A curved beam is generally referred to as a body, the geometric figure of which has been created by the motion of a plane figure in space. The plane figure is termed as the cross-section of the curved beam, wherein the center of gravity of the beam consistently follows an axis or a specific curve. The cross-section of a curved beam can be effectively used to identify the differences in the curved beams. This study focuses on the curvature of beams concerning earthquakes. In addition to that, a comparative analysis is mentioned with regards to the varied behaviour of arches of ancient times with the arches of modern times. For instance, ancient arches may be illustrated through several medieval architectures, while modern construction works, such as bridges may be cited as examples of modern arches.
A masonry arch is known to comprise several components, such as a keystone, a voissour, an impost, an extrados, and an intrados. The other components include a clear span, a rise, and an abutment. Various types and categories of arches have been explored in this regards as well. Since beams and may arches are restrained as well as unrestrained, an occurrence commonly termed as the torsional buckling of beams is found to be prevalent in case the compression flange becomes free and is found to rotate after being displaced laterally. The torsional effect has been found to be a result of the combination of both the tensile and the compressive forces. The reasons for such occurrence have been explored in this study, and the potential actions for the mitigation of the challenges have been identified in this regards as well.
To evaluate the behaviour of curved beams, it is necessary to consider the design of the curved beam in question. Considering that the area of cross-sectional of a beam is symmetrical, it is to be estimated that the impact of the load, which is placed at the plane of symmetry. It may be mentioned in this context that the axis of symmetry is evident on the plane of curvature. Furthermore, the determination of the stress which is normal to the cross-section of the beam may be performed with the aid of the formula.
‘N’ is established to be the longitudinal force, while ‘F’ is the area of cross-section. Furthermore, ‘M’ is the bending moment established in the cross-section, with regards to the axis, Z0, which passes through ‘C’ which may be defined as the center of gravity of the cross section (Wang, Lee & Huang, 2016). On the other hand, ‘y’ is determined to be the distance of the fiber to ‘z,’ the neutral axis. The fiber refers to the one, which is being examined concerning ‘z.’ ‘Sz= Fy0’ is defined as the static moment of the area of cross-section, with regards to ‘z.’ ‘P’ in this respect can be termed as the radius of curvature of the fiber. In addition to the aforementioned parameters, the displacement of the neutral axis ‘Y0’ is estimated to be relative to the center of curvature. It is established that ‘Y0’ is directed towards the center of curvature of the beam to be examined.
The distribution of stress in a curved beam in case of being bent. It is to be noted that h and d indicate the height and diameter of the beam’s cross-section. It may be stated in this regards that the concave edges of the beam have the maximum values for the normal stress. Furthermore, the values of normal stress for the beam may vary in the concave edges, in accordance with hyperbolic law (Ni, Chen, Teng & Jiang, 2015).
To ascertain or study the behaviour of curved beams concerning earthquakes or seismic movements, the seismic resilience of the beam may be taken into consideration. Under experimental conditions, the behaviour of curved beams under seismic activities can be ascertained by studied certain parameters such as the curvature of the beam, ductile cross frames and the seismic isolation (Di Re, Addessi & Sacco, 2018). In addition to that, the live load, column rocking as well as the Abutment-soil interaction are to be taken into account for studying the behaviour of curved beams under seismic activities.
It may be stated in this regards that severe structural damage has been observed in case of past earthquakes on horizontally curved bridges. The bridges are to be considered as the curved beams in this regards. Moreover, one of the major issues noted in this respect has been noted to be the unseating of the deck from the abutment (Di Re, Addessi & Sacco, 2018). This has been identified as a result of the excessive plane-body motion of the decks, resulting from the seismic activities. The irregular geometry of the curved beam along with the seismic poundings observed between the abutments and the decks have been noted to be the prime cause of the failure of the curved beam structures.
Furthermore, it may be stated in this regards that the maximum seismic response of the curved beam has been found to be relevant to the angle of input of the designated earthquake (Miyamoto et al., 2016). The irregularities in the structure of the curved beam or the curved bridge can be attributed to the interaction between the torsion forces and the moment. The response of the curved beam structure in response to the input of the one-way earthquake can be determined through the uniform expression derived from the unfavorable angle of the earthquake. Hence, the maximum response of the curved beam structure corresponding to the seismic activities of the earthquake can be determined.
The various input angles of earthquake acceleration. It depicts a single-direction input, while Figure 3(b) depicts orthotropic dual-direction inputs. On the contrary, Figure 3(c) illustrates Skewed dual-direction inputs relating to the seismic activities of the earthquakes. The input angles are considered concerning a random direction in a plane.
The implementation of the concept of a curved bridge has been widely prevalent in several architectural structures. The constructions of railways have a widespread implementation of the curved beams. However, since the 1970s, multiple devastating earthquakes have been noted by engineers. The impact of the earthquakes on the curved beams or the girder bridge, which constitutes an important variation of the curved beams, is regarded as vital for the study (Hsu & Halim, 2017). For instance, the 1971 earthquake of San Fernando is known to have caused extensive damage to a multi-span girder bridge. Hence, special attention has been paid to the seismic responses generated by curved beams or bridges, as a subject for study and understanding in the field of engineering (Pydah & Sabale, 2017). The Shake table model has been noted in this regards to developing a better understanding and knowledge concerning the matter.
The shake table model has been an extensive aspect of earthquake engineering. The use and application of the shake table model are observed as a technique implemented to test the response of certain structures, such as curved beams, or girder bridges to extensive earthquakes. Furthermore, the study of the seismic performance of rock slopes and soil is also undertaken through the shake table model. This model is typically used for the evaluation of the performance of scales slopes, curved beams, and other structural models with the aid of imitating is simulating earthquakes recorded over some time (Attary et al., 2015). The specimen or the curved beam or the curved bridge to be shaken is experimented on until it reaches the point of ‘failure.’ With the aid of modern devices and other innovative technologies, it is possible to interpret the dynamic behaviour of the curved beams exposed to the artificially simulated earthquakes. The degrees of freedom are generally considered as 6 in this case of the experiment.
However, it may be stated in this context; multiple shake tables are used to determine the seismic response of the curved structure. Figure 4 depicts 5 shake tables used for assessing the seismic response of the curved structure, upon being excited as a result of the artificial seismic waves generated. The figure illustrates the target set for the seismic waves to excite the beam, while the response for each of the tables is noted, to determine the value or the study the behaviour patterns of curved beams. In addition to that, it may be mentioned in this regards that the mechanical model of expansion joints play a crucial role in evaluating the behaviour of curved beams, under the circumstances of earthquakes or seismic excitement (Pydah & Sabale, 2017).
Expansion joints are common inclusions on a bridge which allow expansion as well as the contraction of the metals that constitute the bridge. The expansion and contraction are generally consistent with the temperature variations that are a frequent and common occurrence. Regardless, the absorption of shocks related to the seismic transitions is also regarded as a part of the application for expansion joints (Berardi & De Piano, 2018). Hence the accommodation of any machine as well as thermal changes within the system can be achieved through the implementation of expansion joints in curved beams or curved bridges. The use of reinforced steel and other steel structures, along with prestressed concrete are used for the development of the bridge expansion joints to make it be able to withstand seismic activities, which the curved bridge may be subjected to.
A variety of expansion joints are noted to be in existence. For instance, the expansion joints are prepared such that it can accommodate certain specific changes being made to the curved structure. Considering the example of a bridge, one may note that generally, expansion joints focus on accommodating movement within the range of 30 to 1,200 millimeters (Sun, Cui, Qin & Hou, 2018). These joints are designed to withstand from small to medium, as well as large-scale movement. Since this study would eventually discuss the role of masonry in the resistance towards earthquake on various kinds of arches, it may be suitable to discuss the issues related to masonry and expansion joints. The curved structures, often made out of masonry may also display resilience towards seismic activities. The manufacture of rubber expansion joints has been undertaken to reduce cracks, as well as for shock absorption from the past series of earthquakes.
It may be mentioned in this regards, that despite major attempts at establishing the stability of the expansion joints for the curved beams, a failure mode may be evident. The inability to support or withstand the seismic activities or shocks often results in a failure mode. The phenomena of collision and yielding have often been responsible for the failure mode (Berardi & De Piano, 2018). However, the study of these phenomena under seismic shocks can be used to analyse and evaluate the role or the influence of the expansion joints of the curved bridges. The study chiefly focuses on assessing the response of the curved beam or the curved bridge to gather a better understanding of the events. As initially depicted in Figure 3, the multi-directional inputs of earthquakes have been demonstrated, which aids in the development of an understanding of the SDOF structure as well. SDOF refers to the single degree of freedom, as applied previously for the Shake table model, which implements the use of 6 degrees of freedom, as opposed to one in the SDOF for the study of the multi-directional inputs of earthquakes (Sun, Cui, Qin & Hou, 2018).
Evaluative studies have been performed upon analysing the deformation of the piers, as well as the collision and sliding of the expansion joints involved in the curved bridges. It may be mentioned in this context that among several methods for performing a seismic analysis, the use of the SRSS combination methods has been focused on by various engineers and scientists. SRSS, also known as the square root of the sum of the squares is one of the widely preferred methods for performing the seismic analysis. With the aid of the SRSS method, the use of response spectrum analysis has been implemented, to analyse the factors involved in the generation of seismic response (Berardi & De Piano, 2018). The type of pier connection used, as well as the beam, and the curvature was analysed as being part of the contributing factors. Furthermore, the use of the complete quadratic combination 3 or the CQC3 method was implemented to assess the seismic response of the curved structures used in the case study (Lemos & Campos Costa, 2017).
Regardless, it may be stated in this aspect that the implementation of the Caltrans Seismic Design Criteria, put forward in 2013, have been found to be the most effective regarding generating results for the seismic response. The elastic earthquake response can be calculated from the design above criteria. However, undertaking a study on the better evaluation of a particular type of curved structure, it was noted that the Caltrans method is more appropriate for the study of the curved girder bridge (Lemos & Campos Costa, 2017). The reason for the implementation of the Caltrans method is determined as the suitability of the options regarding the detailed differences in the construction, while the seismic behaviour is observed in accordance with the seismic behaviour.
The complex geometry involved in the construction, as well as the fabrication of curved beams, discussed in this study, provides an example of the wide applicability of curved beams. Furthermore, the study aims at addressing the process of fabrication that implements elastic deformation for the construction of development using curved beams. Expanding control over the curvature of the beams as well as the design surface may aid in the establishment of a network of a quadrilateral mesh, along with a spherical vertex (Sun, Cui, Qin & Hou, 2018). The advantage of the aforementioned construction can be regarded as the geometric structure which provides immense support. A symbiosis of the geometry, the fabrication, as well as the load-bearing behavior of the curved beam structures can be studied in this regards to gathering a better understanding of the constructions which use curved beams.
A prime example of a construction which implements the use of curved beams can be cited as the Asymptotic Gridshell which can be designed for courtyards. Figure 5 depicts various kinds of Gridshell which can be constructed with the aid of curved beams. The figure demonstrates the principal curvature lines which constitute the major framework for the gridshell. Furthermore, a traditional gridshell is depicted in the figure, while illustrating the asymptotic lines present in a gridshell. It may be stated in this regards that gridshells are capable of bearing massive amounts of load, despite the requirements of very few materials for its development or construction (Biondini, Camnasio & Titi, 2015). In addition to that, it may be stated, that the presence of the lamellas aid in the circular development and an elastic assembly via the weak axis.
In addition to that, it may be mentioned in this regards that the network an establishment as a result of the kinetic behaviour which had been determined formerly. Furthermore, curved grid structures can be observed to be used in the Eiffel Tower Pavilions which has been designed by the Moatti Rivière Architects (Biondini, Camnasio & Titi, 2015). Additionally, Frei Otto in 1975 had designed the Multihalle Mannheim which implements a similar use of the curved grid structures, such as the one mentioned in the Eiffel Tower Pavilion. The right node-angles along with the circular lamellas are known to be evident on the surface of CMC, which is commonly known as Constant Mean Curvature (Sarlis et al., 2016).
It may be stated in this regards that the behaviour related to the load-bearing which is demonstrated by the structures, widely constructed through the implementation of the design of the curved beams. The study of the load-bearing behaviour may be pointed out as the key to understanding the seismic response of the structures discussed previously. It may be stated that since the construction of the curved beam structure does not require many materials for the provision of support, the resilience that this aforementioned structure may demonstrate with regards to seismic actions may be considered to be consistent with the load-bearing structures. Additionally, in case of performing a comparative study between two prime structures which uses curved beams, namely, a gridshell and a grillage (Sorace & Terenzi, 2016).
The strong axis of both the structures has a bending stiffness, which compels the lamellas to act as a beam grillage. In addition to that, restraint stresses are noted in the lamellas as a result of the elastic erection process. However, the bending moment in the weak axis of the curved elements is noted to increase as a result of the stress created by the compression of the curvatures present in the curved beams. The combination of the repetitive curvature parameters provides great potential regarding the load-bearing behaviour and the assembly of the gridshells, which have been strained as a result of the construction activities and geometric advantages (Franke, Franke & Harte, 2015).
The modern buildings, bridges, walls of today evolved from the ancient structures. The Roman people were great builders and architects, and they paved the way for the modern day architects to make magnificent buildings using definite methods (Li & Zeng, 2016). The Romans were productive as well as clever and developed span arches, which are smaller to build roofs or tombs. Gradually the size increased, and they made elaborate bridges as well as amphitheaters. They created circular arches, parabolic arches, pointed arches and many more. Based on these models the modern day civil engineers have developed processes to build monuments. While the ancient people used stones and masonries and developed special concrete of the Roman era, the modern builders of today use steel and trusses, concrete that is pre-stressed and beams (Li & Zeng, 2016). The monuments of today have more rigidity and span more than the ancient Roman civilization.
When something is constructed, there is always the possibility that the construction will face erosion and damage from natural causes or might be destroyed or damaged for some structural faults in it (Song, Ming, Wu & Zhu, 2014). To analyze and determine the faults and possible failures in different structures, the failure modes came into existence in the late 20th century. Problems, which might arise from malfunctioning constructions, can be easily analyzed by using a structure, which is high in quality and follows techniques and systems to analyze failure specifically. They are the initial step for the study of structures in a reliable way. It has some aspects, which needs to be followed like subsystems, components, and assemblies to identify the possible modes of failure as well as their effects and causes (Song, Ming, Wu & Zhu, 2014). It can both be devised as a qualitative tool of measuring flaws and be deemed as quantitative when the mathematical and statistical model of failure ratio is combined.
The failure modes are considered tools using forward logic or inductive reasoning and can be used to analyze significant tasks in safety engineering, quality engineering, as well as reliability engineering (LARSSON, 2015). A mode of failure is considered to be successful when it identifies imminent failures based solely on the experiences it had with identical processes and products. The whole concept is based on logic or physics. The failure modes are frequently used in the industries dealing with manufacturing and development to have the proper idea of the systems related to the life cycle of a product. In this paper, the failure modes are considered to understand the modes of failure as well as deformability of materials like concrete (LARSSON, 2015). They are further used in case of masonry to specifically understand the strength of walls of block masonry, which are significant to formulate the behavioral aspects of assembly.
In the contemporary times, to it is quite hard to understand or determine the results of experiments done on the interactive nature of materials like blocks of concrete, mortar as well as the relation between bedding and vertical joints (Costa, Penna, Arêde & Costa, 2015). In this case, one can use functional analysis to evaluate correct modes of failure. They can also be used to mitigate risks or to reduce them so that the effect is not so severe. Failure modes are also used to understand the deformability of masonry walls made of mortar. It has been observed that failure modes when used to identify faults and flaws in walls made of mortar, they go crushing of the mortar of bedding as well as a tensile stress and that cut the vertical joint of mortar and the block as well (Costa, Penna, Arêde & Costa, 2015).
The analysis of reliability can be done using other methods too like the fault tree analysis, which is a logic played backward or deductive form of analyzing failure and can be used at one time to find different failures within some item while involving logistics and maintenance (Leonetti et al. 2018). These modes are used to find an assurance relating to the physical irreversibility and the damage of the functions, which is not continued through the interface as the failure is part of ye units of the interface. However, when someone uses the modes of failure, they have to consider a few things and keep a few things clear (Casapulla & Argiento, 2018). At one point in time, only one mode of failure can be used to determine flaws in the structure of the masonry. The inputs, which will be made or analyzed, have to be in the present and needs to have values that are nominal. All the resources to be consumed have to be present in huge quantities (Leonetti et al. 2018). They have to make sure that there is the availability of the nominal power.
The failure modes are used frequently to generate results, which can be useful. Therefore, they have some specific and continuous advantages when implemented to determine the flaws of a structure or a wall (Greco, Leonetti, Luciano & Trovalusci, 2017). It can provide methods, which can be documented after a design has been, selected which has a high chance of bringing about an operation, which is both safe and successful. A method can be documented which will assess all the possible failure mechanisms, modes as well as the impact of them o the system (Casapulla & Argiento, 2016). It will result in the formation of modes based on their ranks, which can determine in the future of how serious they are to be used and what are the chances of it occurring again. It can be used effectively to determine the effect of the changes, which has been proposed to design and operate the procedures, which will lead to safety and success (Milani & Valente, 2015). The failure modes also help to plan early the tests needed to be done in a specific structure. Through this, one can understand how nonlinearity of masonry will increase deformation later on with an increase in loading. That will cause the mortar to crack extensively and will increase progressively the Poisson’s ratio as well as the cracks, which occurred vertically in the boundary of the joint of blockhead mortar (Fagone, Ranocchiai & Bati, 2015).
The discussion of the former tasks presents a case of curved beams. Arches can be defined as curved beams which are vertical in orientation. Arches have been known to be in existence for several thousands of years. The ancient architecture forms were primarily constructed out of masonry, while modern arches can be observed supporting major structure such as bridges, and so on. The similarity between a curved beam and an arch is quite noticeable. However, defining an arch, one may state that it may be regarded as a soft compression structure. On the contrary, it has been noted that curved beams have rigorous engineering activities involved, along with the calculation derived from the advantageous implementation of the structural geometry (Baek, Sageman-Furnas, Jawed & Reis, 2018).
It may be mentioned in this regards, that it seems evident that both curved beams, as well as an arch, have similar structures, they have subtle differences, which may only be identified through performing a thorough investigation of the same. As discussed in the later stages of this paper, the major forces which contribute to the construction and stability of an arch are tensile and compressive forces. The combination of the action of the tensile and compressive forces establishes and maintains stability of the structure (Lagomarsino, 2015). It may be mentioned in this context that a curved beam may potentially be an arch. However, the construction of an arch may not be performed through the implementation of the principles of a curved beam. Furthermore, it is to be noted that in case of a curved beam, the curve is observed to be incident on a horizontal plane, while the curve in an arch is noted to be vertical in orientation. Therefore, it may be clarified that curved beams and arches are different in their structural orientation, despite the obvious similarities observed.
Regardless of the distinction in their structural orientation, the stability or the resilience observed in both the structures with regards to earthquakes is a significant subject for exploration. The forces which come into play in case of an arch are directed towards the ground. Hence, it may be stated that the arch gradually pushes outward towards the base, which is termed as thrust. The outward thrust and the height of the arch are noted to be inversely proportional to one another. The maintenance of the arch action, along with the assurance to prevent the arch from collapsing, it is crucial to restrain the thrust.
The ancient structures consisting of arches, which are typically made of masonry are regarded as older forms of architecture. However, modern architecture typically comprises architecture which uses metals to bend and give certain shapes which are geometrically correct. Furthermore, the assurance of stability of the structure is provided by both ancient arches as well as modern arches. Regardless, it is to be mentioned in this context that the development or the construction of the historical masonry structures were primarily and predominantly designed to bear gravitation loads (Stochino, Cazzani, Giaccu & Turco, 2016). However, it is to be taken into account for this study that the provision for preparation and dealing with earthquakes were not prioritized in those times. Hence, the capability of the masonry arches with regards to demonstrating resistance towards seismic activities can be regarded as obsolete.
In addition to that, it may be noted that though structures have been structurally stable and self-supporting, the provision for earthquake combat had not been introduced in the former arches made out of masonry. Considering the modern day arches, one may state that there are various considerations to deal and compete with the modern provisions of engineering and architecture. For instance, the inclusions of the provision of safety with regards to earthquake have been made a priority in modern arches. The resistance of the compressive stress has been determined to be the key to the construction of the modern-day arches and relevant structures. In addition to that, the occurrence of any other forms of stress, namely tensile or torsional stress, is deliberately reduced (Stochino, Cazzani, Giaccu & Turco, 2016). The reduction in the stress occurs through the placement of reinforcement fibers or rods.
The stability of the naturally formed aches has not been established. However, certain examples of modern architecture, which implement the principles of the construction of an arch, are prime instances of extensive engineering skills of scientists and engineers. One may discuss the example of ancient arches to better understand the concept implemented in modern engineering. It may be stated in this aspect that the seismic vulnerability of the ancient arches, made out of masonry have been noted to be primarily high (Kassotakis, Sarhosis, Forgács & Bagi, 2017). In addition to that, the properties of the materials used for the construction of the arches have been found to play a vital role in the high seismic vulnerability of the arches. It has been found that lower tensile strength, combined with a high specific mass, along with low ductility as well as moderate shear strength has been observed to be a prime cause of the higher seismic vulnerability demonstrated by older arches or curved structures.
The dependence of the seismic behaviour of the structure has been found to be primarily based on the weak connections between the load-bearing walls and the floors, while the geometry of the structure and the high mass of the masonry wall. These factors have been observed to have a major influence on the patterns of seismic behaviour of the ancient masonry arches. The domes, masonry arches as well as vaults have received attention for quite some time as a result of the factors influencing the seismic resilience of the structures (Sevim, Atamturktur, Altuni?ik & Bayraktar, 2016). Modern architectural reforms have identified the flaws, and the development of the drawbacks with regards to the seismic behaviour can be attributed to the success and stability of the modern structures.
It has been noted that the steepness in the arch can be the cause of increased stability among the prevalent modern arch forms. Furthermore, the achievement of the necessary ‘propping action,’ which signifies the improved stability of the structure, has been identified with regards to the seismic behavioural patterns of modern arches. In addition to that, the combining of the abutments or the bottoms of the arch members are fused may result in the illustration of better stability of the arches. Hence, it becomes evident from the aforementioned discussion that the provision for dealing with earthquakes had not been prominent in the early days (Zampieri, Zanini & Modena, 2015). However, in the modern days, with the progress in the field of science, engineering, and architecture, the development of concept and ideas and the implementation of the ideas have been found to be effective in dealing with seismic behaviour of the arches.
Various examples of ancient arches can be cited through the historical records of churches and cathedrals. Prime architectural examples include the Rhodes Footbridge which can be cited as a significant example of the Greek architecture. The Footbridge has been an early instance of the voussoir arch. In addition to that, the Greek architecture had adapted the techniques as well as the concept of the construction of an arch (Guo, Yuan, Pi, Bradford & Chen, 2016). The Romans had realized that an arch does not have to be a semi-circle. The early identification of this concept led to the rapid and widespread construction of arches throughout their cities. Furthermore, it is to be taken into account that the segmental arch had initially been developed by the Romans. The Romans had also implemented the concept into building several bridges, as well as triumphal arches as military monuments. The Arch of Caracalla is one such instance of a Roman triumphal arch.
In addition to that, it may be stated that the Europeans had developed similar semicircular arches leading to the acceptance of Gothic arches or ogives. The Alconétar Bridge may also be cited as an example of Roman architecture in Spain. The segmental shape of the arches of this particular bridge in question has been noted to the focus of the specifications of the arches. Ponte Santa Trinita can be mentioned in this regards as the semicircular arches mentioned before had been innovated to give the shape of an elliptical structure (Gosden & Malafouris, 2015). Modern architectures are to be mentioned in this regards. The prime examples of modern arches include the Woodrow Wilson Memorial Bridge, which is situated between Virginia and Maryland. In addition to that, the Rainbow Bridge over the river Niagara, which connects the city of New York to Niagara Falls?
Additionally, the mention of the Hell Gate Bridge over the city of New York is essential in this aspect. Furthermore, it may be stated that the bridge is considered to be a through arch bridge. The masonry towers and the steel arch of the Hell Gate Bridge have been left at a gap of approximately 15 feet. Aesthetic girders were added to the bridge to establish that the robustness of the structures (Elsayed, Wille, Al-Akhali & Kern, 2017). The girders had been added to the towers and the upper chord of the arch for additional stability of the structure. Another mention of modern arches can be cited as the Tyne Bridge, which is a through arch bridge, similar to the Hell Gate Bridge. However, the Tyne Bridge is located on the river Tyne in Northeast England.
It may be mentioned in this context that the Juscelino Kubitschek Bridge located in Brazil, can be cited as one of the remarkable works constructed with regards to arches. Furthermore, the stability of the structure with regards to seismic tolerance has been established. The bridge has a series of asymmetric arches made of steel, which consistently criss-cross one another. The diagonal crisscrossing is also considered to be significant reason behind the stability and self-supporting nature of the bridge (Lacidogna & Accornero, 2018). The seismic resilience of the bridge has also been noted to be quite high as a result of the geometric advantages that the structure of the bridge presents. Furthermore, the suspension of the decks with the help of steel cables and wires constitute a twisted plane, which is an outcome of the interlacing of the steel cables at each of the sides of the supporting pillars. It is to be taken into account in this regards that there are four pillars which support the overall structure of the bridge.
Lateral torsional buckling occurs due to unrestrained beam causing displacement in activities. A beam is referred mostly as unrestrained when it subsequently faces compression caused due to the external force (Kala, 2015). Mostly, it witnesses flange caused out of free displacement both laterally and while rotating. For example, when an external load occurs, it causes both lateral displacements. On the other hand, such movement leads to twisting of a section that is referred to as lateral torsional buckling. There can be both horizontal and applied vertical load — both may result in compression and tension causing within the flanges buckling. In such cases, the compression flange effects and deflects laterally away within the concentrated original position (Kala, Z. (2015). The tension flange tries to keep the channel straight so that it does not bend. The lateral movement of the flanges differs based on external and internal force (Kucukler, Gardner & Macorini, 2015). The lateral bending results in displacement in a certain section that is unable to restore forces that oppose the movement that occurs generally. It is important to keep and allow the flange to remain straight. Overall, it is important to restore forces and stops the risks causing the section from deflecting laterally. It is necessary to foresee both lateral component and tensile forces that will help in identifying the buckling resistance of the beam (Ghafoori& Motavalli, 2015). It is important to check the movement of the section and the lateral forces within the flanges. The most important part is to oversee whether the pressure is latitudinal or longitudinal.
The figure above shows, how the twist is created by the torsional stiffness within the section. The torsional stiffness of the section is pressurized by the flange thickness. It is important to accommodate thicker flanges that will help in building and bending strength (PB) (Couto, Real, Lopes & Zhao, 2016). It will help in strengthening the depth of section holding onto the larger and thinner flanges that might cause lateral deflection within the construction beam.
There are certain factors which influence the behavior of torsional buckling of beams:
There are certain degrees of freedom. There are seven degrees of freedom in total in any beam element in every node. These are considered as translation in x-, y-, and z-direction, and mostly rotation about x-, y- and z-axis and also wrapping.
Another most commonly associated situation with lateral torsional buckling is the buckling conditions. It has been seen that to maintain equilibrium, a certain degree of freedom is needed to be always restrained at the boundary. When two ordinary beams are considered which are on two supports then the translations regarding y- and direction regarding z- needs to be always restrained.
Regarding the translation in x, it is always taken as fixed on one side and free on the other side. There needs to be the restraint on the x-axis rotation or else the beam would start to rotate on its axis(Horá?ek and Melcher, 2017). The support condition can be demonstrated through simple fork support. The fork has some boundary conditions. The rotation about the x-axis is fixed and so are the translation in x, y and z, they are fixed as well. However, the rotation about z and y-axis are free so is Wrapping.
An important element of buckling beams is the center of gravity; it is the point in a mass body where the whole weight of the respective body which can be applied in calculations for convenience(Horá?ek and Melcher, 2017). The average of the created gravitational moment from the body of mass, e.g., the static moment, at this particular point it is zero. The center of gravity coincides with the centroid only when the material is homogenous. The center of gravity is generally located in the intersection between the two lines of symmetry in a homogenous I-section which is double-symmetric.
Shear center as discussed above is an important element about the buckling of beams. The shear center is something, to attain in-plane bending the shear center acts as a point the shear force needs to act through the line of action. The beams which have cross-sections with double- symmetry like the I-section, the shear center and the center of gravity coincide(Horá?ek and Melcher, 2017). When non-symmetric or single-symmetric section and considered then there are several instances where the center of gravity and the shear center does not coincide. It can be demonstrated with a u-section which is loaded with sectional forces and marked out resultants. The forces within the flanges will result in twisting the section if in the plane web the section is loaded. Therefore, the shear center can be situated towards the left side of the web where the equilibrium moment is attained.
In different steel construction there are more than one rows which are parallel and are made of supported beams which are braced on the top flange. However, the bracing is the relation to the compression or tension to make sure that the load is transferred toe lateral support that is stiff. This system is known as ‘a bracing plan system.’ The prime utilization of the restraints is to avoid the deflection of the lateral beams art the bracing points(Vild et al., 2017). The buckling length of the beams is dependent on the longitudinal distance among the restraints. For this role, the restraints need a minimum firmness and strength.
The design of a force in regard to a single brace in any location = qdL, along with this when there is addition of forces which happen because of the external actions and QD is regarded as the equivalent stabilizing force and L is regarded as the length of the beam which is restrained(Vild et al., 2017). At the time of using one or more braces throughout the length of the beam, then it needs to be designed in such a way that it can resist a minimum force which is no less than 5qdL/8, in addition to the extra additional forces by the actions which are external. Determining the restraint forces is an iterative procedure, because of the level of bracing system deflection on which the forces are dependent. The typical bracing systems which are applied in the buildings, the deflections of these are improbable to surpass L/2000. When this value is considered to be qDQ, then 2.0% restraint forces in maximum designs force regarding the compression flanges seems to emerge from an equation of a single restraint beam, where again the additional forces because of external actions are to be added(Yang et al. 2016). In this scenario, iteration is not required only if the deflection of the bracing system qDQ is less than L/2000. If L/2000 is exceeded by DQ, then the restraint forces that are gathered with become unsafe. When all these conditions are fulfilled, then the beam can be designed with a length of Lcr which is also equal to the distance among the braced points. It is very important for the lateral support to be capable of bear the force of bracing transferred to it safely. In case of a splice in a particular beam then the bracing system needs to be capable resisting the lateral force impacts which are equivalent to the 1% force of the design at the location of splice within the compression flange.
In the scenario of the multiple beams, the bracing needs to be designed for the average of all the restraint forces for every single beam (Ozbasaran, Aydin and Dogan, 2015). There is universal regulation, the lateral firmness of the whole system of restraining need not exceed 25 times even when combined with the stiffness of the lateral beams which will be braced.
The stability of more than two lateral beams can be improvised by interconnecting the beams at certain intervals throughout their lengths via the bracing plan system. The restraint system solely relies on the connection of a few unstable members which is needed to be ensured that the instability will lead to the deformation of the system of bracing(Ozbasaran, Aydin and Dogan, 2015). Therefore, just connecting the beams with its end pinned with light-crossed members will not give any result as the buckling mode of the beams are not affected in the least. Total lateral restraint regarding the compression flanges is attained at the points which are braced, although the bracing members need to be bonded to the flanges, they also need to be able to resist the equivalent stabilizing force dQD. Lateral torsional buckling of the beams’ resistance is needed to be verified among the points which are braced. The resistance forces which are individual regarding the percentages of the compression forces of the flanges that is maximum are as well as suitable in this perspective.
In several instances, it has been observed that there are beams which support the cavity walls, and their designs are a bit tricky(Ozbasaran and Yilmaz, 2018). When there are two beams which are adjacent to each other which is normally hot rolled I channels, then it generally leads to supporting both the leaves on a cavity wall. These beams are needed to be connected at certain intervals by diaphragms or separators. The separators ensure that among the two beams there is a space or not. They are capable of carrying transverse forces but are incapable of transferring vertical forces among the beams. For the loaded beams which are equally loaded, the separators guarantee that both the beams will buckle in the similar direction, however, there is no evidence of increased capacity for carrying the load above the average of the beams individually.
The function of the diaphragms is to retain the structure and shape of the cross sections at their respective locations, and as a result, the beams are provided and supported with torsional restraint(Naaim, De’nan, Keong and Azar, 2016). The diaphragms are capable of transferring vertical load among the beams however the beams need to contain enough firmness and strength. Hence, the resistance of the beams will be known as the braced pair of beams.
In the steel construction,n there are beams where there is restraint with beams comprised of the tension flange. This is observed mostly in the outer columns and the regions with the hogging moments regarding the rafters mostly in the portal frames of the single story, here the tensions restraints are given through side rails(Lopes, Couto, Vila Real and Lopes, 2016). These elements normally provide complete lateral restraint however no torsional restraint is given to tension flange — there two types of approaches that are followed to understand the member’s stability among the torsional restraint.
In both of the methods, two checks are needed to be done. Firstly, the beam stability about the flexural torsional mode among the flanges’ restraints, secondly, the stability of the beams are needed to be assessed among the restraints of the tension flanges.
There beams in the construction that supports the timber floors which also has the application of torsional buckling beams. It is very common to see that a sequence of steel beams was proving support to the timber floor, it is mostly found in older buildings. In this kind of constructions, it is seen that under the joists of the timber or completely inside the floor the steel beams are located(Lopes, Couto, Vila Real and Lopes, 2016). The level at which both the timber joists provide lateral resistant to steel beams is completely dependent on the connection among these two elements. When there is a positive connection in-between the timber joists and the steel beams, then it can be presumed that the steel beams are lateral and restrained by the timber joists. In this scenario, there are certain recommendations which are provided. The edge beams that are there need a special reflection as being notched by its own does not provide any restraint. In this scenario, steel straps are required. The restraint forces which are transmitted to the timber joists are generally small or less for the normally sized beams which can be resisted with the help of the diaphragms or by attaching or packing the timber joists ends firmly by the walls of the masonry(Kucukler, Gardner and Macorini, 2015). When the steel beams are larger than the and support heavy walls or loads, then a proper position for lateral restraints is needed to be developed with the help of straps and braces. The firm points on the structure like the walls need to bear the restraint forces, or it can be transferred to the floors with the help of diaphragm in the presence of the floorboards and its needed to be made sure that they are attached to the timber joists.
The beams which support concrete slabs also the application of lateral torsional buckling of beams have been found. When a sequence of beams which are parallel to each other are supporting a concrete slab, they will either function compositely of non-compositely(Kucukler, Gardner and Macorini, 2015). It acts compositely simply because of the presence of the shear connectors. In the case of two precast units which are grouted. When composite beams are used then there is the total restraint given to the beams with the help of the action of diaphragm inside the slab and as discussed above it happens through the shear connectors. The composite beams can only be restraint while being under levied load(Jiao, Borchani, Soleimani and McGraw, 2017). At the time of the construction, restraint can be given with the help of steel decking and a reference is needed to be developed. In the perspective of non-composite beams, no positive restraint is found regarding the compression flange. The resistance of lateral buckling only takes place due to the friction which appears among the concrete slab and steel beams. Therefore, the beams which support the concrete slabs can be developed as laterally restrained beams.
Conclusion
This study focuses primarily on the impact of seismic activities on various engineering structures. It may be stated from the above study that the stability of the curved beams has been a part in developing a better comprehension of the occurrence and the behaviour of the curved structures with exposed to seismic disturbances. In addition to that, the Shake Table Model had been identified as a vital model or theory in understanding the response generated by curved beams and such relevant structures. Furthermore, the role of expansion joints has been identified and explored in this study. The scope for the identification of the importance of the presence of the expansion joints within the curved beams has been discussed in this study. The overall capacity of expansion joints, for accommodating seismic compressions, which may result in the net movement of the curved beam from 30 to 1,200 millimeters, was noted in this paper.
SRSS was identified to be an effective technique in identifying and demonstrating the seismic resilience of curved beams and related structures. In addition to that, the differences between a curved beam and an arch have been discussed in this paper as well. A vivid discussion has been made with regards to the former masonry arches as well as the new constructions evident. The stability of the structures and the potential causes behind it has been explored as well. It was noted that Arch of Caracalla along with other notable structures could be cited as major examples of arches. Lastly, a discussion on the torsional buckling of beams of presented. A discussion on the compression flange and lateral displacement with regards to the torsional buckling of beams has been made in this paper as well.
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