Instability Of Arch Research Articles

Mechanics and Stability of Force Chain Arch in Excavated Granular Material

Rock and soil masses in geotechnical engineering projects, such as tunnels, mines and slopes, undergo relative motion, exhibiting mechanical characteristics of solid–fluid transition under critical conditions. This work analyzes the characteristics of the solid–fluid transition interface and the mode of load transfer through biaxial compression particle flow photoelastic experiments on granular materials. The study documents that this interface forms an arch shape, marked by a force chain arch. The granular material exhibits two distinct states depending on its position: below the arch, the granular material is in a solid–fluid transitional state, with bearing capacity reduced, while above the arch, it is in a stable solid state, capable of bearing the overlying rock layer’s load. The presence of the force chain arch alters the direction of the originally downward-transferring load, redirecting it along the trajectory of the arch. Analysis of the force and stability of the force chain arch revealed that the arch shape parameters and boundary loads control the instability of the arch. Changes in the overlying and lateral loads lead to different types of instability of the force chain arch. The findings of the study are crucial for underground engineering construction and for the prevention of geological disasters related to granular material.

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.

Free vibration and snap-through instability of FG-CNTRC shallow arches supported on nonlinear elastic foundation

This paper analytically investigates the nonlinear free vibration and snap-through instability of nanocomposite shallow arches reinforced with carbon nanotubes (CNTs). The functionally graded (FG) carbon nanotube reinforced composite (CNTRC) arch with shallow curvature is analysed under uniformly distributed transverse loading. The arch is assumed to rest on a three-parameter nonlinear elastic foundation in a uniform temperature field. Thermo-mechanical properties of the arch are graded through the thickness and are considered to be temperature-dependent. The equations of motion are established based on a general high-order shear deformation theory. These nonlinear equations are derived by Hamilton’s principle within the framework of the von Kármán assumption. The static equilibrium equations and dynamic equations of motion are solved analytically for the arch with immovable simply supported edges. The two-step perturbation technique and Galerkin method are implemented to obtain the closed-form solutions. The novel results illustrate the influences of CNT distribution pattern, foundation stiffness and geometrical parameters on the linear/nonlinear frequency and snap-through instability of the arch. It is shown that the minimum value of limit loads/frequencies belongs to the FG-O type of nanocomposite arches and the maximum ones are obtained in FG-X pattern.

Analytical and experimental studies on out-of-plane dynamic parametric instability of a circular arch under a vertical harmonic base excitation

The out-of-plane dynamic stability of an arch under a vertical periodical base excitation is investigated by using both analytical and experimental methods. When the excitation frequency is near twice the out-of-plane natural frequency of the arch, an out-of-plane divergent vibration is induced, which will eventually leads to an out-of-plane dynamic parametric instability of the arch. Firstly, Hamiltonian principle is employed to derive the in-plane dynamic equations, and analytical solutions for the dynamic axial compressive force and bending moment in pre-stability state are obtained and verified by numerical simulations. Then the mode shape functions for the out-of-plane dynamic instability are analytically determined and the out-of-plane dynamic instability regions are obtained by Bolotin's method. Factors affecting the critical excitation frequencies of the out-of-plane dynamic instability regions, such as the arch's included angle, the slenderness ratio, and the logarithmic decrement, are thoroughly analyzed and verified via the method of multiple scales. Lastly, laboratory tests are carried out for the out-of-plane parametric resonance instability of fix-ended circular arches with various included angles and slenderness ratios. The critical excitation frequencies of dynamic instability regions of the tested arches are determined by increasing and decreasing the excitation frequency, and the experimental results agree with the theoretical counterparts very well. The main contributions of this work include: (1) obtaining an analytical solution of out-of-plane dynamic instability regions for the arch under a vertical harmonic base excitation; (2) applying a newly-designed experimental system to conduct the out-of-plane dynamic instability tests; (3) proposing a method of time domain analytical solution to determine the critical excitation frequencies of the out-of-plane dynamic instability region, and (4) discussing the dynamic instability mechanism for base-excited arches.

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.

In-Plane Instability of Fixed Arches under Linear Temperature Gradient Field and Uniformly Distributed Radial Load

This paper focuses on an in-plane instability analysis of fixed arches under a linear temperature gradient field and a uniformly distributed radial load, which has not been reported in the literature. Combining a linear temperature gradient field and uniformly distributed radial load leads to the changes in axial expansion and curvature of arches, producing the complex in-plane nonuniform bending moment and axial force. Therefore, it is necessary to explore the in-plane thermoelastic mechanism behavior of fixed arches under a linear temperature gradient field and a uniformly distributed radial load in the in-plane instability analysis. Based on the energy method and the exact solutions of internal force before instability, the analytical solutions of the critical uniformly distributed radial load considering the linear temperature gradient field associated with in-plane thermoelastic instability of arches are derived. Comparisons show that agreements of analytical solutions against FE (finite element) results are excellent. Influences of various factors on in-plane instability are also studied. It is found that the change of the linear temperature gradient field has significant influences on the in-plane instability load. The in-plane instability load decreases as the temperature differential of the linear temperature gradient field increases.

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.

Experimental and analytical investigation on the in-plane dynamic instability of arches owing to parametric resonance

When an arch is subjected to a periodic load, it may lose in-plane stability dynamically owing to parametric resonance. Previous investigations have been concentrated on in-plane dynamic buckling of pin-ended shallow arches. However, in engineering practice, fixed arches with different rise-to-span ratios are often encountered. Little research on in-plane dynamic instability of deep fixed arches has been reported in the literature. This paper is concerned with experimental and analytical investigations for in-plane dynamic instability of fixed circular arches with rise-to-span ratios 1/8–1/2 under a central periodic load owing to parametric resonance. Experiments are carried out to determine the in-plane frequency and damping ratio of arches, to investigate critical regions of frequencies and amplitudes of the periodic load for in-plane dynamic instability of arches, and to explore effects of the rise-to-span ratio and additional weights on dynamic instability. The analytical method for determining the region of excitation frequencies and amplitudes of the periodic load causing in-plane instability of the arch is established using the Hamilton’s principle by accounting for effects of additional concentrated weights. Comparisons of analytical solutions with test results show that they agree with each other quite well. These results show that the rise-to-span ratio significantly influences the bandwidth of regions of critical excitation frequencies for in-plane dynamic instability of arches. The critical frequencies of the periodic load and their bandwidth increase with a decrease of the rise–span ratio of the arch, whereas the corresponding amplitude of the periodic load decreases at the same time. It is also found that the central concentrated weight influences in-plane dynamic instability of arches significantly. As the weight increases, the critical frequencies of excitation and their bandwidth for in-plane dynamic instability of arches decreases, whereas the corresponding amplitude of excitation increases.

Analysis of Out-Of-Plane Stability of Bow-String Arches

Several arch bridge superstructures, designed by the authors of the paper, are presented and analysed. Spatial FEM models were utilised to analyse behaviour of each bridge. Verification of out-of-plane buckling of the arches is discussed in more detail. Comparison of using beam and shell finite elements for the mesh of main beam-arch load-carrying system is given. The variation of top bracing system and its influence on the stability of the arches is presented, as well. A minimum load amplifier to reach the elastic instability of arches is used to estimate the suitability of the bracings. Comparisons of the assessment using results produced by first and second order global analyses are presented. Application of a unique global and local imperfection as the initial geometric imperfection is proposed.

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.

Snap-Through of Arch Model under Multiple Loads

This paper is concerned with the interaction effects of multiple loads on the snap-through instability of arches. Previous work by DaDeppo and Schmidt and by Huseyin involved two independent loading parameters. Several types of interaction curves were obtained by these authors for different arches and loads.

On lateral-torsional instability of arches subjected to motion-dependent loading

The problem of dynamic lateral-torsional instability of circular arches acted upon by generally non-conservative forces is treated herein. The influence of motion-dependency of loading on the stability characteristics is determined in certain cases. It is also found that a semi-circular arch which is only capable of rigid-mode-buckling according to static analysis may lose its stability through non-rigid modes if the dynamic criterion is utilized.