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Latest update February 26, 2012
|Several years ago I posted a much-reduced
version of my book on pulse buckling (description and link below),
which I called The Little Book of Dynamic Buckling
(left click to open, right click to save as a file). I had
intended to add several chapters to this little book but my scanner
broke down and then I lost many files during transitions through at
least five upgrades to new computers. I've now scrounged through my old floppy disks (had
to have them read at six bucks per floppy -- awk!) and found a bit of
what I had intended to include in these added chapters. For now, I'm
just posting papers from these disks as I'm able to put them together
from fragmented data. All of these papers were created with TeX
and most of the figures with raw Postscript, so you can enlarge them
as much as you like and the text and figures will remain crisp.
Paper on Imperfections for
Dynamic Pulse Buckling
Posted March 2, 2012, © Herbert E. Lindberg
I wrote this paper in 1986 to make available procedures to use in finite element calculations of buckling in thin-walled structures. Without a suitable form for imperfections to use in the initial conditions for such calculations there would be no rational way for the calculations to precipitate buckling. Such calculations are common for military structures and in the design of crashworthy civilian road vehicles.
Paper on Dynamic Pulse Buckling of
Posted February 29, 2012, © Herbert E. Lindberg
I wrote this paper in 1989 to make available extensions and corrections to a paper by Hutchinson and Budiansky on dynamic buckling of cylindrical shells from suddenly applied axial loads. This theory is in my 1987 book on pulse buckling but was not more generally available to the applied mathematics and engineering mechanics community. Click below to read the paper on line or print it for detailed study:
Paper on the World's Roughest Roads
Posted February 26, 2012, © Herbert E. Lindberg
I wrote this paper in 2003 to present to the Engineering Mechanics Seminar at Stanford University. It is an application of the relatively new theory of convex modeling of imperfections, in this case road profiles. The substance is given briefly in the slides I used, followed by the complete paper:
Note: The results can be used to show that
"speed bumps" are typically much too narrow to slow the true
speeding culprits -- they go over them fast and feel only a sudden bump, while
the rest of us go over them slowly, which results in maximum
acceleration from the short bump. Speed bumps should be at least 5 feet wide,
so they are tuned to give larger accelerations to the speeders and small
accelerations to those traveling at the prescribed speed.
Textbook on Dynamic Buckling
Latest update September 1, 2007, © Herbert E. Lindberg
For a long time the only technical material on this web site was a small textbook on dynamic buckling, given as a pdf file, and an associated bar buckling movie that can be run as a DOS application. Further material may be added if it proves useful to others.
Little Book of
Dynamic Buckling -- Right click to download book, left click to
display it in your browser.
Bar Impact Buckling
Movie -- Right click to download. Left click action depends on
Instructions for running the movie for various parameters are given on the first few screens. Please note the last instruction -- you must enter time increments less than one in the form 0.2, and not simply .2, which would give a run time error.
by H. E. Lindberg
Dynamic Pulse Buckling
Martinus Nijhoff Publishers
with A. L. Florence, 1987, 384 pp.
Wildflowers of Bridgeport
Special Publication, 2009, 128 pp.
For South Yuba River State Park, CA
Response of Reentry-Vehicle-Type Shells
to Transient Surface Pressures
Poulter Laboratory for High Pressure Research
Stanford Research Institute, 1969, 278 pp.
Sheet Explosive Simulation for Combined
Shock and Structural Response
Poulter Laboratory for High Pressure Research
Stanford Research Institute
with J. D. Colton, 1970, 278 pp.
Handbook of Nuclear Weapon Effects
John Northrup, Editor, 1996, 736 pp.
Chapter 7, X-Ray Radiation, H. E. Lindberg, 62 pp.
A Remark on Books
One of my son Craig's girl friends (she is a girl and a friend, but not a girlfriend -- Craig is married to someone else!) remarked that she found it was amazing that my first book above is on applied mathematics and the second book is on wildflowers, a completely "different area". I didn't find the two books that different in interest but didn't say why. Recently I came across work by Vihart that might explain why, through an exploration of the mathematics of plant growth. Click here to see her work.
Theoretical and experimental structural response investigations of space-vehicle-type structures under suddenly external surface loads are described. The simulation of a simultaneous impulsive load by a traveling load such as produced by an explosive is analyzed for the string and membrane. Three dynamic buckling problems are investigated: (1) dynamic plastic-flow buckling of flat plates due to in-plane flow, (2) dynamic elastic buckling of a thin cylindrical shell under axial impact, and (3) dynamic buckling of cylindrical shells of a strain-rate sensitive material. A scheme for correlating the results of structural response investigations concerned with dynamic failure loads of structures is presented and a brief review of available results is given.
Two types of response are treated, dynamic plastic bending of beams and circular plates under transient lateral loads, and dynamic pulse buckling of bars and cylinders under transient axial thrust or lateral pressure. For each type, the fundamental theory is developed first for the simple beam or bar and then extended to the more complex structural elements. In the bending problems, both simply supported and clamped boundaries are treated. The loads are of the blast type, consisting of a sudden rise to a peak load followed by a decay to zero pressure in an arbitrary duration, including the extremes of an ideal impulse and a step load. Numerical examples are given for rectangular, triangular, and exponential decay shapes. In the buckling problems, simply supported bars under both elastic and plastic thrust are treated for eccentric impact and for bars with random imperfections. Buckling of cylindrical shells is treated for transient lateral blast pressures, again over the entire range of durations. In both the bending and buckling problems, experimental results are given to demonstrate the mechanisms of deformation and the accuracy of the theories in calculating maximum bending deformations or critical buckling loads.. See also Volume 12, AD379848. Prepared in cooperation with Stanford Research Inst., Menlo Park, CA.
Techniques are described for simulating blast-type transient surface loads of nearly exponential pulse shape having characteristic times (impulse/peak pressure) ranging from 10 to 1000 microsec (the quasi-impulsive range for cylindrical shells about 1 foot in diameter). Pressure-time histories are measured at various positions around and along cylindrical models 3.5, 6, and 12 inches in diameter. A basic set of loads is obtained consisting of two limiting pressure distributions, an asymmetric distribution typical of side exposure to a normally incident blast wave, and a symmetric distribution typical of nose-on exposure. All of the loads are obtained using sheet explosive charges of various forms, from flat to semicylindrical to completely cylindrical surrounding the model and flat charges suspended at various standoffs in a shock tube. In support of the experiments, the self-similar solutions for blast waves from intense explosions are used to calculate the range of sheet charges needed to produce loads of interest and to show that the corresponding spherical charges become much too small and much too large for practical application near the extremes in load duration. Approximate formulas are also derived (the Korobeinikov theory) for the variation of peak pressure with distance from plane, cylindrical, and spherical charges. The range of validity of the formulas extends from high pressures, where the self-similar solutions are valid, to acoustic shock pressures. Experimental measurements from the present program and from compiled blast data show excellent agreement with the theory over the entire range. (Author)
Buckling of thin cylindrical shells from axial impact is studied under the assumption that initial imperfections can be approximated by 'white noise'. Linear small deflection theory is used to calculate the resulting growth of the normal modes and a statistical analysis gives the expected values for the 'preferred' axial and circumferential wavelengths. Very high-speed photographs (240,000 frames/sec) of shells buckling under axial impact show excellent agreement with the theory and demonstrate that large deflection buckling follows the pattern established by the early linear motion. (Author)
RESPONSE OF SIMPLIFIED ICBM-TYPE STRUCTURES TO IMPULSIVE LOADING. VOLUME II. DYNAMIC BUCKLING OF RINGS, PLATES, AND CYLINDRICAL SHELLS UNDER IMPULSIVE LOADING. (2005)
Abrahamson, G. R., Florence, A. L., Goodier, J. N., Lindberg, H. E., Vaughan, H.
Structural response of reentry-vehicle-type structures to suddenly applied surface loads, body loads, and combined surface and body loads is investigated. Volume I contains a review of simulation requirements and methods for structural response, theoretical and experimental structural response investigations for body loads and combined surface and body loads, and a method for ABM warhead selection for observable structural kill. Volume II contains five structural response investigations involving dynamic buckling of rings, plates, and cylindrical shells under surface loads. (Author)
An analysis based on a one-dimensional beam-mass model was developed to predict the early-time response of penetrator structures in angle-of-attack impacts. The model was verified by comparison with the strain response measured in a reverse ballistics test and with the strain response measured in idealized scale model penetrators. Loads were simulated by a device that produces the resultant force-time history near the front end of the penetrator. The tests indicated that the loader could be built in a larger size to test full-scale penetrators. The analysis was then used to investigate penetrator response. It was found that the peak compressive strain, which determines whether or not the penetrator casing fails, depends on the magnitude of the lateral load produced by impacts at an angle of attack, the load rise time (which is inversely proportional to impact velocity), and the relative mass of the nose and aft sections. Finally, a procedure was devised to characterize the strength of penetrator structures in terms of impact velocity and angle of attack. The resulting critical impact curves can be used to make tradeoffs among structural dimensions (e.g., length and wall thickness), to select the best structure for a particular applications, and to provide a framework for planning and interpreting experiments and more detailed calculations.
Two new analyses have been developed to predict the surface loads produced by a laser supported detonation (LSD). The first analysis is based on the self-similar theory of expanding blast waves and is very simple to use. It is recommended for systems studies of target damage. The second is a combination of the method of characteristics and self-similar blast wave analysis and provides a more accurate picture of the plasma expansion process. Both models predict pressure pulse and impulse intensity distributions on the target surface with more realistic shapes than those predicted by the model of Pirri. Measurements of surface pressure inside the laser spot show that the pressure histories are close to those predicted by the more complete theory, but with a more rapid decay. This difference between analyses and experiment is attributed to the complex beam profile and pulse shape in the experiments and the comparable uncertainty in measuring laser energy. Analysis of the LSD explosive simulation technique (ESD) showed that it is possible to model both near field and far field effects in the same experiment. However, if near field loads are simulated precisely, far field loads will be 25 percent lower than those of an LSD.
An experimental apparatus was developed to simulate the impact loads on earth penetrating structures. The device has been built to test 1/4-scale model structures, but the design is suitable for fabrication in a larger size for testing full-scale structures. The device uses the controlled flow of high-pressure explosion product gases acting against a piston to apply either an axial load or combined and lateral loads to a model structure. Loading parameters such as rise time and peak load can be easily varied from test to test by simple adjustments of the device. The technique allows hardwired measurement of the applied load and the response of the structure. A series of initial calibration tests conducted without a model structure indicates that the device operates properly. (Author)
The response of monocoque direct-contact and backpacked liners as hardened buried structures was studied through experiments on scale models (5/8-inch diameter tunnel), guided by analyses of structure and rock response assuming a Mohr-Coulomb rock strength characterization. Two loading machines were modified to extend their capabilities and improve their performance. The modifications to the machines produced improvements in tunnel access, in sealing between vertical and lateral pressure chambers, in dynamic pressure pulses, and in general test procedures.
Design and fabrication of large-scale testing machine for static and dynamic testing of rock cavity reinforcement is described. The machine is patterned after a smaller prototype developed previously for DNA. The larger-scale machine can be used to apply static and dynamic triaxial loads on specimens that are 12 inches (0.3 m) in diameter and 12 to 18 inches (0.3 to 0.45 m) high, with rock cavities 2 inches in diameter. A variety of triaxial loadings are possible which permit laboratory simulation of different loadings imposed on deep-based structures in the field. The machine can apply vertical pressures up to 2 kbar (0.2 GPa) statically and to 1 kbar (0.1 GPa) dynamically. The maximum lateral pressure is 1.5 kbar (0.15 GPa) statically or dynamically. The testing machiine can also be used to apply dynamic loading superimposed on a static preload. The static preload pressures can be as large as 0.2 kbar (0.02 GPa). The tunnel cavity is maintained at ambient pressure, with access at both ends for photography and electrical instrumentation. (Author)
Theoretical and laboratory studies were performed to investigate: (1) effects of lateral confinement and rock specimen-to-tunnel diameter ratio in laboratory testing of reinforced tunnels in rock, (2) tunnel response in jointed rock and (3) response to repeat loading of various reinforced tunnels in intact and jointed rock. Results of the laboratory study of effects of lateral confinement and rock specimen-to-tunnel diameter ratio show that: (a) Small deviations from the uniaxial strain lateral confining pressure (10% to 20% over- or underconfinement) cause correspondingly small deviations in the loading needed to produce a critical design crown-invert tunnel closure, (b) The specimen-to-tunnel diameter ratio has a small but measurable effect (less than 20%) on critical loads for tunnel closure in the range used in laboratory testing, and (c) The presence of the tunnel does not cause the laboratory specimen to bulge. Specimen lateral boundaries remain straight to within the accuracy required for uniaxial strain tunnel response (to within 200 microstrain for tunnel closures of interest.
Laboratory tests were performed on 5/8-inch- (16mm) diameter scale models of cylindrical structures fielded in Mighty Epic. The model structures were tested in 4-inch- (0.1-m) diameter specimens of SRI RMG 2C2, a tuff simulant. Results from these tests show that (1) structures are damaged less during dynamic loading than during static loading, (2) if yielding occurs in the free field, tunnel closure is not greatly influenced by structure strength, and (3) structures can effectively resist deformation under repeated loading. Theoretical analyses were performed for axisymmetric loading of a deep-based structure in SRI RMG 2C2. Results show that (1) the analyses can predict tunnel closures measured in the isotropic loading laboratory tests, (2) theoretical tunnel closures under plane strain loading (radical pressure at infinity and axial strains identically zero) and isotropic loading (hydrostatic pressure at infinity) differ only slightly, but (3) tunnel closure and rock stress field under end-on loading differ substantially from closure and stress under isotropic loading; thus substantial theoretical extrapolation of rock response is needed to predict tunnel closures in the field during end-on loading using isotropic loading laboratory data, even though both loading types produce symmetric tunnel response. (Author). See also Volume 1, AD-A063 487.
Laboratory studies were performed in support of DIABLO HAWK structures and cable-hardening experiments to investigate: (1) the influence of loading rate on tunnel closure in both water-saturated and dry specimens of SRI RMG 2C2, a tuff simulant; (2) borehole collapse mechanisms and borehole/cable interaction; and (3) the influence of joints and joint orientation on the closure of circular tunnels in specimens of jointed 16A rock simulant. Results for SRI RMG 2C2 show that greater pressure must be applied in dynamic tests than in static tests to achieve a given tunnel closure. The percentage increase in pressure is twice as large for saturated specimens than for dry specimens. This larger loading rate effect in saturated specimens is due, in part, to no porewater migration and drainage in the dynamic tests. Static and dynamic borehole collapse and borehole/cable interaction test results show that springline collapse is the dominant borehole closure mechanism and that the most severe loading on the cable is from the inward motion of springline rubble. Observed critical loads for cable damage suggest that the TRW fielding ranges in DIABLO HAWK should provide the desired range of damage from none to severe.
The response of re-entry vehicle-type shells to blast loads is investigated and is described in terms of surface loads on the vehicle. Extensive data on both surface pressures and structural response of cylindrical and conical shells are presented. Peak pressure and impulse are identified as the important load characteristics which determine structural response and critical loads are presented in these terms. A theoretical description of dynamic buckling of cylindrical shells subjected to transient surface pressures is given to explain observed modes of failure and to predict critical loads for buckling over a wide range of load and structural parameters. (Author). See also Rept. no. LMSC-B130200-Vol-4-B, AD-369 280. Prepared in cooperation with Stanford Research Inst., Menlo Park, Calif.
A discontinuous variation of coefficients of the differential equation describing the linear control system before nonlinear elements are added is studied in detail. The nonlinear feedback is applied to a second-order system. Simulation techniques are used to study performance of the nonlinear control system and to compare it with the linear system for a wide variety of inputs. A detailed quantitative study of the influence of relay delays and of a transport delay is presented.
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An interesting unrelated paper is included here for those with exceptional curiosity
The Darkon Theory of Light