Journal of Engineering Mechanics

A variational principle is used in conjunction with the finite‐element method to solve the nonlinear coupled field equations of an initial boundary value problem for flow in deformable fractured rock masses. This results in a powerful method for modeling the coupled stress and fluid flow behavior of rocks. Stress and deformation history for both solid and liquid phases, for arbitrary boundary conditions, and within complex geometrical configuration, can be determined. Direct application can be made to fluid flow problems in hydraulically fractured reservoirs and naturally fractured rocks.

The magnitude of the vector sum of two orthogonal, horizontal, wind velocity components is often modelled by the Rayleigh distribution. In the derivation, it is assumed that the components are independent, identically distributed, zero‐mean, Gaussian random variables. The effect of unequal variance components with both zero and nonzero means is investigated and the resulting distribution is found to be closely fitted by the Rayleigh distribution calibrated to the derived mean wind speed. The more general case in which the unequal variance components are correlated, again with both zero and nonzero means, is studied. The Rayleigh distribution again provides a good approximation when calibrated to the mean wind speed. Data collected by the National Weather Service are used to illustrate the approach. Comparison of the derived distribution with the observed histogram and the Rayleigh distribution shows that the Rayleigh provides a good approximation.

Shape optimal design of an elastic solid of revolution under multiple constraints is treated. As a specific example, a device that seals a gun bore and transmits high in‐bore pressure to shear loading on the projectile, is considered. The design objective is minimum weight, with constraints on stress throughout the body, tractions on one surface of the boundary, and dimensions of the body. Methods of the calculus of variations and functional analysis are used to transform the variation of a functional over a variable region as a functional over a fixed region. An adjoint variable method of operator theory is then used to reduce this variation to an explicit function of only design variations. The resulting sensitivity coefficients are used in an iterative optimization algorithm. Numerical results are presented and show that the algorithm is stable and efficient.

A sheet of elastic material, lying on the ground is lifted from one edge by a vertical force F. The nonlinear heavy elastica equations for large deflections are solved by perturbation method for small F and exact numerical integration for general F. An asymptotic solution for large F is also found. The results can be applied to the design and lifting of heavy elastic sheets, such as carpeting and sheet metal, as well as to the design and lifting of heavy beams, rods, and cables.

A Froude‐Mach similitude law for sea‐seabed interaction phenomena is established from the nonlinear Coulomb‐damped poro‐elastic theory recently developed. Generally, the similitude law requires that the three Mach numbers which are the ratios of water wave phase velocity to those of the fast and slow compressional waves and the shear waves in the seabed should be equalized in the prototype situation and the physical scale model in addition to the Froude number and the geometrical similarity. Laboratory wave tank experiments using sand beds and clay beds substantiate the theoretical similitude law. It is found that the response of sand beds to water waves is linear and quasistatic. The effect of the Coulomb friction in sand beds on wave attenuation is usually larger than that of percolation of pore water through sand beds. The response of clay beds to water waves is highly nonlinear and dynamically amplified. The Coulomb friction in clay beds is by far the largest wave damping mechanism compared to other damping mechanisms.

Perturbation methods are employed to determine the dynamic properties of a combined system composed of a multi‐degree‐of‐freedom structure to which is attached a light, less single‐degree‐of‐freedom equipment item. Closed‐form expressions are derived for the natural frequencies, mode shapes, modal dampings, and other modal properties of the combined system in terms of the dynamic properties of the structure alone, and the equipment alone. The effect of tuning and equipment‐structure interaction are included in this analysis. The derived properties can be used in conjunction with a mode‐super‐position procedure to determine the response of the equipment to any excitation of the combined system. For an example structure, the natural frequencies and mode shapes based on the method developed agree well with exact results.

A mode‐superposition method for the evaluation of the dynamic response of light equipment in structures subjected to stochastic excitations is developed. Previously obtained results for modal properties of the combined equipment‐structure system in terms of the modal properties of the two subsystems are used. Both power spectral density and response spectrum descriptions of the input are considered. Results include the effects of tuning, equipment‐structure interaction, and correlation between modal responses. Floor spectra including interaction, as well as the conventional floor spectra, are easily obtained in terms of the input response spectrum. For an example structure, excellent agreement is obtained in a comparison with time‐history results for an ensemble of 20 generated earthquake ground motions.

A procedure based on a direct variational method is presented for the determination of an optimum shape of a shell of revolution connecting a cylinder to a sphere under internal pressure. The optimized configuration of variable thickness has the same strength as that of an unpierced sphere subjected to internal pressure. The design is set up with respect to volume restriction, weight constraint and governing yield criterion. Geometric discontinuity is avoided by representing the midsurface of the shell of revolution by a function differentiable at least twice, up to and including boundaries. Required integrations and iterations are carried out numerically by a digital computer.

A simple upper and lower bound technique developed for the evaluation of the failure probability of a “weakest link” structural system is applied to a structural system having a convex, polyhedral safe set in the space of basic variables. The structure is subjected to a Gaussian vector‐load‐effect process. Conditional second moment calculus, as applied to the set of linear safety margins that completely define the problem, turn out to be an easy and elegant tool for this type of calculations. The upper bound is constructed by bounding the mean outcrossing rate of the vector process while the lower bound is obtained by discretizing the time variable in an optimal way.

A procedure is developed to analyze, under the assumption of linear behavior, the earthquake response of arch dams including hydrodynamic effects. The dam and fluid domain are treated as substructures and modeled with finite elements. The only geometric restriction is that an infinite fluid domain must maintain a constant cross section beyond some point in the upstream direction. For such an infinite, uniform region, a finite element discretization over the cross section is combined with a continuum representation in the upstream direction. The fluid domain model approximately accounts for the effects of foundation flexibility on hydrodynamic pressures. Computed responses of an arch dam to harmonic ground motion are presented.

A solution is presented for the lateral bending of a polar orthotropic sector plate with general boundary conditions. At the present time, solutions for only nine of the 54 possible combinations of boundary conditions are available in the literature; this study provides solutions for all of these boundary conditions. The plate is linearly elastic and subjected to a uniformly distributed lateral load. The principle of minimum potential energy is used to obtain a differential equation which describes the system configuration. This equation is solved to obtain deflections and stresses for the sector. Results compare well with solutions in the literature, and provide additional solutions not previously available.

Under the assumption that strong motion earthquakes result primarily from surface waves in a layered medium resting on a semi‐infinite rock formation, a method is developed to derive the expression for the Rayleigh wave that produces acceleration at the ground surface with a specified power spectral density. The Rayleigh wave characteristics are then used to obtain a corresponding free‐field normal ground strain at any depth in the medium. Other important features of this paper are as follows: (1) The ground acceleration and strain are both modeled as Gaussian stationary processes; (2) the power spectral densities used for these processes are functions of the earthquake's magnitude, epicentral distance (reflecting the attenuation effect), and the N‐values (indicating the site soil conditions); and (3) since the ground strain is modeled as a random process, and since its spectral density function involves a significantly uncertain attenuation relationship, the maximum ground strain is a random variable. The expected value of the maximum ground strain is evaluated as a function of the magnitude, epicentral distance and site soil conditions. The coefficient of variation of the maximum ground strain is also estimated.

A method of analyzing the elastic and inelastic large‐displacement behavior of horizontally curved I‐beams based on the transfer matrix method is presented. The field transfer matrix is derived under the assumption that the elastic core in each segment is constant and the resultant force in the yielded portions of the cross section is replaced by the external force. The point transfer matrix is derived using the continuity of the displacements and the principal of virtual work taking into account the discontinuity of centroidal lines at the nodes. A computer program based on this theory has been developed and the theoretical values for the deformation and the ultimate strengths are shown to be in good agreement with experimental results. Some numerical examples are also presented. The effects of cross‐sectional dimensions, loading conditions, and residual stresses on the ultimate strength are analyzed for specified central angles of curved beams.

The finite element method of analysis is applied to the calculation of hydrodynamic pressures in a reservoir impounded by a gravity dam and subjected to a harmonic ground motion. In the finite element modelling of an infinite reservoir, the fluid region must be truncated at some significant distance from the region of interest. To model the energy loss in outgoing waves in the inifinte reservoir, an appropriate radiation condition must be applied at the truncation boundary. A radiation condition which adequately models the loss over a wide range of excitation frequency is developed and it is shown that this condition gives much better results as compared to the plane wave Sommerfeld condition. Finite element solutions are obtained for several reservoir vibration problems and the result are compared with classical solution where such solutions exist. The method is then applied to the vibration problem of a reservoir with an inclined dam face, for which a classical solution does not exist. It is shown further that the inclination of the dam face significantly reduces the hydrodynamic pressure.

A computational algorithm based upon a strain‐space formulation of plasticity is tested for accuracy in the case of perfect plasticity. A new strategy for updating the position of the relaxation surface is presented and compared with other strategies employed in stress‐space. The algorithm is coupled with a finite element program and used to investigate the response of a vertically loaded foundation resting on an elastic half‐space. The results are compared to those obtained using an accepted stress‐space algorithm.

The velocity field induced in a liquid confined between two coaxial rotating disks of finite radius and contained between two annular disks, rotating at different angular velocities, provides a simple example for both theoretical and experimental study of free shear layers in rotating flows. Also, the differential rotation of the disks, from the rest of the configuration, is of interest in its own right in the modeling of large scale wind‐driven ocean circulation. In the case of antisymmetrically rotating disks, the differences between the theoretical and experimental axial velocity profiles are several times greater than experimental errors. In this paper, without using the equivalent boundary conditions on the effective, flux, the full boundary value problem is solved by a correct treatment of the corner region. This approach leads to a better agreement between the theory and observations of profiles of axial velocity in free shear layers.

An envelope is introduced by using the Hilbert transform to define a complex conjugate to the excitation and response processes of a linear structure. Time‐limited stationary excitation is treated in detail, and the complex correlation function is shown to follow from its stationary equivalent by use of a suitable differential operator. Simple expressions are derived for the case of rational spectral density, and a parametric study of the influence of the frequency content is carried out. It is found that envelope crossings can be predicted by use of stationary measures for the mean crossing frequency and the band‐width of the response combined with the non‐stationary intensity. This is computationally important as these parameters are often available in closed form. Finally, the envelope is used to study the first‐passage probability.

A method for the computation of response spectra for elastoplastic and bilinear hysteretic systems subjected to strong‐motion earthquake records is described and compared to Newmark's method. The technique provides the exact solution to the governing equations of motion assuming that the ground acceleration varies linearly between successive points, A fractional time stepping scheme is incorporated to detect yielding and unloading accurately. The procedure permits the computation of dynamic response in an efficient, arithmetical manner, and provides a two‐ to three‐fold savings in computation time compared to Newmark's method.

The loss of stability by so‐called bifurcation in tension of geometrically nonlinear plates and spherical caps loaded by axisymmetrical cross forces is examined. These phenomena are manifested by a transition from axisymmetrical forms to the nonsymmetrical ones with formation of wrinkles. They may occur on plates, and also on caps when the loading acts on their inner (concave) side. In these cases, the meridians are stretched at every point. It is necessary to distinguish between these phenomena and bifurcation in compression. The latter is impossible for plates and shallow caps and may occur only on relatively deep caps when the loading acts outside (convex side). Experimental observations of the stability loss by bifurcation in tension are described. Qualitative investigations of the writer indicate, in particular, that the considered phenomena cannot occur when the edge supports are fixed.

Full‐scale observation data of wind‐induced behavior of the Golden Gate Bridge from a 1962 study was extensively re‐examined by spectral analysis. The experimental investigation on the same bridge was also conducted using proposed taut strip bridge model with the simulation of natural wind turbulence at laboratory scale. The study shows that the response of the full‐scale bridge at subcritical wind speed range is predominately gust‐induced buffeting in the first asymmetric mode of vibration. As the wind speed increases, the motion shifts to a coupled torsional‐vertical instability in the first asymmetric mode of vibration. The study shows that the taut strip model in turbulent boundary layer flow is capable of simulating the full‐scale motion described in the first and second options within reasonable confidence. Conventional section model wind tunnel tests do not reveal the motion of the full‐scale bridge at subcritical wind speeds. For the prediction of full‐scale bridge behavior from the taut strip model test results, it is important to take the mode of vibration into account.

A general theory and analysis of the nonlinear free coupled vertical‐torsional vibrations of suspension bridges with horizontal decks are presented. Approximate solutions are developed by using the method of multiple scales via a perturbation technique. The amplitude‐frequency relationships for any single set of coupled vertical‐torsional modes are presented for three cases: (1) When the large‐amplitude vertical vibration is dominating the motion; (2) when large‐amplitude torsional vibration is dominating; and (3) when one of the linear natural frequencies of vertical vibration is equal to, or approximately equal to, another linear natural frequency of torsional vibration, and the two modes are strongly coupled; this contrasts with the linear solution, which predicts that the two modes are uncoupled.

The basic characteristics of the nonlinear free flexural‐torsional vibrations of two suspension bridges are examined. The Golden Gate Bridge and the Vincent‐Thomas Bridge were chosen to represent both a relatively long‐and a relatively short‐span suspension bridge's vibrations. The amplitude‐frequency relationships of the first six modes (symmetric and antisymmetric) of both vertical and torsional vibrations for each bridge are presented. The case when one of the linear natural frequencies of vertical vibration is equal to, or approximately equal to, another linear natural frequency of torsional vibration, is considered. This case revealed that the two modes are strongly coupled. Finally, a comparison between the analytical results obtained via the perturbation analysis and those obtained by the numerical integration of the governing coupled nonlinear equations of motion is presented. The agreement is reasonably good.

The relation between the stress intensity of the corner with an arbitrary angle and the stress concentration factor is investigated. In the thin plate bending problem and the plane elastic problem, the relation is concerned with stress conditions, symmetric to the bisector of a re‐entrant corner angle ranging from 180°–360°. If this relation is known, it is possible, from the expression of stress concentation factor, to determine the stress intensity of the corner with an arbitrary angle. If the stress intensity is known, on the contrary, it is possible to determine the coefficient of the first term in the series for the stress concentration factor. A procedure for numerically determining the stress intensity of the corner is also proposed.

A method is presented to determine the power spectral density functions from given smooth response spectra. It is assumed that the underlying excitations constitute a stationary Gaussian random process. The relationship between the response spectrum and the power spectral density function is established by the probability distribution of the extreme values. The power spectral density function is discretized by parameters and piecewise polynomials and the free parameters are determined iteratively by a least square fit. Due to this discretization all required integrations can be carried out analytically. It is also possible to obtain a least square fit value of the duration of the stationary output process. The numerical examples include the calculation of the power spectral density functions for selected response spectra.