Discontinuous Loading Research Articles

Feasibility studies of discontinuous electro-regeneration processes in environmentally-friendly plating for chromate separation from a binary system

Project work was carried out for feasibility and possible design of semi-industrial electro-regeneration units for diluted waters typical for rinse water systems in electroplating industry. Firstly, one subject of interest was the removal of CrO 4 2− ions in the concentration range between 0.06 mol m −3 and 2.3 mol m −3. Ion exchange capacity of several resins under room temperature conditions was tested. A 2-compartment cell with diaphragm was found to be a suitable construction for electrochemical resin regeneration after previous discontinuous loading. The determination of resin conductivities and transport numbers was also part of the studies. Results showed sufficient high current efficiencies up to 70%. Nevertheless, low ion mobility in the resin may limit the removal rate for continuous electrodeionization. In this case, stepwise operation mode in loading and regenerating the resin bed is the only solution. Energy consumption can be lowered by using catholytes of previous electro-regeneration steps. Specific energy consumption until 0.3 kWh per treated cubic meter is to be expected.

Upper and lower bounds in limit analysis: Adaptive meshing strategies and discontinuous loading

AbstractUpper and lower bounds of the collapse load factor are here obtained as the optimum values of two discrete constrained optimization problems. The membership constraints for Von Mises and Mohr–Coulomb plasticity criteria are written as a set of quadratic constraints, which permits one to solve the optimization problem using specific algorithms for Second‐Order Conic Program (SOCP). From the stress field at the lower bound and the velocities at the upper bound, we construct a novel error estimate based on elemental and edge contributions to the bound gap. These contributions are employed in an adaptive remeshing strategy that is able to reproduce fan‐type mesh patterns around points with discontinuous surface loading. The solution of this type of problems is analysed in detail, and from this study some additional meshing strategies are also described. We particularise the resulting formulation and strategies to two‐dimensional problems in plane strain and we demonstrate the effectiveness of the method with a set of numerical examples extracted from the literature. Copyright © 2008 John Wiley & Sons, Ltd.

Macaulay's Method for a Timoshenko Beam

The Macaulay bracket notation is familiar to many engineers for the deflection analysis of a Euler–Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curvature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional deflection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam.

Load Discontinuity in Step-by-step Integration

In this article, both analytical and numerical studies show that a load discontinuity will lead to an amplitude distortion in performing the step-by-step integration. It is also found that the relative amplitude error caused by load discontinuity is closely related to, but not proportional to, the value of the load discontinuity. In general, this amplitude distortion can be reduced if a very small time step is used for a whole step-by-step integration procedure. Hence, a lot of computational efforts will be involved. In order to reduce these computational efforts a remedy is proposed herein. The remedy is proposed to perform a very small single time step immediately after the load discontinuity to reduce amplitude distortion. Since only a very small time step is performed in the step-by-step integration, many computational efforts can be saved. The feasibility of this remedy is analytically proved and numerically confirmed.

Computing singular perturbations for linear elliptic shells

This paper is devoted to the theoretical and numerical study of singular perturbation problems for elliptic inhibited shells. We present a reduction of the classical membrane equations to a partial differential equation with respect to the bending displacement, which is well adapted to the study of singularities of the limit problem. For a discontinuous loading or when the boundary of the loading domain presents corners, we put in a prominent position the existence of two kinds of singularities. One of them is not classical; it reduces to a logarithmic point singularity at the corner of the loading domain. To finish numerical simulations are performed with a finite element software coupled with an anisotropic adaptive mesh generator. They enable to visualize precisely the singularities predicted by the theory with only a very small number of elements.

A technique for overcoming load discontinuity in using Newmark method

Although a closed form solution can be obtained for the shock response to an impulse by using the Duhamel integral method for a single degree-of-freedom system, it is analytically verified herein that an extra amplitude distortion will be introduced at the end of the impulse by using the Newmark method. In fact, the relative amplitude error in the displacement response due to the discontinuity at the end of the impulse can be reliably estimated by the ratio of the area of extra impulse over the area of input impulse, where the area of extra impulse is equal to a half of the product of the time step and load discontinuity. It seems that the momentum equation of motion can be used to easily overcome the discontinuity problem at the end of an impulse. This is because the external force is replaced by its corresponding external momentum, which is obtained from the time integration of external force, and thus the discontinuity problem automatically disappears by using the momentum equation of motion. Numerical studies reveal that this technique is applicable to nonlinear systems. Meanwhile, it is also numerically illustrated that the discontinuity problem can be overcome by performing a very small time step immediately upon termination of the applied impulse.

Approach for Overcoming Numerical Inaccuracy Caused by Load Discontinuity

It was found that the discontinuity at the end of an impulse will lead to numerical inaccuracy as this discontinuity will result in an extra impulse and thus an extra displacement in the time history analysis. In addition, this extra impulse is proportional to the discontinuity value at the end of the impulse and the size of integration time step. To overcome this difficulty, an effective approach is proposed to reduce the extra impulse and hence the extra displacement. In fact, the novel approach proposed in this paper is to perform a single small time step immediately upon the termination of applied impulse, whereas other time steps can be conducted by using the step size determined from accuracy consideration in period. The feasibility of this approach is analytically explored. Further, analytical results are confirmed by numerical examples. Numerical studies also show that this approach can be applied to other step-by-step integration methods. It seems that to slightly complicate the programming of dynamic analysis codes is the only disadvantage of this approach.

Phenomenological modelling of crack generation in brittle crustal rocks using the particle simulation method

The particle simulation method is used in this paper to explicitly simulate spontaneous crack generation in brittle crustal rocks. Although the method was previously used to simulate crack generation in small-scale laboratory specimens and mining sites, it has seldom been used to simulate spontaneous crack generation in large-scale geological systems. If a geological system can be treated as a quasi-static one, the mechanical response of the system should be independent of the loading rate. Based on this understanding, we have compared the commonly-used continuous loading procedure with a newly-proposed discontinuous loading procedure, which is independent of the loading rate within the elastic range of a particle system. The use of the discontinuous loading procedure enables the macroscopic elastic modulus of a two-dimensional particle system to be directly evaluated from the particle stiffness. However, the particle-size sensitivity analysis of at least two different models needs to be conducted for dealing with the particle size-dependent issue when the particle simulation method is applied to solve crack generation problems in geological systems. Through the phenomenological modeling of spontaneous crack generation problems, it has been demonstrated that the particle simulation method is useful for simulating spontaneous crack generation phenomena at geological length scales.

Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings

An efficient scheme, called quasi-linearization finite differences, is developed for large-deflection analysis of prismatic and non-prismatic slender cantilever beams subjected to various types of continuous and discontinuous external variable distributed and concentrated loads in horizontal and vertical global directions. Simultaneous equations of highly nonlinear and linear terms are obtained when casting the derived exact highly nonlinear governing differential equation using central finite differences on the nodes along the beam. A quasi-linearization scheme is used to solve these equations based on successive corrections of the nonlinear terms in the simultaneous equations. The nonlinear terms in the simultaneous equations are assumed constant during each correction (iteration). Several representative numerical examples of prismatic and non-prismatic slender cantilever beams with different loading conditions are analyzed to illustrate the merits of the adopted numerical scheme as well as its validity, accuracy and efficiency. The results of the present scheme are checked using large-displacement finite element analysis by the MSC/NASTRAN program. A comparison between the present secheme, MSC/NASTRAN and available results from the literature reveals excellent agreement. The advantage of the new scheme is that the load can be applied in one step with few iterations (3–6 iterations).

A New Quasi-Linearization Finite Difference Scheme for Large Deflection Analysis of Prismatic and Non-Prismatic Inextensible Slender Beams

A new scheme called quasi-linearization finite differences is developed for large deflection analysis of prismatic and non-prismatic inextensible slender beams with different boundary conditions subjected to various types of continuous and discontinuous external loads in horizontal and vertical global directions. Simultaneous equations of highly nonlinear and linear terms are obtained when casting the derived exact highly nonlinear governing differential equation using central finite differences on the nodes along the beam. A quasi-linearization scheme is used to solve these equations based on successive corrections of the nonlinear terms in the simultaneous equations. The nonlinear terms in the simultaneous equations are assumed constant during each correction (iteration). Several representative numerical examples of prismatic and non-prismatic slender beams with different loading and boundary conditions are analyzed to illustrate the merits of the new adopted numerical scheme as well as its validity, accuracy and efficiency. The results of the present scheme are checked using large displacement finite element analysis of MSC/NASTRAN program. A comparison between present scheme and MSC/NASTRAN results reveals excellent agreements. The advantage of the new scheme is that the load can be applied in one step with few iterations (3 to 6 iterations).

On the use of space-time finite elements in the solution of elasto-dynamic fracture problems

The use of a discontinuous Galerkin (DG) formulation for the solution of dynamic fracture problems in linear elastic media with and without cohesive zones is explored. The results are compared with closed-form as well as numerical solutions available from the literature. The effectiveness of the space-time finite element method in the study of dynamic fracture problems is demonstrated, especially in those cases in which dynamic fracture occurs along with time discontinuous loading.

Performance of a fungal biofilter treating gas-phase solvent mixtures during intermittent loading

Biological treatment processes used to remove and degrade volatile organic compounds (VOCs) from contaminated gases emitted by industrial operations or waste treatment processes are almost always subjected to transient loading conditions because of the inherently unsteady-state nature of contaminant generating processes. In the study presented here, a laboratory-scale biofilter populated by a mixed culture of fungi was used to study the transient response to various periods of no contaminant loading in a system treating a model waste gas stream containing a mixture of commonly used solvents. The biofilter, packed with cubed polyurethane foam media and operated with an empty bed residence time of 15 s, was supplied with a four-component mixture of n-butyl acetate, methyl ethyl ketone, methyl propyl ketone, and toluene at target influent concentrations of 124, 50.5, 174, and 44.6 mg/m 3, respectively. This corresponds to a total VOC loading rate of 94.3 g/(m 3 h). Biofilter performance was evaluated over a 94-day period for three loading conditions intended to simulate processes generating contaminated gases only during daytime operation, daytime operation with weekend shutdown periods, and with long term (9-day) shutdown. Results indicate that fungal biofilters can be an effective alternative to conventional abatement technologies for treating solvent contaminated off-gases even under discontinuous loading conditions.

Yield Point Phenomena During Nanoidentation

The yield point phenomenon observed in nanoindentation of metallic materials has been ascribed to the nucleation of dislocations or multiplication and the film fracture on the surface of the metals. The current study has been undertaken to examine the properties of passivating films on Type 316 austenitic stainless steel. A complementary set of ex situ and in situ experiments showed the presence of deformation prior to a discontinuity in loading with both load-depth sensing techniques and imaging the surface topography. The results of nanoindentation experiments showed that the yield point is dependent on the presence of surface passive film. If the materials, which exhibit permanent deformation prior to a discontinuity in loading, and after the surface films are removed by nanowear and depassivated, Type 316 stainless steel shows no sign of yield point effect; the occurrence of excursions is shown in this material to be dependent on surface passive film. It is more likely that the phenomenon is indeed controlled by film fracture and not the rapid generation of dislocation.

DQEM for free vibration analysis of Timoshenko beams on elastic foundations

A differential quadrature element method (DQEM) based on first order shear deformation theory is developed for free vibration analysis of non-uniform beams on elastic foundations. By decomposing the system into a series of sub-domains or elements, any discontinuity in loading, geometry, material properties, and even elastic foundations can be considered conveniently. Using this method, the vibration analysis of general beam-like structures is to be studied. The governing equations of each element, natural compatibility conditions at the interface of two adjacent elements and the external boundary conditions are developed in a systematic manner, using Hamilton's principle. The present DQEM is to be implemented to Timoshenko beams resting on partially supported elastic foundations with various types of boundary conditions under the action of axial loading. The general versality, accuracy, and efficiency of the presented DQEM are demonstrated having solved different examples and compared to the exact or other numerical procedure solutions.

Theory for Midplane Asymmetric Sandwich Cylindrical Shells

In shells, bending stresses in addition to the membrane stresses exist near each structural discontinuity (such as at edge supports, rings, stringers, holes, etc.) and each load discontinuity. This region is the “bending boundary layer”. In sandwich shells, mid-plane asymmetry can be employed to minimize these bending stresses. Governing equations for mid-plane asymmetric sandwich cylindrical shell subjected to axially symmetric loads are derived herein and solved. The solution involves a mid-plane asymmetry factor ϕ, defined herein such that for any asymmetric sandwich construction, -1 < ϕ < 1. Therefore, perturbation solutions are easily obtained using the solutions for the midplane symmetric cylindrical shell.

Indentation induced film fracture in hard film – soft substrate systems

Nanoindentations were made into oxide films on aluminum and titanium substrates for two cases; one where the metal was a bulk (effectively single crystal) material and the other where the metal was a 1 μm thick film grown on a silica or silicon substrate. In both cases indentation was used to produce discontinuous loading curves, which indicate film fracture after plastic deformation of the metal. The oxides on bulk metals fractures occurred at reproducible loads, and the tensile stresses in the films at fracture were approximately 10 and 15 GPa for the aluminum and titanium oxides, respectively. Fine grained films only exhibited discontinuous loading under cyclic indentations, and demonstrated decreased load at catastrophic film fracture as the amplitude of cycling was decreased. A hypothesis regarding film fracture in these film systems is presented based on plastic deformation in the substrate and the superposition of the load carried by the elastic stretching of the film and the plastic deformation of the substrate.

Further Developments in the MLPG Method for Beam Problems

An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions as in the conventional MLPG method. Linear test functions are also chosen, leading to a variation of the MLPG method that is computationally efficient compared to the conventional implementation. The MLPG method is evaluated by applying the formulation to a variety of patch tests, thin beam problems, and problems with load discontinuities. The formulation successfully reproduces exact solutions to machine accuracy when higher order power and spline functions are chosen as test functions or when the linear test function is used, and when constructing the trial functions, the order of the basis function is properly balanced by the order of the weight function. For mixed boundary value problems, deflections, slopes, moments, and shear forces are calculated to the same accuracy by the MLPG method without the use of elaborate post-processing techniques. Problems with load discontinuities require special care - when a reasonable number of nodes are used, the method yields very accurate results.

Nano-mechano-electrochemistry of passive metal surfaces

In situ nanoindentation tests were performed to evaluate the mechanical properties of the iron (100) and (110) surfaces passivated potentiostatically for 1 h in pH 8.4 borate solution after or without immersion in 5×10 −2 M K 2Cr 2O 7 solution for 24 h. It was found from the measured load–depth curves at a maximum load of 400 μN that the hardness (3.2–3.3 GPa) of the passivated iron (110) surfaces was larger by 10% than that (2.9–3.1 GPa) of the passivated iron (100) surfaces. Moreover, the hardness of the passivated iron (100) and (110) surfaces increased slightly with increasing formation potential of passive film, which was ascribed to the presence of passive film. The dichromate treatment increased the hardness of the iron (100) and (110) surfaces to some extent. The loading discontinuity appeared frequently at a load of about 90 μN in the load–depth curves for the passivated iron (110) surfaces as compared with the passivated iron (100) surfaces. The loading discontinuity in the load–depth curves for the iron (110) surface, however, disappeared completely due to the dichromate treatment.

Analysis of Compression Stress of a Filler and Bending of a Sandwich Panel under Multipoint Loading

The distribution of transverse stresses in the midlayer of a composite sandwich panel under multipoint loading is investigated. The stresses averaged across the thickness of a soft filler are estimated using a discrete model. Finite expressions for the compression of the filler along the length of the panel are derived by means of superposition of the local effects from the bending of face layers under an infinite system of transverse point forces constant across the panel thickness. The effects of compression and transverse extension of the filler, in the case of a high distribution frequency of these forces, i.e., when the distance between the forces is comparable to the panel thickness, are revealed. Compression of the panel by two systems of forces applied symmetrically or nonsymmetrically to the upper and lower faces is considered. The bending characteristics in the cases of loading with point forces and piecewise distributed loads are compared. The formulas obtained are used to determine the length of a small region on the panel surface for which the local effects from the distributed pressure and the point force are equivalent. The corresponding estimates are obtained in a closed form. The analysis, carried out with varied parameters of the structure, allows us to elucidate the peculiarities of the effect of discontinuous loads on the design characteristics in the local zones, using finite expressions derived by the operational method.

Local strain on a leachate collection pipe

Local strain measurements opposite gravel contacts and around a single isolated perforation are reported for a high-density-polyethylene pipe (320 mm outside diameter, 32 mm thick) typical of that commonly used as part of the leachate collection system in municipal solid waste landfills. Emphasis is given to examining the localized effect of coarse gravel contacts on pipe strain and the strain distribution around the perforation. The laboratory testing featured a cylindrical volume of soil with the pipe located in the middle with radially compressive stresses applied along the outer perimeter of the backfill. Two different backfill materials were used. When tested with medium sand backfill the pipe response was essentially uniform, indicative of the near-continuous support and loading provided by the sand. Large variations in local pipe strains were measured with coarse gravel backfill, such as that used in landfill leachate collection systems. Local bending induced by the discontinuous support and loading from the coarse gravel resulted in variations in circumferential and axial strains of over 40%. The local bending effects were not sufficiently large to produce circumferential tension in the pipe tested. A multiplication factor of 1.5 is suggested to account for increases in compressive strain from the coarse gravel. Measurements of surface strain around an isolated 32 mm diameter perforation revealed that a complex three-dimensional response is induced by the presence of the hole. Maximum strains near the hole were found to be 2.7 times larger than those distant from the perforation.Key words: leachate collection pipes, HDPE pipes, leachate collection systems, landfill design.