Global performance indicator (GPI) approach to predict the steel fiber reinforced concrete strength with error analysis

Figures & data

Figure 1. Experimental set up for two point loading flexure test on beam.

Table 1. Mathematical formulae for various statistical tests.

Sr. No	Statistical test	Abbreviation	Ideal Value	Equation
1.	Mean bias error	MBE	Zero	$MBE=\frac{1}{n}\sum _{i=1}^n\left(H_{i,m}-H_{i,c}\right)$
2.	Root mean squared error	RMSE	Zero	$RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(H_{i,m}-H_{i,c}\right)}^2}$
3.	t-Statistic	t-Stat	Near to zero	$t=\sqrt{\frac{\left(n-1\right)xMB{E}^2}{RMS{E}^2-MB{E}^2}}$
4.	Mean percentage error	MPE	Zero	$MPE=\frac{1}{n}\sum _{i=1}^n\left(\frac{H_{i,m}-H_{i,c}}{H_{i,m}}\right)x100$
5.	Uncertaintyat95%.	U95	Zero	$U_{95}=1.96{\left(S{D}^2-RMS{E}^2\right)}^{0.5}$
6.	Maximum absolute relative error	MARE	Zero	$MARE=max\left(\left|\frac{H_{i,m}-H_{i,c}}{H_{i,m}}\right|\right)$
7.	Relative root mean squared error	RRMSE	Zero	$RRMSE=\frac{\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(H_{i,m}-H_{i,c}\right)}^2}}{\sum _{i=1}^n\left(H_{i,m}\right)}x100$
8.	Mean absolute error	MAE	Zero	$MAE=\frac{1}{n}\sum _{i=1}^n\left|\left(H_{i,m}-H_{i,c}\right)\right|$
9.	Coefficient of determination	R2	one	$R{\hspace{0.17em}}_{\hspace{0.17em}}^2=1-\frac{{\sum _{i=1}^n}^{{\left(H_{i,m}-H_{i,c}\right)}^2}}{\sum _{i=1}^n\left(H_{i,m}-H_{m\mu \upsilon \theta }\right)}x100$
10.	Root mean squared relative error	RMSRE	Zero	$RMSRE=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(\frac{H_{i,m}-H_{i,c}}{H_{i,m}}\right)}^2}$

*where (H_{i, c}) and (H_{i, m}) are the calculated and measured values of field data for the prediction of SFRC strength.

Table 2. Statistical tests results for GT: π01.

Models	Exponential Model		Log-Linear Model		Analytical Model		ANN Model
	Predicted	Experimental	Predicted	Experimental	Predicted	Experimental	Predicted	Experimental
Standard Deviation	17.712	17.629	17.521	17.629	17.301	17.629	14.859	17.629
Difference between SD	-0.082		0.108		0.329		2.771

Table 3. Global performance indicator(GPI) for π01 with prediction of SFRC strength.

Statistical Test Model	MBE	RMSE	t-Stat	MPE	U95	MARE	RRMSE	MAE	R 2	RMSRE
Exponential model	0.0114	1.9539	0.0194	–0.3973	3.8331	0.299	0.0201	1.2159	0.9939	0.0914
Log-linear model	0	1.9479	0	–1.1386	3.8238	0.9246	0.0201	1.3556	0.9939	0.2013
Analytical model	0.1745	1.6055	0.3626	0.2499	3.2121	0.219	0.0165	0.9466	0.996	0.0694
ANN model	0.7765	6.3607	0.408	–21.736	13.5984	4.2063	0.0656	4.8735	0.9384	0.8755
MINIMUM	0	1.61	0	–21.74	3.21	0.219	0.0165	0.9466	0.9384	0.0694
MAXIMUM	0.78	6.36	0.41	0.25	13.6	4.2063	0.0656	4.8735	0.996	0.8755

Table 4. Global performance indicator (GPI) for prediction of SFRC strength to determine the median.

Statistical Test Model	MBE	RMSE	t-STAT	MPE	U95	MARE	RRMSE	MAE	R 2	RMSRE
Exponential Model	0.015	0.073	0.047	0.971	0.06	0.0201	0.0733	0.0686	0.9634	0.0273
Log-Linear Model	0	0.072	0	0.937	0.059	0.177	0.072	0.1042	0.963	0.1637
Analytical Model	0.225	0	0.889	1	0	0	0	0	1	0
ANN Model	1	1	1	0	1	1	1	1	0	1
MEDIAN	0.12	0.073	0.468	0.954	0.059	0.0985	0.0726	0.0864	0.9632	0.0955

Table 5. Prediction of SFRC strength parameter depending upon global performance indicator (GPI).

Statistical Test Model	MBE	RMSE	t-STAT	MPE	U95	MARE	RRMSE	MAE	R 2	RMSRE	GPI
Exponential Model	0.105	–0.001	0.421	–0.017	0	0.0785	–6E–04	0.0178	–2E–04	0.0682	0.6717
Log-Linear Model	0.12	0.001	0.468	0.017	0	–0.079	0.0006	–0.018	0.0002	–0.068	0.4418
Analytical Model	–0.105	0.073	–0.421	–0.046	0.059	0.0985	0.0726	0.0864	–0.037	0.0955	–0.0501
ANN Model	–0.88	–0.927	–0.532	0.954	–0.941	–0.902	–0.927	–0.914	0.9632	–0.905	–6.9367

Table 6. Comparative analysis of the models using coefficient of determinants(R²) and GPI.

Model	Exponential Model (1)	Log-Linear Model (2)	Analytical Model (3)	ANN Model (4)
R^2 value	0.99	0.99	1.00	0.94
GPI	0.67	0.44	–0.05	–6.94

Figure 2. Failure pattern of beam under flexure with two point loading.

Figure 3. Compressive strength after 7 days with experimental, theoretical and calculated values.

Figure 4. Compressive strength after 28 days with experimental, theoretical and calculated values.

Figure 5. Flexural strength after 7 days with experimental and theoretical values.

Figure 6. Flexural strength after days with experimental and theoretical values.

Table 7. Results of error analysis and global performance indicator showing the best mode for all response variables.

Statistical TestModel	MBE	RMSE	t-Stat	MPE	U95	MARE	RRMSE	MAE	R^2	RMSRE	GPI
Exponential model	0.01	1.95	0.02	–0.4	3.83	0.3	0.02	1.22	0.99	0.09	0.67
Log-linear model	0	1.95	0	–1.14	3.82	0.92	0.02	1.36	0.99	0.2	0.44
Analytical model	0.17	1.61	0.36	0.25	3.21	0.22	0.02	0.95	1	0.07	–0.05
ANN model	0.78	6.36	0.41	–21.74	13.6	4.21	0.07	4.87	0.94	0.88	–6.94