Journal of Engineering Mechanics

Gravity waves in trapezoidal channels and channels with curved bottoms, including sloshing, longitudinal, and combined modes, are treated. Analytical‐numerical solutions are given for the wave frequency and the velocity potential for waves in trapezoidal channels, and analytical solutions based on the shallow‐water theory are obtained for waves in curved channels.

A heavy semi‐infinite elastic sheet lies on a flat surface. The sheet buckles due to a horizontal compressive edge force. The post‐buckling characteristics are determined by perturbation theory for small deflections and numerical integration for large deflections. As the edge displacement is increased, the sheet lifts upward and begins to fold upon itself in three distinct stages. The results are useful in the handling of heavy elastic sheets, such as carpeting and sheet metal.

The dynamic response of an infinitely long beam on elastic foundation was analyzed by the method of direct analysis. The conditions analyzed included a step sustained loading with two values of spring constant in the foundation, and rectangular pulse loadings with seven pulse durations. The results of the analysis for the sustained loading agreed very well with an available solution. Also, for rectangular pulse loadings the relationship between the dynamic response factor and the ratio of pulse duration to the natural period of the beam‐foundation system was found to be slightly different from that of the single‐degree‐of‐freedom system. The behavior of the beam for the very short time immediately after application of the load is analyzed via some typical cases. The method of direct analysis is a numerical approach which by‐passes the derivation and use of differential equations. Instead, the actual statements of the governing physical laws are applied directly to the beam‐foundation system. The physical laws used are equations of motion and constitutive relations. In spite of some limitations, the method can be used with arbitrary loadings in position and time. It is concluded that the method of direct analysis is a very effective tool for dynamic analysis of beam on elastic foundations.

The Schwarz‐Christoffel transformation formula for flow past an eccentric, normal constriction of zero thickness in a straight channel can be proved to have a simple form of arctangent function from the properties of elliptic integrals. Use of this mapping formula with the aid of another mapping formula between the complex potential and the transformed variable enables one to plot the corresponding flow pattern. Using such transformation formulas, flow graphics for convergent flow are thus developed to plot the flow net and the velocity distributions along the streamlines and across the channel width. To minimize the number of flow patterns in the presentation, the transformation formulas are further normalized by using the channel width and the discharge per unit depth. The normalized flow pattern so plotted is shown to vary only with the two physical parameters, the opening ratio and the eccentricity. Typical convergent flow patterns are plotted to show the validity and applicability of the transformation formulas.

This paper considers the development of an improved constrained time stepping scheme which can efficiently and stably handle the pre‐post‐buckling behavior of general structure subject to high temperature environments. Due to the generality of the scheme, the combined influence of elastic‐plastic behavior can be handled in addition to time dependent creep effects. This includes structural problems exhibiting indefinite tangent properties. To illustrate the capability of the procedure, several benchmark problems employing finite element analyses are presented. These demonstrate the numerical efficiency and stability of the scheme. Additionally, the potential influence of complex creep histories on the buckling characteristics is considered.

In this study, the writers used Tsai and Wu's tensor polynomial theory to rederive a formula originally derived by Cowin for shear strength variation in a plane of material symmetry of orthotropic materials. Experimental work was then performed (based on a method by Arcan, Hashin, and Voloshin) on Sitka spruce specimens to verify the formula. The theory and experiment agreed closely. When the angle between shear force and grain on the longitudinal‐tangential plane increased from 0‐30° shear strength decreased more than 43%. Engineers should take this relationship into account when designing wood structural members.

A simple model is developed to obtain radiation damping coefficients of soil‐foundation systems, for both plane‐strain and axisymmetric loading conditions. Despite the simplifying assumptions made, the obtained closed‐form results are in very good accord with available rigorous solutions for strip footings, circular footings and piles, resting on or embedded in a homogeneous space and subjected to vertical and horizontal vibration. The models are readily extended to a class of realistic inhomogeneous media, for which no exact solutions are presently available. Considerable insight is provided on the nature of radiation damping and its dependence on frequency, and an analogy is drawn with the propagation of sound from a loudspeaker.

A numerical method to simulate discharging processes in mass‐flow silos is presented. The essential point is to formulate the appropriate constitutive law for a granular bulk material, which covers solid‐like as well as fluid‐like behavior during discharging. An elastic‐plastic law is chosen for the former one, which is completed with a simple first approach for fluid‐like behavior. As large and fast deformations occur, geometric nonlinearities and mass properties of the bulk material are considered with respect to an Eulerian frame of reference. The complete set of field equations is numerically solved by the finite element method spatially and by the finite difference method in time. Due to the nature of the finite element method a broad variety of boundary conditions can be studied. The method provides transient velocity and stress fields within the bulk material for a first period of discharging. Remarkable stress redistributions with strong increases of wall pressures are computed.

A numerical step‐by‐step algorithm for the analysis of concrete structures exposed to a periodic history of environmental humidity or temperature is presented. The creep law of concrete is assumed to be linear, and the relationship between humidity and shrinkage is also linear. Cracking is assumed to be absent. The effect of concrete age on creep properties is taken into account. The creep law is considered in a rate‐type form corresponding to the Maxwell chain model. The well‐known exponential algorithm is generalized to complex variables to describe the periodic part of the response. Since this part cannot be separated in advance from the drifting mean response, the standard exponential algorithm in real variables and the new one in complex variables are used simultaneously in each time step to provide the total response. The algorithm allows an arbitrary increase of the time step, and time steps that are orders of magnitude larger than the fluctuation period, as well as the relaxation times, are possible without causing inaccuracies and numerical instability. The algorithm leads to a series of incremental elastic problems in which the stresses, strains, elastic moduli, stiffness matrices, etc., are all complex variables. These spatial problems are solved by finite elements. The proposed algorithm is useful for spectral analysis of the response of concrete structures exposed to random environmental humidity or temperature, and tremendously reduces the computation time when high frequencies are present in the spectral density of environment.

A continuous damage theory is presented for the quasi‐static and dynamic behavior of brittle materials. The fact that the strain‐rate effects observed can be mainly attributed to the rate‐sensitivity of the microcracking process makes the damage concept particularly attractive. Considering flat micro‐cracks a vectorial representation is adopted for the damage variable. A free energy function dependent on the coupled invariants of strain and damage is postulated, and the constitutive equations and the damage evolution equations are derived consistently subject to thermodynamic restrictions. The stress‐strain curves for uniaxial tension and compression resulting from the theory are compared with available experimental results for concrete. A decrease in the non‐linearity of the stress‐strain curves is observed as the strain‐rate is increased. A higher strain‐rate sensitivity in tension, as compared to compression, is also predicted. Further results on uniaxial tension illustrate how pre‐existing damage influences the nonlinearity of the stress‐strain curves and results in a rotation of the damage vector.

Stresses for the top and bottom of an expansive cylinder in a half-space are investigated. The maximum hoop stress is used for controlled cracking of concrete in demolitions. A design formula is given for determination of spacing between holes necessary to fracture when many holes are lying in a straight line.

Post-buckling behavior of inelastic columns is analyzed by taking account of the reversed plasticity which follows elastic unloading in the convex side of the column with an increase of the lateral deflection. The effect of the reversed plasticity on the load-carrying capacity is discussed for a pin-ended column made of the linear hardening plastic body which obeys the isotropic, kinematic or a combined hardening rule for reversed loading. It is shown that just after incipience of the reversed plasticity the load-deflection curves falls significantly compared with the case in which elastic unloading is only considered.

An efficient algorithm for evaluating bounds on the smallest and the largest natural frequencies of discrete linear systems is presented. The algorithm is based on generalized Dunkerley formulas, Danilevsky's method and Newton's identities.