A Comparative Study on the Post-Buckling Behavior of Reinforced Thermoplastic Pipes(RTPs)Under External Pressure Considering Progressive Failure

The collapse pressure is a key parameter when RTPs are applied in harsh deep-water environments.To investigate the collapse of RTPs,numerical simulations and hydrostatic pressure tests are conducted.For the numerical simulations,the eigenvalue analysis and Riks analysis are combined,in which the Hashin failure criterion and fracture energy stiffness degradation model are used to simulate the progressive failure of composites,and the“infinite”boundary conditions are applied to eliminate the boundary effects.As for the hydrostatic pressure tests,RTP specimens were placed in a hydrostatic chamber after filled with water.It has been observed that the cross-section of the middle part collapses when it reaches the maximum pressure.The collapse pressure obtained from the numerical simulations agrees well with that in the experiment.Meanwhile,the applicability of NASA SP-8007 formula on the collapse pressure prediction was also discussed.It has a relatively greater difference because of the ignorance of the progressive failure of composites.For the parametric study,it is found that RTPs have much higher first-ply-failure pressure when the winding angles are between 50°and 70°.Besides,the effect of debonding and initial ovality,and the contribution of the liner and coating are also discussed.

Analysis of nonlinear stability and post-buckling for Euler-type beam-column structure

Based on the assumption of finite deformation, the Hamilton variational principle is extended to a nonlinear elastic Euler-type beam-column structure located on a nonlinear elastic foundation. The corresponding three-dimensional （3D） mathematical model for analyzing the nonlinear mechanical behaviors of structures is established, in which the effects of the rotation inertia and the nonlinearity of material and geometry are considered. As an application, the nonlinear stability and the post-buckling for a linear elastic beam with the equal cross-section located on an elastic foundation are analyzed. One end of the beam is fully fixed, and the other end is partially fixed and subjected to an axial force. A new numerical technique is proposed to calculate the trivial solution, bifurcation points, and bifurcation solutions by the shooting method and the Newton- Raphson iterative method. The first and second bifurcation points and the corresponding bifurcation solutions are calculated successfully. The effects of the foundation resistances and the inertia moments on the bifurcation points are considered.

POST-BUCKLING ANALYSIS AND STRUCTURE OPTIMIZATION OF INTEGRAL FUSELAGE PANEL SUBJECTED TO AXIAL COMPRESSION LOAD

Build-up panels for the commercial aircraft fuselage subjected to the axial compression load are studied by both experimental and theoretical methods.An integral panel is designed with the same overall size and weight as the build-up structure,and finite element models（FEMs）of these two panels are established.Experimental results of build-up panels agree well with the FEM results with the nonliearity and the large deformation,so FEMs are validated.FEM calculation results of these two panels indicate that the failure mode of the integral panel is different from that of the build-up panel,and the failure load increases by 18.4% up to post-buckling.Furthermore,the integral structure is optimized by using the multi-island genetic algorithm and the sequential quadratic programming.Compared with the initial design,the optimal mass is reduced by 8.7% and the strength is unchanged.

BUCKLING AND POST-BUCKLING OF PIEZOELECTRIC PLATES USING FINITE ELEMENT METHOD

The finite element equations considering the geometrical nonlinearity of piezoelectric smart structures are derived based on the total Lagrange method under the assumption of weak coupling between electricity and mechanics. Buckling and post-buckling of piezoelectric-plate with various boundary conditions are investigated. The calculated results show that piezoelectric effects and external voltage can hardly affect the buckling and post-bucking characteristics of piezoelectric-plate under uniaxial pressure while the buckling caused by displacement in-plane has much to do with the electric field.

Buckling and post-buckling analyses of size-dependent piezoelectric nanoplates

This paper attempts to investigate the buckling and post-buckling behaviors of piezoelectric nanoplate based on the nonlocal Mindlin plate model and yon Karman geometric nonlinearity. An external electric voltage and a uniform temperature rise are applied on the piezoelectric nanoplate. Both the uniaxial and biaxial mechanical compression forces will be considered in the buckling and post-buckling analysis. By substituting the energy functions into the equation of the minimum total potential energy principle, the governing equations are derived directly, and then discretized through the differential quadrature （DQ） method. The buckling and post-buckling responses of piezoelectric nanoplates are calculated by employing a direct iterative method under different boundary conditions. The numerical results are presented to show the influences of different factors including the nonlocal parameter, electric voltage, and temperature rise on the buckling and post-buckling responses.

ANALYSIS OF THERMAL POST-BUCKLING OF HEATED ELASTIC RODS

Based on the nonlinear geometric theory of extensible rods, an exact mathematical model of thermal post_buckling behavior of uniformly heated elastic rods with axially immovable ends is developed, in which the arc length s(x) of axial line and the longitudinal displacement u(x) are taken as the basic unknown functions. This is a two point boundary value problem of first order ordinary differential equations with strong non_linearity. By using shooting method and analytical continuation, the nonlinear boundary value problems are numerically solved. The thermal post_buckled states of the rods with transversely simply supported and clamped ends are obtained respectively and the corresponding numerical data tables and characteristic curves are also given.

ON THE POST-BUCKLING ANALYSIS OF PLASTIC STRUCTURES UNDER IMPULSIVE LOADING

An analytical method is suggested to analyze the plastic post-buckling behavior under impulsive loading. The fundamental equation of motion of a cylindrical shell is taken as an example to explain the main concept and procedure. The axial and the radial displacements are decoupled by an approximate scheme, so that only one non-linear equation for the radial buckling displacement is to be solved. By expanding it in terms of an amplitude measure as a time variable, we may get the post-buckling behavior in the form of a series solution. The post-buckling behavior of a rectangular plate used as a special case of cylindrical shell is discussed.

AN EXPERIMENTAL STUDY ON POST-BUCKLING BEHAVIOR OF SLENDER COLUMN WITH FIBER REINFORCED COMPOSITE MATERIAL

Equilibrium paths of post-buckling are measured for large slenderness column specimens made of the fiber reinforced composite material. The influence of the initial curvature is investigated experimentally and compared with the result of the initial post-buckling theory. Both the theoretical and experimental results reveal that the column with the initial curvature has stable post-buckling behaviors and is not sensitive to the imperfection in the form of initial curvature. The experimental results show that when the lateral buckling displacement is less than 20 percent of the column length, the experimental results agree with the results from the theory of initial post-buckling quite well, while they agree with the results from the large deflection theory in a quite large range.

THERMAL POST-BUCKLING OF AN ELASTIC BEAMS SUBJECTED TO A TRANSVERSELY NON-UNIFORM TEMPERATURE RISING

Based on the nonlinear geometric theory of axially extensible beams and by using the shooting method, the thermal post-buckling responses of an elastic beams,with immovably simply supported ends and subjected to a transversely non-uniformly distributed temperature rising, were investigated. Especially, the influences of the transverse temperature change on the thermal post-buckling deformations were examined and the corresponding characteristic curves were plotted. The numerical results show that the equilibrium paths of the beam are similar to what of an initially deformed beam because of the thermal bending moment produced in the beam by the transverse temperature change.

Post-Buckling Behavior of Laminated Composite Cylindrical Shells Subjected to Axial, Bending and Torsion Loads

In present work, post-buckling behavior of imperfect (of eigen form) laminated composite cylindrical shells with different L/D and R/t ratios subjected to axial, bending and torsion loads has been investigated by using an equilibrium path approach in the finite element analysis. The Newton-Raphson approach as well as the arc-length approach is used to ensure the correctness of the equilibrium paths up to the limit point load. Post-buckling behavior of imperfect cylindrical shells with different L/D and R/t ratios of interest is obtained and the theoretical knock-down factors are reported for the considered cylindrical shells.

Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell:an experimental and numerical study

The snap-through behaviors and nonlinear vibrations are investigated for a bistable composite laminated cantilever shell subjected to transversal foundation excitation based on experimental and theoretical approaches.An improved experimental specimen is designed in order to satisfy the cantilever support boundary condition,which is composed of an asymmetric region and a symmetric region.The symmetric region of the experimental specimen is entirely clamped,which is rigidly connected to an electromagnetic shaker,while the asymmetric region remains free of constraint.Different motion paths are realized for the bistable cantilever shell by changing the input signal levels of the electromagnetic shaker,and the displacement responses of the shell are collected by the laser displacement sensors.The numerical simulation is conducted based on the established theoretical model of the bistable composite laminated cantilever shell,and an off-axis three-dimensional dynamic snap-through domain is obtained.The numerical solutions are in good agreement with the experimental results.The nonlinear stiffness characteristics,dynamic snap-through domain,and chaos and bifurcation behaviors of the shell are quantitatively analyzed.Due to the asymmetry of the boundary condition and the shell,the upper stable-state of the shell exhibits an obvious soft spring stiffness characteristic,and the lower stable-state shows a linear stiffness characteristic of the shell.

A Simplified Model for Buckling and Post-Buckling Analysis of Cu Nanobeam Under Compression

Both of Buckling and post-buckling are fundamental problems of geometric nonlinearity in solid mechanics.With the rapid development of nanotechnology in recent years,buckling behaviors in nanobeams receive more attention due to its applications in sensors,actuators,transistors,probes,and resonators in nanoelectromechanical systems(NEMS)and biotechnology.In this work,buckling and post-buckling of copper nanobeam under uniaxial compression are investigated with theoretical analysis and atomistic simulations.Different cross sections are explored for the consideration of surface effects.To avoid complicated high order buckling modes,a stressbased simplified model is proposed to analyze the critical strain for buckling,maximum deflection,and nominal failure strain for post-buckling.Surface effects should be considered regarding critical buckling strain and the maximum post-buckling deflection.The critical strain increases with increasing nanobeam cross section,while themaximumdeflection increases with increasing loading strain but stays nearly the same for different cross sections,and the underlying mechanisms are revealed by our model.The maximum deflection is also influenced by surface effects.The nominal failure strains are captured by our simulations,and they are in good agreement with the simplified model.Our results can be used for helping design strain gauge sensors and nanodevices with self-detecting ability.

POST-BUCKLING AND IMPERFECTION SENSITIVITY OF A COLUMN WITH AN EQUILATERAL TRIANGULAR CROSS-SECTION IN THE PLASTIC RANGE

The asymmetric initial post-buckling corresponding to the lowest bifurcation load and the imperfection sensitivity are analysed in detail for a compressed simply supported column with an equilateral triangular cross-section in the plastic range. The effect of elastic unloading is taken into ao count in the analysis. The asymptotic relations, including high order terms, among the load, the amplitude of imperfection and the amplitude of the bifurcation mode ark realized. The results show that the maximum supported load is very sensitive to imperfection.

Closed Solution for Initial Post-Buckling Behavior Analysis of a Composite Beam with Shear Deformation Effect

This paper is focused on the post-buckling behavior of the fixed laminated composite beams with effects of axial compression force and the shear deformation.The analytical solutions are established for the original control equations(that is not simplified)by applying the Maclaurin series expansion,Chebyshev polynomials,the harmonic balance method and the Newton’s method.The validity of the present method is verified via comparing the analytical approximate solutions with the numerical ones which are obtained by the shooting method.The present third analytical approximate solutions can give excellent agreement with the numerical solutions for a wide range of the deformation amplitudes.What’s more,the effect of shear deformation on the post-bucking configuration of the sandwich beam is also proposed.It can be found that the shear angle has a great influence on the post-buckling load of composite beams.Therefore,the model simplifying the shear formation term as small quantity is not accurate for the case of sandwich beam with soft core.

AN IMPROVED ARC-LENGTH METHOD AND APPLICATION IN THE POST-BUCKLING ANALYSIS FOR COMPOSITE STRUCTURES

Based on the conventional arc-length method, an improved arc-length method with high-efficiency is proposed. The weighted modifications with respect to the variation of structural stiffness and extra-interpolation modification by using the information of known equilibrium points are introduced to improve the incremental arc-length. An approximate expansion method for the accumulated and expected arc-length is used to ensure the convergence at given load levels in large range of applications. Numerical results show that the improved arc-length method has well adaptability and higher efficiency in the post-buckling analysis of plates and shells structures for tracing whole load-deflection path and obtaining the convergence values at any specified load levels.

BUCKLING AND POST-BUCKLING OF ANNULAR PLATES UNDER NON-AXISYMMETRIC PLANE EDGE FORCES

On the basis of both the general theory[1,2]and the finite element method[4]of perforated thin plates with large deflection,the buckling and post-buckling of annular plates under non-axisymmetric plane edge forces are studied.

BUCKLING AND POST-BUCKLING OF ANNULAR PLATES ON AN ELASTIC FOUNDATION

On the basis of von Karnmnequations, the axisymmetric buckling and post-buckling of annular plates on anelastic foundation is so wematically discussed byusing shooting

VARIATIONAL PRINCIPLES OF PERFORATED THIN PLATES AND FINITE ELEMENT METHOD FOR BUCKLING AND POST-BUCKLING ANALYSIS

On the basis of the general theory of perforated thin plates under large deflections, variational principles with deflection w and stress function F as variables are stated in detail.Based on these princi- ples,finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated,owing to the multiple connexity of the region.

Dynamics and vibration reduction performance of asymmetric tristable nonlinear energy sink

With its complex nonlinear dynamic behavior,the tristable system has shown excellent performance in areas such as energy harvesting and vibration suppression,and has attracted a lot of attention.In this paper,an asymmetric tristable design is proposed to improve the vibration suppression efficiency of nonlinear energy sinks(NESs)for the first time.The proposed asymmetric tristable NES(ATNES)is composed of a pair of oblique springs and a vertical spring.Then,the three stable states,symmetric and asymmetric,can be achieved by the adjustment of the distance and stiffness asymmetry of the oblique springs.The governing equations of a linear oscillator(LO)coupled with the ATNES are derived.The approximate analytical solution to the coupled system is obtained by the harmonic balance method(HBM)and verified numerically.The vibration suppression efficiency of three types of ATNES is compared.The results show that the asymmetric design can improve the efficiency of vibration reduction through comparing the chaotic motion of the NES oscillator between asymmetric steady states.In addition,compared with the symmetrical tristable NES(TNES),the ATNES can effectively control smaller structural vibrations.In other words,the ATNES can effectively solve the threshold problem of TNES failure to weak excitation.Therefore,this paper reveals the vibration reduction mechanism of the ATNES,and provides a pathway to expand the effective excitation amplitude range of the NES.

An Analysis of the Post-buckling of Laminated Plates of Symmetric Cross-Ply

In this paper, general equations relating to the stability of laminated plates ofsymmetric cross-ply with initial imperfection factors are derived by using thevariational principle. Taking the deflection as the perturbation parameter, theequilibrium path of the post-buckling path of a simply supported rectangular plate isinvestigated with the perturbation method. An approximation expression and a typicalnumerical example are presented to manifest the post-buekling behavior of therectangular plate.