Buckling of simply supported rectangular plates under combined bending and compression using eigensensitivity analysis

In this paper, an approximate quadratic closed-form expression is presented for the critical buckling analysis of a plate subjected to combined bending and compression. The formula is developed by expanding the eigenvalue, the critical buckling load, for a plate under combined bending and compression in a Mauclaurin's series about a plate subjected only to compression. The general expression can be used for all combinations of simply supported and clamped rectangular plates boundary conditions. An explicit formula in terms of the plate aspect ratio R and plate load parameter α is evaluated for simply supported plates. Compared with the Rayleigh–Ritz method, this approximate expression provides an excellent comparison when the load parameter α1.52 for plate aspect ratio between 0.2R2.8.     

Buckling analysis of trapezoidally corrugated panels using spline finite strip method

Buckling of trapezoidally corrugated panels under in-plane loading is analyzed by a spline finite strip method. The influence on the elastic buckling load of various parameters, such as geometry. loading forms and boundary conditions, etc., is studied. It is found that: (1) for longitudinal compression the buckling load increases with the corrugation angle α, and for a given α the highest buckling load is achieved when the ‘proportion parameter’ γ = 1; (2) for shear loading the buckling load increases as α increases, and for a given α the highest buckling load is obtained when γ = 2; and (3) for a combined loading of compression and shear, interactive curves can be approximated by unit circles when α = 15°, 30°, 45°, 60° and 90°. However, when α = 75° a parabola seems to be a better approximation. Based on the numerical experiments, simplified formulae and interactive curves are suggested for practical design.     

The application of critical state soil mechanics to the mechanical behaviour of porous sandstones

The mechanical properties of three cohesive sandstones of different porosities (φ) and average grain diameters (R) have been investigated. These were Tennessee sandstone (φ=0.07), Darley Dale sandstone (φ=0.12), and Penrith sandstone (φ=0.25). Unconfined uniaxial compression, constant displacement rate triaxial, and hydrostatic experiments were conducted. Yield stress data produced approximately circular envelopes that decreased in size with increasing porosity or grain size when plotted in the differential stress versus effective mean stress (Q−P) space. Normalization of these data with respect to the hydrostatic grain crushing pressure (P^*) resulted in a unique yield envelope for sandstone. Extending these data into the Q–P–φR space allows the principles of critical state soil mechanics to be applied. The critical state line for porous sandstone (the crestal line of the yield surface) appears to correspond to the transition from dilatant behaviour with localized faulting at low effective mean pressures (P/P^*<0.5), to pervasive cataclastic flow at high effective mean pressures (P/P^*>0.5). Post-yield, deformation progresses towards the critical state as observed by constant volume deformation. The critical state model developed for soil mechanics can be applied to make generalizations about the deformation of cohesive, porous sandstones. The expected behaviour of any porous sand appears to be predictable to a useful degree from a knowledge of P^*, which can be estimated from the simple parameters of porosity and mean grain size.Sensitivity to the presence of water, attributed to sub-critical crack growth, was observed in hydrostatic and uniaxial compression tests in all rock types tested. Considerable strength and elastic anisotropy was also observed.     

Effects of boundary conditions on the energy absorption of thin-walled polymer composite tubes under axial crushing

Polymer composite tubes can be designed to absorb high levels of impact energy by progressive crushing. When a tube is crushed onto a flat platen, energy is absorbed by bending failure of the plies, delamination and friction mechanisms. In the present work, significant increases in energy absorption are shown when a shear mode of failure is initiated by crushing the tube onto a radiused plug (or initiator). A study of plug radius, R, normalised with respect to the tube wall thickness, t, in the range of 0R/t5 for circular tube diameter/thickness ratios of 10<D/t<33 was undertaken with continuous filament random mat glass/polyester composite. Different radii plugs lead to significantly different deformed shapes and crush zone morphologies. Large radius initiators (R/t>2) cause the tubes to split and energy is absorbed primarily through friction and axial splitting. As the initiator radius decreases, the amount of through-thickness shear damage in the fronds increases along with specific energy absorption (SEA). When the plug radius becomes small compared to the wall thickness (R/t<0.75) a debris wedge forms between the initiator and the tube and acts like a larger radius initiator. The highest energy absorption was seen to occur at R/t1 when through-thickness shear damage was induced. In this range, under static loading conditions, SEA was seen to be higher than that for tubes crushed onto a flat platen.     

Elastic stability of bi-axially loaded rectangular plates with a single circular hole

The finite-element method has been employed to determine the elastic buckling stresses of a bi-axially loaded perforated rectangular plate with dimensions a and b in the x- and y-directions, respectively. The considered perforation is a single circular hole whose center is located along the longitudinal axis of the plate. The considered plate has simply supported edges in the out-of-plane direction and is subjected to bi-axial uniformly distributed end loads (compressive load σ_x in the x-direction, and compressive or tensile load σ_y in the y-direction). Parameters considered in the study are the plate's aspect ratio a/b, the stress ratio ξ between the applied stresses in the y- and x-directions (ξ=σ_y/σ_x), the circular hole size d and location e_x.The study shows that, in most of the considered cases, the bigger the hole size d, the lesser the plate stability and the lesser the buckling stresses. It also shows that the plate aspect ratios, a/b, between 0.6 and 1.2 should be avoided for plates with large holes and negative ξ, due to the large reduction in the buckling stresses. The hole location should also be selected to be away from the loaded edge of the plate as much as possible (better to have e_x/b>1.0) to increase the buckling stresses and improve stability. The study demonstrates also that the increase in tension in the y-direction in bi-axially loaded plate with large hole (d/b>0.4) reduces its stability. This is in contrary to the expected increase in the stability due to the increase in tension which can be seen clearly in the cases of solid plates and plates with small holes.     

Buckling of thin-walled conical shells under uniform external pressure

Shells are for the most part the deep-seated structures in manufacturing submarines, missiles, tanks and their roofs, and fluid reservoirs; therefore it is a matter of concern to bring about some basic regulations associated with the existing codes. Above all, truncated conical shells (frusta) and shallow conical caps (SCC) subjected to external uniform pressure when discharging liquids or wind loads are discussed closely in this paper concerning and thrashing out their empirical nonlinear responses along with envisaging numerical methods in contrast. The buckling aptitude of shells is contingent upon two leading geometric ratios of “slant-length to radius” (L/R) and “radius to thickness” (R/t). In this paper, developing six frusta and four shallow cap specimens and their relevant FE models, use is made of laboratory modus operandi to enumerate buckling elastic and plastic responses and asymmetric imperfection sensitivity, whose adequacy has been reckoned through comparisons with arithmetical and numerical data correspondingly. These obtained upshots were aimed at validating and generalizing the data for unstiffened truncated cones and SCC in full scale.     

Plastic buckling of complete toroidal shells of elliptical cross-section subjected to internal pressure

The plastic bifurcation buckling pressures of 60 internally-pressurised, perfect, complete toroidal shells of elliptical cross-section are given in the present paper, assuming elastic, perfectly plastic, material behaviour. The shell buckling programs employed in the computations were BOSOR 5 and INCA. Denoting the major-to-minor axis ratio by k, the numerical results show that the plastic buckling pressures are considerably lower than their elastic counterparts in the range 1.25≤k≤1.5 and are approximately equal to them for k=2.5. A limited study of the effects of non-axisymmetric initial geometric imperfections on the buckling pressures of the shells was also carried out using the INCA code. For the four cases studied the post-buckling behaviour was stable. This means that designers can use the buckling pressures given herein for perfect shells as a basis for their initial designs.     

Fracture toughness study for a brittle rock subjected to mixed mode I/II loading

A semi-disk specimen containing an angled edge crack has been used in the past for conducting fracture tests on a brittle rock named Johnstone [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212]. The test specimen is appropriate for investigating brittle fracture when the rock samples are subjected to the combined effects of tension and shear along the crack line. However, the experimental results reported in Lim, Johnston, Choi, Boland [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212.] are inconsistent with all of the well-known theoretical criteria available for predicting mixed mode brittle fracture. In this paper, a modified criterion is used to provide accurate predictions for the reported experimental results. The modified criterion makes use of a three-parameter model (based on K_I, K_II and T) for describing the crack tip stresses. It is shown that the non-singular stress term T has a significant role when the rock fracture tests are conducted on the semi-disk specimens.     

The tolerance averaging model of dynamic stability of thin plates with one-directional periodic structure

A dynamic stability of thin plates with one-directional periodic structure is investigated. An averaged non-asymptotic approach, called the tolerance averaging, is applied to derive governing equations of these plates. Obtained tolerance model equations of the Kirchhoff-type plate take into account the effect of the period lengths on the overall plate behaviour [Jędrysiak J. Dispersive models of thin periodic plates. Theory and applications. Sci. Bul. Łódź Tech. Univ., No. 872, series. Sci Trans 289, Łódź, 2001 [in Polish]]. It is shown that this effect plays a crucial role in some special cases of a dynamic stability of such plates.     

Numerical and experimental investigations of the response of aluminum cylinders with a cutout subject to axial compression

Understanding how a cutout influences the load bearing capacity and buckling behavior of a cylindrical shell is critical in the design of structural components used in automobiles, aircrafts, and marine applications. Numerical simulation and analysis of moderately thick and thin unstiffened aluminum cylindrical shells (D/t=45, 450 and L/D=2, 5, 10), having a square cutout, subjected to axial compression were systematically carried out in this paper. The investigation examined the influence of the cutout size, cutout location, and the shell aspect ratio (L/D) on the prebuckling, buckling, and postbuckling responses of the cylindrical shells.An experimental investigation on the moderately thick-walled shells was also carried out. A good correlation was observed between the results obtained from the finite element simulation and the experiments. Furthermore, empirical equations, in the form of a ‘buckling load reduction factor’ were developed using the least square regression method. These simple equations could be used to predict the buckling capacities of several specific types of cylindrical shells with a cutout.     

Behaviour of cold-formed singly symmetric columns

An experimental investigation into the behaviour of cold-formed plain and lipped channel columns compressed between fixed and pinned ends is presented in this paper. It is shown experimentally that local buckling does not induce overall bending of fixed-ended singly symmetric columns, as it does of pin-ended singly symmetric columns. Consequently, local buckling has a fundamentally different effect on the behaviour of pin-ended and fixed-ended singly symmetric columns. In order to show this fundamental different effect caused by local buckling, a series of tests was performed on plain and lipped channels brake-pressed from high strength structural steel sheets. Four different cross-section geometries were tested over a range of lengths which involved pure local buckling, distortional buckling as well as overall flexural buckling and flexural-torsional buckling. The different effects of local buckling on the behaviour of fixed-ended and pin-ended channels are investigated by comparing strengths, load–shortening and load–deflection curves, as well as longitudinal profiles of buckling deformations. The purpose of the paper is to demonstrate experimentally the different effects of local buckling on the behaviour and strengths of fixed-ended and pin-ended channels.     

Experimental bending tests of partially encased beams at elevated temperatures

This paper presents the result of an experimental research about the lateral torsional buckling instability during bending tests of Partially Encased Beams (PEB) at elevated temperature. A set of twenty seven four-point bending tests, grouped in ten series, were carried out to analyse the influence of relative slenderness, beam temperature and the shear bond conditions between concrete and steel in bending. In addition, this study compares the behaviour of PEB and bare steel beam under bending at room temperature.PEB specimens are based on IPE100 steel profiles, with two different lengths 2.4 m (medium series) and 3.9 m (large series), tested in bending using simple supporting conditions and exposed to different temperatures levels of 200 °C, 400 °C, and 600 °C.Two different shear bond conditions, between steel profile and lateral concrete, were analysed at 400 °C: one series with connectors formed by welded stirrups to the web and another series with natural adherence between steel and concrete, not welded stirrups.PEB attained lateral torsional buckling as deformed failure mode at the ultimate limit state, except for the case of PEB tested at 600 °C that results in a plastic hinge failure. The bending resistance was determined for the maximum load event (F_u) and for the displacement limit corresponding to L/30 (F_L/30) and compared with the results of the Eurocode 3 part 1–2 simple calculation method, considering an adaptation of its formulae to PEB. The expected reduction in bending resistance at elevated temperature is in good agreement with the experimental reduction factor, when the deformation criterion is used.     

Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations

The rock mass failure process is characterized by several distinct deformation stages which include crack initiation, crack propagation and coalescence. It is important to know the stress levels associated with these deformation stages for engineering design and practice.Extensive theoretical, experimental and numerical studies on the failure process of intact rocks exist. It is generally understood that crack initiation starts at 0.3 to 0.5 times the peak uniaxial compressive stress. In confined conditions, the constant-deviatoric stress criterion was found to describe the crack initiation stress level.Here, generalized crack initiation and crack damage thresholds of rock masses are proposed. The crack initiation threshold is defined by σ₁−σ₃=A σ_cm and the crack damage threshold is defined by σ₁−σ₃=B σ_cm for jointed rock masses, where A and B are material constants and σ_cm is the uniaxial compressive strength of the rock masses. For a massive rock mass without joints, σ_cm is equal to σ_cd, the long-term uniaxial strength of intact rock. After examining data from intact rocks and jointed rock masses, it was found that for massive to moderately jointed rock masses, the material constants A and B are in the range of 0.4 to 0.5, 0.8 to 0.9, respectively, and for moderately to highly jointed rock masses, A and B are in the range of 0.5 to 0.6, 0.9 to 1.0, respectively. The generalized crack initiation and crack damage thresholds, when combined with simple linear elastic stress analysis, assist in assessing the rock mass integrity in low confinement conditions, greatly reducing the effort needed to obtain the required material constants for engineering design of underground excavations.     

Influence of intermediate principal stress on rock fracturing and strength near excavation boundaries—Insight from numerical modeling

The influence of the intermediate principal stress on rock fracturing and strength near excavation boundaries is studied using a FEM/DEM combined numerical tool. A loading condition of σ₃=0 and σ₁≠0, and σ₂≠0 exists at the tunnel boundary, where σ₁, σ₂, and σ₃, are the maximum, intermediate, and minimum principal stress components, respectively. The numerical study is based on sample loading testing that follows this type of boundary stress condition. It is seen from the simulation results that the generation of tunnel surface parallel fractures and microcracks is attributed to material heterogeneity and the existence of relatively high intermediate principal stress (σ₂), as well as zero to low minimum principal stress (σ₃) confinement. A high intermediate principal stress confines the rock in such a way that microcracks and fractures can only be developed in the direction parallel to σ₁ and σ₂. Stress-induced fracturing and microcracking in this fashion can lead to onion-skin fractures, spalling, and slabbing in shallow ground near the opening and surface parallel microcracks further away from the opening, leading to anisotropic behavior of the rock. Hence, consideration of the effect of the intermediate principal stress on rock behavior should focus on the stress-induced anisotropic strength and deformation behavior of the rocks. It is also found that the intermediate principal stress has limited influence on the peak strength of the rock near the excavation boundary.     

Experimental investigations on mechanical properties and column buckling behavior of structural bamboo

Bamboo, a fast‐growing natural material with a high strength‐to‐weight ratio, provides an environmentally friendly alternative for building construction. This paper presents experimental studies on natural bamboo and glued laminated bamboo (GLB) products in order to provide fundamental knowledge for the structural design of bamboo buildings. Tensile coupon tests and compressive tests were conducted on small clear specimens, and bending tests were conducted to evaluate the bending strength and modulus of elasticity for GLB beams. It was found that tensile strength in the transverse direction was very low. Therefore, in practical applications, it is recommended that the transverse loading in GLB should be addressed very carefully. Compressive axial tests were conducted to investigate the buckling behavior of natural Julong bamboo (Dendrocalamus giganteus). A sample design procedure against buckling was proposed on the basis of the test results. The mechanical properties and buckling analysis presented in this paper may provide guidance for more structural applications of bamboo in building constructions with enhanced structural performance. Copyright © 2014 John Wiley & Sons, Ltd.     

Static and dynamic analysis of thin plates in bending— A novel finite element model

The static and dynamic behavior of thin flat plates in bending have been studied by means of a recently developed¹ doubly curved triangular shell element. The element's formulation is based on a modified mixed variational principle, wherein the primal variable σ (vector of shell stress resultants) and (boundary displacement vector) are required to satisfy a priori: (1) the complete shallow shell equilibrium equations, and (2) interelement C¹ displacement continuity. Several well-selected plate structures have been analyzed and the numerical results obtained indicate that the new element scheme competes most favorably with recently developed as well as with well-established elements included in commercial general-purpose finite element codes.     

Coefficient of restitution and rotational motions of rockfall impacts

This paper presents experimentally obtained results for the coefficient of restitution for spherical boulders impacting on rock slopes. Plaster modeling material is used for casting both the boulders and slopes. It is observed that the normal component of the coefficient of restitution (R_n) increases with the slope angle α, which agrees with Wu's observations (Trans. Res. Rec. 1–5 (1985) 1031). However, there appears to be no clear correlation between the tangential component of the coefficient of restitution (R_t) and the slope angle α. When the ratio of the resultant velocities and the ratio of the kinetic energies before and after impacts are used to define the coefficient of restitution (i.e. R_V and R_E), a very clear increasing trend in the coefficient of restitution with α is observed. When all data are plotted onto the R_t−R_n space, our laboratory data fall into the rock slope regime proposed by Fornaro et al. (In: D.G. Price (Ed.), Proceedings of the Sixth International Congress IAEG, Amsterdam, Balkema, Rotterdam, 1990, p. 2173) and also agree with those data gleaned from literature. In addition, the rotational kinetic energy E_r, induced at each impact, increases with the slope angle α, achieves a maximum at about α=40°, before decreasing again to a negligible value at α=70°. A simple theoretical model is proposed to explain this observation based on the locking between the boulder and the slope during impact. The α-dependence of E_r differs from the recommendation by the Japanese Railway Association that the induced rotational energy is about 10% of that of the translational kinetic energy.     

Elastic-plastic collapse of non-uniform cylindrical shells subjected to uniform external pressure

The objective of this paper is to derive analytical solutions for the elastic buckling and plastic collapse pressures of a cylindrical shell with reduced thickness over part of its circumference. The section of reduced thickness is used to represent a corroded region in a pipe. The proposed solutions are extensions of Timoshenko's solutions for the elastic-plastic collapse of a linear elastic, perfectly plastic cylindrical shell subjected to uniform external pressure. A modified interaction formula for the fully plastic membrane forces and bending moments in the non-uniform cylinder has been proposed for plastic collapse. A parametric study shows that the elastic buckling pressure decreases smoothly with corrosion angle when the corrosion depth is less than 0.5t. When the corrosion depth is greater than 0.5t, the elastic buckling strength first decreases very rapidly with corrosion angle. Furthermore, the elastic buckling pressure decreases uniformly with corrosion depth when the corrosion angles are greater than 30°, while the elastic buckling strength decreases more rapidly at higher corrosion depths when corrosion angles are less than 30°. Another parametric study on a steel pipe shows that the initial and fully plastic yield pressures both decrease monotonically with corrosion depth for a given corrosion angle and imperfection.     

Numerical and experimental investigations on buckling of steel cylindrical shells with elliptical cutout subject to axial compression

The effect of cutouts on load-bearing capacity and buckling behavior of cylindrical shells is an essential consideration in their design.In this paper, simulation and analysis of thin steel cylindrical shells of various lengths and diameters with elliptical cutouts have been studied using the finite element method and the effect of cutout position and the length-to-diameter (L/D) and diameter-to-thickness (D/t) ratios on the buckling and post-buckling behavior of cylindrical shells has been investigated. For several specimens, buckling test was performed using an INSTRON 8802 servo hydraulic machine and the results of experimental tests were compared to numerical results. A very good correlation was observed between numerical simulation and experimental results. Finally, based on the experimental and numerical results, formulas are presented for finding the buckling load of these structures.     

Experimental investigations on combined resistance of corrugated sheets with strengthened cross-sections under bending and concentrated load

Multi-span corrugated sheets have lower resistance from single-span beams. This effect results from the combined bending moment and bearing load acting at the internal supports. It is a possibility to increase the resistance by means of local strengthening of critical cross-sections over the internal supports. The strengthening can be achieved by doubling of cross-sections (lap connection, applying an additional corrugated sheet). At present there is a lack of knowledge about the resistance of doubled cross-sections of corrugated sheets under concentrated load, bending moment or combined concentrated load and bending moment acting at the strengthened cross-section.The results of experimental investigations on combined M–F relationship for corrugated sheets with doubled cross-section have been presented. For 35 tested models the following parameters have been changed: the length of strengthening a, span distance l, the way of a profile placement, the structure of strengthening, the number and placement of connectors. Obtained results have been compared with those received experimentally for unstrengthened models [Biegus A, Czepiżak D. Research on the interactive resistance of corrugated sheets under combined bending and contact pressure, Thin-Walled Struct 2006;44:825–831]. The conclusions leading to the optimal parameters of strengthening of corrugated sheets have been given.