Journal of Engineering Mechanics

Any structural model must be designed, loaded, and interpreted according to a set of similitude requirements that relate the model to the prototype structure. These similitude requirements are based on the theory of modeling, which in turn is derived from a dimensional analysis of the physical phenomena involved in the behavior of the structure. To accomplish modeling, general similitude theory should be followed properly. This similitude theory is derived from the behavior of the structure using proper boundary conditions. Three theorems exist, of which only the Pi theorem is commonly known to engineers. With the particular problem of plates and shells, the other two theorems are important to derive behaviors of two similar systems at large and at the boundary conditions. Modifications to these theorems are presented in order to simplify the problem of designing the shell model. Methods are presented, with cylindrical shell as a special application, and the accuracy of these methods is verified. Conclusions can be reached in terms of using the theory to model shells and to enhance the results from scaled models by relaxing few conditions.

Analytical expressions are derived for the hydrodynamic pressures generated in reservoirs behind short‐length gravity dams, and resulting from vibrational motions of the dam and ground motions. The dams have vertical upstream faces. The reservoirs are assumed to have rectangular cross‐sections, constant depths and to extend to infinity in the upstream direction. Water compressibility is taken into consideration, and solutions of the three‐dimensional wave equation are obtained under both flexible and rigid boundary assumptions. The formulas obtained are used to calculate the hydrodynamic pressure for a range of geometries, and the results are presented in a form that clearly shows its dependence on the various parameters of the problem. The effect of the length to height ratio of the dam, and the significance of the cross‐stream and vertical components of ground motion on the pressure response are investigated.

A composite characterization and accompanying finite element analysis for layered systems is presented. The model is developed for plane strain and axisymmetric (horizontal layers) configurations with stiff, relatively thin isotropic layers of reinforcement sandwiched between layers of relatively flexible isotropic matrix material. Potential applications include elastomeric bearings and reinforced earth structures. The composite theory, in addition to capturing the overall response characteristics of the layered system, approximately models the important edge effects which occur at boundaries and the stress concentrations at layer interfaces. The primary dependent variables for the composite theory have been selected such that the highest derivatives appearing in the strain energy function are first order, thus requiring only $C_0$ continuity of the finite element approximations. As a result, the accompanying finite analysis is simple and computationally efficient. Numerical examples are presented to verify the theory, compare it to previous results, and to illustrate potential applications of the analysis.

Equations governing the nonlinear phenomenon of fluid flow through deformable porous media are presented in this paper. The proposed formulation is based on a continuum theory of mixtures using a total Lagrangian approach, where large displacement/large strain cases are considered. The incremental finite element equations are obtained using a direct Galerkian approach. Numerical results are presented for several problems assuming infinitesimal strains. Comparisons with known solutions demonstrate the validity of this approach. The effect of nonlinear solid phase behavior is discussed and shown to be very substantial.

The concept of dynamic stability for suddenly‐loaded elastic structural configurations is presented. Moreover, the related criteria and estimates of critical conditions are analyzed. It is shown that dynamic instability of the escaping motion type (unbounded or large amplitude motion) is only possible for those configurations which, under static loading, exhibit limit point instability. On the other hand, this type of dynamic instability is not possible for systems, which under static loading exhibit bifurcational instability with stable postbuckling branch (smooth buckling) and for systems that only exhibit stable equilibrium response (not subject to buckling). Moreover, parametric resonance and its possible occurence under sudden loads is also covered. Finally, some conclusions are presented that address the expected response of real world situations, as opposed to responses predicted by mathematical modeling.

The combined finite element‐transfer matrix method is applied to elastic‐plastic problems with large displacements. In the calculation program developed in this paper, the same procedures as those used in the finite element method based on load increment, are applied except for the estimation of approximate displacements for each specified incremental load. The Prandtl‐Reuss'law obeying the von Mises yield criterion is assumed, and a set of moving coordinate systems is used to take geometric nonlinearity into consideration. Various numerical examples of the geometric and material nonlinear problem are shown to demonstrate the accuracy of this method. These examples' results are in good agreement with those obtained by the finite element method and others.

The shape of suspended tanks equally stressed at all points and in all directions is developed using membrane theory. The equation of equilibrium for the membrane element is nonlinear and hence a numerical solution is resorted to. An iterative procedure to solve them using a simple computer program is explained. Shapes are generated in nondimensional form to facilitate tank designs. The range of shapes suitable for material economy is also indicated. A numerical example is added to bring out the major steps in designing tank dimensions.

The echelon matrix procedure is introduced, as a new base procedure for dimensional analysis, by its application to examples selected by previous authors. It is thereby shown that Buckingham's actual and very perceptive enumeration of the number of nondimensional groups is never in error, despite repeated claims to the contrary. The achievement or the forcing of the echelon matrix formation is shown also to be the natural entry to the stepwise and the proportionality‐stepwise procedures, which, like the basic echelon matrix procedure, offer advantages in straightforward execution, over the traditional, nonintegrated, procedures for dimensional analysis which are ascribed to Rayleigh and to Buckingham. The echelon matrix procedure is also shown to be valuable in examination of various initiatives and claimed improvements to dimensional analysis of the past and present. Vector lengths or “directional analysis” and such concepts are reviewed, and their supposed advantages are shown to be based on misconceptions. It is then shown that the stepwise‐proportionality procedure alone offers an infallible check in the matter of dimensional homogeneity of the variables. An outline is given of the logical sequence of arrangement of variables and execution of analysis by the alternative procedures. Overall, it is shown that dimensional analysis is still a meaningful method for the future provided it is dissociated from past and current misconceptions and some supposed improvements, and is taught and executed in a manner which relates it to the mainstream of basic modern mathematics.

The writer considers finite‐deformation elasticity theory with particular emphasis on the form of variational principles. For the sake of clarity the analysis is first undertaken in two dimensions, with subsequent reformulation in three dimensions. Beginning with a discussion of the known systems of Kirchhoff‐Trefftz and Piola stress components, and of Green and displacement gradient strain components, a case is made for the introduction of generalized, and distinguished generalized Piola components, and for the introduction of generalized displacement gradient components, involving the concept of rotational in addition to translational displacement components. With these various concepts consideration is given to the question of invertability or non‐invertability of constitutive equations and to the significance of this question in connection with the formulation of variational principles. In the discussion of various possible forms of variational statements particular attention is devoted to statements involving independent stress, translational and rotational displacement component variations, and to a statement involving translational and rotational displacement component variations in conjunction with certain “reactive” stress measure component variations.

A general procedure based on polynomial expansion of yield function in terms of invariants of the stress tensor is proposed in the context of associated plasticity for isotropic materials undergoing isotropic hardening. The procedure can be used to evolve one or more models for a material by using appropriate laboratory test results. One of the functions showing invariance at ultimate and a single function to describe continuous yield and ultimate yield behavior is investigated in detail. Based on comprehensive series of bests on cubical specimens for different (geological) materials, a hardening or growth function is defined in terms of the trajectory of plastic strain and the ratio of deviatoric to total plastic strain. The predictions of the proposed model are verified with respect to the observed results from tests with different stress paths. The model provides highly satisfactory predictions for both stress‐strain and volumetric strain responses from various stress paths. The proposed model shows potential for incorporating other characteristics such as nonassociative rule, pore water pressure effect, and induced anisotropy.

This paper investigates some problems related to the ill‐posedness of parameter identification problems which are commonly met with in structural and geotechnical engineering. Viewing the identification problem within a suitable framework, two basic types of ill‐posedness are introduced. The first is inherent to the system (the governing differential equations), while the second is related to the type of data that is acquired at a given location (e.g., velocity, acceleration, etc.) and the basic algorithms used in the identification process. Examples of each type of ill‐posedness are analyzed. Particular attention is given to the recurrent problem of identification of a nonconstant coefficient of a linear differential equation. Such coefficients often model the spatial variation of material properties in media. A method of handling spatially varying parameters for geotechnical applications is proposed. A general stochastic model for the parameter variation is used and it is shown that the identification problem using the Bayesian approach then leads to stable estimates, circumventing the instability problems of the direct inverse approach. Sensitivity studies are included to indicate the robust nature of the estimator. Simultaneous estimation of the “scatter” in the material property variation, and the material property parameter values themselves is next undertaken and it is shown that the method yields good estimates though the estimator now becomes a nonlinear function of the measurement. Simultaneous estimation of the measurement noise characteristics is also shown to be possible.

The motivation for analysis of anisotropic sandwich plates has arisen due to the recent application of filamentary composite materials in building the faces. In this technical note, small deflection theory of orthotropic sandwich plates is extended to the anisotropic case. It is shown that the FRP faced anisotropic sandwich plate exhibits different shear buckling loads under positive and negative shears. In case of isotropic and specially orthotropic sandwich plates, the positive and negative shear buckling loads coincide. Other conclusions are described.