Parabolic Arches Research Articles

In-plane stability behaviours of concrete-filled steel tubular catenary arches under different loading conditions

Concrete-filled steel tubular (CFST) catenary arches, renowned for their superior loading-bearing efficiency due to the alignment of the arch axis with the thrust line, have become the predominant choice for large-span arch bridges over the past decade. However, research on CFST catenary arches remains limited, with most studies focusing on the in-plane strength of circular and parabolic arches. And the existing design methods are tailored to a narrow range of loading conditions, which raises concerns about their applicability to CFST catenary arches. Therefore, this paper addresses this gap by designing and testing six CFST catenary arches under three different loading conditions (quarter-point concentrated load, mid-span concentrated load, and five-point concentrated loads). The failure modes, load-displacement curves, cross-sectional strain and stress development, and confinement effects were compared and analysed among the loading conditions. Additionally, the finite element (FE) models using beam elements were developed and verified against the test results to conduct a parametric analysis. The influence of arch axis coefficient, slenderness ratio, rise-to-span ratio, and steel ratio on the in-plane strength of the CFST catenary arches under seven different loading conditions was discussed. Finally, by accounting for the non-uniform distribution of internal forces along the arch rib, a lower-bound design formula for the in-plane strength of CFST catenary arches under various loading conditions was proposed, and comparisons with numerical results show that the proposed method can provide conservative predictions.

In-plane nonlinear buckling analysis and design method of concrete-filled steel tubular catenary arches

The prevalence of concrete-filled steel tubular (CFST) catenary arches, renowned for the optimal load-bearing efficiency, has shown a marked increase over the past decades. To date, extensive research has been conducted into the nonlinear buckling behaviors and elastic-plastic strength of circular and parabolic CFST arches, grounded on the assumption that the curvature of the arch axis remains a specific constant. However, for CFST catenary arches, this assumption is not appropriate, given the substantial variation in curvature along the arch rib, and the influence of the arch axis is not adequately addressed in the existing design codes. Therefore, in this paper, a more precise strain expression is employed to conduct the nonlinear buckling analysis on CFST catenary arches. The analytic solutions for both symmetric snap-through buckling and anti-symmetric bifurcation buckling loads are obtained. In addition, a validated finite element (FE) model is adopted to investigate the influence of key parameters, including the slenderness ratio, the arch-axis coefficient, the rise-to-span ratio, and the steel ratio, on the in-plane stability bearing capacity of CFST catenary arches subjected to axial compression. Moreover, the obtained numerical results are utilized for comparison with the prediction results calculated by the existing design codes for CFST or steel arches. The comparison results indicate that the existing design codes have tended to overestimate the in-plane strength of CFST catenary arches, mainly due to the ignorance of the influence of pre-buckling deformation. Thus, based on the nonlinear buckling analysis results, a revised lower-bound design formula for predicting the in-plane stability bearing capacity of CFST catenary arches under axial compression is proposed, considering the influence of the rise-to-span ratio and the steel ratio.

Possible Approaches to the Reconstruction of the Vyšehrad Bridge in Prague

Prague has two iconic views. The first is Charles Bridge with the silhouette of Prague Castle in the background, the second is the Vyšehrad Railway Bridge framed by the imposing structure of Vyšehrad. It has been a listed cultural monument since 2004. The Vyšehrad Railway Bridge was opened in its first form on 15 August 1872, its current appearance dates back to 1901. It is a riveted structure with three parabolic arches spanning about 70 m with verticals and diagonals in the shape of segmented cross-sections. The structure is an important technical cultural monument, which is also located in an exposed part of the Prague Monument Reserve. However, due to long neglected basic maintenance, the structure deteriorated to such an extent that the Railway Administration decided to demolish it. Having carried out an inspection and structural assessment of the bridge we are confident that it can be repaired, refurbished, and modified to provide a long and satisfactory additional service life for future operation of the railway and footways. In particular, localized corrosion defects resulting from inadequate maintenance over many years can be repaired and the structural components which fail to meet the assessment criteria in their current condition can be repaired or replaced. Expressed in weight, only 15 % of the structure by mass needs structural repair or improvement; the rest is satisfactory without further intervention, other than renewing the corrosion protection system. The paper is based on a study by Petr Tej, Andreas Galmarini and Ian Firth and will focus on a detailed survey of the bridge’s condition and the possibilities of its reconstruction for rail traffic. It describes the reconstruction system, replacement of individual elements and contact points. Part of the paper is also a proposal for the reconstruction and use of the stone viaduct for trade and services, including static and visual rehabilitation.

Rigid body qualified curved beam approach for lateral torsional buckling of arches

The curved beam finite element approach based on the potential energy formulation is widely recognized as a versatile tool for lateral torsional buckling analysis of arches. However, there exist a variety of potentials, which were developed with distinct assumptions. The diversity of energy-based curved beam theories introduces confusions in practical applications. This paper aims to examine various potential energy formulations as to whether or not they obey the rigid body rule which is a prerequisite for valid second-order analysis. The induced moments at the ends of curved beam element that have been neglected from and/or the distributed moments that should be removed from available potentials are identified to help them become rigid body qualified. Based on the revised potential functional, a curved beam finite element formulation has been developed to effectively predict yet with less effort the out-of-plane critical load for circular and parabolic arches subject to various loading and constraint conditions. The results of numerical examples demonstrate that a potential energy formulation eligible for describing the out-of-plane behavior of curved beam is required to be rigid body qualified by fulfilling two aspects: (i) the neglected end moments that are imperative for satisfying the rigid body rule must be considered unless the virtual work done by them vanishes, and (ii) undesirable distributed moments that may be induced when undergoing the rigid body motion should be removed by considering all essential potential components.

Efficient formulation of a two‐noded geometrically exact curved beam element

AbstractThe article extends the formulation of a 2D geometrically exact beam element proposed by Jirásek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first‐order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier–Bernoulli hypothesis, with a possible extension to shear‐flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral‐shaped beam, and a spring‐like beam with a zig‐zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, that is, the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the Appendix.

Exact Matrix Stiffness Method for Out-of-Plane Buckling Analysis of Funicular Arches Considering Warping Deformations

The out-of-plane buckling behavior of arches is closely related to the element torsional behavior. The traditional 12-degree-of-freedom second-order element stiffness matrix which uses a simplified element torsional stiffness GJ/[Formula: see text] (where [Formula: see text] is the shear modulus, [Formula: see text] the St. Venant torsion constant, [Formula: see text] the element length) may significantly underestimate the out-of-plane buckling loads of funicular arches. This paper presents a simple and effective exact matrix stiffness method (MSM) for the out-of-plane buckling analysis of funicular arches. The developed MSM uses a 14-degree-of-freedom second-order element stiffness matrix of three-dimensional beam-columns considering both torsion and warping deformations. The out-of-plane buckling analysis of funicular arches is performed by using the global structural stability stiffness matrix, which combines the transformed second-order element stiffness matrices. The proposed MSM with the exact 14-degree-of-freedom second-order element stiffness matrix for the out-of-plane buckling analysis is verified by comparing with some classical solutions of funicular circular and parabolic arches with box sections and I-sections. Further discussions show that the 14-degree-of-freedom second-order element stiffness matrix may be reduced to a simplified 12-degree-of-freedom form only by deriving the exact element torsional stiffnesses, which could be significantly larger than GJ/[Formula: see text] for members with large cross-sectional torsional stiffness parameters (especially open cross-sections).

Experimental and numerical study on slender concrete-filled steel tubular arches subjected to tilting loads

Concrete-filled steel tubes (CFST) are economical for arch bridges because of their high compressive strength and efficiency in construction. Compared to the traditional concrete arches, the CFST arches usually have a large slenderness ratio, which makes the arches more prone to buckling. Despite this, limited experimental investigations have been performed on the buckling behaviour of CFST arches. In this study, a buckling test was performed on two 9 m long CFST parabolic fixed arches having a single circular cross-section with one arch subjected to a central concentrated tilting load and the other to half-span three-point tilting loads. Finite element (FE) models using solid elements were built and validated against the test results, with which the influences of the residual stress on the buckling behaviour of the composite arches were analysed and the confining pressure distribution was obtained. The FE models using beam elements were then built to investigate the confinement effects on the buckling behaviour of the CFST arches. Relatively significant confinement effects were found on the arches and were considered in the FE modelling. The residual stress had a fairly limited influence on the load-carrying capacity of the arches and was ignored in the FE models. Both the built FE models using solid and beam elements performed well in predicting the buckling behaviour of the CFST arches.

Vibration of locally cracked pre-loaded parabolic arches

We study linear dynamics of an initially parabolic arch deformed by a uniform ‘dead’ load. The arch is seen as a fully deformable one-dimensional continuum with rigid cross-sections, one of which suffers from a small local crack at its boundary. The crack is simulated by springs, the stiffnesses of which are evaluated via stress intensity factors. By two first-order perturbations we investigate a non-trivial equilibrium adjacent to the reference configuration and small vibration superposed on it. The modulation of the initial load on the natural angular frequencies and its consequences on damage detection is described and commented. It turns out that neglecting the initial load, recalling for actual ‘dead’ structural actions, can be misleading in damage identification, while its inclusion leads to better results.

In‐Plane Instability of Parabolic Arches under Uniformly Distributed Vertical Load Coupled with Temperature Gradient Field

In civil engineering, arches, such as steel arch roofs and arch bridges, are often subjected to linear temperature gradient field. It is known that the in‐plane instability of parabolic arches is caused by the significant axial force. The arch under the linear temperature gradient field produces complex axial force, and so the instability of arches would be affected by temperature gradient field significantly. However, the analytical solutions of in‐plane instability of parabolic arches being subjected to the uniformly distributed vertical load and the temperature gradient field are not solved in the opening literature. In this paper, in‐plane instability of a fixed steel parabolic arch under linear temperature gradient field and vertical uniform load is analyzed theoretically. Firstly, the cross‐sectional effective centroid and effective stiffness of the cross section for arches under the linear temperature gradient field are derived. Secondly, the preinstability internal force analysis of the parabolic arch under the linear temperature gradient field and the vertical uniform load is carried out based on the force methods. Novel theoretical solutions for in‐plane instability load for fixed steel parabolic arches under the linear temperature gradient field and the vertical uniform load are obtained. It is found that the gradient temperature, slenderness, and rise‐span ratio have important influences on the critical in‐plane instability load of the shallow parabolic arch, while there is no significant effect on the deep parabolic arch.

A comparative assessment of developed novel power bridge with conventional power arm for en-masse retraction of maxillary six anterior teeth

• An intra-oral orthodontic appliance called as ‘power bridge’ is developed. • The parabolic power bridge has two arms mounted on the top surfaces of two brackets. • Parameters of power bridge and power arm are compared with finite element analysis. • Power bridge is found to be approximately 10 times safer than the power arm. • Power bridge is found to be 92 times stiffer than the power arm. For en-masse retraction of protruded maxillary six anterior teeth, power arms are attached on the archwire as a general practice to pass the force vector through centre of resistance (C res ) of maxillary six anterior teeth. As a power arm is similar to cantilever beam, it gets more deflected at free end (hook end) where retraction spring is attached. Additionally, the equivalent stresses at base of power arm increases due to deflection of power arm. The developed novel power bridge in this research study has a parabolic shape as parabolic arches can support larger load with less deformation. To verify the performance of the power bridge, finite element analysis was done to compare the results of equivalent (von-mises) stresses and total deformation of this novel power bridge and conventional power arm. The cross-sections of segments of both power bridge and power arm were kept uniform and equal to justify the results of finite element analysis. The maximum equivalent (von-mises) stress in the teeth assembly model with power arm was found to be 10.42 times greater than the teeth assembly model with power bridge. Thus, power bridge structure developed in this study can be considered as approximately 10 times safer than the power arm structure. The maximum total deformation in the teeth assembly model with power arm was found to be 92 times greater than the teeth assembly model with power bridge. Thus, power bridge structure developed in this study can be considered as 92 times stiffer than the power arm structure. So, it is clear that the power bridge structure is significantly safer and stiffer than the power arm structure. Hence, this research study suggests that the power bridge structure should be used instead of power arm once the efficacy of power bridge structure is tested and proved clinically.

A semi analytical method for in-plane free vibrations of arches with a variable curvature

Rayleigh-Ritz method (RRM) is used in this paper in order to investigate the in-plane vibrations of arches with a variable curvature in particular, the frequency parameters and mode shapes. The trial functions are particular solutions of the sixth order differential equation of arch vibrations corresponding to an opening angle equal to one rad. The investigations are made with the follow’s hypotheses: the constant dimensions of the cross-section are small in comparison with the radius, the arch axis is inextensible and the effect of rotary inertia and shear deformation are neglected. The parabolic, catenary, spiral, circular and cycloid arches with different boundary conditions are investigated. The frequency parameters are presented for different types of curved arches with various geometrical properties and boundary conditions and the corresponding mode shapes are plotted. The results are compared with the published literature. The accuracy and relative simplicity of the RRM, applied herein a systematic way with the new choice of trial functions, is established, making it ready to use in more complex situations, such as those of arches with added masses, with non-uniform cross sections or with one or more point supports.

Nonlinear in-plane buckling of shallow parabolic arches with tension cables under step loads

Nonlinear in-plane buckling of shallow parabolic arches with tension cables under step loads

Stability analysis of wooden arches with account for nonlinear creep

Introduction. The paper deals with the calculation of wooden arches taking into account the nonlinear relationship between stresses and instantaneous deformations, as well as creep and geometric nonlinearity, are considered. The analysis is based on the integral equation of the viscoelastoplastic hereditary aging model, originally proposed by A.G. Tamrazyan [1] to describe the nonlinear creep of concrete. Materials and Methods. The creep measure is taken in accordance with the work of I.E. Prokopovich and V.A. Zedgenidze [2] as a sum of exponential functions. The transition from the integral form of the creep law to the differential form is shown. The relationship between stresses and instantaneous deformations for wood under compression is determined from the Gerstner formula, and elastic work is assumed under tension. The solution is carried out using the finite element method in combination with the Newton-Raphson method and the Euler method according to the scheme that involves a stepwise increase in the load with correction of the stiffness matrix taking into account the change in the coordinates of the nodes with the sequential calculation of additional displacements of the nodes, which are due to the residual forces. The proposed approach for increasing the accuracy of determination of creep deformations at each step provides using the fourth-order Runge-Kutta method instead of the Euler method. Results. Based on the Lagrange variational principle, expressions are obtained for the stiffness matrix and the vector of additional dummy loads due to creep. The method developed by the authors is implemented in the form of a program in the MATLAB environment. Calculation examples are given for parabolic arches simply supported at the ends without an intermediate hinge and with an intermediate hinge in the middle of the span under the action of a uniformly distributed load. The results obtained are compared in the viscoelastic and viscoelastic formulation. The reliability of the results is validated through the calculation in the elastic formulation in the ANSYS software package. Discussion and Conclusions. For the arches considered, it is found that even with a load close to the instant critical, the growth of time travel is limited. Thus, the nature of their work under creep conditions differs drastically from the nature of the deformation of compressed rods.

Out-of-plane stability of fixed concrete-filled steel tubular arches under uniformly distributed loads

In recent years, concrete-filled steel tubular (CFST) arches have been widely used and the out-of-plane buckling of CFST arches has become a major concern. So far, few studies have investigated the out-of-plane behaviours of CFST arches, and no codes have given the design formula of the out-of-plane stability bearing capacity. Therefore, in this paper fixed CFST parabolic arches with a circular cross-section under uniformly distributed loads are investigated. Considering the effect of material and geometric non-linearity, the distributions of the internal forces and deformations are observed, and on this basis an equivalent column model is established to derive the out-of-plane effective length coefficient. In addition, based on the stability theory, the nominalised slenderness ratio of CFST arches is modified considering the influence of rise-to-span ratios and can be substituted into any available stability coefficient equations in current design codes to calculate the out-of-plane bearing capacity of CFST parabolic arches. The proposed equation – which has a form of formula similar to the available stability equations in existing designing codes for CFST or steel members and which has an accuracy equal to or greater than the equations in the available literature – would be easier for the designers to accept and use.

In-plane vibration analysis of parabolic arches having a variable thickness

This article deals with free and forced vibration responses of the viscoelastic parabolic arches of variable thickness. Firstly, considering the effects of shear deformation and damping, time-dependent motion equations of in-plane loaded parabolic arches are obtained. Subsequently, the Laplace transform is applied to the obtained equations and solved by a powerful numerical method. Furthermore, the Kelvin viscose model is used to describe the viscoelastic material. Using an effective and suitable inverse numerical Laplace transform method, the results were transferred back to time space. The verification of the presented method is performed by comparing its results with the results of ANSYS. It has been shown in several examples that the proposed method is highly accurate and efficient compared to step-by-step time integration methods.

Internal Force Analysis of Parabolic Arch Considering Shear Effect under Gradient Temperature

This paper theoretical analysis the internal force of the fixed parabolic arches under radient temperature gradient field incorporating shear deformations. The effective centroid of the arch-section under linear temperature gradient is derived. Based on force method and energy method, the analytical solutions of the internal force of fixed parabolic arches at pre-buckling under linear temperature gradient field are derived. A parameter study was carried out to study the influence of linear temperature gradient on the internal force of the fixed parabolic arches with different rise-span ratio and varying slenderness ratio. It is found that the temperature gradient and the rise-span ratio has a significant influence on the internal force of the parabolic arches, the influence of shear deformation causes the bending moment increase while the axial force decreases, and the axial force of parabolic arches decreases as the rise-span ratio increases.

Limit-state analysis of parabolic arches subjected to inertial loading in different gravitational fields using a variational formulation

For thousands of years, arches have been used as durable structures that are easy to build and that rely on gravity for their inherent stability. Since then, many researchers and engineers have studied their stability either when subjected to gravity or inertial loading. Currently, given the Insight mission to Mars and the ambitious Artemis program to the Moon, it has become apparent that there will soon be the need to design and build the first resilient extraterrestrial structures and arches represent an ideal option for such structures. This paper focuses on the stability of parabolic arches with different embrace angles subjected to different levels of equivalent inertial loading in low-gravity conditions. The results are contrasted with the well-studied circular arches. More specifically, this investigation employs variational principles to identify the imminent mechanisms and numerical methods based on the limit thrust line concept in order to estimate the minimum required thickness of parabolic arches for a given loading and in different gravitational fields. The paper shows that although parabolic arches can be much more efficient than their circular counterparts for gravitational-only loading, this is not the case for different combinations of inertial loading and embrace angles where the opposite can be true. It highlights the dominant effect of low-gravity conditions on the minimum thickness requirements for both types of arches and considers the effect of a potential additional infill for radiation shielding. Furthermore, this study reveals a self-similar behaviour, introduces a “universal” inertial loading and showcases, through the use of master curves, the areas where the parabolic arches are more efficient than their circular counterparts and those where the opposite is true. These areas can be used for the preliminary design of such structures. Additionally, the paper identifies hidden patterns associated with the developed mechanisms between the two different geometries for the different gravitational fields. Finally, it presents a case study where the need to optimise the structural form of extraterrestrial structures becomes evident.

Experimental Investigation into In-plane Stability of Concrete-Filled Steel Tubular Parabolic Arches Under Five-Point Concentrated Loads

This paper deals with experimental investigations into in-plane stability of fixed concrete-filled steel tubular (CFST) parabolic arches. Three CFST arches with the same span but different rise-to-span ratios were tested under five-point symmetrical concentrated loads over the full span. All applied loads were controlled in synchronization. The test results show that the test arches buckled in an antisymmetric failure mode and the section positions with the maximum deformation were slightly different for three test arches. It is found that the bearing capacity of CFST arches decreased significantly with the decline of rise-to-span ratios, and the outer steel tube provided significant confinement effects on the core concrete after the load reaches 80% of the load-carrying capacity. Moreover, comparisons between the test and finite element results indicate that the existing beam-element modeling method can predict the in-plane stability performance of CFST arches very well.

Elastic In-Plane Buckling of Funicular Arches

Buckling loads of arches could be significantly affected by the assumptions made on the load behavior during buckling. For a funicular arch whose centerline coincides with the compression line, we may consider two types of load behaviors based on how the line of load action shifts during buckling. This paper presents the governing differential equations for the elastic in-plane buckling problem of funicular circular arches under uniform radial pressure based on the two different load behavior assumptions, as well as analytical and numerical methods for analysis. For the analytical method, buckling criteria of rotationally-restrained ended circular arches with an internal rotational spring are formulated by using the general solution of the governing differential equation. For the numerical method, the Hencky bar-chain model (HBM) and its simple matrix formulations for general funicular arches are established. The buckling loads and mode shapes of funicular circular arches are solved by using HBM and verified against exact solutions obtained from the analytical method. For funicular catenary arches and parabolic arches, the buckling load solutions by HBM with various number of segments are also obtained and compared with the solutions presented by the previous researchers.

New in Old: Simplified Equations for Linear-elastic Symmetric Arches and Insights on Their Behavior

Closed-form equations for determination of reactions and internal forces of linear-elastic symmetric arches with constant cross-sections are derived. The derivation of the equations was initially made for segmental, threehinged, two-hinged, and hingeless arches. Not all derived equations are simple, but still not excessively complex to apply, and they reveal several new insights into the structural behavior of arches. The first is an extremely simple approximate equation for horizontal reactions of a hingeless arch under self-weight, which could be also applied with excellent accuracy to catenary and parabolic arches, and with a desirable level of accuracy to two- and three-hinged arches with a relatively wide range of geometries. The second insight is an approximately linear relationship between reactions and between internal forces of arches with different structural systems, which helps understand the global structural behavior of arches in a new way and enables inference of some other insights presented in the paper. The third insight reflects the relationships between normal force distribution and its eccentricity in different types of arches. Finally, the fourth insight regards the comparison of behavior of arches under the self-weight with those loaded with uniformly distributed load along their span.