Critical combinations of longitudinal and transverse direct stress for an infinitely long flat plate with edges elastically restrained against rotation
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CHARTS-STRESSES-PLATES↗
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Stress, vibration, and buckling of multilayered cylinders
An efficient computational method for lowering the cost of the free vibration, stress, and buckling analyses of multilayered composite cylinders is presented. The analytical methodology is based on the linear three-dimensional theory of elasticity where the cylinders are assumed to have simply supported curved edges, and the fibers of the different layers are either in the circumferential or in the longitudinal direction. The full equations of the finite element model are solved for a single pair of Fourier harmonics, and the response that corresponds to the other Fourier harmonics is generated utilizing a reduced system with considerably fewer degrees of freedom.
Noor, Ahmed K.↗
Consistent Pl Analysis of Aqueous Uranium-235 Critical Assemblies
The lethargy-dependent equations of the consistent Pl approximation to the Boltzmann transport equation for slowing down neutrons have been used as the basis of an IBM 704 computer program. Some of the effects included are (1) linearly anisotropic center of mass elastic scattering, (2) heavy element inelastic scattering based on the evaporation model of the nucleus, and (3) optional variation of the buckling with lethargy. The microscopic cross-section data developed for this program covered 473 lethargy points from lethargy u = 0 (10 Mev) to u = 19.8 (0.025 ev). The value of the fission neutron age in water calculated here is 26.5 square centimeters; this value is to be compared with the recent experimental value given as 27.86 square centimeters. The Fourier transform of the slowing-down kernel for water to indium resonance energy calculated here compared well with the Fourier transform of the kernel for water as measured by Hill, Roberts, and Fitch. This method of calculation has been applied to uranyl fluoride - water solution critical assemblies. Theoretical results established for both unreflected and fully reflected critical assemblies have been compared with available experimental data. The theoretical buckling curve derived as a function of the hydrogen to uranium-235 atom concentration for an energy-independent extrapolation distance was successful in predicting the critical heights of various unreflected cylindrical assemblies. The critical dimensions of fully water-reflected cylindrical assemblies were reasonably well predicted using the theoretical buckling curve and reflector savings for equivalent spherical assemblies.
Fieno, Daniel↗
Differences in Mechanical Properties of the Human and Monkey Tibia
A method which uses an instrument that detects the response of a long bone to a vibratory stimulus to quantify mechanical properties non-invasively was revised and validated for use in the tibia. Stored data from healthy men was reanalyzed and compared with values from non-human primates. The analysis uses the relationship K(sub b) = 48 EI/L(sup 3) where K(sub b) is the lateral stiffness of a beam with force applied midspan, E is the elastic modulus, I the geometric moment of inertia and L, the limb length. Values for stiffness (EI, Nm(sup2)), the Euler buckling load (P(sub cr) = EI (pi/L)(sup 2)), and bone sufficiency (S) which represents the axial load the bone can support, adjusted to BW (S=P(sub cr)/BW) were obtained. The interest precision of the method in relaxed men, 5.8%, and in sedated male monkeys, 4.3%, was based on repeated measures in the same subjects at 1 month intervals. The R tibias of 40 men, aged 38.6 +/- 7.3 yrs with BW 78.9 +/- 7.9 kg, showed average (+/- SD) L to be 35 +/- 2 cm, EI 222 +/- 71 Nm(sup 2), P(sub cr) 18.1 +/- 4.9 kN, and S 23.4 +/- 5.7 N. The R tibias of 24 Rhesus monkeys ranging in age from 2-12 years, BW 4.9 +/- 3 kg, showed L to be 14.7 +/- 1.9 cm, EI 6.0 +/- 4.8 Nm(sup 2), P(sub cr) 2.51 +/- 1.2 kN and S 57.3 N. These measurements indicate that the tibia of a terrestrial non-human primate, M. mulatta, has higher load carrying capacity for the level of body weights in the species than the human bone.
Arnaud, Sara B.↗
Transverse vibration and buckling of a cantilevered beam with tip body under constant axial base acceleration
The planar transverse bending behavior of a uniform cantilevered beam with rigid tip body subject to constant axial base acceleration was analyzed. The beam is inextensible and capable of small elastic transverse bending deformations only. Two classes of tip bodies are recognized: (1) mass centers located along the beam tip tangent line; and (2) mass centers with arbitrary offset towards the beam attachment point. The steady state response is studied for the beam end condition cases: free, tip mass, tip body with restricted mass center offset, and tip body with arbitrary mass center offset. The first three cases constitute classical Euler buckling problems, and the characteristic equation for the critical loads/accelerations are determined. For the last case a unique steady state solution exists. The free vibration response is examined for the two classes of tip body. The characteristic equation, eigenfunctions and their orthogonality properties are obtained for the case of restricted mass center offset. The vibration problem is nonhomogeneous for the case of arbitrary mass center offset. The exact solution is obtained as a sum of the steady state solution and a superposition of simple harmonic motions.
Storch, J.↗
Critical Compressive Stress for Outstanding Flanges
A chart is presented for the values of the coefficient in the formula for the critical compressive stress at which buckling may be expected to occur in outstanding flanges. These flanges are flat rectangular plates supported along the loaded edges, supported and elastically restrained along one unloaded edge, and free along the other unloaded edge. The mathematical derivations of the formulas required for construction of the chart are given.
Lundquist, Eugene E↗
Critical Compressive Stress for Outstanding Flanges
A chart is presented for the values of the coefficient in the formula for the critical compressive stress at which buckling may be expected to occur in outstanding flanges. These flanges are flat rectangular plates supported along the Loaded edges, supported and elastically restrained along one unloaded edge, and free along the other unloaded edge. The mathematical derivations of the formulas required for the construction of the chart are given.
Lundquist, Eugene E.↗
Investigating Local Buckling and Plastic Collapse in Multifunctional Sandwich Composite Cores using a Unit Cell Method
The primary goal of developing analytical models of composite sandwich cores has been to obtain simple expressions for elastic and failure response to improve the rapid analysis of composite structures. An earlier investigation by the authors was made to verify the accuracy of widely-used analytical models that calculate equivalent elastic moduli of a uniform homogeneous single layer of material in place of the complex honeycomb core properties in sandwich composites. This transformation of elastic properties permits a coarsening of the finite element model (FEM) of the core to a fraction of the computational cost in simulating the original complex core architecture. In addition, the core was assumed constructed of multi-layered isotropic walls to support potential additional functionality beyond simple load-carrying capability.
Erik Saether↗
A micromechanics model of the stiffness and strength of laminates with fiber waviness
A mathematical model based on the Euler-Bernoulli beam theory is proposed for predicting the effective Young's moduli of piecewise isotropic composite laminates with local ply curvatures in the main load-carrying layers. Strains in corrugated layers, In-Phase layers, Out-of-Phase layers are predicted for various geometries and material configurations by assuming matrix layers as elastic foundations of different spring constants. The effective Young's moduli measured from corrugated aluminum specimens and aluminum/epoxy specimens with In-Phase and Out-of-Phase wavy patterns coincide very well with the model predictions. Moire fringe analysis of an In-Phase specimen and an Out-of-Phase specimen are also presented confirming the main assumption of the model related to the elastic constraint due to the matrix layers. The present model is also compared with the experimental results and other models, including the micro-buckling models, published in the literature. The results of the present study show that even a very small scale local ply curvature produces a noticeable effect on the mechanical constitutive behavior of a laminated composite.
Harris, Charles E.↗
Local instability of centrally loaded columns of channel section and Z-section
Charts are presented for the coefficients in formulas for the critical compressive stress at which cross-sectional distortion begins in a thin-wall member with either a channel section or a Z-section with identical flanges. The energy method of Timoshenko was used in the theoretical calculations required for the construction of the charts. The deflection equations were carefully selected to give good accuracy. The calculation of the critical compressive stress at stresses beyond the elastic range is briefly discussed. In order to demonstrate the use of the formulas and the charts in engineering calculations, two illustrative problems are included.
ELASTICITY↗
In-Flight Aeroelastic Stability of the Thermal Protection System on the NASA HIAD, Part II: Nonlinear Theory and Extended Aerodynamics
Conical shell theory and a supersonic potential flow aerodynamic theory are used to study the nonlinear pressure buckling and aeroelastic limit cycle behavior of the thermal protection system for NASA's Hypersonic Inflatable Aerodynamic Decelerator. The structural model of the thermal protection system consists of an orthotropic conical shell of the Donnell type, resting on several circumferential elastic supports. Classical Piston Theory is used initially for the aerodynamic pressure, but was found to be insufficient at low supersonic Mach numbers. Transform methods are applied to the convected wave equation for potential flow, and a time-dependent aerodynamic pressure correction factor is obtained. The Lagrangian of the shell system is formulated in terms of the generalized coordinates for all displacements and the Rayleigh-Ritz method is used to derive the governing differential-algebraic equations of motion. Aeroelastic limit cycle oscillations and buckling deformations are calculated in the time domain using a Runge-Kutta method in MATLAB. Three conical shell geometries were considered in the present analysis: a 3-meter diameter 70 deg. cone, a 3.7-meter 70 deg. cone, and a 6-meter diameter 70 deg. cone. The 6-meter configuration was loaded statically and the results were compared with an experimental load test of a 6-meter HIAD. Though agreement between theoretical and experimental strains was poor, the circumferential wrinkling phenomena observed during the experiments was captured by the theory and axial deformations were qualitatively similar in shape. With Piston Theory aerodynamics, the nonlinear flutter dynamic pressures of the 3-meter configuration were in agreement with the values calculated using linear theory, and the limit cycle amplitudes were generally on the order of the shell thickness. The effect of axial tension was studied for this configuration, and increasing tension was found to decrease the limit cycle amplitudes when the circumferential elastic supports were neglected, but resulted in more complex behavior when the supports were included. The nominal flutter dynamic pressure of the 3.7-meter configuration was significantly lower than that of the 3-meter, and it was found that two sets of natural modes coalesce to flutter modes near the same dynamic pressure. This resulted in a significant drop in the limit cycle frequencies at higher dynamic pressures, where the flutter mode with the lower frequency becomes more critical. Pre-buckling pressure loads and the aerodynamic pressure correction factor were studied for all geometries, and these effects resulted in significantly lower flutter boundaries compared with Piston Theory alone. The maximum dynamic pressure predicted by aerodynamic simulations of a proposed 3.7-meter HIAD vehicle was still lower than any of the calculated flutter dynamic pressures, suggesting that aeroelastic effects for this vehicle are of little concern.
Goldman, Benjamin D.↗
An approximation method for configuration optimization of trusses
Two- and three-dimensional elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints. Constraints on member stresses and Euler buckling are imposed and multiple static loading conditions are considered. The method presented here utilizes an approximate structural analysis based on first order Taylor series expansions of the member forces. A numerical optimizer minimizes the weight of the truss using information from the approximate structural analysis. Comparisons with results from other methods are made. It is shown that the method of forming an approximate structural analysis based on linearized member forces leads to a highly efficient method of truss configuration optimization.
Hansen, Scott R.↗
Thermally driven phase transitions in freestanding low-buckled silicene, germanene, and stanene
Low-buckled silicene, germanene, and stanene are group-IV graphene allotropes. They form a honeycomb lattice out of two interpenetrating (A and B) triangular sublattices that are vertically separated by a small distance Δ z . The atomic numbers Z of silicon, germanium, and tin are larger than that of carbon (Z C = 6), making them the first experimentally viable two-dimensional topological insulators. Those materials have a twice-energy-degenerate atomistic structure characterized by the buckling direction of the B sublattice with respect to the A sublattice [whereby the B atom either protrudes above (Δ z > 0) or below (Δ z < 0) the A atoms], and the consequences of that energy degeneracy on their elastic and electronic properties have not been reported thus far. Here, we uncover ferroelastic, bistable behavior on silicene, which turns into an average planar structure at about 600 K. Furthermore, the creation of electron and hole puddles obfuscates the zero-temperature spin-orbit coupling (SOC)-induced band gaps at temperatures as low as 200 K, which may discard silicene as a viable two-dimensional topological insulator for room-temperature applications. Germanene, on the other hand, never undergoes a low-buckled-to-planar two-dimensional transformation, becoming amorphous at around 675 K instead, and preserving its SOC-induced band gap despite band broadening. Stanene undergoes a transition onto a crystalline three-dimensional structure at about 300 K, preserving its SOC-induced electronic band gap up to that temperature. Unlike what is observed in silicene and germanene, stanene readily develops a higher-coordinated structure with a high degree of structural order. Furthermore, the structural phenomena are shown to have far-reaching consequences for the electronic and vibrational properties of those two-dimensional topological insulators.
36 MATERIALS SCIENCE↗
Effects of transverse shearing on cylindrical bending, vibration, and buckling of laminated plates
The displacements for cylindrical bending and stretching of laminated and thick plates are expressed through-the-thickness by a few algebraic terms and a complete set of trigonometric terms. Only a few terms of this series are needed to get sufficiently accurate results for laminated and thick plates. Equations of equilibrium based on a sufficient number of terms of this series for displacements are determined using variational theorems from three-dimensional elasticity. Several examples are worked out. The displacements and stresses are obtained for simply supported isotropic and layered beams with a rectangular cross section and a sinusoidal lateral load distribution. These results are compared to an exact elasticity solution. Results for several approximations are obtained for a simply supported and a clamped beam with a rectangular cross section loaded at the center and results are compared to experimental results. Results are also obtained for isotropic and layered beams with a rectangular cross section for several approximations for the lowest natural frequency and buckling load.
Stein, M.↗
Stability analysis of a reinforced carbon carbon shell
This paper presents the development of a stability analysis for the nose cap of the NASA Space Shuttle Orbiter. Stability is evaluated by the differential stiffness analysis of the NASTRAN finite-element computer code, addressing those nonstandard characteristics in the nose cap such as nonuniform curvature, asymmetrical and nonuniform loads, support fixity, and various combinations of membrane and bending stresses. A full-sized nose cap, thinner than production, was statically tested and stability analyzed. The failing load level correlated to within 30%. The region and mode of buckling that occurred during test was accurately predicted by analysis. The criterion for predicting instability is based on the behavior of the nonlinear deflections. The deflections are nonlinear elastic in that the stresses are well within the elastic range of the material, but the geometry-load relationship produces nonlinear deflections. The load-deflection relationship is well defined by differential stiffness analysis up to the zero-slope portion of the curve, the point of neutral stability or where the shell 'snaps through' just prior to general instability.
Agan, W. E.↗
Observations on the maximum average stress of flat plates buckled in edge compression
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ALUMINUM ALLOYS - ALCOA 14ST↗
Constitutive and Stability Behavior of Granular Materials Under Very Low Effective Stresses
The study of the behavior of granular materials such as sand is of great importance to the understanding of the response of foundations resting on surface and near-surface soil deposits. A clear understanding of the behavior of such materials can also provide insights into other related issues where low effective stresses are encountered such as liquefaction. The main sources of the constitutive and stability properties of cohesionless granular materials is interparticle friction, which in turn under low confinement stress levels is strongly affected by gravitational body forces under terrestrial (1 gravity) conditions. Under moderate-to-high stress levels, the influence of gravity on the behavior of experiments may not be pronounced and therefore the test results in a terrestrial environment may be acceptable for engineering purposes. However, the conduct of experiments on granular materials under very low stress levels and under quasi-static conditions can only be performed in a microgravity environment. A series of displacement-controlled cyclic triaxial compression experiments were performed in a SPACEHAB module on the Space Shuttle during the STS-79 mission to Mir in September, 1996, and the STS-89 mission in January, 1998. The experiments were conducted on six right cylindrical specimens 75 mm in diameter and 150 mm long at effective confining pressures of 0.05, 0.52 and 1.30 kPa. The results show very high peak strength friction angles in the range of 47.6 to 70.0 degrees, which are mainly due to overconsolidation and grain interlocking effects. It was observed that the residual strength levels were in the same range as that observed at higher confining stress levels. The dilatancy angles were unusually high in the range of 30 to 31 degrees. All specimens display substantial initial stiffnesses and elastic moduli during unloading and reloading events, which are nearly an order of magnitude higher than conventional theories predict. A periodic instability phenomenon which appears to result from buckling of multiple internal arches and columnar systems, augmented by stick-slips was observed in the experiments. Computed Tomography (CT) measurements revealed valuable data about the internal fabric and the specimens deformation patterns. Uniform diffuse bifurcation with multiple radial shear bands was observed in the specimens tested in a microgravity environment. In the axial direction, two major conical surfaces were developed. Spatial nonsymmetrical deformations were observed in specimens tested in terrestrial laboratory.
Alshibli, Khalid A.↗
Mechanics of Granular Materials (MGM) Microgravity Experiment
The second series of MGM experiment was conducted during the STS-89 mission in January 1998. The experiment was previously flow on Atlantis's STS-79 mission in September 1996. Six displacement-controlled, drained triaxial compression experiments were performed at very low effective confining stresses. The confining stresses were in the ranges 0.05, 0.52 and 1.30 kPa. Three experiments were subjected to monotonic loading and unloading cycles while the other three experiments were subjected to cyclic loading. The results show very high peak strength friction angles in the range of 47.6 to 70.0 degrees, which are mainly due to overconsolidation and grain interlocking effects. It was observed that the residual strength levels in the monotonic loading experiments were in the same range as that observed at higher confining stress levels. The dilatancy angles were unusually high in the range of 30 to 31 degrees. All specimens display substantial initial stiffnesses and elastic moduli during unloading and reloading events, which are nearly an order of magnitude higher than conventional theories predict. A periodic instability phenomenon which appears to result from buckling of multiple internal arches and columnar systems, augmented by stick-slips was observed in the experiments. Computed Tomography (CT) measurements revealed valuable data about the internal fabric and the specimens deformation patterns. Uniform diffuse bifurcation with multiple radial shear bands was observed in the specimens tested in a microgravity environment. In the axial direction, two major conical surfaces were developed. Spatial nonsymmetrical deformations were observed in specimens tested in terrestrial laboratory.
Alshibli, Khalid A.↗