Asymptotic Matching Analysis of Scaling of Structural Failure Due to Softening Hinges. I: Theory

Propagation of a crack from the tensile face of a beam causes postpeak softening, i.e., the bending moment decreases at increasing rotation of the hinge. Examples are unreinforced concrete beams and plates, foundations plinths, retaining walls, tunnel linings, or arch dams. Softening in an inelastic hinge is also caused by compression crushing of concrete. This happens in reinforced concrete beams that are prestressed, overreinforced, retrofitted by laminates, or subjected to a large enough axial compressive force, which is typical of columns as well as frames or arches with a large enough horizontal thrust. Hinge softening may also be caused by plastic buckling of flanges in deep thin-wall steel beams. An inevitable consequence of inelastic hinge softening in statically indeterminate structures requiring more than one inelastic hinge to fail is a size effect. Although finite element solutions are possible, general analytical formulas for the size effect in such structures do not exist, because of the complexity of response. The idea of this two-part paper is to exploit the technique of asymptotic matching in order to derive approximate formulas for the entire size range. Exact analytical solutions of the nominal strength of structure are derived for the large-size asymptotic case, for which the hinges soften one by one rather than simultaneously, and for the small-size asymptotic case, for which the classical plastic limit analysis applies. Matching these asymptotic solutions by a smooth formula then yields simple, yet general, size effect laws for the peaks and troughs of the load-deflection diagram through the entire size range. The size effect found is very different from the classical size effect in quasibrittle structures failing due to a single dominant crack. The theory is developed in the present Part I, and analysis of its implications is relegated to Part II.     

Why the Observed Motion History of World Trade Center Towers Is Smooth

The collapse of the World Trade Center towers was initiated by the impact of the upper falling part onto the underlying intact story. At the moment of impact, the velocity of the upper part must have decreased. The fact that no velocity decrease can be discerned in the videos of the early motion of the tower top has been recently exploited to claim that the collapse explanation generally accepted within the structural mechanics community was invalid. This claim is here shown to be groundless. Calculations show that the velocity drop is far too small to be perceptible in amateur video records and is much smaller than the inevitable error of such video records.     

Microplane Model M5f for Multiaxial Behavior and Fracture of Fiber-Reinforced Concrete

Despite impressive advances, the existing constitutive and fracture models for fiber-reinforced concrete (FRC) are essentially limited to uniaxial loading. The microplane modeling approach, which has already been successful for concrete, rock, clay, sand, and foam, is shown capable of describing the nonlinear hardening–softening behavior and fracturing of FRC under not only uniaxial but also general multiaxial loading. The present work generalizes model M5 for concrete without fibers, the distinguishing feature of which is a series coupling of kinematically and statically constrained microplane systems. This feature allows simulating the evolution of dense narrow cracks of many orientations into wide cracks of one distinct orientation. The crack opening on a statically constrained microplane is used to determine the resistance of fibers normal to the microplane. An effective iterative algorithm suitable for each loading step of finite element analysis is developed, and a simple sequential procedure for identifying the model parameters from test data is formulated. The model allows a close match of published test data on uniaxial and multiaxial stress–strain curves, and on multiaxial failure envelopes.     

Is the cause of size effect on structural strength fractal or energetic-statistical?

The size effect on structural strength is an important phenomenon with a very old history. Unfortunately, despite abundant experimental evidence, this phenomenon is still not taken into account in most specifications of the design codes for concrete structures, as well as the design practices for polymer composites, rock masses and timber. The main reason appears to be a controversy between two different theories of size effect, namely the theory based on energetic-statistical scaling and the theory based on ideas from fractal geometry. This paper aims to critically analyze these two theories, examine their hypotheses and point out the limitations, in order to help code-writing committees choose a rational basis for their work. The paper begins by reviewing the theory of energetic size effect and the efforts to explain the size effect by fractal geometry. The advantages and disadvantages in modeling the structural size effect by fractals are pointed out. Certain flaws in the fractal theory of size effect are illuminated and it is shown that various aspects of this theory lack a sound physical or mathematical basis, or both. The paper ends by recommending how engineering designers and code writers should take the size effect into account.     

Carbon nanotubes: do they toughen brittle matrices?

The development of a model CNT-brittle matrix composite system, based on SiO₂ glass containing well-dispersed CNTs at up to 15 wt%, allows a direct assessment of the effect of the nanoscale filler on fracture toughness (K _IC). Samples were prepared by colloidal heterocoagulation followed by spark plasma sintering. Detailed K _IC measurements, using both indentation and notched beam techniques, show a linear improvement with CNT content, with up to a twofold increase of fracture toughness at maximum loading. The results from the two methods used in this study show equivalent trends but differing absolute values; the relative merits of these two approaches to measuring nanocomposite toughness are compared. Possible toughening mechanisms associated with CNT pull-out, crack bridging, and crack deflection are identified, and discussed quantitatively, drawing on conventional short-fibre composite theory and the potential effects of scaling fibre diameter.     

Macrocyclic diamide based poly(vinyl chloride) membranes with a high lithium selectivity

Summary Macrocyclic diamides 2, 3 and 4 of 4,4,5,5-tetramethyl-3,6-dioxaoctanedioic acid were prepared. These compounds incorporated in poly (vinyl chloride) membranes show a remarkable selectivity for lithium ions.     

Hydraulic formulae for the added masses of an impermeable sphere moving near a plane wall

Simple formulae for the components of the added-mass coefficient tensor of a sphere moving near a wall with variable velocity in an ideal fluid bounded by a solid surface are derived. The added mass is calculated numerically as a function of the dimensionless distance between the sphere and the wall for both perpendicular and parallel motions. The calculation is performed by the method of successive images. The velocity field is computed as the sum of the velocity fields of sequences of dipoles located along the axis. The obtained dependences of the added-mass tensor components are fitted by simple continuous functions with high accuracy.