Plastic buckling of circular tubes under axial compression—part II: Analysis

In the second part of this study, the evolution of uniform axisymmetric wrinkling in axially compressed cylinders is modeled using the principle of virtual work. A version of this formulation also allows localization of wrinkling. The model domain is assigned an initial axisymmetric imperfection of a chosen amplitude and the wavelength yielded by the first bifurcation check. The solution correctly simulates the growth of wrinkles and results in a limit load instability. The limit strain is influenced by the amplitude of the imperfection. Beyond the limit load, wrinkling tends to localize, eventually leading to local folding.The possibility of bifurcation of the axisymmetric solution to non-axisymmetric buckling modes is examined by using a dedicated bifurcation check. The bifurcation check was found to yield such buckling modes correctly. The evolution of such buckling modes is simulated by a separate non-axisymmetric model assigned imperfections with axisymmetric and nonaxisymmetric components. The domain analyzed is one characteristic wavelength long (2λ_C). Initially, compression activates mainly axisymmetric deformation. In the neighborhood of the bifurcation point, non-axisymmetric deformation starts to develop, eventually leading to a limit load instability. Experimental responses were simulated with accuracy by assigning appropriate values to the two imperfection amplitudes. Prediction of the limit strains for the whole range of diameter-to-thickness ratios (D/t) considered in the experiments was achieved by making the amplitude of the non-axisymmetric imperfection proportional to (D/t)²/m³ (m is the circumferential wavenumber). Matching all aspects of the experiments required inclusion of the anisotropy measured in the tubes tested through Hill's yield criterion in all models.     

Plastic buckling of circular tubes under axial compression—part I: Experiments

Elastic buckling of cylindrical shells due to axial compression results in sudden and catastrophic failure. By contrast, for thicker shells that buckle in the plastic range, failure is preceded by a cascade of events, where the first instability and failure can be separated by strains of 1–5%. The first instability is uniform axisymmetric wrinkling that is typically treated as a plastic bifurcation. The wrinkle amplitude gradually grows and, in the process, reduces the axial rigidity of the shell. This eventually leads to a limit load instability, beyond which the cylinder fails by localized collapse. For some combinations of geometric and material characteristics, this limit load can be preceded by a second bifurcation that involves a non-axisymmetric mode of deformation. Again, this buckling mode localizes resulting in failure.The problem is revisited using a combination of experiments and analysis. In Part I, we present the results of an experimental study involving stainless steel specimens with diameter-to-thickness ratios between 23 and 52. Fifteen specimens were designed and machined to achieve uniform loading conditions in the test section. They were subsequently compressed to failure under displacement control. Along the way, the evolution of wrinkles was monitored using a special surface-scanning device. Bifurcation buckling based on the J₂ deformation theory of plasticity was used to establish the onset of wrinkling. Comparison of measured and calculated results revealed that the wrinkle wavelength was significantly overpredicted. The cause of the discrepancy is shown to be anisotropy present in the tubes used. Modeling of the postbuckling response and the prediction of the limit load instability follows in Part II.     

Energy absorption of pressurized thin-walled circular tubes under axial crushing

By pressurizing cellular materials, honeycombs, or thin-walled structures, their energy absorption can be greatly enhanced, and this enhancement can be controlled by the applied pressure. This concept shines light on the possibility of achieving adaptive energy absorption. To investigate the effect of internal pressure on energy absorption of thin-walled structures, this paper presents a study of axial crushing of pressurized thin-walled circular tubes. In the experiments, three groups of circular tubes with radius/thickness ratio R/t=120-200 were axially compressed under different pressurizing conditions. The results show that with an increase of internal pressure, the deformation mode switches from diamond mode with sharp corners to that with round corners, and eventually to ring mode. In diamond mode, the mean force of the tubes increases linearly with internal pressure. The enhancement comes from two mechanisms: direct effect of pressure and indirect effect due to interaction between pressure and tube wall. After the deformation switches to ring mode, the enhancement resulting from the second mechanism becomes weaker. Based on experimental observations, the deformation mode, energy dissipation mechanisms as well as interaction between internal pressure and tube wall are analyzed theoretically and the theoretical results are in good agreement with the experimental ones.     

Plastic buckling of tubes under axial compression and internal pressure

The plastic buckling and collapse of long cylinders under combined internal pressure and axial compression was investigated through a combination of experiments and analysis. Stainless-steel cylinders with diameter-to-thickness values of 28.3 and 39.8 were compressed to failure at fixed values of internal pressure up to values 75% of the yield pressure. The first effect of internal pressure is a lowering of the axial stress–strain response. In addition, at some plastic strain level, the cylinder develops uniform axisymmetric wrinkling. Under continued compression, the wrinkles grow stably, gradually reducing the axial rigidity of the structure and eventually lead to a limit load instability. All pressurized cylinders remained axisymmetric until the end of the test past the limit load.The critical stress and wavelength were established using classical plastic bifurcation theory based on the deformation theory of plasticity. The evolution of wrinkling, and the resultant limit state, were established by modeling a periodic domain that is one half of the critical wavelength long. The domain was assigned an initial imperfection corresponding to the axisymmetric buckling mode calculated through the bifurcation check. The inelastic material behavior was modeled through the flow theory of plasticity with isotropic hardening. The variations of the axial response and of the limit strain with pressure observed in the experiments were reproduced well by the model. Inclusion of Hill-type anisotropic yielding in all constitutive models was required for good agreement between predictions and experiments.     

Plastic mechanism analysis of circular tubes under pure bending

There are a number of solutions available to predict the response of a circular steel tube under pure bending. However, most of these solutions are based on an elasto-plastic treatment, which is complex and difficult to use in any routine design. This paper describes a theoretical treatment to predict the moment-rotation response of circular hollow steel tubes of varying D/t ratios under pure bending. The Mamalis et al. (J. Mech. Sci. 1989;203:411–7) kinematics model for a circular tube under a controlled moment gradient was modified to include the effect of ovalisation along the length of the tube. Inextensional deformation and rigid plastic material behaviour were assumed in the derivation of the deformation energy. The plasticity observed in the tests was assumed to spread linearly along the length of the tube. Two local plastic mechanisms (Star and Diamond shapes) were studied to model the behaviour observed in the tests especially during the unloading stage. The theoretical predictions are compared with the experimental results recently obtained by Elchalakani et al. (Quartral. J. Struct. Eng. 2000;3(3):1–16). Good agreement was found between the theoretical predictions and experimental moment-rotation responses, particularly for the Star shape mechanism. A closed-form solution is presented suitable for spreadsheet programming commonly used in routine design.     

Initial collapse of braced elliptical tubes under lateral compression

The initial collapse behavior of elliptical tubes subjected to lateral compression is investigated. Both unbraced and braced tubes are analysed, and the main factors which affect the initial collapse behavior are discussed and interpreted. Lower and upper bound theorems are applied to bracket the collapse load in a direct manner. The initial collapse loads for braced circular tubes, obtained as a special case of elliptical tubes, are compared to previous research results. The results and concepts presented herein can be useful in the development of design criteria for energy dissipating devices. Elliptical shapes, when loaded parallel to their major axes, can have an advantage over their circular counterparts in that they possess larger collapse strokes and an increased energy dissipation potential per unit mass.     

On the axial splitting and curling of circular metal tubes

The present paper investigates the axial splitting and curling behaviour of circular metal tubes. Mild steel and aluminum circular tubes were pressed axially onto a series of conical dies each with different semi-angle. By pre-cutting eight 5 mm slits which were distributed evenly at the lower end of each tube, the tube split axially and the strips curled outward. Experiments showed that this mechanism results in a long stroke and a steady load. An approximate analysis is presented which successfully predicts the number of propagated cracks, the curling radius and the force applied. This analysis takes into account ductile tearing of the cracks, plastic bending/stretching and friction. Effects of tube dimensions, semi-angle of the die and friction are discussed in detail.     

Dynamic axial crushing of short to long circular aluminium tubes with agglomerate cork filler

Cork is a complex natural cellular material with quite unknown or not well understood properties. It is available in the natural and in the agglomerate form and it is an ecological and very durable material. That is why it is used today as thermal and acoustic insulator, as a seal and as an energy-absorbing medium in flooring, shoes and packaging, among others. However, the application of agglomerate cork as filler inside structural thin-walled sections, in order to increase the energy absorption, has not been much explored. Dynamic experimental tests were carried out on empty and micro-agglomerate cork-filled tubes with 22 and 50 mm in internal diameter (D) and length (L), respectively, and numerical simulations were performed with the finite element method software LS-DYNA^TM, showing good agreement in terms of load–displacement curves and deformation patterns. Having validated the numerical model with experiments, the finite element model was used to undertake a systematic study of circular tubular structures impacted at 10 m/s. The load-deformation characteristics, energy-absorption response and collapse mode transitions of empty and cork-filled aluminium tubes with varying diameters and thicknesses (t), lengths of 25, 300 and 350 mm, but with constant slenderness ratios D/t and D/L were thus studied. Relevant comparisons were raised, showing that the slenderness ratios are very important parameters that globally govern the percentage increase in energy absorbed by tubular structures after cork-filling during an impact loading.     

A numerical study on the impact response and energy absorption of tapered thin-walled tubes

Tapered tubes have been considered desirable impact energy absorbers due to their relatively stable mean load–deflection response under dynamic loading. Relatively few studies have been reported on the energy absorption performance of tapered tubes compared with straight tubes. This paper compares the energy absorption response of straight and tapered thin-walled rectangular tubes under both quasi-static and dynamic axial impact loading, for variations in wall thickness, taper angle, impact mass and impact velocity. It is found that the dynamic response of tapered tubes is more sensitive to impact velocity and wall thickness than taper angle for lower impact velocities. Inertia effects influenced the dynamic response for both straight and tapered tubes, yet were less significant for the latter. Overall, the results indicate that the energy absorption response of tapered tubes can be controlled via their wall thickness and taper angle, and this highlights their potential for use as energy absorbers. Analysis has been undertaken using a finite element model, validated using existing theory.     

Quasi-static compression of electric resistance welded mild steel tubes with axial gradient-distributed microstructures

This paper presents the deformation behavior and crashworthiness of electric resistance welded mild steel tubes with axial gradient microstructures in quasi-static compression. Three sets of tubes were prepared, and regions of each tube were Induction heated and directly quenched (IH-DQ). The effect of the length to diameter (L/D) ratio, and length of the IH-DQ region on crushing characteristics was investigated, and compared with untreated tubes. The compression tests revealed that improved energy absorption can be obtained in IHDQ tubes if the collapse is controlled by the formation of a concertina buckling mode. However, there was a tendency to produce mixed or Euler buckling modes as the ratio of L/D increased. Meanwhile, the results of the crush experiments and the FEM models showed that the heat-treatment process should be precisely controlled to produce the correct type of microstructure, and circumferential uniformity of microstructure distribution.     

The inhomogeneous deformation of polycarbonate circular honeycombs under in-plane compression

In-plane uniaxial and equi-biaxial compression tests were conducted quasi-statically on polycarbonate circular honeycombs. While the uniaxial compression tests were easy to implement, a special test rig was designed to carry out the in-plane (x–y) equi-biaxial compression tests in a conventional universal testing machine. The deformation characteristics of a honeycomb block under uniaxial compression are quantitatively described by tracking the variations of the cells’ parameters, such as the area strain and the angle of cells, during the deformation process of the honeycomb. As the distribution of the deformation within the honeycomb block under biaxial compression is much more complex, a series of color maps based on the area strain of each cell are produced to demonstrate the inhomogeneous deformation among the cells in the honeycomb block, by which the initiation of deformation inhomogeneity and its evolution in the honeycomb block are characterized. To identify the role of the friction between the honeycomb sample and the test rig, finite element analysis is conducted to simulate the collapse process of the circular honeycomb under equi-biaxial compression. Furthermore, an inhomogeneity index, I_inh, is defined as a function of the overall compression of a sample to quantify the severity of the deformation inhomogeneity of the honeycomb; this index facilitates quantitative comparisons among the results under various loading conditions. It is found that the value of I_inh of a sample is closely related to some internal factors, such as the localization band of deformation and the deformation pattern of cells.     

轴心受压高强钢圆管稳定承载性能研究

对770MPa级高强钢管件的初始几何缺陷、残余应力和本构关系进行了测试,完成了66根770MPa级不同规格高强钢管件的轴压极限承载力试验,结合仿真分析,验证了添加初始缺陷的一致模态法分析高强钢管件极限承载力的准确性。     

Parametric study and numerical analysis of empty and foam-filled thin-walled tubes under static and dynamic loadings

In this paper the crushing behavior of thin-walled tubes under static and dynamic loading is investigated. First, a finite element (FE) model for empty thin-walled tube was constructed and validated by available experimental and numerical data. The comparison between the FE results and the existing numerical solutions as well as the available experimental results showed good agreements. Next, a model for the foam was adopted and implemented in an in-house FE code. The implemented isotropic foam model was then used to simulate the behavior of foam-filled tubes under both static and dynamic loadings. Good agreement was observed between the results from the model with those obtained by analytical relations and experimental test data. The validated FE model was then used to conduct a series of parametric studies on foam-filled tapered tubes under static and dynamic loadings. The parametric studies were carried out to determine the effect of different parameters such as the number of oblique sides, foam density and boundary conditions on crushing behavior of rectangular tubes. The characteristic included deformed shapes, load–displacement, fold length and specific energy absorptions.     

Torsional crushing of foam-filled thin-walled square columns

Torsional crushing behavior of foam-filled thin-walled square columns were investigated analytically, numerically and experimentally. The lower and upper bounds on the torsional resistance of foam-filled columns were established analytically. Numerical simulations were carried out and showed that the presence of the filler changes the torsional collapse mechanism and gives rise to higher order sectional collapse modes, which results in a higher torsional resistance. Torsional experiments were performed and results were compared to the analytical and numerical solutions with reasonably good agreement. It was found that bonding of the foam to the walls changes the deformation mode by spreading deformation over the whole length. The corresponding torsional resistance is also larger for the first 40° of rotation. It is concluded that fitting prismatic members with the aluminum foam of a density ranging from 0.14 to 0.28 g/cm³ can double the energy absorption of a given member.     

薄壁管复合载荷下应变增量的数值算法

薄壁管复合载荷实验是研究材料在一般应力状态下力学属性的有效手段。本文以 Hill非二次屈服准则以及塑性增量理论为基础 ,给出了薄壁管复合载荷 (拉扭、拉胀 )下应变增量的数值解法 ,分析了 Hill非二次屈服准则的次数 (m值 )对计算结果的影响。通过与解析解的对比以及简单加载下的实验验证 ,本文的算法正确、有效。     

Experimental evaluation of the strain field history during plastic progressive folding of aluminium circular tubes

The axisymmetric collapse by plastic progressive folding of a circular tube submitted to axial loading is considered by an experimental approach. The strain field history is measured by means of electric strain gages properly placed on the external surface of the tube so that more than one fold is covered and both axial and circumferential strains are measured. The measured strains are examined both as time-histories and as a deformation field. The formation and development of circumferential plastic hinges are pointed out. The strain histories, reported as a function of the displacement of the testing machine cross-head, are then correlated with the crushing force diagram, leading to a better understanding of the folding mechanics. In particular, the formation of each fold develops through three subsequent phases: the initialization at the closure of the previous fold, the flattening of the upper conical surface, and the flattening of the lower conical surface. While most of the tube wall is pushed outwards of the original cylindrical surface, a portion is pushed inwards of that surface. Moreover, there is a small portion of the wall that is pulled inward during the fold initialization and then pushed outward during the fold closure. The analysis of these histories lead to the validation of the basic assumption of our and other recent kinematical models of the plastic progressive folding.     

Three-dimensional effects of the bend–stretch forming of aluminum tubes

Bend–stretch forming is commonly used to shape extruded tubular aluminum parts for automotive applications. The tubes are pre-stretched, pressurized and bent over rigid dies. Tension prevents buckling of the compressed side and significantly reduces springback during unloading. An unwanted byproduct of the process is distortion of the cross section. Small amounts of pressure applied during forming can reduce this distortion. A systematic study of how to select the appropriate amounts of tension and pressure for accurate forming with minimal distortion has been conducted. The problem was first studied experimentally using a custom forming facility. An efficient 2-D model of the process was previously developed which was shown capable of capturing the main deformation features of interest. Its efficiency made this model a useful design tool for optimally selecting the forming parameters. In this paper, a 3-D finite element model of the forming process is used to simulate the complete forming process. By using a specially calibrated non-quadratic yield function, the model accurately reproduces all aspects of the process. The model is used to study 3-D features of the problem such as variations of distortion and springback along the length, the effect of friction, the lifting of a section of the tube around the mid-span off of the die, the effect of post-tension, and forming over a variable radius die.     

Effect of an ultralight metal filler on the bending collapse behavior of thin-walled prismatic columns

The effect of low-density metal filler, such as aluminum foam or honeycomb, is studied on the bending collapse resistance of thin–walled prismatic columns. A combination of analytical and numerical results is used to predict the initial and post collapse response of empty and filled columns. Closed-formed solutions for the bending-rotation characteristics are constructed in terms of the geometrical parameters and the filler strength. The low-density metal core retards sectional collapse of the thin-wall column, and increases bending resistance for the same rotation angle. Numerical simulations show that, in terms of achieving the highest energy absorption to weight ratio, columns with aluminum honeycomb or foam core are preferable to thickening the column wall. Moreover, the presence of adhesive improved the specific energy absorption significantly.