Study on comprehensive evaluation and decision-making method for all-composite bridge structure design scheme

Abstract

A comprehensive evaluation and decision-making method for all-composite bridge structure design scheme with multiple structural parameters as variables was proposed. First, four types of structure layouts were designed according to the designed conditions. Combining grey correlation analysis and entropy weight method, a comprehensive efficiency evaluation model for the design scheme was established, and the I-beam bridge was selected as the optimal scheme. Then, the detail-structure optimization and lay-up design of the I-beam bridge were carried out by using equi-strength design and stiffener design. Finally, all-composite bridges including box-girder bridge, I-beam bridge, truss bridge, and arch bridge, as well as stiffened I-beam bridge, were manufactured by using molding method. The four point bending test was used to verify the optimum design. The results show that I-beam bridge has highest structural efficiency, then followed by box-girder bridge, and truss bridge has the least structural efficiency, which is consistent with the comprehensive efficiency evaluation results. The damage location of the test and numerical analysis are basically the same. The ratio of load to mass for stiffened I-beam bridge reaches 48.88 N g⁻¹, which is 38.16% higher comparing with that of the initial design. The effectiveness of the optimum method was verified by the experimental results.

Introduction

Composite materials have a series of advantages, such as light weight, high strength, and strong designability, which make its application gradually expand from aerospace field to bridge engineering.^1–4 Compared with the traditional steel structure bridge and reinforced concrete structure bridge, fiber-reinforced composite bridge has the characteristics of corrosion resistance, fatigue resistance, not easy to crack, salt and alkali resistance, and low thermal expansion coefficient. Based on the superior mechanical properties, composite materials in the field of bridge engineering have been widely studied. Gosteev et al. ⁵ conducted numerical and experimental studies by using fiber-reinforced polymers in long-span suspension bridges, which was the first time that it was directly established that the strength, stiffness, and aerodynamic stability of a suspension hybrid bridge. Banerjee et al. ⁶ proposed the design of an integral bridge with reinforced and prestressed fiber-reinforced composite tendons alternative to the traditional simply supported bridge reinforced with corrosion-prone steel rebars. It has proven helpful in reducing computational cost and time requirements. Akimov et al. ⁷ created four options of the FRP bridge deck which showed better mechanical behavior in terms of vertical deflection by 20% on average in comparison to the reinforced concrete deck. Wu et al. ⁸ developed the fiber-reinforced polymer grid reinforcement technology, which had been successfully applied in the reinforcement and repair of Nanjing Yangtze River Bridge. Zhu et al. ⁹ designed an experimental program to investigate the fatigue flexural performance of concrete beams reinforced with a combination of steel and glass fiber-reinforced polymer bars. Siwowski et al. ¹⁰ proposed an all-composite structural system for road bridges, which consisted of four U-shaped girders bonded with sandwich deck slab. Wei et al. ¹¹ studied the dynamic characteristics of footbridge composed of fiber-reinforced polymers, steel, and concrete. It can be seen that the current research focuses on the bridge reinforcement and composite components, rarely involving the whole bridge structure. With the fast development of technology for fiber-reinforced composite, further application of composite materials in the bridge engineering will gradually develop from bridge repair and reinforcement to bridge manufacturing.

Comparing with the traditional bridge, the design of fiber-reinforced composite bridge is more complex due to many variables and constraints. In the design of fiber-reinforced composite bridge, technicians need to carry out detailed calculation and complex structure design according to the use conditions and construction requirements to accurately reflect the mechanical response of composite materials to external loads, and finally select the appropriate all-composite bridge structure. The structure design of all-composite bridge includes structure layout design, detail-structure optimization, and lay-up design. In view of the above problems, Smits ¹² reviewed the use of fiber-reinforced polymers in architectural and structural bridge design in the Netherlands. Tole et al. ¹³ conducted experiment on CFRP retrofitted RC T-beams using four-point loading to fail in shear. The results show that there is an improvement in shear capacity when retrofitted with CFRP wraps. Liu et al. ¹⁴ summarized the research progress of fiber-reinforced composite bridge from the aspects of materials, technology, basic components, mechanical theory, and durability. Ding et al. ¹⁵ calculated the initial failure load of three ply schemes of CFRP box-girder, and obtained the best ply scheme. Majumdar et al. ¹⁶ designed a composite bridge structure with lightweight fiber reinforced polymer deck based on the given conditions. In structure design, the structure layout determines the stiffness and bearing efficiency of the bridge, which is the primary problem to be solved in the bridge design. However, the existing research mostly focuses on the design and optimization of single structure layout, and rarely involves the optimization of multiple schemes.

There are two evaluation methods for the structural efficiency, including single factor efficiency evaluation and comprehensive efficiency evaluation. Single factor efficiency evaluation can only choose the most important factor as the evaluation index through experience, and then make the best choice by considering other indexes artificially. However, the failure modes of composite components are diverse. Different failure modes will cause different structural stiffness reductions, and even directly cause the components to lose the bearing capacity. Moreover, the structural efficiency of composite components often affects each other, therefore it is difficult to use a single index to evaluate the structure layout. Thus, it is necessary to carry out a comprehensively efficiency evaluation considering all the main performance indicators to provide a quantitative reference for the optimization design of the bridge. For multi-objective and multi-disciplinary comprehensive optimization problems, the grey correlation analysis is simple and practical, and has been widely used in optimization evaluation.^17,18 Grey relational analysis is used to understand the real world through the generation of some known information. It focuses on the objects with clear denotation and unclear connotation. It breaks through the constraints of traditional precise mathematics, and is often used as a multi-objective optimization method. However, the traditional grey relational analysis usually uses simple arithmetic average method to determine the weight, which will cause large errors in the calculation results. There are subjective method and objective method for determining the weight. The entropy weight method determines the index weight by a matrix composed of evaluation index values. ¹⁹ This method can avoid the subjectivity of weight calculation and make the evaluation result more realistic. Based on the above analysis, this paper designs four types of all-composite bridges with different structural layouts under designed conditions. Combining the grey correlation analysis and the entropy weight method, the comprehensive efficiency evaluation for different types of bridges was carried out, and the optimal structure layout was determined. Moreover, detail-structure optimization and lay-up design for the optimal structure layout were carried out to improve the ratio of load to mass for all-composite bridge. Finally, the bridges were manufactured and tested, and a full-composite bridge with high bearing efficiency was obtained. The work re-emphasizes and applies existing technologies to the field of optimum design for all-composite bridge structure, which can promote engineering production. Moreover, the work may have general implications in the overall design and manufacturing of composite bridges, which has high engineering value.

Design conditions and structure layout

Fiber-reinforced composites are used in bridge construction for a short time. At present, most bridges using fiber-reinforced composites are small-span pedestrian bridges. The small-span pedestrian bridges using fiber-reinforced composites mainly have the following design standards: (1) the AASHTO LRFD bridge design specifications ²⁰ ; (2) the design manual for roads and bridges: design of FRP bridges and highway structure ²¹ ; and (3) the code for design of the municipal bridge. ²²

The model used in the experiment is replicated based on a certain similarity relationship with the actual structure, which has all or part of the characteristics of the actual structure. As long as the designed model meets similar conditions, the results obtained through model experiment can be directly extrapolated to similar prototype structure. Based on three design specifications and using the similarity theorem, this study designed the similarity ratio and experimental conditions of the bridge model, ultimately determining appropriate similarity constants for physical quantities, and then designing a simplified bridge model. The geometric dimensions of the bridge model are mainly designed based on the small span pedestrian overpass, and the loading method is designed according to the actual load of the small span pedestrian overpass, distributing the concentrated load evenly on the supporting beams.

Geometrical dimension and loading method of the bridge are shown in Figure 1. The minimum size of bridge is 600 mm long and 100 mm wide. The bridge deck must be smooth and impermeable, so that the loading car can pass continuously. The bridge may be arched, but the arch height cannot change more than 50 mm. The net height of the bridge cannot exceed 120 mm. In order to meet the effective size requirements of the loading car, the bridge width cannot exceed 150 mm. The middle of the bridge must be unobstructed so that the loading car can be positioned. There shall be no supporting structure at the bottom of the bridge. The loading form is four point bending on a given span, and the ratio of load to mass is taken as the standard. Material uses 3K plain weave carbon fiber prepreg, and its mechanical performance parameters are shown in Table 1. In the table: X_t is the longitudinal tensile strength; E₁ is the longitudinal elastic modulus; Y_t is the transverse tensile strength; E₂ is the transverse elastic modulus; Y_c is the transverse compressive strength; X_c is the longitudinal compressive strength; S is the in-plane shear strength; G₁₂ is the in-plane shear modulus; τ₀ is the interlaminar shear strength; ν is the Poisson’s ratio.

Figure 1.

Geometrical dimension and loading method of the bridge.

Table 1.

Properties of 3K plain carbon fiber prepreg.

Mechanical properties	X t (MPa)	E 1 (GPa)	Y t (MPa)	E 2 (GPa)	Y c (MPa)	X c (MPa)	S (MPa)	G 12 (GPa)	τ 0 (MPa)	ν
Value	884.34	68.13	742.12	55.96	354.11	342.96	66.94	42.85	59.84	0.10

The structure design of all-composite bridge mainly includes three aspects: structure layout design, detail-structure optimization, and lay-up design. Among them, the structure layout design is the key, which determines the stiffness and bearing efficiency of the bridge. Therefore, the structure layout should be determined first, and then the detail-structure optimization and lay-up design should be carried out. A large amount of literature has studied different types of composite bridges. Based on the classic laminate theory and the strength criterion of maximum stress, Nguyen and Nguyen ²³ carried out the optimization design of I-beam. Zhang et al. ²⁴ studied the load-deflection curves and bending failure modes of box-girder bridge. Bacinskas et al. ²⁵ studied the structural performance of glass fiber-reinforced truss bridge under static loads. Xiao et al.^26,27 conducted a lot of research on truss bridges. Dambrisi et al. ²⁸ proposed the design criteria of arch bridge strengthened with CFRP. Dong and Wadley ²⁹ analyzed the mechanical properties and predicted the failure modes of lattice bridge. According to previous researches, ³⁰ the bridge structures that have used composite materials include beam bridge, truss bridge, arch bridge, and lattice bridge. Beam bridge includes I-beam bridge and box-girder bridge, of which I-beam is the most widely used. Considering the practicability of fiber-reinforced composite bridges, four types of structure layout schemes are designed as shown in Figure 2, including box-girder bridge, I-beam bridge, truss bridge, and arch bridge. This study adopted a unified rule to design four types of composite bridge structures. The geometric dimensions of four types of composite bridge structures are mainly designed based on the small span pedestrian overpass, all of which adopt a similar structural scheme of double support beams and bridge deck. Through the analysis of four types of bridges, the influence of different structure layout on the stiffness and bearing efficiency is investigated. Finally, the optimal structure layout scheme is selected.

Figure 2.

All-composite bridges with different structure layouts: (a) box-girder bridge, (b) I-beam bridge, (c) truss bridge, and (d) arch bridge.

In order to compare the different structure layouts for all-composite bridges, ABAQUS is used for analysis and calculation. The finite element models for four types of bridges are established, as shown in Figure 3. The two backup rollers are fixed. The wheels of the loading car and the bridge deck, the bottom surface of the bridge and the backup rollers are all in surface-to-surface contact. By loading the car, the four point bending test of the bridge is simulated. Since the bridge deck and the supporting structures on both sides are thin, the shell element model is adopted. In addition, the supporting structures on both sides for four types of bridges are constructed with a lay-up of [(0°,90°)/(±45°)]₆. Due to the small stress on the bridge deck, it is constructed with a lay-up of [(±45°)].

Figure 3.

Finite element models for four types of bridges under four point bending: (a) box-girder bridge, (b) I-beam bridge, (c) truss bridge, and (d) arch bridge.

Structure layout optimization of bridges

In this paper, the structural efficiency evaluation index system of the bridge design is constructed to unify the bridge mechanical performance parameters, and then the comprehensive efficiency evaluation is carried out by combining the grey correlation analysis and entropy weight method.

Construction of structural efficiency evaluation index system

The structure design of the bridge mainly considers the factors of stiffness, strength, and stability. The following variables are defined as the criteria for evaluating the stiffness and stability bearing efficiency of bridges.

$$\{\begin{array}{c}{X}_1=\frac{p_{\mathrm{cr}}}{m}\hfill \\ X_2=\frac{F{U}_{max}}{\mathrm{mU}}\hfill \end{array}$$

(1)

Where: X₁ is the buckling efficiency (N g⁻¹), that is, the buckling load per unit mass. The larger the X₁ is, the higher the anti-buckling efficiency is. X₂ is the stiffness efficiency (N g⁻¹), and the larger the X₂ is, the better the stiffness is. P_cr is the buckling load (N). m is the bridge mass (g). U is the displacement (mm). U_max is the limited maximum displacement, where U_max = 25 mm. F is the external load, where F = 5 kN.

Through the finite element analysis, the buckling efficiency and stiffness efficiency for four types of bridges are calculated, as shown in Table 2.

Table 2.

Buckling efficiency and stiffness efficiency for four types of bridges.

Scheme	m (g)	P cr (kN)	U (mm)	X 1 (N g−1)	X 2 (N g−1)
Box-girder bridge	342.6	9.96	5.25	29.07	69.49
I-beam bridge	325.3	11.51	5.55	35.38	69.23
Truss bridge	311.7	7.75	6.05	24.90	66.28
Arch bridge	330.0	10.37	6.42	31.42	59.00

Considering the strength, the failure modes of the bridge under four-point bending mainly include bridge deck failure, beam web failure, and bridge failure. Different structural forms of bridges have different failure modes. For this reason, the following variables are defined as criteria for evaluating the static strength bearing efficiency of bridges.

$$\{\begin{array}{c}{X}_3=\frac{F{X}_{\mathrm{t}}}{m{\sigma }_{max}}\hfill \\ X_4=\frac{\mathrm{FS}}{m{\tau }_{max}}\hfill \end{array}$$

(2)

Where: X₃ is the bearing efficiency of bridge deck under tension and compression failure (N g⁻¹). X₄ is the bearing efficiency of beam web under in-plane shear failure (N g⁻¹). σ_max is the maximum tensile and compressive normal stress of bridge deck (MPA). X_t is the longitudinal tensile strength of composite material (MPa). τ_max is the maximum in-plane shear stress of beam web (MPa). S is the in-plane shear strength of composite material (MPa).

Through the finite element analysis, the static strength bearing efficiency for four types of bridges are calculated, as shown in Table 3.

Table 3.

The static strength bearing efficiency for four types of bridges.

Scheme	m (g)	σ max (MPa)	τ max (MPa)	X 3 (N g−1)	X 4 (N g−1)
Box-girder bridge	342.6	352	29.16	36.67	33.51
I-beam bridge	325.3	346	33.56	39.29	30.66
Truss bridge	311.7	528	57.63	26.87	18.63
Arch bridge	330.0	342	33.89	39.18	29.93

Grey relational analysis

As shown in Table 2 and Table 3, the I-beam bridge is better in terms of stability and deck bearing. However, in terms of stiffness and beam web bearing, the effect of box-girder bridge is better. In addition, the design effect for different types of bridges is close. If a single factor is used to measure the bridge design, it is impossible to make accurate judgment. Therefore, it is necessary to comprehensively evaluate all indexes to optimize the structure layout.

For multi-objective problems, the grey relational analysis proposed by Deng ³¹ has significant advantages. The analysis steps are as follows: firstly, the evaluation indexes of bridge structure layout are selected, which can be expressed as the original sequence

$${\mathit{X}}_i=[{x_i}_1{x_i}_2\cdots x_{\mathrm{in}}]$$

Where: i (i = 1, 2, …, m) is the sequence number of structural schemes; n is the number of structural efficiency indexes, and then the original sequences of all design schemes constitute the decision matrix

$$X=[X_1{X}_2\cdots X_m{]}^{\mathrm{T}}$$

Since the unit and magnitude of each data are different, it is necessary to carry out non-dimensional standard preprocessing for different dimension factors in the comparative analysis. Through data preprocessing, the original sequence can be converted into a comparable sequence. If the target value of the evaluation index is infinite, then it has a characteristic of “the-larger-the-better.” However, if the target value of the evaluation index is finite, then it has a characteristic of “the-smaller-the-better.”

Due to the presence of both positive index and negative index, we used normalization method (MMS) and negative normalization method (NMMS) for the normalization treatment. The MMS method is used to perform non-dimensional treatment for the positive index, while the NMMS method is used to perform non-dimensional treatment for the negative index. Ultimately, all index are compressed within the range of [0, 1], and have a property that the larger is better. The preprocessing formulas for the original sequence are as follows:

$$\{\begin{array}{c}{x}_{\mathrm{ij}}^{*}=\frac{x_{\mathrm{ij}}-\underset{i}{min{x}_{\mathrm{ij}}}}{\underset{i}{max{x}_{\mathrm{ij}}}-\underset{i}{min{x}_{\mathrm{ij}}}}\hfill \\ x_{\mathrm{ij}}^{*}=\frac{\underset{i}{max{x}_{\mathrm{ij}}}-x_{\mathrm{ij}}}{\underset{i}{max{x}_{\mathrm{ij}}}-\underset{i}{min{x}_{\mathrm{ij}}}}\hfill \end{array}$$

(3)

where: $\underset{i}{max{x}_{\mathrm{ij}}}$ and $\underset{i}{min{x}_{\mathrm{ij}}}$ are the maximum and minimum values of the index j in all design schemes. The x _ij * generated by the dimensionless standardized data preprocessing is the comparability sequence of the data.

When calculating the grey correlation coefficient, the absolute difference between the comparability sequence and the reference sequence is calculated first, and then the grey correlation coefficient can be calculated by the following formulas

$$\{\begin{array}{c}{\xi }_{\mathrm{ij}}=\phi (x_j^0,x_{\mathrm{ij}}^{*})=\frac{{\mathrm{\Delta }}_{min}+\zeta {\mathrm{\Delta }}_{max}}{{\mathrm{\Delta }}_{\mathrm{ij}}+\zeta {\mathrm{\Delta }}_{max}}\hfill \\ {\mathrm{\Delta }}_{\mathrm{ij}}=|x_j^0-x_{\mathrm{ij}}^{*}|\hfill \\ {\mathrm{\Delta }}_{min}=\underset{i}{min}\underset{j}{min}{\mathrm{\Delta }}_{\mathrm{ij}}\hfill \\ {\mathrm{\Delta }}_{max}=\underset{i}{max}\underset{j}{max}{\mathrm{\Delta }}_{\mathrm{ij}}\hfill \end{array}$$

(4)

Where: ξ _ij is the correlation coefficient between the comparability sequence of the structure layout scheme j and the reference sequence; x _j ⁰ = [x₁⁰x₂⁰⋯x _n ⁰] is the reference sequence, which is composed of the optimal values of each index; Δ _ij is the absolute difference of the structure layout scheme j; Δ_min and Δ_max are the minimum difference and maximum difference corresponding to all the difference sequences. ζ is the distinguishing coefficient, which reflects the correlation integrity of various influencing factors in the target. Deng ³¹ pointed out based on experience that the coefficient ζ is generally between 0.1 and 0.5, usually 0.5, which can effectively increase the resolution between the correlation degrees and make the sorting results more accurate.

In order to improve the evaluation accuracy of grey correlation analysis, the average value of correlation coefficient between comparability sequence and reference sequence is calculated as grey correlation degree.

$$\delta (x^0,x_i^{*})=\frac{1}{n}\sum _{j=1}^n\phi (x_j^0,x_{\mathrm{ij}}^{*})$$

(5)

For this research object, the weighted sum of grey correlation coefficients is the grey correlation degree, and the calculation formula is as follows:

$$\delta (x^0,x_i^{*})=\sum _{j=1}^n{\omega }_j\phi (x_{\mathrm{ij}}^0,x_j^{*});\sum _{j=1}^n{\omega }_j=1$$

(6)

where, ω _j is the weight value of the response variable j. Since the role and influence of each indicator are not the same, different weights must be reasonably assigned according to the importance of each indicator. Therefore, this paper adopts entropy method to assign target value.

Entropy weight method

Entropy weight method is an objective weighting method that determines the weight of indicators based on the amount of information contained in each indicator. The entropy weight method has simple calculation steps, effectively utilizes index data, and excludes the influence of subjective factors. Firstly, according to the x _ij * generated by the dimensionless standardized data preprocessing, the proportion of the index value of the evaluation object i under the evaluation index j is calculated.

$$R_{\mathrm{ij}}=\frac{x_{\mathrm{ij}}^{*}}{\sum _{i=1}^n{x}_{\mathrm{ij}}^{*}}$$

(7)

Then, the entropy value of the evaluation index j is calculated.

$$\{\begin{array}{c}{\mathrm{\Gamma }}_j=-H\sum _{i=1}^m{R}_{\mathrm{ij}}\ln R_{\mathrm{ij}}(j=1,2···,m)\hfill \\ H=\frac{1}{\ln n}(H>0,0\le R_{\mathrm{ij}}\le 1)\hfill \end{array}$$

(8)

According to Shannon’s information theory, when $R_{\mathrm{ij}}=0$ , $R_{\mathrm{ij}}\ln R_{\mathrm{ij}}=0$ . Finally, the difference coefficient and weight of the evaluation index j is calculated.

$$\{\begin{array}{c}{\epsilon }_j=1-{\mathrm{\Gamma }}_j\hfill \\ {\omega }_j=\frac{{\epsilon }_j}{\sum _{j=1}^n{\epsilon }_j},\sum _{j=1}^n{\omega }_j=1\hfill \end{array}$$

(9)

Comprehensive efficiency evaluation

• (1) Through the finite element analysis, the decision-making data of bridge design evaluation indexes is constructed. The original index data for four types of bridges is as follows in Table 4.

• (2) The original decision data is processed with dimensionless standardized data, and the comparability sequence x _ij * is generated, as shown in Table 5.

• (3) The correlation coefficient is calculated as shown in Table 6.

• (4) Calculation of the weight of each index. Firstly, the data x _ij * is normalized to get the data R _ij shown in Table 7.

Table 4.

The original index data for four types of bridges.

Scheme	X 1 (N g−1)	X 2 (N g−1)	X 3 (N g−1)	X 4 (N g−1)
Box-girder bridge	29.07	69.49	36.67	33.51
I-beam bridge	35.38	69.23	39.29	30.66
Truss bridge	24.90	66.28	26.87	18.63
Arch bridge	31.42	59.00	39.18	29.93

Table 5.

The comparability sequence for four types of bridges.

Scheme	X 1	X 2	X 3	X 4
Box-girder bridge	0.4	1	0.79	1
I-beam bridge	1	0.98	1	0.81
Truss bridge	0	0.04	0	0
Arch bridge	0.62	0	0.99	0.76

Table 6.

The correlation coefficient for four types of bridges.

Scheme	X 1	X 2	X 3	X 4
Box-girder bridge	0.45	1	0.7	1
I-beam bridge	1	0.96	1	0.72
Truss bridge	0.33	0.62	0.33	0.33
Arch bridge	0.57	0.33	0.98	0.68

Table 7.

The R _ij for four types of bridges.

Scheme	X 1	X 2	X 3	X 4
Box-girder bridge	0.198	0.375	0.284	0.389
I-beam bridge	0.495	0.367	0.360	0.315
Truss bridge	0	0.258	0	0
Arch bridge	0.307	0	0.356	0.296

The R _ij is calculated using equation (7) based on the comparability sequence x _ij * in Table 5 generated by the dimensionless standardized data preprocessing.

Then, the weight of each index is calculated.

$$W={[{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4{]}^{\mathrm{T}}=[0.282,0.239,0.246,0.233]}^{\mathrm{T}}$$

• (5) Calculation of correlation degree.

$${[\mathrm{\delta }(x^0,{x_i}^{*}){]}^{\mathrm{T}}=[0.771,0.925,0.399,0.639]}^{\mathrm{T}}$$

It can be seen that δ (x⁰, x₂*) > δ (x⁰, x₁*) > δ (x⁰, x₄*) > δ (x⁰, x₃*). Therefore, the structural efficiency level of I-beam bridge is the highest, followed by box-girder bridge, and truss bridge is the worst. Due to the addition of the bottom surface, the quality of the box girder bridges has significantly increased, resulting in a decrease in structural efficiency. Due to limitations in the manufacturing process of composite materials, truss bridges cannot use the same nodes as metal materials, which weakens the load-bearing capacity at the nodes. The strength of nodes has a significant impact on the overall strength of truss bridges. Based on the above analysis, the structural efficiency of truss bridges is relatively low.

Detail-structure optimization and lay-up design of I-beam bridge

Detail-structure optimization and lay-up design are important factors to improve the performance of composite components. Large specific strength and stiffness can be achieved by optimizing the structural details of composite components. The main design methods include equi-strength design, equi-stiffness design, variable thickness design, stiffener design, and slotting design.^32,33 According to the bridge load and structure form, this study adopts equi-strength design and stiffener design to realize the bridge detail-structure optimization.

Equi-strength design is an important design method to improve the bearing efficiency of components. It considers that the strength deviation of different parts of the components should not be too large, and the strength value matching with different parts of the components can be achieved by changing the shape and thickness of the components, so as to reduce the weight of the structure without sacrificing the strength and improve the utilization rate of materials. According to the geometric dimensions and loading method of the bridge, and simplifying the bridge as a simply supported beam, the bending moment of the bridge can be calculated as follows

$$\begin{array}{c}M\left(x\right)=0.25\mathrm{Fx},0<x<250\mathrm{mm}\hfill \\ M\left(x\right)=62.5F,250\mathrm{mm}\le x\le 325\mathrm{mm}\hfill \\ M\left(x\right)=143.75F-0.25\mathrm{Fx},325\mathrm{mm}<x<575\mathrm{mm}\hfill \end{array}$$

Where, M(x) is the bending moment of the beam, F is the load, x is the distance from the center of the left support roller.

It can be seen that the bending moments on both sides of the bridge vary linearly. If the shape and thickness of the I-beam are the same, the bending stress of the bridge varies greatly. Therefore, in order to further improve the structural efficiency of I-beam bridge, it is necessary to carry out the equi-strength design of I-beam. Due to the complex forming process of variable thickness composite I-beam, I-beam with uniform web thickness is used. Through the above analysis, this paper adopts changing the shape of I-beam web to meet the equi-strength design principle. If $\overline{\sigma }$ is taken as the allowable bending stress, then the equi-strength design requirements of I-beam are as follows:

$$\frac{|M(x)|}{W(x)}=\overline{\sigma }$$

(10)

Where, M(x) is the bending moment of the beam, W(x) is the torsional section coefficient of the beam. In order to conveniently calculate the shape of the web, I-beam is equivalent to a rectangular beam, then $W(x)=\frac{1}{6}\mathrm{bh}(x{)}^2$ , where b is the thickness, h(x) is the height. After calculation, the function expression of the beam height h(x) is obtained as follows:

$$\{\begin{array}{c}y={(6325+18.69x)}^{\frac{1}{2}};(-305\le x\le 37.5)\hfill \\ y=75;(-37.5<x<37.5)\hfill \\ y={(6325-18.69x)}^{\frac{1}{2}};(37.5\le x\le 305)\hfill \end{array}$$

In order to avoid stress concentration on the upper edge of I-beam, the three curves on the upper edge of I-beam are approximately treated as a smooth curve. In this paper, the straight line in the middle is approximately treated as a smooth curve, which is consistent with the slope of other curves at the junction. Finally, the approximate curve of the second straight line is obtained, and its function expression is as follows

$$y=1.66e^{-3}{x}^2+77.33;(-37.5<x<37.5)$$

After equi-strength optimization, I-beam bridge with arched deck is designed. By analyzing the stress of the I-beam, it can be seen that the upper edge bar is mainly compressed, the lower edge bar is mainly tensile, and the web is mainly shear. According to the stress conditions of different parts of I-beam, the web is mainly constructed with a lay-up of [(±45°)], the edge strip is mainly constructed with a lay-up of [(0°,90°)], and the lay-up scheme is shown in Table 8.

Table 8.

Lay-up scheme of I-beam.

Scheme	Web and stiffener	Edge strip
I-beam	Web: [(±45°)2/(0°,90°)2/(±45°)2/(0°,90°)2/(±45°)2]S	[(0°,90°)3/(±45°)]
I-beam (arched deck)	Web: [(±45°)2/(0°,90°)2/(±45°)2/(0°,90°)2/(±45°)2]S	[(0°,90°)3/(±45°)]
I-beam (stiffener)	Web: (±45°)2/(0°,90°)/(±45°)2/(0°,90°)/(±45°)2]S	[(0°,90°)3/(±45°)]
	Stiffener: [(±45°)]4

Through the finite element analysis, the stress nephogram of the I-beam bridge after the equi-strength design is obtained, as shown in Figure 4. It can be seen that the stress is mainly concentrated in the loading and support. Therefore, in order to alleviate the local stress concentration of the bridge, a group of stiffeners are symmetrically designed from the trolley loading position to the backup rollers on the web of I-beam. In addition, the stiffener design method can further improve the stability of the bridge.

Figure 4.

Stress nephogram of I-beam bridge based on equi-strength design.

There are many variables in the optimization design of composite stiffened structures, ³⁴ including continuous variables and discrete variables. These variables are coupled with each other, which makes it difficult to optimize the design. The traditional mathematical criterion method is difficult to solve this kind of problem. The concurrent subspace optimization (CSSO) is an optimization technology to solve multidisciplinary design optimization problems. ³⁵ It can disperse the multidisciplinary computing tasks, and alleviate the computational complexity and organizational complexity in multidisciplinary design. Therefore, this paper uses CSSO to optimize the stiffened web of I-beam. The optimization problem is divided into three parts: the optimization of stiffeners layout, the optimization of stiffeners size, and the optimization of lay-up thickness. After optimization, all results are coordinated, then the design variables are updated, and finally the optimization process is repeated until the convergence condition is satisfied. The optimal design variables of stiffened I-beam are the number of stiffeners n, the width of stiffeners b, the number of layers for stiffeners t, and the number of layers for web T. The constraint conditions include static strength constraint and stability constraint. The optimization objective is the lightest weight. The optimization model is as follows:

$$\{\begin{array}{c}minW(X)\hfill \\ s.t.\lambda (X)\ge {\lambda }^{*}\hfill \\ \sigma (X)\le \left[\sigma \right]\hfill \\ \epsilon (X)\le \left[\epsilon \right]\hfill \\ X_{\mathrm{L}}\le X\le X_{\mathrm{M}}\hfill \end{array}$$

(11)

Where: X = (n, b, t, T), X_L and X_M are the lower and upper limits of all variables in X; W(X) is the weight of the stiffened web; λ(x) is the buckling factor; λ* is the design value of the buckling factor; and [σ] and [ε] are the allowable stress and strain of the material. The optimization process adopts the method mentioned by Yao et al. ³⁶

The layout optimization problem of stiffened I-beam is divided into four sub-problems. Correspondingly, the design variables are divided into four categories. The four sub-problems are optimized in their respective subspaces. The initial value is selected within the variation range of design variables, and then the iterative calculation is carried out. In each iteration, the average value of weight after parallel optimization is recorded, and the iteration process is shown in Figure 5. Finally, the optimization approximate result is X = (4, 15, 4, 8).

Figure 5.

Weight iteration process of the stiffened I-beam bridge.

After stiffener design, I-beam web with “M” stiffener is designed. According to the stress conditions of different parts of I-beam, the stiffeners adopt [(±45°)] lay-up along the rod direction, and the lay-up scheme is shown in Table 8. The finite element analysis is carried out, and the stress nephogram of stiffened I-beam bridge is shown in Figure 6. It can be seen that the stress concentration at the loading and support is significantly reduced.

Figure 6.

Stress nephogram of I-beam bridge based on stiffener design.

By calculating the weight of each efficiency evaluation index of the bridge, it can be seen that the weight of index X₁ is the highest. Therefore, this paper uses single efficiency evaluation index X₁ to evaluate I-beam bridge. Through the finite element analysis, the buckling efficiency of the optimized I-beam bridge is obtained, as shown in Table 9. The results show that the quality of I-beam bridge is reduced and the buckling efficiency is improved after equi-strength design. The weight of stiffened I-beam bridge is greatly reduced, and the buckling efficiency is greatly improved. Therefore, the layout for I-beam bridge with detail-structure of arched deck and stiffeners can greatly improve the bearing efficiency of the bridge.

Table 9.

Buckling efficiency of three types of I-beam bridges.

Scheme	m (g)	P cr (kN)	X 1 (N g−1)	P cr increase (%)	X 1 increase (%)
Initial bridge	325.3	11.51	35.38	–	–
Equi-strength design	315.3	12.32	39.07	7.30	10.43
Stiffener design	301.7	14.11	46.77	22.59	31.82

Bridge manufacturing and mechanical verification

The co-curing molding method has fewer processes and good bonding quality, which can ensure the continuity of the fiber and avoid the transfer of load at the bonding interface. The main bearing beam of the bridge is manufactured by this method. After the main bearing beam is cured, it is bonded and cured with the bridge deck. Finally, this study manufactured box-girder bridge, I-beam bridge, truss bridge, arch composite bridge, and stiffened I-beam bridge. Among them, the forming process of stiffened I-beam bridge is the most complex. The forming process includes stiffened I-beam forming and bridge deck forming. The stiffened I-beam is formed by a molding process. After grinding to remove the residual material and flash, I-beam and bridge deck are cured together to get the final bridge, as shown in Figure 7. The bridge is clamped on the testing machine according to the loading method shown in Figure 8, and the four-point bending test is conducted on the bridge. The loading is controlled by displacement. The loading rate is 20 mm/min. When the deformation is measured, continuous loading is adopted. The testing machine automatically records the load-displacement curve. Figure 9 shows the load-displacement curves of five bridges.

Figure 7.

Stiffened I-beam bridge after curing.

Figure 8.

Clamping and test of the bridge.

Figure 9.

Load-displacement curves of five bridges.

It can be seen from Figure 9 that the I-beam bridge has the largest damage load, followed by the arch bridge, and the truss composite bridge has the smallest damage load. In addition, compared with initial I-beam bridge, stiffened I-beam bridge has greatly improved structural rigidity and failure load, which is consistent with the previous detail-structure optimization conclusions. Table 10 shows the ratios of load to mass for five bridges. It can be seen that among four types of bridges, the bearing efficiency of I-beam bridge is higher than that of the other three bridges, followed by arch bridge, box-girder bridge, and truss bridge. The efficiency ranking for four types of bridges is completely consistent with the ranking estimated by the grey relational analysis, which proves the correctness and effectiveness of the comprehensive efficiency evaluation model used in this paper. The stiffened I-beam bridge has a mass of 305.2 g, a failure load of 14.92 kN, and a ratio of load to mass of 48.88 N g⁻¹. Compared with the initial design, the failure load increases by 29.6%, and the ratio of load to mass increases by 38.16%, which proves the effectiveness of the detail-structure optimization and lay-up design method.

Table 10.

Mass, failure load, and the ratio of load to mass of five bridges.

Scheme	M (g)	P (kN)	U (mm)	P/M (N g−1)
Box-girder bridge	350.7	10.02	11.57	28.57
I-beam bridge	329.5	12.36	12.01	37.51
Truss bridge	303.9	8.12	12.85	26.72
Arch bridge	341.1	11.05	14.89	31.28
Stiffened I-beam bridge	305.2	14.92	9.23	48.88

Figure 10 shows the experiment failure location and simulated stress concentration location of stiffened I-beam bridge under four-point bending loading. It can be seen that the damage location of the experiment and the numerical analysis are basically the same, which proves that the finite element simulation results are effective.

Figure 10.

Comparison of experiment and simulation results of bridge: (a) experiment and (b) finite element analysis.

The failure location of I-beam bridge appears at the end of the bridge and the backup rollers. The main failure modes are fiber fracture, matrix cracking, and interface delamination between the bridge deck and the upper edge strip of I-beam. This is the local buckling failure caused by the compression, bending, and shear of the bridge. This failure location needs to be further optimized and strengthened. Compared with the other three types of bridges, the box-girder bridge has one more bottom surface, so its mass is relatively large. In addition, local buckling and instability of the two side panels cause the overall damage of the bridge. Therefore, the optimization of the box-girder bridge is mainly to prevent instability, such as adopting reinforcement on both side panels. The bearing capacity of truss bridge is the worst, and its forming is relatively complex, especially, the joint between web member and chord is difficult to ensure the forming quality. It is easy to produce stress concentration at the joint, and the whole lower chord is seriously damaged. Moreover, due to the complex stress, the bridge deck has a wavy deformation. The failure location of arch bridge is mainly at both sides of loading car and backup rollers, which needs to be strengthened locally. In conclusion, through the optimization of the main bearing beam structure, the bearing load and bending stiffness of the bridge can be improved. Through the detail-structure optimization, the weight of the bridge can be reduced, and the structure efficiency can be improved. Combined with the structure layout design, detail-structure optimization, and lay-up design, the material efficiency of each part of the bridge can be fully improved, so as to obtain higher bearing efficiency.

Conclusion

Focusing on the optimum design of all-composite bridge structure, this study uses grey relational analysis and entropy weight method to conduct a structural efficiency assessment, and draws the following conclusions:

• (1) Considering the efficiency of the strength, stiffness, and stability of the bridge, combined with the grey relational analysis and entropy weight method, the comprehensive efficiency evaluation of the bridge structure layout is carried out, which overcomes the human factors of the single factor evaluation. The evaluation results show that the I-beam bridge has the highest structural efficiency, followed by box-girder bridge, and truss bridge has the worst structural efficiency. Through the experiment, it is found that the efficiency ranking for four types of bridges is completely consistent with that estimated by the grey correlation analysis. The correctness and effectiveness of the evaluation model are proved.

• (2) The I-beam bridge adopts the equi-strength design and stiffener design to carry out the detail-structure optimization and lay-up design. It is found that the ratio of load to mass for stiffened I-beam bridge is increased by 38.16% compared with the initial I-beam bridge. The correctness and effectiveness of detail-structure optimization and lay-up design methods are proved.

• (3) The stress concentration position of the bridge obtained by the finite element analysis is consistent with the test failure position, which proves that the established finite element model is correct. In addition, combined with numerical analysis and experiment, the measures of structural improvement for different types of bridges are presented.

In the future, this approach can be applied to the structure optimization design for other fields, and the efficiency index system needs be adjusted accordingly.