A prediction model for flexural strength of corroded prestressed concrete beam using artificial neural network

Figures & data

Table 1. Comparison of existing ANN models used in RC beam analysis

Scheme	Model	Input Size	Training,Testing Size (in %)	I-H-O Layers	I-H-O Neurons
Congro et al. (Congro et al., 2021)	^1BRNN	400	70,30	1-1-1	6–42-3
Fu and Feng (Fu & Feng, 2021)	GBRT	158	70,30	–	–
Murad et al. (Murad et al., 2021)	GEP	116	75,25	–	–
Hung et al. (Nguyen et al., 2021)	BPNN,BRNN	125	70,30	1-1-1	9–17–1
Cai et al. (Cai et al., 2020)	GA-BPNN	480	75,25	–	7-10-1
Abuodeh et al. (Abuodeh et al., 2020)	RBPNN	120	70,30	1-2-1	15–17,7-1
Alabduljabbar et al. (Alabduljabbar et al., 2020)	MBNN	107	90,10	1-1-1	6-12-1
Erdem (Erdem, 2010)	BPNN	294	70,30	1-1-1	7-8-1
Mansour et al. (Mansour et al., 2004)	MBNN	176	80,20	1-1-1	9-3-1

¹BRNN = Bayesian Regularization Neural Network

Table 2. Comparison of existing ANN models used in RC beam analysis (Continued)

Scheme	Epoch Count	Input Dataset	Normalization	Activation Function	Cost Function	Output Parameters
Congro et al. (Congro et al., 2021)	1000	literature	–	hyperbolic tangent, sigmoid	R^2 = 0.9372	three flexural strengths
Fu and Feng (Fu & Feng, 2021)	–	literature	–	—	R^2 = 0.955, RMSE = 19.19, MAE = 12.84	shear strength
Murad et al. (Murad et al., 2021)	–	literature	–	—	R^2 = 0.96, RMSE = 11.2%, MAE = 8.6%	flexural strength
Hung et al. (Nguyen et al., 2021)	1000	literature	–	sigmoid	R^2 = 0.9577, RMSE = 19.97, MAE = 14.98	shear strength
Cai et al. (Cai et al., 2020)	1000	ABAQUS	min-max	sigmoid	R^2 = 0.99617 RMSE = 4.7	post fire flexural capacity
Abuodeh et al. (Abuodeh et al., 2020)	–	–	min-max	sigmoid	R^2 = 0.885 RMSE = 8.1	shear strength
Alabduljabbar et al. (Alabduljabbar et al., 2020)	–	literature	–	sigmoid	R = 0.99934	compression strength, steel reinforcement area in corroded beam
Erdem (Erdem, 2010)	20	simulation	–	sigmoid	R ${ }^2$=0.9975, MSE = 0.00017	moment capacity of slab
Mansour et al. (Mansour et al., 2004)	45,000	ABAQUS	sigmoid	sigmoid	—	shear strength

R = correlation coefficient, R² = coefficient of determination

Figure 1. The workflow of the proposed methodology.

Table 3. The input datasets taken from various literatures (Belletti et al., 2020; Dai et al., 2020; Dai, Chen, Wang, Ma et al., 2021, 2021; Dai et al., 2016; ElBatanouny et al., 2015; Jeon et al., 2020; Liu & Fan, 2019; Lu et al., 2021; Menoufy & Soudki, 2014; Moawad, El-Karmoty et al., 2018; Moawad, Mahmoud et al., 2018; Papãľ & Melchers, 2011; Rinaldi et al., 2010; Saraswathy et al., 2017; Zeng et al., 2010; Zhang, Wang, Zhang, Liu et al., 2017)

$\mathit{B}$	$\mathit{D}$	${\mathit{A}}_{\mathit{c}}$	$\mathit{c}$	${\mathit{d}}_{\mathit{p}}$	$\mathit{c}/{\mathit{d}}_{\mathit{p}}$	$\mathit{L}$	${\mathit{D}}_{\mathit{p}}$	${\mathit{A}}_{\mathit{p}}$	${\mathit{f}}_{\mathit{pu}}$	${\mathit{f}}_{\mathit{pr}}$	${\mathit{A}}_{\mathit{s}}$	${\mathit{f}}_{\mathit{y}}$	${\mathit{f}}_{\mathit{ck}}$	${\eta }_{\mathit{corr}}$	${\mathit{P}}_{\mathit{u}}$	Δ
600	680	194,000	85	595	0.14	12,660	50	3926.9	1670	1252.5	240.3	450	34	7	112	–
600	680	194,000	85	595	0.14	12,660	50	3926.9	1670	1252.5	240.3	450	34	76.5	57	–
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	20	90.6	–
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	25	80.9	–
200	300	60,000	45	255	0.18	3000	12	279	1976	1300	314.15	450	34	20	115	–
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
150	300	45,000	62	238	0.26	2600	15.2	139	1950	1014	628.3	435	44	5.8	276.7	–
150	300	45,000	62	238	0.26	2600	15.2	139	1950	994.5	628.3	435	44	0	211.5	–
130	150	19,500	42	108	0.39	2000	15.2	139	1910	1395	157.07	400	43.9	9.41	48	–
130	150	19,500	42	108	0.39	2000	15.2	139	1910	0	157.07	400	41.6	10.45	28.3	–
$\mathit{B}$	$\mathit{D}$	${\mathit{A}}_{\mathit{c}}$	$\mathit{c}$	${\mathit{d}}_{\mathit{p}}$	$\mathit{c}/{\mathit{d}}_{\mathit{p}}$	$\mathit{L}$	${\mathit{D}}_{\mathit{p}}$	${\mathit{A}}_{\mathit{p}}$	${\mathit{f}}_{\mathit{pu}}$	${\mathit{f}}_{\mathit{pr}}$	${\mathit{A}}_{\mathit{s}}$	${\mathit{f}}_{\mathit{y}}$	${\mathit{f}}_{\mathit{ck}}$	${\eta }_{\mathit{corr}}$	${\mathit{P}}_{\mathit{u}}$	${\mathit{M}}_{\mathit{u}}$
150	220	33,000	52	168	0.3	2000	15.2	139	1830	1395	326.7	335	34.1	0	126	37.8
150	220	33,000	52	168	0.3	2000	15.2	139	1830	1395	326.7	335	32.4	12.1	110	33
130	150	19,500	42	108	0.39	2000	15.2	139	1910	0	157.07	400	41.6	10.45	28.3	8.4
130	150	19,500	42	108	0.39	2000	15.2	139	1910	943	157.07	400	43.4	12.48	33.5	10.2
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	0	65	37.8
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	2.7	55	33
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	6.2	52.5	31.5
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	9.1	48	28.8
$\mathit{B}$	$\mathit{D}$	${\mathit{A}}_{\mathit{c}}$	$\mathit{c}$	${\mathit{d}}_{\mathit{p}}$	$\mathit{c}/{\mathit{d}}_{\mathit{p}}$	$\mathit{L}$	${\mathit{D}}_{\mathit{p}}$	${\mathit{A}}_{\mathit{p}}$	${\mathit{f}}_{\mathit{pu}}$	${\mathit{f}}_{\mathit{pr}}$	${\mathit{A}}_{\mathit{s}}$	${\mathit{f}}_{\mathit{y}}$	${\mathit{f}}_{\mathit{ck}}$	${\eta }_{\mathit{corr}}$	${\mathit{P}}_{\mathit{u}}$	$\mathrm{\Delta }$
600	680	194,000	85	595	0.14	12,660	50	3926.9	1670	1252.5	240.3	450	34	44.5	78	52
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	0	120.6	4.8
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	15	98.5	6.3
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	25	80.9	5.2
200	300	60,000	45	255	0.18	3000	12	279	1976	1300	314.15	450	34	20	95	7
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
400	300	45,000	50	250	0.2	3600	12.7	95.7	1860	1302	100.53	435	40	10	48.3	33.34
130	150	19,500	42	108	0.39	2000	15.2	139	1910	465	157.07	400	43.5	7.47	42	22
130	150	19,500	42	108	0.39	2000	15.2	139	1910	1395	157.07	400	43.9	9.41	48	18
610	381	92,720	35	346	0.1	4980	13	200	1860	1264.8	314.15	435	40.7	6.3	101	86.1

Figure 2. Correlation plots of input parameters v/s output parameters.

Figure 3. Marginal histograms between corrosion, compressive strength and output parameters.

Table 4. The statistical measurements of input datasets

Parameter	Dataset1					Dataset2					Dataset3
	Min	Max	Mean	Median	SD	Min	Max	Mean	Median	SD	Min	Max	Mean	Median	SD
$\mathit{B}$	80	610	178.7	150	106.4	80	600	157.2	150	88.5	130	610	205.3	150	129.4
$\mathit{D}$	150	680	271.5	250	87.2	150	680	256.9	240	93	150	680	292.7	300	104.2
$A_c$	19,500	194,000	46,470.2	41,235	26,687.5	19,500	194,000	42,922.3	37,500	30,298.4	19,500	194,000	52,194.6	45,000	32,946.6
$\mathit{c}$	25	85	49.2	50	12.5	40	85	46.9	43	10	25	85	53.1	52	14.5
$d_p$	95	595	222.3	210	84.4	108	595	210	200	85.3	95	595	239.6	238	102.5
$\mathit{c}/d_p$	0.07	0.57	0.2	0.236	0.1	0.14	0.39	0.2	0.22	0.1	0.07	0.57	0.3	0.26	0.1
$\mathit{L}$	1000	12,660	2941.9	2500	1682.7	2000	12,660	2860	2500	1968.8	1000	12,660	3286.1	2600	2100.2
$D_p$	7	50	15	15.2	5.5	12.7	50	16	15.2	6.7	7	50	15.2	15.2	7
$A_p$	76.97	3926.9	231.5	139	552.5	95.7	3926.9	280.3	139	708.9	76.97	3926.9	292.7	139	707.5
$f_{\mathit{pu}}$	622.83	1976	1797.1	1910	308.8	622.83	1951.5	1743.2	1910	380.5	1470	1976	1853.7	1910	124.5
$f_{\mathit{pr}}$	0	1432.5	1136.2	1250	313.9	0	1408	1122.7	1250	314	0	1432.5	1147.4	1300	336.2
$A_s$	39.3	986	384.1	327	178.4	39.3	986	368.5	327	170.7	39.3	838.8	368.9	326.7	204.7
$f_y$	235	470	379.6	375	61.5	235	450	358.4	360	59.4	335	470	410.1	435	47.3
$f_{\mathit{ck}}$	31.8	87	42.4	43.5	8.6	31.8	55.7	41.9	44	7	31.8	87	42.4	41.7	9.1
${\eta }_{\mathit{corr}}$	0	100	15.9	7.47	22.1	0	100	16.7	7	24.8	0	100	16.6	9.41	20.8
$P_u$	16.4	276.7	99.6	93.4	54.5	16.5	276.7	82.4	79	45.2	16.4	276.7	118	111.36	59.8
$M_u$	—	—	—	—	—	8.4	490	54.7	42.03	71.3	—	—	—	—	—
$\mathrm{\Delta }$	—	—	—	—	—	—	—	—	—	—	4.66	120.7	31	20.2	28.6

SD = Standard Deviation

Table 5. The statistical measurements of input datasets (Continued …)

Parameter	Dataset1			Dataset2			Dataset3
	Variance	Skewness	Kurtosis	Variance	Skewness	Kurtosis	Variance	Skewness	Kurtosis
$\mathit{B}$	11,323.5	3.3	10.5	7833.4	4.6	21.0	15,453.0	2.5	5.1
$\mathit{D}$	7596.0	2.3	8.5	8646.9	3.4	3.6	10,849.7	1.7	4.9
$A_c$	712,223,310.3	3.8	18.3	917,993,658.1	4.5	6.0	1,085,480,806.9	2.9	10.4
$\mathit{c}$	156.6	0.7	0.4	99.5	2.2	2.1	208.8	0.0	−0.3
$d_p$	7116.3	2.0	6.5	7274.2	3.3	3.4	10,497.9	1.4	3.2
$\mathit{c}/d_p$	0.0	0.9	1.8	0.0	0.9	1.0	0.0	0.5	0.3
$\mathit{L}$	2,831,562.5	4.3	22.1	3,876,030.9	4.6	6.1	4,410,851.2	3.2	12.3
$D_p$	29.9	5.7	34.6	44.9	4.8	6.7	49.1	4.5	20.4
$A_p$	305,301.2	6.6	41.9	502,604.2	5.0	7.2	500,503.7	5.0	23.7
$f_{\mathit{pu}}$	95,368.4	−2.9	7.7	144,792.5	−2.2	−2.2	15,504.9	−1.8	2.5
$f_{\mathit{pr}}$	98,541.0	−1.5	1.5	98,603.8	−1.6	−1.6	113,000.0	−1.5	1.6
$A_s$	31,832.7	0.5	0.4	29,135.1	0.3	0.3	41,900.3	0.4	−0.5
$f_y$	3781.5	−0.6	0.0	3524.3	−0.6	−0.7	2240.0	−0.6	−1.0
$f_{\mathit{ck}}$	73.4	1.9	8.1	49.6	0.1	0.1	82.6	2.6	11.3
${\eta }_{\mathit{corr}}$	487.5	2.2	4.8	613.3	2.0	2.0	431.9	2.3	5.5
$P_u$	2970.6	1	0.8	2039.6	1.6	1.6	3578.0	0.6	−0.3
$M_u$	—	—	—	5088	4.8	4.3	—	—	–
$\mathrm{\Delta }$	—	—	—	—	—	–	820.6	1.7	2.1

Figure 4. The proposed RBPBTNN models for ultimate load, ultimate moment and deflection predictions.

Figure 5. The proposed RBPBTNN training performance comparison when K-fold and training repetitions are varied.

Table 6. Parametric details of the proposed ANN models during training and validation phases

Parameter	Details
	UltimateLoad	UltimateMoment	Deflection
Input neurons	15	16	16
Output neurons	1	1	1
Hidden layer count	2	2	2
Hidden neurons	13,13	15,15	16,16
Activation function	sigmoid	sigmoid	sigmoid
Training, testing data	155,30	91,20	91,20
RMSE	3.1373	2.1441	0.6795
R ${ }^2$	0.9967	0.9991	0.9995
K-fold	3	3	3
No. of repetition	10	10	10
No. of simulations	1	1	1

sigmoid $\Rightarrow \mathit{f}$ ($\mathit{z}$) = 1/(1 +$e^{-\mathit{z}}$)

Table 7. Performance comparison of the proposed RBPBTNN with other ANN models on testing data

Model	Cost	Ultimate	Ultimate	Deflection
	Function	Load	Moment
BPNN	R^2	0.9668	0.9760	0.8880
	RMSE	19.3695	15.0696	10.7425
	MAE	16.3099	8.7368	6.0040
	MAPE	19.22%	18.94%	32.86%
	NSE	0.8738	0.9523	0.7820
GCM-BPNN	R^2	0.9765	0.9466	0.9090
	RMSE	15.9672	24.8600	10.3240
	MAE	12.0127	9.2977	5.8680
	MAPE	15.82%	13.38%	23.31%
	NSE	0.9142	0.8712	0.7990
RBPWBTNN	R^2	0.9080	0.9272	0.8914
	RMSE	38.3460	30.4650	11.3150
	MAE	23.1595	9.3636	6.2180
	MAPE	39.45%	10.62%	24.10%
	NSE	0.5050	0.8066	0.7570
RBPBTNN	R^2	0.9860	0.9691	0.9180
	RMSE	13.2020	17.7800	10.3060
	MAE	10.5340	5.6900	5.8250
	MAPE	12.96%	7.79%	19.24%
	NSE	0.9414	0.9341	0.8060

GCM-BPNN = Globally Convergent Modification of BPNN

RBPWBTNN = Resilient Back Propogation Without Backtracking Neural Network

Figure 6. Parametric analysis of input parameters on ultimate load, ultimate moment and deflection predictions.

Figure 7. RMSE comparison of different ANN algorithms when K-fold is varied during training phase.

Figure 8. Linear fit of predicted ultimate load, ultimate moment and deflection.

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