Continuously reinforced concrete pavement with basalt fibre (BFRP) is provided with continuous BFRP reinforcement in the concrete slab. This material eliminates the transverse shrinkage joints and adds a certain thickness of asphalt mixture to the concrete slab to significantly improve the integrity of the pavement structure and driving comfort. Controlling the longitudinal reinforcement ratio of concrete slabs and determining the reasonable spacing and width of transverse cracks to reduce the impact of concrete slabs are actual contents of this type of pavement research and design. Some scholars used the nonlinear finite element semi-discrete method to analyse the shrinkage strain of concrete and proposed a method for calculating the early crack width of CRCP. Some scholars have developed continuous reinforcement design software based on the stress analysis of steel bars under different conditions. Some scholars combined the finite element analysis and test data to obtain the concrete temperature stress equation . According to the theory of composite slabs, some scholars used trigonometric series and Fourier transform to analyse the internal force and stress of concrete slabs on Winkler’s foundation. Some scholars established a single-reinforced slat model and used spring elements to perform finite element simulation analysis of the interface between steel and concrete. Some scholars used the CRC layer uniform temperature drop model to analyse the mechanism of cracks but did not consider the friction resistance of the base layer.
Studies have shown that BFRP bars have corrosion resistance characteristics, ageing resistance, and high strength and can realise longitudinal mechanised continuous production. We use BFRP bars to reinforce the CRC layer, which can prolong the pavement service life . This effectively reduces construction overlap points and connection lengths. However, there are few research pieces on the reinforcement design method of BFRP bars in pavement structures.
In this paper, the uniform temperature drop model of the CRC layer is established with the continuously reinforced concrete plus asphalt overlay composite pavement (CRC+AC) as the research object. The article considers the drying shrinkage of concrete materials and the restraint effect of the grassroots level and derives the analytical solution formula of crack control index . At the same time, we use ABAQUS finite element software to verify the validity of the formula. The AC asphalt layer can weaken the temperature gradient. So this article ignores the warpage stress of the CRC layer. The article uses the analytical method to study different BFRP bar material characteristics and reinforcement schemes on pavement cracks.
The shrinkage of concrete materials due to temperature drops will cause relative slippage between the CRC and base layers. According to the shear test and regression statistics results, the distribution of frictional shear stress caused by slip is usually divided into linear, bilinear and hyperbolic models . They are different due to different types of grassroots. We use a bilinear slip model closer to the actual situation, as shown in Figure 1. Comprehensively consider the critical displacement value
Relative slippage occurs between the concrete and the reinforcement due to the uncoordinated deformation. We use the shear stiffness
The CRC board temperature drop and shrinkage model is shown in Figure 3. We analyse the micro-element section with the length of the concrete slat
In the formula
From the force balance analysis, it is easy to know that the stress of the reinforcement is the smallest at the midpoint of the slab and the stress of the concrete is the largest here. The stress of the reinforcement increases gradually away from the midpoint . It reaches the maximum value at the crack end
The shrinkage of the concrete slab on both sides of the crack is the same. The crack width
The analytical expressions of crack width, crack spacing and stress of the reinforcement in the continuous reinforcement design control index of the CRC layer can be obtained . For cement stabilised bases, since the frictional shear stress can easily reach the limit value, we should judge the correctness of the base slip relationship assumption based on the calculation result of
To verify the validity of the analytical method, we established a finite element model to simulate the temperature field cooling. The base layer and the CRC layer adopt a linear shear-slip relationship. Based on the known crack spacing, we obtain the concrete stress in the slab, the crack width and the BFRP tendons’ control stress.
According to the symmetry, we use the plate length L for analysis. We arrange two BFRP ribs symmetrically in the transverse direction to consider the interaction between the ribs. We use an 8-node reduced integral solid element (C3D8R) for the BFRP rib, CRC and base layers. At the same time, we encrypt the BFRP rib mesh, as shown in Figure 4. Assuming that each part of the material is uniform linear elasticity, the concrete and BFRP tendons have orthotropic expansion characteristics, and only the longitudinal expansion coefficient is given. We use cohesive contact to simulate the bond-slip characteristics between BFRP bars and concrete and ignore the lateral deformation caused by the embedding of the bars . We use general static analysis to achieve uniform temperature drop simulation by setting different temperature field values. At the bottom of the model, the intermediate phase displacement constraints are imposed on both lateral sides and the longitudinal ends of the base layer.
The length of the slats used by the finite element and analytical methods is L = 1.8 m. The longitudinal BFRP ribs in the board are arranged at 1/2 the height of the CRC board. The horizontal spacing and board margin are 8 cm. The longitudinal bending stiffness between BFRP bar and concrete is 20 GPa/m. The bottom friction coefficient of the CRC layer is 40 MPa/m. The temperature field decreases evenly by 30°C. Compared with the finite element results, we use the analytical method to calculate separately . The shrinkage strain
Material parameters of finite element simulation
|Concrete||28||2.7 × 1010||10−5||0.15|
|BFRP rib||1.8||4 × 1010||9 ×10−6||0.3|
|Grassroots||20||1.3 × 103||_||0.25|
BFRP, basalt fibre.
Due to the limited range of influence of the base layer and BFRP bars, the stress and displacement of each element at the same section of the CRC layer are not uniform. For example, the longitudinal displacement value at the bottom of the CRC layer is more significant than that near the BFRP ribs. The CRC layer is 1/4 height from the bottom and 1/2 of the horizontal element node as the output position of the representative value of
Comparison of analytical method and finite element results
|Analytical method ( = 0||0.83||171||2.01 < 2.54|
|Analytical method ( = 1||1.1||229||2.69 > 2.54|
|Finite element ( = 0||0.88||167||2.14 < 2.54|
From the results in Table 2, it can be seen that the crack control index results of the analytical method and the finite element method are consistent without accounting for shrinkage strain. The error is <7%. The displacement value of the concrete at the crack, which is 1/2 of the width value, does not exceed the critical slip value of most base courses. This is in line with the assumption of the slip relationship at the grassroots level . When we use the analytical method to calculate the shrinkage strain, each index value has increased by about 35%. The concrete stress in the slab has exceeded the ultimate tensile strength, indicating that the effect of concrete drying shrinkage on the transverse cracks of the CRC layer cannot be ignored.
In summary, the analytical crack calculation proposed by us can more accurately predict the actual road conditions of the BFRP bar continuously reinforced pavement under the conditions of uniform temperature drop and dry shrinkage. This can provide a reference for the continuous reinforcement design of BFRP bars.
We use the analytical method to analyse the influence of the elastic modulus of BFRP bars on the crack mode of the CRC layer. The BFRP bars used are two types of thread bars with the elastic modulus of 40 and 60 GPa, and the yield strength is 510 MPa. In addition, the comparative calculation of the steel bar shows that the elastic modulus of the steel bar is 200 GPa and the yield strength is 335 MPa. The linear expansion coefficient of the bar is 9×10−6°C, the diameter is 12 mm and the bending stiffness is 20 GPa/m. The horizontal spacing of longitudinal bars adopts eight types between 8 cm and 22 cm, and the reinforcement ratio is 0.18–0.51%. The shrinkage factor of concrete is 0.0002. The rest of the parameters are the same as the values taken while verifying the above-mentioned finite element model. The calculation results of crack width, spacing and BFRP tendon control stress are shown in Figure 5.
It can be seen from Figure 5 that the three indexes of BFRP tendons are the same as the steel bars along with the change of the longitudinal bar spacing, and the crack width and spacing are more significant than the calculated values of the bars . For the same type of reinforcement, the crack spacing and width of the BFRP bars are more significant than the calculated values of the steel bars. Taking the longitudinal rib spacing of 16 cm as an example, when the elastic modulus of BFRP ribs changes from 60 MPa to 40 MPa, the crack width increases by about 15.3%, the maximum crack spacing increases 14.8% and the control stress decreases by 3.4%. When the spacing of longitudinal bars is <16 cm, the restraining effect of BFRP bars on the concrete gradually weakens with the spacing increase, and the effect of elastic modulus on the crack spacing and width becomes more significant. When the longitudinal rib spacing is >16 cm, the calculation result of the control stress is only a theoretical value. This can be used to reflect the overall trend of change.
The characteristics of low elastic modulus of BFRP tendons result in large pavement crack spacing and width, and its joints have inadequate load transfer capacity and are prone to breakage. We take the crack width as the primary control target, so we need to choose a material with a higher elastic modulus.
Let’s take a diameter of 12 mm and a modulus of elasticity of 40 GPa as an example. The bonding stiffness of BFRP tendons was selected as 15, 20, 25 and 30 GPa/m to analyse the influence on the control index, as shown in Figure 6. The parameter settings of the transverse spacing of longitudinal ribs and other materials are the same as the elastic modulus analysis in Section 4.1.
The results in Figure 6 show that under the condition that the reinforcement is not changed, the bonding stiffness decreases and the crack spacing and width increase, and the increase in the two is equivalent. When the reinforcement ratio is low, the bonding stiffness has a more significant impact on the stress of the BFRP tendons. For example, when the spacing is 8 cm (reinforcement ratio is 0.50%), the crack spacing and width increase of the remaining three bonding stiffness are 9%, 20% and 36%. On the whole, the stress increase of BFRP tendons is less affected by the bonding stiffness.
According to the results of laboratory tests, it can be known that the bonding stiffness of BFRP bars is related to the strength of concrete, the diameter of the bars and the thickness of the protective layer. The bonding stiffness is usually about 25 GPa/m. Therefore, to control the width and spacing of cracks, BFRP bar material and surface structure choice should ensure that the bending stiffness is not <20 GPa/m in actual engineering.
We analyse the influence of the reinforcement ratio on the three major indicators. The article takes three kinds of BFRP bars with diameters of 12, 14 and 16 mm as examples, and the lower value of 40 GPa is selected for the modulus of elasticity. The transverse spacing of the longitudinal ribs and the material parameters are the same as the elastic modulus analysis. The reinforcement ratio
As shown in Figure 5, the crack spacing, width and control stress of the two types of BFRP tendons increase approximately linearly with the spacing of longitudinal tendons. For example, when the spacing is 8–22 cm, the crack spacing, width and control stress of BFRP tendons are increased by 1.4 times. The three major indicators of BRFP reinforcement in Figure 6 change with the fluctuation of the reinforcement ratio, and the overall trend becomes smaller as the reinforcement ratio increases. The diameter of the reinforcement limits the size of the reinforcement ratio. This shows that simply controlling the reinforcement ratio in the crack control design cannot achieve the desired effect. This requires careful consideration of the design of the reinforcement scheme. Three sets of data are selected from the analytical calculation results, and the reinforcement ratio of each set is close. It was shown in Table 3. Through analysis, it is found that the crack width, spacing and control stress are smaller under the reinforcement scheme with small diameter and small spacing. And the smaller the reinforcement ratio, the more significant the difference. Therefore, the diameter and spacing of the control ribs should not be too large to obtain a better crack control effect.
Comparative analysis of BFRP reinforcement schemes
BFRP, basalt fibre.
The current ‘Code’ mainly provides three control standards for reinforcement design for continuous reinforced cement concrete pavement. (1) The average spacing of transverse cracks is not >1.8 m. (2) The average width of the cracks at the embedding depth of the longitudinal steel bars is not >0.5 mm. (3) The tensile stress of the steel bar does not exceed the yield strength of the steel bar.
According to the conclusion that the limit slip of BFRP bar and concrete linear bonding is about 0.5 mm according to the experiment in this paper, we use the analytical method to calculate the crack width limit to be about 1.0 mm. Considering the absence of rain corrosion in BFRP and the reduction of road cost, we can appropriately relax the cracks and spacing limits given in the existing ‘Code’ in the design of BFRP reinforcement. Therefore, we propose to increase the limit of crack width to 1.0 mm, the corresponding limit of average crack spacing is 2.0 m and the reinforcement ratio of the CRC layer is not <0.5%. The actual project must consider various factors such as specific traffic volume and traffic axle load, pavement functional requirements, pavement structure composition, and short-term and long-term economic rationality to select a reasonable reinforcement ratio.
According to the calculation and analysis results of the analytical method in this paper and taking into account the absence of rain corrosion and cost of BFRP bars, we recommend a limit of the average crack spacing of 2.0 m, a limit of crack width of 1.0 mm and reinforcement ratio in the design of BFRP bar reinforcement.
Comparison of analytical method and finite element results
|Analytical method ( = 0
||0.83||171||2.01 < 2.54|
|Analytical method ( = 1
||1.1||229||2.69 > 2.54|
|Finite element ( = 0
||0.88||167||2.14 < 2.54|
Material parameters of finite element simulation
|Concrete||28||2.7 × 1010||10−5||0.15|
|BFRP rib||1.8||4 × 1010||9 ×10−6||0.3|
|Grassroots||20||1.3 × 103||_||0.25|
Comparative analysis of BFRP reinforcement schemes
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of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation 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on vectorized image processing technology BIM Engineering Management Oriented to Curve Equation Model Relationship Between Physical Activity Sedentary Behavior and Body Mass Index in Children and Adolescents Based on FIT Modelo Green supply chain innovation management strategy based on the combination of low carbon economy and e-commerce with big data technology Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Grid data asset relationship and intelligent classification integrating knowledge graph and Internet of Things Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's 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and character intelligence analysis based on semantic ontology and Poisson equation Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G) on the set of maximal subgroups of p-groups Digital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour Algebraic Equations in 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game review IoT-based interaction design of smart home products for elderly families Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression A multidimensional interpretation analysis of Marxism globalization theory based on distributed Internet Application of Nonlinear Differential Equation in Electric Automation Control System Research on the operation and management mechanism of cross-border e-commerce shared practice training base Strategy for financing mode optimization in international trade supply chain based on deep learning model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Strategies for improving Chinese language proficiency based on artificial intelligence technology Early warning and detection of geological disasters based on intelligent genetic algorithm The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Exploring the Application of Modern Fine Arts Artistic Modeling and Color in Modern Art Design Based on Information Fusion Technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Research on China’s tourism marketing model in the data perspective: based on ecological perspective Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations Research on human resource management performance evaluation based on BP algorithm The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in 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processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on urban energy Internet planning based on fuzzy comprehensive method Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the intelligent path of college students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory Virtual Museum Scene Design Based on VRAR Realistic Interaction under PMC Artificial Intelligence Model A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Research on innovative cultivation mode of English and American literature translation talents based on internet environment Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Analysis of quantitative management of online intelligent monitoring of tailing ponds based on the perspective of safety prevention and control Optimization of college physical education curriculum based on G-DINA Optimization of local ethnic music teaching transmission path based on logistic regression model RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Retraction: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406