Journal of Engineering Mechanics

In this paper, Biot's consolidation of layered soil is solved by the finite layer (strip) method based on quasi‐variational as well as least square approaches. The two‐ (three‐) dimensional medium is idealized by as many strips (layers) as is necessary to achieve the required accuracy, and the formulation of the relevant matrices requires only simple mathematics involving polynomials and Fourier series, and therefore is much simpler to compare with other techniques using intergral transforms. Examples are computed and the results compare favorably with known solutions.

The mixed mode fracture failure of eastern red spruce (Picea rubens) was investigated by means of single edge notch and center crack specimens with various crack inclinations in the tangential‐longitudinal plane with propagation in the longitudinal direction (i.e., TL system of crack propagation). These results demonstrate there is a definite interaction between failure stress intensity factors $K_I$ and $K_{\mathrm{II}}$ during the mixed mode fracture of wood. These mixed mode data were supplemented by the fracture toughness $\left(K_{\mathrm{IC}}\right)$ and pure mode II critical stress intensity factor $\left(K_{\mathrm{IIC}}\right)\text{.}$ Several mixed mode fracture failure criteria were compared. The criterion that could not be rejected is $K_I/K_{\mathrm{IC}}+(K_{\mathrm{II}}^2/K_{\mathrm{II}}^C)=1\text{.}$

A semi‐active tuned mass damper (TMD) is proposed to control wind induced vibrations in tall buildings. The semi‐active TMD uses a small amount of external power to modulate the damping. The net result is the equivalent of a passive TMD with hydraulically controlled time‐varying characteristics. Simulation studies show the proposed system is superior to conventional passively controlled and comparable to actively controlled systems.

The plane problem of a layer of finite length under the action of a frictionless indenter is studied. The solution obtained accounts for both the local contact behavior near the indenter and the overall beam behavior as well. The analysis superposes an infinite layer solution derived through the use of integral transforms with a pure bending beam theory solution. The problem is reduced to a Fredholm integral equation of the second kind, which is solved numerically. Clamped and simply supported end conditions are studied; overall compliances and local indenter stresses are computed and plotted for various ratios of contact length to layer thickness. The results are compared to beam theory compliances and Hertz theory contact stresses.

A universal relationship between the frequency of oscillation induced in a restrained fluid body by an external free flow, the velocity of the latter, and a typical width of the body, is established. Evidence is offered of the accuracy of this “Strouhal law” in such phenomena as wing autorotation, intermittent formation of vortices behind weirs and orifices, jet flapping and puffing, wake vortex shedding, oscillation of shock waves, cavitation bubbles and vortex breakdowns. A corollary of the Strouhal law shows that travelling waves of a well‐defined length may associate with the fluid‐body oscillation. Corroborations are offered concerning the spacing of wind waves, internal waves and dunes and Taylor‐cell waves. The usefulness of the law as a heuristic tool is stressed.

A three‐dimensional oscillatory model is proposed to represent the main features of the transition from laminar to turbulent flow and of turbulent flow along a wall. The model is based on the Strouhal law $f=u/2\pi \delta \text{,}$ giving the frequency f of periodic perturbations within the boundary layer as a function of the boundary‐layer thickness δ and of the outer velocity u. The validity of this law for the main boundary‐layer features is proved. Experimental evidence suggests the presence of a traveling wave and an associated standing wave, both of length $\lambda =2\pi \delta \text{,}$ by which bursting activity is controlled and coherent turbulent structures are shaped and timed. Other waves at wall appear to account for the production, spacing and length of viscous‐sublayer longitudinal streaks. The model provides means for a quantum‐mechanics approach to turbulent motion. This is shown through an example in which the turbulent energy‐spectrum is obtained by deterministic considerations.

The physically important effects of material anisotropy and nonlinearity on a fracture toughness parameter, the J‐integral, are investigated numerically for some representative testing geometries. An anisotropic nonlinear model, calculating stress as a function of strain to second order, was computer coded for implementation in a finite element program. Analyses were then performed for two common testing geometries, namely the standard “compact tension” specimen and the internally pressurized radially‐cracked jacketed annulus. After obtaining the resulting states of stress and strain by finite element discretization, the J‐integral was calculated along several paths around the crack, to confirm path‐independence. For a linear transversely isotropic material it was discovered that crack orientation can strongly affect the toughness calculated for given loading conditions, and can even remove geometric domains of stable crack growth or relevance of conventional formulas; appreciable nonlinear effects are also found. The two chosen geometries, exemplifying stable vs. unstable crack growth, also provide an interesting contrast in their response to these various material characteristics.

The Stokes flow problems of an incompressible viscous fluid in corrugated channels are considered. The corrugations are of small amplitude and are represented by stationary random functions. Perturbation solutions are developed leading to expressions for the flow rate enhancements as functions of the statistics of the corrugations. At high enough frequencies of corrugations, a negative flow enhancement is always obtained.

A method for analyzing in‐plane deflections of elastic framed structures by numerical integration of the exact differential equations governing the finite bending and axial deformations of beams has been developed and applied to nonrectangular rigid frames. In principle, the method leads to exact results. Load‐deflection curves, up to buckling, were calculated for gabled bents with and without overhangs, under symmetric and asymmetric distributed loads and under combinations of independently prescribed distributed and concentrated loads. Asymmetric as well as symmetric gabled bents and a two‐story, two‐bay frame were analyzed. Load‐deflection curves, interaction diagrams, and curves showing the results of parametric studies are presented. Symmetric gabled bents under symmetric loads were found to buckle in either a symmetric snap‐through mode or by sidesway according to rise‐to‐span ratio and column height‐to‐span ratio. In virtually all cases considered, it was concluded that the members of a frame would have to be unrealistically thin for the frame to buckle elastically.

Energy equations governing the flexural, torsional, and lateral instability of thin walled bars of open cross section are derived. Shear deformations due to nonuniform bending, and distortion of the cross section are not considered, but the influence of pre‐buckling displacements is included in the analysis. Solutions of the governing energy equations are obtained, by the finite element method, which agree with closed form solutions based on the governing differential equations.

Based on the existence of a yield surface and a memory surface which represents the maximum stress state the material ever experienced, a plasticity model is proposed to describe the inelastic behavior of metal. By choosing the initial locations and sizes of these two surfaces based on the internal structure and the loading history of materials, theoretical results have been obtained. These results compare well with available experimental observations on cyclic creep, cyclic stress relaxation and other transitory phenomena of strain softening materials under generalized cyclic loading conditions. The inelastic behavior of annealed copper subjected to nonproportional strain paths can also be predicted by the proposed model.

A long elastic sheet or cable lies horizontally on equally spaced supports. The sheet sags due to its own weight. The problem is governed by the distance between the supports, $\mathrm{2a}\text{,}$ and a nondimensional parameter, K, which represents the relative importance of density and length to flexural rigidity. The heavy elastica equations are solved both analytically and numerically. The design parameters, horizontal force, and maximum moment are found to be sensitive to both a and K.

The design of large flat‐roof systems necessitates a means of analyzing the required strength of bracing members. If the braced beam is initially perfectly straight, it will remain straight until it buckles; the prebuckling stresses in the braces are theoretically zero. If the braced beam has initial imperfections, they will grow under load and so will the stresses in the braces. Thus, there is need to analyze the buckling of beams with initial imperfections. This was done in the context of linear small‐deflection theory; the results are simple and suitable for use in design where loads are well below buckling and initial deformations are small. This paper presents a variational derivation of a set of linear differential equations and boundary conditions and applies them to the calculation of forces in midspan diagonal bracing members on a simply supported beam under constant bending. A sample calculation for a typical wood roof application shows the forces in the bracing to be quite small. Only rectangular beams are treated in detail, but the extension to singly symmetric I‐beams is sketched.

The equations of the bending and stability theories for the orthotropic shell are solved using the finite element method (FEM). A biaxial material law for concrete and a nearly bilinear stress‐strain diagram for reinforcing steel were considered. Taking a layered ring element the influence of bending moments together with the membrane forces can be followed under increasing load up to the failure of concrete or steel. At each level the buckling factor can be calculated considering the stress dependent orthotropic buckling stiffness. The method of calculation developed is applied to two large cooling tower shells, one under wind load according to German code DIN 1055 and the other one under unusually high wind pressures, as those considered in the U.S.A. According to the wall thickness taken the buckling load factor drops to half or less of the value obtained assuming a linear elastic behavior. This high reduction is mainly due to orthotropy and the descending tangent modulus near the ultimate load.

Incrementally linear constitutive equations that are characterized by an orthotropic tangential stiffness or compliance matrix have recently become widely used in finite element analysis of concrete structures and soils. It does not seem to be, however, widely appreciated that such constitutive equations are limited to loading histories in which the prinicipal stress directions do not rotate, and that a violation of this condition can sometimes have serious consequences. It is demonstrated that in such a case the orthotropic models do not satisfy the form‐invariance condition for initially isotropic solids, i.e., the condition that the response predicted by the model must be the same for any choice of coordinate axes in the initial stress‐free state. An example shows that the results obtained for various such choices can be rather different. The problem cannot be avoided by rotating the axes of orthotropy during the loading process so as to keep them parallel to the principal stress axes, first, because this would imply rotating against the material, the defects that cause material anisotropy, such as microcracks, and, second, because the principal directions of stress and strain cease to coincide. The recently popular cubic triaxial tests do not give information on loading with rotating principal stress directions.

A uniaxial constitutive equation describing the deviations from the linear principle of superposition of aging concrete at constant moisture content and temperature is presented. Both the creep increase (flow) at high stress and the stiffening (adaptation) due to low sustained compressive stress are modeled, the latter being of principal interest. The constitutive equation expresses the creep rate as a history integral with a singular kernel involving the time lag of creep strain. The integral has the property that the strain response is proportional to the stress history but depends nonlinearly on the stress history when nonproportional stress histories are superposed. The double power law for aging creep is a special limiting case. The constitutive relation is also explained in terms of the rate‐process (activation energy) theory for the rate of bond ruptures causing creep. For structural creep problems a corresponding step‐by‐step integration algorithm which correctly captures the asymptotic properties of the integral is also developed. A good agreement with test data from the literature is achieved.

A solution of Navier's equations in spherical coordinates is obtained through a direct and simple method rather than the usual Papkovich‐Neuber approach. Both the axisymmetric and non‐axisymmetric problems are treated. Also presented is a pair of governing equations for thin spherical shells subject to axisymmetric deformations. These equations are expressed in terms of displacement components instead of the commonly used transverse shearing force and an auxiliary function.

An imposed strain approach is used for the reliability analysis of brittle parallel systems with arbitrary stress‐strain behavior of its components. It leads to a formal description of the failure event as a parallel system. The failure probability of the strongest component overestimates the system failure probability, so that the consideration of the other components can be recommended for which first‐order reliability techniques supply an efficient tool. Numerical comparisons with certain special exact results show that the correlation structure is taken fairly well into account by this first‐order method. Essential improvements result from better estimates of the component reliability. This induces the definition of so called equivalent components, which might also be useful for the evaluation of more general systems. The accuracy of the method appears sufficient for many engineering applications.

The Bernoulli equation approach for predicting the variation of internal pressure induced by wind in a single‐room building with a single opening is extended to multi‐room buildings of any number of openings. Examples are given to illustrate this new general approach. The result shows that following the breakage of a windward window during a windstorm, the internal pressure of a multi‐room building oscillates in a manner similar to that in a set of interconnected Helmholtz oscillators.