Journal of Engineering Mechanics

Static analysis of orthotropic functionally graded (FG) elastic, rectangular, and simply supported (diaphragm) plates under transverse loads is presented based on a higher order shear and normal deformation theory (HOSNT). Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with mechanical properties changing smoothly with respect to the spatial coordinates. The material properties of FG plates are assumed here to be varying through the thickness of the plate in a continuous manner. The Poisson’s ratios of the FG plates are assumed to be constant, but their Young’s moduli vary continuously in the thickness direction according to the volume fraction of constituents, which are mathematically modeled as an exponential function. The governing equations of equilibrium for the FG plates are derived on the basis of a HOSNT assuming varying material properties. Numerical solutions are obtained by the use of the Navier solution method. Several examples of isotropic, orthotropic, and FG plates are presented. The accuracy of the numerical solutions has been compared with the solutions obtained by other models and the exact three-dimensional (3D) elasticity solutions.

Many highway bridges in the United States are reaching their intended design lives and are in need of attention. To avoid replacing these bridges simultaneously, methodologies to safely extend their lives are important to help avoid high replacement costs and to allow for bridge replacement to occur over a longer time window. This paper expands an approach to extend the fatigue life of vulnerable steel bridges through a response modification (RM) apparatus, consisting of a mechanical amplifier and a RM device, which provide efficient supplemental stiffness and damping to the bridge. Because of the relatively small deflections encountered under typical service loads, the use of a mechanical amplifier allows for a more efficient and less intrusive apparatus to provide the required RM forces imparted to the bridge for increased bridge life. This paper presents a parameter study exploring the flexibility of the elements in the RM apparatus, the length of the apparatus, and the characteristics of the RM device. The analyses also add superstructure damping and other model improvements to a comprehensive numerical model of a realistic in-service bridge. The paper carries out dynamic analyses in the frequency domain to ensure the robustness of the RM apparatuses for various loading frequencies. The most important characteristic of the RM apparatus is found to be the cross-sectional area of the members. Additionally, the parameter study showed that a smaller apparatus is most cost effective. Finally, the dynamic analyses showed that the use of a semiactive device may be beneficial to the effectiveness of the RM approach.

A theoretical model of in-plane and out-of-plane rain-wind-induced vibrations (RWIVs) of stay cables is developed in this paper. The proposed scheme models the cable, akin to sectional models of a bridge deck, as a segmental rigid element with two degrees of freedom (DOF) (in plane and out of plane) and the rivulet as a single DOF oscillator. The interaction between the cable surface and the water rivulet is first analyzed by using a wetting theory, which suggests a combined application of linear and Coulomb damping forces between the rivulet and the cable. Based on the model, the response of the combined cable-rivulet system is numerically evaluated. The results show that the in-plane and out-of-plane DOF of the cable are coupled. For the equilibrium of the rivulet on the cable surface, the gravitational and aerodynamic forces on the rivulet have similar time-averaged values with opposite signs. The interaction between the cable surface and the rivulet leads to energy dissipation, which helps to stabilize the rivulet. Like galloping, RWIV appears to be triggered by a sudden drop in the mean aerodynamic force coefficient. However, it is not a divergent type of vibration because the rivulet oscillates on the cable surface and its equilibrium position changes with wind velocity. Finally, the turbulence effect on the RWIVs is investigated. As the longitudinal turbulence intensity approaches 15%, the RWIV abates concomitantly.

The presented work investigates the deformed shape and tensile state of inflated membranes partially submerged in fluid. The understanding of membrane behavior is of primary concern in marine-related problems, as well as other fields. This analysis is motivated by the flexible seals of surface effect ships and inflatable pontoons. A new exact solution for the governing differential equations and a new finite-element approach are developed. The stiffness influence coefficient method is used to derive the stiffness of the curved elements in which the end nodal displacements define the degrees of freedom. The methods of analysis are shown to provide two-dimensional results that are in excellent agreement in terms of both deformed geometry and membrane tension.

Nakamura’s model is proposed to describe lateral vibrations of a footbridge induced by pedestrians. The model is found to be a slow-fast system if the number of pedestrians on a footbridge increases slowly. Then, based on the delayed bifurcation theory, Nakamura’s model can be used to describe the delayed increase in the lateral vibration of the footbridge. The critical number of pedestrians on the footbridge is calculated by using the delayed bifurcation theory to quantitatively demonstrate our analysis. By comparison with a previously obtained formula, the rationality of the authors’ formula for predicting the critical number of pedestrians is illustrated. Based on the authors’ formula, the experimental parameter $k=300$ used in the previously obtained formula does not seem to be always applicable to other bridges. Based on Nakamura’s model, the value of the parameter $k$ seems to be affected mainly by the properties of the pedestrians on the bridge rather than those of the bridge.

Mathematical modeling of the nonlinear triaxial behavior and damage of such a complex material as concrete has been a long-standing challenge in which progress has been made only in gradual increments. The goal of this study is a realistic and robust material model for explicit finite-element programs for concrete structures that computes the stress tensor from the given strain tensor and some history variables. The microplane models, which use a constitutive equation in a vectorial rather than tensorial form and are semimultiscale by virtue of capturing interactions among phenomena of different orientation, can serve this goal effectively. This paper presents a new concrete microplane model, M7, which achieves this goal much better than the previous versions M1–M6 developed at Northwestern University since 1985. The basic mathematical structure of M7 is logically correlated to thermodynamic potentials for the elastic regime, the tensile and compressive damage regimes, and the frictional slip regime. Given that the volumetric-deviatoric (V-D) split of strains is inevitable for distinguishing between compression failures at low and high confinement, the key idea is to apply the V-D split only to the microplane compressive stress-strain boundaries (or strain-dependent yield limits), the sum of which is compared with the total normal stress from the microplane constitutive relation. This avoids the use of the V-D split of the elastic strains and of the tensile stress-strain boundary, which caused various troubles in M3–M6 such as excessive lateral strains and stress locking in far postpeak uniaxial extension, poor representation of unloading and loading cycles, and inability to represent high dilatancy under postpeak compression in lower-strength concretes. Moreover, the differences between high hydrostatic compression and compressive uniaxial strain are accurately captured by considering the compressive volumetric boundary as dependent on the principal strain difference. The model is verified extensively in the companion paper.

The microplane material model for concrete, formulated mathematically in the companion paper, is calibrated by material test data from all the typical laboratory tests taken from the literature. Then, the model is verified by finite-element simulations of data for some characteristic tests with highly nonuniform strain fields. The scaling properties of model M7 are determined. With the volumetric stress effect taken from the previous load step, the M7 numerical algorithm is explicit, delivering in each load step the stress tensor from the strain tensor with no iterative loop. This makes the model robust and suitable for large-scale finite-element computations. There are five free, easily adjustable material parameters, which make it possible to match the given compressive strength, the corresponding strain, the given hydrostatic compression curve, and certain triaxial aspects. In addition, there are many fixed, hard-to-adjust parameters, which can be taken to be the same for all concretes. The optimum values of material parameters are determined by fitting a particularly broad range of test results, including the important tests of compression-tension load cycles, mixed-mode fracture, tension-shear failure of double-edge-notched specimens, and vertex effect when axial compression is followed by torsion. Because of the lack of information on the material characteristic length or fracture energy, which can be obtained only by size effect tests on the same concrete, and on the precise boundary conditions and precise gauge locations, the finite-element fitting of the present test data can hardly be expected to give better results than single-point simulations of specimens with approximately homogeneous strain states within the gauge length. Nevertheless, tensile test data with severe localization are delocalized on the basis of assumed material length. Model M7 is shown to fit a considerably broader range of test data than the preceding models M1–M6.

The Hertzian contact model, which is usually used in analyzing the response of vibroimpact system with elastic contact, cannot deal with the dissipated contact problem in general. A modified Hertzian contact model that was developed through recent experiments, however, provides an opportunity to break this limitation. The random response of vibrating system with dissipated impact subjected to Gaussian white noise is investigated in this paper, and the dissipated impact is incorporated into random analysis through this modified model. The inelastic contact force can be separated into two parts: elastic restoring component and dissipative component. The restoring component is expressed as the gradient of the potential energy preserved in the system and the dissipative component is approximated by a damping force with energy-dependent damping coefficient through energy dissipation balance technique. The Itô stochastic differential equation with respect to the total energy of the system, which is a one-dimensional Markov process, is derived through the stochastic averaging of the energy envelope. The stationary probability density of system total energy, the joint probability density of system displacement and velocity, and the statistics of system response are analytically obtained from the associated Fokker-Planck-Kolmogorov equation. The agreement between the analytical results and Monte-Carlo simulations validates the effectiveness of the proposed technique.

The emergence of performance-based wind engineering calls for improved probabilistic modeling of wind effects on buildings. This paper focuses on the development of probabilistic models of the peak distribution and peak factor for non-Gaussian processes and explores the applications of this development in wind engineering. The closed-form expressions for the mean, SD, and fractile levels of extremes are derived for a random process whose peaks are modeled by the parametric Weibull distribution. A new translated-peak-process method is then developed for the estimation of the peak distribution, peak factor, and variability of extremes, based on the Weibull distribution and point-to-point mapping procedure. The proposed translated-peak-process method is validated by wind-tunnel pressure measurements on a standard tall building and is shown to be more robust and practical than many existing methods in analyzing non-Gaussian wind pressure data.

An analytical method is introduced to deal with the coupling problem for Euler-Bernoulli beams on elastic bidimensional foundations by considering the horizontal and vertical displacements of the beam-foundation system. The approach is an extension of the modified Vlasov model. With separation of the variables, the horizontal and vertical displacements are expressed as the displacement function at the ground surface and the attenuation function along the depth of the foundations, respectively. The governing equations and the corresponding boundary conditions of the model are obtained via the variational principle. Then, the differential operator method is used to uncouple the governing equations and boundary conditions. An iterative procedure is executed to accomplish the numerical implementation. A parametric study is conducted to illustrate the effects of the applied loadings and the physical and geometry properties on the static responses of the beam and foundations. The numerical results show that the coupling elasticity should be taken into account in cases of flexible beams and high soil Poisson ratios. Moreover, the horizontal loads on the beam significantly affect the response of the beam-foundation system.

In previous studies by the authors, a substructure identification method for shear structures was proposed to identify all structural story stiffness and damping parameters from top to bottom inductively. In the method derivation, the structural responses were required to be wide sense stationary to convert structural dynamic equations to differential equations in the correlation functions of structural responses, which are used to formulate substructure identifications. In this paper, this method is extended to accommodate nonstationary structural responses. A different derivation procedure is adopted to formulate substructure identifications directly from the Fourier transform of the structural dynamic equations, resulting in a formulation nearly the same as its stationary response predecessor. An identification error analysis for the substructure identification method reveals how structural responses affect the identification accuracy. On the basis of this result, a smart reference selection rule is proposed to choose the best reference response candidate, needed for forming the substructure identifications, to improve the identification accuracy. A 10-story shear building structure is used to illustrate the effectiveness of the substructure identification method with two kinds of nonstationary excitations: a long nonstationary response and a group of short earthquake excitations. The simulation results show that this substructure identification method provides very accurate identification results even with very large measurement noise and nonstationary responses.

To meet the requirements of performance-based earthquake engineering, the influence of damage should be considered when conducting a structural performance assessment. In recent years, significant effort has gone toward developing accurate and efficient damage models for seismic damage analysis. In this paper, a framework for seismic damage analysis of RC frame structures considering stiffness degradation is established based on the force analogy method (FAM). A damaged hinge (DH) model, which is located at the ends of columns and beams, is proposed for modeling damage behavior owing to concrete cracking. The DH, together with the plastic hinge (PH), arising from structural materials and presented in prior research on the FAM, forms a complete inelastic mechanism including stiffness degradation behavior for RC frame structures. As damage effect is implemented by introducing the damage indexes as internal variables, the real-time structural performance and damage level can be evaluated during the computation process. Additionally, because only initial stiffness is used throughout the dynamical computation analysis, the usage of the state space formulation (an outstanding advantage of the FAM) is retained and makes the real-time damage analysis more accurate, efficient, and stable. The inelastic mechanism, which takes into account the damage effect on the stiffness degradation behavior in the FAM, is verified against experimental data, and the proposed method is applied in a RC frame structure to demonstrate its potential and efficiency on the seismic damage analysis.

For medium-thick cylindrical pressure vessels, or the pressure vessels with radius ratio $k$ less than the value determined by the equation $k^2\ln k/\left(k^2-1\right)=1$, or $k\le 2.2184574899167\dots$, the theoretical problems on autofrettage are studied according to the third strength theory including the overstrain, the distributions of residual stresses and the total stresses, and load bearing by comprehensively controlling various stresses, such as residual stresses, the total stresses, and their equivalent stresses; the safe conditions of load bearing are found. The results show that for this kind of pressure vessel, the overstrain when they are treated with autofrettage can be 100%, with no compressive yield taking place. As long as the pressure that this kind of pressure vessel contains is not higher than the entire yield pressure $p_y/{\sigma }_y=\ln k$, the total stresses and their equivalent stress cannot exceed the yield strength ${\sigma }_y$, and the safety of the pressure vessels can be assured. The theory established in this research can be applied to engineering directly or with some minor modifications.

Development of a model for the analysis of strain transfer mechanism in Brillouin-based sensors with strain singularities is provided in this study. The main objective of the research pertained to the development of a method for accurate detection of cracks and their locations in sensing with Brillouin-based fiber optic distributed sensors. The work involved formulation of a shear lag–based model considering the elastic as well as elastoplastic stages of the fiber optic coating strains. Feasibility of the proposed approach is evaluated through an experimental program. The experimental program involved use of a Brillouin optical time domain reflectometer (BOTDR) for distributed measurement of strain and detection of simulated cracks in a 15-m-long beam. The results indicate that the discontinuities in the strain distribution based on the theoretical analysis provide the means to accurately pinpoint the location of simulated cracks. On the other hand, the distortion effect of the BOTDR system due to averaging of the strains over the spatial resolution of the system masked the influence of strain discontinuities. In contrast to the strains acquired through the averaging process in BOTDR, the theoretical computations pertain to the actual distribution of strain along the length of the beam. Optomechanical relationships were also used to simulate the effect of spatial resolution on the theoretical results. The apparent strains obtained in this way were compared with the BOTDR measured values. Because of the measurement noise, the apparent strains based on the theoretical computations provided better information about the location of the cracks.

At an early age, hydration of cement leads to a reduction in volume (caused by Le Chatelier contraction) that induces autogenous shrinkage. In addition, hydration is an exothermic reaction, and an increase in temperature occurs (followed by a decrease). Because autogenous shrinkage arises only in cement paste, and because the coefficient of thermal expansion may be different between cement paste and aggregates, strain incompatibilities lead to an internal self-equilibrated state of stress. Depending especially on the concrete mix, initial cracking may occur at the cement-paste scale leading to a modification of the global concrete behavior. In this paper, finite-element simulations using a mesoscopic mesh are performed to access to the internal stresses and damage state resulting from the hydration process. Calculations take into account the autogenous shrinkage of the cement paste and the differential thermal behavior between cement paste and aggregates. The influences of basic creep strains, thermal boundary conditions, and concrete mix (by a simplified approach) are studied, and it is shown that for ordinary concrete, hydration leads to a slight reduction in elastic stiffness and tensile strength when creep is taken into account. However, for high-performance concrete, a significant reduction in the elastic stiffness and tensile strength is expected (with respect to the potential tensile strength) even if creep is taken into account.

In this paper, Bayesian linear finite-element (FE) model updating is applied for uncertainty quantification (UQ) in the vibration-based damage assessment of a 7-story RC building slice. This structure was built and tested at full scale on the University of California at San Diego-Network for Earthquake Engineering Simulation shake table: progressive damage was induced by subjecting it to a set of historical earthquake ground motion records of increasing intensity. At each damage stage, modal characteristics, such as natural frequencies and mode shapes, were identified through low-amplitude vibration testing; these data are used in the Bayesian FE model updating scheme. To analyze the results of the Bayesian scheme and gain insight into the information contained in the data, a comprehensive uncertainty and resolution analysis is proposed and applied to the 7-story building test case. The Bayesian UQ approach and subsequent resolution analysis are shown to be effective in assessing uncertainty in FE model updating. Furthermore, it is demonstrated that the Bayesian FE model updating approach provides insight into the regularization of its often ill-posed deterministic counterpart.

A brief overview of vortex-induced vibration (VIV) of bridge decks is presented, highlighting special VIV features concerning bridge decks. A popular VIV model (Van der Pol–type model) for bridge decks is examined in detail. Alternatively, a truncated Volterra series–based nonlinear oscillator is introduced to model the VIV system. Typical features of VIV such as the limit cycle oscillation (LCO), frequency shift, hysteresis, and beat phenomenon are parsimoniously and accurately captured in the proposed nonlinear model. As a functional expansion of a nonlinear system, the Volterra series is convenient for estimating the linear and nonlinear contributions to VIV. It is demonstrated that the relative contribution of nonlinear effects in VIV is around 50% of the total response for a range of bridge cross sections. The efficacy of the Volterra series as a reduced-order model (ROM) in capturing aerodynamic nonlinearities eliminates the need for reliance on conventional phenomenological models as it promises to offer a unified framework for nonlinear wind effects on long-span bridges—for example, VIV, buffeting, and flutter.

An adaptive linear FEM for modal analysis of structures is presented in this paper. The method uses frequency-dependent shape functions in addition to conventional low-order polynomial interpolation functions to improve accuracy in natural-frequency and mode-shape estimation. The proposed technique requires the iterative computation of the mass and stiffness matrices because shape functions are adjusted recursively as functions of estimated natural frequencies without mesh adaptation or refinement. Applied to frame structures, the method uses conventional polynomial interpolating functions for axial, bending, and torsional displacements and additional generalized coordinates multiplied by frequency-dependent functions derived from modal analysis of continuous beam models. Because these additional functions or their derivatives do not vanish at nodes located at the element boundaries, linear kinematic constraints are imposed to the augmented set of displacement coordinates to ensure displacement field continuity and compatibility conditions at element boundaries. Applications of the methodology to straight rod elements in longitudinal vibration and beams in flexural vibration are presented to compare accuracy obtained using conventional finite-element (FE) meshes with the proposed method.

A nonlinear theoretical model is developed in this paper for investigating the time-dependent behavior of one-way high-strength RC panels, with particular emphasis on the combined effects of creep and shrinkage on the buckling capacity and its degradation with time. A rheological generalized Maxwell chain model is used for modeling the creep of the concrete including its cracking, tension stiffening, and aging through strain- and time-dependent springs and dashpots. The incremental governing equations of the panel are derived and solved through a step-by-step time analysis that takes into account the variation of the internal stresses and deformations with time. A smeared cracking model is adopted, and an iterative procedure is conducted at each time step for the determination of the unknown rigidities of the cracked section, as well as the length of the cracked region. The capabilities of the proposed model are demonstrated through numerical and parametric studies, which show the important roles of creep and shrinkage in the buckling of high-strength concrete panels and which reveal the sensitivity of the nonlinear response to the magnitude and eccentricity of the sustained load and to the reinforcement ratio.

The deformable discrete-element method (DEM) was employed to study the effect of particle shape on the macroscopic mechanical behavior of rockfill materials. Relationships among the micromechanical parameters and macroscopic response of rockfill materials were identified by simulating triaxial compression tests on numerical samples. The numerical experiment results showed that the particle shape has a significant impact on the macroscopic response of rockfill. For instance, the peak strength and residual strength of the numerical sample and the axial strain corresponding to the peak strength increased with an increase in the aspect ratio of the particle.

A problem-invariant $M_c$-integral is proposed as an energy parameter for describing the degradation of structural integrity caused by irreversible evolution of multiple cracks in three-dimensional (3D) elastic solids. The physical meaning for 3D $M_c$, which is related to the surface energy corresponding to creation of the cracks, does not hold in the same manner as that for two-dimensional (2D) $M_c$ and needs to be properly reformulated. Also, the 3D integration is shown to be surface-independent in a modified sense. With this property, by choosing a closed surface remote from the crack fronts, the 3D $M_c$ can then be accurately evaluated with finite-element (FE) solutions even when the near-front areas are not simulated with very fine grids.

For partial-interaction composite beams, two beam theories (i.e., the Euler-Bernoulli and Timoshenko beam theories) are usually used to investigate their deflections, slips, and stress resultants. However, the relationships between the solutions of partial-interaction composite beams based on the two beam theories have not been discussed while the corresponding relationships for homogeneous beams have been investigated in detail for several years. By analyzing the constitutive relationships and equations of equilibrium, the authors derived the relationships of the solutions between single-span Euler-Bernoulli and Timoshenko partial-interaction composite beams. The integral constants are also presented for various boundary conditions. Through the presented relationships, the solutions of the Timoshenko partial-interaction composite beams could be readily obtained from the solutions of the corresponding Euler-Bernoulli counterparts. As a result, the more complicated flexural-slip-shear deformation analysis based on Timoshenko beam theory may be avoided for engineering designers.

Modern bridge decks exhibit nonlinear behavior in wind tunnel experiments; this has placed increasing importance on nonlinear bridge aerodynamics/aeroelasticity considerations for long-span bridges. In this context, various observed nonlinear phenomena in wind tunnels are discussed, and their identification through high-order spectrum is illustrated.

Direct numerical simulation of two-dimensional flow past a normal flat plate having an aspect ratio of 0.125 has been carried out in the Reynolds number range of 30–175 to investigate the effect of various transitions of the wake with increasing Reynolds number reported previously on the mean and RMS flow field. The time-averaged and RMS flow field, particularly the far-wake, has been observed to be sensitive to various transitions. However, a marginal effect is observed on the mean integral parameters such as the drag coefficient and the Strouhal number.