Particle swarm optimization model to predict scour depth around a bridge pier

doi:10.1007/s11709-020-0619-2

Front. Struct. Civ. Eng.

2020, Vol. 14

Issue (4) : 855-866 https://doi.org/10.1007/s11709-020-0619-2

RESEARCH ARTICLE

Particle swarm optimization model to predict scour depth around a bridge pier

Shahaboddin SHAMSHIRBAND^1,²(

), Amir MOSAVI^3,^4,^5,⁶, Timon RABCZUK⁶

¹. Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam
². Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam
³. Institute of Automation, Kando Kalman Faculty of Electrical Engineering, Obuda University, Budapest 1034, Hungary
⁴. Department of Mathematics and Informatics, J. Selye University, Komarno 94501, Slovakia
⁵. School of the Built Environment, Oxford Brookes University, Oxford OX30BP, UK
⁶. Institute of Structural Mechanics, Bauhaus-Universität Weimar, Weimar 99423, Germany

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Abstract

Scour depth around bridge piers plays a vital role in the safety and stability of the bridges. The former approaches used in the prediction of scour depth are based on regression models or black box models in which the first one lacks enough accuracy while the later one does not provide a clear mathematical expression to easily employ it for other situations or cases. Therefore, this paper aims to develop new equations using particle swarm optimization as a metaheuristic approach to predict scour depth around bridge piers. To improve the efficiency of the proposed model, individual equations are derived for laboratory and field data. Moreover, sensitivity analysis is conducted to achieve the most effective parameters in the estimation of scour depth for both experimental and filed data sets. Comparing the results of the proposed model with those of existing regression-based equations reveal the superiority of the proposed method in terms of accuracy and uncertainty. Moreover, the ratio of pier width to flow depth and ratio of d50 (mean particle diameter) to flow depth for the laboratory and field data were recognized as the most effective parameters, respectively. The derived equations can be used as a suitable proxy to estimate scour depth in both experimental and prototype scales.

Keywords scour depth bridge design and construction particle swarm optimization computational mechanics artificial intelligence bridge pier

Corresponding Author(s): Shahaboddin SHAMSHIRBAND

Just Accepted Date: 27 May 2020 Online First Date: 28 June 2020 Issue Date: 27 August 2020

Cite this article:

Shahaboddin SHAMSHIRBAND,Amir MOSAVI,Timon RABCZUK. Particle swarm optimization model to predict scour depth around a bridge pier[J]. Front. Struct. Civ. Eng., 2020, 14(4): 855-866.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-020-0619-2
https://academic.hep.com.cn/fsce/EN/Y2020/V14/I4/855

parameter	laboratory				field
parameter	min	max	mean	std	min	max	mean	std
D (m)	0.01585	0.915	0.107	0.1435	0.3048	28.7	2.797	3.674
V (m/s)	0.149	2.16	0.512	0.3216	0.088	4.084	1.366	0.75
V_c(m/s)-L (m)	0.222	1.27	0.443	0.2213	0.975	38.1	10.705	4.255
y (m)	0.0201	1.9	0.269	0.2513	0.1524	22.524	4.163	3.518
d50 (m)	0.0002	0.008	0.00118	1.37	0.000008	0.108	0.01675	25.15
S (m)	0.0039	1.41	0.1357	0.149	0	10.393	1.0528	1.404
$σ$	1.1	5.5	1.454	0.692	1.2	20.34	3.358	2.806
$V / V c − L / y$	0.4148	5.38	1.254	0.846	0.5142	81.818	5.3487	6.575
$D / y$	0.0477	19.16	0.704	1.724	0.0722	50.297	1.251	2.805
$d 50 / y$	0.0001	0.107	0.007	0.0107	4.9e–7	0.2264	0.0106	0.0226
Fr	0.067	1.498	0.377	0.248	0.0269	1.184	0.2703	0.177
$S / y$	0.0198	6.87	0.754	0.805	0	3.4	0.3066	0.2948

Tab.1 Statistical analysis of laboratory and field data sets

Fig.1 Schematic layout of PSO algorithm.

model	equation	type of data set
Laursen and Toch [²⁶]	$S / y = 1.35 (D / y) 0.7$	laboratory
El-Said [²⁷]	$S / y = 3.4 (F r) 0.67 (D / y) 0.67$	laboratory
Riahi-Madvar et al. [²⁸]	$S / D = 2.42 (2 V V c − 1) (V g D) 1 / 3$	laboratory
Melville and Sutherland [²⁹]	$S / D = K I K D K y K α K S$ ^*	laboratory
Johnson [⁴]	$S / y = 2.02 (σ) − 0.98 (F r) 0.21 (D / y) 0.98$	laboratory
Richardson and Davis [³]	$S / y = 2.6 (F r) 0.65 (D / y) 0.43$	field
HEC-18 (Mohamed et al.) [³⁰]	$S / y = 2.1 (F r) 0.43 (D / y) 0.65$	laboratory and field
Azamathulla et al. [⁷]	$S / y = 1.82 (σ) − 0.03159 (F r) 0.42 (d 50 / y) 0.042 (D / y) − 0.28 (L / y) − 0.37$	field
Sharafi et al. [¹⁰]	$S / y = 0.28 (σ) 0.13 (F r) 0.47 (d 50 / y) − 0.1 (D / y) 0.44 (L / y) 0.23$	field

Tab.2 Empirical equations for estimation of scour depth around bridge piers

model no.	input variables	model no.	input variables
L1	$σ, F r, D y, d 50 y, V V c$	F1	$σ, F r, D y, d 50 y, L y$
L2	$F r, D y, d 50 y, V V c$	F2	$F r, D y, d 50 y, L y$
L3	$σ, D y, d 50 y, V V c$	F3	$σ, F r, D y, d 50 y$
L4	$σ, F r, d 50 y, V V c$	F4	$σ, F r, d 50 y, L y$
L5	$σ, F r, D y, V V c$	F5	$σ, F r, D y, L y$
L6	$σ, F r, D y, d 50 y$	F6	$σ, D y, d 50 y, L y$

Tab.3 Model specifications for sensitivity analysis of scour depth

Fig.2 Schematic layouts of the study.

model no.	derived equation	$R 2$	$R M S E$ (m)	$e ¯$ (m)	MAE (m)
L1	$S y = 1.282 (σ) − 0.397 (F r) 0.679 (D y) 0.610 (d 50 y) − 0.142 (V V c) − 0.476$	0.877	0.046	0.009	0.029
L2	$S y = 0.893 (F r) 1.016 (D y) 0.625 (d 50 y) − 0.262 (V V c) − 0.836$	0.861	0.049	0.008	0.030
L3	$S y = 2.235 (σ) − 0.434 (D y) 0.587 (d 50 y) 0.111 (V V c) 0.212$	0.897	0.043	0.010	0.029
L4	$S y = 5 (σ) − 0.372 (F r) − 0.119 (d 50 y) 0.354 (V V c) 0.305$	0.119	0.161	0.059	0.098
L5	$S y = 1.777 (σ) − 0.383 (F r) 0.323 (D y) 0.595 (V V c) − 0.111$	0.886	0.045	0.009	0.029
L6	$S y = 1.874 (σ) − 0.421 (F r) 0.226 (D y) 0.595 (d 50 y) 0.029$	0.891	0.044	0.009	0.029

Tab.4 Derived equations and their performance for laboratory data set during testing period

Fig.3 Width of uncertainty band for the derived equations (laboratory data set).

model	$R 2$	$R M S E$ (m)	$e ¯$ (m)	MAE (m)	$1.96 S e$ (m)
Laursen and Toch [²⁶]	0.894	0.061	0.032	0.0398	0.052
El-Saiad [²⁷]	0.616	0.105	0.059	0.069?	0.087
Riahi-Madvar et al. [²⁸]	0.009	0.909	0.546	0.579?	0.726
Johnson [⁴]	0.677	0.082	−0.011???	0.037?	0.081
Richardson and Davis [³]	0.513	0.106	0.055	0.0708	0.091
HEC-18 (Mohamed et al.) [³⁰]	0.829	0.057	0.025	0.038?	0.051
L3 (this study)	0.897	0.043	0.010	0.029?	0.042

Tab.5 Results of the best derived equations against existing equations

Fig.4 Scatter plot of estimated scour depth versus measured values. (a) Laursen and Toch [26]; (b) Johnson [⁴]; (c) HEC-18; (d) L3.

model no.	derived equation	$R 2$	$R M S E$ (m)	$e ¯$ (m)	MAE (m)
F1	$S y = 0.095 * (σ) 0.116 (F r) 0.178 (D y) 0.189 (d 50 y) − 0.136 (L y) 0.324$	0.727	0.822	0.014	0.547
F2	$S y = 0.1 * (F r) 0.154 (D y) 0.219 (d 50 y) − 0.145 (L y) 0.308$	0.774	0.753	0.030	0.520
F3	$S y = 0.241 * (σ) 0.075 (F r) 0.265 (D y) 0.349 (d 50 y) − 0.099$	0.770	0.766	0.153	0.513
F4	$S y = 0.05 * (σ) 0.237 (F r) 0.231 (d 50 y) − 0.162 (L y) 0.553$	0.629	0.966	-0.086	0.599
F5	$S y = 0.193 * (σ) 0.222 (F r) − 0.012 (D y) 0.191 (L y) 0.202$	0.658	1.018	-0.054	0.629
F6	$S y = 0.086 * (σ) 0.081 (D y) 0.193 (d 50 y) − 0.110 (L y) 0.352$	0.722	0.849	-0.018	0.570

Tab.6 Derived equations and their performance for field data set during testing period

Fig.5 Width of uncertainty band for the derived equations (field measurements).

model	$R 2$	$R M S E$ (m)	$e ¯$ (m)	MAE (m)	$1.96 S e$ (m)
Laursen and Toch [²⁶]	0.540	3.898	2.802	2.802	5.310
El-Saiad [²⁷]	0.468	2.871	2.295	2.320	3.384
Johnson [⁴]	0.336	1.372	0.284	0.881	2.631
Richardson and Davis [³]	0.624	2.226	1.839	1.893	2.459
HEC-18 (Mohamed et al.) [³⁰]	0.567	2.434	1.932	1.942	2.902
Azamathulla et al. [⁷]	0.448	3.519	1.985	2.182	5.696
Sharafi et al. [¹⁰]	0.710	0.869	0.123	0.607	1.687
L6	0.599	1.324	0.854	1.022	1.982
F2	0.774	0.753	0.030	0.520	1.474

Tab.7 Results of the best derived equations against existing equations for field measurements

Fig.6 Scatter plot of estimated values versus field measurements of scour depth. (a) Richardson and Davis [³]; (b) Sharafi et al. [¹⁰]; (c) L6; (d) F2.

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