Buckling Values Research Articles

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.

Vibration behavior of thin-walled steel members subjected to uniform bending

Abstract This article reports the results of an investigation on the effects of internal moments on the vibration behavior of thin-walled steel members. The analyses are based on the Generalized Beam Theory (GBT), a thin-walled bar theory accounting for cross-section in-plane deformations - its main distinctive feature is the representation of the member deformed configuration by means of a linear combination of cross-section deformation modes, multiplied by their longitudinal amplitude functions. The study concerns a simply supported T-section (with unequal flanges) members exhibiting a wide range of lengths and subjected to uniform internal moment diagrams - their magnitudes are specified as percentages of the corresponding critical buckling values. After providing a brief overview of the main concepts and procedures involved in performing a GBT-based structural analysis, the vibration behavior of load-free and loaded T-section members is addressed - the influence of the applied loadings is assessed in terms of (i) the fundamental frequency difference and (ii) the change in the corresponding vibration mode shape. For validation purposes, some GBT results are compared with values yielded by shell finite element analysis performed in the code ABAQUS (Simulia, 2008).

A Computer Model Assessment of the Effect of Hindfoot Alignment on Mechanical Axis Deviation and Ankle Fractures

Category: Basic Sciences/Biologics Introduction/Purpose: Conventional mechanical axis is calculated from the center of the femoral head to the center of the ankle. Mechanical axis deviation of the lower limb can be associated with a pes planus hindfoot. Malalignment of the lower limb has been shown to increase progression of osteoarthritis of the knee and ankle and decrease joint arthroplasty longevity. Clinically, a pes planus hindfoot has also been seen with patients who present with a stress fracture of the lateral malleolus. This biomechanical study aims to utilize computer modeling to evaluate the hypothesis that altered force transmission on the lateral malleolus with resultant stress fractures in a pes planus model is attributable to mechanical axis deviation. Methods: A free-body diagram of the fibula in single leg stance was generated by modeling the fibula as a uniform cylinder. It includes the axially applied load and a single evertor muscle force as an eccentric load applied to the mid-diaphysis . Previously derived relationships between body weight (BW = 667 N, 150lbs) and a) normal axial fibula load (BW*0.17) and b) muscle force (BW*0.25) were used. Fibula length (286.5 mm) and diameter (8 mm) were derived from anthropological data. Mechanical axis deviation in pes planus was simulated in two manners: 1) increased (2 and 3 times normal) axial fibula load and 2) increased evertor muscle force. The compressive stress along the length of the bone was determined through static analysis and the total applied load was compared to theoretical Euler buckling load. Results: Increasing the load on the fibula, either by increasing the axial load (Figure 1A) or the muscle load (Figure 1B), increases the maximum compressive stress below the lateral muscle origins, namely the section between the distal tibiofibular ligaments and the evertor muscles. The compressive stress for both cases was less than the compressive yield stress of cortical bone (200 MPa) and cancellous bone (100 MPa) even as the force was increased to the critical buckling value. This model serves as a first attempt to relate the spatial distribution of stress in the fibula with muscle force, axial load, and compressive stress in light of distal fibular fractures associated with pes planus. Conclusion: The importance of lower extremity mechanical axis deviation is well established in the progression of arthritis in the knee and ankle. The role of the mechanical axis in the predisposition of stress fractures around the ankle has not been evaluated in the literature. This biomechanical study represents the first attempt to understand how deviation of the mechanical axis can result in stress fractures of the lateral malleolus. Future studies including a finite element analysis will provide further information and the results of these studies may alter how clinicians treat patients with stress fractures of the fibula.

Corneal Subbasal Nerve Density and Sensitivity After Pars Plana Vitrectomy Using Contact or Noncontact Wide-Angle Viewing Systems.

To assess corneal subbasal nerve density (SBND) and corneal sensitivity (CS) after pars plana vitrectomy (PPV) for rhegmatogenous retinal detachment (RRD) without intraoperative 360-degree laser treatment using contact or noncontact panoramic viewing systems. This is a prospective study of 34 eyes affected by RRD, which underwent PPV using contact (18 eyes) and noncontact (16 eyes) panoramic viewing systems; 12 eyes which underwent scleral buckling, and 17 eyes which underwent cataract surgery. SBND and CS were assessed before surgery and 3 and 6 months postoperatively by in vivo confocal microscopy and Cochet-Bonnet esthesiometry. Compared with baseline values, at 6 months, SBND and CS decreased in both contact PPV (SBND preoperative value: 19.1 ± 3.7 mm/mm; SBND postoperative value: 3.5 ± 1.3 mm/mm; CS preoperative value: 5.1 ± 0.5 cm; CS postoperative value: 1.5 ± 0.4 cm) and in noncontact PPV groups (SBND preoperative value: 19.5 ± 3.8 mm/mm; SBND postoperative value: 8.7 ± 2.3 mm/mm; CS preoperative value: 5.3 ± 0.5 cm; CS postoperative value: 2.5 ± 0.7 cm) (P < 0.001 for all comparisons). SBND reduction was greater in the contact PPV group than in the noncontact PPV group (P < 0.001). By contrast, the scleral buckling and cataract surgery group values were unchanged (P > 0.1 for all comparisons). In multivariate analysis, no significant effect was found for cataract surgery associated with PPV, pseudophakia, surgical time, intraocular pressure, or for laser in horizontal sectors for PPV groups. SBND was highly correlated with CS (r = 0.93). A contact viewing system reduces SBND after PPV more than a noncontact system does.

A priori calculation of the thermal neutron die-away time of large samples irradiated in the facility MEDINA

An analytical method for the a priori calculation of the thermal neutron die-way time was developed in order to optimize the detection of prompt and delayed gamma rays induced in the neutron activation of large samples irradiated in the facility MEDINA. The method was tested against experimental data collected in measurements of large homogeneous and heterogeneous samples using two different ways of calculating the neutron buckling value. The overall results are very satisfactory considering the simplifications performed to derive an analytical representation of the thermal neutron die-away time; the relative deviations for the experimental values are about 20%. However, when an effective buckling value is used the analytical die-away times show deviations not higher the 4% highlighting how the neutron buckling is the main source of error.

The stability of cracked rectangular plate with variable thickness using phase field method

This study focuses on the investigation of the stability in a rectangular FGM plate with central crack. The plate thickness is changed linearly following the length of the plate. Using the Reissner-Mindlin first order shear deformation theory (FSDT), phase field theory and finite element method (FEM), the stability of fracture of the plate is determined. In order to ensure the reliability of the study, the obtained numerical results in this paper are compared with results reported in other publications. The work also presents the analysis of critical buckling computation for plate that have variation in thickness, the length of the crack on plate as well as the inclined angle of the crack. The numerical results show that the crack length impacts significantly to the critical buckling values of the plate, whereas the impact of inclined angle is less.

Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory

Fast improvements in technology causes to provide the ability to manipulate porous graded materials for using in directed design of micro/nano-structures. The prime objective of the current study is to anticipate the size-dependent nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets, and subjected to the uniform distributed load together with an axial compressive load. Via the nonlocal strain gradient theory of elasticity, two entirely different features of size effects are incorporated in the third-order shear deformable beam model. In addition to the uniform distribution of porosity, three different functionally graded porosity distributions along the thickness of micro/nano-beams are supposed in such a way that the relationship between coefficients related to the relative density and porosity is considered for an open-cell metal foam. On the basis of Hamilton's principle, the non-classical governing differential equations of motion are established. After that, the Galerkin method in conjunction with an improved perturbation technique is utilized to attain explicit analytical expressions for nonlocal strain gradient load-deflection paths of the functionally graded porous micro/nano-beams reinforced with the graphene nanofillers. It is observed that the type of porosity dispersion pattern has no considerable influence on the significance of size effects. However, by approaching the axial compressive load to the critical buckling value, the significance of the both nonlocality and strain gradient size dependencies on the nonlinear bending behavior of functionally graded porous micro/nano-beams increases, and this increment in the nonlocal small scale effect is more considerable than in the strain gradient one.

Influence of Grain Inclination Angle on Shear Buckling of Laminated Timber Sheathing Products

Recent advances in timber production industries have enabled production of new innovative laminated timber products having layers with grain inclination angle. This paper is aimed to study influence of grain inclination angle in the laminated veneer lumber (LVL) and plywood sheathings on their shear buckling loads. Two extreme edge conditions of simply supported and clamped edges are considered. First, an accurate differential quadrature (DQ) computational code is developed using MAPLE programming software to obtain eigen buckling values and their corresponding eigen mode shapes. Next, for convenience of engineering calculations, approximate algebraic formulae are presented to predict critical shear buckling loads and mode shapes of LVL and plywood panels having layers with grain inclination angle, with adequate accuracy. Furthermore, finite element (FE) modelling is conducted for several cases using ANSYS software to show validity and accuracy of the predicted results for the problem. It is shown that the highest shear buckling loads of LVL sheathings is achievable when the inclination angle of about 30° with respect to the shorter edges is considered for production of LVL panels, whereas the same angle with respect to the long edges of the LVL sheathings results in a relatively lower buckling load. Considering similar inclination angle with respect to any edges of a plywood sheathings will also results in its highest pre-buckling capacity. It is also demonstrated that, under optimal design and certain loading circumstances, LVL shows a higher shear buckling capacity compared to a similar plywood sheathing.

Reliability-based design optimization and uncertainty quantification for optimal conditions of composite structures with non-linear behavior

An approach to reliability-based design (RBDO) of beam reinforced composite structures with non-linear geometric behavior is proposed. A unified approach following both buckling and first-ply failure (FPF) is used to verify the integrity of beam reinforced shallow shell laminated structures. A new RBDO methodology using a genetic algorithm and a hierarchical decomposition searches the global most probable failure point (MPP). For the reliability analysis, the random parameters are the mechanical properties of laminates. Simultaneously the optimal design based on weight minimization under prescribed reliability and buckling constraints is searched through this hierarchical genetic algorithm (HGA). The design variables are the ply angle, the ply thickness, the height and the width of the cross sections of the stiffeners. Numerical results show the capabilities of the proposed approach using the MPP search inner loop integrated in a HGA scheme. Based on a sensitivity methodology the uncertainty for the optimal solution obtained from HGA is analyzed. In the neighborhood of critical buckling values of the structural response the asymptotic behavior of uncertainty propagation is observed. The influence of uncertainties from random parameters and design variables are studied on critical load factor and critical displacement. The variability of these structural response functions are measured by their coefficients of variation and Sobol indices. The most important influences for uncertainty propagation are obtained from ply angle of shell laminates and from longitudinal elastic modulus group.

Modified fractional neutron point kinetics equations for finite and infinite medium of bar reactor core

Fractional order neutron point kinetics equations are generalization of the classical neutron point kinetics equations. The system of the fractional neutron point kinetics equations is treated in different context by many authors ignoring an important term related to the derivative of the reactivity. In this paper, we derive a time fractional neutron point kinetics equations (FNPK) model considering a new term for the time derivative of the reactivity. Fractional Leibenz rule is used for such derivation. The effect resulting from the additional term of the reactivity derivative is discussed. It has been found that without this additional term, the modified model will be compliant with the earlier published models for the FNPK. The proposed model of FNPK describes infinite and finite bar reactor cores according to the nonleakage probability and the geometric buckling values. We have applied an effective implicit difference approximation to solve the modified model for finite cylindrical reactor core for both average one group and six groups of delayed neutron precursors. The neutron density results with different values of fractional order for step, ramp and oscillatory reactivities are shown and compared with the classical neutron point kinetics equations.

Web crippling of beams under ITF loading: A novel DSM-based design approach

A new approach to estimate the web crippling failure load of cold-formed steel beams under Internal Two Flange (ITF) loading using the Direct Strength Method (DSM) is proposed in this paper. After the description of the existing test data to calibrate the DSM-based formula, the accuracy of the analytical expressions given in Eurocode 3 [1] and the North-American Specification [2] is briefly assessed. In order to obtain additional information on the web crippling behaviour of each test specimen, several types of analyses are performed: (i) quasi-static non-linear analyses (using finite elements), (ii) elastic buckling analyses (using finite elements and the GBTWEB software), and (ii) rigid-plastic analyses (using yield-line theory). The coefficients of a DSM-based formula are determined on the basis of the set of 85 experimental results available in the literature and the corresponding buckling and plastic load values. In spite of the different cross-sections, fastening conditions and test set-ups considered in the calibration procedure, it is possible to establish a clear relationship between the web crippling slenderness and the nominal-to-plastic load ratio. Finally, it is shown that the proposed DSM-based formula for ITF loading yields reasonable predictions of web crippling loads and provides safe estimates of flange crushing loads.

Flow sensing by buckling monitoring of electrothermally actuated double-clamped micro beams

We report on a flow sensing approach based on deflection monitoring of a micro beam buckled by the compressive thermal stress due to electrothermal Joule's heating. The air stream, convectively cooling the device, affects both the critical buckling values of the electric current and the postbuckling deflections of the structure. After calibration, the flow velocity can be obtained from the deflection measurements. The quasi-static responses of 1000 μm and 2000 μm long, 10 μm wide, and 30 μm high single crystal silicon beams were consistent with the prediction of the model, which couples thermoelectric, thermofluidic, and structural domains. The deflection sensitivity of up to 1.5 μm/(m/s) and the critical current sensitivity of up to 0.43 mA/(m/s) were registered in the experiments. Our model and experimental results demonstrate the feasibility of the sensing approach and further suggest that simple, robust, and potentially downscalable beam-type devices may have use in flow velocity and wall shear stress sensors.

Buckling effects on electronic and optical properties of BeO monolayer: First principles study

Using ab initio study, we apply full potential augmented plane waves plus local orbital and generalized gradient approximation for expanding wave functions and exchange–correlation functional term to investigate the electronic and optical properties of beryllium oxide nanosheet. The different values of planar buckling are considered to study. Also the results for fluorinated BeO nanosheet is compared to corresponding non-fluorinated buckled nanosheet. A red-shift in optical spectra is observed obviously as a result of weakening of σ bonds and increasing the number of π electrons. Also optical transparency property for visible range of the electromagnetic spectrum and vanishing energy-band degeneracy is concluded from our data.

Supercritical and subcritical buckling bifurcations for a compressible hyperelastic slab subjected to compression

We study the buckling bifurcation of a compressible hyperelastic slab under compression with sliding–sliding end conditions. The combined series-asymptotic expansions method is used to derive the simplified model equations. Linear bifurcation analysis yields the critical stress value of buckling, which gives a non-linear correction to the classical Euler buckling formula. The correction is due to the geometrical non-linearities coupled with the material non-linearities. Then through non-linear bifurcation analysis, the approximate analytical solutions for the post-buckling deformations are obtained. The amplitude of buckling is expressed explicitly in terms of the aspect ratio, the incremental dimensionless engineering stress, the mode of buckling and the material constants. Most importantly, we find that both supercritical and subcritical buckling could occur for compressible materials. The bifurcation type depends on the material constants, the geometry of the slab and the mode numbers.

Analytical solutions for the size dependent buckling and postbuckling behavior of functionally graded micro-plates

In this study, the buckling and postbuckling analysis of FG micro-plates under different kinds of traction on the edges is investigated based on the modified couple stress theory. The static equilibrium equations of an FG rectangular micro-plate as well as the boundary conditions are derived using the principle of minimum total potential energy. The analytical solutions are developed for three case studies including: simply supported micro-plates subjected to uniform transverse load and biaxial tractions, clamped-simply supported micro-plates under uniform transverse load and axial traction, and simply supported micro-plates subjected to shear traction. All plate properties except the length scale parameter are assumed to vary through the thickness according to the simple power law. The numerical results present the effects of the aspect ratio, material length scale parameter and power index on the critical values of traction and the curves of static equilibrium paths. Findings indicate that the length scale parameter is the most effective factor on the critical buckling values. Meanwhile, the aspect ratio has the most influence on the forms of the static equilibrium paths.

Direct strength prediction of web crippling failure of beams under ETF loading

This paper proposes a new approach to predict the web crippling failure load of cold-formed steel beams under External Two Flange (ETF) loading using the Direct Strength Method (DSM). After the description of the existing test data, Eurocode 3 (Comité Européen de Normalisation (CEN), 2006) [1] and the North-American Specification (American Iron and Steel Institute (AISI), 2012) [2] are used to predict the corresponding web crippling failure loads and the accuracy of such predictions is briefly assessed. In order to obtain additional information on the web crippling behaviour of each test specimen, non-linear numerical simulations are performed. Since the calibration of the DSM-based design expressions requires the previous calculation of (i) elastic buckling loads and (ii) plastic strengths/loads, two procedures are presented to achieve these goals. While the buckling loads are determined using the GBTWEB software, specifically developed for this purpose, the plastic loads are calculated by means of analytical expressions based on yield-line models. By adopting a non-linear regression procedure, the coefficients of a DSM-based formula are determined on the basis of the set of 128 experimental results available and the corresponding buckling and plastic load values. In spite of the different cross-sections, fastening conditions and test set ups considered in the calibration procedure, it is possible to establish a clear relationship between the web crippling slenderness and the nominal-to-plastic load ratio. Finally, it is shown that the developed/proposed DSM-based formula for ETF web crippling design yields reliable predictions, since they are associated with a LRFD (Load and Resistance Factor Design) resistance factor ϕ=0.81, which is located within the range of values currently adopted in the 2012 NAS (American Iron and Steel Institute (AISI), 2012) [2].

A semi-flexible model prediction for the polymerization force exerted by a living F-actin filament on a fixed wall.

We consider a single living semi-flexible filament with persistence length ℓp in chemical equilibrium with a solution of free monomers at fixed monomer chemical potential μ1 and fixed temperature T. While one end of the filament is chemically active with single monomer (de)polymerization steps, the other end is grafted normally to a rigid wall to mimic a rigid network from which the filament under consideration emerges. A second rigid wall, parallel to the grafting wall, is fixed at distance L < < ℓp from the filament seed. In supercritical conditions where monomer density ρ1 is higher than the critical density ρ1c, the filament tends to polymerize and impinges onto the second surface which, in suitable conditions (non-escaping filament regime), stops the filament growth. We first establish the grand-potential Ω(μ1, T, L) of this system treated as an ideal reactive mixture, and derive some general properties, in particular the filament size distribution and the force exerted by the living filament on the obstacle wall. We apply this formalism to the semi-flexible, living, discrete Wormlike chain model with step size d and persistence length ℓp, hitting a hard wall. Explicit properties require the computation of the mean force f̄i(L) exerted by the wall at L and associated potential f̄i(L)=-dWi(L)/dL on a filament of fixed size i. By original Monte-Carlo calculations for few filament lengths in a wide range of compression, we justify the use of the weak bending universal expressions of Gholami et al. [Phys. Rev. E 74, 041803 (2006)] over the whole non-escaping filament regime. For a filament of size i with contour length Lc = (i - 1) d, this universal form is rapidly growing from zero (non-compression state) to the buckling value fb(Lc,ℓp)=π(2)kBTℓp4Lc (2) over a compression range much narrower than the size d of a monomer. Employing this universal form for living filaments, we find that the average force exerted by a living filament on a wall at distance L is in practice L independent and very close to the value of the stalling force Fs (H)=(kBT/d)ln(ρˆ1) predicted by Hill, this expression being strictly valid in the rigid filament limit. The average filament force results from the product of the cumulative size fraction x=x(L,ℓp,ρˆ1), where the filament is in contact with the wall, times the buckling force on a filament of size Lc ≈ L, namely, Fs (H)=xfb(L;ℓp). The observed L independence of Fs (H) implies that x ∝ L(-2) for given (ℓp,ρˆ1) and x∝lnρˆ1 for given (ℓp, L). At fixed (L,ρˆ1), one also has x∝ℓp (-1) which indicates that the rigid filament limit ℓp → ∞ is a singular limit in which an infinite force has zero weight. Finally, we derive the physically relevant threshold for filament escaping in the case of actin filaments.

Investigation of innovative steel runway beam in industrial building

The design of a runway beam for overhead cranes is of great importance when constructing steel structures, as is the lateral-torsional buckling (LTB) value obtained for I-beam sections. Therefore, engineers must always consider the optimal design of these beams under overhead cranes loads. In this study, runway beams of three overhead crane groups were analyzed for LTB, and a sinusoidal runway beam body was developed. These runway beams with different sinusoidal angles were analyzed by using the finite element method (ABAQUS) and compared to each other and other runway IPE-IPN beam sections; furthermore, all models were compared with the Canadian Institute of Steel Construction standard. As a result, a new method was proposed for developing sinusoidal runway beams for overhead cranes. The sinusoidal runway beam has lower weight than runway IPE-IPN beam sections. Ultimately, a sinusoidal beam body was developed practically with a scale of 1:1.

A Review Paper on Comparison of Numerical Techniques for Finding Approximate Solutions to Boundary Value Problems on Post-Buckling in Functionally Graded Materials

The use of finite element models as research tools in biomechanics and orthopedics grew exponentially over the last two decades. However, the attention to mesh quality, model validation and appropriate energy balance methods and the reporting of these metrics has not kept pace with the general use of finite element modeling. Therefore, the purpose of this review was to develop the nonlinear filter and thermal buckling of an FGM panel under the combined effect of elevated temperature conditions and aerodynamic loading is investigated using a finite element model based on the thin plate theory and von Karman strain-displacement relations to account for moderately large deflection. It is found that the temperature increase has an adverse effect on the FGM panel flutter characteristics through decreasing the critical dynamic pressure. Decreasing the volume fraction enhances flutter characteristics, but this is limited by the structural integrity aspect. Structural finite element analysis has been employed to determine the FGM panel's adaptive response while under the influence of a uniaxial compressive load in excess of its critical buckling value. By increasing the applications of using composite materials inside aviation stages, it is visualized that the versatile FGM plate setup will broaden the operational execution over traditional materials and structures, especially when the structure is presented to a raised temperature. The vicinity of air motion facilitating stream brings about delaying the locking temperature and in stifling under loads, while the temperature build gives route for higher thermal-cycle abundance.

Buckling Behaviors of Single-Layered Lattice Dome under Radial Uniform Loads

This paper presented the nonlinear behaviors of the single-layered lattice dome, which is widely used for the long-span structure system. The behaviors were analysed through the classical shell buckling theory as the single-layered lattice dome behaves like continum thin shell due to its geometric characteristics, and finite element analysis method using the software program Nastran. Shell buckling theory provides two types of buckling loads, the global- and member buckling, and finite element analysis provides the ultimate load of geometric nonlinear analysis as well as the buckling load of Eigen value solution. Two types of models for the lattice dome were analysed, that is rigid- and pin-jointed structure. Buckling load using the shell buckling theory for each type of lattice dome, governed by the minimum value of global buckling or member buckling load, resulted better estimation than the buckling load with Eigen value analysis. And it is useful to predict the buckling pattern, that is global buckling or member buckling.