Journal of Engineering Mechanics

In this study new exact Green’s functions and a new exact Poisson-type integral formula for a boundary-value problem (BVP) in thermoelasticity for a half-wedge with mixed homogeneous mechanical boundary conditions are derived, in which the boundary angle is rigidly fixed and the normal displacements and tangential stresses or the normal stresses and tangential displacements are prescribed on the boundary quarter-planes. The thermoelastic displacements are subjected to a heat source applied to the inner points of the half-wedge and to mixed nonhomogeneous boundary heat conditions, in which the temperature is prescribed to the boundary angle or to one boundary quarter-plane and the heat flux is given on the other boundary quarter-plane. When the thermoelastic Green’s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by the $\delta$-Dirac function. All results are obtained in elementary functions that are formulated in a special theorem. Analogous results for an octant and for a quarter-space as particular cases of the angle of the thermoelastic half-wedge are also obtained. The main difficulties in obtaining these results are in deriving the functions of the influence of a unit concentrated force onto elastic volume dilatation ${\Theta }^{(q)}$ and, also, in calculating a volume integral of the product of function ${\Theta }^{(q)}$ and the Green’s function in heat conduction. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for a half-wedge, using the derived Poisson-type integral formula and the influence functions ${\Theta }^{(q)}$, also are included. The proposed approach may be extended not only to many various BVPs for half-wedges but also to many canonical cylindrical and other orthogonal domains.

An anisotropic elastic-viscoplastic constitutive model is developed for $K_0$-consolidated soft clays in general stress space. Two surfaces are assumed to exist for any loading stress as the loading surface and the reference surface. The scalar multiplier is established by the viscoplastic volumetric strain rate under a one-dimensional straining condition. This model can adequately simulate the stress–strain behavior of undrained triaxial constant strain rate shear tests and undrained creep tests for $K_0$-consolidated clays. The strain-rate dependencies of preconsolidation pressure and undrained shear strength are investigated in analytical formulations, and the similarities and differences of their corresponding strain-rate parameters are examined.

A transformation method is presented for the response of yielding oscillators to dynamic loading. The method employs a translation in the ordinates and the abscissa of the excitation function by means of a pair of parameters uniquely dependent on the yielding resistance and the vibrational characteristics of the system. By this approach: (1) the differential operator becomes linearlike, with the nonlinearity transferred to the right-hand side; (2) the initial conditions are simplified; and (3) the modified forcing term becomes uniquely associated with the development of plastic deformation. The theory is applied to various yielding oscillators subjected to idealized earthquake pulses, with the modified excitation function termed plastic input motion (PIM). A procedure for applying the method to general waveforms is provided. The coordinates of PIM may be discontinuous and significantly smaller than those of the original excitation function, as a considerable amount of ground acceleration is devoted to overcoming the elastic resistance of the system. The theory can be useful in earthquake engineering by offering a replacement to physical ground motions with system-dependent PIMs for establishing demand indices.

The dynamic responses of a structure with different foundations, including the so-called periodic foundation, are studied. Based on the calculation and optimization regarding the band of frequency gap of a periodic structure, a new kind of periodic isolated foundation is designed. To test the validity of this periodic isolated foundation, the dynamic responses of a seven-storey frame structure with three different foundations are conducted and compared. The analysis assumes that seismic wave input is from the arbitrary direction. This investigation shows that the seismic waves cannot propagate in the periodic foundation without being attenuated when the frequencies of the seismic wave fall within the band of frequency gap of the foundation. Thus, the dynamic responses of the supported structure will be greatly reduced. The investigation supports this new and effective method to block seismic waves for structures in civil engineering.

Rubber blocks (or springs) are structural components widely used in many applications. The design characteristics of a rubber block (or spring) under vertical loading are an apparent modulus (or compression modulus) and normal and shear stresses on the contact surfaces. The problem of a rubber block bonded to two rigid plates has been solved using various approaches. In this study, improved approximate solutions for rectangular blocks in plane strain and for axisymmetric disks are derived using a combination of an energy method and linear elasticity approach when the rubber is treated as a linear elastic incompressible material. The closed-form results are compared with previous results for the apparent modulus and contact stress distributions. Similarly, the problem of a rubber block with a surface bonded to a rigid plate while another surface is in contact with a zero frictional rigid plate is also solved using the same method. Finite-element analysis and experiment results for axisymmetric disks of several shape factors ranging from 1 to 6 are compared for validation of the analytical results for both contact conditions.

Steady amplitude vibration on cylinder structures caused by dynamic ice loading has been observed during the measurements on prototype structures. Further analysis indicates that it belongs to self-excited vibration, in which the vibration amplitude stays constant and the frequency of the dynamic ice load is locked by the frequency of the structural vibration. Based on the conclusions from ice mechanics, this study proposes a physical mechanism to describe the ice-induced self-excitation process: during the loading phase within a single vibration cycle, the compressive strain rate in the ice sheet close to the structure enters a ductile-brittle transition range, and thereby the steady formation of micro cracks maintains the resistance of the ice sheet as the ice load steadily increases; when the density of cracks in the ice reaches a critical level, cracks coalesce and collapse, resulting in the unloading phase. In this way structure vibration dominates the time-variation of the ice load. The proposed mechanism is verified using field data.

This paper introduces a novel modeling approach for wood composites using concepts of numerical homogenization employed in synthetic composites. It describes a multiscale model based on a unit cell that incorporates both the wood and resin phases for simulating structural composite lumber made of strands. In this approach, constant resin thickness and strand geometry, elastic properties of constituents, and perfect bonding between wood and resin are assumed. The multiscale modeling is composed of two steps. The first step estimates the effective elastic properties of a unit cell based on the numerical homogenization with periodic boundary conditions. The second step consists of a macroscopic finite element structural analysis of a beam (assembly of several unit cells) under three-point bending. Random distribution of strand orientation that may be encountered in an actual composite beam is introduced at this stage. Results indicate a significant influence of the resin. The first step of the approach provides an initial illustration when comparing effective properties of unit cells with different resin volume fractions and/or elastic properties. The resin decreases the Young’s modulus of the unit cell in the fiber direction while strengthening the transverse and shear moduli depending on the contrast between the resin and wood properties. The final results obtained for the beam show that the bending modulus decreases with increasing resin content, which is a combined effect of the micromechanical phenomena. The key contribution of this paper to modeling wood composites is the addition of the resin phase to a numerical model and inclusion of random distribution of strand orientation.

In full-scale ambient vibration tests, many situations exist where it is required to obtain a detailed mode shape of a structure with a limited number of sensors. A common feasible strategy is to perform multiple setups with each one covering a different part of the structure while sharing some reference degrees of freedom (DOF) in common. Methods exist that assemble the mode shapes identified in individual setups to form a global one covering all measured DOF. This paper presents a fast Bayesian method for modal identification capable of incorporating the fast Fourier transform information in different setups consistent with probability logic. The method allows the global mode shape to be determined, taking into account the quality of data in different setups. A fast iterative algorithm is developed that allows practical implementation even for a large number of DOF. The method is illustrated with synthetic and field test data. Challenges of the mode shape assembly problem arising in field applications are investigated through a critical appraisal.

Vehicle-bridge interaction has been studied for a long time to investigate the structural behavior of bridges and vehicle ride comfort. An original frequency domain method is presented where the vehicle-bridge interaction problem is solved in a frame of reference that moves with the vehicle. The Fourier transform of the interaction force is computed directly from the vehicle compliance and bridge compliance, without requiring any iterations. The method is particularly useful when a closed-form solution of the bridge compliance is available, as in the case of a simply supported Euler-Bernoulli beam model for the bridge. The solution is, therefore, well-suited for parametric studies on the bridge and vehicle response characteristics and offers a reference for more detailed models of the bridge and the vehicle or more complicated bridge configurations (e.g., continuous beam on multiple supports). The frequency domain approach also leads to enhanced physical understanding, because it shows how the interaction force decomposes into a term resulting from the dynamic response of the bridge to the constant moving load component and a term because of road surface unevenness. An efficient solution procedure based on random vibration analysis is presented, which allows for the computing of the statistical characteristics of the bridge and vehicle response from the power spectral density function of the unevenness. The procedure is validated by means of Monte Carlo simulation results for the case where the passage of a heavy vehicle on a highway bridge is considered.

The lattice discrete particle model (LDPM) is a mesoscale model for heterogeneous materials. Developed for concrete, it simulates material mesostructure by modeling coarse aggregate particles and their surrounding mortar as polyhedral cells. A tetrahedralization of the particle centers generates a lattice framework where each lattice member is associated with a triangular-shaped plane of contact (facet) between two cells. Compatibility equations are formulated by describing the deformation of an assemblage of particles through rigid-body kinematics. Equilibrium equations are obtained through the force and moment equilibrium of each cell. The material behavior is assumed to be governed by a vectorial constitutive law imposed at the facets. A natural extension for this discrete model is to include the effect of dispersed fibers as discrete entities within the mesostructure. The LDPM incorporates this effect by modeling individual fibers randomly placed within the framework according to a given fiber volume fraction. The number and orientation of the fibers crossing each facet is computed and the contribution of each fiber to the facet response is formulated on the basis of a previously established micromechanical model for fiber-matrix interaction. The theory for the developed model, entitled the LDPM-F, is discussed herein. A subsequent companion paper will address model calibration and validation through the numerical simulation of experimental test results.

In Part I of this two-part study, a theory is provided for the extension of the lattice discrete particle model (LDPM) to include fiber reinforcing capability. The resulting model, LDPM-F, is calibrated and validated in the present paper by comparing numerical simulations with experimental data gathered from the literature. The analyzed experiments include direct tension, confined and unconfined compression, and notched three-point bending tests.

Seismic performance of extended pile-shafts depends not only on the intensity of the earthquake ground motion, but also on the pile structural properties and the soil conditions surrounding the pile. For assurance of good foundation performance, damage can be controlled by limiting the curvature ductility demand in the yielding region of the pile. However, curvature ductility demand is influenced by the strength and stiffness of the pile and the soil, and the ability of the soil-pile system to distribute the inelastic deformation in the pile. The pile curvature distribution can be written in the form of a nonlinear differential equation, including the effects of the pile yielding, soil nonlinearity, and secondary $P-\Delta$ moment arising from the axial load. A solution to the differential equation lends itself to the pile deflection, pile internal forces, soil pressure distribution, and, more importantly, the inelastic curvature distribution in the pile. The overall response in terms of a lateral force versus lateral displacement curve, and a kinematic relationship between the displacement ductility and curvature ductility factors, are computed as part of the solution. The proposed model is shown to provide good agreement with a finite-element model.

Extended UH model is a three-dimensional elastoplastic constitutive model that builds on a modification of the UH model with a unified hardening parameter, to account for the effect of anisotropy and the influence of $K_0$ consolidation on the stress-strain-strength behavior of clays. The combined effects of anisotropy and stress magnitude are considered through the minimum value of angles ($\alpha$) made by the spatially mobilized planes (SMPs) and soil depositional plane. The original transformed stress tensor is revised by incorporating the anisotropic peak stress ratio $M_{\alpha }$, which is defined as a function of $\alpha$, with the stress tensor ${\sigma }_{ij}$. The comparison with test results on San Francisco Bay Mud demonstrates the capability of the proposed anisotropic criterion. The UH model is extended to account for the combined effects of inherent anisotropy and $K_0$ conditions on the stress-strain-strength response of anisotropic clays by incorporating the revised transformed stress tensor $\widetilde{{\sigma }_{ij}}$, the potential strength ratio $M_f$, the characteristic stress ratio $M$, and the state stress ratio ${\tilde{\eta }}_a$. A series of undrained triaxial tests on isotropically and anisotropically consolidated specimens with different overconsolidation ratios (OCRs) are successfully predicted using the proposed extended UH model. New parameters for anisotropic soils can be determined conveniently from the conventional triaxial compression tests on vertical and horizontal specimens.

Substructure hybrid simulation of engineering structures subjected to multiple-support excitation (MSE) is investigated. In time-history analysis of MSE problems, the equations of motion are generally posed in relative coordinates (designated EOM-rel), which only require the ground acceleration as an input. However, the suitability of EOM-rel for hybrid simulation with MSE is unclear. Because estimating the tangent stiffness of the experimental substructure during testing is challenging, constant stiffness is often adopted in widely used integration schemes for hybrid simulation (e.g., the operator-splitting method); the equivalent force is not updated during experimentation. In previous studies, the equations of motion in absolute coordinates (designated EOM-abs) have been employed for hybrid simulation considering MSE. With EOM-abs, nonlinearity in the stiffness matrix is naturally considered in the measured restoring force—no approximation is involved. One drawback to EOM-abs is that the displacement and velocity of the ground motion must be computed prior to the experiment. This paper carefully formulates the hybrid simulation problem for structures subjected to MSE and examines the tradeoffs in the two approaches. First, the EOM-rel approximation and the EOM-abs are formulated and compared for a nonlinear single degree of freedom structure with nonlinearities modeled by a Bouc-Wen model subjected to two ground inputs. Significant error is found in the EOM-rel approximation because of the constant stiffness matrices used. Both formulations are then evaluated in hybrid simulation of a four-span highway ramp bridge. This study demonstrates that the EOM-rel approximation can provide results with good accuracy if the nonlinear components of the experimental substructure do not significantly affect the global stiffness matrix. Otherwise, the EOM-abs is recommended for hybrid simulation with MSE.

A combined experimental and computational study shows that the pullout capacity of anchors embedded at small depths in prestressed concrete is associated with the strongest possible (linear elastic fracture mechanics) size effect. A design formula is proposed that reflects the effects of embedment depth and the nondimensional parameters that quantify the level of prestressing and the characteristic length of the matrix.

Engineering structures must be designed for an extremely low failure probability, $P_f<{10}^{-6}$. To determine the corresponding structural strength, a mechanics-based probability distribution model is required. Recent studies have shown that quasibrittle structures that fail at the macrocrack initiation from a single representative volume element (RVE) can be statistically modeled as a finite chain of RVEs. It has further been demonstrated that, based on atomistic fracture mechanics and a statistical multiscale transition model, the strength distribution of each RVE can be approximately described by a Gaussian distribution, onto which a Weibull tail is grafted at a point of the probability about ${10}^{-4}$ to ${10}^{-3}$. The model implies that the strength distribution of quasibrittle structures depends on the structure size, varying gradually from the Gaussian distribution modified by a far-left Weibull tail applicable for small-size structures, to the Weibull distribution applicable for large-size structures. Compared with the classical Weibull strength distribution, which is limited to perfectly brittle structures, the grafted Weibull-Gaussian distribution of the RVE strength makes the computation of the strength distribution of quasibrittle structures inevitably more complicated. This paper presents two methods to facilitate this computation: (1) for structures with a simple stress field, an approximate closed-form expression for the strength distribution based on the Taylor series expansion of the grafted Weibull-Gaussian distribution; and (2) for structures with a complex stress field, a random RVE placing method based on the centroidal Voronoi tessellation. Numerical examples including three-point and four-point bend beams, and a two-dimensional analysis of the ill-fated Malpasset dam, show that Method 1 agrees well with Method 2 as well as with the previously proposed nonlocal boundary method.

Shake table tests were undertaken on a full-scale wind turbine (65-kW rated power, 22.6-m hub height, and 16-m rotor diameter) using the Network for Earthquake Engineering Simulation Large High Performance Outdoor Shake Table at the University of California, San Diego. Structural response characteristics and modal parameters are presented for base shaking imparted in two configurations, both parallel (configuration 1) and perpendicular (configuration 2) to the axis of rotation of the rotor. Results are summarized for a series of progressively stronger motions imparted in configuration 1, with analysis identifying damage sources leading to an overall loss in stiffness. Two sources of observed softening are identified and quantified: (1) degradation of grout at the tower base, and (2) loss of bolt torque at the connections between tower segments. Results showed that the two configurations had little difference in structural response and demand parameters. For the tested turbine, with appropriate consideration of boundary conditions and modal characteristics, linear theory for a single degree-of-freedom system can explain most of the observed dynamics. Although not significant for the tested turbine, it was observed that higher mode behavior may be important for large turbines.

This article presents some aspects of continuous assumed gradient (CAG) methods applied to explicit structural dynamics. CAG elements are finite elements in which the strain (i.e., the deformation gradient) is replaced by a $C^0$-continuous interpolation. Examples provide measures for the accuracy and numerical efficiency of CAG methods. First-order hexahedral and tetrahedral elements are tested. The improvements in accuracy are even larger than in static examples, and increased numerical costs can be balanced by a larger critical time step.