Design Research of 600 M Scale Concrete Arch Bridge Based on Mathematical Optimization Method

Hongjiang Yu; Jiuqun Mao; Qian Zhou; Boxu Cheng; Xuesong Liu; Chunlin Du

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Design Research of 600 M Scale Concrete Arch Bridge Based on Mathematical Optimization Method

| 27 ago 2020

Applied Mathematics and Nonlinear Sciences

Volume 5 (2020): Numero 2 (July 2020)

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Pubblicato online: 27 ago 2020

Pagine: 503 - 518

Ricevuto: 14 gen 2020

Accettato: 22 feb 2020

DOI: https://doi.org/10.2478/amns.2020.2.00030

Parole chiave
The mathematical algorithm of zero order optimization, 600 m scale concrete arch bridge, optimal design, Arch axis coefficient, Section size

© 2020 Hongjiang Yu et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Relation between feasible solution and objective function

Relationship between design variables and tensile stress

Relationship between design variables and compressive stress

Strength results under wind load and dead load

Optimization results of arch axis coefficient

Number of solutions	4		3		7		13
Arch axis coefficient	1.6		1.8		2		2.2
Scope of solutions	lower limit	upper limit	lower limit	upper limit	lower limit	upper limit	lower limit	upper limit
Z1/m	22.12	23.68	19.40	21.05	17.19	23.48	17.90	23.48
Z2/m	10.14	11.75	10.02	10.15	10.02	13.00	10.02	13.00
L1/m	139.04	155.30	175.13	182.22	177.60	251.79	177.60	254.41
H1/m	12.72	13.30	13.59	13.81	10.24	15.00	10.24	15.00
H2/m	7.79	8.33	7.87	8.06	7.27	9.17	7.08	9.17

Results of overall optimization

Feasible solutions	1	2	3	4	5	6	7
L1(DV)/m	200.00	200.37	234.87	259.61	218.37	242.73	269.43
N(DV)/m	0.92	0.99	1.01	1.15	1.08	2.57	2.10
Z1(DV)/m	24.00	17.93	16.76	16.45	25.78	15.70	19.27
T1(DV)/m	0.60	0.58	0.51	0.51	0.55	0.47	0.54
T2(DV)/m	0.60	0.46	0.43	0.42	0.58	0.40	0.51
Z2(DV)/m	12.00	13.53	12.92	12.98	11.08	14.22	14.24
H1(DV)/m	14.00	12.51	14.51	14.85	13.47	11.59	10.02
H2(DV)/m	10.00	8.47	7.71	7.55	7.53	8.07	7.35
VOLUME(DV)	10.406	8.264	6.369	6.281	6.722	6.265	7.355
DISP(SV)/mm	−423	−487	−483	−475	−377	−533	−475

Optimization results

Design variable	Symbol	Optimum solution	Feasible solution range
Width of foot/m	Z1	22	15–22
Width of constant section /m	Z2	13	13–15
Height of foot /m	H1	15	12–15
Height of constant section /m	H2	7.5	6–8
variability index of width	N	2.2	2–3
Length of variable section /m	L1	220	200–250
Thickness of roof and floor /m	T1	0.5	0.42–0.6
Thickness of web /m	T2	0.55	0.55–0.6

Optimization results under different arch axis coefficient

Arch axis coefficient	1.6	1.8	2.0	2.20
FREQN	7.11	6.44	6.04	6.25
Vault deformation/mm	−401.67	−398.24	−429.90	−463.15
Z1/m	17.39	23.15	24.05	23.76
Z2/m	10.26	11.13	11.00	11.07
H1/m	14.90	12.91	12.86	12.65
H2/m	8.36	7.60	7.03	7.40
Volume/m³	14462	14480	14014	14493

Limits of state variables

State variables	Symbol	Lower limit	Upper limit
Vault deformation/mm	DISP	−700	−400
First order elastic stability coefficient	FREQN	6
Tensile stress of arch foot/MPa	SMAX1	0	1.7
Pressure stress of arch foot/MPa	SMAX2	−27	0
Tensile stress of L/8/ MPa	SMAX3	0	1.7
Pressure stress of L/8 / MPa	SMAX4	−27	0
Tensile stress of L/4/ MPa	SMAX5	0	1.7
Pressure stress of L/4/ MPa	SMAX6	−27	0
Tensile stress of L3/8/ MPa	SMAX7	0	1.7
Pressure stress of L3/8/ MPa	SMAX8	−27	0
Tensile stress of vault/MPa	SMAX9	0	1.7
Pressure stress of vault/MPa	SMAX10	−27	0

The feasible solution of each design variable in local optimization

Feasible solution	1	2	3	4	5	6	7	8

L1(DV)/m	269.43	164.06	242.73	255.04	280.17	218.37	286.29	277.66
Z1(DV)/m	23.47	19.57	18.88	21.92	20.38	21.18	23.48	15.03
Z2(DV)/m	14.24	13.46	13.09	14.24	13.96	12.91	13.48	14.95
N1(DV)	3.96	2.10	1.86	2.92	2.38	2.66	3.47	0.51
H1(DV)/m	14.24	12.29	11.94	13.46	13.09	13.96	10.02	11.60
H2(DV)/m	9.24	7.29	6.94	8.46	7.69	8.09	9.24	8.96
T1(DV)/m	0.57	0.49	0.48	0.54	0.51	0.52	0.57	0.40
T2(DV)/m	0.57	0.54	0.49	0.57	0.56	0.54	0.42	0.60

feasible solution	9	10	11	12	13	14	15	16

L1(DV)/m	245.32	241.76	285.64	211.01	254.46	293.25	199.18	200.37
Z1(DV)/m	19.69	18.20	22.93	20.82	21.95	15.70	24.90	23.04
Z2(DV)/m	13.16	13.86	14.02	14.18	13.80	13.57	12.95	12.39
N1(DV)	2.14	1.62	2.54	3.27	0.75	0.89	1.03	3.00
H1(DV)/m	10.56	14.95	12.65	13.16	14.02	13.80	13.57	14.18
H2(DV)/m	9.24	7.29	6.94	8.46	7.69	8.09	9.24	8.96
T1(DV)/m	0.49	0.46	0.56	0.54	0.47	0.43	0.42	0.52
T2(DV)/m	0.55	0.56	0.57	0.54	0.55	0.41	0.52	0.42

Material parameters

Material	Modulus of elasticity/(N/mm²)	Mass density (kg/m³)	Coefficient of linear expansion	Poisson ratio
C80	3.80E10	2600	1.0E-5	0.2
C40	3.25E10	2500	1.0E-5	0.2

Limits of design variables

Design variables	Symbol	Lower limit	Upper limit
Width of foot/m	Z1	15	25
Width of constant section/m	Z2	10	15
Height of foot/m	H1	8	15
Height of constant section/m	H2	6	10
variability index of width	N	0.5	5
Length of variable section/m	L1	100	300
Thickness of roof and floor/m	T1	0.4	0.6
Thickness of web/m	T2	0.4	0.6

Results of optimization

Arch axis coefficient	1.6	1.8	2	2.2
L1/m	137.84	163.33	180.98	217.00
N	3.39	3.60	3.83	4.80
Volume/m³	14259	14698	14978	15622
FREQN	6.58	6.60	6.60	6.79
Vault deformation/mm	−398	−400	−417	−420

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Design Research of 600 M Scale Concrete Arch Bridge Based on Mathematical Optimization Method

Pubblicato online: 27 ago 2020

Pagine: 503 - 518

Ricevuto: 14 gen 2020

Accettato: 22 feb 2020

DOI: https://doi.org/10.2478/amns.2020.2.00030

Parole chiave
The mathematical algorithm of zero order optimization, 600 m scale concrete arch bridge, optimal design, Arch axis coefficient, Section size

© 2020 Hongjiang Yu et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

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Fig. 11

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Fig. 13

Optimization results of arch axis coefficient

Results of overall optimization

Optimization results

Optimization results under different arch axis coefficient

Limits of state variables

The feasible solution of each design variable in local optimization

Material parameters

Limits of design variables

Results of optimization

Design Research of 600 M Scale Concrete Arch Bridge Based on Mathematical Optimization Method

Pubblicato online: 27 ago 2020

Pagine: 503 - 518

Ricevuto: 14 gen 2020

Accettato: 22 feb 2020

DOI: https://doi.org/10.2478/amns.2020.2.00030

Parole chiaveThe mathematical algorithm of zero order optimization, 600 m scale concrete arch bridge, optimal design, Arch axis coefficient, Section size

© 2020 Hongjiang Yu et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Optimization results of arch axis coefficient

Results of overall optimization

Optimization results

Optimization results under different arch axis coefficient

Limits of state variables

The feasible solution of each design variable in local optimization

Material parameters

Limits of design variables

Results of optimization

Parole chiave
The mathematical algorithm of zero order optimization, 600 m scale concrete arch bridge, optimal design, Arch axis coefficient, Section size