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Technical Material
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.

Imperfections for Dynamic Pulse Buckling

Paper on Dynamic Pulse Buckling of
Imperfection-Sensitive Shells
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:

Dynamic Pulse Buckling of Imperfection-Sensitive Shells

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:

The World's Roughest Roads -- Seminar Slides

The World's Roughest Roads -- 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.
This is a 1113-KB file, so if you have a slow Internet connection downloading before viewing is recommended.

Bar Impact Buckling Movie -- Right click to download.  Left click action depends on your browser.
This is a small 66-KB file so either option is reasonable even with a slow Internet connection.  It is an executable file, so your security software will ask your permission to open it.  I vouch it is bug free and will play properly if you have a Windows PC with DOS capability.

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.


Representative Publications
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.
Click on above link to view complete 2013 edition.
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.


Representative Peer-Reviewed Papers

Only an incomplete list of papers in the ASME Journal of Applied Mechanics is given. A similar number has been published here and elsewhere on a variety of topics. The title and author links are now broken (Forbidden), I suppose by the NSA because so many hits were made from Russia, China, and so on. I defer to their (the NSA) superior wisdom on such matters.

In addition, the paper
    Critical Loads to Destroy Reentry Vehicles with X-Rays, Blast, or Directed Beams
    Journal of Defense Research
was selected as the Best Paper of 1977.  The award was presented by then Deputy Secretary of Defense William J. Perry, who became Secretary of Defense and later a research fellow at the Hoover Institute, Stanford University.

All of the papers below are on dynamic pulse buckling, a subject pioneered at Stanford Research Institute (now SRI International) beginning about 1960. The papers are simply in the order in which they appear in the ASME list obtained by clicking on the heading above to go to the ASME digital library, and then searching on lindberg.. The publication dates span from 1964 to 1992.

A Case and Elastomer Annulus Under Lateral Impulse
Herbert E. Lindberg
and Yvonne D. Murray
Appl. Mech. Rev. 42, S142 (1989)

Fourier series and finite element solutions are given for stresses in a cylindrical case filled with an annulus of elastomeric material. The Fourier series solutions are for membrane stresses, which dominate at early time, and are given for three case-elastomer models: (1) a slide boundary model in which the case wall moves as a unit with the elastomer in radial motion but, with a weak bond between the case and elastomer, is free to slide relative to the elastomer in tangential motion, (2) a unit motion model for a well-bonded elastomer in which the case wall and elastomer are assumed to move together as a unit in both radial and tangential motion, and (3) a radiation boundary model in which tangential motion of the case wall radiates energy into a well-bonded elastomer. For typical case, elastomer and bond mechanical properties, the radiation boundary model gives the most appropriate solution, which differs substantially from the other solutions even for very soft elastomers. Finite element solutions agree closely with and support the validity of all three analytical models, which were used to guide the finite element “experiments” and interpret and generalize their results.

Dynamic Plastic Pulse Buckling Beyond Strain-Rate Reversal
H. E. Lindberg
and T. C. Kennedy
J. Appl. Mech. 42, 411 (1975)

A large deflection, elastic-plastic numerical theory (SABOR/DRASTIC 6) is used to investigate dynamic buckling motion beyond strain-rate reversal in cylindrical shells under radial impulse. It is found that the mode number of the most amplified harmonic is smaller than predicted by the simple analytic tangent modulus theory because of the increased stiffness after strain-rate reversal, neglected in the simple theory. Nevertheless, the simple theory gives reasonable estimates for threshold buckling impulses. In buckling well beyond threshold (about three times the threshold impulse in a thin shell, radius-to-thickness ratio of 120) plastic deformation becomes more localized at the buckle crests, but compressive strain continues to dominate so that a simple plastic hinge postbuckling analysis would be inadequate. Accuracy of the numerical theory is confirmed by comparison with experiment (asymmetric loading below buckling threshold and symmetric loading above) and with the tangent modulus theory prior to strain-rate reversal.

Stress Amplification in a Ring Caused by Dynamic Instability
H. E. Lindberg
J. Appl. Mech. 41, 392 (1974)

An analysis is presented for stress amplification in a ring caused by the dynamic instability of symmetric in-and-out breathing oscillations, which results in energy transfer to flexural modes. Stress amplification is shown to depend on a ring stability parameter p that is proportional to the unperturbed hoop strain and inversely proportional to the thickness-to-radius ratio. The analysis is a generalization of earlier work by Goodier and McIvor. They showed that, with p tacity assumed small, stress is amplified by approximately the factor sqrt(6), independent of the wave number of the flexural mode into which the breathing energy is transferred. The present analysis shows that for large p (increased instability), higher-order nonlinear terms must be included in the differential equations in order to give bounded solutions. With these terms included, stress amplification departs from sqrt(6) as p is increased, the peak compressive stress becoming larger and the peak tensile stress becoming smaller. With damping also included, the amplification approaches unity (no amplification) as p approaches zero, and passes through a maximum as p is increased. An experimental example of flexural fracture caused by such amplification is given.

Impulse Buckling of an Elastic-Plastic Cylinder Containing an Elastic Core
H. Vaughan
and H. E. Lindberg
J. Appl. Mech. 35, 827 (1968)

Abstract not available.

Dynamic Plastic Buckling of Sandwich Shells
Henry Vaughan
and H. E. Lindberg
J. Appl. Mech. 35, 539 (1968)

The formation of small, flexural wrinkles in the walls of a sandwich shell under uniform radial impulse is examined. A buckling criterion based on the magnifications of initial wall imperfections is used to evaluate threshold impulses. Using a given weight of material, this criterion is used to determine the optimum ratios of the inner and outer wall thicknesses for various core thicknesses. It is shown that the optimum shell for a given weight of material and outer diameter is a single-wall shell. The loss in efficiency obtained by using typical sandwich shells is about 30 percent. These results are supported by experimental evidence.

Dynamic Buckling of a Thin Cylindrical Shell Under Axial Impact
H. E. Lindberg
and R. E. Herbert
J. Appl. Mech. 33, 105 (1966)

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 wave-lengths. 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.

Dynamic Plastic Buckling of a Thin Cylindrical Shell Containing an Elastic Core
H. E. Lindberg
J. Appl. Mech. 32, 803 (1965)

Impact Buckling of a Thin Bar
H. E. Lindberg
J. Appl. Mech. 32, 315 (1965)

Experiments are presented which demonstrate the statistical nature of the buckling of thin strips under very high axial compression. Probability distributions of wavelengths determined theoretically from assumed “white-noise” perturbations in initial shape are compared favorably with the experimental distributions. In other experiments, a new optical-lever method of recording lateral motions is used to observe bending waves and buckling in an impacted strip. From these and experiments on large-strain buckling of rubber strips it is concluded that effects of axial-wave propagation do not significantly influence the wavelengths at which the buckles form.

Buckling of a Very Thin Cylindrical Shell Due to an Impulsive Pressure
H. E. Lindberg
J. Appl. Mech. 31, 267 (1964)

This paper shows that when very thin cylindrical shells are subjected to an external impulsive pressure, the interaction between the radial, purely extensional mode and the flexural modes is sufficient to precipitate permanent wrinkles. The theory predicts that these wrinkles occur at a wavelength which depends on the magnitude of the pressure pulse as well as on the cylinder parameters; experiments on cylindrical shells and thin strips confirm this prediction. Framing camera photographs showing the formation of wrinkles of this type are also presented.

An Evaluation of Convex Modeling for Multimode Dynamic Buckling
Herbert E. Lindberg
J. Appl. Mech. 59, 929 (1992)

Convex and probabilistic model solutions are obtained for multimode dynamic buckling of cylindrical shells with uncertain imperfections under symmetric radial impulsive loads. It is found that the maximum possible buckling deformations for any imperfection within uniform bounds ± delta-hat can be made comparable to the buckling deformations from the probabilistic models at a reliability of about 99.5 percent. Numerical evaluation and interpretation of the convex model is much simpler than for the probabilistic models, and the convex model solution provides means for quality control of each and every shell by simply recording delta-hat from appropriately filtered imperfection measurements. The power and simplicity of convex modeling for multimode buckling shows promise for more complex problems than require finite element modal analysis.

Dynamic Pulse Buckling of Imperfection-Sensitive Shells

Herbert E. Lindberg
J. Appl. Mech. 58, 743 (1991)

The theoretical basis of two related but distinctly different dynamic buckling criteria are summarized with the objective of demonstrating the range of applicability of each, so that together they cover the entire range of dynamic pulse loads from nearly impulsive loads to step loads of infinite duration. The example chosen is a cylindrical shell under elastic axial loads but the approach is applicable more generally. A critical amplification-of-imperfections criterion with a linear shell theory is shown to be applicable for short duration loads, for which a threshold nonlinear divergence criterion gives loads an order of magnitude too conservative. Conversely, the linear theory is inapplicable for long duration loads, for which critical loads are lower than the linear static buckling load because of imperfection sensitivity. In this range the threshold nonlinear divergence criterion is used. For loads of intermediate duration, an extended critical amplification criterion is used with equations that conservatively assume zero static buckling load but give an unchanged formula for critical load amplitude-duration combinations.

Convex Models for Uncertain Imperfection Control in Multimode Dynamic Buckling
H. E. Lindberg
J. Appl. Mech. 59, 937 (1992)

Control of uncertain imperfections by means of convex bounds on finite Fourier transforms is shown to be more direct and not as overly conservative as control based on uniform bounds, i.e., bounding maximum and minimum imperfections. With either method, conservatism in bounds on buckling response is reduced by filtering the imperfection measurements. Extraction of the needed filtered information by operating directly on the Fourier coefficients is straightforward and allows use of additional information on the variation of the coefficients with mode number. Use of this information in example multimode buckling problems gives a bound on maximum possible buckling response that is a factor of1.6 larger than the response at a reliability of 99.5 percent for hypothetical (but reasonably representative) probabilistic imperfections. The bounding response itself, of course, does not depend on any assumptions concerning the probabilistic distribution of imperfections. Two additional combined uniform and Fourier ellipsoid bound models further reduce this factor to 1.1 and 0.5, and require only a simple, unfiltered imperfection bound measurement during quality control inspection.

Dynamic Response and Buckling Failure Measures for Structures With 
Bounded and Random Imperfections

H. E. Lindberg
J. Appl. Mech. 58, 1092 (1991)

Comparisons between an unknown-but-bounded imperfection model and a random imperfection model show that for simple pointwise failure measures, at least, the two models give the same expressions for their measures of response, but each measure has a distinctly different interpretation. The former gives the maximum possible response for any imperfection within a specified bound. The latter gives the standard deviation of response, which, together with the statistical distribution, can be used to specify the maximum response at a specified confidence level. However, since the statistical distributions of imperfections, and hence of the response are often unknown, confidence levels are difficult to define, especially in the tail of the distribution at high confidence levels. The unknown-but-bounded model requires less information about the imperfections to come to a well-defined bound on response. It is further shown that, while the maximum possible response might seem to be a severe failure avoidance criterion, it can be less constricting than having to impose artificially high confidence levels with poorly known statistical distributions.


Representative Technical Reports
Gleaned Quickly from the Internet


H. E. Lindberg,  Details der Publikationsliste

Zeitraum   1958 - 2005
Anzahl   15

Abrahamson, G. R.
, Florence, A. L., Lindberg, H. E.

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.

Abrahamson, G. R.
, Florence, A. L., Lindberg, H. E.

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.

Lindberg, H. E.
, Firth, R. D.

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)

Lindberg,H. E.
, Herbert,R. E.

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)

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)

Response of Earth Penetrator Structures in Angle-of-Attack Impacts (2003)
Colton, J. D.
, Lindberg, H. E., Emerson, R. E.

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.

The Mechanical Loads from LSD Waves and Their Simulation. Volume I. Analysis and Pressure Measurements. (2002)
Holmes,B. S.
, Targer,C., Erlich,D., Lindberg,H. E.

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.

Simulation of Earth Penetration Loads. (2002)
Colton,J. D.
, Emerson,R. E., Lindberg,H. E.

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)

Laboratory Investigation of Rock Cavity Reinforcement. (2002)
Kennedy,T. C.
, Lindberg,H. E.

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.

Theoretical and Laboratory Study of Deep-Based Structures. Volume I. Triaxial Machine for Static and Dynamic Testing of 12-Inch Diameter Rocks. (2002)
Senseny,P. E.
, Lindberg,H. E.

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 Experimental Study of Deep-Based Structures in Intact and Jointed Rock. (2002)
Senseny,P. E.
, Lindberg,H. E.

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.

Theoretical and Laboratory Study of Deep-Based Structures. Volume II. Model Tests and Analyses of Mighty Epic Structures. (2002)
Senseny,P. E.
, Lindberg,H. E.

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 Study of Deep-Based Structures in Support of DIABLO HAWK. (2002)
Seneny,P. E.
, Lindberg,H. E.

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.

Lindberg, H. E.
, Anderson, D. L., Firth, R. D., Parker, L. V.

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.

Investigation of a nonlinear control system (1958)
I. Flugge-Lotz
, C. F. Taylor, H. E. Lindberg

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