Journal of Engineering Mechanics

An elasticity solution for sandwich plates assembled with nonrigid bondings is presented. The solution satisfies the equilibrium equations of the face and core elements, the compatibility equations of stresses and strains at the interfaces, and the relevant boundary conditions. To investigate the effects of bonding stiffnesses on the responses of sandwich plates, numerical evaluations are conducted. The results obtained have shown that the bonding stiffness, up to a certain level, has a strong effect on the plate deflection. Beyond this level, the usual assumption of perfect bonding used in classical theories is quite acceptable. An answer to what constitutes “perfect” bonding is found in terms of the ratio of the core stiffness to the bonding stiffness.

A method of solution for the transient response of nonlinear fluid‐structure systems is presented. Finite element discretization is applied to the nonlinear hydrodynamic equations describing water. Performance of five equations of state describing water in the presence of strong shocks are compared for the case of shocks propagating underwater and reflecting from a rigid wall. Differences in the ratios of reflected to incident pressure indicate the need for correlation with experimental data. Results also differ from those obtained by using an acoustic (linear) fluid model. The dynamic response of a one‐dimensional water‐linear structure system is obtained for shock wave loading. The tendency of the fluid to damp the structural response is illustrated and shown to be greatly dependent on the structural stiffness. Analysis of a one‐dimensional water‐nonlinear structure system is performed to investigate dynamic limit point buckling.

The fracture front in concrete, as well as rock, is blunted by a zone of microcracking, and in ductile metals by a zone of yielding. This blunting causes deviations from the structural size effect known from linear elastic fracture mechanics (LEFM). The size effect is studied first for concrete or rock structures, using dimensional analysis and illustrative examples. Fracture is considered to be caused by propagation of a crack band that has a fixed width at its front relative to the aggregate size. The analysis rests on the hypothesis that the energy release caused by fracture depends on both the length and the area of the crack band. The size effect is shown to consist in a smooth transition from the strength criterion for small sizes to LEFM for large sizes, and the nominal stress ${\sigma }_N$ at failure is found to decline as $(1+\lambda /{\lambda }_0{)}^{-1/2}$ in which ${\lambda }_0=\mathrm{constant}$ and $\lambda =\mathrm{relative}$ structure size. This function is verified by Walsh's test data. If reinforcement is present at the fracture front and behaves elastically, the decline of ${\sigma }_N$ is of the same type but is shifted to larger sizes; however, if the reinforcement yields, the decline of ${\sigma }_N$ stops. It is also noted that some known size effects which have been attributed to random strength variations within the structure should be explained by fracture mechanics, which gives a very different extrapolation to large structures. Finally, exploiting the fact that in metals the size of the yielding zone at the fracture front is approximately constant, it is shown by dimensional analysis that elastic‐plastic fracture causes a similar size effect.

When design factors are considered as random variables and the failure condition cannot be expressed by a closed form algebraic inequality, computations of risk (or probability of failure) may become extremely difficult or very inefficient. This study suggests using a simple and easily constructed second degree polynomial to approximate the complicated limit state in the neighborhood of the design point; a computer analysis relates the design variables at selected points. Then a fast probability integration technique (i.e., the Rackwitz‐Fiessler algorithm) can be used to estimate risk. The capability of the proposed method is demonstrated in an example of a low cycle fatigue problem for which a computer analysis is required to perform local strain analysis to relate the design variables. A comparison of the performance of this method is made with a far more costly Monte Carlo solution. Agreement of the proposed method with Monte Carlo is considered to be good.

A solution for the bending of polar orthotropic pie and ring‐sector plates is presented for three cases of boundary conditions along the straight edges. The classical method of separation of variables is employed and, the basic function in the angular direction satisfies the boundary conditions of the radial edges. Satisfaction of the governing differential equation of the plate is achieved by the derivation of appropriate radial functions. The deflection expression is generated. Isotropic and Orthotropic sector plates of various end conditions have been analyzed and compared with other methods wherever comparisons have been possible.

Equations governing small‐amplitude free vibrations of extensible cables are determined. The resulting eigenvalue problem is solved using the Galerkin procedure. Numerical results agree with previous results in the inex‐tensible limit and provide useful extension to extensibility for a wide range of geometrical and material parameters.

The paper presents two numerical methods for estimating the experimental critical load and initial imperfection of rectangular plates loaded in edge compression from recordings of load and deflection. The Three Point technique makes use of three sets of such readings to form a system of nonlinear simultaneous equations, the solution of which yields the critical load, initial imperfection, and the constant governing the curvature of the load‐deflection curve. The Least Square technique employs the principle of least square curve fitting to calculate these values. Although the techniques were developed primarily for stable bifurcation, it is equally applicable to neutral bifurcation. Both techniques were applied to various experimental results and the calculated values were found to match the recorded values very well.

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 analytical solution. Results for auto‐spectral densities, mean values, standard deviations, peak and gust response factors are obtained for a cooling tower subject to random wind loads at three different wind velocities, based on the quasi‐steady aerodynamic theory and Davenport's spectrum for wind fluctuations.

Mean upcrossing rates are determined for translation processes obtained from normal processes by univariate, nonlinear transformations. Monotonic and more general transformations are studied. It is shown that translation processes can have any marginal distribution and autocorrelation function and that approximations proposed previously for the mean upcrossing rate of non‐Gaussian processes can be unsatisfactory. These approximations assume that the process and its time‐derivative, considered to follow a Gaussian distribution, are independent. Theoretical findings are applied to determine crossing characteristics of wind speeds, river flows, and other non‐Guassian processes.

Measurements are presented of the vortex shedding frequency from a pair of smooth parallel circular cylinders in both tandem and staggered arrangements for diameter ratios of 0.5, 1 and 2. No discontinuity was observed in the variation of the Strouhal number of the downstream cylinder with respect to the streamwise separation. For a pair of identical cylinders in tandem arrangement, the downstream body inhibited the vortex formation behind the upstream one up two diameters spacing. It was generally inferred that the Reynolds number and the turbulence characteristics of the free-stream flow as well as the cylinders aspect ratio are significant factors in the vortex shedding behavior.

The Strawley equations are commonly used to determine the stress-intensity value for a single-edge-notch beam subjected to three-point, four-point, or pure bending. These equations were derived using assumptions treating the concentrated loads that limited the results to certain ranges of span to depth. A well known method for expressing boundary conditions—the bending analogy—is utilized along with the William's Stress Function to calculate stress-intensity values. This approach requires no special treatment on the boundary in dealing with concentrated loads. This method was applied to determine the influence of the span to depth ratio on stress-intensity values. It was found that the stress-intensity approaches a constant value for a unit midspan moment when the span to depth approaches 12. The present method is very economical to use for any span to depth ratio and three-point or four-point bending. Pure bending and other loadings such as direct tension are handled easily. Results are also presented in the form of engineering equations for K_I for various ratios of span to depth which are accurate within 3% based on least squares.

The distribution of sums of independent variables can generally be obtained from convolution integrals by numerical analysis. This paper develops closed-form approximations of and bounds on convolution integrals based on pragmatical and analytical considerations. It shows that convolution integrals can be approximated simply and that the Chernoff inequality provides a satisfactory bound on these integrals.

The paper presents a generalized Lagrangian interpolation formula which generates polynomial shape functions of any desired order. Examples are given to show that polynomial shape functions can be generated for simple one to three dimensional elements commonly used in finite element analysis. The higher order functions are of interest in problems where large strain gradients may be expected and are frequently used in isolated ``super elements'' as in stress intensity calculations. Computational advantages exist due to better matching of the physical phenomena being modeled and consequent reduction in number of elements for a given level of accuracy. The formula presented facilitates selection of appropriate polynomials to represent shape functions associated with a given element type and physical problem.

For the prediction of response of footings subjected to horizontal vibration, different types of contact shear distributions and displacement conditions are to be considered. Solutions using elastic half-space theory are not available for all the cases of shear distribution and displacement conditions. In this paper, solutions are obtained for the cases in which solutions are not available and the relevant coefficients are presented in tables which could be used in the appropriate equations for the prediction of dynamic response. Spring constants are evaluated and tabulated for different displacement and shear distribution conditions.

In engineering structures circular holes, or those of other shapes generally occur in groups, thus introducing significant stress concentrations. Minimizing the stress concentration around these holes is an important consideration in engineering design. One of the methods to minimize the stress concentration is to change the hole shape itself until the optimized shape with minimum stress concentration factor (SCF) is reached within specified geometric constraints and loading condition. Using a two-dimensional photoelastic technique, hole shapes have been optimized leading to minimum SCF in diametrically loaded circular disks with two neighboring holes located symmetrically side by side having the load axis perpendicular to the hole axis. Such optimum hole shapes are found to be quasi-trapezoidal with rounded corners. Results are given for a range of disk diameter/hole diameter ratios (11.44=D/d=4.16). In comparison with circular holes, SCF has been reduced up to about 14% with quasi-trapezoidal holes at regions of peak stresses (compressive) and up to 23% at peak tensile stress regions