LOCAL BUCKLING AND POSTBUCKLING BEHAVIOUR OF THINWALLED SHELLS
(TEST DATA & THEORY)
G.D. Gavrylenko
S.P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev,
Nesterova str. 3, 03057 Kiev, Ukraine
[email protected]
ABSTRACT
An investigation of stability cylindrical shells with initial local dents by experimental and theoretical ways was undertaken.
Subsequent progress in the theory of stability in past years took two directions. The first was determination of the critical loads
and evaluation of the load-carrying capacity of structures (including shells) within the framework of Euler’s concepts, with
allowance for the main factors affecting their values. The second direction involved analysis of the behaviour of structures in
the supercritical region, i.e., in the presence of large deflections and strains. Despite of successes of theory the full
coincidence of the theoretical data and experimental ones is absent. Here an approach for analysis of the stability and loadcarrying capacity of imperfect smooth shells is developed. This method is based on the finite-difference method. The emphasis
in on shells with local dents. The numerical results are successively corrected and compared with available experimental data
for shells with a single dent. The method enables us to discover new features in the behaviour of thin-walled structures under
loading : development of precritical state, change in the dent shape, and exhaustion of load-carrying capacity. The lower local
critical loads and upper stresses also are determined. They correspond to general buckling and agree well with available
experimental data.
Introduction
Stability of smooth elastic nonideal cylindrical shells was considered in many articles and monographs [1 – 7]. A procedure for
determination of critical loads on nonideal shells is proposed with allowance for the technique of stability analysis of ribbed
shells in an inhomogeneous stress-strain state developed earlier [4]. Different forms of regular and local initial imperfections
are analyzed, and the dependences of dimensionless parameters of critical loads on the dimensionless amplitude of
imperfections are constructed. At the Institute of Mechanics in Kiev, new algorithms have been developed based upon a finite
difference discretisation of the nonlinear differential equations. The behaviour of shells in the subcritical state is described by
nonlinear differential equations of Mushtari-Donnell-Vlasov type, presented in [4, 8, 9]. The approach proposed to finding the
buckling load has been based on the generalized Euler criterion. The new data presented here are compared with the wellknown theoretical results and test data.
Experimental Procedure and Test Specimens
The experimental results fully belong to Prof. Krasovskii [10] Was investigated the behaviour of circular cylindrical shells at
all stages of deformation in the area of artificial initial local dents that trigger the process of the loss of stability. Other
perturbing factors (initial stresses, nonuniformity of loading, variation of the shell thickness, external actions) were
minimized, wherever possible. This allowed us to obtain, along with qualitative results, sufficiently reliable quantitative
estimates of the phenomena and effects studied.
All specimens were manufactured from the same material (Khl8N9n steel sheet) using the same technique (contact spot
welding with one longitudinal weld) and tested with the fulfillment of unified requirements to the observations and
measurements made. The specimen had the inside diameter 2r = 143 mm, relative length of the working portion l/r = 2,
and three radius-to-thickness ratios r/t = 150, 260, and 360. The weld width was 1.5% of the shell perimeter. Welding was
conducted on a special mandrel with the accuracy of the inside diameter being not higher than 1.0t. A double-row weld
inspection provided high reliability of the weld. Since the load transfer to the shell was realized along the plane, the shell
ends were treated carefully using a lapping plate. The variation of the end surfaces in plane did not exceed 0.05t.
The material used had a high offset yield stress (σ 0.2 ≈ 800 MPa, Young's modulus E = 191 GPa, and Poisson's ratio
ν = 0.32), which restricted plastic effects not only at the stages of subcritical loading of the shells but also at the stage of
initial postcritical deformation (with local postcritical configurations). The manufacturing technique and pre-testing
procedures used ensured a high quality of shells, which was defined by the level of initial imperfections, i.e., by their depth
and zone of extension, and was estimated in reference to the critical loads of ideal shells.
Prior to testing, the shells were dented with spherical segments to produce a local dent of certain dimensions, whose depth
was controlled by the pressure force. The dent was located at the midlength of the shell generatrix, in the longitudinal section
diametrically opposite to the weld. Prior to denting, the shell was put on a special elastic drum. In the course of testing, the
shell surface was carefully measured in the region of the artificial imperfections.
One of peculiar features of these experimental investigations was studying the shell behaviour at all stages of its
deformation: subcritical deformation, in the process of the loss of stability, and postcritical deformation, including complete
collapse. In this case, a unified procedure was used for loading the shell by an axial compressive force via centering spheres
and special flat testing devices. The compressive force N was generated by the press of an UMM-50 multipurpose
testing machine of hydraulic type. The load from the press was transferred to the shell ends via the testing devices that
provided the invariance of the boundary conditions, which were close to the conditions of the hinged support. The
displacement of the shell ends toward the curvature center was restricted by guiding disks, which maintained the circular
shape of the shell in the course of testing. Aligning spheres made of steel were placed between the plates of the machine
press and the testing devices.
Prior to the beginning of the tests, the specimen quality control was carried out. The shape of the shells was measured by a
contact method using an indicating gauge with a graduation mark of 0.01 mm. The indicating gauge was fastened on a
slide, which moved along a steel rod mounted parallel to the shell axis and connected rigidly to the lower plate of the press.
The indicating gauge plunger was located normally to the shell surface. To avoid a possible initiation of buckling, the spring
pressing the plunger to the shell was slackened. The measurements were made in both longitudinal and transverse
directions. In measurements of the generatrix shape, the slide with the indicating gauge moved along the rod. In the
process of measurements around the periphery and in passing to measurements of generatrices of other sections, the
indicating gauge remained in position, whereas the shell turned on the aligning spheres.
Initial imperfections were measured at a load of approximately 0.1% of the critical one. Subsequent measurements were
made step-by-step at various load levels. The loads, at which the last measurement was made, were as high as 80 to
90% of the critical one. Based on the discrete measurements made with different pitches depending on the size and
shape of the initial imperfections, the curves of the initial deflections were plotted, and the subcritical deformation history was
traced. In the course of loading, N vs ∆l (the specimen shortening) and N vs w (radial displacements) diagrams were
recorded, critical loads (Ncr) were registered, and strain measurements of the shell surface were made.
Main Results of the Experimental Investigation
The loss of stability of the analogous specimens without artificial imperfections occurred suddenly as a sharp snap. As this
took place, two or three zones of specific diamond-shaped dents were formed that covered approximately 75% of the shell
surface in the circumferential direction. The number of dents in this direction n (considering wave formation over the entire
surface) was practically independent of the shell geometry and varied within rather narrow limits (n = 9-10). The mean
values of the relative critical loads in axial compression P = Ncr / Ncl (where N cr and N cl are the respective actual and
classical values of the critical load in a smooth shell) for specimens with r/t = 150, 260, and 360 were 0.83, 0.72, and 0.60,
respectively. Evidently, a decrease in
shells.
P
with increasing r/t is due to a corresponding reduction in the quality of the
In the studies of the influence of the localized initial geometrical imperfections (artificial dents) on the stability of shells,
the following results were obtained. It was found that even a single initial dent affects strongly the magnitude of the critical
load. Its defining parameters are the depth, the extent along the directrix, and the length of the highly curved portion of the
generatrix. The changes in the dent extent in the longitudinal direction and in the curvature in the circumferential
direction as well as the site of the dent location on the shell (at a distance of 2 rt from the ends) have an insignificant
effect on the magnitude of N cr .
For the above-mentioned geometry of the shells with artificial local dents that are identical in shape, the dependences of
the critical loads on the relative depth of a dent ( w0
= w0 / t ),
which is the most important characteristic of the
latter, were obtained. Experiments with small shallow dents were found to be representative. The dimensions of those dents
in the longitudinal and circumferential directions were almost similar and did not exceed e magnitude of a subcritical dent.
The shell curvature in the dent area varied insignificantly, which is characteristic of the natural initial deflection of actual
shells. Figure 1 presents the relationships P = f(|w 0 |/t) for specimens of various thickness with small shallow dents,
which describe the peculiarities of the behavior and buckling of shells with local imperfections. Here are data for
cylindrical shells with single shallow dents and three radius-to-thickness ratios r / t = 150, 260, and 360. In Figure 1 are
used the following symbol for test data: (○, ●) r / t = 150; (∆, ▲) r / t = 260;
solid symbols correspond to the overall buckling and open ones to the local buckling.
(◊, ♦) r / t = 360.
In this figure,
The total buckling occurred in the same way as in the case where artificial imperfections were absent, namely, as an
instantaneous change from the initial nondeformed shape to a postcritical configuration with two or three zones of typical
diamond-shaped dents. In the case of local buckling, a local postcritical dent formed at the site of the initial dent at the
same load. The character of transformation of the initial dent into the postcritical one was determined by the quantity |w0|/t.
For small values of |w0|/t, the transition to the local configuration occurred as a snap, and for high |w0|/t values, this was
a gradual increase in the extent of the initial dent. The shell with a local postcritical dent took up the increasing
longitudinal load. As this took place, all the dimensions of such a dent increased. The process of the dent development
resulted in the above-mentioned overall buckling at the upper local critical load. Note that the upper local critical load (the
upper bound of the occurrence of the postcritical configuration with a single local dent) is a rather stable quantity, which is close
to 0.5 Ncl for a wide range of variation of the shell parameters (150 ≤ r/t ≤ 700, 2.0 ≤ l/r ≤ 8.0) [10].
Analysis of the relationships given in Figure 1 revealed that there are three regions of the parameter w 0 variation within
which the effect of the geometry perturbation on the behavior and load-carrying capacity of shells differs considerably: 1) a
P
0,8
0,6
3
2
0,4
4
1
0,2
0
0,5
1
1,5
2
2,5
3
Figure 1. Experimental & theoretical relations
3,5
w0 / t
P = f ( w0 / t )
region where the artificial dent does not affect the Ncr value since the cause of buckling is the dents from natural deflection
w0 ); 2) a region where the artificial dent results in a continuous process of the overall loss of stability
(medium values of w0 , which, being increased, lead to a drastic decrease in Ncr); and 3) a region where the overall
buckling is preceded by one stable local postcritical dent (high values of w0 that do not affect the load-carrying capacity of
(small values of
the shell since it is governed by the upper local critical load).
It is obvious that the dimensions of the first region depend on the level of the initial imperfections and increase with the
extension of the latter. The boundary between the second and third regions depends on the dent shape. For shallow
dents under consideration, it is approximately equal to
w0
= 0.8. In terms of the value of the relative load, the upper
local critical loads are the boundary between these very important regions. From below, the third region is limited by the
lower local critical loads (the lower boundary of the occurrence of the postcritical configuration with one local dent), whose
magnitude amounts to (0.35–0.40)Ncl for high-quality shells with the following geometry: 150 ≤ r/t ≤ 700 and 2.0 ≤ l/r ≤ 8.0.
However, considering that a large initial dent was located in the area of the postcritical dent formation, the magnitude of
the lower local critical load was reduced to 0.3N c!.
It should be noted that as the critical load decreased, the dimensions of the postcritical dents responsible for the overall
loss of stability increased appreciably in the second region of the parameter
w0
variation. Thus, at the Ncr value
corresponding to the upper local critical loads, this increase reached 20 – 25%. In addition, the dimensions of the
postcritical dent in the region of the initial forced perturbation of the configuration were always from 15 to 20% higher than
those of the remaining dents. This testifies that after the loss of the shell stability, the region, wherein the process of overall
buckling was initiated, can be determined with reasonable probability.
The difference between the experimental and calculated data for r/t =360 was primarily caused by the way the dent shape
was described, which, not resulting in great differences, permits, however, determining the critical load with a safety margin at
higher amplitudes of dents. Note that the attempt to use the Koiter approach [7] for assessing the critical load by considering
a single dent as an axisymmetrical regular deflection results in a considerable underestimation of the load-carrying capacity
of an imperfect shell (curve 1 in Figure 1). At the same time, in the second region of | w0 | / t variation, a rather good
agreement with the experimental data is observed when the calculation is made by the formula derived as a result of the
asymptotic analysis of the nonlinear functional of the total potential energy of an elastic cylinder with a localized dent [11].
Curve 2 in Figure 1 corresponds to this calculation. However, it is not valid for estimating the upper local critical loads
for | w0 |/t >l.
Theoretical Predictions of Buckling Loads & Comparison with the Test Observations
To determine p1min of a shell having initial deflections, it is necessary to analyze the critical loads for various number working
regions. Here, p1min is the minimum value of the buckling load p1 = pcr / pcl, where pcr is the buckling load of the imperfect shell,
pcl = 0,605EFt / r, and F is the cross-sectional area of the casing. A working region is l /2 or l /4 of the shell development with
conditions of symmetry, antisymmetry, or their combination specified at the edges. To improve the result, any number of
regions that is a multiple of the number of stringer or one or two neighboring panels are considered. The accuracy of
determination of the critical loads is estimated by making the mesh finer for a fixed imperfection localization zone.
Based upon a Donnell-type nonlinear theory of shells, theoretical modelling of this problem is realized by using a finite
difference method (FDM) [4, 12]. With just 2 or 3 unknowns at each node point, it is suggested that this FDM has considerable
practical advantage as compared with the usual 5 degrees of freedom at each node for the finite element method. This
modelling allows incorporation of the effects of nonuniform pre and post-buckling nonlinearities, including the effects of
arbitrary initial geometric imperfections, and has been extended to handle isotropic, as well as orthotropic continuous and
discrete rib-reinforced shells.
The test data [10] was fulfilled earlier irrespective of presented here theoretical results. The initial imperfections of form will be
created by pressing in surface of shell on special rolling press of spherical or cylindrical segments of different sizes. Tests on
nominally identical cylindrical shell specimens, containing at mid-length single local dents of varying amplitude, illustrate the
severe sensitivity of axial buckling loads to both the amplitude and shape of the localized dents.
Figure 1 shows typical comparisons between test observations and predictions from the FDM. Buckling loads are normalized
with respect to the classical axial critical load pc1 for the perfect cylinder, and are shown as functions of the amplitude of the
local dent imperfections, wo, normalized in terms of the shell thickness, t.
Theoretical local dents in centre of shell surface are taken as a single lobe of the form
fo = −
wo
nπy
πx
sin
,
∑ sin
t
l 1 n =1,3
l2
(1)
with wavelengths ( l 1 , l 2 ,) chosen to model as closely as possible the local dents of the test specimen. The values of buckling
load will be estimated with using of
f1 = −
wo
nπy
πx
sin
∑ sin
l 1 n =1,3,5
l2
t
.
(2)
A row of test shells with w0 / t = 0,817 and 1,95 had both as local buckling load determining the beginning of process loss
stability so general buckling load, when load – carrying capacity may be lost in full.
Received theoretical results show a good qualitative and quantitative accordance with test data. In diapasons of considered
amplitude w0 / t ≥ 0,75 influence of artificial dents are overpowering and initial background have no deciding influence to
behaviour of shell under loading. The estimation p1 theoretical without taking into account of small initial deflections leads to
close coincidence with test data. The severe sensitivity of buckling loads to small amplitude dents (those less than the shell
thickness) is evident in both the test data and the theoretical predictions. Both are also shown to exhibit plateau for larger
amplitudes of dent. Theoretical predictions are shown to closely follow the test observations.
The full analysis of theoretical and experimental dependencies is given in Figure 1 for isotropic shell having l/r = 2, r = 143
mm, r/t = 360. The finite difference mesh used had 71 × 230 divisions in the axial and circumferential directions respectively,
and 14 × 22 divisions covering the area of the local dent imperfection. Figure 1 includes the following theoretical information.
The upper thick continuous line (3) represents the maximum loads on the load deflection plots at which a general buckling
occurs. The lower thin continuous line (4) indicates the load at which a local buckle is initiated in the region of the dent
imperfection.
The newest results on stability and carrying capacity of incomplete shells are represented in [13 – 22]. An approach suggested
is used for the analysis load-carrying capacity of shell with symmetric and asymmetric, local and regular imperfections of form
[13]. The data presented have been compared with well-known theoretical and experimental data. A technique for numerical
determination of the critical loads of imperfect ribbed shells is described in [14, 22]. The method takes into account the
discreteness in the rib arrangement in two directions and nonuniform subcritical state of the shells. A finite difference method is
used in [19] to analyse critical loads of cylindrical shells with initial imperfections of different type, both regular and local. A
technique for calculation of load-carrying capacity of elastic shells with periodic dents is represented in [22].
Conclusions
The proposed method for testing shells with dents has been realized using specimens with an artificial dent [10]. A
procedure of numerical calculation has been used for the theoretical analysis of the critical loads and the load-carrying
capacity of the shell tested. The theoretical-and-experimental investigation into stability of smooth cylindrical shells with a
single local dent has been accomplished. For the shells with localized small dents, we obtained a good qualitative
agreement between the calculated and experimental values of the critical loads in all the regions of the parameter w0 / t
variation. In the case where 150 ≤ r / t ≤ 360 a good agreement between those values is also reached. Localized
imperfections can be used as one of the criteria by which the quality of the shell manufacture is evaluated. This approach
calls for a careful consideration and analysis of these as well as other specific imperfections.
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