Shallow Arch Research Articles

Sensitivity of FGM shallow arches to loading imperfection when loaded by a concentrated radial force around the crown

The non-linear in-plane stability of pin-ended circular shallow arches is analytically investigated. The arches are made of functionally graded material with the material properties varying along the thickness. The external load is a radial concentrated force in the direct vicinity of the crown. The effect of the bending moment on the membrane strain is included in the one-dimensional model. The equations of the pre-buckling and buckled equilibrium are derived using the principle of minimum potential energy. Analytical solutions are found for both bifurcation and limit point buckling. Extensive parametric studies are performed to find and demonstrate the effect of various parameters on the buckling load and in-plane behaviour. It is found that most arches have multiple stable and unstable equilibria and the number of equilibria increases with the modified slenderness. When the load is applied exactly at the crown, the lowest buckling load is related to bifurcation buckling for most geometries and material compositions but when there is a small imperfection in the load position, only limit point buckling is possible. The position of the load can have quite a huge influence on the buckling load, therefore pin-ended functionally graded shallow arches are sensitive to small imperfections in the load position.

Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables’ external excitations

The nonlinear dynamic behaviors of a double cable-stayed shallow arch model are investigated under the one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. The in-plane governing equations of the system are obtained when the harmonic excitation is applied to cables. The excitation mechanism due to the angle-variation of cable tension during motion is newly introduced. Galerkin’s method and the multi-scale method are used to obtain ordinary differential equations (ODEs) of the system and their modulation equations, respectively. Frequency- and force-response curves are used to explore dynamic behaviors of the system when harmonic excitations are symmetrically and asymmetrically applied to cables. More importantly, comparisons of frequency-response curves of the system obtained by two types of trial functions, namely, a common sine function and an exact piecewise function, of the shallow arch in Galerkin’s integration are conducted. The analysis shows that the two results have a slight difference; however, they both have sufficient accuracy to solve the proposed dynamic system.

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.

Finite element analysis of functionally graded hyperelastic beams under plane stress

An alternative finite element formulation to analyze highly deformable beams composed of functionally graded (FG) hyperelastic material is presented. The 2D beam element adopted has a general order and seven degrees of freedom per node, allowing both axial and shear effects, as well as cubic variation of transverse strains. The constitutive law employed is the hyperelastic compressible neo-Hookean model for plane stress conditions. The material coefficients vary along the beam thickness according to the power law. The nonlinear system of equilibrium equations is solved numerically by the Newton–Raphson iterative technique. Full integration scheme and division in load steps are employed to obtain accuracy and stability, respectively. Four illustrative examples involving highly deformable elastic beams under plane stress and static conditions are analyzed: cantilever beam under free-end shear force, semicircular cantilever beam, column under buckling and shallow thin arch. The effects of the mesh refinement, the FG power coefficient and the transverse enrichment scheme on the beam behavior are investigated. In general, mesh refinement provides more accurate results, the power coefficient has a more significant effect on displacements and the second enrichment rate is needed to correctly predict the nonlinear variation of transverse strains across the thickness. This nonlinear variation, together with the moderate values of strains and stresses achieved, reinforces the need of adopting a nonlinear hyperelastic model. Finally, the determination of the out-of-plane strain from the plane components is solved numerically by a proposed Newton’s method.

Geometrically nonlinear rapid surface heating of temperature-dependent FGM arches

Based on the nonlinear dynamic analysis, thermally induced vibrations of the FGM shallow arches subjected to different sudden thermal loads are studied. Temperature and position dependence of the material properties are taken into account. Based on the uncoupled thermoelasticity assumptions, The non-linear one-dimensional transient heat conduction equation is solved numerically by a hybrid iterative GDQ method and Crank-Nicolson time marching scheme. A first order shear deformation arch theory (FSDT) is also combined with the von Kármán type of geometrical non-linearity and the Donnell kinematic assumption to obtain the equations of motion employing the Hamilton principle. Discretization of the highly coupled non-linear equations of motion is done by using the GDQ method in the arch domain. The solution of the system of the ordinary differential equations is established by means of a hybrid iterative Picard-Newmark scheme. Comparison is also made with the existing results for the case of isotropic homogeneous shallow arches, where good agreement is obtained. Also, parametric studies are proposed to show the effects of temperature dependency, geometrical non-linearity, arch thickness, power law index, and the type of thermal-mechanical boundary conditions upon the arch deflection.

Nonlinear Dynamics of MEMS Arches Assuming Out-of-Plane Actuation Arrangement

Abstract In this work, the nonlinear dynamics of a microbeam shallow arch actuated through an out-of-plane electrostatic force arrangement is investigated. A reduced order model is developed to analyze the static, free vibration, and nonlinear dynamic response of the microstructure under different direct current and alternating current load conditions. A numerical investigation is conducted by comparing the response of the arch near primary and secondary resonances using a nonparallel plates actuation scheme where the arch itself forms a moving electrode. The results show that the nonparallel excitation can be efficient for primary and secondary resonances excitation. Moreover, unlike the classical parallel plates method, where the structure is vulnerable to the dynamic pull-in instability, this nonparallel excitation arrangement can provide large amplitude motion while protecting the structure from the so-called static and dynamic pull-in instabilities. In addition to primary resonance, secondary resonances are demonstrated at twice and one-half the primary resonance frequency. The ability to actuate primary and/or secondary resonances without reaching the dynamic pull-in instability can serve various applications where large strokes increase their performance, such as for resonator-based sensitive mass sensors.

An Efficient Strategy for Solving Structural Nonlinear Equations by Combining the Orthogonal Residual Method and Normal Flow Technique

This paper presents a new procedure for solving structural nonlinear problems by combining the orthogonal residual method (ORM) and normal flow technique (NFT). The perpendicularity condition to the Davidenko flow, introduced by the NFT, which must be satisfied during the iterative process, overcome the difficulties, i.e. the poor convergence and inefficiency of the ORM close to the limit points, particularly the displacement limit points (snap-back behavior). Basically, the idea of the proposed strategy is to adjust the load parameter, which is treated as a variable in the nonlinear incremental-iterative solution process, assuming that the unbalanced forces (residual forces) must be orthogonal to the incremental displacements. This constraint is used together with the NFT perpendicularity condition. The proposed procedure is tested, and its efficiency is corroborated through the analyses of slender shallow and nonshallow arches and an L-frame since they exhibit highly nonlinear behaviors under certain loading conditions. It is concluded that the proposed procedure can overcome the numerical instability problems in the neighborhood of critical points when using only the conventional OR process, and the procedure compares favorably with the arc-length method, minimum residual displacement method, and generalized displacement control method.

Planar nonlinear dynamic behavior of a cable-stayed bridge under excitation of tower motion

Based on a double cable-stayed shallow arch model of the cable-stayed bridge, the novel dynamic theory and analysis of nonlinear dynamic behavior of the system when cables' upper ends are subjected to harmonic excitation introduced by motion of tower are established and carried out. A set of partial differential equations governing the motion of present system are derived firstly according to the classic dynamic theories of cables and the shallow arch. Then, they are used to obtain the ordinary differential equations of the system by Galerkin's integral method. The corresponding modulation equations are derived by implementing the standard process of perturbation method of multiple scales when the 1:1:1 internal resonance among the lowest modes of cables and the shallow arch and external resonance of the system occur simultaneously. Frequency- and force-response curves are plotted to explore the rich dynamic behaviors of the system. The research shows the asymmetric harmonic excitations can cause the different jump phenomenon of cables, even the reverse jump is observed when the subharmonic resonance occurs.

Thermally induced large deflection of FGM shallow micro-arches with integrated surface piezoelectric layers based on modified couple stress theory

Based on the modified couple stress theory, nonlinear thermally induced large deflection analysis of shallow sandwich arches is studied. A functionally graded material (FGM) micro-arch with piezoelectric layers integrated into the surfaces and with immovable pinned and fixed edges is analyzed. Temperature and position dependence of the thermomechanical properties for an FGM micro-arch are taken into account. The piezo-FGM arches are subjected to different types of thermal loads such as uniform temperature, linear temperature, and heat conduction. A modified couple stress theory is combined with the uncoupled thermoelasticity assumptions to derive the governing equations of the arch by using the virtual displacement principle. The von Karman type of geometrical nonlinearity and first-order shear deformation theory are also used to obtain the equilibrium equations. The nonlinear governing equilibrium equations of the piezo-FGM sandwich arch under different thermal loads are solved analytically. The solutions of the system of ordinary differential equations for both cases of boundary conditions are established by employing the two-step perturbation technique. Comparison is made with the existing results for the cases of FGM arch without couple stress and piezoelectric layers under uniform temperature rise, and good agreement is obtained. Also, parametric studies are proposed to show the effects of couple stress, piezoelectric layers, volume fraction index, geometrical parameters, and temperature dependence, the thermally induced deflection of the piezo-FGM sandwich arch.

Nonlinear Buckling and Postbuckling of Shallow Arches With Vertical Elastic Supports

This paper presents an analytical method to investigate the effects of symmetric and asymmetric elastic supports on the nonlinear equilibria and buckling responses of shallow arches. It is found that arches with symmetric elastic supports can bifurcate into secondary paths with high-order symmetric modes. When a small asymmetry exists in the elastic supports, the equilibria of the arch may abruptly split and lead to the occurrence of remote unconnected equilibria. Such unconnected equilibria can be obtained experimentally or numerically using typical path following controls only with prior knowledge of location of these paths. A small asymmetry in the elastic supports may also make a secondary branch shrink into points connecting surrounding equilibria, resulting in the appearance of more limit points. The analytical solutions are also derived to directly calculate critical loads. We find that the magnitude of the stiffness of symmetric elastic supports has no influence on limits loads and bifurcation loads at branching into secondary paths with symmetric configurations, but greatly affect the bifurcation loads of secondary paths with asymmetric configurations. All critical loads are very sensitive to the degree of asymmetry in the elastic supports. The asymmetry in the supports reduces the top values of all pairs of critical loads compared to the case of symmetric elastic supports. The results obtained from the analytical derivations are confirmed using finite element analysis (FEA).

Static response and free vibration of MEMS arches assuming out-of-plane actuation pattern

In this research work, the nonlinear structural behavior of a non-parallel plates micro-actuator design is investigated via a reduced-order modeling. The micro-actuator is considered to be made of an initially flexible curved doubly-clamped microbeam and two evenly arranged stationary rectangular shaped out-of-plane electrodes of different lengths. The subsequent actuating attractive electrostatic force is mainly formed by the unevenness of the electric fringing-fields (non-parallel) electrodes arrangement. Results of this numerical investigation show that by increasing the actuation voltage the shallow arch behavior tend to reach a softening like behavior, mainly due to its dominant flexibility effect. With a further increase in the actuation voltage, the micro-arch starts to stretch and thus develops a hardening like behavior. Furthermore, and for certain values of the design parameters (mainly the shape and the length of the stationary electrodes), the shallow arch presented a snap-through like bi-stability behavior. Indeed, when the voltage approaches a certain critical value, the micro-arch static profile alters from a symmetrical shape to an asymmetric one, thus showing a symmetry breaking like behavior that depends a lot on the shape and length of the stationary non-parallel actuating electrodes. It is also demonstrated that by varying few of the micro-actuator design parameters, the variation of the normalized fundamental frequency showed rises and declines for certain ranges of the applied bias voltage. This approves that with such electrostatically actuated micro-actuator design, and with a smart tuning of its geometrical design parameters, one can obtain softening as well as hardening like behaviors. Modes frequency variations curves showed also that for certain actuation DC loads, it is possible to achieve a one-to-one internal resonance state involving both the first symmetric and the first antisymmetric modes of the shallow micro-arch.

Nonlinear rapid heating of shallow arches

The analysis of large amplitude thermally induced vibrations of a shallow curved beam is considered in this article. The beam is made from an isotropic homogeneous material. One surface of the arch is subjected to rapid surface heating while the other surface is kept at reference temperature. The temperature profile is evaluated by means of the 1D heat conduction equation across the thickness of the beam. The thermally induced moments and forces are evaluated from the temperature profile. These resultants are inserted into the equations of motion of the arch. To establish the equations of motion, kinematic assumptions of a shallow arch with considerations of the von Kármán type of geometrical nonlinearity, Euler–Bernoulli beam assumptions, and linear stress–strain–temperature law are employed. The equations of motion are discreted using the conventional polynomial Ritz formulation. The resulting equations are nonlinear and at each time step the Newton–Raphson technique is used to obtain the solution. The equations of motion of the arch are traced in time domain using the Newmark time marching scheme, which results in the temporal evolution of lateral displacement at arbitrary point of the arch. Numerical results are provided to explore the effects of different parameters. It is shown that thermally induced vibrations indeed exist especially for thin arches. As the beam becomes thicker or shorter, the thermally induced vibrations fade up and for sufficiently thick arches, quasi-static and dynamic responses of the arch become identical.

Planar multimodal 1:2:2 internal resonance analysis of cable-stayed bridge

Abstract This study focuses on the analysis of the linear and nonlinear dynamics of a reduced nonlinear coupling model of cable-stayed bridge consisting of two cables and a shallow arch, considering the effect of geometric nonlinearity of cables and the shallow arch. The planar 1:2:2 internal resonance among three first modes of cables and the shallow arch is firstly investigated with the external harmonic excitation applied on the shallow arch. Partial differential equations that govern motion of the system are derived. The piecewise mode function of the shallow arch, satisfying continuous and mechanical conditions simultaneously, is newly derived and chosen as the trial function in Galerkin’s method rather than commonly trigonometric function. Moreover, Galerkin’s method is applied to get the corresponding ordinary differential equations of the system and the method of multiple scales is used to obtain modulation equations for exploring the nonlinear dynamic behaviors. Lastly, the frequency- and force-response curves are given to investigate the planar dynamic behaviors of the system. Some interesting and novel conclusions have been obtained.

Thermomechanical nonlinear in-plane analysis of fix-ended FGM shallow arches on nonlinear elastic foundation using two-step perturbation technique

An attempt is made in this research to analyse the nonlinear response of functionally graded material shallow arches with both edges clamped. The arch is resting on a three parameter nonlinear elastic foundation during deformation and is subjected to uniform lateral pressure and uniform temperature rise. Material properties are expressed according to a power law function and are assumed to be temperature dependent. The governing equilibrium equations of the arch are established with the aid of third order shear deformation curved beam theory of Reddy and von Karman type of strain–displacement relations. The obtained equations contain three coupled and nonlinear equations in terms of circumferential displacement, lateral displacement and cross section rotation. Considering the immovable type of edge supports, the equations are reduced to two new coupled and nonlinear equations. These equations are solved using the two step perturbation technique for the case of clamped boundary conditions. Explicit expressions are resulted which yield the deflected shape of the arch as a function of temperature elevation and uniform pressure. It is shown that the arch reveals the snap-through type of instability under certain conditions. The response of the arch is highly affected by the power law index, thermal environment, side to thickness ratio and stiffnesses of the foundation.

Multiple internal resonances in MEMS arch resonators

Micromachined shallow arch resonant beams have attracted significant attention thanks to their rich dynamical behavior, inherent nonlinearities, and the potential to excite various internal resonances. Currently, there is a lack of comprehensive experimental studies for the various types of internal resonances in arches and particularly at the micro and nano scales. Here, we aim to investigate and identify different types of internal resonances of an initially curved beam, electrothermally actuated and electrostatically driven, by electrical characterization techniques. Upon changing the electrothermal voltage of silicon micromachined arches, the second symmetric natural frequency of an arch is adjusted to near twice, three times, and four times the fundamental natural frequency, which gives rise to 2:1, 3:1, and 4:1 autoparametric resonances between the two modes. These resonances are demonstrated experimentally. We show various frequency-response curves of the total response around the excitation frequency and highlight the contribution of each mode before, during, and after the internal resonances. Allan-deviation results are also shown indicating enhanced frequency stabilization of the arch oscillation when experiencing internal resonances. These studies motivate further research in this direction to exploit internal resonances of micromachined resonators for practical applications, such as sensors and mechanical amplifier.

Instability of imperfect non-uniform shallow arch under uniform radial pressure for pinned and fixed boundary conditions

The instability of imperfect non-uniform circular shallow arch under radial pressure for both pinned-pinned and fixed-fixed boundary conditions is discussed in the article. The imperfection includes the symmetric initial displacement inducing no stress and the non-uniformity which represented by three piecewise constant-stiffness segments. Analytical theoretical formulation is derived by the least potential energy principle. Through mathematical analysis, it's shown rigorously that the effect of imperfection still doesn’t vanish as the mode number of imperfection approaching to infinity. The stiffer end case and stiffer center case are discussed concerning equal potential energy load and snap-through criterion. Distinct snap-through behavior is observed for stiffer end case with fixed boundary condition by a proposed index plot. Moreover the two limiting cases including rigid end case and rigid center case are investigated by employing the augmented potential energy with Lagrangian multipliers considering the imperfection effect. Snap-through criteria are obtained in a closed form for rigid center case. And for the rigid end case with pinned BC, when the center segment's size is tending to zero, an asymptotical solution is derived and by this solution, the importance of multiplicity of roots of some equations governing the snap-through behavior is illustrated clearly. Besides, the equal potential energy load of two limiting cases is analyzed in detail with emphasis on imperfection effect. This paper aims to extend the understanding of imperfect non-uniform circular shallow arch under radial pressure.

Detection of symmetry breaking bifurcations using finite element analysis packages

Trusses with geometric and loading symmetries have been used in many structures to reduce the complexity of the design. Slight asymmetries in geometry and in loading could lead to bifurcation in the structural response. Failure of such structures still occur occasionally causing major damage to the property and to the human lives. A fully nonlinear structural analysis is expected to detect such symmetry breaking bifurcations. In this paper, we conducted the bifurcation analysis of a two-bar truss and a shallow arch structure with seven bars. Two program packages Gesa and Ansys based on finite elements method have been used to detect the symmetry breaking bifurcation points. However, unlike typical bifurcation analysis packages they cannot detect the bifurcation point without inserting a small perturbation in the initial geometry or the loading of the structure. Such a bifurcation can be easily missed in finite element-based analysis. The theoretical analysis reveals that the bifurcation leads to a much lower critical load in the presence of small asymmetry compared to the symmetric case. The results are verified by using Gesa program in Matlab for fully nonlinear analysis and with results by using Ansys commercial program. The two structural examples serve to illustrate the limitations of widely used finite element analysis packages for nonlinear bifurcation analysis.

A Semianalytical Approach for Nonlinear Dynamic System of Shallow Arches Using Higher Order Multistep Taylor Method

This study aimed at obtaining a semianalytical solution for nonlinear dynamic system of shallow arches. Taylor method was applied to find the analytical solution, and an investigation of their dynamic characteristic was carried out to verify the applicability of this methodology for the shallow arches under step or periodic excitation. A polynomial solution can be obtained from this multistep approach with respect to time, and direct buckling as well as indirect buckling of the shallow arches can be observed, also. The results indicated that the dynamic buckling load level was higher with higher shape factor. Additionally, a change of attractor in phase space was investigated. Coupling in symmetric mode as well as asymmetric mode was observed in case of indirect buckling, and a sensitive response was also manifested during sinusoidal and beating excitation. These results of applying multistep Taylor series for the investigation of displacement response and attractor change revealed that this analytical approach was valid in explaining the dynamic buckling behavior of shallow arches under direct and indirect snapping.

Stability of a half-sine shallow arch under sinusoidal and step loads in thermal environment

Abstract The complex structural behavior of shallow arches can be remarkably affected by many parameters. In this paper, the structural responses of a half-sine pin-ended shallow arch under sinusoidal and step loadings are accurately calculated. Additionally, the effects of environmental temperature changes are considered. Three types of sinusoidal loadings are separately investigated. Displacements, load-bearing capacity, the magnitude of the axial force and the locus of critical points (including limit and bifurcation points) are directly obtained without tracing the corresponding equilibrium path. Furthermore, the boundaries identifying the number of critical points are investigated. All mentioned structural responses are formulized based on the rise of the arch and the environmental temperature change, which are introduced in a dimensionless form. The proposed formulation is also developed for generalized sinusoidal loadings. Additionally, the structural behavior of the shallow arch under two types of step loadings is investigated. Finally, the accuracy of the suggested approach is examined by a non-linear finite element method.

Symmetric snap-through and equal potential energy load of non-uniform shallow arch under a concentrated load considering imperfection effect

This paper deals with analytical investigation of non-uniform circular shallow arch under a concentrated load considering imperfection effect under pinned-pinned boundary conditions. After characterizing the non-uniformity by three piecewise constant stiffness segments, governing equilibrium equations including imperfection effect are derived via least potential energy principle and criteria for possible occurrence of symmetric snap-through are explicitly presented by using L'Hospital's rule after identifying two modified slenderness parameters. The existence of an equal potential energy load is rigorously shown in a straightforward manner and its parametric dependence on imperfection and non-uniformity is discussed. The relationship between equal potential energy load and buckle propagation load is studied by 3D FEA of buckle propagation of corresponding non-uniform shallow panel. Moreover, two limiting cases including rigid end case and rigid center case are analyzed by employing augmented potential energy with Lagrangian multipliers. An asymptotical solution shows that a finite modified slenderness (independent of imperfection) is sufficient to ensure the occurrence of symmetric snap-through when dimension of center segment is approaching to zero in rigid end case and this conclusion holds when there is some rotational restraint at ends. For rigid center case, the closed-form criterion considering imperfection for symmetric snap-through occurrence is presented. Equal potential energy load for two limiting cases is derived and analyzed in detail. This paper is expected to enhance the understanding of stability of imperfect non-uniform shallow arch.