Buckling of shallow torispherical domes subjected to external pressure — a comparison of experiment, theory, and design codes

1987 ◽  
Vol 22 (3) ◽  
pp. 163-175 ◽  
Author(s):  
G D Galletly ◽  
J Kruzelecki ◽  
D G Moffat ◽  
B Warrington
Keyword(s):  
External Pressure ◽  
Test Results ◽  
Spherical Cap ◽  
Initial Shape ◽  
Geometric Imperfections ◽  
Shell Buckling ◽  
Theoretical Predictions ◽  
Design Values ◽  
Initial Geometric Imperfections ◽  
Thickness Measurements

The test results obtained on 24 externally-pressurised torispherical steel shells are given in this paper. The knuckle radius-to-diameter ratio of the domes varied from 0.06 to 0.18 and the spherical cap radius-to-thickness ratios were between 75 and 335. Initial shape and thickness measurements were carried out on all the torispheres and a summary of this information is given. The BOSOR 5 shell buckling program was employed to predict the buckling/collapse pressures of all the domes; both perfect domes and those with axisymmetric imperfections were considered. The correlation between the theoretical predictions and the experimental results was, in general, very good. The main conclusions of the present investigation are: (i) that some of the experimental buckling pressures were lower than those obtained by multiplying the BS 5500 design values by a safety factor of 1.5, and (ii) that those torispheres with sharp knuckle radii failed by plastic collapse in the knuckle region and the collapse pressures were not very sensitive to initial geometric imperfections. It thus appears that the BS 5500 rules relating to the strength of shallow torispheres subjected to external pressure need to be amended, and that the tolerances on geometric shape for cases which are likely to be imperfection-insensitive should be reconsidered.

Clamped torispherical shells under external pressure — some new results

1988 ◽  
Vol 23 (1) ◽  
pp. 9-24 ◽  
Author(s):  
J Blachut ◽  
G D Galletly
Keyword(s):  
Failure Mode ◽  
External Pressure ◽  
Spherical Cap ◽  
Geometric Imperfections ◽  
Collapse Pressure ◽  
Modes Of Failure ◽  
Shell Buckling ◽  
Bifurcation Buckling ◽  
Initial Geometric Imperfections ◽  
The One

Perfect clamped torispherical shells subjected to external pressure are analysed in the paper using the BOSOR 5 shell buckling program. Various values of the knuckle radius-to-diameter ratio ( r/D) and the spherical cap radius-to-thickness ratio ( Rs/ t) were studied, as well as four values of σyp, the yield point of the material. Buckling/collapse pressures, modes of failure and the development of plastic zones in the shell wall were determined. A simple diagram is presented which enables the failure mode in these shells to be predicted. The collapse pressures, pc, were also plotted against the parameter Λs (√( pyp/ pcr)). When the controlling failure mode was axisymmetric yielding in the knuckle, the collapse pressure curves depended on the value of σyp, which is unusual. However, when the controlling failure mode was bifurcation buckling (at the crown/knuckle junction), the collapse pressure curves for the various values of σyp all merged, i.e., they were independent of σyp. This latter situation is the one which normally occurs with the buckling of cylindrical and hemispherical shells. A limited investigation was also made into the effects of axisymmetric initial geometric imperfections on the strength of externally-pressurised torispherical shells. When the failure mode was axisymmetric yielding in the knuckle, initial imperfections of moderate size did not affect the collapse pressures. In the cases where bifurcation buckling at the crown/knuckle junction occurred, small initial geometric imperfections at the apex did not affect the buckling pressure, but axisymmetric imperfections at the buckle location did influence it. With the other failure mode (i.e., axisymmetric yielding collapse at the crown of the shell), initial geometric imperfections caused a reduction in the torisphere's strength.

Buckling Design of Imperfect Welded Hemispherical Shells Subjected to External Pressure

1991 ◽  
Vol 205 (3) ◽  
pp. 175-188 ◽  
Author(s):  
G D Galletly ◽  
J Blachut
Keyword(s):  
Experimental Information ◽  
External Pressure ◽  
Design Stage ◽  
Arc Length ◽  
Spherical Shells ◽  
Geometric Imperfections ◽  
Collapse Pressure ◽  
Buckling Resistance ◽  
Initial Geometric Imperfections ◽  
Hemispherical Shells

Welded hemispherical or spherical shells in practice have initial geometric imperfections in them that are random in nature. These imperfections determine the buckling resistance of a shell to external pressure but their magnitudes will not be known until after the shell has been built. If suitable simplified, but realistic, imperfection shapes can be found, then a reasonably accurate theoretical prediction of a spherical shell's buckling/collapse pressure should be possible at the design stage. The main aim of the present paper is to show that the test results obtained at the David Taylor Model Basin (DTMB) on 28 welded hemispherical shells (having diameters of 0.75 and 1.68 m) can be predicted quite well using such simplified shape imperfections. This was done in two ways. In the first, equations for determining the theoretical collapse pressures of externally pressurized imperfect spherical shells were utilized. The only imperfection parameter used in these equations is δ0, the amplitude of the inward radial deviation of the pole of the shell. Two values for δ0 were studied but the best overall agreement between test and theory was found using δ0 = 0.05 ✓ (Rt). This produced ratios of experimental to numerical collapse pressures in the range 0.98–1.30 (in most cases the test result was the higher). The second approach also used simplified imperfection shapes, but in conjunction with the shell buckling program BOSOR 5. The arc length of the imperfection was taken as simp = k ✓ (Rt) (with k = 3.0 or 3.5) and its amplitude as δ0 = 0.05√(Rt). Using this procedure on the 28 DTMB shells gave satisfactory agreement between the experimental and the computer predictions (in the range 0.92–1.20). These results are very encouraging. The foregoing method is, however, only a first step in the computerized buckling design of welded spherical shells and it needs to be checked against spherical shells having other values of R/t. In addition, more experimental information on the initial geometric imperfections in welded spherical shells (and how they vary with R/t) is desirable. A comparison is also given in the paper of the collapse pressures of spherical shells, as obtained from codes, with those predicted by computer analyses when the maximum shape deviations allowed by the codes are employed in the computer programs. The computed collapse pressures are frequently higher than the values given by the buckling strength curves in the codes. On the other hand, some amplitudes of imperfections studied in the paper give acceptable results. It would be helpful to designers if agreement could be reached on an imperfection shape (amplitude and arc length) that was generally acceptable. Residual stresses are not considered in this paper. They might be expected to decrease a spherical shell's buckling resistance to external pressure. However, experimentally, this does not always happen.

Elastic buckling of imperfect circular toroidal shells under external pressure

1998 ◽  
Vol 212 (3) ◽  
pp. 197-209 ◽  
Author(s):  
G D Galletly
Keyword(s):  
External Pressure ◽  
Mode Shapes ◽  
Buckling Mode ◽  
Geometric Imperfections ◽  
Buckling Modes ◽  
The North ◽  
Buckling Resistance ◽  
Maximum Decrease ◽  
Initial Geometric Imperfections ◽  
Toroidal Shells

When perfect, externally pressurized complete circular toroidal shells buckle, the minimum buckling pressure pcr usually occurs in the axisymmetric n = 0 mode, with pcr for n = 2 being only slightly larger. In the present paper, the effects of axisymmetric initial geometric imperfections on reducing pcr for the perfect shell are investigated. Various types of imperfection are studied, i.e. localized flat spots, smooth dimples, sinusoids and buckling mode shapes. The principal geometry investigated was R/b = 10, b/t = 100, although other geometries were also considered. The maximum decrease in buckling resistance, Δ pcr, was found to be about 16 per cent at δ 0/t = 1 and it occurred with smooth dimples at the north (φ = 180°) and south (φ=0°) poles. This value of Δ pcr is not large. Circular toroidal shells thus do not appear to be very sensitive to axisymmetric initial geometric imperfections. The reductions in the buckling pressure of the above shell, arising because of initial imperfections having the shape of the n = 0 and the n = 2 buckling modes, were 12 and 9 per cent respectively for wo/t = 1. These decreases in the buckling resistance are smaller than that for the ‘two smooth dimple’ case mentioned above.

Comparison of Numerical Results With Experiments on the Ultimate Strength of Long Stiffened Panels

2011 ◽  
Author(s):  
Mingcai Xu ◽  
C. Guedes Soares
Keyword(s):  
Numerical Analysis ◽  
Test Results ◽  
Buckling Mode ◽  
Geometric Imperfections ◽  
Geometric Nonlinearities ◽  
Stiffened Panels ◽  
Initial Imperfections ◽  
Linear Buckling ◽  
Initial Geometric Imperfections ◽  
Collapse Behavior

The behavior of long stiffened panels are simulated numerically and compared with test results of axial compression until collapse, to investigate the influence of the stiffener’s geometry. The material and geometric nonlinearities are considered in the simulation. The initial geometric imperfections, which affect the collapse behavior of stiffened panels, are also analyzed. The initial imperfections are assumed to have the shape of the linear buckling mode. Four types of stiffeners are made of mild or high tensile steel for bar stiffeners and mild steel for ‘L’ and ‘U’ stiffeners. To produce adequate boundary conditions at the loaded edges, three bays stiffened panels were used in the tests and in the numerical analysis.

On the Collapse of Inelastic Thick-Walled Tubes Under External Pressure

1986 ◽  
Vol 108 (1) ◽  
pp. 35-47 ◽  
Author(s):  
M. K. Yeh ◽  
S. Kyriakides
Keyword(s):  
External Pressure ◽  
Thickness Variation ◽  
Nonlinear Formulation ◽  
Geometric Imperfections ◽  
Collapse Pressure ◽  
Tube Cross Section ◽  
Wall Thickness Variation ◽  
Initial Geometric Imperfections ◽  
T Values ◽  
Good Agreement

The collapse of long thick-walled tubes under external pressure is studied both experimentally and analytically. A two-dimensional nonlinear formulation of the problem is presented. The formulation is general enough to include initial geometric imperfections of the tube cross section such as initial ovality and wall thickness variation. In addition the effects of residual stresses and of initial inelastic anisotropy are considered. Experiments on tubes with D/t values between 10 and 40 were carried out. Good agreement between experiments and theory is shown to occur provided all parameters are modeled correctly. A study of the effect of the various parameters of the problem on the collapse pressure is also presented.

ELASTIC BUCKLING OF EXTERNALLY-PRESSURIZED IMPERFECT TOROIDAL SHELLS

2001 ◽  
Vol 01 (01) ◽  
pp. 31-45 ◽  
Author(s):  
GERARD D. GALLETLY
Keyword(s):  
Initial Imperfection ◽  
External Pressure ◽  
Elastic Buckling ◽  
Geometric Imperfections ◽  
Buckling Modes ◽  
Initial Imperfections ◽  
Numerical Studies ◽  
Initial Geometric Imperfections ◽  
Sinusoidal Buckling ◽  
Toroidal Shells

This paper summarizes the results of numerical studies into the effects of initial geometric imperfections on the elastic buckling behaviour of steel circular and elliptic toroidal shells subjected to follower-type external pressure. The types of initial imperfection studied are (a) axisymmetric localized ones and (b) sinusoidal buckling modes. The principal localized imperfections studied are (i) circular increased-radius "flat spots" and (ii) smooth dimples. The buckling pressures pcr of circular toroidal shells were not very sensitive to initial imperfections. With elliptic toroids, whether the shell was sensitive to initial imperfections or not depended on the ratio k(≡ a/b) of major to minor radii of the section. The shells on the ascending part of the pcr versus k curve behaved like circular toroidal shells, i.e. they were not sensitive to initial imperfections. However, the behaviour of elliptic toroids on the descending part of the versus k curve was very different. The numerical results quoted in the paper are for limited ranges of the geometric parameters. It would be useful to extend these ranges, to explore the effects of plasticity and to conduct model tests on imperfect steel models to verify the conclusions of the numerical studies.

Calculation of Working Pressure for Cylindrical Vessel Under External Pressure

2010 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Monte Carlo ◽  
Cylindrical Shell ◽  
Pressure Vessel ◽  
External Pressure ◽  
Working Pressure ◽  
Geometric Imperfections ◽  
The Monte Carlo Method ◽  
Cylindrical Pressure Vessel ◽  
Section Viii ◽  
Initial Geometric Imperfections

Initial geometric imperfections have a significant effect on the load carrying capacity of asymmetrical cylindrical pressure vessels. This research paper presents a comparison of a reliability technique that employs a Fourier series representation of random asymmetric imperfections in a defined cylindrical pressure vessel subjected to external pressure. Evaluations as prescribed by the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 rules are also presented and discussed in light of the proposed reliability technique presented herein. The ultimate goal of the reliability type technique is to statistically predict the buckling load associated with the cylindrical pressure vessel within a defined confidence interval. The example cylindrical shell considered in this study is a fractionating tower for which calculations have been performed in accordance with the ASME B&PV Code. The maximum allowable external working pressure of this tower for the shell thickness of 0.3125 in. is calculated to be 15.1 psi when utilizing the prescribed ASME B&PV Code, Section VIII, Division 1 methods contained within example L-3.1. The Monte Carlo method as developed by the current authors and published in the literature is then used to calculate the maximum allowable external working pressure. Fifty simulated shells of geometry similar to the example tower are generated by the Monte Carlo method to calculate the nondeterministic buckling load. The representation of initial geometric imperfections in the cylindrical pressure vessel requires the determination of appropriate Fourier coefficients. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion that appears in the form of a double Fourier series. Multi-mode analyses are expanded to evaluate a large number of potential buckling modes for both predefined geometries and the associated asymmetric imperfections as a function of position within a given cylindrical shell. The method and results described herein are in stark contrast to the dated “knockdown factor” approach currently utilized in ASME B&PV Code.

Behaviour of cold-formed steel built-up battened columns composed of four lipped angles: Tests and numerical validation

2019 ◽  
Vol 23 (1) ◽  
pp. 51-64 ◽  
Author(s):  
M Anbarasu
Keyword(s):  
Cross Section ◽  
Finite Element Models ◽  
Test Results ◽  
Design Standards ◽  
Geometric Imperfections ◽  
The North ◽  
Compression Behaviour ◽  
Cold Formed Steel ◽  
Initial Geometric Imperfections ◽  
Box Columns

This article mainly investigates the behaviour and strength of built-up battened box column composed of lipped angles under axial compression. Ten specimens were fabricated and tested under pinned with warping-restrained end condition including two different cross-section dimensions of columns with five different geometric lengths. Three material tensile coupon tests were conducted to obtain the material properties of the steel used for fabricating the test specimens. The overall initial geometric imperfections were measured. The plate slenderness, member slenderness, chord slenderness and slenderness of batten plates may affect the compression behaviour of cold-formed steel built-up battened box columns and were accordingly investigated. It was found that the chord slenderness significantly affects the compressive strength of the built-up columns. Test results, including the compression resistances, the load versus displacement responses and the deformed shapes were presented. The test strengths were compared with the design strengths predicted using the North American Specifications (AISI-S100:2016), EuroCode (EN1993-1-3:2006) and design equations proposed by EI Aghoury et al. The design strengths predictions by these two design standards were unconservative, with EI Aghoury et al.’s standard performing better. Finite-element models were developed and verified against the test results.

scholarly journals Nonlinear dynamics of spherical shells buckling under step pressure

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180884 ◽  
Author(s):  
Jan Sieber ◽  
John W. Hutchinson ◽  
J. Michael T. Thompson
Keyword(s):  
Nonlinear Dynamics ◽  
External Pressure ◽  
Dynamic Buckling ◽  
Spherically Symmetric ◽  
Imperfection Sensitivity ◽  
Spherical Shells ◽  
Buckling Mode ◽  
Geometric Imperfections ◽  
Pressure Threshold ◽  
Initial Geometric Imperfections

Dynamic buckling is addressed for complete elastic spherical shells subject to a rapidly applied step in external pressure. Insights from the perspective of nonlinear dynamics reveal essential mathematical features of the buckling phenomena. To capture the strong buckling imperfection-sensitivity, initial geometric imperfections in the form of an axisymmetric dimple at each pole are introduced. Dynamic buckling under the step pressure is related to the quasi-static buckling pressure. Both loadings produce catastrophic collapse of the shell for conditions in which the pressure is prescribed. Damping plays an important role in dynamic buckling because of the time-dependent nonlinear interaction among modes, particularly the interaction between the spherically symmetric ‘breathing’ mode and the buckling mode. In general, there is not a unique step pressure threshold separating responses associated with buckling from those that do not buckle. Instead, there exists a cascade of buckling thresholds, dependent on the damping and level of imperfection, separating pressures for which buckling occurs from those for which it does not occur. For shells with small and moderately small imperfections, the dynamic step buckling pressure can be substantially below the quasi-static buckling pressure.

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