Moderate Displacements Research Articles

Shake table testing and finite element modeling of a modular prefabricated concrete bridge‐like specimen accounting for geometry imperfections and additional damping

AbstractThis paper presents the shake table testing and finite element (FE) modeling of a modular prefabricated concrete bridge‐like specimen. The specimen comprised four equal‐height cylindrical reinforced concrete (RC) columns capped with an RC slab. The structural connections were non‐monolithic. Hence, controlled relative motion of the members, including rocking (uplift) of the piers, was allowed. The columns were connected to the slab with stiff tendons that provided positive post‐uplift stiffness. The specimen was subjected to 184 triaxial shake table tests, so that a statistical validation of numerical models can be performed. Subsequently, a detailed three‐dimensional FE model of the bridge was developed. The objectives of the present study were to: i) investigate the shake table response of a modular bridge with positive post‐uplift stiffness under multiple ground motions, ii) develop an FE model of the proposed structural system, iii) investigate the influence of geometrical imperfections on rocking bridges, and iv) evaluate the efficiency of using additional dissipative rebars. After being subjected to 184 shake table tests, the specimen showed zero damage, moderate displacements and tendon forces (TFs), low slab torsion, and zero residual displacements. The shake table tests were practically repeatable. The proposed FE model accurately captured the experimental results. Geometrical imperfections heavily affect the response of negative stiffness systems. However, they have a marginal influence on positive stiffness systems. When comparing systems with equivalent uplift resistance and post‐uplift stiffness, the use of additional dissipative rebars results in lower slab torsion and TFs, provided that the rebars do not fracture.

Reliability-based optimization of supported pendulum TMDs’ nonlinear track shape using Padé approximants

Supported Pendulum Tuned Mass Dampers (PTMDs) are vibration absorbers whose mass is constrained to oscillate along a curved path, enabling to use gravity as a restoring force. Through a proper track shape selection, they can adapt to the so-called Nonlinear Energy Sinks (NES) concept, consolidated for spring–mass–damper TMDs. Accordingly, NESs are recognized for their ability to resonate in a wide range of frequencies due to their nonlinear nature, becoming more effective when variations on the host structure’s natural frequencies occur. However, regardless of the recent advances in the last few years, gaps remain to be explored in the track shape optimization of supported PTMDs. For instance, a comprehensive literature review shows that Reliability-Based Design Optimization (RBDO) studies to determine the track shape of supported PTMDs cannot be found to the best of the author’s knowledge. Within this context, this paper proposes an original RBDO approach to define the track shape of supported PTMDs in buildings subjected to seismic excitation. For this purpose, a generic rational function with an arbitrary curvature, determined from the Padé approximation of the circle equation, is constructed to allow the PTMD’s path shape search. This procedure allows optimizing fewer parameters than required in Taylor series expansions while providing a wider range of alternatives for numerical optimization. Finally, an active-learning Kriging-based Efficient Global Optimization (EGO) procedure is employed to find the optimum solutions with few objective function evaluations. The application case consists of a typical soft story building subjected to seismic loadings. The optimal track differs from the circle equation, with a higher slope for moderate displacements.

Evaluation of structural alterations in actin by computational simulation and high-speed AFM to study the mechanism of myosin force generation.

In this study, we used High-Speed Atomic Force Microscopy (HS-AFM) and computational simulation to evaluate the effect of actin oxidation on myosin force generation. For computational analysis, we implemented oxidation of certain tyrosine residues in the PDB structures of G-actin monomer and F-actin filaments. The selection of oxidized residues was based on our previous work, where oxidized residues in F-actin were identified by mass spectrometry. Computationally oxidized and non-oxidized PDB structures of the G-actin monomers were used to evaluate the electrostatic potential energy as well as for the simulation of the HS-AFM images. The comparison of the G-actin electrostatic maps revealed several spots affected by oxidation. A cross-section analysis around oxidized and non-oxidized residues in the simulated HS-AFM images of G-actin monomers revealed a shift in the height values (∼0.2 nm to 1 nm in magnitude). These differences were confirmed in the computationally oxidized F-actin structures and in experiments conducted with HS-AFM using F-actin oxidized in-vitro by an NO and O2 donor, SIN-1 (∼1.5 nm in magnitude). To better understand actin oxidation-induced impairment in contractility we determined the myosin head displacements while attached to non-oxidized or SIN-1-treated F-actin. We observed that adding myosin molecules to SIN-1-treated F-actin in the presence of ATP resulted in a change of the size of myosin head displacement. The myosin heads were able to produce short (below 1 nm), moderate (2-3 nm), and long (over 4 nm) displacements on the non-oxidized F-actin, but the frequency of long displacements was significantly decreased in the SIN-1-treated F-actin. These results suggest that oxidation decreases the pool of the weak bound myosin molecules and shortens the long displacements related to the Pi release step, thus reducing the force generation by myosin motors.

Influence of rheologically weak layers on fault architecture: insights from analogue models in the context of the Northern Alpine Foreland Basin

We present a series of analogue models inspired by the geology of the Zürcher Weinland region in the Northern Alpine Foreland Basin of Switzerland to explore the influence of rheological weak, i.e. (partially) ductile layers on the 3D evolution of tectonic deformation. Our model series test the impact of varying weak layer thickness and rheology, as well as different kinematics of an underlying “basal fault”. Model analysis focuses on deformation in the weak layer overburden and, uniquely, within the weak layer itself. We find that for low to moderate basal fault displacements, the above-mentioned parameters strongly influence the degree of coupling between the basal fault and the weak layer overburden. Coupling between the basal fault and overburden decreases by reducing the strength of the weak layer, or by increasing the weak layer’s thickness. As a result, basal fault displacement is less readily transferred through the weak layer, leading to a different structural style in the overburden. By contrast, increasing the amount, or rate, of basal fault slip enhances coupling and leads to a more similar structural style between basal fault and overburden. Moreover, dip-slip displacement on the basal fault is more readily transferred to the overburden than strike-slip displacement of the same magnitude. Our model results compare fairly well to natural examples in the Northern Alpine Foreland Basin, explaining various structural features. These comparisons suggest that rheological weak layers such as the Jurassic Opalinus Clay have exerted a stronger control on fault zone architecture than is commonly inferred, potentially resulting in vertical fault segmentation and variations in structural style. Furthermore, the novel addition of internal marker intervals to the weak layer in our models reveals how complex viscous flow within these layers can accommodate basal fault slip. Our model results demonstrate the complex links between fault kinematics, mechanics and 3D geometries, and can be used for interpreting structures in the Alpine Foreland, as well as in other settings with similar weak layers and basal faults driving deformation in the system.

Mesh-moving arbitrary Lagrangian–Eulerian three-dimensional technique applied to sloshing problems

Validation of three-dimensional (3D) sloshing models is a relevant task in order to guarantee the correctness of numerical predictions. In this work, attention is focused on assessing the 3D free surface (FS) evolution when an Arbitrary Lagrangian–Eulerian method is applied to solve sloshing problems. The computational scheme consists of three stages: (i) solve the incompressible Navier–Stokes equations over a single fluid domain with a FS; (ii) determine the FS displacements from the velocity field computed at (i); and (iii) update the mesh with a computational mesh dynamic problem. The study aims to evaluate the global mass-conservation strategy reported in this work for long term simulations. The forced sloshing cases present moderate FS displacements with complex wave pattern behaviours. Specifically, sloshing in a tank with a stepped shape in the bottom and two sloshing cases of a rectangular tank are presented. The numerical results are compared with experimental data to evaluate the performance of the proposed model.

Effect of pressure on pull-off of flat 1-D rectangular punch adhered to membrane

ABSTRACT Pressure can sometimes be utilized to increase or decrease the strength of adhesion. This phenomenon is analyzed for pull-off of a long, flat, rectangular punch from a long rectangular membrane. A one-dimensional nonlinear analysis is conducted. The pressure is applied to the membrane, and then the adhered punch is pulled upward. Both force control and displacement control are considered, and the pressure may be positive (acting upward) or negative (acting downward) on the membrane. The membrane is assumed to be linearly elastic, and a von Kármán-type theory is used to allow moderate displacements. A JKR-type theory of adhesion is adopted. The effects of the work of adhesion, residual stress, and prestretch are investigated. With an increase in pressure, the maximum force decreases, the pull-off deflection of the punch relative to the initial flat membrane level increases, and the total punch deflection from the pressurized membrane may increase or decrease.

Large-displacement symmetrical foldable cones.

A foldable cone, or f-cone, is a discrete vertex folded from flat with curved faces, and is an exemplar of nonrigid origami. We determine the exact equilibrium shape of a well-folded symmetrical f-cone by geometrical considerations alone, by treating as cones intersecting along original fold lines expressing equal fold angles. When moderate displacements prevail, the shape is found alternatively from the spherical image of the vertex, from deploying Gauss's mapping. The analytical working is much less compared to direct linearization of the exact solution framework; expressions for shape parameters are availed in closed form, and some of their variation is accurately retained compared to the exact case.

Effect of pressure on pull-off of punch adhered to circular or 1-D rectangular plate

Pressure can sometimes be utilized to increase or decrease the strength of adhesion. This phenomenon is analyzed for upward pull-off of (i) a flat, circular cylindrical punch from a circular plate and (ii) a long, flat, rectangular punch from a long rectangular plate. The pressure is applied, and then the punch is pulled upward. The pressure may act upward (positive) or downward (negative) on the plate. Both force control and displacement control are considered. The plate is assumed to be linearly elastic, and nonlinear von Kármán theory is used to allow moderate displacements. A transversality (debonding) condition is obtained, and the effects of the work of adhesion, the relative size of the punch, and residual stress are investigated. With an increase in pressure, the pull-off force decreases, the pull-off deflection of the punch relative to the initial flat plate level increases, and the pull-off deflection with respect to the pressurized plate may increase or decrease. Some of the results are quite different from those for a membrane (where bending stiffness is neglected).

Effect of pressure on pull-off of flat cylindrical punch adhered to circular membrane

ABSTRACT Pressure can sometimes be utilized to increase or decrease the strength of adhesion. This phenomenon is analyzed for pull-off of a flat, circular cylindrical punch from a circular membrane. The pressure is applied, and then the punch is pulled upward. Both force control and displacement control are considered, and the pressure may be positive (acting upward) or negative (acting downward) on the membrane. The membrane is assumed to be linearly elastic, and Föppl-von Kármán theory is used to allow moderate displacements. The JKR theory of adhesion is adopted. A transversality (debonding) condition is derived, and then the effects of the work of adhesion, Poisson’s ratio, residual stress, and the relative radius of the punch to that of the membrane are investigated. With an increase in pressure, the pull-off force decreases, the pull-off deflection of the punch relative to the initial flat membrane level increases, and the total pull-off punch deflection from the pressurized membrane may increase or decrease.

The Effect of Backfilling Stiffness on the Lateral Response of Deeply Buried Pipelines: an Experimental Study

Subsea pipelines passing through the shallow area are physically protected against the environmental, accidental, and operational loads by trenching and backfilling. Depending on construction methodology, environmental loads, and seabed soil properties, the stiffness of backfilling material may become largely different from the native ground (softer than native ground in most of the cases). The different stiffness between the backfill and native ground affects the soil failure mechanisms and lateral soil resistance against large pipeline displacements that may happen due to ground movement, landslides, ice gouging, and drag embedment anchors. This important aspect is not considered by current design codes. In this paper, the effect of trench-backfill stiffness difference on lateral pipeline-backfill-trench interaction was investigated by performing centrifuge tests. The soil deformations and failure mechanisms were obtained by particle image velocimetry (PIV) analysis. Three experiments were conducted by using three different backfills including loose sand, slurry, and chunky clay that represent the purchased, natural in-fill, and pre-excavated materials, respectively. The study shows that the current design codes underestimate the lateral soil resistance for small to moderate pipe displacements inside the trench and overestimate it for large lateral displacement, where the pipeline is penetrating into the trench wall.

Evaluation of Various Geometrical Nonlinearities in the Response of Beams and Shells

This paper investigates the consistency and compatibility of various assumptions and strain measurements in a large displacements/geometric nonlinear analysis of beams and thin-walled structures. For this purpose, a refined beam model with enhanced three-dimensional accuracy is employed in a total Lagrangian scenario. This model is developed in the domain of the Carrera unified formulation, which allows expression of the nonlinear governing equations in terms of fundamental nuclei. These nuclei are independent of the theory approximation order; thus, low- to high-order theories of structures can be implemented with ease. Various numerical problems are addressed, and solutions are provided by using a classical finite element method and a Newton–Raphson linearization scheme. Given the intrinsic scalable nature of Carrera unified formulation, investigating the effects of the various nonlinear strain components is straightforward. It is demonstrated that the full Green–Lagrange strain tensor produces good approximation in case of large rotations, postbuckling, and nonlinear couplings. In contrast, approximations may be reasonable as moderate displacements and simpler problems (for example, slender beams under flexure) are considered.

The curve shape of a bent ruler—analytical, numerical and experimental studies

A ruler, where one of its ends is axially displaced a known distance, is studied using analytical, numerical and experimental methods. The exact nonlinear differential equation for the bent ruler is derived and solved numerically. This exact differential equation is then simplified to a fourth order differential equation using a small angle approximation. Using this equation together with appropriate boundary conditions, an approximate curve shape for the ruler is determined, which turned out to be a sine function with no dependence on material parameters or cross-sectional data. An expression for the length of the bent ruler is derived and using the constraint that this length is constant, an equation for determination of a relationship between the midpoint transverse displacement (deflection) and the known axial displacement is obtained. This equation is solved numerically. In addition a simple algebraic relationship between the midpoint deflection and the axial displacement is derived. A comparison between measured data for a bent ruler, the numerical solution of the exact differential equation, the numerical solution of the approximate equation and the algebraic expression is presented. It is found that the algebraic sine-shaped solution gives reasonable results for moderate axial displacements. The experimental investigations, the algebraical expressions and the numerical simulations can be useful in high school teaching and at undergraduate university level.

Using 3D kinematic models in subduction channels. The case of the Chañaral tectonic mélange, Coastal Cordillera, northern Chile

The Chañaral tectonic mélange (northern Chile) is a local unit within the late Paleozoic accretionary complex formed at the southwestern margin of Gondwana. The structural characteristics of the studied mélange were mostly developed during a first deformation phase (D1) and include a block-in-matrix fabric, lineations (L1) and foliations (S1), tight to intrafoliar folds, S-C and S-C-C′ composite planar fabrics, and a conspicuous spatial separation of domains with predominantly linear and linear-planar fabrics. Folding during a second stage (D2) modified the orientation of the previous fabrics and structures. The eastern boundary of the Chañaral mélange is N-S to NNW-SSE oriented, moderately dipping to the east. Its western boundary is not exposed. The plane showing the maximum structural asymmetry (the vorticity normal section) is ENE-WSW directed, and sub-vertical. Kinematic criteria consistently reveal top-to-the-WSW displacement. A kinematic model of triclinic transpression with inclined extrusion has been applied to evaluate the D1 structural features of the Chañaral mélange. The pitch of the simple shear direction on the deformation plane ranged from 60°N to 90°. The pitch of the estimated extrusion direction was of 30°-40°S. The coaxial component was clearly constrictional (logarithmic K value of 2 to 5). The vorticity number has not been constrained by the model, but its spatial variation can explain the domainal distribution of the fabrics in the mélange. The simulated particle paths show the predominance of material displacement parallel to the margin, with low to moderate down-dip displacements, which is in accordance with the low-pressure metamorphic assemblages found in the mélange. The convergence direction between the blocks separated by the mélange unit was N50°-60°E. Kinematic blocking of the Chañaral mélange, probably related to the accretion of an oceanic volcanic domain, allowed the D2 folding of the previous structures, a process that, at least initially, proceeded without a change in the convergence direction.

Modeling the non-linear elastic response of periodic lattice materials

Periodic elastic lattice materials are a class of cellular materials with unique properties that cannot be achieved with fully uniform solids. In this work we employ moderate-rotation theory, which is an expansion of linear elasticity that takes into account moderate displacements and rotations, to derive a fundamental multi-scale model that captures the non-linear response of infinitely periodic elastic lattice materials and allows the imposition of local constraints on lattice junctions. We begin by modeling the response of a single strut. Next, an infinitely periodic lattice structure is considered. The periodicity of the structure is accounted for through a unit cell (UC) and periodic lattice directions. A representative volume element (RVE) comprising one or more UCs is then defined. It is assumed that the RVE experiences the macroscopic displacement gradient. To determine the stress, we employ the principle of virtual work and homogenization methods. To illustrate the predictions of the model, we analyze two infinitely periodic lattice materials - a diamond lattice and a triangular lattice with rigid junctions. To validate our predictions, we compute the response for RVEs of several sizes until convergence is reached.

Nonlinear Theory of Sandwich Shells with a Transversely Soft Core Containing Delamination Zones and Edge Support Diaphragm

Sandwich shells with transversely soft cores and skin layers supported along the edges by a thin deformable diaphragm with a ruled middle surface are considered. Delamination zones are assumed on the contact surfaces between the core and skin layers. The refined geometrically nonlinear theory has been constructed for these type of structures at small deformations and moderate displacements, which can describe prebuckling deformation and reveal all possible buckling mode shapes of skin layers (antisymmetric, symmetric, mixed bending and mixed bending-shear, as well as arbitrary mode shapes comprising all listed modes) and of the reinforcing diaphragm. This theory is based on the introduction of interaction forces between the core and skin layers and between the core and reinforcing diaphragm at each point of the contact surfaces as unknowns. The previously proposed variant of the generalized Lagrangian variational principle has been utilized in the derivation of the governing equations, static boundary conditions for the shell and reinforcing diaphragm, and kinematic coupling conditions between the core and skin and between the core and support diaphragm.

Axial-transversal coupling in the nonlinear dynamics of a beam with an inclined roller

The nonlinear oscillations of an Euler–Bernoulli beam hinged at one end and having a roller support sliding on a inclined line on the other end are investigated for the first time (to the best of the authors’ knowledge). Free and forced dynamics are studied, and the backbone curve and frequency response curve are obtained by the multiple time scale method. These results allow us to detect analytically the main features of the nonlinear dynamics, in particular the hardening/softening dichotomy, and how it is influenced by the various beam parameters. Furthermore, the analytical results are compared with finite element simulations to check their reliability. The main goal of the paper is that of investigating the coupling between axial and transversal displacements, which appears yet in the linear regime (first order solution) due to the slope of the constraint. An element of novelty consists of showing the qualitative and quantitative changes that the coupling undergoes passing from small (linear regime) to moderate and large (nonlinear regime) displacements.

Dynamic linear response of a shock/turbulent-boundary-layer interaction using constrained perturbations

Comprehensive experimental and computational investigations have revealed possible mechanisms underlying low-frequency unsteadiness observed in spanwise homogeneous shock-wave/turbulent-boundary-layer interactions (STBLI). In the present work, we extend this understanding by examining the dynamic linear response of a moderately separated Mach 2.3 STBLI to small perturbations. The statistically stationary linear response is analysed to identify potential time-local and time-mean linear tendencies present in the unsteady base flow: these provide insight into the selective amplification properties of the flow at various points in the limit cycle, as well as asymmetry and restoring mechanisms in the dynamics of the separation bubble. The numerical technique uses the synchronized large-eddy simulation method, previously developed for free shear flows, significantly extended to include a linear constraint necessary for wall-bounded flows. The results demonstrate that the STBLI fosters a global absolute linear instability corresponding to a time-mean linear tendency for upstream shock motion. The absolute instability is maintained through constructive feedback of perturbations through the recirculation: it is self-sustaining and insensitive to external forcing. The dynamics are characterized for key frequency bands corresponding to high–mid-frequency Kelvin–Helmholtz shedding along the separated shear layer$(St_{L}\sim 0.5)$, low–mid-frequency oscillations of the separation bubble$(St_{L}\sim 0.1)$and low-frequency large-scale bubble breathing and shock motion$(St_{L}\sim 0.03)$, where the Strouhal number is based on the nominal length of the separation bubble,$L$:$St_{L}=fL/U_{\infty }$. A band-pass filtering decomposition isolates the dynamic flow features and linear responses associated with these mechanisms. For example, in the low-frequency band, extreme shock displacements are shown to correlate with time-local linear tendencies toward more moderate displacements, indicating a restoring mechanism in the linear dynamics. However, a disparity between the linearly stable shock position and the mean shock position leads to an observed asymmetry in the low-frequency shock motion cycle, in which upstream motion occurs more rapidly than downstream motion. This is explained through competing linear and nonlinear (mass depletion through shedding) mechanisms and discussed in the context of an oscillator model. The analysis successfully illustrates how time-local linear dynamics sustain several key unsteady broadband flow features in a causal manner.

Finite element modeling of cable galloping vibrations—Part I: Formulation of mechanical and aerodynamic co-rotational elements

Cables are widely used lightweight and efficient structural members affected by various nonlinearities. This paper is devoted to the development of a finite element approach to the study of the nonlinear dynamic behavior of cable structures under wind loading. Firstly, the formulation of a co-rotational beam-cable element is presented to account for cable geometrical nonlinearities within the context of an Updated Lagrangian approach. The Euler–Bernoulli kinematics is adopted in the co-rotated frame assuming small or moderate displacements and strains. Secondly, the formulation of an aerodynamic element, meant to be superimposed to the mechanical beam element, is developed to fully account for the nonlinearity of the aerodynamic forces. The wind interaction forces are computed within the framework of the quasi-steady theory. Appropriate procedures for applying the aforementioned elements in static and dynamic analyses are presented and applied to the study of the static configuration and the galloping vibrations of a suspended cable under steady and turbulent wind conditions. The results point out the importance of the first anti-symmetric in-plane mode on the galloping response of the iconic selected structure, and reveal the role of the mechanical and aerodynamic model by comparison with a different discretization based on cable finite elements. A mechanical model, such as the proposed co-rotational finite element formulation, which includes the description of torsional rotations is required to properly model the aerodynamic loads in dynamic galloping analyses under steady wind. The mechanical and aerodynamic assumptions, instead, play a less important role in the case of turbulent wind conditions, leading to an eminently buffeting response entailing a strong effect of the swing of the structure.

Numerical and Experimental Investigations on Preload Effects in Air Foil Journal Bearings

A detailed elastogasdynamic model of a preloaded three-pad air foil journal bearing is presented. Bump and top foil deflections are herein calculated with a nonlinear beamshell theory according to Reissner. The two-dimensional pressure distribution in each bearing pad is described by the Reynolds equation for compressible fluids. The assembly preload is calculated by simulating the assembly process of top foil, bump foil, and shaft. Most advantageously, there is no need for the definition of an initial radial clearance in the presented model. With this model, the influence of the assembly preload on the static bearing hysteresis as well as on the aerodynamic bearing performance is investigated. For the purpose of model validation, the predicted hysteresis curves are compared with measured curves. The numerically predicted and the measured hysteresis curves show a good agreement. The numerical predictions exhibit that the assembly preload increases the elastic foil structural stiffness (in particular for moderate shaft displacements) and the bearing damping. It is observed that the effect of the fluid film on the overall bearing stiffness depends on the assembly preload: For lightly preloaded bearings, the fluid film affects the overall bearing stiffness considerably, while for heavily preloaded bearings the effect is rather small for a wide range of reaction forces. Furthermore, it is shown that the assembly preload increases the friction torque significantly.

On limit cycle oscillations of typical aeroelastic section with different preset angles of incidence at low airspeeds

Helicopter blades and wind turbines are examples of aeroelastic systems that can reach high angles of attack and can vibrate due to the effects of the dynamic stall, thereby leading to fatigue problems or performance loss. Structural and aerodynamic nonlinearities influence the aforementioned behavior and their modeling is crucial for phenomena characterization. Such system modeling requires proper knowledge of the physical events during the stall, which can be better achieved by validating the model with experimental data. This work investigates the nonlinear dynamics of a NACA 0012 airfoil under the influence of structural and aerodynamic nonlinearities due to dynamic stall effects at high angles of attack. Experimental and numerical analyses are carried out. Moreover, different preset incidence angles for the typical aeroelastic section are also considered. The aeroelastic signals are used for estimating the Hopf bifurcation onset and to build the bifurcation diagrams. By using a typical section model with two degrees of freedom coupled to the Beddoes–Leishman aerodynamic model, numerical results have been able to capture with good precision experimental features. The onset of the Hopf bifurcations allows the determination of the flutter critical airspeed. Results for zero preset angle show that limit cycle oscillations from small to moderate displacements are mostly driven by the hardening nonlinearity. After reaching larger angles of incidence the dynamic stall nonlinearities supplant those from structural sources. For higher preset angles, the dynamic stall effects tend to increase the energy associated with pitching motion and to reduce amplitudes in plunge motion. Another effect related to the aerodynamic nonlinearities relies on the increase of the flutter velocity by around 10% for preset angles ranging from zero up to ten degrees. For higher preset angles an abrupt reduction in the flutter onset velocity is observed.