperature exposure, the regression models given by using equations (7) and (8) (i.eastward., Model 3 and Model 4) are transformed to the empirical pattern models such that 95% of the data lies to a higher place the proposed equations. The proposed design models are therefore given by Model v: and Model half-dozen:
The T versus R plots of the two proposed design models (i.e., Model 5 and Model vi) are shown in Effigy 6, which shows that most of the data points lie in a higher place the curves given past the two models. For temperatures of thou°C and in a higher place, the values of R are taken as zero for the two models, which is a conservative gauge for some of the experimental results. Figure 9 depicts a comparison between the normalized force of HSC assessed by proposed design models with experiments. As seen from the figure, nonconservative predictions of 5.6% and iv.seven% were calculated for Model 5 and Model 6, respectively. Figures 10 and 11 show a comparing between the normalized strength of HSC assessed by models of codes and researchers with experiments, respectively.
(a)
(b)
Figure 12 presents the spread of quartiles of the deviations in the cess of rest concrete strength. A global view of the scatter in the cess of residuum concrete force using different models is provided by this plot. In this effigy, nonconservative assessment (with predicted values being greater than the experimental ones) is given by the positive deviation. The upper and lower bounds of the deviations are represented, respectively, past the top and bottom ends of the bars. This plot will assist in identifying the all-time performing model for which the full height of divergence bar should exist nether the line of cipher divergence, and the 2nd and 3rd quartiles as well equally the total height of vertical bar are minimum. The proposed pattern models (Model five and Model half dozen) are the ones to meet all these favorite ideal features equally axiomatic from Figure 12.
The design models were evaluated on the basis of the mean, SD, CV, CC, and R 2 . Tabular array seven provides the values of these parameters. The table likewise enlists 5th percentile value and the pct of nonconservative data. For the best design model, SD and CV should be small, whereas CC and R 2 should exist close to unity. Moreover, the mean value should be more one; nevertheless, it should be close to unity. As well, the nonconservative information points should be negligible and the 5th percentile should exist at least one.0.
Amidst the three different lawmaking models, ACI 216.1-07 [5] is better than others. Moreover, all the three lawmaking models are nonconservative; yet, the nonconservative data for the model developed past ACI 216.1-07 [5] are low which is 17.9%. The model of the ASCE Manual [6] is the almost nonconservative with 66.0% nonconservative information. Among the researchers' design models, the model of Phan and Carino [33] is the best with the nonconservative data of 7.half-dozen%, whereas the nonconservative information of other models vary from 12.4% to 41.4%. Although the nonconservative data for Phan and Carino [33] model are close to the proposed models, the error estimates of the model are significantly higher. Information technology is explicable from Tabular array 7 that the proposed regression-based design models (i.e., Model 5 and Model half-dozen) are meliorate than other models. The value of the 5th percentile for both of the proposed blueprint models is 1.00, thus satisfying the target.
As just six.two% of the data is for greater than 100 MPa, the proposed pattern formulae should be applicable only to HSC of compressive force not exceeding 100 MPa. Moreover, the maximum heating charge per unit of the experimental information (xx °C/min) is quite less than that of standard fire [86–88]. Although the effect of the heating rate is found to be insignificant, this determination is based on the information of low heating rate. In time to come, these models may however be reviewed and revised once more than data for higher concrete force and higher heating rate become available.
7. Conclusions
The post-obit conclusions tin can be derived from the electric current report almost the effect of high temperatures on the residue compressive forcefulness of HSC: (i) As expected, the maximum temperature plays by far the major role in controlling rest compressive forcefulness of HSC. The remaining parameters prove near the same level of sensitivity and the outcome of these parameters is quite small every bit compared to the elevated temperature. (2) Amidst the formulae of dissimilar codes, ACI 216.i-07 [5] is meliorate with least nonconservative data, whereas the model of the ASCE Manual [6] is the nigh nonconservative with 66.0% nonconservative data. Amidst the researchers' blueprint models, the model of Phan and Carino [33] is the best with the nonconservative information of 7.6%. (iii) 4 regression-based models take been developed for the assessment of the residual compressive force of HSC. These models propose single formula for whole range of elevated temperature. Two of the models comprise all identified parameters, whereas the remaining two are a office of the elevated temperature lone. The models evidence small errors and acceptable correlation coefficients. Based on the models that are a function of the elevated temperature lonely, two new design models are proposed. These models are cubic polynomial and power function of temperature, respectively. The proposed design models show low fault estimates, and the nonconservative data points do not exceed six%. (iv) The proposed models are applicable for high-force apparently concrete produced using OPC and mineral additives such equally fly ash or silica smoke not exceeding 15% past weight of the cement. The apply of the proposed models should be restricted to HSC of compressive strength up to 100 MPa. (v) Although there is marginal effect of charge per unit of heating on the remainder compressive strength of physical, the future research should focus on higher heating rate respective to the fire because the current data are only for the maximum heating rate of twenty °C/min. (vi) The test information used in this study were obtained from different test programs by testing HSC specimens of different size and geometry using dissimilar heating rates and curing weather condition. These differences may lead to inconsistent examination results. Therefore, there is a necessity for a standardized examination protocol, especially for HSC equally the wet escape path and the rate of pore pressure buildup have a major effect on the beliefs of the examination specimen. For the sake of comparison of data obtained from dissimilar sources, information technology is highly recommended to create a set of standard examination methods. Also, the furnishings of previous load histories in addition to specimen size and geometry on measured properties should securely be investigated. Moreover, other material characteristics such as moisture transport properties, pore pressure buildup, and water release during the aridity procedure take to be measured as the functions of temperature and heating charge per unit to give input data for theoretical models.
Nomenclature
| a/b: | aggregate-to-binder ratio |
| BP: | back propagation |
| C1 to C7: | model parameters |
| d: | bore of the concrete cylinder or size of the concrete cube in mm |
| : | compressive force of the standard 150 × 300 mm concrete cylinder |
| : | compressive strength of concrete at room temperature |
| : | residuum compressive strength of concrete afterwards exposure to temperature |
| : | compressive forcefulness of the full general cube with size d in mm |
| : | compressive strength of the general cylinder of diameter d in mm and acme h in mm |
| h: | height of the concrete cylinder in mm |
| H r: | heating rate |
| n: | number of specimens |
| n 1: | number of neurons |
| T: | elevated temperature in °C |
| west/b: | water-to-binder ratio |
| R: | normalized concrete forcefulness (ratio of residuum compressive strength to the compressive strength of concrete at room temperature) |
| R exp: | measured normalized concrete strength |
| R th: | predicted normalized concrete strength |
| R 2: | coefficient of determination |
| CC: | coefficient of correlation |
| CV: | coefficient of variation |
| MAPE: | mean absolute per centum error |
| MPE: | mean percent mistake |
| RMSE: | root mean square error |
| SD: | standard deviation. |
Data Availability
The experimental data used to support the findings of this written report are available from the author upon asking.
Conflicts of Interest
The writer declares no conflicts of interest.
Acknowledgments
This research was supported by Deanship of Scientific Research Chairs at King Saud University, Saudi Arabia, for Chair of Research and Studies in Strengthening and Rehabilitation of Structures at Ceremonious Engineering Department.
Copyright
Copyright © 2019 Hussein Thou. Elsanadedy. This is an open admission article distributed under the Creative Commons Attribution License, which permits unrestricted employ, distribution, and reproduction in any medium, provided the original work is properly cited.
strongallontention.blogspot.com
Source: https://www.hindawi.com/journals/amse/2019/6039571/
0 Response to "Peer-reviewed Stress-strain Curves for High Strength Concrete at Elevated Temperatures"
Post a Comment