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The vibrations and buckling of continuously supported beams resting on elastic foundation impact the design of aircraft structures, highway pavements, and railroad tracks. This article demonstrates the efficacy of Chebychev polynomials in the Rayleigh Ritz method when studying the torsional vibrations and buckling of beams on elastic foundation. Simple polynomials and orthogonal functions serve as displacement functions to illustrate the advantages of orthogonal functions. Fixed-fixed and fixed-simply supported represent the two treated sets of boundary conditions. The functions are chosen so that, most often, the kinematic boundary conditions are satisfied. In some cases, however, a penalty-type approach is used in which appropriate springs with large stiffness coefficients simulate the kinematic boundary conditions. The numerical results for natural frequencies and buckling loads are obtained and compared to the values of other researchers; good agreement is observed.

A finite element formulation and Gaussian quadrature procedure, using both the direct complex matrix inversion and the modal superposition methods, are presented for studying the stationary random response of shell structures, such as a cooling tower. The random distributed loads are assumed as stationary in time but can be nonhomogeneous in space. A 48 d.o.f. quadrilateral shell element with bi‐cubic Hermitian polynomial interpolation functions as displacement shape functions is adopted. The shape functions are used to form the matrix of cross‐spectral densities of the generalized nodal forces for distributed loads. The shape functions are also used to interpolate the response quantities at an arbitrary pair of points located within two different elements. Cross‐spectral densities of displacement and stresses are first obtained for a simply supported cylindrical panel subjected to purely random load, using both the direct and modal superposition methods, which are in excellent agreement with an earlier a...

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 a...

Time history response of an elevator rail and counterweight model is studied using numerical and experimental methods. Beam finite elements are used to model the rail and the counterweight. When in contact, the rail and the counterweight are assumed to be connected by very stiff springs. Wilson‐θ method is used to perform the forced response computations. The long computing times result from the need for extremely short time steps used to integrate the nonlinear equations. In addition to the development of simple models and the use of Guyan reduction method, two techniques are introduced to reduce computing time. The numerical and experimental results compare both favorably and qualitatively.

Intending to achieve optimum finite element modeling of column supported cooling towers for seismic response studies according to the distributions of dominating bending and membrane stresses and intending to model the vulnerable shell‐column region using discrete column elements and quadrilateral shell elements, a set of elements is adopted, modified or extended. The set includes: (1) A 16 d.o.f. column element; (2) a 48 d.o.f. doubly curved quadrilateral general shell element; (3) a 42 d.o.f. doubly curved general‐membrane transition element; (4) a 21 d.o.f. and 39 d.o.f. doubly curved triangular membrane filler elements; and (5) a 28 d.o.f. doubly curved quadrilateral membrane element. Examples are given to evaluate a single type, combined types, and the whole set of elements with results in good agreement with alternative solutions. These elements may be used for accurate and efficient free vibration analysis of column supported cooling towers. The modeling may be used in seismic response analysis of ...