An experimental investigation is carried out to study the effects of non-proportional loading in the plastic range on the buckling load. The discrepancy between experimental and theoretical results indicates some major flaw in the analysis.
INTRODUCTION 1.1 GENERAL
A more suitable form of the theory (for an introduction to BOSOR5) can be found in Lam [Ref. Compressive stresses are observed whenever the radius (r) of the torispherical vessel in the joint area is smaller than the radius v.
GENERAL DESCRIPTION
THE TEST CHAMBER (PRESSURE SLEEVE)
END PLUGS AND END PLATES
An exploded view of the end plug and test piece is shown in Figure 2.2. Observation of the buckling phenomenon is also possible as the end plugs extend from the test chamber and allow viewing of the inner surface of the test shell.
BONDING THE TEST SHELL TO THE END PLUGS
The length of the bonding surface between the test shell and the end studs was also increased from 1" to 1.5" for these tests. Removal of the test shell from the end plugs after completion of the test is carried out with the help of a soldering torch.
WALL THICKNESS AND IMPERFECTIONS
After application of the two-component adhesive, the (glued) interface between the specimen and the end plug is cured at a temperature of 80°F for one day. During low to medium stress test runs (tensile stress less than 16,000 psi), EPOXI-PATCH bonding material performed satisfactorily. SCOTCH-WELD 2216 B/A epoxy adhesive with extensive pretreatment of the ends of the test section (priming and bathing in a hot sulfuric acid solution) provided the desired strength for these high loading conditions.
BUCKLING DETECTION SYSTEM
- DSD SCANNING SYSTEM
- CALIBRATION OF THE PROBE
- POSITION SENSING CIRCUITRY
- DESCRIPTION OF THE SERVO-HYDRAULIC AXIAL LOADING DEVICE (SLD)
- DESCRIPTION OF SERVO-HYDRAULIC PRESSURE DEVICE (SPD) A second system of accumulators, servo-controller and actuator, feeding from the
- FUNCTION GENERATOR CONTROL
- SIGNAL INTERRUPTION SYSTEM (SIS)
A simplified hydraulic and electrical schematic of the servo hydraulic loading device (SLD) is presented in Figure 2.15. The mechanical parts of the SLD are contained in four basic modules: 1) the hydraulic power supply, 2) a flow control module containing valves and filters, 3) accumulators and 4) the load frame (MTS), to which the servo valve, actuator, load cell and LVDT (Linear Variable Differential Transformer) is attached. Because of this problem, it was necessary to use a separate fluid to pressurize the test chamber. The schematic shown in Figure 2.14 depicts the dual actuator system, which uses the second actuator to force a hydraulic fluid into the test chamber for pressurization.
MATERIAL PROPERTIES
- END FIXTURES AND TESTING EQUIPMENT
- CIRCUMFERENTIAL PROPERTIES
- RAMBERG-OSGOOD APPROXIMATION
- UNIAXIAL TEST PROGRAM (STRESS-STRAIN)
This material curve (power law) is then used for the analysis phase of the research. First, all available stress-strain curves of the same pipe are displayed on a single graph.
RUN CONDITIONS
Included in this section of the program is the ability to generate Southwell plots from load versus displacement data. Normally, one would not be able to see the buckling waveform with the naked eye due to the load interruption of the SIS.
EXPERIMENTAL RESULTS 3.1 MATERIAL PROPERTIES
STRESS-STRAIN EXPERIMENTS
AXIAL MATERIAL BEHAVIOR "SET A"
Some of the differences between the various stress-strain curves shown in Figure 3.1 are most likely due to inaccurate determination of the cross-sectional areas of these samples. A significant deviation of the Ramberg-Osgood fit from the average material curve occurs above 1.5% strain, as shown in Figure 3.2.
AXIAL AND CIRCUMFERENTIAL MATERIAL BEHAVIOR "SET B"
Differences appear small enough to justify the use of isotropic assumptions made for "set A" after determining only axial behavior. Both material curves, although only slightly different, are used in the analysis for "set B." In this way, the effects of material behavior on the overall results can be studied.
PRELIMINARY RESULTS
VARIABLE LENGTH SPECIMENS
The yield stress is slightly lower, while the hardening behavior of these samples appears to be somewhat higher. It is important to realize that the main purpose of the investigation is to study the effects of plasticity, not geometry, on bifurcation.
AXIAL LOAD RELAXATION AND BIAXIAL RESULTS
Problems such as inaccurate branch detection and load relaxation led to the development of the Displacement Sensing Device (DSD) for buckling detection. This method of determining buckling load is highly repeatable and is not very sensitive to the proximity of the last recorded load to the actual split point (as shown in the next section).
11EXEEBIMEKΕAL RESULTS "SET A"
CONSTANT TENSION EXPERIMENTS
The Southwell process begins with maximum and minimum load displacement plots in the bending waveform. In this case, where the axial tensile stress is zero, the bending load obtained using the Southwell method is 978 psi.
CONSTANT PRESSURE EXPERIMENTS
This "normalizing" profile change is required since high axial load significantly changes the initial imperfection scan. This may explain why Southwell's method is not as effective when tensile stress instead of external pressure is used as the load parameter in the plot.
COMBINED RESULTS "SET A"
Suffice it to say that the test shells failed under this type of loading and the curve showing these split points appears to be lower (closer to the elastic region) than the curve for constant stress experiments, but not by much. It is important to recognize that Southwell's results are not plotted for the constant pressure experiments. Again, this is due to the previous arguments that axial stress is not really the loading condition that causes the sample to buckle and therefore can cause some problems in the Southwell plot.
EXPERIMENTAL RESULTS "SET B"
- CONSTANT PRESSURE EXPERIMENTS
This is also manifested in the Southwell plots in that the linear regions tend to be more restricted than those seen in "set A". The effectiveness of the Southwell plot is reduced to the point where it is almost impossible to distinguish a linear region in the plot and determine the associated bending pressure for the "perfect" shell. However, it is noted that for the few experiments in which the Southwell plot appeared to work well, more data points were obtained near the buckling load.
BOSOR5 ANALYSIS
BOSOR5 AND SHELL PARAMETERS
Various assumptions regarding the type of end conditions applied to the shell will also be discussed, the most important condition being the in-plane distortion of the shell during bending. This means that axial displacements in the bending calculation are free and the edges of the shell are allowed to take any shape in the plane. The welded end of the shell has limitations imposed on the axial displacement. u=0) and momentum (M=0), while the center of the shell has constraints applied to the surface slope (dw∕dx=0) and circular displacement (v=0).
NUMERICAL ANALYSIS ,,SET A"
- CONSTANT TENSION "SET A”
The deformation theory, although independent of the path, captures the weakening behavior of the material with increasing axial load as observed in the experiment. When the ends of the short shell are unconfined (i.e., the studs are neglected), both theories predict the same buckling pressures and adhere to the well-known result that the deformation theory and the incremental theory are the same for proportional loading. The shell wall thickness of the model in the BOSOR5 analysis was taken as the average thickness measured in actual experiments.
NUMERICAL ANALYSIS "SET B"
- CONSTANT TENSION "SET B
It is important to realize that underprediction as observed for part of the loading problem remains a clear indication that the deformation theory is also incorrect. Because more waves form around the periphery of the shell, it is likely that the material behavior at the periphery may be of greater importance in the buckling analysis than the axial behavior. It is important to realize that the circumferential properties are obtained under uniaxial "hoop" stress resulting from the internal pressure of the test shell.
CHRISTOFFERSEN-HUTCHINSON CORNER THEORY
This may explain why, in the case where the axial loads are small, there is little dependence on the material behavior. When the axial loads are very large, the sudden change in the load path causes some stress components to be overpredicted as observed. Advent theory seems to have some positive aspects that cannot be found in J2.
BOSOR5 OPERATION
NUMERICAL CONVERGENCE AND SHEAR RESPONSE USING BOSOR5
- ANALYTIC PLANE STRESS SOLUTION
- BOSOR5 RESULTS
Due to the asymmetric nature of the problem, no shear occurs in the prebuckling stage. The foregoing discussion regarding the plastic behavior of the material can be found in any basic text on plasticity and will not be further developed here. Overall, it can be said that the inclusion of the subincremental method in the BOSOR5 analysis significantly reduces the load step control problem.
BIFURCATION BUCKLING IN THE PLASTIC RANGE
- FLOW THEORY AND DEFORMATION THEORY
Closer examination of the BOSOR5 analysis leads to the observation that the displacement response in the incremental bifurcation analysis is not the usual elastic response. In the prebuckling analysis shear is absent due to the nature of the problem and. In contrast, if deformation theory is used in the bifurcation analysis, the instantaneous shear modulus [Ref.
SOUTHWELL METHOD AND PLASTIC BUCKLING
THE SOUTHWELL PLOT
- SOUTHWELL FOR PLASTICALLY LOADED SHELLS
In previous sections, the discussion centered on the application of the Southwell method to inelastic column problems. However, limited theoretical justification for the use of the Southwell plot in the analysis of elastic cylinders under axial compression was presented by Donnell [Ref. The application of the Southwell method in this research basically assumes an extension of the theory to plastic buckling of shells under the prescribed load.
RELIABILITY AND PATH DEPENDENCE
The Southwell plot has a pronounced "smoothing" effect on the assembly of bending results in this research. The "smoothing" effect derived from the Southwell plot is a highly desirable feature when comparing experimental data with analyzes in a parametric study. The external pressure load parameter in Southwell's chart is the buckling load, as it introduces compressive circle stresses into the test shell.
CHRISTOFFERSEN-HUTCHINSON J2 CORNER THEORY 7.1 CORNER THEORY
A theory that allows newcomers to develop on an initially smooth yield surface is the focus of investigation in this chapter. 53], A special version of the plastic model of the CH arrival theory was introduced in the BOSOR5 code and compared with the experimental results. The special problems and issues of the process of applying the CH theory to the BOSOR5 analysis will be addressed in the following sections.
CH CORNER THEORY IN BOSOR5
- BOSOR5 MODIFICATION
This is one of the simplest versions of the theory and has the added attraction that Poh -Sang Lam [Ref. Introducing the chosen corner theory into an axisymmetric shell code turned out to be a challenging but simple task. First, it was necessary to thoroughly understand the details of the operational parameters in the plasticity subroutine.
J2 CORNER THEORY
- IMPLEMENTATION OF THE J2 CORNER THEORY
- OBSERVATIONS OF THE J2 CORNER THEORY
The angle constraint limits the deviation of the yield surface from a smooth J2 incremental yield surface. This again made the implementation of the angle theory much simpler and is not considered a major problem. A schematic of the J2 angle theory model, which is implemented in the BOSOR5 shell code, is shown in Figures 7.4a and b.
DISCUSSION AND CONCLUSIONS
Probe sensitivity can be better understood by studying the effects of shell (material) imperfection and curvature of the scanned surface. Preliminary investigation into the effects of shell curvature led to calibration of the probe on an actual curved-shell segment. Some of the irregular spikes occasionally observed in the displacement plots may be due to wall thickness variation or material imperfection, not a result of wall deformation.
EXPERIMENTAL RESULTS
The contour profile graphs add significantly to the understanding of the formation of the buckling waveform. Path dependence of the buckling loads of the test shells appears to be less in the thinner shells. This may be a consequence of the bifurcation points being closer to the elastic region in which path independence prevails.
ANALYSIS RESULTS
This behavior is expected, but apparently does not change the analysis enough to predict the "meaning" of the experimental results. The introduction of the comer theory shows some progress in that it predicts the "sense" of the experimental data, while the bifurcation wavenumber is also correctly predicted. Corner theory retains the desirable features of incremental theory (normality, convexity, yield surface), but the definition of a corner is an unsolved problem.
APPENDIX (A)
3] Bushnell, D., "Nonsymmetric Buckling of Internally Pressurized Ellipsoidal and Torispherical Elastic-Plastic Pressure Vessel Heads," ASME J. 14] Singer, J., Arbocz, J., og Babcock, C.D., "Buckling of Imperfect Stiffened Cylindrical Shells Under Axial Compression," AIAA Journal, Vol. 15] Arbocz, J., og Babcock, C.D., "The Effect of General Imperfections on the Buckling of Cylindrical Shells," Transactions of the ASME, Ser.
23] Bushnell, D., "BOSOR5 - A Computer Program for Buckling Elastic-Plastic Complex Shells of Revolution Including Large Deflections and Creep," Vol. 24] Bushnell, D., "Bifurcation Buckling of Shells of Revolution, Including Large Deflections, Plasticity and Creep," International Journal of Solids Structures, Vol.
