Holistic design of pre-tensioned concrete beams based on Artificial Intelligence

Figures & data

Figure 1. Bonded pre-tensioned beams with pin-pin ends for an AI-based design.

Table 1. Fifteen input parameters to generate big datasets using AutoPTbeam for pre-tensioned beams.

15 INPUTS
No	Symbol	Definition	No	Symbol	Definition
1	$b$	Beam width (mm)	9	${\varphi }_s$	Preferred rebar diameter (mm)
2	$h$	Beam depth (mm)	10	$\rho p$	Tendon ratio
3	$fck$	Concrete strength (MPa)	11	$fpy$	Yielding stress of tendon (MPa)
4	$fck$	Concrete strength at transfer stage (MPa)	12	$\varphi P$	Preferred tendon diameter (mm)
5	$\rho st$	Bottom rebar ratio	13	$PTlosslt$	Long-term prestress losses
6	$\rho sc$	Top rebar ratio	14	$L$	Beam length (mm)
7	$fsy$	Yielding stress of rebar (MPa)	15	$Kc$	Creep coefficient
8	$fsw$	Yielding stress of stirrup (MPa)

Figure 2. Three load stages of PT beams.

Table 2. 18 output of forward designs of a PT beam.

(a) Seven outputs at transfer load stages
No	Symbol	Definition
1	$\sigma c/fcki$	Top concrete stress per concrete strength at transfer stage
2	$\epsilon st$	Bottom rebar strain at transfer stage
3	$\epsilon sc$	Top rebar strain at transfer stage
4	$\epsilon p$	Tendon strain at transfer stage
5	$\sigma ct/fcki$	Bottom concrete stress per concrete strength at transfer stage
6	$cw$	Crack width at transfer stage
7	$\mathrm{\Delta }$	Defection at transfer stage
(b) Five output parameters at service load limit stage (Quasi, frequent, characteristic)
8	$qL/250$	Nominal strength when deflection reaches to L/250
9	$q0.2mm$	Nominal strength when crack width reaches to 0.2 mm
10	$q0.6fck$	Nominal strength when concrete stress reaches to stress limitation ($0.6f_{ck}$)
11	$q0.8fsu$	Nominal strength when rebar stress reaches to stress limitation ($min\left[0.8f_{su};f_{sy}\right])$
12	$q0.75fpu$	Nominal strength when tendon stress reaches to stress limitation ($min\left[0.75f_{pu};f_{py}\right]$)
(c) Six output parameters at ultimate load limit stage, considering both strength and ductility of member
13	$qstr$	Maximum nominal strength that a PT beam
14	$\epsilon st$	Bottom rebar strain at ultimate stage
15	$\epsilon sc$	Top rebar strain at ultimate stage
16	$\epsilon p$	Tendon strain at ultimate stage
17	$\rho sw$	Stirrup ratio
18	${\mu }_{\Delta }$	Deflection ductility

Figure 3. Histogram of 15 inputs based on 50.000 data.

Figure 4. Histogram of seven outputs of a transfer stage based on 50.000 data.

Figure 5. Histogram of five outputs of a service stage based on 50.000 data.

Figure 6. Histogram of six outputs of an ultimate stage based on 50.000 data (a) Algorithm for designing bonded pre-tensioned beams (AutoPTbeam) (b) Generation of large datasets.

Figure 7. Generation of large datasets of bonded pre-tensioned beams for AI-based design (Nguyen and Hong 2021)

Table 3. List of random variables and corresponding ranges of the big structural datasets.

(a) Big datasets for bonded pre-tensioned beams; non-normalized fifteen inputs
	$b\left(mm\right)$	$h\left(mm\right)$	$f_{ck}\hspace{0.17em}\left(MPa\right)$	$f_{cki\left(MPa\right)}$	${\rho }_{st}$	${\rho }_{sc}$	$f_{sy\left(MPa\right)}$	$f_{sw}\left(Mpa\right)$
Large data	555.27	837	45	33.43	0.0300	0.0225	447	329
	1034.07	646	44	26.11	0.0199	0.0214	524	341
	395.70	893	42	20.70	0.0329	0.0106	545	322
	2970.61	980	35	23.94	0.0148	0.0174	526	486
	…
MAX	3977.23	1000	50	40.00	0.0400	0.0400	600	500
MIN	203.58	500	30	12.00	0.0012	0.0005	400	300
MEAN	1645.86	749.84	40.04	23.97	0.0208	0.0203	499.46	400.11
STANDARD DEVIATION	857.44	144.67	6.07	5.92	0.0110	0.0113	57.99	58.27
VARIANCE	735210.02	20,931.70	36.89	35.03	0.0001	0.0001	3363.86	3396.45
	${\varphi }_s$	${\rho }_p$	$f_{py\left(MPa\right)}\hspace{0.17em}$	${\varphi }_p$	$PTlos{s}_{lt}$	$L\left(mm\right)$	$K_c$
Large data	29	0.0020	1693	20.2	0.11	14,224	3.28
	22	0.0012	1662	19.5	0.16	12,262	2.93
	14	0.0024	1600	10.7	0.14	9115	2.94
	25	0.0028	1615	18.2	0.14	11,109	2.88
	…
MAX	32	0.0032	1700	21.5	0.20	19,892	3.50
MIN	10	0.0006	1600	9.5	0.10	4058	2.50
MEAN	21	0.0018	1650.08	15.5	0.15	10,476	3.00
STANDARD DEVIATION	6.35	0.0006	28.98	3.4	0.029	3332	0.288
VARIANCE	40.41	4.7675	839.88	11.9	0.0008	11,106,931	0.083
(b) Big datasets for bonded pre-tensioned beams; non-normalized eighteen outputs
(b-1) Non-normalized seven outputs of a transfer stage
TRANSFER STAGE
	${\sigma }_c/f_{cki}$	${\epsilon }_{st}$	${\epsilon }_{sc}$	${\epsilon }_p$	${\sigma }_{ct}/f_{cki}$	$cw$	$\Delta \left(mm\right)$
Large data	−0.039	−0.00014	0.00002	0.006953	0.151	0.000	−3.74
	−0.025	−0.00010	0.00000	0.007001	0.103	0.000	−1.02
	−0.108	−0.00023	0.00005	0.006685	0.239	0.000	−3.42
	0.000	−0.00036	0.00014	0.006621	0.354	0.018	−7.44
	…
MAX	0.035	−0.00001	0.00105	0.007086	0.797	0.323	20.63
MIN	−0.129	−0.00089	−0.00003	0.006107	0.010	0.000	−32.59
MEAN	−0.044	−0.00020	0.00005	0.006862	0.204	0.005	−3.57
STANDARD DEVIATION	0.034	0.0001	6.6e-05	0.00012	0.107	0.013	3.249
VARIANCE	0.001	1.18e-08	4.36e-09	1.49e-08	0.0115	0.00017	10.55
(b-2) Non-normalized five outputs of a service stage
SERVICE STAGE
	$q_{L/250}\left(kN/m\right)\hspace{0.17em}$	$q_{0.2mm}\left(kN/m\right)$	$q_{0.6fck\left(kN/m\right)}$	$q_{0.8fsu}\left(kN/m\right)$	$q_{0.75fpu\left(kN/m\right)}$
Large data	49.17	106.10	95.44	161.74	81.18
	44.50	92.81	111.31	172.16	97.73
	164.26	258.75	179.56	396.24	253.55
	814.41	1004.05	963.57	1585.56	1015.59
	…
MAX	3219.04	4194.60	3589.44	6130.08	5236.87
MIN	2.45	2.65	8.28	7.11	2.87
MEAN	300.80	441.55	426.48	686.60	429.37
STANDARD DEVIATION	341.78	439.91	387.76	664.14	430.99
VARIANCE	116819.81	193,525.337	150,362.42	441,092.31	185,759.85
(b-3) Non-normalized six outputs of a ultimate stage
ULTIMATE STAGE
	$q_{str\left(kN/m\right)}$	${\epsilon }_{st}$	${\epsilon }_{sc}$	${\epsilon }_p$	${\rho }_{sw}$	${\mu }_{\Delta }$
Large data	178.02	0.0099	−0.00239	0.01641	0.0052	1.66
	188.93	0.0150	−0.00158	0.02127	0.0031	1.64
	208.93	0.0011	−0.00121	0.00703	0.0274	1.00
	1698.61	0.0154	−0.00220	0.02172	0.0034	1.87
	…
MAX	4622.31	0.0526	0.00105	0.05913	0.0334	10.26
MIN	7.80	0.0003	−0.00313	0.00598	0.0009	1.00
MEAN	595.59	0.0135	−0.00140	0.01971	0.0065	1.97
STANDARD DEVIATION	527.73	0.0113	0.00064	0.01137	0.0064	1.0037
VARIANCE	278503.34	0.000128	4.106e-07	0.000129	4.096e-05	1.007

Table

A	-	Target values in test datasets
P	-	Predicted values in test datasets
E	-	Errors in test procedures; $E=A-P$
n	-	Number of data in test datasets
${\mu }_A$	-	Mean value of target values in test datasets
${\sigma }_A$	-	Standard deviation of target values in test datasets
${\mu }_P$	-	Mean value of predicted values in test datasets
${\mu }_P$	-	Standard deviation of predicted values in test datasets

Figure 8. Design scenarios; one forward and one reverse designs.

Figure 9. Design accuracies; tendon ratio of 0.001 based on PTM.

Figure 10. Design accuracies; tendon ratio of 0.002 based on PTM.

Table 4. Training summary based on PTM for a forward design.

(a) Training on Output 16–22
FORWARD PROBLEM:
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – (7 + 6 + 5) Outputs:
(Transfer 7 Outputs: $\sigma c/fcki,\epsilon st,\epsilon sc,{\epsilon }_p,\sigma ct/fcki,cw,\mathrm{\Delta }$)
(Service Load 5 Outputs: $q_{L/250},q_{0.2mm},q_{0.6f_{ck}},q_{0.8f_{su}},q_{0.75f_{pu}}$)
(Ultimate Load 6 Outputs: $qstr,\epsilon st,\epsilon sc,\epsilon p,\rho sw,\mu \mathrm{\Delta }$)
No.			Data	Layers	Neurons	Validation check	Required Epoch	Best Epoch for Training	Stopped Epoch	Mean Square Error	R at Best Epoch	Comments
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Transfer$\sigma c/fcki$) – PTM
FW.PTM.	16	a	50,000	15	15	500	50,000	1,071	1,571	0.004782	0.944072	Insufficient
FW.PTM.	16	b	50,000	20	20	500	50,000	986	1,486	0.005467	0.940664	Insufficient
FW.PTM.	16	c	50,000	25	25	500	50,000	1,322	1,822	0.006158	0.937434	Insufficient
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Transfer$\epsilon st$) – PTM
FW.PTM.	17	a	50,000	15	15	500	50,000	8,397	8,897	3.18E-05	0.998924	Good
FW.PTM.	17	b	50,000	20	20	500	50,000	3,935	4,435	4.21E-05	0.998705	Good
FW.PTM.	17	c	50,000	25	25	500	50,000	12,330	12,830	3.90E-05	0.998705	Good
15 Inputs ($,h,fc,{ }_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Transfer$\epsilon sc$) – PTM
FW.PTM.	18	a	50,000	15	15	500	50,000	3,708	4,208	9.75E-05	0.983229	Neutral
FW.PTM.	18	b	50,000	20	20	500	50,000	6,081	5,581	1.34E-04	0.979253	Neutral
FW.PTM.	18	c	50,000	25	25	500	50,000	4,425	4,925	1.32E-04	0.982499	Neutral
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Transfer$\epsilon p$) – PTM
FW.PTM.	19	a	50,000	15	15	500	50,000	7,784	8,284	2.78E-05	0.999136	Good
FW.PTM.	19	b	50,000	20	20	500	50,000	10,014	10,514	4.16E-05	0.998751	Good
FW.PTM.	19	c	50,000	25	25	500	50,000	15,211	15,711	2.38E-05	0.999307	Good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Transfer$\sigma ct/fcki$) – PTM
FW.PTM.	20	a	50,000	15	15	500	50,000	5,531	6,031	4.75E-05	0.99888	Good
FW.PTM.	20	b	50,000	20	20	500	50,000	4,262	4,762	5.99E-05	0.998598	Good
FW.PTM.	20	c	50,000	25	25	500	50,000	14,267	14,767	5.57E-05	0.998716	Good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Transfer$cw$) – PTM
FW.PTM.	21	a	50,000	15	15	500	50,000	1,690	2,190	8.41E-05	0.966521	Neutral
FW.PTM.	21	b	50,000	20	20	500	50,000	4,241	4,741	1.06E-04	0.958348	Neutral
FW.PTM.	21	c	50,000	25	25	500	50,000	8,261	8,761	9.61E-05	0.962456	Neutral
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Transfer$\mathrm{\Delta }$) – PTM
FW.PTM.	22	a	50,000	15	15	500	50,000	6,270	6,770	9.37E-06	0.998423	Good
FW.PTM.	22	b	50,000	20	20	500	50,000	3,303	3,803	2.97E-05	0.993811	Good
FW.PTM.	22	c	50,000	25	25	500	50,000	4,559	5,059	5.53E-05	0.992654	Good
(b) Training on Output 23–27
FORWARD PROBLEM:
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – (7 + 6 + 5) Outputs:
(Transfer 7 Outputs: $\sigma c/fcki,\epsilon st,\epsilon sc,{\epsilon }_p,\sigma ct/fcki,cw,\mathrm{\Delta }$)
(Service Load 5 Outputs: $q_{L/250},q_{0.2mm},q_{0.6f_{ck}},q_{0.8f_{su}},q_{0.75f_{pu}}$)
(Ultimate Load 6 Outputs: $qstr,\epsilon st,\epsilon sc,\epsilon P,\rho sw,\mu \mathrm{\Delta }$)
No.			Data	Layers	Neurons	Validation check	Required Epoch	Best Epoch for Training	Stopped Epoch	Mean Square Error	R at Best Epoch	Comments
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Service$q_{L/250}$) – PTM
FW.PTM.	23	a	50,000	15	15	500	50,000	4,892	5,392	2.30E-05	0.998656	Good
FW.PTM.	23	b	50,000	20	20	500	50,000	10,102	10,602	2.61E-06	0.99986	Very good
FW.PTM.	23	c	50,000	25	25	500	50,000	14,343	14,843	2.32E-06	0.999879	Very good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTos{s}_{lt},L,K_c$) – 1 Output (Service$q_{0.2mm}$) – PTM
FW.PTM.	24	a	50,000	15	15	500	50,000	18,517	19,017	2.37E-07	0.999989	Very good
FW.PTM.	24	b	50,000	20	20	500	50,000	16,097	16,597	2.63E-06	0.999876	Very good
FW.PTM.	24	c	50,000	25	25	500	50,000	17,348	17,848	8.69E-07	0.999958	Very good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Service$q_{0.6f_{ck}}$) – PTM
FW.PTM.	25	a	50,000	15	15	500	50,000	15,311	15,811	7.29E-07	0.999,962	Very good
FW.PTM.	25	b	50,000	20	20	500	50,000	15,522	16,022	7.52E-07	0.999962	Very good
FW.PTM.	25	c	50,000	25	25	500	50,000	18,775	19,275	9.69E-07	0.999954	Very good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Service$q_{0.8f_{su}}$) – PTM
FW.PTM.	26	a	50,000	15	15	500	50,000	13,024	13,524	3.92E-06	0.999,809	Very good
FW.PTM.	26	b	50,000	20	20	500	50,000	5,021	5,521	2.11E-05	0.999004	Good
FW.PTM.	26	c	50,000	25	25	500	50,000	20,844	21,344	7.11E-06	0.999717	Very good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Service$q_{0.75f_{pu}}$) – PTM
FW.PTM.	27	a	50,000	15	15	500	50,000	4,490	4,990	1.85E-05	0.999126	Good
FW.PTM.	27	b	50,000	20	20	500	50,000	9,634	10,134	5.49E-06	0.999703	Very good
FW.PTM.	27	c	50,000	25	25	500	50,000	6,148	6,648	2.24E-05	0.999177	Good
(c) Training on Output 28–33
FORWARD PROBLEM:
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – (7 + 6 + 5) Outputs:
(Transfer 7 Outputs: $\sigma c/fcki,\epsilon st,\epsilon sc,{\epsilon }_p,\sigma ct/fcki,cw,\mathrm{\Delta }$)
(Service Load 5 Outputs: $q_{L/250},q_{0.2mm},q_{0.6f_{ck}},q_{0.8f_{su}},q_{0.75f_{pu}}$)
(Ultimate Load 6 Outputs: $qstr,\epsilon st,\epsilon sc,\epsilon P,\rho sw,\mu \mathrm{\Delta }$)
No.			Data	Layers	Neurons	Validation check	Required Epoch	Best Epoch for Training	Stopped Epoch	Mean Square Error	R at Best Epoch	Comments
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Ultimate$q_{str}$) – PTM
FW.PTM.	28	a	50,000	15	15	500	50,000	17,103	17,603	1.01E-05	0.99962	Good
FW.PTM.	28	b	50,000	20	20	500	50,000	8,466	8,966	1.36E-05	0.999481	Good
FW.PTM.	28	c	50,000	25	25	500	50,000	8,643	9,143	2.35E-05	0.99908	Good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Ultimate$\epsilon st$) – PTM
FW.PTM.	29	a	50,000	15	15	500	50,000	2,002	2,502	1.19E-04	0.999265	Neutral
FW.PTM.	29	b	50,000	20	20	500	50,000	4,198	4,698	8.08E-05	0.999451	Good
FW.PTM.	29	c	50,000	25	25	500	50,000	11,860	12,360	2.60E-05	0.999857	Good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Ultimate$\epsilon sc$) – PTM
FW.PTM.	30	a	50,000	15	15	500	50,000	5,690	6,190	1.00E-04	0.998833	Neutral
FW.PTM.	30	b	50,000	20	20	500	50,000	8,964	9,464	8.03E-05	0.999406	Good
FW.PTM.	30	c	50,000	25	25	500	50,000	10,401	10,901	4.00E-05	0.999559	Good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Ultimate${\epsilon }_p\hspace{0.17em}$) – PTM
FW.PTM.	31	a	50,000	15	15	500	50,000	7,733	8,233	2.79E-05	0.99979	Good
FW.PTM.	31	b	50,000	20	20	500	50,000	5,486	5,986	4.92E-05	0.999644	Good
FW.PTM.	31	c	50,000	25	25	500	50,000	8,574	9,074	6.39E-05	0.999697	Good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Ultimate$\rho sw$) – PTM
FW.PTM.	32	a	50,000	15	15	500	50,000	10,874	11,374	1.56E-05	0.999823	Good
FW.PTM.	32	b	50,000	20	20	500	50,000	12,317	12,817	1.76E-05	0.999865	Good
FW.PTM.	32	c	50,000	25	25	500	50,000	13,555	14,055	1.30E-05	0.999914	Good
15 Inputs ($b,h,fck,f_{cki},\rho st,\rho sc,f_{sy},f_{sw},{\varphi }_S,{\rho }_p,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c$) – 1 Output (Ultimate${\mu }_{\mathrm{\Delta }}$) – PTM
FW.PTM.	33	a	50,000	15	15	500	50,000	6,413	6,913	6.33E-05	0.998257	Good
FW.PTM.	33	b	50,000	20	20	500	50,000	9,294	9,794	4.27E-05	0.998717	Good
FW.PTM.	33	c	50,000	25	25	500	50,000	7,693	8,193	7.33E-05	0.99824	Good
FW.PTM.16a

Note: Three PTMs for each output:• PTM a: 15 layers, 15 neurons, 500 validation checks• PTM a: 20 layers, 20 neurons, 500 validation checks• PTM a: 25 layers, 25 neurons, 500 validation checks

Figure 11. Design charts corresponding to ρ_p.

Figure 12. Design results based on 20 layers-20 neurons for Step 1;q_L/250 = 80 kN/m at SLL (at a deflection of L/250), q_str = 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 1.50 at ULL.

Table 5. Training accuracies of a reverse scenario based on PTM; reverse network of Step 1.

REVERSE NETWORK OF STEP 1-REVERSE SCENARIO 15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – (4) Outputs ($b,\hspace{0.17em}\rho st,\rho sc,{\rho }_p)$
No.			Data	Layers	Neurons	Validation check	Required Epoch	Best Epoch for Training	Stopped Epoch	Mean Square Error	R at Best Epoch	Comments
15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – 1 Output ($b$) – PTM
R3.PTM.	1	a	50,000	20	20	500	50,000	31,407	31,907	0.001083	0.990338	Neural
R3.PTM.	1	b	50,000	30	30	1000	70,000	67,507	68,507	0.000714	0.994255	Neural
R3.PTM.	1	c	50,000	40	40	5000	100,000	55,324	60,324	0.000764	0.994578	Neural
15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – 1 Output ($\rho st$) – PTM
R3.PTM.	5	a	50,000	20	20	500	50,000	27,052	27,552	1.67E-03	0.989345	Insufficient
R3.PTM.	5	b	50,000	30	30	1000	70,000	47,114	48,114	0.001197	0.994218	Neural
R3.PTM.	5	c	50,000	40	40	5000	100,000	48,774	53,774	9.18E-04	0.995224	Neural
15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – 1 Output ($\rho sc$) – PTM
R3.PTM.	6	a	50,000	20	20	500	50,000	18,532	19,032	1.45E-02	0.911693	Insufficient
R3.PTM.	6	b	50,000	30	30	1000	70,000	26,751	27,751	0.013178	0.929903	Insufficient
R3.PTM.	6	c	50,000	40	40	5000	100,000	35,302	40,302	1.29E-02	0.9375	Insufficient
15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – 1 Output (${\rho }_p$) – PTM
R3.PTM.	10	a	50,000	20	20	500	50,000	18,887	19,387	7.82E-03	0.937977	Insufficient
R3.PTM.	10	b	50,000	30	30	1000	70,000	30,611	31,611	0.004052	0.976151	Neural
R3.PTM.	10	c	50,000	40	40	5000	100,000	37,338	42,338	3.41E-03	0.9796	Neural

Figure 13. Design results based on 20 layers-20 neurons for Step 1; q_L/250 = 80 kN/m at SLL (at a deflection of L/250), q_str = 150 kN/m at ULL, and deflection ductility ratio (μ_Δ) of 1.50 adjusted to 1.95 at ULL.

Figure 14. Design results based on 20 layers-20 neurons for Step 1; q_L/250 = 80 kN/m at SLL (at a deflection of L/250), q_str = 150 kN/m at ULL, and deflection ductility ratio (μ_Δ) of 2.00 and 2.50 at an ultimate load limit state (ULL).

Figure 15. Design results 30 layers-30 neurons for Step 1 neurons for Step 1; q_L/250 = 80 kN/m at SLL (at a deflection of L/250), q_str = 150 kN/m at ULL, deflection ductility ratio (μ_Δ) of 2.00 and 2.50 at an ultimate load limit state (ULL).

Figure 16. Design results 40 layers-40 neurons for Step 1 neurons for Step 1; q_L/250 = 80 kN/m at SLL (at a deflection of L/250), q_str = 150 kN/m at ULL, deflection ductility ratio (μ_Δ) of 2.00 and 2.50 at an ultimate load limit state (ULL).

Table 6. Training sequence of a reverse scenario based on CRS (DNN).

(a) Feature scores based on NCA
(b) Training accuracies of a reverse Scenario based on CRS (DNN)
REVERSE NETWORK OF STEP 1-REVERSE SCENARIO
15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – (4) Outputs ($b,\hspace{0.17em}\rho st,\rho sc,{\rho }_p)$
No.			Data	Layers	Neurons	Validation check	Required Epoch	Best Epoch for Training	Stopped Epoch	Mean Square Error	R at Best Epoch	Comments
15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – 1 Output ($b$) – PTM
R3.PTM.	1	a	50,000	20	20	500	50,000	31,407	31,907	1.1E-03	1.1E-03	0.99034
R3.PTM.	1	b	50,000	30	30	1000	70,000	67,507	68,507	7.1E-04	7.3E-04	0.99425
R3.PTM.	1	c	50,000	40	40	5000	100,000	55,324	60,324	7.6E-04	7.5E-04	0.99458
16 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }},b$) – 1 Output ($\rho st$) – CRS
R3.CRS.	5	a	50,000	20	20	500	50,000	24,914	25,414	1.7.E-04	1.7.E-04	0.99908
R3.CRS.	5	b	50,000	30	30	1000	100,000	40,601	41,601	9.9.E-05	1.1.E-04	0.99947
R3.CRS.	5	c	50,000	40	40	5000	100,000	51,497	56,497	7.3.E-05	9.1.E-05	0.99969
17 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }},b,\rho st$) – 1 Output ($\rho sc$) – CRS
R3.CRS.	10	a	50,000	20	20	500	50,000	26,261	26,761	2.3.E-04	2.3.E-04	0.99835
R3.CRS.	10	b	50,000	30	30	1000	100,000	35,490	36,490	1.2.E-04	1.2.E-04	0.99919
R3.CRS.	10	c	50,000	40	40	5000	100,000	99,850	100,000	3.2.E-05	3.2.E-05	0.99982
18 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }},b,\rho st,,\rho sc$) – 1 Output ($\rho p$) – CRS
R3.CRS.	6	a	50,000	20	20	500	50,000	5,096	5,596	2.4.E-03	2.2.E-03	0.98852
R3.CRS.	6	b	50,000	30	30	1000	100,000	7,630	8,630	2.5.E-03	2.2.E-03	0.98743
R3.CRS.	6	c	50,000	40	40	5000	100,000	7,887	12,887	2.5.E-03	3.0.E-03	0.98848
Note: R3.CRS.10a

Figure 17. Design accuracies of a reverse scenario based on CRS (DNN); q_L/250 = 80 kN/m at SLL (at a deflection of L/250), q_str = 150 kN/m at ULL, deflection ductility ratio (μ_Δ) of 2.00 at ULL; 20 layers-20 neurons for Step 1.

Figure 18. Design accuracies of a reverse scenario based on CRS (DNN); q_L/250 = 80 kN/m at SLL (at a deflection of L/250), q_str = 150 kN/m at ULL, deflection ductility ratio (μ_Δ) of 1.50 adjusted to 1.75 at ULL; 20 layers-20 neurons for Step 1.

Figure 19. Design accuracies of a reverse scenario based on CRS (DNN); q_L/250 = 80 kN/m at a deflection of L/250 at SLL, q_str = 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 2.00, 2.50 at ULL; 20 layers-20 neurons for Step 1.

Figure 20. Design accuracies of a reverse scenario based on CRS (DNN); q_L/250 = 80 kN/m at a deflection of L/250 at SLL, q_str = 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 2.00, 2.50 at ULL; 30 layers-30 neurons for Step 1.

Figure 21. Design accuracies of a reverse scenario based on CRS (DNN); q_L/250, 80 kN/m at a deflection of L/250 at SLL, q_str 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 2.00, 2.50 at ULL; 40 layers-40 neurons for Step 1.

Figure 22. Reciprocal design charts, errors indicated.

Figure 23. Design accuracies of Reverse Scenario based on CRS (DNN); qL/250 = 80 kN/m at a deflection of L/250 at SLL, qstr = 150 kN/m at ULL, and deflection ductility ratios (μΔ) of 2.75, 1.50 at ULL; 30 layers-30 neurons for Step 1.

Figure 24. Design accuracies of Reverse Scenario based on CRS (DNN); q_L/250 = 80 kN/m at a deflection of L/250 at SLL, q_str = 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 2.75, 1.50 at ULL; 40 layers-40 neurons for Step 1.

Figure 25. Design accuracies of Reverse Scenario based on CRS (DNN); q_L/250 = 80 kN/m at a deflection of L/250 at SLL, q_str = 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 2.50, 3.00 at ULL; 30 layers-30 neurons for Step 1.

Figure 26. Design accuracies of Reverse Scenario based on CRS (DNN); q_L/250 = 80 kN/m at a deflection of L/250 at SLL, q_str = 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 2.50, 3.00 at ULL; 40 layers-40 neurons for Step 1.

Figure 27. Design accuracies of Reverse Scenario based on CRS (DNN); q_L/250 = 80 kN/m at a deflection of L/250 at SLL, q_str = 150 kN/m at ULL, and deflection ductility ratios (μ_Δ) of 1.75, 2.00 at ULL; 40 layers-40 neurons and 30 layers-30 neurons for Step 1.

Table 7. Training accuracies of a reverse scenario based on deep and shallow ANNs; training accuracies obtained by mapping b based on PTM and ρ_st,ρ_sc, and ρ_p based on CRS.

REVERSE NETWORK OF STEP 1-REVERSE SCENARIO 3 15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – (4) Outputs ($b,\hspace{0.17em}\rho st,\rho sc,{\rho }_p)$
No.			Data	Layers	Neurons	Validation check	Required Epoch	Best Epoch for Training	Stopped Epoch	Mean Square Error	R at Best Epoch	Comments
15 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }}$) – 1 Output ($b$) – PTM
R3.PTM.	1	a	50,000	20	20	500	50,000	31,407	31,907	1.1E-03	1.1E-03	0.99034
R3.PTM.	1	b	50,000	30	30	1000	70,000	67,507	68,507	7.1E-04	7.3E-04	0.99425
R3.PTM.	1	c	50,000	40	40	5000	100,000	55,324	60,324	7.6E-04	7.5E-04	0.99458
R3.PTM.	1	d	50,000	1	20	500	100,000	23,555	24,055	3.2E-03	3.3E-03	0.96889
R3.PTM.	1	e	50,000	1	30	500	100,000	27,598	28,098	2.8E-03	2.8E-03	0.97271
R3.PTM.	1	f	50,000	1	40	500	100,000	34,205	34,705	2.8E-03	2.7E-03	0.9747
R3.PTM.	1	g	50,000	2	20	500	100,000	55,209	55,709	1.2E-03	1.2E-03	0.98866
R3.PTM.	1	h	50,000	2	30	500	100,000	23,228	23,728	1.2E-03	1.1E-03	0.98954
R3.PTM.	1	i	50,000	2	40	500	100,000	31,943	32,443	1.1E-03	1.2E-03	0.99055
16 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }},b$) – 1 Output ($\rho st$) – CRS
R3.CRS.	5	a	50,000	20	20	500	50,000	24,914	25,414	1.7.E-04	1.7.E-04	0.99908
R3.CRS.	5	b	50,000	30	30	1000	100,000	40,601	41,601	9.9.E-05	1.1.E-04	0.99947
R3.CRS.	5	c	50,000	40	40	5000	100,000	51,497	56,497	7.3.E-05	9.1.E-05	0.99969
R3.CRS.	5	d	50,000	1	20	500	100,000	50,946	51,446	4.1.E-04	3.9.E-04	0.99756
R3.CRS.	5	e	50,000	1	30	500	100,000	36,977	37,477	3.3.E-04	3.3.E-04	0.99809
R3.CRS.	5	f	50,000	1	40	500	100,000	26,860	27,360	2.9.E-04	3.0.E-04	0.99824
R3.CRS.	5	g	50,000	2	20	500	100,000	29,700	30,200	2.2.E-04	2.2.E-04	0.99875
R3.CRS.	5	h	50,000	2	30	500	100,000	14,328	14,828	1.9.E-04	1.9.E-04	0.99891
R3.CRS.	5	i	50,000	2	40	500	100,000	6,637	7,137	2.6.E-04	2.7.E-04	0.99855
17 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }},b,\rho st$) – 1 Output ($\rho sc$) – CRS
R3.CRS.	10	a	50,000	20	20	500	50,000	26,261	26,761	2.3.E-04	2.3.E-04	0.99835
R3.CRS.	10	b	50,000	30	30	1000	100,000	35,490	36,490	1.2.E-04	1.2.E-04	0.99919
R3.CRS.	10	c	50,000	40	40	5000	100,000	99,850	100,000	3.2.E-05	3.2.E-05	0.99982
R3.CRS.	10	d	50,000	1	20	500	100,000	43,054	43,554	3.1.E-04	3.3.E-04	0.99776
R3.CRS.	10	e	50,000	1	30	500	100,000	28,699	29,199	2.7.E-04	2.8.E-04	0.99806
R3.CRS.	10	f	50,000	1	40	500	100,000	54,119	54,619	1.4.E-04	1.5.E-04	0.99902
R3.CRS.	10	g	50,000	2	20	500	100,000	42,064	42,564	1.4.E-04	1.5.E-04	0.99901
R3.CRS.	10	h	50,000	2	30	500	100,000	21,752	22,252	1.1.E-04	1.3.E-04	0.99921
R3.CRS.	10	i	50,000	2	40	500	100,000	42,061	42,561	4.1.E-05	4.8.E-05	0.99972
18 Inputs ($h,fck,f_{cki},f_{sy},f_{sw},{\varphi }_S,f_{py},{\varphi }_P,PTlos{s}_{lt},L,K_c,q_{L/250},q_{0.2mm},qstr,{\mu }_{\mathrm{\Delta }},b,\rho st,,\rho sc$) – 1 Output ($\rho p$) – CRS
R3.CRS.	6	a	50,000	20	20	500	50,000	5,096	5,596	2.4.E-03	2.2.E-03	0.98852
R3.CRS.	6	b	50,000	30	30	1000	100,000	7,630	8,630	2.5.E-03	2.2.E-03	0.98743
R3.CRS.	6	c	50,000	40	40	5000	100,000	7,887	12,887	2.5.E-03	3.0.E-03	0.98848
R3.CRS.	6	d	50,000	1	20	500	50,000	14,455	14,955	3.1.E-03	3.0.E-03	0.982
R3.CRS.	6	e	50,000	1	30	500	100,000	58,525	59,025	1.3.E-03	1.5.E-03	0.99171
R3.CRS.	6	f	50,000	1	40	500	100,000	23,233	23,733	1.2.E-03	1.4.E-03	0.99306
R3.CRS.	6	g	50,000	2	20	500	50,000	2,527	3,027	2.7.E-03	2.8.E-03	0.98471
R3.CRS.	6	h	50,000	2	30	500	100,000	12,373	12,873	1.1.E-03	1.2.E-03	0.99442
R3.CRS.	6	i	50,000	2	40	500	100,000	3,959	4,459	2.5.E-03	2.5.E-03	0.99007

Figure 28. Design accuracies of Reverse Scenario 3 based on deep and shallow ANNs.

Nguyen, M. C., and W. K. Hong. 2021. “Analytical Prediction of Nonlinear Behaviors of Beams post-tensioned by Unbonded Tendons considering Shear Deformation and Tension Stiffening Effect.” Journal of Asian Architecture and Building Engineering 1–22. doi:10.1080/13467581.2021.1908299.

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