Flexural performance tests and machine learning analysis of prestressed high strength steel reinforced concrete beams

Figures & data

Figure 1. Workflow of this study.

Table 1. Meanings of nouns.

Nouns	Meaning
PHSSRC	prestressed high strength steel reinforced concrete
ML	machine learning
SRC	steel reinforced concrete
PSRC	prestressed steel reinforced concrete
ANN	Artificial Neural Network
XGBoost	extreme gradient boosting
ECC	Engineered Cementitious Composites （ECC-LWC)
LWC	Lightweight Concrete
MAPE	mean absolute percentage error
RMSE	root mean squared error
R^2	coefficient of determination
BP	backpropagation
LM	Levenberg-Marquardt
SHAP	Shapley Additivity interpretation method

Figure 2. Cross-sectional reinforcement diagram.

Table 2. Parameter of specimens.

Specimen	Steel strength grade	Steel ratio (%)	Steel shape(mm) bf×h×t1×t2	Longitudinal reinforcement ratio (%)
B1Q3S3L1	Q355	3.03%	100×150×8×5	1.67%
B2Q5S3L1	Q550	3.03%	100×150×8×5	1.67%
B3Q6S2L1	Q690	2.27%	100×150×5×5	1.67%
B4Q6S3L1	Q690	3.03%	100×150×8×5	1.67%
B5Q6S3L2	Q690	3.03%	100×150×8×5	2.29%
B6Q6S3L0	Q690	3.03%	100×150×8×5	1.09%
B7Q6S4L1	Q690	4.4%	100×150×10×10	1.67%

Table 3. Mechanical properties of steel.

Grade	Thicknesses (Diameters)/mm	Yield strength fy/MPa	Ultimate strength fu/MPa	Elongation at break δ/%
Q355	5	397	527	42.9
Q355	8	391	540	22.6
Q550	5	626	713	26.5
Q550	8	633	756	19.4
Q690	5	740	819	22.6
Q690	8	787	845	17.7
Q690	10	690	766	21.7
Prestressed wire	15.2	1820	1910	20.5
HRB400	14	453	591	25.6
HRB400	18	415	586	28.7
HRB400	20	433	566	29.5
HPB300	8	358	537	23.9

Figure 3. Test device and instrumentation.

Figure 4. Arrangement of strain gauge and displacement meter.

Figure 5. Damage pattern of some test beams.

Table 4. Yield load, ultimate load of the test.

Specimen	$P_{\mathrm{y}}^{\hspace{0.17em}}$/kN	$P_{\mathrm{u}}^{\hspace{0.17em}}$/kN
B1Q3S3L1	140	190
B2Q5S3L1	170	223
B3Q6S2L1	160	210
B4Q6S3L1	190	240
B5Q6S3L2	240	264
B6Q6S3L0	200	224
B7Q6S4L1	240	277

Figure 6. Load-deflection curves for different beams.

Figure 7. Strain distribution along the depth of the steels.

Figure 8. Strain distribution along the depth of the beams.

Figure 9. Architecture of the ANN model.

Figure 10. Performance of models with different numbers of neurons.

Table 5. Optimized values of the hyperparameters for the XGBoost model.

Hyperparameter	Optimal value
Maximum number of estimators (CARTs)	100
Maximum depth of tree	5
Learning rate	0.1

Figure 11. Stress distribution diagram of beam section of JGJ369–2016.

Figure 12. Comparison of the predicted and experimental values of the ANN model.

Figure 13. Comparison of the predicted and experimental values of the XGBoost model.

Figure 14. Comparison of models and codes.

Figure 15. Shap values of input parameters, (a) each point (b) average value.

Figure 16. Interaction plot of factors.

Table A1. Details of all specimens

number	b (mm)	h (mm)	E (mm)	fp0 (kN)	fc (Mpa)	fa (Mpa)	fy (Mpa)	Ea (Mpa)	As (mm^2)	Aa ((mm^2)	Ap ((mm^2)	M (kN⋅m)	reference
1	210	280	0	270	24.396	275	347.2	27440	610	3910	280	152.04	(Wang, Wu, and Zheng 2009)
2	235	310	0	270	24.396	275	347.2	27440	610	4997.5	280	177.52
3	235	310	0	270	24.396	275	35.2	27440	864	4997.5	280	200.32
4	260	330	0	270	24.396	275	347.2	27440	610	6208	280	255.31
5	260	330	0	270	24.396	275	351.9	27440	1118	6208	280	278.11
6	200	300	0	0	26.5	611	200	35000	252	2328	0	147.2	(Li 2005)
7	200	300	0	0	26	611	200	35000	396	2328	0	168.4
8	200	300	0	0	24.1	611	200	35000	252	2328	0	147.6
9	200	300	0	0	26.4	611	200	35000	396	2328	0	166.4
10	200	300	0	0	3.8	600	300	35000	252	2892	0	175.6
11	200	300	0	0	31.4	600	300	35000	396	2892	0	195.6
12	200	300	0	0	31.7	600	300	35000	252	2892	0	176.8
13	200	300	0	0	31.4	600	300	35000	396	2892	0	192.8
14	200	300	0	0	32.2	300	300	35000	396	1728	0	130.4
15	200	300	0	0	32.4	815	300	35000	396	1728	0	214.8
16	200	300	0	0	31	600	300	35000	144	2892	0	156.4
17	200	300	0	0	3.6	611	300	35000	396	2328	0	160.8
18	200	300	0	0	34.1	600	300	35000	396	2892	0	181.6
19	200	300	0	0	35.6	605	300	35000	396	3456	0	219.6
20	200	300	0	0	37.1	600	300	35000	540	2892	0	186
21	200	300	0	0	3.9	300	300	35000	396	2892	0	146.4
22	200	300	0	0	36.4	500	300	35000	252	2892	0	184
23	200	300	0	0	37	840	300	35000	396	2892	0	256
24	200	300	0	107.8	34.3	284	434	35000	308	2580	98	124.8	(Tang, Zhou, and Zhang 2018)
25	200	300	0	154	32	284	434	35000	308	2580	140	132.6
26	200	300	0	215.6	35.9	284	434	35000	308	2580	196	147.6
27	200	300	0	261.8	32.1	284	434	35000	308	2580	238	151.2
28	200	300	0	308	35.4	284	434	35000	308	2580	280	166.2
29	220	400	40	0	31.17	325	365	35000	1531.2	2612	0	245	(J, M, and Z 2014)
30	220	400	40	262.5	36.55	325	365	35000	1531.2	2612	240	320
31	220	400	40	281.2	3.23	325	365	35000	1531.2	2612	240	324
32	220	320	0	0	35.05	325	365	35000	1070.1	2612	0	155
33	220	320	0	262.5	35.21	325	365	35000	1070.1	2612	240	205
34	220	320	0	281.2	31.1	325	365	35000	1070.1	2612	240	203
35	220	320	0	0	38.5	331	320	35000	1013.8	2612	0	151.9	(L et al. 2020)
36	220	320	0	22.3	38.5	331	320	35000	1013.8	2612	240	200.9
37	220	320	0	22.3	39.6	331	340	35000	1013.8	2612	240	199.9
38	220	400	40	0	39.93	331	320	35000	1531.2	2612	0	240.1
39	220	400	40	237.1	39.93	331	320	35000	1531.2	2612	240	313.6
40	220	400	40	237.1	39.6	331	340	35000	1531.2	2612	240	312.6
41	200	300	0	99.6	34.3	283.7	434	34500	615	2580	98	124.8	(T et al. 2022)
42	200	300	0	142.2	32	283.7	434	34500	615	2580	140	132.6
43	200	300	0	199.1	35.9	283.7	434	34500	615	2580	196	147.6
44	200	300	0	241.8	32.1	283.7	434	34500	615	2580	238	151.2
45	200	300	0	284.5	35.4	283.7	434	34500	615	2580	280	166.2
46	200	300	0	142.2	33.8	283.7	434	34500	615	2580	140	100.8
47	200	400	0	0	31.6	315	365	35000	628	1840	0	140	(Kabir et al. 2020)
48	200	400	0	0	31.6	315	365	35000	989	1840	0	158.8
49	200	400	0	0	31.6	315	365	35000	628	4240	0	185.2
50	200	400	0	0	31.6	315	365	35000	989	4240	0	227.5
51	200	260	0	0	26.8	235	400	32500	603	1789.2	0	75	(Zu-qiang, Beng-you, and Jian-yang 2023)
52	180	300	0	0	28.3	335.4	362.7	32500	314	2428	0	168	(Wakjira et al. 2022)
53	200	300	50	0	4.6	365	521	18200	452	1255.9	0	97	(Golafshani et al. 2012)
54	200	300	50	0	4.6	365	521	18200	452	1255.9	0	96
55	180	250	0	0	22.1	270	552	32000	452	2104	0	58.9	(Wakjira et al. 2022)
56	180	250	0	0	22.1	270	552	32000	452	2104	0	58.9
57	180	250	0	0	22.1	270	552	32000	452	2104	0	58.1
58	180	250	0	0	22.1	270	552	32000	452	2104	0	58.8
59	180	280	0	0	26.8	325	400	32500	628	2354	0	106.5	(Li et al. 2022)
60	200	200	0	0	2.3	272	320	30000	452	1297.4	0	47.18	(Wakjira et al. 2022)
61	200	200	0	0	2.8	272	320	30000	452	1297.4	0	47.32
62	200	250	25	0	2.3	272	320	30000	452	1297.4	0	63.35
63	200	250	25	0	24.2	272	320	30000	452	1297.4	0	66.5
64	200	300	50	0	2.3	272	320	30000	452	1297.4	0	96.6
65	200	300	50	0	25	272	320	30000	452	1297.4	0	103.14
66	200	300	0	0	24.3	272	320	30000	452	1297.4	0	75.78
67	200	300	0	0	2.3	272	320	30000	452	1297.4	0	71.91
68	200	350	75	0	2.3	272	320	30000	452	1297.4	0	119.88
69	200	350	75	0	27	272	320	30000	452	1297.4	0	133.2
70	200	350	25	0	2.3	272	320	30000	452	1297.4	0	93.78
71	200	350	25	0	23	272	320	30000	452	1297.4	0	96.3
72	250	300	0	28.6	37.1	400	430	35500	1250	2270	280	190	This paper
73	250	300	0	223.4	37.1	630	430	35500	1250	2270	280	223
74	250	300	0	268.3	37.1	740	430	35500	1250	1450	280	210
75	250	300	0	26.5	37.1	780	430	35500	1250	2270	280	240
76	250	300	0	282.9	37.1	780	430	35500	1718	2270	280	264
77	250	300	0	29.7	37.1	780	430	35500	817	2270	280	224
78	250	300	0	27.8	37.1	690	430	35500	1250	3300	280	277
79	200	300	0	249.2	86.18	254	410	48700	465	2125.9	280	157.3	(Wang, Wu, and Zheng 2009)
80	200	300	0	289.8	86.18	254	410	48700	465	2125.9	280	165.1
81	200	300	0	312.5	86.18	254	410	48700	465	2125.9	280	165.1
82	200	300	0	292.9	86.18	254	410	48700	666	2125.9	280	183.3
83	200	300	0	282.2	86.18	254	410	48700	920	2125.9	280	222.3
84	200	300	0	203.3	86.18	254	410	48700	920	2125.9	197	184
85	200	300	30	295.7	86.18	254	410	48700	465	2125.9	280	204.8
86	200	300	30	287	86.18	254	410	48700	666	2125.9	280	224.3
87	200	300	30	297.1	86.18	254	410	48700	920	2125.9	280	243.8
88	200	300	0	285.3	86.18	254	410	48700	465	2613.1	280	171.6
89	200	300	0	29.9	86.18	254	410	48700	465	2796.4	280	180.7
90	200	350	−5	0	37.21	272	365	35100	534	2612	0	115.35	(Wang, Shi, and Zhou 2019)
91	200	350	−25	0	37.21	272	365	35100	534	2612	0	108.61
92	200	350	−5	0	37.21	272	390	35100	854	2612	0	147.61
93	200	350	−5	0	4.99	272	365	36300	534	2612	0	127.29
94	200	350	−5	0	47.97	272	365	37100	534	2612	0	122.31
95	200	350	−5	279.61	37.21	272	365	35100	534	2612	280	172.58
96	200	350	−5	222.15	37.21	272	365	35100	534	2612	280	184.72
97	200	350	−5	269.53	37.21	272	365	35100	534	2612	280	174.31
98	200	350	−5	261.77	37.21	272	470	35100	854	2612	280	202.48
99	200	350	−5	283.88	4.99	272	365	36500	534	2612	280	189.79
100	200	350	−5	293.77	47.97	272	365	36500	534	2612	280	196.65
101	200	350	−5	271.66	37.21	272	365	36500	534	2612	280	172.58
102	180	250	0	0	22.1	270	552	32000	452	2104	0	53.9	(Wakjira et al. 2022)
103	180	250	0	0	22.1	270	552	32000	452	2104	0	54.6
104	180	250	0	0	22.1	270	552	32000	452	2104	0	59.4
105	180	250	0	0	22.1	270	552	32000	452	2104	0	58.5
106	180	240	0	0	26.8	325	410	32500	452	2354	0	96.6	(Li et al. 2022)
107	180	240	0	0	26.8	325	400	32500	628	2354	0	91.6
108	180	240	0	0	26.8	325	400	32500	628	2354	0	96.5
109	180	240	0	0	26.8	325	400	32500	628	2354	0	91.9
110	180	240	0	0	26.8	325	400	32500	628	2354	0	92.2
111	180	280	0	0	26.8	325	400	32500	628	2354	0	114.9
112	180	280	0	0	26.8	325	400	32500	628	2354	0	106.5

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