- Email: [email protected]
Prediction of the moment capacity of reinforced concrete slabs in fire using artificial neural networks
Advances in Engineering Software 41 (2010) 270–276
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Prediction of the moment capacity of reinforced concrete slabs in fire using artificial neural networks Hakan Erdem * Department of Civil Engineering, Faculty of Engineering and Architecture, Nigde University, Nigde 51245, Turkey
a r t i c l e
i n f o
Article history: Received 20 May 2009 Received in revised form 7 July 2009 Accepted 17 July 2009 Available online 14 August 2009 Keywords: Fire Reinforced concrete Slab Ultimate moment capacity Artificial neural networks
a b s t r a c t In this study, the application of artificial neural networks (ANN) to predict the ultimate moment capacity of reinforced concrete (RC) slabs in fire is investigated. An ANN model is built, trained and tested using 294 data for slabs exposed to fire. The data used in the ANN model consists of seven input parameters, which are the distance from the extreme fiber in tension to the centroid of the steel on the tension side of the slab (d0 ), the effective depth (d), the ratio of previous parameters (d0 /d), the area of reinforcement on the tension face of the slab (As), the fire exposure time (t), the compressive strength of the concrete (fcd), and the yield strength of the reinforcement (fyd). It is shown that ANN model predicts the ultimate moment capacity (Mu) of RC slabs in fire with high degree of accuracy within the range of input parameters considered. The moment capacities predicted by ANN are in line with the results provided by the ultimate moment capacity equation. These results are important as ANN model alleviates the problem of computational complexity in determining Mu. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Slabs are affected from fire as all other reinforced concrete (RC) members which are exposed to fire. As material strength of RC decreases with high temperature, the moment capacity (Mu) of RC slab decreases. In order to calculate the moment capacity of RC slab exposed to fire, the parameters such as the temperature increase depending on the fire exposure time, the temperature-distribution inside slab, the cross section and material properties, the tension and compression forces in slices, the depth equivalent rectangular stress block and the balance of the internal forces must be determined. As it is known, a large number of calculations are required to obtain these parameters. In order to avoid such computational complexity ANN approach may be used to predict effectively those parameters. However, to the best of author’s knowledge there is no research reported in literature on modeling of moment capacity using ANN method yet. A neural network model is a computer model, whose architecture essentially mimics the learning capability of the human brain. ANN is a technique which can be applied to complex problems described with a large amount of data. It does not require knowledge of the physical processes involved. However, it identifies the relationships in a set of data. Therefore ANN approach may be applied * Tel.: +90 388 2252306; fax: +90 388 2250112. E-mail addresses: [email protected], [email protected] 0965-9978/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2009.07.006
to problems where conventional mathematical solutions are not applicable. In recent years, ANN method has been used in variety of subjects in the structural engineering by many researchers. Some of these subjects are about prediction of the various properties of concrete [1–3], the properties of concrete aggregate [4], the strength of cements produced with pozzolans [5], the modeling of compressive strength of cement mortar [6], the rebar corrosion damage in RC structures [7], the performance of fiber reinforced RC beams and slabs [8–10], the moment and shear capacity [11], the shear strength [12–14], and the moment capacity and area of reinforcement [15] of RC beams, the ultimate deformation capacity of RC columns [16], the optimum cost [17] and the optimum depth [18] of RC slabs etc. Al-Khaleefi et al. [19] represented a functional relationship, using ANN method, between the fire resistance of a concrete filled steel column and the fire resistance parameters. However, there is no intensive research in determining the moment capacity of the RC slab using ANN yet. In this study, a predictive model for the ultimate moment capacity of the RC slab in fire was developed by using ANN. The calculated moment capacities of the RC slabs in fire were compared with the moment capacities predicted by the ANN model to verify reliable application of the artificial neural network model. This research explores the applicability of using ANN to create an intelligent model for prediction of the ultimate moment capacity of RC slabs in fire.
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
2. Ultimate moment capacity of RC slabs in fire
Q¼
In order to calculate the moment capacity of RC slab exposed to fire, the rising temperature by the fire exposure time, the temperature-distribution inside slab and the decrease in material strengths must be determined. After they are obtained, the tension and compression forces inside slab section are examined. The moment when the forces are balanced is the ultimate moment capacity of RC slab in fire. In the following subsections, these subjects are introduced. 2.1. The ISO834 temperature–time curve In this study, ISO834 is used to define fire time–temperature relation [20]. The equation for the ISO834 temperature–time curve is as follows:
T ¼ 345 log10 ð8t þ 1Þ þ T a
ð CÞ
ð1Þ
where t is the fire exposure time and Ta is the ambient temperature (°C). 2.2. Concrete strength at high temperature Reduction in concrete compressive strength at high temperatures has to be taken into consideration. The local concrete compressive strength rcT can be calculated using the temperature at each position by the relationship given in Eurocode2 [21]:
rcT ¼ kc rc20 C kc ¼ 1 for T 6 100 kc ¼ ð1:067 0:00067 TÞ for 100 6 T 6 400
ð4Þ
where Rtotal is the total thermal resistance, expressed as
Rtotal ¼ Rconv ;1 þ Rslab þ Rconv ;2 1 L1 1 ¼ þ þ h1 A kA h2 A
ð5Þ
where k is thermal conductivity coefficient, h1 and h2 the convection heat transfer coefficients, A the surface area and h is the thickness of slab. Once Q is known, T1 and T2 can be calculated. An unknown temperature T(y) inside slab at any slice can be determined as:
TðyÞ ¼ T 2 þ ðT 1 T 2 Þ
y h
ð6Þ
where T1 is temperature in surface exposed to fire and T2 is temperature in surface unexposed to fire. 2.5. Moment capacity of RC slab exposed to fire Once the temperature variations are known, the effects of temperature on the material properties and moment capacity of the slab can be examined. As can be understood from the reduction coefficients given in previous section, rising temperature results in both the corruption of the material properties and decrease in the residual moment capacity of the slab. Temperature distribution in the slab must be known for obtaining the change of material properties inside the slab. The cross section of RC slab is divided into M slices and for each slice the material temperature, reduction factors and mechanical characteristics are specified [23]. The tension force in the steel rebars can be derived by (Fig. 2):
ð2Þ Fs ¼
kc ¼ ð1:44 0:0016 TÞ for 400 6 T 6 900
T 11 T 12 Rtotal
271
M X
ksi fyd Asi
ð7Þ
i¼1
kc ¼ 0 for 900 6 T where rc20 °C and rcT are the concrete compressive strengths at 20 °C and the rising temperature °C, respectively, kc is the temperature reduction factor for the compressive strength and T is temperature.
The compressive force for unit width in the concrete can be calculated by summing all the compressive forces on the compressive side of lumped units. a
F c ¼ 0:85
Dy X
kci fcd Dy
ð8Þ
i¼1
2.3. Steel tensile strength at high temperature The values of reduced ultimate tensile strength of rebars due to temperature can be obtained from the following equations [21]:
fsuT ¼ ks fsu20 C ks ¼ 1 for 0 6 T 6 350 ks ¼ 1:899 0:00257T for 350 6 T 6 700
If the forces of the cross section are in static equilibrium, Eqs. (7) and (8) should be equal. If not, the value of a is increased progressively, and the calculation is repeated. The process continues until Eqs. (7) and (8) are equal. When the forces in slab are in equilibrium, the residual moment capacity of the slab Mu can be calculated as: a
ð3Þ
ks ¼ 0:24 0:0002T for 700 6 T 6 1200
Mu ¼ 0:85
Dy X
i¼1
Dy kci fcd Dy d iDy 2
ð9Þ
where fsu20 C and fsuT are the ultimate tensile strengths of rebars at 20 °C and the rising temperature °C, respectively, and ks is the temperature reduction factor for the tensile strength.
where fcd is the compressive strength of the concrete at temperature 20 °C, fyd is the yield strength of the reinforcement at temperature 20 °C, Fc and Fs are compressive and tensile forces in slab, respectively, kci and ksi are the reduction factors for each segment of the material temperature in slab.
2.4. Steady heat conduction in plane slab
3. Artificial neural network
Heat transfer through the slab can be modeled as steady and one-dimensional. The thermal resistance concept can be used to determine the rate of steady heat transfer through slabs. The rate of steady heat transfer through slab can be expressed using two surfaces at known temperatures by the total thermal resistance as shown in Fig. 1 [22–23].
Artificial neural networks are a computational tool that attempts to simulate the architecture and internal features of the human brain and nervous system. ANNs are consisting of a large number of simple processing elements called as neurons. Artificial neurons connected together form a network. The structure of artificial neural networks is layered. These are input, hidden and out-
ks ¼ 0 for 1200 6 T
272
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
Fig. 1. The thermal resistance network for heat transfer through a plane slab subjected to convection on both sides.
Fig. 2. Temperatures and forces in plane RC slab.
put layers. The back-propagation (BP) is one of the most popular learning algorithms. The back-propagation learns by comparing its output of each input pattern with a target output of that pattern, then calculating the error and propagating an error function backward through the net. To run the network after it is trained, the values for the input parameters are presented to the network. The network then calculates the node outputs by using the existing weight values and thresholds developed in the training process. If the neural network correctly determines the training data and correctly identifies the testing data, it is considered as trained. Usually a large amount of reliable data is required for training of ANN model. The mean square error (MSE) and the correlation coefficient (R) can be used to test the accuracy of the trained network. Each layer in ANNs has a number of processing neurons and each neuron is fully interconnected with weighted connections to units in the subsequent layer. A multi-layered perception (MLP) transforms i inputs into k outputs through non-linear mapping functions. The output of a neuron is determined by the activation of the neurons in the previous layer as follows [24]:
xo ¼ f
X
! xh who
training process, the MLP starts with a random set of initial weights and then training continues until a set of wih and who are optimized, so that a predefined error threshold is met between the output of network xo and the corresponding target value to. According to the BP algorithm, each interconnection between the neurons is adjusted by the amount of the weight update value as follows:
@E ¼ gdo xh @who @E Dwih ¼ g ¼ gdh xi @wih
Dwho ¼ g
do ¼ x0o ðto xo Þ X do who do ¼ x0h
N X L 1X 2 ðt ðsÞ xðsÞ o Þ 2 s o o
ð10Þ
ð11Þ
ð12Þ
where N is the number of data set patterns and L is the number of output neurons. The aim is to reduce the error by adjusting the interconnections between layers. The weights are adjusted using a gradient descent BP algorithm. The algorithm requires training data that consist of a set of corresponding input and target pattern values to. During the
ð16Þ
where x0o ¼ xo ð1 xo Þ and x0h ¼ xh ð1 xh Þ when a sigmoid activation function is used.
Based on the differences between the calculated output xo and the target value to an error is defined as follows:
E¼
ð15Þ
o
P where f( h xh who ) is the activation function, xh the activation of the hth neuron in the previous layer and who is the interconnection between hth layer neuron and oth layer neuron. The most widely used activation function is the sigmoid and it is given as follows:
1 P 1 þ expð xh who Þ
ð14Þ
where E is the error function given in Eq. (11), g learning rate, and
h
xo ¼
ð13Þ
Fig. 3. The architecture of the system on back-propagation networks.
273
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
Moment capacity, kNm
60
Table 1 The statistical values of ANN model.
training validation testing
50 Training Validation Testing
40 30
Data number
MSE
R
206 44 44
1.21326E4 1.42537E4 1.70820E4
0.99775 0.99795 0.99750
20 10 0 25
50
75
100 125 150 175 200 225 250 275 300
Number of data Fig. 4. Training, validation, and testing data used for prediction of the ultimate moment capacity.
Mean Squared Error (MSE)
1 training validation testing
0.1
0.01
0.001
0.0001 0
2
4
6
8
10
12
14
16
18
20
Epochs Fig. 5. Performance of ANN for training, testing, and validation processes.
4. ANN model used for the prediction of moment capacity
60
Predicted moment capacity, kNm
Predicted moment capacity, kNm
The objective of this work is to develop a neural network model for prediction of the ultimate moment capacity of slabs in fire. The data are obtained for different fire time, material and cross section
ANN training y = 0,9922x + 0,0526 R = 0,99775
50 40 30 20 10 0 0
10
20
30
40
50
60
40 30 20 10 0 0
Predicted moment capacity, kNm
Calculated moment capacity, kNm 60
ANN testing y = 0,9867x + 0,1715 R = 0,99750
50 40 30 20 10 0 0
10
20
30
40
50
60
Calculated moment capacity, kNm
ANN validation y = 1,0253x - 0,4406 R = 0,99795
50
60
10
20
30
40
50
60
Calculated moment capacity, kNm
Predicted moment capacity, kNm
0
properties. Temperature, concrete and steel material properties in each slice are obtained with slicing. Forces in all slices are determined by using the deteriorated properties of material. Heat transfer through slab is modeled as steady and one-dimensional. Heat transmission inside RC slab is calculated using a simulation of electrical flow. The ultimate moment capacity of RC slab is also obtained from tensile and compression forces equilibrium. The neural network developed in the investigation has seven neurons in the input layer, eight neurons in the hidden layer and one neuron in the output layer. The input parameters are the distance from the extreme fiber to the centroid of the steel on the tension side of the slab (d0 ), the effective depth (d), the ratio of previous parameters (d0 /d), the area of reinforcement on the tension side of the slab (As), the fire exposure time (t), the compressive strength of concrete (fcd), and the yield strength of the reinforcement (fyd). In this study, the Levenberg–Marquardt network (LM) is chosen as learning algorithm. A architecture of the artificial neuron is shown in Fig. 3. Using the process given in previous section, the data are obtained for prediction of the ultimate moment capacity of RC slabs exposed to fire with ANN. k = 1.30 W/m K and h1 = h2 = 10 W/ m2 K values are taken as constant during process obtained data. The values in the input layer are taken as d0 (30–40 mm), d (110– 130 mm), d0 /d (0.23,0.27 and 0.36), As (678.58, 923.63 and1231.50 mm2), t (0, 5, 10,20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min), fcd (20, 25 and 30 N/mm2), and fyd (220 and 420 N/mm2). For the presented problem, the ANN models are developed using the Matlab software. The 294 data were used in the analyses and they were divided into data for training (70% of the total data), validation (15% of the total data) and testing (15% of the total data). The values selected randomly for three stages
60
ANN all input y = 0,9964x - 0,0084 R = 0,99765
50 40 30 20 10 0 0
10
20
30
40
50
60
Calculated moment capacity, kNm
Fig. 6. Comparison of calculated and predicted moment capacity.
274
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
70 60
Moment capacity, kNm
5. Analysis results
target predicted
In this study, the ANN model has been developed to predict of the moment capacity of RC slabs in fire using the 294 data. The variation of Mean Squared Error (MSE) against the number of epochs for LM network training, validation and testing stages are shown in Fig. 5. It is evident that the MSE decreases rapidly with the increasing number of epochs. The result in terms of MSE here is reasonable, because the test set error and the validation set error have similar characteristics, and it does not appear that any significant overfitting has occurred. The relationship between calculated moment capacity using the developed program by author and predicted moment capacity by the ANN method is illustrated in Fig. 6. The training of ANN is completed with a high correlation coefficient of 0.99775. In the testing stage, the correlation coefficient is obtained as 0.99750. It is clear to point out from the performance and generalization capacity of ANN that the proposed model is extremely suitable in determining the moment capacity. Table 1 shows the correlation coefficient (R) and the mean squared error (MSE) for the obtained results. Comparison of the calculated moment capacity with the predicted moment capacity is made and shown in Fig. 7. In addition, the values analyzed in terms of differential between the calculated and predicted moment capacities and they are shown in Fig. 8. It has been observed from Figs. 7 and 8 that the ANN model is able to predict the moment capacity. Hence, the results of ANN can be directly used for the prediction of the moment capacity of slabs. For the first 28 data in the present problem, the values of the calculated moment capacity and the predicted moment capacity with ANN method is given Table 2. The results indicate that neural network was successful in learning and testing.
50 40 30 20 10 0 0
25
50
75
100
125
150
175
200
225
250
275
300
Number of data Fig. 7. The calculated and predicted moment capacity.
4 3
Error, kNm
2 1 0 0
25
50
75
100 125 150 175 200 225 250 275 300
-1 -2 -3
Number of data Fig. 8. The differential moment capacity value distribution in the analyses.
5.1. Effect of effective slab depth on the moment capacity are shown in Fig. 4. They were normalized by maximum values of parameters. The training patterns were randomly input into the network to train it. In this way, the neural network was applied to identify the moment capacity of a given data.
The ultimate moment capacities for the different effective slab depths are examined by means of ANN method. In calculations, the input parameters are taken as d = 110, 120, and 130 mm,
Table 2 The calculated and predicted moment capacity. Data
d0
d
d0 /d
As (mm2)
t (min)
fcd (MPa)
fyd (MPa)
Mr (kNm)
Mpredicted (kNm)
Mpredicted–Mr
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30
110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110
0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27
678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63
0 5 10 20 30 40 50 60 70 80 90 100 110 120 0 5 10 20 30 40 50 60 70 80 90 100 110 120
25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25
420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420
29.44 28.91 24.91 20.16 17.37 15.39 14.18 12.89 11.8 10.86 10.02 9.28 8.61 7.99 39.13 38.4 33.1 26.79 23.64 20.95 18.85 17.14 15.69 14.44 13.65 12.63 11.71 10.88
31.05 27.95 25.25 20.92 17.70 15.39 13.83 12.76 11.86 10.95 10.05 9.21 8.47 7.83 40.46 36.70 33.39 28.00 23.97 21.00 18.89 17.37 16.09 14.84 13.61 12.45 11.40 10.46
1.61 0.96 0.34 0.76 0.33 0.00 0.35 0.13 0.06 0.09 0.03 0.07 0.14 0.16 1.33 1.70 0.29 1.21 0.33 0.05 0.04 0.23 0.40 0.40 0.04 0.18 0.31 0.42
275
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
70
Moment capacity (kNm)
d0 = 30 mm, As = 1231.5 mm2, fcd = 30 MPa, fyd = 420 MPa and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. The results for the different d are shown in Fig. 9. While the effect of effective slab depth on the moment capacity exists in short fire time, the difference between the moment capacities for different effective slab depths decrease with increasing fire time rapidly. In addition, as seen in Fig. 9, the effect of slab dept does not exist for the fire exposure time more than 70 min. This reason is that the slab thickness has not enough to prevent temperature-distribution inside cross section. The generation of high temperature inside slab apparently decreases the compressive strength of the concrete.
fyd=220 Mpa fyd=420 Mpa
60 50 40 30 20 10 0 0
20
40
60
80
100
120
Fire exposure time (minute)
5.2. Effect of different rebar areas on the moment capacity
Fig. 11. The effect of the different rebar strengths on the moment capacity.
70 fcd=20 Mpa fcd=25 Mpa fcd=30 Mpa
60
Moment capacity (kNm)
In order to determine the ultimate moment capacities for the different rebar areas the ANN analyses are performed. The analyses results are shown in Fig. 10. In calculations, the input parameters are taken as d = 130 mm, d0 = 30 mm, As = 700, 900 and 1100 mm2, fcd = 30 MPa, fyd = 420 MPa, and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. As shown in Fig. 10, while effect of rebar areas on the moment capacity exists in short fire time, the difference between moment capacities for the different rebar areas decreases with the increasing fire time slowly. Although the effect of rebar decrease with rising fire exposure time, it does not vary for the fire exposure time more than 60 min.
50 40 30 20 10 0 0
5.3. Effect of different rebar strengths on the moment capacity
20
40
60
80
100
120
Fire exposure time (minute)
The ultimate moment capacities for the different yield strengths of the reinforcement are determined and are shown in Fig. 11. In calculations, the input parameters are taken as d = 130 mm,
d0 = 30 mm, As = 1231.5 mm2, fcd = 30 MPa, fyd = 220 and 420 MPa, and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. As shown in Fig. 11, while the effect of yield strengths of the reinforcement on the moment capacity exists in short fire time, the difference between the moment capacities for the different yield strengths of the reinforcement decreases with increasing fire time. The increase of fire exposure time causes the increase in temperature of rebar, and thus the material strength of steel decreases.
Moment capacity (kNm)
70 d=110mm d=120mm d=130mm
60
Fig. 12. The effect of the different concrete strengths on the moment capacity.
50 40 30 20
5.4. Effect of different concrete strengths on the moment capacity
10 0 0
20
40
60
80
100
120
Fire exposure time (minute) Fig. 9. The effect of the effective slab depth on the moment capacity.
Moment capacity (kNm)
70 As=700mm^2 As=900mm^2 As=1100mm^2
60 50 40 30 20
The ultimate moment capacities for the different compressive strengths of concrete are examined and shown in Fig. 12. In calculations, the input parameters are taken as d = 130 mm, d0 = 30 mm, As = 1231.5 mm2, fcd = 20, 25 and 30 MPa, fyd = 220 and 420 MPa, and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. As shown in Fig. 12, while the effect of compressive strengths of concrete on the moment capacity exists in short fire time, the difference among the moment capacities for the different compressive strengths of concrete decreases with the increasing fire time, and is negligible. The variation in concrete strength does not significantly affect the moment capacity, in contrast to the variation in steel yield strength. Because the compressive strength of concrete is lower than steel. 6. Conclusions
10 0 0
20
40
60
80
100
120
Fire exposure time (minute) Fig. 10. The effect of the different rebar areas on the moment capacity.
The ultimate moment capacity of RC slabs exposed to fire for the different slab effective depths, the areas of reinforcement, the compressive strengths of the concrete and the yield strengths of the reinforcement is examined and predicted by using ANN method and the effect of these parameters are investigated. The
276
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
calculated 294 data by author are used for this study. The correlation coefficient is obtained as 99.775% for training stage and 99.750% for testing stage. These values indicate that the proposed ANN model is highly successful. The test results also show that the generalization ability of ANN is well. It is seen that the increase of the slab effective depth is not enough to prevent rising temperature inside slab, and thus the constructive effect of the slab effective depth on the moment capacity with rising temperature vanish. Although the constructive effect of the area of the reinforcement and the yield strengths of the reinforcement on the moment capacity decrease with the fire exposure time, this effect goes on. The effect of the compressive strengths of the concrete on the moment capacity is also very little and is vanishing with the increase of the fire exposure time completely. Without calculating several computations in calculation stage, the trained ANN model can be used to provide the moment capacity with high accuracy. References [1] Yeh IC. Modeling slump flow of concrete using second-order regressions and artificial neural networks. Cem Concr Compos 2007;29:474–80. [2] Karahan O, Tanyıldızı H, Atısß CD. An artificial neural network approach for prediction of long-term strength properties of steel fiber reinforced concrete containing fly ash. J Zhejiang Univ Sci A 2008;9(11):1514–23. [3] Hola J, Schabowicz K. Application of artificial neural networks to determine concrete compressive strength based on non-destructive tests. J Civ Eng Manage 2005;1:23–32. _ [4] Topçu IB, Sarıdemir M. Prediction of properties of waste AAC aggregate concrete using artificial neural network. Comput Mater Sci 2007;41:117–25. _ [5] Topçu IB, Karakurt C, Sarıdemir M. Predicting the strength development of cements produced with different pozzolans by neural network and fuzzy logic. Mater Des 2008;29:1986–91. [6] Akkurt S, Özdemir S, Tayfur G, Akyol B. The use of GA-ANNs in the modelling of compressive strength of cement mortar. Cem Concr Res 2003;33:973–9. [7] Ukrainczyk N, Pecur IB, Bolf N. Evaluating rebar corrosion damage in RC structures exposed to marine environment using neural network. Civ Eng Environ Syst 2007;24(1):15–32.
[8] Flood I, Muszynski L, Nandy S. Rapid analysis of externally reinforced concrete beams using neural networks. Comput Struct 2001;79:1553–9. [9] Pannirselvam N, Raghunath PN, Suguna K. Neural network for performance of glass fibre reinforced polymer plated RC beams. Am J Eng Appl Sci 2008;1:83–8. [10] Bisby LA, Kodur VKR. Evaluating the fire endurance of concrete slabs reinforced with FRP bars: considerations for a holistic approach. Compos Part B: Eng 2007;38:547–58. [11] Rao HS, Babu BR. Hybrid neural network model for the design of beam subjected to bending and shear. Sadhana 2007;32(5):577–86. [12] Mansour MY, Dicleli M, Lee JY, Zhang J. Predicting the shear strength of reinforced concrete beams using artificial neural networks. Eng Struct 2004;26:781–99. [13] Ahn N, Jang H, Park DK. Presumption of shear strength of steel fiber reinforced concrete beam using artificial neural network model. J Appl Polym Sci 2007;103:2351–8. [14] Cladera A, Mari AR. Shear design procedure for reinforced normal and highstrength concrete beams using artificial neural networks. Part I: beams without stirrups. Eng Struct 2004;26:917–26. [15] Kelesßog˘lu Ö. The analyses of the reinforced concrete beam sections by artificial neural networks. Tech J Turk Chamber Civ Eng 2006:3935–42 [in Turkish]. [16] Inel M. Modeling ultimate deformation capacity of RC columns using artificial neural networks. Eng Struct 2007;29:329–35. [17] Ahmadkhanlou F, Adeli H. Optimum cost design of reinforced concrete slabs using neural dynamics model. Eng Appl Artif Intell 2005;18:65–72. [18] Rafıq MY, Bugmann G, Easterbrook DJ. Neural network design for engineering applications. Comput Struct 2001;79:1541–52. [19] Al-Khaleefi AM, Terro MJ, Alex PA, Wang Y. Prediction of fire resistance of concrete filled tubular steel columns using neural networks. Fire Safety J 2002;37:339–52. [20] ISO-834. Fire resistance tests-elements of building construction. Part 1– 9. Geneva: International Standards Organisation; 1975. [21] Eurocode 2: Design of concrete structures. ENV 1992. Part 1–2: General rulesstructural fire design. European Committee for Standardization, Brussels; 1995. [22] Çengel YH. Heat transfer: a practical approach. McGraw-Hill; 1998. p. 57–111. [23] Erdem H. Effect to reinforced concrete slab moment capacity of high temperature. In: The 30th anniversary of Çukurova University, engineering and architecture faculty symposium, Adana; 2008 [in Turkish]. [24] Kahraman S, Altun H, Tezekeci BS, Fener M. Sawability prediction of carbonate rocks from shear strength parameters using artificial neural networks. Int J Rock Mech Min 2006;43:157–64.
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Prediction of the moment capacity of reinforced concrete slabs in fire using artificial neural networks Hakan Erdem * Department of Civil Engineering, Faculty of Engineering and Architecture, Nigde University, Nigde 51245, Turkey
a r t i c l e
i n f o
Article history: Received 20 May 2009 Received in revised form 7 July 2009 Accepted 17 July 2009 Available online 14 August 2009 Keywords: Fire Reinforced concrete Slab Ultimate moment capacity Artificial neural networks
a b s t r a c t In this study, the application of artificial neural networks (ANN) to predict the ultimate moment capacity of reinforced concrete (RC) slabs in fire is investigated. An ANN model is built, trained and tested using 294 data for slabs exposed to fire. The data used in the ANN model consists of seven input parameters, which are the distance from the extreme fiber in tension to the centroid of the steel on the tension side of the slab (d0 ), the effective depth (d), the ratio of previous parameters (d0 /d), the area of reinforcement on the tension face of the slab (As), the fire exposure time (t), the compressive strength of the concrete (fcd), and the yield strength of the reinforcement (fyd). It is shown that ANN model predicts the ultimate moment capacity (Mu) of RC slabs in fire with high degree of accuracy within the range of input parameters considered. The moment capacities predicted by ANN are in line with the results provided by the ultimate moment capacity equation. These results are important as ANN model alleviates the problem of computational complexity in determining Mu. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Slabs are affected from fire as all other reinforced concrete (RC) members which are exposed to fire. As material strength of RC decreases with high temperature, the moment capacity (Mu) of RC slab decreases. In order to calculate the moment capacity of RC slab exposed to fire, the parameters such as the temperature increase depending on the fire exposure time, the temperature-distribution inside slab, the cross section and material properties, the tension and compression forces in slices, the depth equivalent rectangular stress block and the balance of the internal forces must be determined. As it is known, a large number of calculations are required to obtain these parameters. In order to avoid such computational complexity ANN approach may be used to predict effectively those parameters. However, to the best of author’s knowledge there is no research reported in literature on modeling of moment capacity using ANN method yet. A neural network model is a computer model, whose architecture essentially mimics the learning capability of the human brain. ANN is a technique which can be applied to complex problems described with a large amount of data. It does not require knowledge of the physical processes involved. However, it identifies the relationships in a set of data. Therefore ANN approach may be applied * Tel.: +90 388 2252306; fax: +90 388 2250112. E-mail addresses: [email protected], [email protected] 0965-9978/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2009.07.006
to problems where conventional mathematical solutions are not applicable. In recent years, ANN method has been used in variety of subjects in the structural engineering by many researchers. Some of these subjects are about prediction of the various properties of concrete [1–3], the properties of concrete aggregate [4], the strength of cements produced with pozzolans [5], the modeling of compressive strength of cement mortar [6], the rebar corrosion damage in RC structures [7], the performance of fiber reinforced RC beams and slabs [8–10], the moment and shear capacity [11], the shear strength [12–14], and the moment capacity and area of reinforcement [15] of RC beams, the ultimate deformation capacity of RC columns [16], the optimum cost [17] and the optimum depth [18] of RC slabs etc. Al-Khaleefi et al. [19] represented a functional relationship, using ANN method, between the fire resistance of a concrete filled steel column and the fire resistance parameters. However, there is no intensive research in determining the moment capacity of the RC slab using ANN yet. In this study, a predictive model for the ultimate moment capacity of the RC slab in fire was developed by using ANN. The calculated moment capacities of the RC slabs in fire were compared with the moment capacities predicted by the ANN model to verify reliable application of the artificial neural network model. This research explores the applicability of using ANN to create an intelligent model for prediction of the ultimate moment capacity of RC slabs in fire.
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
2. Ultimate moment capacity of RC slabs in fire
Q¼
In order to calculate the moment capacity of RC slab exposed to fire, the rising temperature by the fire exposure time, the temperature-distribution inside slab and the decrease in material strengths must be determined. After they are obtained, the tension and compression forces inside slab section are examined. The moment when the forces are balanced is the ultimate moment capacity of RC slab in fire. In the following subsections, these subjects are introduced. 2.1. The ISO834 temperature–time curve In this study, ISO834 is used to define fire time–temperature relation [20]. The equation for the ISO834 temperature–time curve is as follows:
T ¼ 345 log10 ð8t þ 1Þ þ T a
ð CÞ
ð1Þ
where t is the fire exposure time and Ta is the ambient temperature (°C). 2.2. Concrete strength at high temperature Reduction in concrete compressive strength at high temperatures has to be taken into consideration. The local concrete compressive strength rcT can be calculated using the temperature at each position by the relationship given in Eurocode2 [21]:
rcT ¼ kc rc20 C kc ¼ 1 for T 6 100 kc ¼ ð1:067 0:00067 TÞ for 100 6 T 6 400
ð4Þ
where Rtotal is the total thermal resistance, expressed as
Rtotal ¼ Rconv ;1 þ Rslab þ Rconv ;2 1 L1 1 ¼ þ þ h1 A kA h2 A
ð5Þ
where k is thermal conductivity coefficient, h1 and h2 the convection heat transfer coefficients, A the surface area and h is the thickness of slab. Once Q is known, T1 and T2 can be calculated. An unknown temperature T(y) inside slab at any slice can be determined as:
TðyÞ ¼ T 2 þ ðT 1 T 2 Þ
y h
ð6Þ
where T1 is temperature in surface exposed to fire and T2 is temperature in surface unexposed to fire. 2.5. Moment capacity of RC slab exposed to fire Once the temperature variations are known, the effects of temperature on the material properties and moment capacity of the slab can be examined. As can be understood from the reduction coefficients given in previous section, rising temperature results in both the corruption of the material properties and decrease in the residual moment capacity of the slab. Temperature distribution in the slab must be known for obtaining the change of material properties inside the slab. The cross section of RC slab is divided into M slices and for each slice the material temperature, reduction factors and mechanical characteristics are specified [23]. The tension force in the steel rebars can be derived by (Fig. 2):
ð2Þ Fs ¼
kc ¼ ð1:44 0:0016 TÞ for 400 6 T 6 900
T 11 T 12 Rtotal
271
M X
ksi fyd Asi
ð7Þ
i¼1
kc ¼ 0 for 900 6 T where rc20 °C and rcT are the concrete compressive strengths at 20 °C and the rising temperature °C, respectively, kc is the temperature reduction factor for the compressive strength and T is temperature.
The compressive force for unit width in the concrete can be calculated by summing all the compressive forces on the compressive side of lumped units. a
F c ¼ 0:85
Dy X
kci fcd Dy
ð8Þ
i¼1
2.3. Steel tensile strength at high temperature The values of reduced ultimate tensile strength of rebars due to temperature can be obtained from the following equations [21]:
fsuT ¼ ks fsu20 C ks ¼ 1 for 0 6 T 6 350 ks ¼ 1:899 0:00257T for 350 6 T 6 700
If the forces of the cross section are in static equilibrium, Eqs. (7) and (8) should be equal. If not, the value of a is increased progressively, and the calculation is repeated. The process continues until Eqs. (7) and (8) are equal. When the forces in slab are in equilibrium, the residual moment capacity of the slab Mu can be calculated as: a
ð3Þ
ks ¼ 0:24 0:0002T for 700 6 T 6 1200
Mu ¼ 0:85
Dy X
i¼1
Dy kci fcd Dy d iDy 2
ð9Þ
where fsu20 C and fsuT are the ultimate tensile strengths of rebars at 20 °C and the rising temperature °C, respectively, and ks is the temperature reduction factor for the tensile strength.
where fcd is the compressive strength of the concrete at temperature 20 °C, fyd is the yield strength of the reinforcement at temperature 20 °C, Fc and Fs are compressive and tensile forces in slab, respectively, kci and ksi are the reduction factors for each segment of the material temperature in slab.
2.4. Steady heat conduction in plane slab
3. Artificial neural network
Heat transfer through the slab can be modeled as steady and one-dimensional. The thermal resistance concept can be used to determine the rate of steady heat transfer through slabs. The rate of steady heat transfer through slab can be expressed using two surfaces at known temperatures by the total thermal resistance as shown in Fig. 1 [22–23].
Artificial neural networks are a computational tool that attempts to simulate the architecture and internal features of the human brain and nervous system. ANNs are consisting of a large number of simple processing elements called as neurons. Artificial neurons connected together form a network. The structure of artificial neural networks is layered. These are input, hidden and out-
ks ¼ 0 for 1200 6 T
272
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
Fig. 1. The thermal resistance network for heat transfer through a plane slab subjected to convection on both sides.
Fig. 2. Temperatures and forces in plane RC slab.
put layers. The back-propagation (BP) is one of the most popular learning algorithms. The back-propagation learns by comparing its output of each input pattern with a target output of that pattern, then calculating the error and propagating an error function backward through the net. To run the network after it is trained, the values for the input parameters are presented to the network. The network then calculates the node outputs by using the existing weight values and thresholds developed in the training process. If the neural network correctly determines the training data and correctly identifies the testing data, it is considered as trained. Usually a large amount of reliable data is required for training of ANN model. The mean square error (MSE) and the correlation coefficient (R) can be used to test the accuracy of the trained network. Each layer in ANNs has a number of processing neurons and each neuron is fully interconnected with weighted connections to units in the subsequent layer. A multi-layered perception (MLP) transforms i inputs into k outputs through non-linear mapping functions. The output of a neuron is determined by the activation of the neurons in the previous layer as follows [24]:
xo ¼ f
X
! xh who
training process, the MLP starts with a random set of initial weights and then training continues until a set of wih and who are optimized, so that a predefined error threshold is met between the output of network xo and the corresponding target value to. According to the BP algorithm, each interconnection between the neurons is adjusted by the amount of the weight update value as follows:
@E ¼ gdo xh @who @E Dwih ¼ g ¼ gdh xi @wih
Dwho ¼ g
do ¼ x0o ðto xo Þ X do who do ¼ x0h
N X L 1X 2 ðt ðsÞ xðsÞ o Þ 2 s o o
ð10Þ
ð11Þ
ð12Þ
where N is the number of data set patterns and L is the number of output neurons. The aim is to reduce the error by adjusting the interconnections between layers. The weights are adjusted using a gradient descent BP algorithm. The algorithm requires training data that consist of a set of corresponding input and target pattern values to. During the
ð16Þ
where x0o ¼ xo ð1 xo Þ and x0h ¼ xh ð1 xh Þ when a sigmoid activation function is used.
Based on the differences between the calculated output xo and the target value to an error is defined as follows:
E¼
ð15Þ
o
P where f( h xh who ) is the activation function, xh the activation of the hth neuron in the previous layer and who is the interconnection between hth layer neuron and oth layer neuron. The most widely used activation function is the sigmoid and it is given as follows:
1 P 1 þ expð xh who Þ
ð14Þ
where E is the error function given in Eq. (11), g learning rate, and
h
xo ¼
ð13Þ
Fig. 3. The architecture of the system on back-propagation networks.
273
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
Moment capacity, kNm
60
Table 1 The statistical values of ANN model.
training validation testing
50 Training Validation Testing
40 30
Data number
MSE
R
206 44 44
1.21326E4 1.42537E4 1.70820E4
0.99775 0.99795 0.99750
20 10 0 25
50
75
100 125 150 175 200 225 250 275 300
Number of data Fig. 4. Training, validation, and testing data used for prediction of the ultimate moment capacity.
Mean Squared Error (MSE)
1 training validation testing
0.1
0.01
0.001
0.0001 0
2
4
6
8
10
12
14
16
18
20
Epochs Fig. 5. Performance of ANN for training, testing, and validation processes.
4. ANN model used for the prediction of moment capacity
60
Predicted moment capacity, kNm
Predicted moment capacity, kNm
The objective of this work is to develop a neural network model for prediction of the ultimate moment capacity of slabs in fire. The data are obtained for different fire time, material and cross section
ANN training y = 0,9922x + 0,0526 R = 0,99775
50 40 30 20 10 0 0
10
20
30
40
50
60
40 30 20 10 0 0
Predicted moment capacity, kNm
Calculated moment capacity, kNm 60
ANN testing y = 0,9867x + 0,1715 R = 0,99750
50 40 30 20 10 0 0
10
20
30
40
50
60
Calculated moment capacity, kNm
ANN validation y = 1,0253x - 0,4406 R = 0,99795
50
60
10
20
30
40
50
60
Calculated moment capacity, kNm
Predicted moment capacity, kNm
0
properties. Temperature, concrete and steel material properties in each slice are obtained with slicing. Forces in all slices are determined by using the deteriorated properties of material. Heat transfer through slab is modeled as steady and one-dimensional. Heat transmission inside RC slab is calculated using a simulation of electrical flow. The ultimate moment capacity of RC slab is also obtained from tensile and compression forces equilibrium. The neural network developed in the investigation has seven neurons in the input layer, eight neurons in the hidden layer and one neuron in the output layer. The input parameters are the distance from the extreme fiber to the centroid of the steel on the tension side of the slab (d0 ), the effective depth (d), the ratio of previous parameters (d0 /d), the area of reinforcement on the tension side of the slab (As), the fire exposure time (t), the compressive strength of concrete (fcd), and the yield strength of the reinforcement (fyd). In this study, the Levenberg–Marquardt network (LM) is chosen as learning algorithm. A architecture of the artificial neuron is shown in Fig. 3. Using the process given in previous section, the data are obtained for prediction of the ultimate moment capacity of RC slabs exposed to fire with ANN. k = 1.30 W/m K and h1 = h2 = 10 W/ m2 K values are taken as constant during process obtained data. The values in the input layer are taken as d0 (30–40 mm), d (110– 130 mm), d0 /d (0.23,0.27 and 0.36), As (678.58, 923.63 and1231.50 mm2), t (0, 5, 10,20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min), fcd (20, 25 and 30 N/mm2), and fyd (220 and 420 N/mm2). For the presented problem, the ANN models are developed using the Matlab software. The 294 data were used in the analyses and they were divided into data for training (70% of the total data), validation (15% of the total data) and testing (15% of the total data). The values selected randomly for three stages
60
ANN all input y = 0,9964x - 0,0084 R = 0,99765
50 40 30 20 10 0 0
10
20
30
40
50
60
Calculated moment capacity, kNm
Fig. 6. Comparison of calculated and predicted moment capacity.
274
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
70 60
Moment capacity, kNm
5. Analysis results
target predicted
In this study, the ANN model has been developed to predict of the moment capacity of RC slabs in fire using the 294 data. The variation of Mean Squared Error (MSE) against the number of epochs for LM network training, validation and testing stages are shown in Fig. 5. It is evident that the MSE decreases rapidly with the increasing number of epochs. The result in terms of MSE here is reasonable, because the test set error and the validation set error have similar characteristics, and it does not appear that any significant overfitting has occurred. The relationship between calculated moment capacity using the developed program by author and predicted moment capacity by the ANN method is illustrated in Fig. 6. The training of ANN is completed with a high correlation coefficient of 0.99775. In the testing stage, the correlation coefficient is obtained as 0.99750. It is clear to point out from the performance and generalization capacity of ANN that the proposed model is extremely suitable in determining the moment capacity. Table 1 shows the correlation coefficient (R) and the mean squared error (MSE) for the obtained results. Comparison of the calculated moment capacity with the predicted moment capacity is made and shown in Fig. 7. In addition, the values analyzed in terms of differential between the calculated and predicted moment capacities and they are shown in Fig. 8. It has been observed from Figs. 7 and 8 that the ANN model is able to predict the moment capacity. Hence, the results of ANN can be directly used for the prediction of the moment capacity of slabs. For the first 28 data in the present problem, the values of the calculated moment capacity and the predicted moment capacity with ANN method is given Table 2. The results indicate that neural network was successful in learning and testing.
50 40 30 20 10 0 0
25
50
75
100
125
150
175
200
225
250
275
300
Number of data Fig. 7. The calculated and predicted moment capacity.
4 3
Error, kNm
2 1 0 0
25
50
75
100 125 150 175 200 225 250 275 300
-1 -2 -3
Number of data Fig. 8. The differential moment capacity value distribution in the analyses.
5.1. Effect of effective slab depth on the moment capacity are shown in Fig. 4. They were normalized by maximum values of parameters. The training patterns were randomly input into the network to train it. In this way, the neural network was applied to identify the moment capacity of a given data.
The ultimate moment capacities for the different effective slab depths are examined by means of ANN method. In calculations, the input parameters are taken as d = 110, 120, and 130 mm,
Table 2 The calculated and predicted moment capacity. Data
d0
d
d0 /d
As (mm2)
t (min)
fcd (MPa)
fyd (MPa)
Mr (kNm)
Mpredicted (kNm)
Mpredicted–Mr
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30
110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110
0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27
678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 678.58 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63 923.63
0 5 10 20 30 40 50 60 70 80 90 100 110 120 0 5 10 20 30 40 50 60 70 80 90 100 110 120
25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25
420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420
29.44 28.91 24.91 20.16 17.37 15.39 14.18 12.89 11.8 10.86 10.02 9.28 8.61 7.99 39.13 38.4 33.1 26.79 23.64 20.95 18.85 17.14 15.69 14.44 13.65 12.63 11.71 10.88
31.05 27.95 25.25 20.92 17.70 15.39 13.83 12.76 11.86 10.95 10.05 9.21 8.47 7.83 40.46 36.70 33.39 28.00 23.97 21.00 18.89 17.37 16.09 14.84 13.61 12.45 11.40 10.46
1.61 0.96 0.34 0.76 0.33 0.00 0.35 0.13 0.06 0.09 0.03 0.07 0.14 0.16 1.33 1.70 0.29 1.21 0.33 0.05 0.04 0.23 0.40 0.40 0.04 0.18 0.31 0.42
275
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
70
Moment capacity (kNm)
d0 = 30 mm, As = 1231.5 mm2, fcd = 30 MPa, fyd = 420 MPa and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. The results for the different d are shown in Fig. 9. While the effect of effective slab depth on the moment capacity exists in short fire time, the difference between the moment capacities for different effective slab depths decrease with increasing fire time rapidly. In addition, as seen in Fig. 9, the effect of slab dept does not exist for the fire exposure time more than 70 min. This reason is that the slab thickness has not enough to prevent temperature-distribution inside cross section. The generation of high temperature inside slab apparently decreases the compressive strength of the concrete.
fyd=220 Mpa fyd=420 Mpa
60 50 40 30 20 10 0 0
20
40
60
80
100
120
Fire exposure time (minute)
5.2. Effect of different rebar areas on the moment capacity
Fig. 11. The effect of the different rebar strengths on the moment capacity.
70 fcd=20 Mpa fcd=25 Mpa fcd=30 Mpa
60
Moment capacity (kNm)
In order to determine the ultimate moment capacities for the different rebar areas the ANN analyses are performed. The analyses results are shown in Fig. 10. In calculations, the input parameters are taken as d = 130 mm, d0 = 30 mm, As = 700, 900 and 1100 mm2, fcd = 30 MPa, fyd = 420 MPa, and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. As shown in Fig. 10, while effect of rebar areas on the moment capacity exists in short fire time, the difference between moment capacities for the different rebar areas decreases with the increasing fire time slowly. Although the effect of rebar decrease with rising fire exposure time, it does not vary for the fire exposure time more than 60 min.
50 40 30 20 10 0 0
5.3. Effect of different rebar strengths on the moment capacity
20
40
60
80
100
120
Fire exposure time (minute)
The ultimate moment capacities for the different yield strengths of the reinforcement are determined and are shown in Fig. 11. In calculations, the input parameters are taken as d = 130 mm,
d0 = 30 mm, As = 1231.5 mm2, fcd = 30 MPa, fyd = 220 and 420 MPa, and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. As shown in Fig. 11, while the effect of yield strengths of the reinforcement on the moment capacity exists in short fire time, the difference between the moment capacities for the different yield strengths of the reinforcement decreases with increasing fire time. The increase of fire exposure time causes the increase in temperature of rebar, and thus the material strength of steel decreases.
Moment capacity (kNm)
70 d=110mm d=120mm d=130mm
60
Fig. 12. The effect of the different concrete strengths on the moment capacity.
50 40 30 20
5.4. Effect of different concrete strengths on the moment capacity
10 0 0
20
40
60
80
100
120
Fire exposure time (minute) Fig. 9. The effect of the effective slab depth on the moment capacity.
Moment capacity (kNm)
70 As=700mm^2 As=900mm^2 As=1100mm^2
60 50 40 30 20
The ultimate moment capacities for the different compressive strengths of concrete are examined and shown in Fig. 12. In calculations, the input parameters are taken as d = 130 mm, d0 = 30 mm, As = 1231.5 mm2, fcd = 20, 25 and 30 MPa, fyd = 220 and 420 MPa, and t = 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 min. As shown in Fig. 12, while the effect of compressive strengths of concrete on the moment capacity exists in short fire time, the difference among the moment capacities for the different compressive strengths of concrete decreases with the increasing fire time, and is negligible. The variation in concrete strength does not significantly affect the moment capacity, in contrast to the variation in steel yield strength. Because the compressive strength of concrete is lower than steel. 6. Conclusions
10 0 0
20
40
60
80
100
120
Fire exposure time (minute) Fig. 10. The effect of the different rebar areas on the moment capacity.
The ultimate moment capacity of RC slabs exposed to fire for the different slab effective depths, the areas of reinforcement, the compressive strengths of the concrete and the yield strengths of the reinforcement is examined and predicted by using ANN method and the effect of these parameters are investigated. The
276
H. Erdem / Advances in Engineering Software 41 (2010) 270–276
calculated 294 data by author are used for this study. The correlation coefficient is obtained as 99.775% for training stage and 99.750% for testing stage. These values indicate that the proposed ANN model is highly successful. The test results also show that the generalization ability of ANN is well. It is seen that the increase of the slab effective depth is not enough to prevent rising temperature inside slab, and thus the constructive effect of the slab effective depth on the moment capacity with rising temperature vanish. Although the constructive effect of the area of the reinforcement and the yield strengths of the reinforcement on the moment capacity decrease with the fire exposure time, this effect goes on. The effect of the compressive strengths of the concrete on the moment capacity is also very little and is vanishing with the increase of the fire exposure time completely. Without calculating several computations in calculation stage, the trained ANN model can be used to provide the moment capacity with high accuracy. References [1] Yeh IC. Modeling slump flow of concrete using second-order regressions and artificial neural networks. Cem Concr Compos 2007;29:474–80. [2] Karahan O, Tanyıldızı H, Atısß CD. An artificial neural network approach for prediction of long-term strength properties of steel fiber reinforced concrete containing fly ash. J Zhejiang Univ Sci A 2008;9(11):1514–23. [3] Hola J, Schabowicz K. Application of artificial neural networks to determine concrete compressive strength based on non-destructive tests. J Civ Eng Manage 2005;1:23–32. _ [4] Topçu IB, Sarıdemir M. Prediction of properties of waste AAC aggregate concrete using artificial neural network. Comput Mater Sci 2007;41:117–25. _ [5] Topçu IB, Karakurt C, Sarıdemir M. Predicting the strength development of cements produced with different pozzolans by neural network and fuzzy logic. Mater Des 2008;29:1986–91. [6] Akkurt S, Özdemir S, Tayfur G, Akyol B. The use of GA-ANNs in the modelling of compressive strength of cement mortar. Cem Concr Res 2003;33:973–9. [7] Ukrainczyk N, Pecur IB, Bolf N. Evaluating rebar corrosion damage in RC structures exposed to marine environment using neural network. Civ Eng Environ Syst 2007;24(1):15–32.
[8] Flood I, Muszynski L, Nandy S. Rapid analysis of externally reinforced concrete beams using neural networks. Comput Struct 2001;79:1553–9. [9] Pannirselvam N, Raghunath PN, Suguna K. Neural network for performance of glass fibre reinforced polymer plated RC beams. Am J Eng Appl Sci 2008;1:83–8. [10] Bisby LA, Kodur VKR. Evaluating the fire endurance of concrete slabs reinforced with FRP bars: considerations for a holistic approach. Compos Part B: Eng 2007;38:547–58. [11] Rao HS, Babu BR. Hybrid neural network model for the design of beam subjected to bending and shear. Sadhana 2007;32(5):577–86. [12] Mansour MY, Dicleli M, Lee JY, Zhang J. Predicting the shear strength of reinforced concrete beams using artificial neural networks. Eng Struct 2004;26:781–99. [13] Ahn N, Jang H, Park DK. Presumption of shear strength of steel fiber reinforced concrete beam using artificial neural network model. J Appl Polym Sci 2007;103:2351–8. [14] Cladera A, Mari AR. Shear design procedure for reinforced normal and highstrength concrete beams using artificial neural networks. Part I: beams without stirrups. Eng Struct 2004;26:917–26. [15] Kelesßog˘lu Ö. The analyses of the reinforced concrete beam sections by artificial neural networks. Tech J Turk Chamber Civ Eng 2006:3935–42 [in Turkish]. [16] Inel M. Modeling ultimate deformation capacity of RC columns using artificial neural networks. Eng Struct 2007;29:329–35. [17] Ahmadkhanlou F, Adeli H. Optimum cost design of reinforced concrete slabs using neural dynamics model. Eng Appl Artif Intell 2005;18:65–72. [18] Rafıq MY, Bugmann G, Easterbrook DJ. Neural network design for engineering applications. Comput Struct 2001;79:1541–52. [19] Al-Khaleefi AM, Terro MJ, Alex PA, Wang Y. Prediction of fire resistance of concrete filled tubular steel columns using neural networks. Fire Safety J 2002;37:339–52. [20] ISO-834. Fire resistance tests-elements of building construction. Part 1– 9. Geneva: International Standards Organisation; 1975. [21] Eurocode 2: Design of concrete structures. ENV 1992. Part 1–2: General rulesstructural fire design. European Committee for Standardization, Brussels; 1995. [22] Çengel YH. Heat transfer: a practical approach. McGraw-Hill; 1998. p. 57–111. [23] Erdem H. Effect to reinforced concrete slab moment capacity of high temperature. In: The 30th anniversary of Çukurova University, engineering and architecture faculty symposium, Adana; 2008 [in Turkish]. [24] Kahraman S, Altun H, Tezekeci BS, Fener M. Sawability prediction of carbonate rocks from shear strength parameters using artificial neural networks. Int J Rock Mech Min 2006;43:157–64.