Journal of Engineering Mechanics

Discontinuous deformation analysis (DDA) is a discrete element method that was developed for computing large deformation in fractured rock masses. In this paper we present details of the DDA time integration scheme, where the acceleration is taken constant over the time step, equal to the acceleration at the end of the time step (“right Riemann”). The integration scheme has several advantages: (1) Self-starting, (2) accelerations never need to be computed which reduces implementation complexity, (3) unconditionally stable, and (4) dissipative, contains algorithmic damping which may be important considering the penalty formulation of DDA. However, the right Riemann scheme is implicit, requiring expensive factorization or iteration to solve the resulting system of equations, and is accompanied by a bifurcation in the spectrum when the time step is large with respect to the period. This bifurcation has important ramifications for controlling spurious resonance in DDA simulations due to linear scaling in system stiffness compared to cubic scaling of the system mass as the characteristic length of the domain increases.

Two energy balance equations widely used to describe simultaneous transfer of heat and mass in porous media are inconsistent with control volume energy conservation. Potential energy, enthalpy, and internal energy terms are involved in the discrepancies. Energy within a volume is properly counted as the sum of internal, potential, and kinetic energy. However, one equation uses enthalpy where internal energy should have been used. In the other, potential energy and shifts in internal energy associated with heat of wetting are not included. Energy conservation for a control volume dictates summing convective fluxes of internal, potential, and kinetic energy at the control volume surface along with conducted heat and work crossing the boundary. The pressure–volume (pv) work at the volume surface may be combined with internal energy convection so that flow of enthalpy is used in the flux term. Examples of energy change versus work input in adiabatic processes illustrate the error introduced when enthalpy rather than internal energy is used to compute control volume energy content. For porous media flows kinetic energy can be dropped. A consistent equation based on the control volume approach is presented. It includes effects due to internal energy, potential energy, heat of wetting, conducted heat, non-pv work, enthalpy, and mass flow. Substantial temperature changes due to heat of wetting have been found experimentally in a separate work. A comparison is needed of the experiments and a numerical simulation based on the new equation.

A linear failure criterion is widely used in slope stability analyses. However, the strength envelope of almost all geomaterials has the nature of nonlinearity. This paper computes rigorous upper bounds on slope stability factors under the condition of plane strain with a nonlinear yield criterion by employing the upper bound theorem of plasticity. A stability factor (or a limit load) computed using a linear Mohr-Coulomb (MC) failure criterion which circumscribes the actual nonlinear failure criterion is an upper bound value of the actual stability factor (or limit load). In this paper, an improved method using a “generalized tangential” technique to approximate a nonlinear failure criterion is proposed to estimate the stability factor of a slope on the basis of the upper bound theorem of plasticity. Using the “generalized tangential” technique, the curve of the nonlinear failure criterion is simplified as a set of straight lines according to the linear MC failure criterion. The straight line is tangential to the curve of the nonlinear failure criterion. The set of straight lines of the linear MC failure criterion is employed to formulate the slope stability problem as a classical optimization problem. The objective function formulated in this way is minimized with respect to the location of sliding body center and the location of tangency point. Two typical slope stability problems (a homogeneous soil slope with two slope angles and a vertical cut slope with a tension crack) are analyzed using the proposed method. For the soil slope with two slope angles, the computed results are compared with published solutions by others. The comparison shows that the proposed method gives reasonable and consistent values of the stability factor of the slope. For the vertical cut slope with a tension crack, a statically admissible stress field is constructed for the slope. The stress field does not violate the nonlinear failure criterion. Lower bound solutions are obtained by satisfying stress equilibrium conditions. The upper bound solutions obtained from the proposed method are equal to the lower bound solutions for the vertical cut slope. The agreement further supports the validation of the proposed approach. The influences of the strength parameters in the nonlinear criterion on the stability of slopes are also studied and discussed in this paper.

The large-deflection elastic analysis of slender beam-columns of symmetrical cross sections with semirigid connections under end loads (forces and moments) including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions are utilized in the proposed method which can be used in the large-deflection stability analysis of slender beam-columns with rigid, semirigid, and simple connections under any combination of end loads (conservative and nonconservative). The proposed method consisting of a closed-form solution of the Elastica can also be utilized in the large deflection analysis of beam-columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The main limitation of the Elastica is that only flexural strains are considered (the effects of axial and shear strains are neglected). Therefore results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for plane beam-columns under bending actions. The large-deflection analysis of a beam-column with flexible connections at both ends becomes a complex problem requiring the simultaneous solution of at least two highly nonlinear equations with elliptical integrals. The solution of this problem becomes even more complex when the end connections are nonlinear or the direction of the applied end load changes (like “follower” loads). The validity and effectiveness of the proposed method and equations are verified against available solutions of very large deflection elastic analysis of beam-columns. Four comprehensive examples are included for verification and easy reference.

A novel mesoscale granular continuum model is derived by homogenizing the mechanical behavior of six particles in contact forming an octahedron structure. The elastic behavior at the particle contacts is simulated in accordance with Hertz theory and the frictional behavior is taken into account by means of an evolution law for the contact shear stiffness. The mesoscale continuum model is calibrated on discrete element simulations of a cuboidal volume filled with identical spheres and loaded under axisymmetric compression. By imposing a set of radial deviatoric stress paths, the failure contour is computed in the deviatoric plane of the principal stress space.

In order to carry out nonlinear dynamic analyses of fixed offshore structures in the frequency domain, polynomial approximations of the distributed drag term, $u\left|u\right|\text{,}$ and inundation drag term, $\eta u\left|u\right|\text{,}$ of the Morison forces are studied. The methods of least-squares and moment-based approximations are considered for cases with and without current. Numerical results and analytical expressions of the polynomial coefficients are presented for the cubic approximations of $u\left|u\right|$ and quartic approximations of $\eta u\left|u\right|\text{.}$ The curve shapes, first four central moments, and probability density functions of the different approximations are evaluated and compared with the corresponding exact solutions. For the nonmonotonic inundation drag term with the current effect included, a hybrid polynomialization, based on the least-squares approximation for the odd-order polynomial coefficients and the moment-based approximation for the even-order coefficients, is proposed.

A probabilistic approach for failure analysis is presented in this paper, which investigates the probable scenarios that occur in case of failure of engineering systems with uncertainties. Failure analysis can be carried out by studying the statistics of system behavior corresponding to the random samples of uncertain parameters that are distributed as the conditional distribution given that the failure event has occurred. This necessitates the efficient generation of conditional samples, which is in general a highly nontrivial task. A simulation method based on Markov Chain Monte Carlo simulation is proposed to efficiently generate the conditional samples. It makes use of the samples generated from importance sampling simulation when the performance reliability is computed. The conditional samples can be used for statistical averaging to yield unbiased and consistent estimate of conditional expectations of interest for failure analysis. Examples are given to illustrate the application of the proposed simulation method to probabilistic failure analysis of static and dynamic structural systems.

This paper explores the use of mass/stiffness eccentricity in simple buildings to damp the response to excitation due to base motion. Examples include one- and two-story buildings when discussing low-rise structures and shear beams when discussing taller structures. Base motions include a simple step function and earthquake input. A broad range of efficiencies is achieved depending upon the system eccentricity and the available viscous damping. Asymptotic efficiencies of 50% are indicated in some cases. Optimal parameters are discussed.

Propagation of shallow water waves in viscous open-channel flows that are convectively accelerating or decelerating under gradually varying water surface profiles is theoretically investigated. Issues related to the hydrodynamics of wave propagation in a rectangular open channel are studied: the effect of viscosity in terms of the Manning coefficient; the effect of gravity in terms of the Froude number; wave translation and attenuation characteristics; nonlinearity and wave shock; the role of tailwater in wave propagation; and free surface instability. A uniformly valid nonlinear solution to describe the unsteady gradually varying flow throughout the complete wave propagation domain at and away from the kinematic wave shock as well as near the downstream boundary that exhibits the tailwater effect is derived by employing the matched asymptotic method. Different scenarios of hydraulically spatially varying surface profiles such as M1, M2, and S1 type profiles are discussed. Results from the nonlinear wave analysis are further interpreted and the influence of the tailwater effect is identified. In addition to the nonlinear wave analysis, a linear stability analysis is introduced to quantify the impact from such water surface profiles on the free surface instability. It is shown that the asymptotic flow structure is composed of three distinct regions: an outer region that is driven by gravity and channel resistance; a near wave shock region dominated by the convective inertia, pressure gradient, gravity and channel resistance; and a downstream boundary impact region where the convective inertia, pressure gradient, gravity and channel resistance terms are of importance. The tailwater effect is demonstrated influential to the flow structure, free surface stability, wave transmission mechanism, and hydrostatic pressure gradient in flow.

Natural hazards, such as earthquakes and strong wind events, place large forces on tall, slender structures and on long-span bridges. In view of the numerous uncertainties due to model errors, stress calculations, material properties, and environmental loads, the structural system is uncertain. Here, the Lagrangian representation is modeled as an uncertain state-space model. The paper develops a robust active control approach with uncertainties in not only the system and control input matrices, but also the disturbance input matrices. Robust active control provides both robust relative stability and $H_{\infty }$ disturbance attenuation. The $H_{\infty }$ norm of the transfer function from the external disturbance forces (e.g., earthquake, wind, etc.) to the observed system states is restricted by a prescribed attenuation index. The uncertainties considered herein are norm-bounded unstructured uncertainties. Preservation of $H_2$ optimality of robust structural control is also revealed. The results may be further extended to structured uncertainties. A numerical example illustrates that the approach may be applied to robust control of structural systems under earthquake excitation.

This paper explores the use of energy concepts in the analysis of base-isolated structures subject to severe earthquake ground motions. We formulate the energy balance equations in moving- and fixed-base coordinate frames and provide new physical insight into the time-dependent behavior of individual terms. Conventional wisdom in earthquake engineering circles is that systems with base isolation devices should be economically competitive and designed to: (1) minimize input energy, and (2) maximize the percentage of input energy dissipated by damping and inelastic mechanisms. Through the nonlinear time-history analysis of a base-isolated mass-spring system subject to an ensemble of severe ground motion inputs, we demonstrate that improvements in objective (2) often need to be balanced against increases in input energy. Hence, by itself, objective (1) presents an overly simplified view of desirable behavior.

The emphasis of this note is placed on finding out whether dynamic snap-through will occur when a shallow arch resting on an elastic foundation with elastic constant β is stretched or compressed with constant speed. It is found that for $\beta >8/7$ no dynamic snap-through will occur when the arch is stretched. However, as long as $\beta <4\text{,}$ dynamic snap-through is still possible when the arch is compressed.