Calculation Method and Its Numerical Verification of Mid-span Deflection of Fish-Bellied Beam Support

doi:10.12068/j.issn.1005-3026.2025.20240085

Abstract

Abstract:

The deformation calculation of fish-bellied beam support is a highly statically indeterminate problem， involving complex relationships between stress and strain. Therefore， a theoretical method suitable for the deformation of the fish-bellied beam support structure was needed. Based on the reduction of the force and deformation characteristics of the fish-bellied beam support， a calculation method for the mid-span deflection of the fish-bellied beam support structure was proposed. By comparing various working conditions using finite element software， the theoretical assumption error was corrected. A highly accurate calculation method for the mid-span deflection of fish-bellied beam support was obtained. This method can be used for calculating the deformation of fish-bellied beam support and can efficiently predict the mid-span deformation of fish-bellied beam support. Verification by various methods shows that this method has small errors and conforms to the rules， providing a reference for related engineering projects.

Key words: fish-bellied beam support, statically indeterminate structure, mid-span deflection, calculation method, numerical verification

CLC Number:

TU 31

Cheng-fei WANG, Shu-hong WANG, Shi-hui JIAO, Ming-zhu REN. Calculation Method and Its Numerical Verification of Mid-span Deflection of Fish-Bellied Beam Support[J]. Journal of Northeastern University(Natural Science), 2025, 46(11): 106-114.

Figures/Tables 21

Fig.1 Schematic diagram of fish-bellied beam support structure

Fig.2 Schematic diagram of three types of

Fig.3 Force characteristic diagram of fish-bellied beam support

Fig.4 Method derivation process

Fig.5 Simplified fish-bellied beam support structure

Fig.6 Simplified fish-bellied beam deformation without pre-stress

Fig.7 Simplified stress diagram of steel wale

Fig.8 Simplified fish-bellied beam deformation with pre-stress

Table 1 Fish-bellied beam support working condition table

工况编号	自变量			常数
工况编号	$L 1$ /m	$L 2$ /m	$F P$ /kN	$E 1$ /GPa	$E 2$ /GPa	I/m⁴	$A 1$ /m²	$A 2$ /m²	q/(kN·m^-1)
1	12	5	0	206	206	0.006 69	0.021 95	0.003 04	100
2	12	8	1 000	206	206	0.009 30	0.029 62	0.007 33	125
3	12	10	2 000	206	206	0.011 90	0.036 14	0.009 78	150
4	19	5	0	206	206	0.009 30	0.029 62	0.009 78	150
5	19	8	1 000	206	206	0.011 90	0.036 14	0.003 04	100
6	19	10	2 000	206	206	0.006 69	0.021 95	0.007 33	125
7	26	5	1 000	206	206	0.011 90	0.036 14	0.007 33	150
8	26	8	2 000	206	206	0.006 69	0.021 95	0.009 78	100
9	26	10	0	206	206	0.009 30	0.029 62	0.003 04	125
10	12	5	2 000	206	206	0.009 30	0.029 62	0.007 33	100
11	12	8	0	206	206	0.011 90	0.036 14	0.009 78	125
12	12	10	1 000	206	206	0.006 69	0.021 95	0.003 04	150
13	19	5	1 000	206	206	0.006 69	0.021 95	0.009 78	125
14	19	8	2 000	206	206	0.009 30	0.029 62	0.003 04	150
15	19	10	0	206	206	0.011 90	0.036 14	0.007 33	100
16	26	5	2 000	206	206	0.011 90	0.036 14	0.003 04	125
17	26	8	0	206	206	0.006 69	0.021 95	0.007 33	150
18	26	10	1 000	206	206	0.009 30	0.029 62	0.009 78	100

Table 1 Fish-bellied beam support working condition table

工况编号	自变量			常数
工况编号	$L 1$ /m	$L 2$ /m	$F P$ /kN	$E 1$ /GPa	$E 2$ /GPa	I/m⁴	$A 1$ /m²	$A 2$ /m²	q/(kN·m^-1)
1	12	5	0	206	206	0.006 69	0.021 95	0.003 04	100
2	12	8	1 000	206	206	0.009 30	0.029 62	0.007 33	125
3	12	10	2 000	206	206	0.011 90	0.036 14	0.009 78	150
4	19	5	0	206	206	0.009 30	0.029 62	0.009 78	150
5	19	8	1 000	206	206	0.011 90	0.036 14	0.003 04	100
6	19	10	2 000	206	206	0.006 69	0.021 95	0.007 33	125
7	26	5	1 000	206	206	0.011 90	0.036 14	0.007 33	150
8	26	8	2 000	206	206	0.006 69	0.021 95	0.009 78	100
9	26	10	0	206	206	0.009 30	0.029 62	0.003 04	125
10	12	5	2 000	206	206	0.009 30	0.029 62	0.007 33	100
11	12	8	0	206	206	0.011 90	0.036 14	0.009 78	125
12	12	10	1 000	206	206	0.006 69	0.021 95	0.003 04	150
13	19	5	1 000	206	206	0.006 69	0.021 95	0.009 78	125
14	19	8	2 000	206	206	0.009 30	0.029 62	0.003 04	150
15	19	10	0	206	206	0.011 90	0.036 14	0.007 33	100
16	26	5	2 000	206	206	0.011 90	0.036 14	0.003 04	125
17	26	8	0	206	206	0.006 69	0.021 95	0.007 33	150
18	26	10	1 000	206	206	0.009 30	0.029 62	0.009 78	100

Fig.9 Model diagram of single-span fish-bellied beam support structure

Table 2 Theoretical calculation results and numerical

工况编号	理论值	模拟值	差值
1	85.047	64.855	20.192
2	14.340	6.929	7.411
3	-0.301	-5.066	4.765
4	299.261	220.351	78.910
5	153.508	120.942	32.566
6	56.032	45.085	10.947
7	1 072.694	823.985	248.709
8	247.918	198.288	49.630
9	676.887	549.720	127.167
10	5.991	1.021	4.970
11	22.622	13.714	8.908
12	29.275	19.164	10.111
13	244.159	195.207	48.952
14	209.547	180.852	28.695
15	82.194	62.518	19.676
16	1 401.509	1 243.367	158.142
17	616.749	500.453	116.296
18	174.866	138.439	36.427

Fig.10 Comparison diagram of theoretical and simulated values

Fig.11 Relationship diagram of theoretical and simulated values

Fig.12 Comparison diagram of error between two theoretical values

Fig.13 Finite element simulation results

Fig.14 Comparison diagram of theoretical and simulation results

Table 3 Engineering geological stratification

土层	重度	黏聚力 C/kPa	弹性模量 E/GPa	内摩擦角 φ_in/(°)
土层	γkN·m^-3	黏聚力 C/kPa	弹性模量 E/GPa	内摩擦角 φ_in/(°)
1-杂填土	18.0	15.0		10.0
2-中粗砂	19.7	3.6	23.2	34.0
3-砾砂	19.9	5.2	19.0	36.5
4-砾砂	20.2	7.3	29.4	37.4
5-圆砾	20.2		41.6	39.0
6-粗砂	20.1		29.6	37.8

Fig.15 A certain subway station’s foundation

Table 4 Simulated deformation situation of each

监测点编号	工况编号
监测点编号	1	2	3	4
监测点1	31.27	36.41	41.79	46.03
监测点2	34.74	39.36	44.72	49.15
监测点3	38.25	42.32	48.65	54.27
监测点4		43.97	50.42	55.76
监测点5		46.92	51.81	56.89
监测点6		47.88	53.74	58.01
监测点7			54.51	59.93
监测点8			56.97	61.62
监测点9			57.37	62.74

Fig.16 Simulation results of horizontal displacement of intermediate-span soil using software

Table 5 Theoretical calculation of deformation at

监测点编号	工况编号
监测点编号	1	2	3	4
监测点1	35.36	40.28	50.14	53.43
监测点2	38.64	43.57	53.43	56.72
监测点3	41.93	46.86	56.72	60.01
监测点4		48.68	51.97	55.26
监测点5		51.97	56.90	58.54
监测点6		55.26	60.19	61.83
监测点7			55.44	57.08
监测点8			57.08	60.37
监测点9			62.01	63.66

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