Span Bridges Research Articles

Dynamic amplification factor of multi-span simply supported beam bridge under traffic flow

Simply supported bridges occupy the majority of bridges. Compared with the flexible large span bridges, Dynamic Amplification Factor (DAF) of them is relatively large and attracts lots of studies. However, most of these studies are focused on the simplified condition that a single vehicle passes through a single span bridge. In general, the bridge bears complex traffic flow rather than a single vehicle. The vehicle type and dynamic loads on the bridge are thus dispersed. It is necessary to explore the multi-vehicle effects on the DAF caused by the complex traffic flow. Also, adjacent spans of a multi-span simply supported beam bridge may have continuous dynamic effects on the vehicles. In this regard, the DAF of a simply supported bridge considering the multi-vehicle and multi-span effects are explored numerically based on the coupling dynamic analysis of traffic flow and bridge, which can also consider the acceleration and deceleration of the vehicles. Cellular Automata (CA) method is used to simulate the traffic flow. It is found that the worse the driving condition of the road roughness, the greater the DAF. When road roughness is Class C, the DAF exceeds the values of American and Chinese codes. Under the action of traffic flow, DAF of the first span of the multi-span simply supported beam is larger than that of the span under the single vehicle. Only considering the effect of a single vehicle may underestimate the dynamic impact on the bridge. Sparse traffic flow has a larger DAF than moderate traffic flow and dense traffic flow in the statistic meaning of the multi-span, because the driving speed of sparse traffic flow is closer to the resonant speed of the bridge and the vibration disharmony of multi-vehicles is smaller. The greater the deceleration of the traffic, the greater the dynamic impact on the bridge.

Unified method for fully automated modal identification and tracking with consideration of sensor deployment

Automated modal identification and tracking provides the key technique for modal parameters based structural condition evaluation. However, the existing automated modal identification and tracking methods have limitations for situation with very few sensors deployed and require manual intervention. This paper proposes a unified fully automated modal identification and tracking method that can be used for structures with many or very few sensors deployed, for closely spaced modes with severe spatial aliasing and for structures with high flexibility or rigidity. A new index, which is called Modified Modal Observability Correlation, is proposed to estimate the similarity of two modes. New spurious mode elimination method and fully automated mode separation and tracking methods are proposed to realize the automated modal identification and tracking. The feasibility and novelty of the proposed method are validated using in-field monitoring data from a long span suspension bridge with high flexibility and a short span bridge with high rigidity and closely spaced modes. Automated modal identification and tracking for situations with many and very few sensors deployed for the two bridges are conducted, and extreme situation of closely spaced modes with spatial aliasing in mode shape is also considered to prove the novelty of the proposed method. Comparisons with existing mainstream methods are also conducted. The results show that the proposed method yields the right identification and tracking results in all situations, showing novelty in automated modal identification and tracking for structures with closely spaced modes having severe spatial aliasing when only very few sensors are deployed.

Life cycle dynamic sustainability maintenance strategy optimization of fly ash RC beam based on Monte Carlo simulation

Small and medium span bridges account for 86% of the bridges in China, and maintenance is necessary to ensure their safety and serviceability. The infrastructure sector causes substantial CO2 emissions, and it is one of the most polluting sectors. Fly ash (FA) is often used in bridges to reduce CO2 emissions. Moreover, the lack of temporal information is an important limitation of the life cycle CO2 assessment of bridges. Taking the upper structure of small and medium span bridges as the research object, this paper proposes a Monte Carlo simulation-based life cycle multi-objective dynamic maintenance strategy optimization model to analyze the life-cycle carbon emissions, costs, and life cycle reliability of different preventive maintenance (PM) and essential maintenance (EM) methods of FA reinforced concrete (RC) beams. Considering the continuous effect of carbon emissions and the net present value of the cost, the time effect of carbon emissions and costs of maintenance and reinforcement at different times is analyzed. Through 50,000 simulations of the case beam, the parameter characteristics of different optimal reinforcement schemes and the relationship between the reliability, life cycle CO2 emissions, and costs are studied. When introducing the discount rate, the essential maintenance with higher carbon emissions and costs has a marginal effect, i.e., the longer that future carbon emissions and costs occur, the smaller the net present value. The proposed method is demonstrated to be effective at supporting decision making for bridge managers. It would be helpful for identifying the effectiveness of PM and EM, selecting optimal maintenance time interval, and providing a rational estimate of the CO2 emissions and costs of bridges.

Improvement of inspection and maintenance gantries in the Honshu Shikoku Bridges

Currently the importance of bridge inspection and maintenance is widely recognized. It seems reasonable to mention that a gantry is essential for the inspection and maintenance of a long span bridge. Therefore, most long span bridges are equipped with a gantry as a permanent access tool. Honshu Shikoku Bridges compose three highways and one railway between Honshu and Shikoku islands in Japan. Three routes consist of 17 long span bridges with a main span of more than 200 m. Since the bridges had been constructed with accumulating knowledge for construction and maintenance from 1975 to 1999, a gantry had also been improved for gaining better access. Despite the improvement, there were still a certain number of inaccessible elements and components, so further improvement is being made to increase access capability due to growing requirements of arms-length inspection and maintenance. There are 152 gantries in total, consisting of four types of gantries: a suspended gantry, an internal gantry, a tower gantry, and a cable gantry. This article presents the gantries installed in the Honshu-Shikoku Bridges, containing improvement histories for each type of gantry in the construction and several efforts to improve access capability in the maintenance.

Reference-free damage identification method for highway continuous girder bridges based on long-gauge fibre Bragg grating strain sensors

Damage information is important for bridge condition assessment and maintenance. Extensive research has been conducted to extract this information from the responses of bridges (e.g., displacement and strain). However, among these methods, the responses of intact bridges are required for damage identification, which is difficult to acquire for most existing bridges. For such circumstances, a reference-free damage identification method for highway continuous girder bridges is presented in this paper, which takes advantage of long-gauge fibre Bragg grating strain sensors. A 1:10 indoor model experiment is conducted to verify the practicability of the method, which is effective under various damage and vehicle parameters. The extents of damage identification errors can reach 50% because of experimental errors. Furthermore, the number of bridge spans and road surface roughness are proven to have no effect on the method with a bridge-vehicle coupling system, and the extents of damage identification errors are below 10%.

ОСЛІДЖЕННЯ НАПРУЖЕНО-ДЕФОРМОВАНОГО СТАНУ ЗАЛІЗОБЕТОННОЇ МОНОЛІТНОЇ ПРОГОНОВОЇ БУДОВИ МОСТУ ПІСЛЯ ТРИВАЛОЇ ЕКСПЛУАТАЦІЇ

Introduction. The vast majority of reinforced concrete bridges built in Ukraine until the 1950s were mostly monolithic. Spans were built of monolithic concrete with a length of 2, ..., 6 m and ribbed structures with a length of 8, ..., 15 m. Monolithic spans were provided with good spatial work and considerable durability. Problems Statement. The main disadvantage of reinforced concrete monolithic spans of bridges built before the 50s of the last century was the low design strength of concrete, which over time led to defects and thus reduce their durability. In addition, the vast majority of such bridges with monolithic spans were built according to technical standards of 1962 and earlier, and at this time the load capacity does not meet the requirements of modern regulations. The need to study the stress-strain state, to determine the bearing capacity of the spans of bridges after long-term operation, given the above, is an important factor in deciding on the feasibility of their repair. Purpose. Research of the stress-strain state of the structures of the monolithic spans of the bridge after its long operation. Materials and Methods. A survey of the monolithic reinforced concrete span of the bridge, built in 1940, was carried out. It was established that the span of the bridge, in accordance with the identified defects, is in a limited working condition (condition 4). And given the low load capacity of the span, condition 5 (inoperable) was finally adopted. The main defects found during the inspections of the span significantly reduce its durability and bearing capacity. Destruction of the protective layer of beams of the span with exposure and intensive corrosion of reinforcement with weakening of the area by more than 10% was revealed. Based on the data from the inspection of the bridge, it was decided to conduct additional tests to determine the stress-strain state of the spans, the availability of strength reserves and the possibility of strengthening them or the need for replacement. Results. Based on the test of the bridge, the parameters of spatial work and stress-strain state of the span were determined by measuring the absolute deformations (deflections) under the action of static load. According to the test results, the value of the design coefficient k, which is in the range of 0.1… 0.79, was determined. The values of the transverse distribution coefficients (CRC) for each beam were experimentally determined and the numerical analysis of the field and analytical spatial work of the span of the bridge on the transverse distribution coefficients was performed. Conclusion. The results of the analysis of experimentally and analytically obtained test data of the bridge indicate the normal spatial operation of the bridge spans, which corresponds to the nature of the idealized finite element scheme, good spatial rigidity and the presence of strength reserves in the bridge spans. Key words: bridges, span, tests, deformability, technical condition.

Dynamic behaviour of railway poles built on bridges under train passage in high-speed railways and a simple evaluation method

There have been studies on increased dynamic response of bridges for high-speed railways (HSRs) and damage to the overhead contact line system (OCS), including wires supported by railway poles placed on resonant bridges. However, the dynamic behavior of the poles installed on bridges under train passages, which is considerably related to the damage to the OCS, has not been systematically examined. This study aims to examine the dynamic behaviour of poles installed on bridges and provide a simple calculation method or guidelines for bridge design and OCS maintenance considering pole vibration. First, a field test of a HSR bridge with poles was conducted, and numerical simulations using a finite element model of pole and bridge were validated. Next, a parametric study using the validated numerical model was conducted. The results demonstrated that the multiple resonances observed when the excitation frequency of travelling train matches the bridge and pole frequencies can cause dominant vibration of the poles built at the bridge edge, thus posing a risk of OCS wire fatigue. In addition to the understanding of the basic dynamic behavior, a simple calculation approach based on a theoretical model considering the interaction between the bridge and poles installed at the bridge edge is developed for practical use. The proposed model clarifies train-induced pole top displacements with standard bridge and pole specifications in Japanese HSRs. Using the bridge/pole frequency ratio and bridge span length, the results are presented in the form of a pole response map that can be used to calculate the pole top displacement. Finally, an example of bridge frequency guidelines is provided for extracting and evaluating poles that may require attention for maintaining the OCSs.

Boundary Conditions Matter-Impact of Test Setup on Inferred Construct Mechanics in Plated Distal Femur Osteotomies.

The mechanics of distal femur fracture fixation has been widely studied in bench tests that employ a variety of approaches for holding and constraining femurs to apply loads. No standard test methods have been adopted for these tests and the impact of test setup on inferred construct mechanics has not been reported. Accordingly, the purpose of this study was to use finite element models to compare the mechanical performance of a supracondylar osteotomy with lateral plating under conditions that replicate several common bench test methods. A literature review was used to define a parameterized virtual model of a plated distal femur osteotomy in axial compression loading with four boundary condition sets ranging from minimally to highly constrained. Axial stiffness, fracture gap closure, and transverse motion at the fracture line were recorded for a range of applied loads and bridge spans. The results showed that construct mechanical performance was highly sensitive to boundary conditions imposed by the mechanical test fixtures. Increasing the degrees of constraint, for example, by potting and rigidly clamping one or more ends of the specimen, caused up to a 25× increase in axial stiffness of the construct. Transverse motion and gap closure at the fracture line, which is an important driver of interfragmentary strain, was also largely influenced by the constraint test setup. These results suggest that caution should be used when comparing reported results between bench tests that use different fixtures and that standardization of testing methods is needed in this field.

Multilayer bamboo composite hollow-core: Lab-scale and non-destructive testing of full-scale concrete bridges considering transverse bamboo pole as shear reinforcement

Due to bamboo's superior mechanical qualities, it has been often used in rural construction, however research on bamboo poles as structural reinforcement has been fairly limited. The purpose of this paper is to introduce a novel composite system made of several layers of bamboo hollow-core, named Mulbamcomphol concrete system for simplicity. On that premise, this paper analyzes the implementation of a composite slab system on a short span bridge consisting of concrete and multilayer bamboo arranged in a cross-section, without any utilization of steel reinforcement. Mulbamcomphol is so termed because to the presence of entire bamboo in this composite system (multilayer cross-ply bamboo composite hollow-core). Non-destructive testing will be utilized to analyze the flexural behavior of a full-scale vehicle under an actual service load. Due to the difficulties of obtaining exact results and the potential for bias, the study is conducted using a FEM-based model (thick shell and solid element). Bamboo slabs with a weak axis can sustain less stress than regular slabs, but they deflect by 81% less when subjected to the same 21 kN force. Simultaneously, when tested on a rigid axis, multilayer bamboo pole slabs can withstand a force 52 % more than conventional slabs at the same concrete volume, resulting in a 73 % reduction in deflection. Bridges constructed using Mulbamcomphol cross-ply bamboo composite concrete slab systems, on the other hand, demonstrate higher resistance to flexural loads in terms of ductility and maximum load that can be carried in both analysis and experimental testing.

Comparative study of microimage strain sensing system and FBG sensing system for strain measurement

Strain is one of the conventional parameters that evaluate structural performance in the field of bridge health monitoring. Among the various techniques that have been utilized to measure strain, fiber Bragg grating (FBG) sensing system has been broadly applied for long-term strain monitoring in civil engineering, whereas microimage strain sensing (MISS), with its corresponding sensors designed and fabricated, is a new technology for strain measurement. Furthermore, the MISS system is developed to monitor the strain of small and medium span bridges. To verify the performance of the MISS system in strain measurement, a comprehensive and comparative study of the MISS system and FBG strain sensor system is provided. First, the technical features of the two technologies were described. Moreover, more details were expounded in terms of service life, deployment, coverage, cost, and temperature dependence. Then, assisted data comparisons were carried out through component testing. Results show that the proposed MISS system is effective and robust. Therefore, the proposed MISS system will serve as a promising strain measuring alternative in the field of bridge structural health monitoring.

Influence of Permanent Deflections on The Vibrations of Bridge Spans in Operating Conditions

Abstract The paper presents the method and results of numerical simulations of the mutual dynamic interactions between vehicles and the bridge structure, which has defect in the form of excessive permanent deformations. The parametric analyses were carried out taking into account the following parameters: permanent deflections of spans, with a maximum value ranging from 0 to 150 mm, vehicle speed from 10 m/s (36 km/h) to 30 m/s (108 km/h), and two types of heavy vehicle suspension systems, each with very different properties. The presented analyses are based on the characteristics of real motorway bridge structure with permanent deflections of the multi-span main girders, constructed of steel beams and reinforced concrete slab. The proposed procedure of dynamic numerical analysis can also be useful in the assessment of the influence of permanent deflections on the interactions between vehicles and other types of bridge structures.

Determining the effect of reinforcing a cement-concrete coating of bridges on the stressed-strained state of structures

This paper reports a study of the cement-concrete coating on bridges using FRP reinforcement. That has made it possible to design optimal structures by selecting the height for reinforcement arrangement in the layers of a roadbed in order to ensure strength characteristics. An engineering method for calculating a hard roadbed with composite reinforcement has been devised, which makes it possible to take into consideration its work both in a joint package of the structure with a slab and separately – when it exfoliates from the slab of the bridge's span structure. Underlying this research are effort-determining methods, estimation dependences from the theory of bending layered structures, as well as dependences from elasticity theory to assess the strength of materials for a roadbed. The consideration of shear strains when designing slabs has helped establish that the deflections according to the devises method were 1.4 times larger than those in the classical approach. The method was tested by a numerical experiment, which confirmed the need to use composite reinforcement in the upper layers of a road surface on bridges, which improves its durability by 1.2 times. The results of the numerical experiment indicate that the equivalent stresses in the lower layers of a free-moving roadbed were 2.91 MPa, and, when operating in a joint assembly with a slab, they took a negative value (–0.2 MPa). Practical application of the devised calculation method makes it possible to determine the refined normal stresses in the layers of a roadbed, taking into consideration the characteristics of structure operation. Owing to this, additional opportunities open up for calculating the roadbeds of bridges whose design utilizes the most common types of span structures in the bridge industry.

Comparison of the Behavior of Haunched Single Span Bridges with Other Types of Bridges

A novel method for the design of variable depth single span bridges having maximum depth at the ends and minimum depth at midspan has been presented in a previous paper by the author. This design is possible by “fixing” the ends to “lock-in” negative moments, so that the positive moments at midspan are greatly reduced, allowing a shallow section to be used. The negative moments are locked-in by having short end cantilevers with tie-downs (rock/soil anchors) at their ends. The present paper expands on this work by comparing the static behavior of haunched single span bridges to that of other well-known bridge types, notably: (1) inclined leg and V-frame bridges, (2) three span continuous bridges, and (3) deck tied arch bridges. It is shown that the behavior of haunched single span bridges with tie-downs is very much the same as these three other well-known types of bridges. These haunched single span bridges are intended to be used for long span river crossings or highway overpasses, where only a single span is required, and an aesthetically pleasing graceful shape is desired, and/or the vertical clearance below needs to be maximized.

An analytical solution to traffic loads on long span bridges

The paper deals with the prediction of extreme values of traffic loads and load effects of long span bridges. The traffic states are idealized a series of spatially stationary stochastic fields, which correspond to free, congested and full stop traffic states. For each traffic state, the Poisson point field is invoked to model the spatial distribution of vehicles. The vehicle density is then determined initially by the empirical formula based on microsimulation and modified further by a Poisson point model with a limited minimum space lag. With the modified vehicle density and influence functions of the bridge response, the expressions of the mean values and variances of the traffic loads or load effects are derived based on Poisson model, which are very simple compared with Ditlevsen’ formulas. Because the bridge loads are the sum of a large number of independent vehicle weights, the normal distribution is a reasonable model for loads and load effects. Correspondingly, the long term characteristic values and distribution functions of traffic loads and load effects are estimated easily with the derived mean and variance . The numerical analyses show that all results calculated are very close to those by the popular microsimulation methods. Hence, an analytical method is developed for predicting the traffic loads and load effects, which can be easily used to develop a design code for a long span bridge. Moreover, the analytical method based on Poisson models incorporated with the normal distribution of loads or load effects would make it more easily combine the traffic loads with wave and wind loadings for a floating bridge.

ESTIMATION OF THE LOAD-CARRYING CAPACITY OF THE CONSTRUCTIONSOF THE CULTURAL HERITAGE OBJECT«THE OLDENBURG’S COMPLEX. LOVERS BRIDGE. 1900»

Statement of the problem. A lot of years of uncontrolled operation of the bridge have resulted in its supporting structures to an emergency condition. For further safe operation of the bridge, structural strengthening of almost all elements is required. Since their direct strengthening is impossible without violating the original appearance, we set the task of strengthening the structures indirectly by choosing permissible schemes and limit values of loads on the span on mathematical and physical models.Results. To solve this problem, we carried out and analyzed the results of qualitative and quantitative theoretical studies of the object of cultural heritage «The Oldenburg’s complex. Lovers Bridge. 1900». We carried out an instrumental inspection of the monument, designed and tested a laboratory model of the original structure.Conclusions. The data obtained as a result of comprehensive studies made it possible to assess the load-bearing capacity of the structures of the cultural heritage object, analyze the possibilities of loading the bridge span and make a conclusion about its further safe operation in modern conditions.

Influence Analysis of the Antifriction Layer Materials and Thickness on the Contact Interaction of Spherical Bearings Elements

Bearings are the supporting elements of bridges. They perceive vertical and horizontal loads from the bridge span. Spherical bearings are one of the construction common types. The material and configuration of the anti-friction layers determine the bearing performance. The paper performed the contact deformation analysis of spherical bearing elements at a nominal vertical load of 1000 kN. The six types of the spherical sliding layer material are considered: ultra-high molecular weight polyethylene (UHMWPE) from three different manufacturers, modified polytetrafluoroethylene (PTFE), and composite materials by PTFE with two different forms of reinforcing bronze inclusions. Young’s modulus, Poisson’s ratio, and strain curve are obtained experimentally for spherical sliding layer materials. Paper considered the influence of the sliding layer material on the contact parameters and deformation characteristics of the structure with a standard interlayer thickness by 4 mm. Research observed significant the composite interlayer deformation and the appearance of “no contact” zones on the mating surfaces. The option of increasing the sliding layer thickness up to 6–8 mm is considered. A decrease is observed in the maximum level of contact parameters by increase of the sliding layer thickness. The influence of the anti-friction layer materials becomes insignificant on the bearing deformation with an increase of the spherical sliding layer thickness.

A Review Paper on Lithely Arch Bridge

Abstract: In this we are studying modern technology of arch bridges i.e “Lithely Arch technology”. In this technology the blocks are precasted, blocks are jointed with screed concrete and lifted to a location by transporting them on site and again lifted to form arch bridge and then working towards the finishing. It also includes the study of reading the drawing, Bearing capacity, prepared all over detailed estimate and stability checks. Various IRC codes, IS codes is studied. The masonry arch is one of the oldest structural elements. Brick masonry arches have been used for hundreds of years. This technical note is an introduction to brick masonry arches. Many of the different types of brick masonry arches are discussed and a glossary of arch terms is provided. Material selection, proper construction methods, detailing and arch construction recommendations are discussed to ensure proper structural support, durability and weather resistance of the brick masonry arch. Lithely arch method is innovative, modern, economic and fastest method of construction of arch bridges. This method is invented in Maharashtra, India. Bridges constructed using this method are purely compressive in nature. This method utilizes precast units for construction of arch bridges. This method can be utilized for rehabilitation of old arch bridges in India very economically and with less time. The lithely arch is India's only Precast, Unreinforced Bridge solution and is developed on the lines of the conventional stonemason built arch bridges. Arch bridges are the most reliable and time tested solution for medium span (5m to 20m) bridges. These bridges have been standing strong for centuries and while they were initially designed for 8-10 tons of load are carrying modern vehicular load without any distress. Keywords: Lithely Arch, Wire Rope, Concrete, Spandrel Wall, Sockets, Screed, Moulds, Strip .

The definition of «ship» by international and national regulations

The article analyzes the definition of «ship» by international and national regulations. It is noted that there are many examples of the legal definition of «ship». There are attempts in international law to define the term «ship».
 The definition of «ship» under the International Convention for the Prevention of Pollution from Ships contains a tautology, as the term «ship» is defined as «ship». Thus, for this Convention, a ship is understood to mean not only ships in the usual sense of the term but also «stationary or floating platforms». In other international legal acts, there is also no unambiguous (universal) interpretation of the concept of «ship». And in one of the most important sources of international maritime law - the comprehensive universal The United Nations Convention on the Law of the Sea 1982, although the term «ship» is often used, there is no norm-definition that should explain it. Based on the study, several features that are inherent in the vessel, namely: the concept of the vessel, is applied only to the Merchant shipping code of Ukraine, and the vessel is understood as a «self-propelled or non-self-propelled floating structure». However, the restriction of the vessel as a «floating structure» from other «floating objects» can be carried out for their intended purpose. Thus, «floating objects», in contrast to ships, are not intended for permanent stay in a floating state but are brought into such a state for their one-time movement or specific work. Therefore, they should not be considered as vessels - rafts, sealed tanks, which are towed to the place of their installation (pipes, tanks, pontoons), delivered on a tug in a floating state of metal structures (oil rigs, bridge spans, etc.).
 Based on the study, it was concluded that the Merchant shipping code of Ukraine uses concepts of the same meaning: ship (so-called section II Merchant shipping code of Ukraine and this concept is used in the title of many articles), merchant ship (Part 1 of Article 15 of the Merchant shipping code of Ukraine), sea vessel (Part 1 of Article 13 of the Merchant shipping code of Ukraine). Therefore, to improve the Merchant shipping code of Ukraine, it should be borne in mind that the concepts of «ship», «merchant ship», «sea vessel» are differentiated and have different meanings and relate to each other as general and special.

Effect of adhesion failure and temperature on the mechanical behavior of orthotropic steel bridge deck

Orthotropic steel bridge decks (OSBDs) have been widely used in long span bridges with many advantages, however, the long-term performance of the asphalt wearing course is still questionable. There is a lack of understanding the causes of cracking and adhesion failure between the tack coat and the steel plate, etc. Therefore, this paper focuses on the adhesion failure between the asphalt wearing course and the steel plate, and the temperature influence on the mechanical behavior of the orthotropic deck. The five-point bending beam test model using the finite element model is used in this investigation in order to clarify the effect of ratio of adhesion failure and the temperature on the mechanical behavior of the OSBD. The ratio of adhesion failure varies from 0.1 to 0.5, and the temperature changes from 30oC to 60oC. The results reveal the influence of temperature and adhesion failure on the mechanical behavior of the OSBD

The use of bystal elements in bridge spans

The object of the research is the structures of spans made of two types of steel of different strengths. The article examines the effectiveness of the use of the bisteel beams of bridge structures based on a technical and economic analysis, which consists of determining the value of construction and structural coefficients for beams of the same spans in order to identify metal savings. In the course of the study, the effectiveness of using steels of various strengths in a set was confirmed. The material described in the article can be used as the basis for the practical application of bisteel spans.