Initial Shape Imperfections Research Articles

Slender metal web stability analysis: Impact of initial imperfection on the mode of buckling

The post buckling of a rectangular slender web in compression has been analyzed. Shapes of a buckling area obtained from the nonlinear analysis have been compared with buckling modes from the linearized problem for various aspect ratios. Effects of initial shape imperfections upon the analysis have been investigated using nonlinear approach. To trace the complete nonlinear equilibrium curves, specialized code based on FEM was created. The Newton-Raphson iteration algorithm was used, load versus displacement control was changed during the process of calculation. Obtained results were verified using Ansys system, in this case arc-length method was activated for overcoming critical points.

Effect of spatially correlated initial geometric imperfection on reliability of spherical latticed shell considering global instability

Latticed shells are extensively used for large-span structures because of their good structural behaviour, lightweight and ease of construction. Their codified design is often governed by the global instability requirement for gravity load and specified shape and amplitude of maximum initial geometric imperfection (IGI). The reliability basis for the assigned shape and maximum IGI is unknown. In the present study, it is proposed that IGI is modelled using the simultaneous autoregressive model. It is shown that this model is flexible in modelling spatially varying correlated imperfection and can cope with irregular grid system. By using this model, the effect of the spatially correlated IGI on the probability distribution of the load capacity for global instability and the reliability of spherical latticed shell designed according to the Chinese design code is assessed. It is shown that the use of the prefixed shape of the IGI to assess the probability distribution of the load capacity is unconservative as compared to that by using spatially varying random imperfection. The shape of the probability distribution for the former differs from that for the latter. The reliability analysis results indicate that the use of a critical load factor of 2 implemented in the code leads to a failure probability less than about 10−6 for global instability if the specified maximum IGI defined in the code equals about two to three times the standard deviation of IGI. This failure probability becomes less than about 10−5 if a critical load factor of 1.5 is used for design. It is suggested that the code recommended critical load factor may be reduced to 1.5 while achieving target reliability index often adopted for design code calibration.

A new formula to predict buckling pressure of steel ellipsoidal head under internal pressure

Abstract Buckling is a critical failure mode of ellipsoidal head under internal pressure. This study focuses on the prediction of buckling pressure of steel ellipsoidal heads subjected to internal pressure. First, finite element model of ellipsoidal head is generated, taking into account the effects of material and geometrical nonlinearity. The plastic bifurcation buckling pressure of ellipsoidal head is determined by nonlinear FEM. Second, we study the effects of diameter-thickness ratio, radius-height ratio, yield strength, strain hardening and attached cylinder on buckling pressure. Based on FE results of the models with elastic-perfectly plastic material, we propose a formula to predict buckling pressure of perfect ellipsoidal head. In addition, we summarize experimental results on the buckling of twenty-one ellipsoidal heads which cover different geometrical parameters, material types, and methods of fabrication. Eleven out of twenty-one ellipsoidal heads were tested by the authors recently. Finally, based on the formula for perfect ellipsoidal heads, a new formula is developed for the buckling pressure of actual ellipsoidal heads after considering a reduction factor that accounts for the effect of initial shape imperfection. The reduction factor depends on the methods of head fabrication, i.e. whether cold pressing and spinning or assembly of formed segments. It is shown that the new formula has comprehensive advantage in accuracy and applicability in comparison with other formulae.

Nonlinear parametric oscillations of a viscoelastic shallow shell of variable thickness

The problem of parametric oscillations of an isotropic viscoelastic shallow shell of variable thickness under periodic load is considered. It is believed that under the influence of specified load, the shallow shell allows displacements (in particular, deflections), commensurate with its thickness. In a geometrically nonlinear statement, taking into account the viscoelastic properties of material, a mathematical model of the problem has been developed using the classical Kirchhoff-Love hypothesis. Using the Bubnov-Galerkin method based on the polynomial approximation of the deflections, the problem is reduced to the study of the system of integro-differential equations, where time is the independent variable. The solution of the system of integrodifferential equations is determined by the proposed numerical method. Based on this method, a numerical solution algorithm is described. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. At the same time, the effect of geometric nonlinearity, viscoelastic properties of material, as well as other physicomechanical and geometric parameters and factors (rheological parameters, thickness, initial shape imperfections, aspect ratios, boundary conditions, excitation coefficient) on the area of dynamic instability is taken into account. The results obtained in this study are in good agreement with the results and data obtained by other authors.

Influence of geometrical imperfection on critical temperature evaluation for steel corrugated arch sheets exposed to fire

The critical temperature for an arched corrugated steel sheet exposed to fully developed fire is determined in this paper. The analysis is performed on 3D numerical model developed within the ANSYS computational environment with parameters calibrated experimentally on an experiment performed for persistent design scenario. During the simulations performed the self-supporting steel sheet with an assumed initial shape imperfection is at first loaded with static load, up to a predetermined level of material effort, and subsequently evenly heated until the capability to safely resist the applied load is completely exhausted. Because of the nonlinearity of the analysis having several sources, the location of the limit point on the equilibrium path determined during calculations represents the temperature sought. Three imperfection patterns, denoted as A, B and C are considered. In the case A the imperfection has the shape and magnitude measured in situ, selected as representative for a set of 10 steel sheets of the same type. In the case B the imperfection is a substitute one with size analogous to that assumed in the case A, but with a different shape, corresponding to the first antisymmetric buckling mode identified in conventional Local Buckling Analysis (LBA). In the case C the shape of imperfection B has been kept, but the positive and negative deviation from the perfect circular shape have been increased twofold.

Lateral torsional buckling of steel I-beams: Effect of initial geometric imperfection

In the current study, the influence of the initial lateral (sweep) shape and the cross-sectional twist imperfection on the lateral torsional buckling (LTB) response of doubly-symmetric steel I-beams was investigated. The material imperfection (residual stress) was not considered. For this objective, standard European IPN 300 beam with different unbraced span was numerically analyzed for three imperfection cases: (i) no sweep and no twist (perfect); (ii) three different shapes of global sweep (half-sine, full-sine and full-parabola between the end supports); and (iii) the combination of three different sweeps with initial sinusoidal twist along the beam. The first comparison was done between the results of numerical analyses (FEM) and both a theoretical solution and the code lateral torsional buckling formulations (EC3 and AISC-LRFD). These results with no imperfection effects were then separately compared with three different shapes of global sweep and the presence of initial twist in these sweep shapes. Besides, the effects of the shapes of initial global sweep and the inclusion of sinusoidal twist on the critical buckling load of the beams were investigated to unveil which parameter was considerably effective on LTB response. The most compatible outcomes for the perfect beams was obtained from the AISC-LRFD formulation; however, the EC-3 formulation estimated the Pcr load conservatively. The high difference from the EC-3 formulation was predicted to directly originate from the initial imperfection reduction factor and high safety factor in its formulation. Due to no consideration of geometric imperfection in the AISC-LFRD code solution and the theoretical formulation, the need to develop a practical imperfection reduction factor for AISC-LRFD and theoretical formulation was underlined. Initial imperfections were obtained to be more influential on the buckling load, as the unbraced length of a beam approached to the elastic limit unbraced length (Nr). Mode-compatible initial imperfection shapes should be taken into account in the design and analysis stages of the I-beam to properly estimate the geometric imperfection influence on the Pcr load. Sweep and sweep.twist imperfections led to 10% and 15% decrease in the Pcr load, respectively, thus; well-estimated sweep and twist imperfections should considered in the LTB of doubly-symmetric steel I-beams.

An unified formula for the critical force of lateral buckling of imperfect submarine pipelines

Unburied submarine pipelines might buckle laterally under the conditions of high temperature and high pressure. As an important design parameter of submarine pipelines, the lateral buckling critical force (LBCF) is affected by the maximum amplitude, wavelength, shape of the initial imperfection, and etc. However, the exact relationship between the LBCF and the initial imperfection shape is not clear at present. In this paper, the lateral buckling behaviors of submarine pipelines with different initial imperfection shapes are studied using 3D finite element (FE) analysis. A parametric study is conducted and the characteristic parameter of the initial imperfection shape is found. It is the characteristic parameter that makes the LBCF greatly affected by the initial imperfection shape. A unified formula is proposed to reveal the exact relationship between the LBCF and the initial imperfection shape. An application of the unified formula is carried out and the results show the accuracy of the unified formula.

Lateral buckling critical force for submarine pipe-in-pipe pipelines

The lateral buckling critical force of a submarine pipe-in-pipe pipeline with symmetrical initial imperfection on soft foundation is determined. The effects of initial imperfection shapes, out-of-straightness, soil parameters as well as lateral nonlinear pipe-soil interaction, and structure parameters (stiffness ratio of inner pipe to outer pipe and clearance between inner and outer pipes) are discussed. A simple formula considering the above factors is proposed to determine the lateral buckling critical force of submarine pipe-in-pipe pipelines based on dimensional analysis and numerical results. The accuracy of the presented formula is shown by a case study.

On the mechanics of local-distortional interaction in thin-walled lipped channel columns

The objective of this paper is to present and discuss numerical results concerning the geometrically non-linear behaviour of thin-walled lipped channel columns experiencing local-distortional (L-D) interaction that will shed fresh light on the mechanics underlying this coupling phenomenon. These results, obtained through Generalised Beam Theory (GBT) elastic post-buckling analyses, provide the evolution, along given equilibrium paths, of the column (i) deformed configuration, expressed in modal form, (ii) relevant displacement profiles, and (iii) modal participation diagrams. Taking full advantage of the GBT modal features, it becomes possible to unveil the most relevant behavioural/mechanical aspects associated with the occurrence of L-D interaction. Particular attention is devoted to the structural interpretation and explanation of the quantitative and qualitative differences exhibited by the column post-buckling behaviours associated with (i) “pure”/individual local and distortional behaviours and (ii) the occurrence of L-D interaction stemming from either the closeness between the local and distortional critical buckling loads (“true L-D interaction”) or from a secondary bifurcation phenomenon (“secondary distortional or local bifurcation L-D interaction”). The columns analysed exhibit (i) the aforementioned different types of L-D interaction, (ii) two support conditions, namely fixed (mostly) or pinned end cross-sections, and (iii) several critical-mode initial geometrical imperfection shapes.

07.02: Distortional‐global interaction in lipped channel beams: Part I: Mechanics and elastic behaviour

ABSTRACTThis work aims at presenting and discussing Generalised Beam Theory (GBT) numerical results concerning the elastic geometrically non‐linear behaviour of simply supported lipped channel (LC) beams under uniform major‐axis bending and experiencing distortional‐global (D‐G) interaction, making it possible to shed fresh light on the mechanics underlying this coupling phenomenon. Two LC beams geometries are considered, each exhibiting a different type of D‐G interaction, namely (i) “true D‐G interaction” (close global, McrG, and distortional, McrD, critical buckling moments), containing three critical‐mode initial geometrical imperfection shapes (two global, due to the lack of symmetry, and one distortional) and (ii) “secondary global‐bifurcation interaction – SGI” (McrD«McrG), containing a critical‐mode distortional initial imperfection. In order to clarify the surprising behaviour of the beam undergoing a SGI, an additional beam is analyzed, deemed to exhibit a “pure” distortional post‐buckling behaviour. The GBT results provide the evolution, along given equilibrium paths, of the beam deformed configuration (expressed in modal terms), relevant displacement profiles and modal participation diagrams. The knowledge acquired has visible impact on the development of rational design rules for CFS beams affected by D‐G interaction, a topic addressed in a companion paper.

Shape and Size of Initial Geometrical Imperfections of Structural Glass Members

Abstract In this paper measuring and determining of initial geometrical imperfections of set of glass members are presented. Measured specimens had the different dimensions and different composition – monolithic glass or laminated glass using different types of interlayer foils and glass panes. The shape of initial overall bow was measured using different methods and results were statistically evaluated and compared with allowable values defined in standards.

Experimental compressive strength analyses of high tensile steel thin-walled stiffened panels with a large lightening opening

This work investigates the compressive load carrying capability of experimentally tested reinforced steel panels with a large lightening opening. A series of experiments have been carried out on reinforced panels with circular and elongated circular opening of different sizes. The effect of different opening shapes, different opening sizes as well as different initial imperfection shapes and amplitudes, on the loading carrying capacity are studied. The resultant collapse modes are presented and both initial and post-collapse shapes are analysed and the sequential occurrence of the post-collapse deformation shapes have been analysed. The experimental results of force-vertical/lateral displacements relationships, strain energy density and stress-strain relationships are presented and analysed. Finally, the experimental results have been compared with other experimental ones for unstiffened plates with elongated circular opening subjected to uniaxial compressive load.

Composite spheroidal shells under external pressure

ABSTRACTBuckling, collapse, and first ply failure pressures have been obtained for a range of spheroidal FRP shells under static external pressures. Comparisons of the load carrying capacities are made for like geometries. It is shown that some prolate ellipsoids can be stronger, for up to 70% than the equivalent spheres. The effect of boundary conditions in the equatorial plane on the buckling strength is assessed for the diameter-to-wall- thickness ratio ranging from 100 to 500. Regions where the strength (FPF) rather than asymmetric bifurcation control the failure have been identified. Small perturbations in ply orientations, ±5°, have been introduced in order to assess their influence on the buckling strength. Both symmetric and asymmetric stacking with respect to the mid-plane have been analyzed (including random stacking).Initial shape imperfection positioned at the apex of a hemispherical head has been examined. The latter aimed at assessing the sensitivity of buckling pressure to deviations from perfect geometry. Results have been obtained for the lower bound approach based on the local flattening using the increased-radius. Imperfections in ellipsoids have also been studied. Axisymmetric modeling of all shells was used throughout in this study. Only the upper half of shells was modeled, and the issue of boundary conditions at the equatorial plane was also explored.

Influence of Small Added Mass on Splitting Flexural Frequency Spectrum of Circular Ring with Initial Imperfection of Shape

The paper presents the results of the research into the problem of vibration of the infinitely long circular cylindrical shell, which has a symmetrical initial shape imperfection. The infinitely long shell is viewed as the ring of a single width in situations of planar deformation. It was discovered that the effect of splitting the bending spectrum frequency of the isolated ring not only occurs when the number of waves of asymmetric imperfection coincides with the number of waves of dynamic deformations, but also where the number of imperfections is twice as big as the number of the forming waves. This conclusion is correct provided that the number of waves of shape imperfection is even. An opportunity to remove the splitting frequency of the bending spectrum of a thin imperfect circular ring by acceding to it little linear mass is studied. The research has shown that neither the selection of attaching point of added mass nor its size allow discarding the unwanted in terms of a dynamic strength and reliability of the constructions, effect of staggering frequency. During the research a range of qualitative effects caused by the ring having inevitable deviations from the perfect geometrical form were identified.

Bearing Capacity of Aluminum Bars with Built-Up Cross Section and Glued or Welded Connections

The paper is focused on numerical analysis and experimental test of aluminum bars with glued and welded connections between separate members. The connection properties were tested experimentally. The numerical analysis were conducted for the beam and shell model of the structure. The shape and magnitude of initial geometric imperfection was taken into account.

Buckling and Post-buckling Behavior of Beams With Internal Flexible Joints Resting on Elastic Foundation Modeling Buried Pipelines

The buckling and post-buckling behavior of axially loaded Winkler beams with flexible internal hinges is addressed, aiming to provide a background for the investigation of upheaval buckling of buried pipelines equipped with flexible joints for their protection against activation of reverse seismic faults. In order to acquire qualitative understanding of the interaction between the hinge stiffness and the soil stiffness for different cases, the beams under investigation are considered as either simply-supported or clamped. At first, elastic critical buckling loads and corresponding eigenmodes are numerically obtained using linearized buckling analysis, and eigenmode cross-over is investigated considering soil and hinge rotational stiffness. Geometrically nonlinear analyses with imperfections (GNIA) are then performed, indicating for most cases descending post-buckling paths, thus unstable post-buckling behavior, with the exception of cases of very soft soil. The sensitivity of the response to initial imperfection shape and magnitude is also addressed, to identify their impact on the post-buckling behavior. Beam buckling behavior is moreover examined by considering the beam being surrounded by soil exhibiting different stiffness in the upward and the downward direction. The results are compared to the case of a continuous beam, in order to highlight the impact of internal hinges on the beam overall buckling behavior.

Experimental strength analysis of steel plates with a large circular opening accounting for corrosion degradation and cracks subjected to compressive load along the short edges

This work studies the combined effect of both geometrical characteristics (opening and initial imperfections) and age related damage (corrosion and cracks) on the local and global responses of thin rectangular steel plates. A series of experimental tests has been conducted for in-service steel plates with a circular opening and subjected to several damage actions and uni-axial compression. Different initial imperfection shapes and amplitudes, corrosion degradation levels and locked crack lengths are considered. Several collapse modes were observed and the reasons for their occurrence are discussed. The experimental results of force-displacement and stress––strain relationships are presented and the relevant dissipated energies, resilience and toughness are estimated. The experimental results are compared with published experimental results, confirming the complexity of the plate behaviour accounting for different damage scenarios.

Numerical investigation and direct strength design of cold-formed steel lipped channel columns experiencing local–distortional–global interaction

This paper reports the results of a numerical investigation concerning the relevance and Direct Strength Method (DSM) prediction of the ultimate strength erosion caused by local-distortional-global (LDG) interaction in cold-formed steel fixed-ended lipped channel columns. The geometries of the columns analysed (cross-section dimensions and lengths) were carefully selected to ensure that the three competing critical buckling loads are not more than 20% apart, thus guaranteeing a fairly high level of LDG coupling, and ordered in all the various possible sequences. In order to cover a wide slenderness range, several yield stresses are considered, falling below, in-between and above the lowest and highest critical buckling stresses. After providing a brief description of the column selection procedure, which is based on buckling analyses performed with Generalised Beam Theory (GBT), the methodology adopted to identify the most detrimental initial geometrical imperfection shape (in the sense it minimises the column strength) is addressed – it employs Koiter's asymptotic method along with a Monte Carlo simulation. Then, columns containing those initial geometrical imperfections are compressed up to failure, by means of Abaqus shell finite element analyses (SFEA), making it possible to acquire in-depth knowledge on the behaviour of lipped channel columns undergoing LDG interaction and gather considerable failure load data. Finally, the last part of the paper is devoted to the DSM prediction of those failure loads and uses the obtained “data bank” to assess whether the available design approaches are able to handle adequately the ultimate strength erosion caused by the triple interaction phenomenon under investigation – if this is not the case, new design curves must be developed.

Prediction of the upheaval buckling critical force for imperfect submarine pipelines

Upheaval buckling behavior of submarine pipelines under high temperature and high pressure conditions is a primary concern for structural integrity. The critical axial force is a key factor governing the buckling behavior. There have been already some formulas to calculate critical axial force for some particular initial imperfection shapes. However, there is no universal formula to express the effects of initial imperfection shape and Out-of-Straight (OOS) on the critical axial force. In this paper, the upheaval buckling behaviors of eight groups of pipeline segments with different imperfection shapes and different OOS have been studied using the finite element method. A new parameter is defined to express the differences of imperfection shapes. An approximation and universal formula is proposed to calculate the critical axial force which covers the new parameter and the OOS of pipeline. A case study is presented which illustrates the application of the formula. Finally, comparison between this study and previous research results is conducted, and it manifests that this formula has a greater precision.

Locally flattened or dented domes under external pressure

The paper provides comparisons of sensitivity of buckling pressures to initial shape imperfections, for the case of externally pressurised steel domes. A priori defined deviations from perfect shape include: Legendre polynomials, increased-radius patch, and localised inward dimple. The latter is created by a concentrated force acting radially. In this case the effects of spring back and/or annealing of the dented patch and the surrounding area on the load carrying capacity are assessed. Some of the elastic solutions related to the lower-bound approach are compared with the available experimental data.