Short-Term Structural Instability under Random Dynamic Loads

In the technical note a single-degree-of-freedom system with stationary random temporal variations of its stiffness is considered. The problem of short-term instability is studied. The analysis is based on a parabolic approximation of the randomly varying apparent stiffness within the negative domain achieved by invoking a Slepian process model     

Transient Response of Supported Beams to Moving Forces with Sinusoidal Time Variation

Moving forces are a common loading pattern for flexible beams, found in many applications in both civil and mechanical engineering. These forces give rise to a transient response, the nature of which depends on the time variation of the amplitude of the force and its position along the beam. In addition to the possibility of numerical evaluation, closed form solutions of the beam response are beneficial for their simplicity of use, and because they allow an understanding of the system behavior. On the other hand, these prove to be rather complicated in most cases, and only a limited number of cases are available in the literature. This paper studies the simple but common case of a supported beam loaded by a force with sinusoidal time variation moving at a constant speed. Simple equations are presented for the approximated responses at and away from resonance, and their accuracy is discussed. Transient frequency response functions are also shown. Finally, as an example, the results are applied to an evaluation of the response of a beam footbridge to the action of a walker, and compared to code specifications.     

Planar Bending of Sandwich Beams with Transverse Loads off the Centroidal Axis

Conventional analysis methods for beams do not distinguish between transverse loads that are applied at the beam centroidal axis and those acting either above or below the centroidal axis. In contrast, this paper formulates a sandwich beam finite element solution which models the effect of load height relative to the centroidal axis. Towards this goal, the governing equilibrium equations and associated boundary conditions are derived based on a Timoshenko beam formulation for the core material. Special shape functions satisfying the homogeneous form of the equilibrium equations are derived and subsequently used to formulate exact stiffness matrices. By omitting the stiffness terms related to the faces, the formulation for a homogeneous Timoshenko beam can be recovered. Also, the Euler–Bernouilli counterpart of the formulation is recovered as a limiting case of the current Timoshenko beam formulation. Effects of load height relative to the centroid are observed to have similarities with those induced by axial forces in beam-columns. For a simply supported beam, downward acting loads located below the centroidal axis are found to induce a stiffening effect while those acting above the centroidal axis are found to induce a softening effect, resulting in higher transverse displacements.     

Vibro-Impact Behavior of Two Orthogonal Beams

Structural interaction between two beam-like structures is a situation that occurs in piping systems, among other applications relevant to the nuclear, petroleum, biomedical, and automotive industries, for instance. This paper analytically investigates the repetitive impact dynamics of two orthogonal pinned–pinned beams subjected to base excitation at specified frequency and acceleration. The orthogonal beam configuration restricts the contact to a single point, and the contact interface is modeled by a spring. Although many approaches have been developed for multibody dynamics, the constraint and modal mapping method is efficiently applied herein to obtain the forced response through modal analysis. The vibration is described in a piecewise fashion as switching between the linear in-contact and not-in-contact states, and compatibility conditions are applied at their junctions. The development of the conjoined mode shapes and their orthogonality is derived in detail. The contact impulse is used to describe the structures’ complex interacting behavior through repetitive impact frequency-response functions. In order to determine major response factors, parameter studies are performed on contact stiffness, relative beam stiffness, contact location, modal damping, and stand-off gap.     

Multicrack Detection on Semirigidly Connected Beams Utilizing Dynamic Data

The problem of crack detection has been studied by many researchers, and many methods of approaching the problem have been developed. To quantify the crack extent, most methods follow the model updating approach. This approach treats the crack location and extent as model parameters, which are then identified by minimizing the discrepancy between the modeled and the measured dynamic responses. Most methods following this approach focus on the detection of a single crack or multicracks in situations in which the number of cracks is known. The main objective of this paper is to address the crack detection problem in a general situation in which the number of cracks is not known in advance. The crack detection methodology proposed in this paper consists of two phases. In the first phase, different classes of models are employed to model the beam with different numbers of cracks, and the Bayesian model class selection method is then employed to identify the most plausible class of models based on the set of measured dynamic data in order to identify the number of cracks on the beam. In the second phase, the posterior (updated) probability density function of the crack locations and the corresponding extents is calculated using the Bayesian statistical framework. As a result, the uncertainties that may have been introduced by measurement noise and modeling error can be explicitly dealt with. The methodology proposed herein has been verified by and demonstrated through a comprehensive series of numerical case studies, in which noisy data were generated by a Bernoulli–Euler beam with semirigid connections. The results of these studies show that the proposed methodology can correctly identify the number of cracks even when the crack extent is small. The effects of measurement noise, modeling error, and the complexity of the class of identification model on the crack detection results have also been studied and are discussed in this paper.     

Solution Method for Beams on Nonuniform Elastic Foundations

This technical note presents an analytical method, accompanied by a numerical scheme, to evaluate the response of beams on nonuniform elastic foundations, namely, when the foundation modulus is k = k(x). The method employs a Green’s function formulation, which results in a system of nonsingular integral equations for the distributed reaction q(x). These equations can be discretized in a straightforward manner to yield a system of linear algebraic equations that can be solved by elementary numerical techniques.     

Nonclassical Modes of Unrestrained Shear Beams

This paper considers the effect that a rotational motion has on the normal modes of a shear beam that is free to rotate, either because it is free in space or it is pivoted at one end. It is shown that the classical solutions for these two cases violate the principle of conservation of angular momentum, and that this is true even when the rotational inertia of the beam vanishes or is neglected.     

Interface Behavior in Fiber-Reinforced Polymer-Strengthened Beams Subjected to Transverse Loads: Maximum Transferred Force

Experimental results of externally bonded strengthened beams, available to date in the literature, have shown the existence of some types of brittle failures that involve the premature laminate debonding before reaching the design load. Since the design procedure to obtain the laminate area to strengthen a reinforced concrete element should avoid these premature failures, there is a need to understand the mechanism of the debonding process in order to prevent it. This paper studies the evolution of the debonding process through the propagation of an interfacial crack by using nonlinear fracture mechanics assuming a bilinear constitutive law for the interface. The transference of stresses between the laminate and the concrete through the interface is first analyzed in a simple case of a pure shear specimen. Later on, this formulation is extended to a general case of a beam subjected to transverse loads. As shown in the analysis performed, the transferred force between both adherents should be limited to a maximum value to prevent laminate debonding. This theoretical maximum transferred force has been obtained for each case studied and will be the basis for the development of a future design proposal.     

Behavior of FRP Strengthened Reinforced Concrete Beams under Torsion

The use of fiber reinforced plastics (FRPs) for flexural and shear strengthening of reinforced concrete beams has been scrutinized to a considerable depth by researchers worldwide. The area of torsional strengthening however has not been as popular. This paper presents the results of an experimental investigation together with a numerical study on reinforced concrete beams subjected to torsion that are strengthened with FRP wraps in a variety of configurations. In the experimental study, the increase in the ultimate torque for different strengthening configurations, failure mechanisms, crack patterns, and ductility levels are monitored and presented. Experimental results show that FRP wraps can increase the ultimate torque of fully wrapped beams considerably in addition to enhancing the ductility. The experimental results upgrade the weak archival data on torsional strengthening by application of FRP. The numerical section reports on analyses performed by the ANSYS finite element program. Predictions are compared with experimental findings and are in reasonable agreement.     

Analysis of Snap-Back Instability due to End-Plate Debonding in Strengthened Beams

The problem of end-plate debonding of the external reinforcement in strengthened concrete beams is analyzed in this paper. As experimentally observed, this mode of failure is highly brittle and poses severe limitations to the efficacy of the strengthening technique. A numerical analysis of the full-range behavior of strengthened beams in bending is herein proposed to study the stages of nucleation and propagation of interfacial cracks between the external reinforcement and the concrete substrate. This is achieved by modeling the nonlinear interface behavior according to a cohesive law accounting for Mode Mixity. The numerically obtained load versus midspan deflection curves for three- or four-point bending beams show that the process of end-plate debonding is the result of a snap-back instability, which is fully interpreted in the framework of the Catastrophe Theory. To capture the softening branch with positive slope, the interface crack-length control scheme is proposed in the numerical simulations. The results of a wide parametric study exploring the effect of the relative reinforcement length, the mechanical percentage of fiber-reinforced polymer sheets, the beam slenderness, and the ratio between Mode II and Mode I fracture energies are collected in useful diagrams. Finally, an experimental assessment of the proposed model completes the paper.     

Numerical Evaluation of Uniform Beam Modes

The equation for calculating the normal modes of a uniform beam under transverse free vibration involves the hyperbolic sine and cosine functions. These functions are exponential growing without bound. Tables for the natural frequencies and the corresponding normal modes are available for the numerical evaluation up to the 16th mode. For modes higher than the 16th, the accuracy of the numerical evaluation will be lost due to the round-off errors in the floating-point math imposed by the digital computers. Also, it is found that the functions of beam modes commonly presented in the structural dynamics books are not suitable for numerical evaluation. In this paper, these functions are rearranged and expressed in a different form. With these new equations, one can calculate the normal modes accurately up to at least the 100th mode. Mike’s Arbitrary Precision Math, an arbitrary precision math library, is used in the paper to verify the accuracy.     

Numerical Efficiency in Monte Carlo Simulations—Case Study of a River Thermodynamic Model

Trade-offs between precision of numerical solutions to deterministic models of the environment, and the number of model realizations achievable within a framework of Monte Carlo simulation, are investigated and discussed. A case study of a model of river thermodynamics is employed. It is shown that the tractability of Monte Carlo simulation relies on adaptation of the numerical solution time-step, giving results with a guaranteed error in the time domain as well as near-optimum speed of calibration under any chosen accuracy criteria. Time-step control is implemented using two adaptive Runge–Kutta methods: a second order scheme with first order error estimator, and an embedded fourth-fifth order scheme. In the case study, where the effects of sparse and imprecise data dominate the overall modeling error, both the schemes appear adequate. However, the higher order scheme is concluded to be generally more reliable and efficient, and has wide potential to improve the value of applying the Monte Carlo method to environmental simulation. The problem of reconciling spatial error with the specified temporal error is discussed.     

Techniques for Numerical Simulations of Concrete Slabs for Demolishing by Blasting

The subject of this article is the numerical simulation of concrete under explosive loading using a meshbased and a meshfree discretization technique. The presented techniques are verified by experimental data. Experimental evidence suggests that the complete stress–strain history relation must be considered as a basis for constitutive modeling if concrete is subjected to high loading rates. These dynamic phenomena cause a retardation of damage activation which must be taken into account when constitutive modeling is pursued on mesolevel instead of microlevel. By including a dynamic relaxation formulation within the framework of a general three-dimensional coupled continuum damage-plasticity law, it is shown that the solution of the wave propagation problem in materials with strain-softening becomes independent of mesh size. As the simulation of concrete under contact detonation causes severe numerical problems because of the large deformations, special numerical spatial discretization techniques have to be used. In this article we compare the results of a concrete slab under contact detonation using the finite element method code LS-DYNA with an arbitrary Lagrangian Eulerian coupling and the results obtained by a MLSPH code developed at our institute with experimental data. The same constitutive model for concrete and the same equation of state for the explosive is implemented in the two codes. The results of the different numerical simulations and the experimental data agree with each other well.     

Dynamic Load Simulator: Actuation Strategies and Applications

The development of a multiple-actuator dynamic load simulator (DLS), for the simulation of correlated dynamic loads on small-scale structural components and substructures, or on bench-scale system assemblage is presented in this paper. Conceptually, the DLS employs actuators to simulate a desired dynamic loading environment due to wind, waves, or earthquakes, which in special cases may serve as a replacement for conventional facilities such as wind tunnels, wave tanks and shaking tables. The actuation strategy of the DLS is based on force-control rather than the customary motion control (displacement/velocity) scheme. The load simulator is ideal for structural components and for systems that can be idealized as lumped mass systems. An actuation strategy for the DLS based on an innovative scheme that utilizes the coupled control system is developed. For implementation of this scheme, the nonlinear control system toolbox in MATLAB is used. In this scheme, the tuning of control parameters in the time domain is carried out by solving a constrained optimization problem. A suite of loading protocols that includes sinusoidal, two-point correlated fluctuations in wind loading, earthquake induced loading and loads characterized by strong non-Gaussian features is simulated by employing the control scheme introduced here. The load simulation examples presented here demonstrate that the loading time histories generated by utilizing the DLS matched the target values with high fidelity.     

Lateral-Torsional Buckling of Stepped Beams with Continuous Bracing

Continuous span multibeam steel bridges are common along the state and interstate highways. The top flange of the beams is typically braced against lateral movement by the deck slab, and in many bridges the cross section is stepped at discrete points along the span. Design equations for lateral–torsional buckling (LTB) resistance in the American Association of State Highway and Transportation Officials “Load and resistance factor design bridge design specifications” are for prismatic beams and ignore the lateral restraint provided by the bridge deck. A new design equation is proposed that can be applied to I-shaped stepped beams with continuous top flange lateral bracing. By including the effects of the change in cross section size and the continuous top flange bracing, the calculated LTB resistance is significantly increased. Critical bending moment values from the proposed equation are compared to values from finite element method buckling analyses. The new equation is sufficiently accurate for use in design and in the evaluation of existing bridges.     

Upper-Bound Solutions for Rigid-Plastic Beams and Plates of Large Deflections by Variation Principles

This paper demonstrates deriving upper-bound solutions of geometrically nonlinear problems for beams and plates from rigid perfectly plastic material by the principles of virtual work in general form and stationary of total energy. Presented noncomplicated examples justify that the first is more appropriate when a kinematically admissible displacement field is defined by several generalized displacements. The second can serve as effective means for comparison in accuracy solutions corresponding to different displacement fields playing the same role as the upper-bound theorem in the limit analysis. Procedures of the latter for obtaining upper-bound solutions mainly remain valid. Solutions for a beam and rectangular plate subjected to uniformly distributed load illustrate importance of taking into account transformation forms of displacements in loading process.     

Identification of Concentrated Damages in Euler-Bernoulli Beams under Static Loads

An identification procedure of concentrated damages in Euler-Bernoulli beams under static loads is presented in this work. The direct analysis problem is solved first by modeling concentrated damages as Dirac’s delta distributions in the flexural stiffness. Closed-form solutions for both statically determinate and indeterminate beams are presented in terms of damage intensities and positions. On this basis, for the inverse damage identification problem, a nonquadratic optimization procedure is proposed. The presented procedure relies on the minimization of an error function measuring the error between the analytical model response and experimental data. The procedure allows to recognize “a posteriori” some sufficient conditions for the uniqueness of the solution of the damage identification problem. The influence of the instrumental noise on the identified parameters is also explored.     

Dynamic Instability of Nanorods/Nanotubes Subjected to an End Follower Force

This paper presents an investigation on the dynamic instability of cantilevered nanorods/nanotubes subjected to an end follower force. Eringen’s nonlocal elasticity theory is employed to allow for the small length scale effect in the considered dynamic instability problem. The general solution for the governing differential equation is obtained and the dynamic instability characteristic equation is derived by applying the boundary conditions. Exact critical load factors are obtained. These nonlocal solutions are compared with the classical local solutions to assess the sensitivity of the small length scale effect on the critical load factors and flutter mode shapes. It is found that the small length scale effect decreases the critical load and the corresponding frequency parameters as well as reduces the severity of the double-curvature flutter mode shape.     

Eulerian-Lagrangian Numerical Scheme for Simulating Advection, Dispersion, and Transient Storage in Streams and a Comparison of Numerical Methods

Past applications of one-dimensional advection, dispersion, and transient storage zone models have almost exclusively relied on a central differencing, Eulerian numerical approximation to the nonconservative form of the fundamental equation. However, there are scenarios where this approach generates unacceptable error. A new numerical scheme for this type of modeling is presented here that is based on tracking Lagrangian control volumes across a fixed (Eulerian) grid. Numerical tests are used to provide a direct comparison of the new scheme versus nonconservative Eulerian numerical methods, in terms of both accuracy and mass conservation. Key characteristics of systems for which the Lagrangian scheme performs better than the Eulerian scheme include: nonuniform flow fields, steep gradient plume fronts, and pulse and steady point source loadings in advection-dominated systems. A new analytical derivation is presented that provides insight into the loss of mass conservation in the nonconservative Eulerian scheme. This derivation shows that loss of mass conservation in the vicinity of spatial flow changes is directly proportional to the lateral inflow rate and the change in stream concentration due to the inflow. While the nonconservative Eulerian scheme has clearly worked well for past published applications, it is important for users to be aware of the scheme’s limitations.