A crucial element of structural and continuum mechanics, stability theory has limitless applications in civil, mechanical, aerospace, naval and nuclear engineering. This text of unparalleled scope ...presents a comprehensive exposition of the principles and applications of stability analysis. It has been proven as a text for introductory courses and various advanced courses for graduate students. It is also prized as an exhaustive reference for engineers and researchers.The authors' focus on understanding of the basic principles rather than excessive detailed solutions, and their treatment of each subject proceed from simple examples to general concepts and rigorous formulations. All the results are derived using as simple mathematics as possible. Numerous examples are given and 700 exercise problems help in attaining a firm grasp of this central aspect of solid mechanics.The book is an unabridged republication of the 1991 edition by Oxford University Press and the 2003 edition by Dover, updated with 18 pages of end notes.Sample Chapter(s)Introduction (1,516 KB)Chapter 1: Buckling of Elastic Columns by Equilibrium Analysis (14,524 KB)Contents: Buckling of Elastic Columns by Equilibrium AnalysisBuckling of Elastic Frames by Equilibrium AnalysisDynamic Analysis of StabilityEnergy MethodsEnergy Analysis of Continuous Structures and Approximate MethodsThin-Walled BeamsPlates and ShellsElastoplastic BucklingCreep BucklingStability of Inelastic Structures; Bifurcation and Thermodynamic BasisThree Dimensional Continuum Instabilities and Effects of Finite Strain TensorFracture as a Stability ProblemDamage and Localization InstabilitiesReadership: Advanced undergraduates and graduate students, academia and researchers in structural stability.
Quasibrittle (or brittle heterogeneous) materials are becoming increasingly important for modern engineering. They include concretes, rocks, fiber composites, tough ceramics, sea ice, bone, wood, ...stiff soils, rigid foams, glass, dental and biomaterials, as well as all brittle materials on the micro or nano scale. Their salient feature is that the fracture process zone size is non-negligible compared to the structural dimensions. This causes intricate energetic and statistical size effects and leads to size-dependent probability distribution of strength, transitional between Gaussian and Weibullian. The ensuing difficult challenges for safe design are vanquished in this book, which features a rigorous theory with detailed derivations yet no superfluous mathematical sophistication; extensive experimental verifications; and realistic approximations for design. A wide range of subjects is covered, including probabilistic fracture kinetics at nanoscale, multiscale transition, statistics of structural strength and lifetime, size effect, reliability indices, safety factors, and ramification to gate dielectrics breakdown.
Recent investigations prompted by a disaster in Palau revealed that worldwide there are 69 long-span segmental prestressed-concrete box-girder bridges that suffered excessive multi-decade ...deflections, while many more surely exist. Although the excessive deflections were shown to be caused mainly by obsolescence of design recommendations or codes for static creep, some engineers suspect that cyclic creep might have been a significant additional cause. Many investigators explored the cyclic creep of concrete experimentally, but a rational mathematical model that would be anchored in the microstructure and would allow extrapolation to a 100-year lifetime is lacking. Here it is assumed that the cause of cyclic creep is the fatigue growth of pre-existing microcracks in hydrated cement. The resulting macroscopic strain is calculated by applying fracture mechanics to the microcracks considered as either tensile or, in the form of a crushing band, as compressive. This leads to a mathematical model for cyclic creep in compression, which is verified and calibrated by laboratory test data from the literature. The cyclic creep is shown to be proportional to the time average of stress and to the 4th power of the ratio of the stress amplitude to material strength. The power of 4 is supported by the recent finding that, on the atomistic scale, the Paris law should have the exponent of 2 and that the exponent must increase due to scale bridging. Exponent 4 implies that cyclic creep deflections are enormously sensitive to the relative amplitude of the applied cyclic stress. Calculations of the effects of cyclic creep in six segmental prestressed concrete box girders indicate that, because of self-weight dominance, the effect on deflections absolutely negligible for large spans (>150m). For small spans (<40m) the cyclic creep deflections are not negligible but do not matter since the static creep causes in such bridges upward deflections. However, the cyclic creep is shown to cause in bridges with medium and small spans (<80m) a significant residual tensile strain which can produce deleterious tensile cracking at top or bottom face of the girder.
•Multi-specimen set of test data for the size effect on Paris law and fatigue lifetimes.•Improved and more reliable estimates of cyclic fracture process zone size.•Conclusive evidence that the cyclic ...FPZ is smaller than the monotonic one.•Finding a new form of the size effect in Paris law, via updated dimensional analysis.•Analysis leading to a non-monotonic size effect on quasibrittle fatigue lifetimes.
Unlike brittle materials, quasibrittle materials exhibit a structure size effect on the fatigue crack growth rate, particularly on the Paris law coefficient (prefactor). This size effect is strong for specimens not much larger than the dominant material inhomogeneities (or aggregate sizes in concrete), and vanishes for very large structures. It can be quantified by a size adjustment of Paris law which is similar to the size effect law for monotonic loading. But the transitional size D0c at which the transition is centered is not the same. Previous experiments aimed at quantitative analysis of this size effect involved only one or a few specimens per size. Thus the huge scatter, inevitable in fatigue tests, distorted the estimates of D0c and, thereby, also of the size of the cyclic fracture process zone (FPZ), to which D0c is proportional. Here, more reliable estimates of D0c and the cyclic FPZ size are obtained by conducting, on concrete, multiple fatigue tests per size and taking the average. Furthermore, these length characteristics are also estimated numerically using the latest version of the microplane constitutive damage model for concrete (model M7), extended to quasibrittle fatigue. It is conclusively shown that the cyclic FPZ is smaller than the monotonic FPZ. Further, the numerical simulations of the cyclic deformations within the FPZ reveal that the D0c obtained from the previous form of the size-adjusted Paris law is not proportional to the cyclic FPZ size. A new form is proposed and verified by updated dimensional analysis. It involves the transitional sizes for both the monotonic and cyclic size effects and is seen to yield values of D0c that are proportional to the cyclic FPZ size. The ensuing size effect on fatigue lifetimes is simulated using both the previous and new forms. Both forms are found to predict a non-monotonic size effect on the lifetimes, initially decreasing and, after a minimum, eventually increasing with increasing size. It is also shown that, similar to Paris law for brittle fatigue, an extension to the fatigue threshold at nearly vanishing amplitude is impossible because of a transition to Charles–Evans law for static fatigue. Finally, a possible ramification to fatigue of micrometer-scale metallic devices is pointed out.
The failure probability of engineering structures such as aircraft, bridges, dams, nuclear structures, and ships, as well as microelectronic components and medical implants, must be kept extremely ...low, typically <10⁻⁶. The safety factors needed to ensure it have so far been assessed empirically. For perfectly ductile and perfectly brittle structures, the empirical approach is sufficient because the cumulative distribution function (cdf) of random material strength is known and fixed. However, such an approach is insufficient for structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared with the structure size. The reason is that the strength cdf of quasibrittle structure varies from Gaussian to Weibullian as the structure size increases. In this article, a recently proposed theory for the strength cdf of quasibrittle structure is refined by deriving it from fracture mechanics of nanocracks propagating by small, activation-energy-controlled, random jumps through the atomic lattice. This refinement also provides a plausible physical justification of the power law for subcritical creep crack growth, hitherto considered empirical. The theory is further extended to predict the cdf of structural lifetime at constant load, which is shown to be size- and geometry-dependent. The size effects on structure strength and lifetime are shown to be related and the latter to be much stronger. The theory fits previously unexplained deviations of experimental strength and lifetime histograms from the Weibull distribution. Finally, a boundary layer method for numerical calculation of the cdf of structural strength and lifetime is outlined.
Recent earthquakes revealed poor performance of very tall shear walls. This is no surprise because the design for size effect has long been hampered by the lack of large-size tests and by mingling of ...different concretes and different test parameters among the existing test data. Fortunately, recent progress in material modeling and computer power permits overcoming this obstacle by extending the data to large sizes computationally. Herein, a large classical set of reduced-scale shear wall tests performed (at ETH) are selected and used to verify and calibrate the finite element simulations with the crack band model, in which a powerful constitutive damage model--microplane model M7--is implemented. Then the calibrated computer code is used to predict the strength and ductility of much larger shear walls. As expected, the size effect is found to occur if the shear in concrete controls the failure load but not if the steel bar yielding does. Increasing the reinforcement ratio may compromise ductility; it may cause the shear in concrete to control the peak load and the failure to occur before the steel bars begin to yield. Increasing the height-to-width aspect ratio of the wall is shown to lead to flexural failure, which does not show size effect on the shear wall strength. Increasing the wall size may change the failure mechanism, cause the concrete to fail at a lower stress, and shorten or remove the yield plateau. Adding horizontal reinforcing bars to vertical ones tends to prevent inclined shear cracks in the web, although it has little effect on sliding shear. Computer simulations of the present type are helpful for checking the design of large shear walls. Keywords: crack band model; cracking; failure mechanism; finite element analysis; fracture; microplane model; shear wall; size effect.
Engineering structures must be designed for an extremely low failure probability such as 10
−6, which is beyond the means of direct verification by histogram testing. This is not a problem for ...brittle or ductile materials because the type of probability distribution of structural strength is fixed and known, making it possible to predict the tail probabilities from the mean and variance. It is a problem, though, for quasibrittle materials for which the type of strength distribution transitions from Gaussian to Weibullian as the structure size increases. These are heterogeneous materials with brittle constituents, characterized by material inhomogeneities that are not negligible compared to the structure size. Examples include concrete, fiber composites, coarse-grained or toughened ceramics, rocks, sea ice, rigid foams and bone, as well as many materials used in nano- and microscale devices.
This study presents a unified theory of strength and lifetime for such materials, based on activation energy controlled random jumps of the nano-crack front, and on the nano-macro multiscale transition of tail probabilities. Part I of this study deals with the case of monotonic and sustained (or creep) loading, and Part II with fatigue (or cyclic) loading. On the scale of the representative volume element of material, the probability distribution of strength has a Gaussian core onto which a remote Weibull tail is grafted at failure probability of the order of 10
−3. With increasing structure size, the Weibull tail penetrates into the Gaussian core. The probability distribution of static (creep) lifetime is related to the strength distribution by the power law for the static crack growth rate, for which a physical justification is given. The present theory yields a simple relation between the exponent of this law and the Weibull moduli for strength and lifetime. The benefit is that the lifetime distribution can be predicted from short-time tests of the mean size effect on strength and tests of the power law for the crack growth rate. The theory is shown to match closely numerous test data on strength and static lifetime of ceramics and concrete, and explains why their histograms deviate systematically from the straight line in Weibull scale.
Although the present unified theory is built on several previous advances, new contributions are here made to address: (i) a crack in a disordered nano-structure (such as that of hydrated Portland cement), (ii) tail probability of a fiber bundle (or parallel coupling) model with softening elements, (iii) convergence of this model to the Gaussian distribution, (iv) the stress-life curve under constant load, and (v) a detailed random walk analysis of crack front jumps in an atomic lattice. The nonlocal behavior is captured in the present theory through the finiteness of the number of links in the weakest-link model, which explains why the mean size effect coincides with that of the previously formulated nonlocal Weibull theory. Brittle structures correspond to the large-size limit of the present theory. An important practical conclusion is that the safety factors for strength and tolerable minimum lifetime for large quasibrittle structures (e.g., concrete structures and composite airframes or ship hulls, as well as various micro-devices) should be calculated as a function of structure size and geometry.
AbstractThe segmental prestressed concrete box girder of Koror-Babeldaob (KB) Bridge in Palau, which had a record span of 241 m (791 ft), presents a striking paradigm of serviceability loss because ...of excessive multidecade deflections. The data required for analysis have recently been released and are here exploited to show how the analysis and design could be improved. Erected segmentally in 1977, this girder developed a midspan deflection of 1.61 m (5.3 ft) compared with the design camber after 18 years, and it collapsed in 1996 as a consequence of remedial prestressing, after a 3-month delay. Compared with three-dimensional analysis, the traditional beam-type analysis of box girder deflections is found to have errors up to 20%, although greater errors are likely for bridges with higher box-width-to-span ratios than the KB Bridge. However, even three-dimensional finite-element analysis with step-by-step time integration cannot explain the observed deflections when the current American Concrete Institute, Japan Society of Civil Engineers, Comité Euro-International du Béton (or Comité Euro-International du Béton—Fédération internationale de la précontrainte), and Gardner and Lockman prediction models for creep and shrinkage are used. These models give 18-year deflection estimates that are 50–77% lower than measured and yield unrealistic shapes of the deflection history. They also predict the 18-year prestress loss to be 46–56% lower than the measured mean prestress loss, which was 50%. Model B3, which is the only theoretically based model, underestimates the 18-year deflection by 42% and gives a prestress loss of 40% when the default parameter values are used. However, in Model B3, several input parameters are adjustable and if they are adjusted according to the long-time laboratory tests of Brooks, a close fit of all the measurements is obtained. For early deflections and their extrapolation, it is important that Model B3 can capture realistically the differences in the rates of shrinkage and drying creep caused by the differences in the thickness of the walls of the cross section. The differences in temperature and possible cracking of the top slab also need to be taken into account. Other paradigms on which data have recently been released are four bridges in Japan and one in the Czech Republic. Their excessive deflections can also be explained. The detailed method of analysis and the lessons learned are presented in Part II.
The comprehensive fracture tests of notched and unnotched beams presented in the preceding paper are evaluated to clarify the size effects and fracture energy dissipation. The test results for beams ...with notches of various lengths agree closely with Bažant’s Type 2 size effect law and yield the value of the initial fracture energy Gf. Since nearly complete post-peak softening load–displacement curves have been obtained, the total energy dissipation by fracture can be accurately evaluated and used to determine the RILEM total fracture energy GF. For the present concrete, GF/Gf≈2. The transition between the Type 1 and 2 size effects are determined and is found to be very different from that assumed in Hu and Duan’s boundary effect model. Determination of a universal size effect law fitting the beam strengths for all the sizes and all notch depths is relegated to a subsequent paper.
Recent compilation of data on numerous large-span prestressed segmentally erected box girder bridges revealed gross underestimation of their multi-decade deflections. The objective of this paper is ...to provide a systematic and comprehensive presentation, appropriate not only for bridges but also for any large creep-sensitive structure. For each time step, the solution is reduced to an elastic structural analysis with generally orthotropic elastic moduli and eigenstrains. This analysis should normally be three-dimensional. The rate-type creep analysis is computationally more efficient than the classical integral-type analysis. More importantly, though, it makes it possible to take into account the evolution of various inelastic and nonlinear phenomena such as tensile cracking, cyclic creep, and stress relaxation in prestressing tendons at variable strain, as well as the effects of humidity and temperature variations, and the effect of wall thickness variation on drying creep and shrinkage. Finally, the advantages compared to the existing commercial programs, based on step-by-step integration of memory integrals, are pointed out and illustrated by a simple example.
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