Journal of Engineering Mechanics

Nonlocal continuum, in which the (macroscopic smoothed‐out) stress at a point is a function of a weighted average of (macroscopic smoothed‐out) strains in the vicinity of the point, are of interest for modeling of heterogeneous materials, especially in finite element analysis. However, the choice of the weighting function is not entirely empirical but must satisfy two stability conditions for the elastic case: (1) No eigenstates of nonzero strain at zero stress, called unresisted deformation, may exist; and (2) the wave propagation speed must be real and positive if the material is elastic. It is shown that some weighting functions, including one used in the past, do not meet these conditions, and modifications to meet them are shown. Similar restrictions are deduced for discrete weighting functions for finite element analysis. For some cases, they are found to differ substantially from the restriction for the case of a continuum if the averaging extends only over a few finite elements.

Dynamic response of a pipeline buried in a semi‐infinite elastic medium is investigated. The pipeline is modeled as a circular cylindrical shell of small thickness. The incident disturbances are assumed to be plane waves moving perpendicular to the axis of the pipeline. Thus, the problem is either one of plane strain, which will be examined herein, or antiplane strain. Two problems are considered: (1) The pipe is surrounded by a homogeneous soft soil; and (2) the pipe lies in a cylinder of soft soil, which is surrounded by a rock‐like material. It is shown by comparison that the stresses and displacements are significantly modified in the latter case. The response is also found to be considerably influenced by the frequency of the incident wave and depth of the embedment.

The earthquake‐induced torsional response of suspension bridges when subjected to multiple support excitations is analyzed in the frequency domain by means of random vibration theory. Appropriate rocking and torsional ground motion inputs are defined from finite Fourier transforms of recorded translational motions by using a simplified approach based upon wave propagation theory. An example is presented in which the torsional response of the Golden Gate Bridge to earthquake ground motions (the 1979 Imperial Valley earthquake) with characteristics significantly different at each support point is investigated. It is observed that the participation of higher modes in the total response is essential to assess the torsional seismic behavior of such structures. It is also observed that both the vibrational displacement and vibrational cable tension induced by torsional vibration are very small but the flexural stresses induced by the torsional vibration are relatively large as live loads.

In the numerical modeling, solutions are highly subject to the boundary conditions that should mathematically represent as much physical phenomena as possible. However, in the predictions of a turbulent flow associated with a well‐developed turbulent plane jet, the currently insufficient similarity solutions of plane jet profiles cause so much trouble in the starting points that several test runs have to be performed in order to examine the exactness of the guessed profiles of some turbulence variables. In the present study, analytical solutions of the stream function, vorticity, shear stress, and viscosity are derived from Tollmein's solution for a plane turbulent jet. Mean‐while, a numerical study based on the two‐equation, $K‐\epsilon \text{,}$ turbulence model is used to examine this set of similarity solutions. Except for the eddy viscosity, good agreement is obtained between the numerical and analytic solutions. From the comparisons, there also results a simple modification to the analytic solution of the eddy viscosity profile.

Jet impingement on a solid wall is a common problem for engineers. Near solid boundaries, flow situations depend highly on the impingement geometries. Therefore, it is almost impossible for engineers to labor through experiments on a case‐by‐case basis. In this study, the major effort was concentrated on a turbulence numerical model of a well‐developed normal plane jet impingement on a solid wall. As a result, different jet impingement geometries can be expressed as different boundary conditions. The two‐equation, $K‐\epsilon \text{,}$ turbulence model combined with the continuity equation and the transport equation of vorticity is adopted to solve this problem. An iterative finite difference technique is utilized in the computations. It has been found that the best prediction can be achieved when the wall turbulence Reynolds numbers fall within the range 100–150. The peak levels of turbulent kinetic energy occur not only in the zone of impingement but also in the intermingling zone between the edges of the free jet and the wall jet.

Arbitrary modes of vertical vibrations of a circular disk on an elastic half space is investigated mathematically. The problem is formulated with an integral equation, which describes explicitly the relation between the vertical displacement of the disk and the normal stress of contact. This equation has long remained difficult to handle, because its integral kernel has been represented as an integral of a singular function over an infinite interval. This integral kernel, the Green's function of the problem, can be reduced to definite integrals of some higher analytic functions over finite intervals. These higher functions, also defined as a definite integrals of trigonometric functions over a finite interval, are closely similar to the Bessel function and are easily evaluated. The integral equation accompanied by the new Green's function affords a basis of analysis; one can readily take the dynamic interaction into account by solving this equation simultaneously with the equation of motion of the disk or some other structure whose bottom is circular. Complicated structures can be treated if an appropriate discretization procedure is introduced. If the method presented here for the first time is applied to the problem of a rigid disk for demonstration, it gives identical solutions to the conventional theories.

Random wind response analysis of a cooling tower is performed using the Monte‐Carlo simulation approach. A 48 d.o.f. quadrilateral shell element is used. The response analysis capability is first verified by finding the deterministic response of a cooling tower due to an artificial wind load. Five different wind simulation models are then tested, based on quasi‐steady aerodynamics and ARIMA models derived from measured data, all with different correlation assumptions in the meridional and circumferential directions. Time domain responses of a cooling tower for the five wind models are obtained and compared. For generating random shocks an equation relating the variances of pressure and velocity, and other quantities to the variance of random shocks is derived to provide input to the ARIMA velocity‐pressure transfer function model. The assumption of ergodicity is made. Results are presented in the form of time‐history response, mean, RMS values, peak values, and gust factors. Frequency domain results are also obtained for the most realistic example to compare with the time‐domain results. Favorable agreement is found.

The effect of initial stress on the transient response of a beam subjected to impulsive end loading is investigated. This loading can be prescribed in the form of stress, strain, velocity or acceleration boundary conditions. The Timoshenko equations, modified to include the presence of initial stress, are used to model the beam. The analysis is based on the concept of a wave as a carrier of discontinuities in the field variable and its derivatives. These discontinuities are determined from a set of recurrence relations that are, in turn, generated by using time‐harmonic asymptotic series solutions to the equations of motion. The numerical results obtained confirm the influence of prestress on the transient velocity, bending moment, and shear force distributions in the beam. The results for zero initial stress compare favorably with those from a closed form solution, while the other results follow a particular trend. Solutions to problems with other boundary conditions can be obtained by using the present results.

The critical uniform lateral pressure of a simply supported beam initially compressed into a buckled state is derived for the cases of static loading, step‐loading, and impulse loading. The results are calculated for different values of buckle amplitudes and are found to be at most about 75% of the corresponding values for shallow sinusoidal arches having the same amplitude. The method of analysis of the stability of the beam under dynamic loading is similar to that of Hoff and Bruce who used an energy criterion to determine the critical pressures of laterally loaded shallow arches.

The contribution of the nonlinear wind velocity term to the wind induced dynamic response of compliant offshore structures, e.g., a tension leg platform is evaluated. The analysis is carried out both in time and frequency domains. Analytical and numerical schemes are used to evaluate the autocorrelation and the spectrum of the nonlinear velocity term. Expectations of the product of jointly normally distributed random variables and the convolution of the spectrum of the wind fluctuations are used for autocorrelation and spectral estimates. For illustrating the procedure, the fluctuations are idealized as a sine wave, a band limited white noise, and a filtered white noise process. A general expression for the response of an offshore structure to linear and nonlinear velocity terms is formulated. The results indicate that for typical values of turbulence intensity, the contribution of the nonlinear velocity term to the structural response is negligible. Therefore, ignoring the nonlinear velocity term in the formulation of the wind loads, results in an acceptable approximation.