Buckling Mode Shapes Research Articles

In-plane and out-of-plane buckling of architected cellular plates: Numerical and experimental study

Abstract In this article, the nature of instability in architected cellular materials is explored through computational modelling and experimental testing. The in-plane and out-of-plane buckling as two major instability mechanisms in cellular materials are distinguished and the effects of their microarchitectural parameters on the buckling behavior are demonstrated. Different architected cellular materials in the form of cellular plates are analysed with a finite element method (FEM) by eigen buckling and nonlinear analyses, while a few samples of elastomeric architected cellular materials are prototyped and experimentally tested to corroborate our numerical predictions for the buckling load. We show that an appropriate gradient of cell architecture within the cellular plates could lead to an increase in both out-of-plane and in-plane buckling loads. The experimental tests conducted on the casted cellular plates with uniform and graded microarchitectures, made of a two-component silicone elastomer , confirm that different buckling mode shapes can be observed by tailoring the microarchitectures of cellular plates. This idea can pave the path for devising advanced reconfigurable materials, for shape matching and energy absorption applications, whose mode shapes and resistance to buckling can be programmed based on the desirable functionalities without compromising their total mass.

OPTIMUM BUCKLING DESIGN OF CYLINDRICAL STIFFENER SHELL UNDER EXTERNAL HYDROSTATIC PRESSURE

This paper present an investigation of the collapse load in cylinder shell under uniformexternal hydrostatic pressure with optimum design using finite element method viaANSYS software. Twenty cases are studied inclusive stiffeners in longitudinal and ringstiffeners. Buckling mode shape is evaluated. This paper studied the optimum designgenerated by ANSYS for thick cylinder with external hydrostatic pressure. The primarygoal of this paper was to identify the improvement in the design of cylindrical shell underhydrostatic pressure with and without Stiffeners (longitudinal and ring) with incorporativetechnique of an optimization into ANSYS software. The design elements in this researchwas: critical load, design variable (thickness of shell (TH), stiffener’s width (B) andstiffener’s height (HF). The results obtained illustrated that the objective is minimizedusing technique of numerical optimization in ANSYS with optimum shell thickness andstiffener’s sizes. In all cases the design variables (thickness of shell) was thicker than themonocoque due to a shell’s thicker is essential to achieve the strength constraints. It can beconcluded that cases (17,18,19, and 20) have more than 90% of un-stiffened critical load.The ring stiffeners causes increasing buckling load than un-stiffened and longitudinalstiffened cylinder.

A Parametric Analysis on Hull Penetration Position and Frame Spacing Leading to Minimal Welding Distortion

Hull penetration and openings are governed by standards, which ensure the structural integrity of the assembly. However, a number of stiffeners are usually welded to the panel, giving rise to welding residual stresses and distortion. This study identifies the optimal position and shape of penetration relative to various stiffener spacing configurations, leading to the least welding distortion. A numerical parametric study is adopted where an uncoupled thermo-elastoplastic technique based on the inherent strain method is used. The results have shown that the final distortion is highly dependent on the buckling mode shape predicted by the numerical models that in turn is influenced by the structural stiffness and support position of the assembly. When the models follow a similar buckling mode, the results have shown that smaller stiffener spacing leads to less distortion. Centrally positioning penetrations results in symmetric panel stiffness leading to near symmetric out-of-plane distortion and ultimately minimal distortion, when compared to positioning penetrations next to the stiffener. The difference in the predicted distortion for square and circular penetrations is relatively small when placed in the middle of the assembly. Nonetheless, when penetrations are placed close to the stiffener, circular penetrations are preferred to reduce distortion. Small circular penetrations gave rise to less overall distortions but more penetration residual twisting and rotations, particularly when placed close to the stiffener. 1. Introduction Hull penetrations and openings are governed by standards such as DNV GL SE (2015) and ASTM F994-86 (2011). Their prime focus ensures that the panels under consideration have significant stiffness to withstand loads. These panels are generally stiffened by means of stiffeners that are welded to the plate at different positions and frame spacing. The inherent thermal strains developed during welding give rise to residual stresses, that coupled with the structural stiffness of the welded assembly will manifest themselves in welding distortions (Masubuchi 1980; Radaj 2003). If the assembly is significantly stiff to withstand the welding contraction forces, the developed distortion will be small. However, welding residual stresses and consequently distortion will inevitably still develop (Gray et al. 2014). A number of techniques are possible to reduce the welding contraction forces, such as thermal tensioning (Masubuchi 1980; Michaleris & Sun 1997) and cryogenic cooling (Gabzdyl et al. 2002; Camilleri et al. 2008). Another possible strategy that can be adopted to reduce welding distortions involves the optimal identification of assembly designs that possibly provide higher stiffness to counter the inherent welding stresses. Different panel assemblies and configurations will give rise to different magnitudes of welding distortion and thus the optimal welded assembly that gives rise to minimal distortion can be identified prior to fabrication and established during the design phase (Gray et al. 2014). This study focuses on investigating and identifying the optimal penetration position and frame spacing leading to minimal welding distortion and aims to answer the following research hypothesis:Does central positioning of penetrations lead to minimal distortion?Placing penetrations close to stiffeners gives rise to less welding distortion?Will a square penetration create more distortion than a circular penetration?Does the frame spacing have an effect on the final welding distortion?How does the size of the openings influence the out-of-plane distortion?

Decoupled stability equation for buckling analysis of FG and multilayered cylindrical shells based on the first-order shear deformation theory

Based on the first-order shear deformation and Donnell's shell theory with von Karman non-linearity, one decoupled stability equation for buckling analysis of functionally graded (FG) and multilayered cylindrical shells with transversely isotropic layers subjected to various cases of combined thermo-mechanical loadings is developed. To this end, the equilibrium equations are uncoupled in terms of the transverse deflection, the force function and a new potential function. Using the adjacent equilibrium method, one decoupled stability equation which is an eighth-order differential equation in terms of transverse deflection is obtained and conveniently solved to present analytical expressions for buckling loads of cylindrical shells under any type of loading. These analytical expressions can be used in design and as a benchmark in numerical studies. For numerical purpose, the formulation is applied to FG shells, a three-layer shell laminated of transversely isotropic layers, and sandwich shells with an isogrid lattice core and transversely isotropic face sheets. The results are validated with the existing ones in the literature. Finally, the effect of different parameters on the buckling loads in the presence of combined loadings is discussed in detail. Numerical results show that the existence of an initial hoop stress as a main type of initial imperfection has significant influence on the buckling behavior (including buckling values and mode shapes) of long cylindrical shells subjected to axial loading and this effect reduces for cylindrical shells with a lattice core. Furthermore, the critical values of torsion significantly reduce under combined loading.

Study of thermal buckling behavior of plain woven C/SiC composite plate using digital image correlation technique and finite element simulation

In this study, the thermal buckling behavior of plain woven C/SiC composite plate is investigated by a noncontact measurement based on the three-dimensional digital image correlation (DIC) technique and finite element analysis. The plain woven C/SiC composite plate is fixed by a water-cooling steel frame and one-side heated by quartz lamp array heating apparatus. The buckling temperature and the first buckling mode shape of the C/SiC composite plate are determined from the temperature-displacement curves and full-field deformation that are obtained from the DIC-based experiment. A nonlinear finite element buckling analysis with initial imperfection is performed using the ANSYS software. In order to improve the accuracy of the numerical simulation, a clamping frame model is further proposed for simulating the real clamping boundary in experiment. The results of the finite element simulation and DIC-based measurement coincide well regarding the temperature-displacement curve tendency and critical buckling temperature. Finally, a parametric study is performed using the presented numerical model to investigate the thermal buckling behavior of plain woven C/SiC composite plates with various dimension sizes.

Nonlinear Theory of Sandwich Shells with a Transversely Soft Core Containing Delamination Zones and Edge Support Diaphragm

Sandwich shells with transversely soft cores and skin layers supported along the edges by a thin deformable diaphragm with a ruled middle surface are considered. Delamination zones are assumed on the contact surfaces between the core and skin layers. The refined geometrically nonlinear theory has been constructed for these type of structures at small deformations and moderate displacements, which can describe prebuckling deformation and reveal all possible buckling mode shapes of skin layers (antisymmetric, symmetric, mixed bending and mixed bending-shear, as well as arbitrary mode shapes comprising all listed modes) and of the reinforcing diaphragm. This theory is based on the introduction of interaction forces between the core and skin layers and between the core and reinforcing diaphragm at each point of the contact surfaces as unknowns. The previously proposed variant of the generalized Lagrangian variational principle has been utilized in the derivation of the governing equations, static boundary conditions for the shell and reinforcing diaphragm, and kinematic coupling conditions between the core and skin and between the core and support diaphragm.

Nonlinear Stability Analysis of Sandwich Wide Panels—Part I: Buckling Behavior

This is a series of two papers in which the nonlinear stability behavior of sandwich panels is investigated. This part presents the buckling behavior and focuses on the critical load and the buckling mode. The buckling analysis is based on the extended high-order sandwich panel theory (EHSAPT) which takes transverse compressibility and axial rigidity of the core into account. It allows for the interaction between the faces and the core. The geometric nonlinearity, i.e., large displacement with moderate rotation, is considered in both faces and core. The weak form governing equations are derived based on the EHSAPT-based element. Detailed formulations and analysis procedures are provided. It presents a general approach for arbitrary buckling type without decoupling it into isolated global buckling and wrinkling. There are no additional assumptions made about the prebuckling state and buckling mode shape, which are commonly presumed in the literature. In addition, edge effects which are also commonly neglected are included. The prebuckling state is determined via a nonlinear static analysis. Solving an eigenvalue problem yields the critical load and the corresponding eigenvector gives the buckling mode. Sandwich panels with different lengths are studied as examples. Both global buckling and wrinkling are observed. It shows that the axial rigidity of the core has a pronounced effect on both the critical load and the buckling mode.

Nonlinear Buckling Analysis of H-Type Honeycombed Composite Column with Rectangular Concrete-Filled Steel Tube Flanges

This paper was concerned with the nonlinear analysis on the overall stability of H-type honeycombed composite column with rectangular concrete-filled steel tube flanges (STHCC). The nonlinear analysis was performed using ABAQUS, a commercially available finite element (FE) program. Nonlinear buckling analysis was carried out by inducing the first buckling mode shape of the hinged column to the model as the initial imperfection with imperfection amplitude value of L/1000 and importing the simplified constitutive model of steel and nonlinear constitutive model of concrete considering hoop effect. Close agreement was shown between the experimental results of 17 concrete-filled steel tube (CFST) specimens and 4 I-beams with top flanges of rectangular concrete-filled steel tube (CFSFB) specimens conducted by former researchers and the predicted results, verifying the correctness of the method of FE analysis. Then, the FE models of 30 STHCC columns were established to investigate the influences of the concrete strength grade, the nominal slenderness ratio, the hoop coefficient and the flange width on the nonlinear stability capacity of SHTCC column. It was found that the hoop coefficient and the nominal slenderness ratio affected the nonlinear stability capacity more significantly. Based on the results of parameter analysis, a formula was proposed to predict the nonlinear stability capacity of STHCC column which laid the foundation of the application of STHCC column in practical engineering.

Accurate thermal buckling analysis of functionally graded orthotropic cylindrical shells under the symplectic framework

Accurate thermal buckling analysis of functionally graded orthotropic cylindrical shells is presented based on the Reissner's shell theory under the symplectic framework. By introducing a full-state vector, the high-order governing differential equation is reduced into a set of low-order ordinary differential equations. The fundamental unknowns are expanded in terms of the symplectic eigensolutions without any trial function. The buckling equations and buckling mode shapes are analytically obtained. The present study demonstrates that the expressions of displacements have different forms and strongly depend on the end conditions and thickness. Some new results are presented in numerical examples.

Buckling Behavior of Non-Uniformly Heated Tapered Laminated Composite Plates with Ply Drop-Off

The thermal buckling characteristics of non-uniformly heated tapered laminated composites plates with ply drop-off have been investigated numerically. Detailed parametric studies have been carried out for the effects of taper configuration, temperature variation, aspect ratio and structural boundary conditions on critical buckling temperatures and buckling mode shapes. It is found that the nature of taper as well as the applied temperature field have considerable effects on the critical buckling temperatures of laminated composite tapered plates. Square plates buckle at the highest temperature when subjected to the decreasing temperature profile. Additionally, it is noted that Taper B and Taper C plates show the best behavior under buckling for most structural boundary conditions. Moreover, the change in buckling mode shapes with respect to temperature profile and taper configuration is significant for rectangular plates in comparison with square plates.

A bilateral relationship between stable profiles of pinned–pinned bistable shallow arches

Arch-profiles in the two force-free stable equilibrium states of shallow bistable arches are related to each other. We derive a two-way, i.e., bilateral, relationship between stress-free initial profile and stressed toggled profile so that pinned–pinned bistable arches of arbitrary profiles can be efficiently analyzed and designed. The derivation relies on representing the initial and toggled profiles with two sets of mode weights corresponding to the buckling mode shapes of a pinned–pinned column. Furthermore, we prove that the fundamental mode weights should be non-zero for an arch to be bistable. The following corollaries arise from the aforementioned relation: (1) symmetry in initial and toggled profiles remains unchanged; (2) all the mode weights other than the fundamental mode weight have the same sign in both stable states; (3) magnitudes of corrugations in stable force-free arch-profiles are approximately equal. Derivations and proofs of the principal relationship and its corollaries as well as examples of analysis and design of bistable arches of arbitrary arch-profiles are presented in the paper.

A nonlinear resonance (eigenvalue) approach for computation of elastic collapse pressures of harmonically imperfect relatively thin rings

A nonlinear resonance (eigenvalue) based semi-analytical approach is employed here for computation of the elastic mode 2 collapse pressures of moderately-thick to thin isotropic rings, weakened by harmonic or modal type imperfections. The mode 2 collapse pressure is, by definition, associated with the buckled mode shape of cos(2θ) type, and is the harmonically imperfect ring counterpart to the Euler type buckling pressure of a hydrostatically pressurized thin perfect ring. A von Karman type iterative nonlinear analysis, which is based on the assumptions of transverse inextensibility and first-order shear deformation theory (FSDT), is utilized for computation of hydrostatic collapse pressure of a harmonically imperfect ring. Interesting and hitherto unavailable numerical results pertaining to the effects of harmonic imperfections on the hydrostatic collapse pressures of imperfect metallic rings are also presented.

Suitable radial grading may considerably increase buckling loads of FGM circular plates

In this paper, we study buckling of radially FGM circular plates. In a previous study, a fourth-order polynomial expressing the exact solution of a linear elastic problem was used as buckling mode shape. To generalise such investigation, in this contribution the buckling mode is postulated to take the shape of a fifth-order polynomial function of the radial coordinate. The flexural rigidity is consequently sought as a polynomial of suitable order, expressing the functional grading. New solutions in closed form are then obtained by a semi-inverse method. It is found that suitable choices of functional grading may increase the buckling load up to 246% with respect to the homogeneous and uniform cases.

On the nonlinear dynamic responses of FG-CNTRC beams exposed to aerothermal loads using third-order piston theory

The purpose of this study is to analyze the nonlinear dynamic responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams exposed to axial supersonic airflow in thermal environments. The dynamic model of the FG-CNTRC beam is developed with regard to the first-order shear deformation theory incorporating the von Karman geometrical nonlinearity. The thermomechanical properties of the constituents are assumed to be temperature dependent. The third-order piston theory is adopted to estimate the nonlinear aerodynamic pressure induced by the supersonic airflow. Harmonic differential quadrature method is implemented to discretize the equations of motion in the spatial domain. A comprehensive parametric study is performed to expatiate on the effect of the distribution type and volume fraction of CNTs, boundary condition, slenderness ratio, and thermal environments on the aerothermoelastic responses of the FG-CNTRC beam. Simulation results indicate that the presence of the aerodynamic pressure not only increases the critical buckling temperature of the FG-CNTRC beam, but also changes the buckling mode shapes of the beam. Furthermore, the results show that aerothermoelastic characteristics of FG-CNTRC beams may be remarkably improved by the selection of a proper distribution of CNTs. Besides, it is found that FG-CNTRC beams with intermediate CNT volume fraction do not have an intermediate critical buckling temperature.

Strength of Thin-Walled Lipped Channel Section Columns with Shell-Shaped Curved Grooves

Thin-walled member is structurally superior to a construction member. However, by reason of complexity in structure the stress and the deformation to yield the cross section are complicated. Specially, in case thin-walled members, such as thin-walled channel section columns, which are subjected to compressive force, these members produce the local buckling, distortional buckling and overall buckling. A number of experimental and theoretical investigations subjected to axial compressive force are generated for thin-walled channel section columns with triangle-shaped folded groove by Hancock [1] and with complex edge stiffeners and web stiffeners by Wang [2]. In case thin-walled channel section column with folded groove which is subjected to axial compressive force, it is cleared that the buckling mode shapes are ordinarily generated for local buckling mode shape of plate-panel composing cross section of member in short member aspect ratio and overall buckling mode shape as column and distortional buckling mode shape interacting between local buckling and overall buckling similarly normal thin-walled member. It is cleared analytically and experimentally that buckling strength and critical strength of thin-walled channel section column with folded groove can increase sharply in comparison with that of normal thin-walled member composing only plate-panel. In this paper a new cross section of shell-shaped curved groove [3] was proposed instead of the thin-walled lipped channel section column with triangle- and rectangle-shaped folded grooves used ordinarily, and therefore the comparison and the examination of buckling strength and buckling behavior were generated in the case of preparing triangle-shaped folded and shell-shaped curved grooves to web and flange of thin-walled channel section column. And then in order to investigate the buckling behavior on the thin-walled channel section column with folded and curved grooves, exact buckling strength and the buckling mode shapes are generated by using the transfer matrix method. The analytical local distortional and overall elastic buckling loads of thin-walled channel section column with folded and curved grooves can be obtained simultaneously by use of the transfer matrix method. Furthermore, a technique to estimate the buckling mode shapes of these members is also shown.

Buckling of axially compressed composite cylinders with geometric imperfections

Cylindrical shell structures buckle at service loads which are much lower than their associated theoretical buckling loads. The main source of this discrepancy is the presence of various imperfections which are created on the cylinder body during different processes as manufacturing, handling, assembling and machining. Many cylindrical shell structures are still designed against buckling based on the experimental data introduced by NASA SP-8007 as conservative lower bound curves. This study employed the numerical based Linear Buckling mode shape Imperfection (LBMI) method and modified it using a stochastic method to assess the effect of geometrical imperfections in more details on the buckling of cylindrical shells with and without the cutout. The comparison of results with those obtained from the numerical Simcple Perturbation Load Imperfection (SPLI) method for cylinders with and without cutout revealed a good correlation. The effect of two parameters of size and number of cutouts on the buckling load was investigated using the linear buckling and Modified LBMI methods. Results confirmed that in cylinders with a small cutout inserting geometrical imperfection using either SPLI or modified LBMI methods significantly reduced the value of the predicted buckling load. However, in cylinders with larger cutouts, the effect of the cutout is dominant, thus considering geometrical imperfection had a minor effect on the buckling loads predicted by both SPLI and modified LBMI methods. Furthermore, the modified LBMI method was employed to evaluate the combination effect of cutout numbers and size on the buckling load. It is shown that in small cutouts, an increasing in the cutout size up to a certain value resulted in a remarkable reduction of the buckling load, and beyond that limit, the buckling loads were constant against D/R ratios. In addition, the cutout number shows a more significant effect on decreasing the buckling load at small D/R ratios than large D/R ratios.

On the mechanics of distortional-global interaction in fixed-ended columns

This work reports the results of a numerical investigation concerning the elastic post-buckling behaviour of fixed-ended thin-walled lipped channel and zed-section columns affected by distortional-global (D-G) interaction – “unexpected” features associated with the global buckling nature are unveiled in both cases. The columns analysed (i) exhibit cross-section dimensions and lengths ensuring the simultaneous occurrence of distortional and global critical buckling, and (ii) contain critical-mode initial geometrical imperfections (linear combinations of the critical distortional and global buckling mode shapes). For comparison and clarification purposes, the post-buckling behaviour of lipped channel columns buckling in an “isolated” distortional mode and containing “pure” distortional and global (flexural-torsional) initial geometrical imperfections are also analysed. The results presented and discussed are obtained through geometrically non-linear Generalised Beam Theory (GBT) analyses and provide the evolution, along given equilibrium paths, of the column deformed configuration (expressed in GBT modal form), relevant displacement profiles and modal participation diagrams, making it possible to acquire in-depth insight on the mechanics underlying D-G interaction in columns. Finally, particular attention is paid to interpreting the differences exhibited by the various column post-buckling behaviours investigated.

GBT buckling analysis of generally loaded thin-walled members with arbitrary flat-walled cross-sections

This paper deals with extending the domain of applicability of a recently developed Generalised Beam Theory (GBT) formulation intended to perform elastic linear buckling analyses of thin-walled members (i) exhibiting arbitrary flat-walled cross-sections (including those combining closed cells and open branches), and (ii) acted by general loadings. These loadings, which include transverse forces acting away from the member shear centre axis, are termed “general” in the sense that they may involve the presence of pre-buckling stress distributions associated with any possible combination of all the stress tensor membrane components (σxx, σss and τxs, for a plane stress state), including cell shear flows – therefore, all the relevant geometrically non-linear effects need to be taken into consideration. After briefly presenting the main concepts and procedures involved in the development and implementation of the above GBT formulation, this same formulation is employed to analyse the buckling behaviour of beams with different types of cross-section geometry (containing closed cells) and exhibiting different loading and support conditions. In particular, they consist of (i) a RHS cantilever acted by two tip point loads, (ii) a closed-flange I-section simply supported beam subjected to a uniformly distributed load and (iii) a two-cell RHS section cantilever acted by tip transverse forces and couples. In all cases, the loads are applied both along the shear centre axis and also along axes parallel to it and located at the beam top and bottom surfaces. The results presented and discussed, which consist of pre-buckling stress fields, buckling curves and buckling mode shapes, are obtained by means of the newly released code GBTul 2.0 and validated by means of the comparison with shell finite element values obtained with the code Ansys.

Identification of buckling modes in generalized spline finite strip analysis of cold-formed steel members

In this paper, a mode identification technique in the context of spline finite strip method (SFSM) is presented to compute the contribution of primary (global, distortional and local) and secondary (shear/transverse extension) buckling modes. The base vectors corresponding to individual buckling modes are developed based on the principles of generalized beam theory. The buckling mode shape in SFSM is approximated as a linear combination of these orthonormal base vectors to evaluate the participation of individual buckling mode. The proposed mode identification technique is able to successfully quantify the participation of different buckling modes and the mode participation is comparable with mode identification using finite strip method (FSM) and generalized beam theory (GBT). Illustrative examples are presented to calculate the participation of individual modes in cold-formed steel sections under different loading and boundary conditions. Also the specific application of mode identification in SFSM is demonstrated.

Postbuckling analysis of smart FG-CNTRC annular sector plates with surface-bonded piezoelectric layers using generalized differential quadrature method

The purpose of the present study is to introduce the application of carbon nanotubes (CNTs) and piezoelectric layers in suppressing the postbuckling deflection of functionally graded carbon nanotube reinforced composite (FG-CNTRC) annular sector plates. To achieve this objective, a structural model is developed based on the first-order shear deformation theory (FSDT) along with the von Karman geometrical nonlinearity. The distribution of electric potential across the thickness of piezoelectric layers is modeled by a combination of linear and sinusoidal functions and the closed circuit electrical boundary condition is taken into account for the top and bottom surfaces of the piezoelectric layers. Generalized differential quadrature method (GDQM) is implemented to discretize the nonlinear stability equations, boundary conditions and Maxwell equation. The nonlinear system of equations is solved via a direct iterative method. A detailed parametric study is conducted to explore effects of geometrical parameters, boundary conditions, piezoelectric materials, thickness of the piezoelectric layers, external electric voltage, CNT distribution, and volume fraction on the postbuckling responses of FG-CNTRC annular sector plates with surface-bonded piezoelectric layers. Results indicate that both volume fraction and distribution of CNTs play a key role in enhancing the postbuckling strength of CNTRC annular sector plates. It is found that the type of piezoelectric materials, thickness of piezoelectric layers and the external electric voltage have a significant effect on suppressing the postbuckling deflection of FG-CNTRC annular sector plates. It is also found that the distribution of CNTs plays a pivotal role in changing buckling mode shapes of CNTRC annular sector plates. Besides, the proposed solution procedure has shown clear advantages over existing methods and demonstrated itself as a general, stable and accurate numerical method in solving strongly coupled nonlinear partial differential equations.