Shallow Circular Arches Research Articles

Instability of shallow arches with possible end-movements and rotations

The article addresses the stability of shallow circular arches. In the majority of the available works, ideal circumstances are considered. It is now assumed that due to support or foundation imperfection, at one end, small displacements and rotations could occur. The one-dimensional beam model is geometrically non-linear and in an attempt to improve and extend the applicability of the existing solutions, the model incorporates the effect of axial force on the curvature change. The static equilibrium equations are found from the principle of virtual work. These relations are then solved analytically. The aim is to find the instability load when the arch is under the action of a concentrated force. The possible influence of the support displacements and rotations are evaluated in terms of the geometry to find out the potential benefits or disadvantages of these effects on the lowest buckling load.

Experimental path-following of equilibria using Newton’s method. Part I: Theory, modelling, experiments

Modern numerical path-following techniques provide a comprehensive suite of computational tools to study the bifurcation behaviour of engineering structures. In contrast, experimental testing of load-bearing nonlinear structures is still performed using simple force control (dead loading) or displacement control (rigid loading). This means that established experimental methods cannot trace equilibrium manifolds in their entirety because structures snap to alternative equilibria at limit points in the forcing parameter and because branch switching to alternative equilibria cannot be controlled and performed reliably. To extend current testing methods, in Part I of this paper, we implement an experimental path-following method that uses tangent quantities (stiffness and residual forces) and Newton’s method to continue along stable and unstable equilibrium paths and traverse limit points. In addition to enforcing the displacement at primary load-introduction points, the overall shape of the structure is controlled via secondary actuators and sensors. Small perturbations of the structure using the secondary actuators allow an experimental tangent stiffness to be computed, which is then used in a control algorithm. As a pertinent test case, the experimental method is applied to a transversely loaded shallow circular arch. Due to the complexity of the test setup, the experiment is first designed using a virtual testing environment based on a surrogate finite element model. Experimental results demonstrate the robustness of the proposed experimental method and the usefulness of virtual testing as a surrogate, but also highlight that experimental efficiency and the effects of noise and sensor uncertainty is of particular concern. In Part II, we present perspectives on future research directions and novel testing capabilities that are enabled by extending the methodology to pinpointing of critical points, tracing of critical boundaries, and branch switching.

Stability of pinned-rotationally restrained arches

The article aims to find the buckling loads for pinned?rotationally restrained shallow circular arches in terms of the rotational end stiffness, geometry and material distribution. The loading is a concentrated vertical force placed at the crown. A geometrically nonlinear model is presented which relates not only the axial force but also the bending moment to the membrane strain. The nonlinear load-strain relationship is established between the strain and load parameters. This equation is then solved and evaluated analytically. It turns out that the stiffness of the end-restraint has, in general, a significant effect on the lowest buckling load. At the same time, some geometries are not affected by this. As the stiffness becomes zero, the arch is pinned-pinned and as the stiffness tends to infinity, the arch behaves as if it were pinned-fixed and has the best load-bearing abilities.

In-plane dynamic instability of a shallow circular arch under a vertical-periodic uniformly distributed load along the arch axis

This paper presents an analytical investigation on the in-plane dynamic instability of a shallow circular arch subjected to a vertical-periodic load uniformly distributed along the arch axis, which has not been previously reported in the literature hitherto. Equations of motion of the arch in the radial and tangential directions are derived by employing the Hamilton's principle. The trigonometric and hyperbolic functions are utilized as the mode shapes for the tangential and radial displacements. A method for determining critical instability regions bounded by the periodic solutions with the period T or 2T is presented for the antisymmetric and symmetric dynamic instability of the arch. Frequency sweep transient finite element analyses (FEA) are conducted to verify the analytical solution. Through comparisons, it is observed that analytical solutions agree well with the FEA results. It is shown that the bandwidth of instability regions bounded by the periodic solutions with a period 2T is significantly greater than that bounded by the periodic solutions with a period T. Effects of the rise-span ratio, damping ratio and static component of the periodic load on the dynamic instability are explored. It is found that the dynamic instability behavior of an arch is significantly influenced by the rise-span ratio and damping ratio and their increases lead to the decrease of the bandwidth of the dynamic instability region. It is also found that when the static component of the periodic load is considered, the dynamic instability region undergoes significant variations and moves towards to the lower excitation frequency owing to the decrease of vibration frequency of the arch. The bandwidth of the instability regions, however, increases with an increase of the static component. In addition, it is shown that dynamic instability of modal superposition may occur under a periodic load with a period between T and 2T and symmetric and antisymmetric instability modes are alternately changed during the dynamic instability of modal superposition.

Snap-Through of Shallow Circular Arches with Variable Horizontal Supports under Unilateral Displacement Control

AbstractThis study proposes a numerical method to solve the snap-through problem of elastic shallow inextensible circular arches. Specifically, the snap-through behavior of arches subjected to a do...

In-plane stability of shallow concrete arches under fire

PurposeConcrete arch structures are commonly constructed for various civil engineering applications. Despite their frequent use, there is a lack of research on the response and performance of concrete arches when subjected to fire loading. Hence, this paper aims to investigate the response and in-plane failure modes of shallow circular concrete arches subjected to mechanical and fire loading.Design/methodology/approachThis study is conducted through the development of a three-dimensional finite element (FE) model in ANSYS. The FE model is verified by comparison to a non-discretisation numerical model derived herein and the reduced modulus buckling theory, both used for the non-linear inelastic analysis of shallow concrete arches subjected to uniformly distributed radial loading and uniform temperature field. Both anti-symmetric and symmetric buckling modes are examined, with analysis of the former requiring geometric imperfection obtained by an eigenvalue buckling analysis.FindingsThe FE results show that anti-symmetric bifurcation buckling is the dominant failure mode in shallow concrete arches under mechanical and fire loading. Additionally, parametric studies are presented which illustrate the influence of various parameters on fire resistance time.Originality/valueFire response of concrete arches has not been reported in the open literature. The authors have previously investigated the stability of shallow concrete arches subjected to mechanical and uniform thermal loading. It was found that temperature greatly reduced the buckling loads of concrete arches. However, this study was limited to the simplifying assumptions made which include elastic material behaviour and uniform temperature loading. The present study provides a realistic insight into the fire response and stability of shallow concrete arches. The findings herein may be adopted in the fire design of shallow concrete arches.

Non-linear in-plane multiple equilibria and buckling of pin-ended shallow circular arches under an arbitrary radial point load

• Load location influences multiple limit points and non-linear equilibria of pin-ended arch. • Criteria distinguishing limit points and multiple equilibria are established. • Special modified slenderness corresponding to point of inflexion or cusp for switching multiple limit points and equilibria. • Load, axial force and displacement at limit points, points of inflexion and cusps are derived. • In some cases, mathematical solutions of nonlinear equilibria do not represent mechanical equilibrium paths. This paper analytically investigates effects of the load location on the non-linear in-plane multiple equilibria, limit points, stationary points of inflexion, cusp, and buckling behavior of a pin-ended elastic shallow circular arch under a radial point load at an arbitrary location along the arch length. Theoretical solutions for the non-linear response of the arch to the arbitrary radial point load including the limit points, stationary points of inflexion, cusp and multiple equilibria are derived. The major findings are: (1) there exists special modified slenderness corresponding to an arch, whose non-linear equilibrium path has stationary point of inflexion or a cusp and which can distinguish the number of multiple limit points and equilibria; (2) criteria distinguishing multiple limit points and equilibria are developed by relating the special modified slenderness to the load location; (3) theoretical solutions for the load, axial force and displacement at stationary points of inflexion and at cusps are also deduced; (4) the load location and the modified slenderness of an arch significantly influence the non-linear multiple equilibria of the arch; and (5) the load location has significant influence on the buckling pattern of an arch, and when the point load is applied at a location away from the apex of the arch, the arch can buckle only in a limit point instability pattern, but not in a bifurcation pattern.

Nonlinear in-plane elastic buckling of a laminated circular shallow arch subjected to a central concentrated load

This paper investigates the in-plane elastic buckling of a laminated shallow circular arch subjected to a central concentrated load based on the classical laminated theory (CLT). Nonlinear equilibrium equation is established by using the principle of minimum potential energy, from which analytical solutions of the nonlinear in-plane elastic equilibrium path and buckling load are derived for laminated circular shallow fixed arches fixed. The critical modified slenderness ratio which switches the buckling characteristics is also obtained. The finite element (FE) simulation and experiments employing displacement control are conducted to verify the accuracy of the analytical solutions. A parametric study is then carried out and results are presented in both tabular and graphical forms to analyze the effects of the ply angle and thickness of laminates, and rise-span ratio of the arch on the critical bucking load and nonlinear equilibrium path of laminated arches.

Static and dynamic symmetric snap-through of non-uniform shallow arch under a pair of end moments considering critical slowing-down effect

This paper presents analytical analysis of static and dynamic symmetric snap-through of non-uniform shallow circular shallow arch under a pair of end moments with the same magnitude. The non-uniformity is characterized by dividing the arch into three piecewise constant-stiffness segments. Hamilton's principle is used to derive the governing differential equations by assuming negligible axial inertia. The snap-through load and snap-through criterion are analyzed in detail and by an index plot, the stiffer center case is shown to behave distinctly when stiffer end case is compared. The dynamic snap-through when the moment is slightly higher than the snap-through moment is analyzed analytically by a perturbation method, and a critical slowing effect is observed when the moment is approaching to the snap-through moment. Comparison with dynamic FEA shows a good agreement with the analytical result and analysis on theoretical finite time blow-up phenomenon reveals that when geometric parameters are corresponding to the critical snap-through condition, the initial quadratic phase's motion is slow and relatively blow-up phase's motion is fast. The analytical formulations have been extended to include two limiting cases including rigid end case and rigid center case by using the constrained Hamilton's principle by Lagrangian multipliers. Snap-through criteria analysis reveals closed-form criterion for rigid center case and an asymptotic result for rigid end case. This paper serves to enhance the knowledge on snap-through and critical slowing down for shallow arches with non-uniformity under end moments.

Nonlinear stability analysis of FGM shallow arches under an arbitrary concentrated radial force

The nonlinear in-plane buckling behaviour of pinned-pinned shallow circular arches made of functionally graded material (FGM) is investigated using a one-dimensional Euler–Bernoulli model. The effect of the bending moment on the membrane strain is included. The material properties vary along the thickness of the arch. The external load is a radial concentrated force at an arbitrary position. The pre-buckling and buckled equilibrium equations are derived using the principle of virtual work. Analytical solutions are given for both bifurcation and limit point buckling. Comprehensive studies are performed to find the effect of various parameters on the buckling load and on the behaviour. The major findings: (1) arches have multiple stable and unstable equilibria; (2) when the load is at the crown, the lowest buckling load is related to bifurcation buckling for most geometries and material compositions but when the external force is replaced, only limit point buckling is possible; (3) the position of the load has huge influence on the buckling load; (4) such arches are sensitive to small loading imperfections when loaded in the vicinity of the crown. The paper intends to improve and extend the existing knowledge about the behaviour of pinned-pinned shallow FGM arches under arbitrary concentrated radial load.

Localized loading and nonlinear instability and post-instability of fixed arches

In the engineering practice, arches, such as arch bridges, are often subjected to localized loading. It is known that the nonlinear in-plane instability of shallow arches is caused by the significant in-plane bending action. The different localized loading produces very different in-plane bending distributions and so the instability of arches under localized loading would be quite different from that under a central point load or under a radial load uniformly distributed over the entire arch. However, there are no studies about the nonlinear instability and post-instability of arches under localized loading reported in the literature. This paper presents nonlinear in-plane analyses for fixed shallow circular arches under a localized uniform radial load. Novel theoretical solutions for nonlinear equilibrium, limit point instability, and bifurcation instability for arches under a localized uniform load are obtained, and new theoretical solutions of the certain modified slenderness switching the instability patterns of an arch under localized uniform loading are also obtained. It is found that the length of localized loading segment significantly influences the in-plane instability and post-instability responses of an arch. The limit point and bifurcation instability loads decrease with a decrease of the length of the localized loading segment. However, the modified slenderness switching instability patterns is found to increase with a decrease of the length of the localized loading segment. Comparisons against the finite element results show the theoretical solutions derived in this paper are accurate. These theoretical solutions provide comprehensive solutions for nonlinear instability and post-instability of fixed arches and they can reduce to those under a central point load or to extend to those under a uniform load over the entire arch.

On collapse of non-uniform shallow arch under uniform radial pressure

Analytical investigation of collapse of a pinned non-uniform shallow circular arch under uniform radial pressure is presented in this paper. The non-uniformity is introduced by dividing the arch into three regions of constant stiffness. The equilibrium equations are obtained by using least potential energy principle. By proper nondimensionalization, the presented solution is independent of total length of the arch. And two modified slenderness parameters are identified which enables the result to be valid for any symmetric cross-section shape. Parametric study on different geometric parameters has been carried out on the snap-through buckling load and anti-symmetric bifurcation load. The validity of the analytical result is verified by comparison with FEA results. Criteria concerning four deformation modes are identified. We show there is an equal potential energy load for some non-uniform shallow arches by a straightforward deduction, the existence of which load is a necessary condition for the buckle propagation of a corresponding long shallow panel. Finally two limiting cases (rigid center case and rigid end case) with extreme non-uniformity are analytically studied by using augmented potential energy with Lagrangian multipliers. For rigid center case, closed-form condition for the possible occurrence of symmetric snap-through is presented. For rigid end case, an asymptotical analysis leads to a somewhat surprising critical value of slenderness which makes the snap-through buckling just possible. This paper intends to improve the understanding of the effect of non-uniformity on collapse of shallow arch under radial uniform pressure.

Nonlinear Dynamic Buckling of Fixed Shallow Arches under an Arbitrary Step Radial Point Load

AbstractThis paper investigates the nonlinear elastic in-plane dynamic buckling of a fixed shallow circular arch that was subjected to an arbitrarily located step radial point load, which has not b...

Nonlinear Equilibrium and Buckling of Fixed Shallow Arches Subjected to an Arbitrary Radial Concentrated Load

This paper is concerned with an analytical study of the nonlinear in-plane equilibrium and buckling of fixed shallow circular arches that are subjected to an arbitrary radial concentrated load. The structural behavior of an arch under an arbitrary radial concentrated load is quite different from that of an arch under a central concentrated load. It is shown that a fixed arch under an arbitrary radial concentrated load can buckle in a limit point instability mode, but cannot buckle in a bifurcation mode, which is different from that of a fixed arch under a central concentrated load that can buckle in a bifurcation mode or in a limit point instability mode. Analytical solutions for the nonlinear equilibrium path and limit point buckling load of shallow circular arches under an arbitrary radial concentrated load are derived. It is found that the load position influences the buckling load significantly and the influence is much related to the modified slenderness of the arch defined in the paper. It is also found that when the modified slenderness of an arch is smaller than a specific value, the arch has no typical buckling behavior. The analytical solution for the relationship of the specific modified slenderness with the load position is also derived. Comparisons with finite element (FE) results show that the analytical solutions can accurately predict the nonlinear equilibrium and buckling load of shallow fixed arches under an arbitrary radial concentrated load.

In-plane nonlinear multiple equilibria and switches of equilibria of pinned–fixed arches under an arbitrary radial concentrated load

Nonlinear in-plane multiple equilibria and buckling of pinned–fixed shallow circular arches under an arbitrary radial concentrated load are investigated. Analytical solutions for the multiple nonlinear equilibria, buckling and limit points are derived. New findings are: (1) pinned–fixed shallow arches under the arbitrary concentrated load have multiple stable and unstable equilibria; (2) the position of the arbitrary concentrated load and the modified slenderness of the arch influence the number of multiple equilibria and limit points as well as the first buckling load significantly; (3) a pinned–fixed arch under the arbitrary concentrated load can buckle in a limit point instability mode, but not in a bifurcation mode; (4) when the load is located between the crown and the pinned end, the buckling load is lower than that when the load is located between the crown and the fixed end, and (5) in addition to limit points, the nonlinear equilibria of pinned–fixed arches under the arbitrary concentrated load have inflexion points, which corresponds to specific modified slenderness switching the number of equilibria and limit points or switching buckling and no buckling behaviour. The analytical solutions for inflexion points and specific modified slenderness, and for the corresponding load, axial force and displacement are also derived for the first time in the literature. Comparisons with the finite element results have shown that the analytical solutions can accurately predict the multiple equilibria, limit points, inflexion points, and buckling load of shallow pinned–fixed arches under the arbitrary concentrated load.

Nonlinear buckling analysis of shallow arches with elastic horizontal supports

The stiffness of the supports has an influence on the buckling modes and buckling loads of the arch. This paper investigates the in-plane nonlinear behavior and stability of shallow circular arches with elastic horizontal supports that are uniformly subjected to a radial load by the principle of virtual work. The three limiting shallowness values and the critical flexibility of the elastic horizontal supports are derived to differentiate the buckling mode. As the flexibilities of the elastic horizontal supports of an arch increase, the buckling load of the arch decreases, and the central radial displacement increases. And the buckling mode of the arch would then be changed from asymmetric bifurcation buckling to asymmetric bifurcation buckling after the occurrence of snap-through buckling. Then, snap-through buckling occurs, and finally, there is no buckling. Then the parameters including rise-to-span ratio(f/L) and load type are further investigated. An experimental model is designed to verify the analytical results. Finally, a design method for a shallow circular arch with elastic horizontal supports is proposed with the requirement of the elastic-plastic buckling load carrying capacity and the maximum horizontal displacement of the supports for engineering reference.

Analytical and experimental studies on out-of-plane dynamic instability of shallow circular arch based on parametric resonance

Little research about the out-of-plane dynamic stability of arches under in-plane loading has been reported in the literature hitherto. This paper presents analytical and experimental investigations of the out-of-plane dynamic instability of elastic shallow circular arches under an in-plane central concentrated periodic load owing to parametric resonance. Differential equations of out-of-plane motion of shallow arches are established using the Hamilton principle by accounting for the effects of geometric nonlinearity, additional concentrated weights and damping. The analytical solutions of the critical excitation frequencies of the concentrated periodic load for out-of-plane dynamic instability of arches are obtained. The corresponding experimental investigations are also carried out to verify the analytical solutions. Agreements between the analytical and experimental results are very good. In addition, the effects of the central concentrated weight and the in-plane excitation amplitude on out-of-plane dynamic instability of arches are investigated. It is found that as the weight increases, the bandwidth of the critical in-plane excitation frequencies for out-of-plane dynamic instability of the arch decreases. It is also found that the bandwidth of critical frequencies increases with an increase in the excitation amplitude. Furthermore, the nonlinear inertial force is derived, which is essential in determining the out-of-plane parametric resonance. It is shown that the curve of the excitation frequency versus amplitude of out-of-plane vibration bends toward the low-frequency region and that the “traction” out-of-plane instability may occur owing to “amplitude” perturbation. To authors’ knowledge, the analytical solutions and experimental investigations for out-of-plane dynamic instability of arches owing to parametric resonance presented in the paper are first time reported in the literature. The new findings in the paper can provide an in-depth understanding of out-of-plane dynamic instability behavior of arches under a periodic load.

EFFECTS OF FOUNDATION SETTLEMENT ON IN-PLANE BUCKLING OF SHALLOW ARCHES

Foundation settlement of structures may take place and it may influence the structural behavior of structures. Shallow arches are used in engineering structures such as large span roofs and bridges. This paper is concerned with the effects of the foundation settlement on the in-plane buckling of shallow circular arches under a uniform radial load. Because the deformations of a shallow arch prior to buckling become nonlinear, their effects on the buckling also need to be considered. Differential equations of equilibrium for the nonlinear analysis are derived using the principle of virtual work. Analytical solutions for the nonlinear behavior and for the nonlinear buckling load are obtained. It is found that the foundation settlement plays an important role in the nonlinear structural response of the arch to the external load and that significantly influence the nonlinear buckling load of the arch.

Revisiting Nonlinear In-Plane Elastic Buckling and Postbuckling Analysis of Shallow Circular Arches under a Central Concentrated Load

AbstractThis paper revisits the nonlinear in-plane analysis of a shallow pin-ended circular arch under a central concentrated load. It is found that the arch may have multiple equilibrium branches and limit points, and a modified slenderness of the arch defined in the paper plays an important role in the number of the limit points and equilibrium branches. The analytical solution for the specific modified slendernesses that switch the number of equilibrium branches and limit points is derived. The analytical solutions for the load, axial force, and radial displacements corresponding to the specific modified slendernesses are also derived. This paper deepens the understanding for the geometric nonlinear analysis of circular arches and their geometric softening and stiffening behavior and provides useful benchmarks for the analyses of shallow arches. In addition, this paper extends the current solutions of nonlinear equilibrium of arches associated with axial compression to those associated with axial tensi...

Effects of approximations on non-linear in-plane elastic buckling and postbuckling analyses of shallow parabolic arches

In the non-linear in-plane elastic buckling and postbuckling analyses of shallow parabolic arches, to overcome the difficulty in deriving an accurate expression for the non-linear normal strain, an approximation assumption that the derivative of the vertical coordinate with respect to the horizontal coordinate satisfies (dy/dz)1≪1 has been adopted in many investigations. The merit of the assumption is that it leads to the same differential equations of equilibrium and the same solutions as those for shallow circular arches. However, the accuracy of the assumption and the limitation of the analytical solutions have not been examined and because of the approximation, the analytical solutions may lead to significant errors for the buckling loads of shallow parabolic arches in some cases. This paper investigates the effects of the approximation assumption on the accuracy of in-plane buckling and postbuckling analyses of pin-ended and fixed shallow parabolic arches by comparing the analytical solutions with their finite element counterparts. It is found that the analytical solutions based on the assumption have some limitations because the assumption holds approximately only for extremely shallow parabolic arches, but is not valid for most shallow parabolic arches. The analytical solutions for the buckling loads based on the assumption are larger than the corresponding finite element results for parabolic arches with a rise-to-span ratio greater than 0.08, the error of the analytical solution increases with an increase of the rise-to-span ratio of the arch, and the sources for the errors are identified and discussed. Hence, caution should be exercised when using the analytical solutions to predict the buckling load of shallow parabolic arches, particularly of those with a rise-to-span ratio greater than 0.08.