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Today high-performance computing systems and new analytical and numerical techniques enable engineers to explore the use of advanced materials for shell design. This paper reviews some of the historical developments of shell buckling analysis and design. The paper concludes by identifying key research directions for reliable and robust methods development in shell stability analysis and design.
A number of issues involving the postbuckling of doubly-curved panels exposed to a non-uniform temperature field and subjected to a system of mechanical loads are considered in this paper. Effects played by transverse shear, initial geometric imperfections, membrane and thickness-wise temperature gradient, tangential edge constraints, linear/non-linear elastic foundation are incorporated and their implications on the non-linear response behavior of shells are discussed. Issues related with the delay of the occurrence of the snap-through buckling, alleviation of its intensity and even with its removal are addressed and remarks on conditions rendering a curved panel insensitive to initial geometric imperfections are included. Results on small vibration of shells about sub- and post-critical ranges are provided and structural modeling issues playing a non-negligible role towards a reliable prediction of the postbuckling behavior of shells under complex loading conditions are discussed. Results on non-linear response of flat reinforced panels subjected to thermomechanical loads are also presented. Finally, problems not covered by this article are briefly indicated.
This study is an extension of a previous investigation of the combined effect of axisymmetric thickness variation and axisymmetric initial geometric imperfection on buckling of isotropic shells under uniform axial compression. Here the anisotropic cylindrical shells are investigated by means of Koiter's energy criterion. An asymptotic formula is derived which can be used to determine the critical buckling load for composite shells with combined initial geometric imperfection and thickness variation. Results are compared with those obtained by the software packages BOSOR4 and PANDA2.
Multi-field mechanics are presented for curvilinear piezoelectric laminates interfaced with distributed passive electric components. The equations of motion for laminated piezoelectric shell structures with embedded passive electric networks are directly formulated and solved using a finite element methodology. The modal damping and frequencies of the piezoelectric shell are calculated from the poles of the system. Experimental and numerical results are presented for the modal damping and frequency of composite beams with a resistively shunted piezoceramic patch. The modal damping and frequency of plates, cylindrical shells and cylindrical composite blades with piezoelectric-resistor layers are predicted. Both analytical and experimental studies illustrate a unique dependence of modal damping and frequencies on the shunting resistance and show the effect of structural shape and curvature on piezoelectric damping.
Progressive damage and fracture of large shell structures is investigated. A computer model is used for the assessment of structural response, progressive fracture resistance, and defect/damage tolerance characteristics. Critical locations of a stiffened conical shell segment are identified. Defective and defect-free computer models are simulated to evaluate structural damage/defect tolerance. Safe pressurization levels are assessed for the retention of structural integrity at the presence of damage/ defects. Damage initiation, growth, accumulation, and propagation to fracture are included in the simulations. Damage propagation and burst pressures for defective and defect-free shells are compared to evaluate damage tolerance. Design implications with regard to defect and damage tolerance of a large steel pressure vessel are examined.
The Penalized Discrete Least-Squares (PDLS) stress recovery (smoothing) technique developed for two dimensional linear elliptic problems is adapted here to three-dimensional shell structures. The surfaces are restricted to those which have a 2-D parametric representation, or which can be built-up of such surfaces. The proposed strategy involves mapping the finite element results to the 2-D parametric space which describes the geometry, and smoothing is carried out in the parametric space using the PDLS-based Smoothing Element Analysis (SEA). Numerical results for two well-known shell problems are presented to illustrate the performance of SEA/PDLS for these problems. The recovered stresses are used in the Zienkiewicz-Zhu a posteriori error estimator. The estimated errors are used to demonstrate the performance of SEA-recovered stresses in automated adaptive mesh refinement of shell structures. The numerical results are encouraging. Further testing involving more complex, practical structures is necessary.
Two 61-cm-diameter eight-ply quasi-isotropic IM7/PEEK cylindrical shells were fabricated by automated fiber placement the NASA Langley Research Center using only infrared radiant heat to preheat the substrate and incoming composite uni-tape. The shells were characterized by ultrasonic c-scans for overall consolidation quality, and by optical microscopy and acid digestion for void content. Compression tests were also performed. Although the material used in the study was of generally poor quality due to numerous splits and dry fiber regions, the process was able to achieve a net reduction in void content in the as-placed component. Microscopy of the composite shells revealed well-consolidated, void-free interfaces. The two cylinders were then tested in uni-axial compression in a 1334 kN-capacity hydraulic test machine until buckling occurred. A geometrically nonlinear finite element analysis was conducted, and the differences between the predicted and measured values were 18.0 and 25.8%, respectively. Inclusion of measured imperfections of the cylinder into the analysis is expected to reduce these differences.
The four large satellites of Jupiter are famous for their planet-like diversity and complexity, but none more so than ice-covered Europa. Since the provocative Voyager images of Europa in 1979, evidence has been mounting that a vast liquid water ocean may lurk beneath the moon's icy surface. Europa has since been the target of increasing and sometimes reckless speculation regarding the possibility that giant squid and other creatures may be swimming its purported cold, dark ocean. No wonder Europa tops everyone's list for future exploration in the outer solar system (after the very first reconnaissance of Pluto and the Kuiper belt, of course). Europa may be the smallest of the Galilean moons (so-called because they were discovered by Galileo Galilei in the early 17th century) but more than makes up for its diminutive size with a crazed, alien landscape. The surface is covered with ridges hundreds of meters high, domes tens of kilometers across, and large areas of broken and disrupted crust called chaos. Some of the geologic features seen on Europa resemble ice rafts floating in polar seas here on Earth-reinforcing the idea that an ice shell is floating over an ocean on this Moon-size satellite. However, such features do not prove that an ocean exists or ever did. Warm ice is unusually soft and will flow under its own weight. If the ice shell is thick enough, the warm bottom of the shell will flow, as do terrestrial glaciers. This could produce all the observed surface features on Europa through a variety of processes, the most important of which is convection. (Convection is the vertical overturn of a layer due to heating or density differences-think of porridge or sauce boiling on the stove.) Rising blobs from the base of the crust would then create the oval domes dotting Europa's surface. The strongest evidence for a hidden ocean beneath Europa's surface comes from the Galileo spacecraft's onboard magnetometer, which detected fluctuations in Jupiter's magnetic field consistent with a conductor inside Europa. The most likely conductor: a somewhat salty ocean.
It is shown that the T(sub 1) operator used in a previous study to define the open-shell D(sub 1) diagnostic is invalid, and leads to an arbitrary definition of the open-shell D(sub 1) diagnostic. A new definition is proposed that eliminates this ambiguity and approximately restores the mathematical relationship previously noted between the closed-shell D(sub 1) and T(sub 1) diagnostics. Statistical comparison of the T(sub 1) and D(sub 1) diagnostics shows a very high degree of correlation between them, although it is argued that both diagnostics used together can provide more information than either can separately.
We present an analysis of the X-ray spectrum of the Galactic shell-type SNR G347.3-0.5 (RX 51713.7-3946). This SNR is a member of a growing class of SNRs which are dynamically young, shell-type sources that emit non-thermal X-rays from specific regions on their outer shells. By performing a joint spectral analysis of data from observations made of G347.3-0.5 using the ROSAT PSPC, the ASCA GIS and the RXTE PCA, we have fit the spectra of particular regions of this SNR (including the bright northwestern and southwestern rims, the northeast rim and the interior diffuse emission) over the approximate energy range of 0.5 through 30 keV. We find that fits to the spectra of this SNR over this energy range using the SRCUT model were superior to a simple power law model or the SRESC model. We find that the inclusion of a thermal model with the SRCUT model helps to improve the fit to the observed X-ray spectrum: this represents the first detection of thermal X-ray emission from G347.3-0.5. Thermal emission appears to be more clearly associated with the diffuse emission in the interior of the SNR than with the bright X-ray emitting rims. A weak emission feature seen near 6.4 keV in the RXTE PCA spectrum most likely originates from diffuse X-ray emission from the surrounding Galactic Ridge rather than from G347.3-0.5 itself. We have analyzed our RXTE PCA data to search for pulsations from a recently discovered radio pulsar (PSR 51713-3949) which may be associated with G347.3-0.5, and we do not detect any X-ray pulsations at the measured radio period of 392 ms. Using the best-fit parameters obtained from the SRCUT model, we estimate the maximum energy of cosmic-ray electrons accelerated by the rims of G347.3-0.5 to be 19-25 TeV (assuming a magnetic field strength of B = 10muG), consistent with the results of Ellison et al. We present a broadband (radio to gamma ray) photon energy-flux spectrum for the northwestern rim of G347.3-0.5, where we have fit the spectrum using a more sophisticated synchrotron-inverse Compton model with a variable magnetic field strength. Our fit derived from this model yields a maximum energy of only 8.8 (+4.1) (-3.4) TeV for the accelerated cosmic-ray electrons and a much greater magnetic field strength of 150 (+250) (-80). Finally, we compare the gross properties of G347.3-0.5 with other SNRs known to possess X-ray spectra dominated by non-thermal emission.
We consider in this report the determination of the upper limit of critical loads in the case of simultaneous action of a compressive force, uniformly distributed over plane cross sections, and of isotropic external normal pressure on cylindrical or conical shells of circular cross section. As a starting point we use the differential equations for neutral equilibrium of conical shells which have been used for the solution of the problem of stability of conical shells under torsion and under axial compression; upon solution of the problem it is possible to satisfy all boundary conditions, in contrast to the report where no attention is paid to the fulfillment of the boundary conditions, and to the report where only part of the boundary conditions are satisfied by solution of the problem according to Galerkin's method. Approximate formulas are used for the determination of the critical external normal pressure with simultaneous action of longituninal compression. Let us note that the formulas suggested in reference 5 are not well founded and may lead, in a number of cases, to a substantial mistake in the magnitude of the critical load.
A computationally effective method is described to evaluate the non-deterministic dynamic instability (probabilistic dynamic buckling) of thin composite shells. The method is a judicious combination of available computer codes for finite element, composite mechanics, and probabilistic structural analysis. The solution method is incrementally updated Lagrangian. It is illustrated by applying it to thin composite cylindrical shell subjected to dynamic loads. Both deterministic and probabilistic buckling loads are evaluated to demonstrate the effectiveness of the method. A universal plot is obtained for the specific shell that can be used to approximate buckling loads for different load rates and different probability levels. Results from this plot show that the faster the rate, the higher the buckling load and the shorter the time. The lower the probability, the lower is the buckling load for a specific time. Probabilistic sensitivity results show that the ply thickness, the fiber volume ratio and the fiber longitudinal modulus, dynamic load and loading rate are the dominant uncertainties, in that order.
The shell shall be considered as a three-dimensional continuous medium; for the coordinate surface, the middle surface of the shell shall be assumed parallel to the bounding surfaces. Let alpha and beta be the curvilinear orthogonal coordinates of this surface, coinciding with the lines of principal curvatures, and gamma the distance along the normal from the point (alpha,beta) of the coordinate surface to any point (alpha,beta,gamma) of the shell.
"Smart" structures composed of piezoelectric materials may significantly improve the performance of aeropropulsion systems through a variety of vibration, noise, and shape-control applications. The development of analytical models for piezoelectric smart structures is an ongoing, in-house activity at the NASA Glenn Research Center at Lewis Field focused toward the experimental characterization of these materials. Research efforts have been directed toward developing analytical models that account for the coupled mechanical, electrical, and thermal response of piezoelectric composite materials. Current work revolves around implementing thermal effects into a curvilinear-shell finite element code. This enhances capabilities to analyze curved structures and to account for coupling effects arising from thermal effects and the curved geometry. The current analytical model implements a unique mixed multi-field laminate theory to improve computational efficiency without sacrificing accuracy. The mechanics can model both the sensory and active behavior of piezoelectric composite shell structures. Finite element equations are being implemented for an eight-node curvilinear shell element, and numerical studies are being conducted to demonstrate capabilities to model the response of curved piezoelectric composite structures (see the figure).
A simplified nonlinear numerical model for the development of incompressible magnetohydrodynamics in the presence of a strong magnetic field B|| and stratification, nicknamed 'Shell-Atm,' is presented. In planes orthogonal to the mean field, the nonlinear incompressible dynamics is replaced by two-dimensional shell models for the complex variables u and b, allowing one to reach large Reynolds numbers while at the same time carrying out sufficiently long integrations to obtain good statistics at moderate computational cost. The shell models of different planes are coupled by Alfve'n waves propagating along B||. The model may be applied to open or closed magnetic field configurations where the axial field dominates and the plasma pressure is low; here we apply it to the specific case of a magnetic loop of the solar corona heated by means of turbulence driven by photospheric motions, and we use statistics for its analysis. The Alfven waves interact nonlinearly and form turbulent spectra in the directions perpendicular and, through propagation, also parallel to the mean field. A heating function is obtained and shown to be intermittent; the average heating is consistent with values required for sustaining a hot corona and is proportional to the aspect ratio of the loop to the -1.5 power, and haracteristic properties of heating events are distributed as power laws. Crosscorrelations show a delay of dissipation compared with energy content.
The Multi-Purpose Logistics Module (MPLM) is the primary carrier for "pressurized" logistics to and from the International Space Station (ISS). The MPLM is transported in the payload bay of the Space Shuttle and is docked to the ISS for unloading, and reloading, of contents within the ISS shirt sleeve environment. Foil heaters, controlled originally with bi-metallic thermostats, are distributed across the outside of the MPLM structure and are utilized to provide energy to the structure to avoid exposure to cold temperatures and prevent condensation. The existing bi-metallic, fixed temperature set point thermostats have been replaced with Programmable Thermostats Modules (PTMs) in the Passive Thermal Control Subsystem (PTCS) 28Vdc shell heater circuits. The goal of using the PTM thermostat is to improve operational efficiency of the MPLM on-orbit shell heaters by providing better shell temperature control via feedback control capability. Each heater circuit contains a programmable thermostat connected to an external temperature sensor, a Resistive Temperature Device (RTD), which is used to provide continuous temperature monitoring capability. Each thermostat has programmable temperature set points and control spans. The data acquisition system uses a standard RS-485 serial interface communications cable to provide digital control capability. The PTM system was designed by MSFC, relying upon ALTEC support for their integration within the MPLM system design, while KSC performed the installation and ground checkout testing of the thermostat and RS-485 communication cable on the MPLM FM1 flight module. The PTMs were used for the first time during the STS-121/ULF1.1 mission. This paper will describe the design, development and verification of the PTM system, as well as the PTM flight performance and comparisons with SINDA thermal model predictions.
A comprehensive development of nondimensional parameters and equations for nonlinear and bifurcations analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory for thin anisotropic shells, is presented. A complete set of field equations for geometrically imperfect shells is presented in terms general of lines-of-curvature coordinates. A systematic nondimensionalization of these equations is developed, several new nondimensional parameters are defined, and a comprehensive stress-function formulation is presented that includes variational principles for equilibrium and compatibility. Bifurcation analysis is applied to the nondimensional nonlinear field equations and a comprehensive set of bifurcation equations are presented. An extensive collection of tables and figures are presented that show the effects of lamina material properties and stacking sequence on the nondimensional parameters.
Stereo-derived topographic mapping of 50% of Enceladus reveals at least 6 large-scale, ovoid depressions (basins) 90-175 km across and 800-to-1500 m deep and uncorrelated with geologic boundaries. Their shape and scale are inconsistent with impact, geoid deflection, or with dynamically supported topography. Isostatic thinning of Enceladus ice shell associated with upwellings (and tidally-driven ice melting) can plausibly account for the basins. Thinning implies upwarping of the base of the shell of 10-20 km beneath the depressions, depending on total shell thickness; loss of near-surface porosity due to enhanced heat flow may also contribute to basin lows. Alternatively, the basins may overly cold, inactive, and hence denser ice, but thermal isostasy alone requires thermal expansion more consistent with clathrate hydrate than water ice. In contrast to the basins, the south polar depression (SPD) is larger (~350 wide) and shallower (~0.4-to-0.8 km deep) and correlates with the area of tectonic deformation and active resurfacing. The SPD also differs in that the floor is relatively flat (i.e., conforms roughly to the global triaxial shape, or geoid) with broad, gently sloping flanks. The relative flatness across the SPD suggests that it is in or near isostatic equilibrium, and underlain by denser material, supporting the polar sea hypothesis of Collins and Goodman. Near flatness is also predicted by a crustal spreading origin for the "tiger stripes (McKinnon and Barr 2007, Barr 2008); the extraordinary, high CIRS heat flows imply half-spreading rates in excess of 10 cm/yr, a very young surface age (~250,000 yr), and a rather thin lithosphere (hence modest thermal topography). Topographic rises in places along the outer margin of the SPD correlate with parallel ridges and deformation along the edge of the resurfaced terrain, consistent with a compressional, imbricate thrust origin for these ridges, driven by the spreading.