Time-independent Loading Research Articles

Transverse vibrations of cantilever beams: Analytical solutions with general steady-state forcing

Novel theoretical approach for characterization of transverse vibrations of cantilever beams under continuous spatially distributed load is proposed. Four beam theories (Euler–Bernoulli, Euler–Bernoulli with dissipation, shear and shear with dissipation) are considered. Explicit analytical solutions are obtained for three of them and simplification algorithm for the fourth is proposed. Despite several limitations implied (time-independent loading, homogeneous boundary conditions) the solution developed is general enough and is applicable for a wide class of space-dependent loads. The approach is verified with available experimental data and shown to be applicable in a wide scale range.

Demand response-integrated economic dispatch incorporating renewable energy sources using ameliorated dragonfly algorithm

This paper presents an integrated demand response (DR) and dynamic economic dispatch (DED) model consisting of renewable energy sources (RESs) and shunt FACTS (flexible AC transmission systems) devices with focus on reducing thermal energy consumption. RESs are environment friendly and can be located close to the load centers, which helps in reducing T&D losses as well as harmful emissions. DR helps in reducing peak power consumption and generation cost of thermal power plants. Shunt FACTS device, static synchronous compensator (STATCOM), helps in improving voltage profile of the network. Analyses are carried out using the proposed ameliorated dragonfly algorithm (ADFA). The efficiency of developed ADFA is proved by comparing the obtained results with many other methods. Static economic dispatch is carried out on 6-, 17- and 40-thermal generator test systems using ADFA, and the results are compared with other existing methods available in the literature. Further, DED is carried out on 6- and 17-thermal generator test systems. Later, a DED is performed on a real-time 17-thermal generator (South Indian 86-bus system) system for four seasons. Solar irradiance and wind speed are modeled using beta distribution and Weibull distribution functions by collecting the historical data of the test system. The developed demand response program is implemented on a real-time 17 thermal unit and 86-bus test system located in Tamil Nadu, southern part of India. The developed DR model alters time-independent loads so that burden on generating stations during the peak load is minimized. The effect of DR, RES and STATCOM in the test system is studied individually. Further, a comparative analysis considering DR, RES and STATCOM together is presented. The results obtained are promising, and it can be interpreted that by using the developed model there will be improvement in the penetration levels of RES which in turn reduces the demand for thermal energy.

Free vibration analysis of functionally graded double-tapered beam rotating in thermal environment considering geometric nonlinearity, shear deformability, and Coriolis effect

The present work investigates the free vibration behavior of double-tapered functionally graded beams rotating in thermal environment, using an improved mathematical model. The functional gradation for ceramic–metal compositions, following power-law, is considered to be symmetric with respect to the mid-plane, leading to metal-rich core and ceramic-rich outer surfaces of the beam. The temperature dependence of the material properties are considered using Touloukian model. The nonlinearity in strain–displacement relationships for both the axial and transverse shear strains are considered. Firstly, the governing equations for deformed beam configuration under time-independent centrifugal loading are obtained using minimum total potential energy principle, and the solution is obtained following Ritz method. Then the free vibration problem of the centrifugally deformed beam is formulated employing Lagrange’s principle and considering tangent stiffness of the deformed beam configuration. Coriolis effect is considered in the mathematical model, and the governing equations are transformed to the state-space for obtaining an eigenvalue problem. The results for the first two modes of both chord-wise and flap-wise vibrations are presented in nondimensional plane to show the effects of taperness parameter, root-offset parameter, volume fraction exponent, operating temperature, and functionally graded material composition. The results in comparative form are presented for both temperature-dependent and temperature-independent material properties.

Elastic and Viscoelastic Poisson’s Ratios: The Theoretical Mechanics Perspective

A recent review publication presented an extensive and comprehensive assessment of the phenomenological relations of Poisson’s ratios (PRs) to the behavior and responses of contemporary materials under specific loading conditions. The present review and analysis paper is intended as a theoretical mechanics complement covering mathematical and physical modeling of a single original elastic and of six time and process (i.e. path and stress) dependent viscoelastic PR definitions as well as a seventh special path independent one. The implications and consequences of such models on material characterization are analyzed and summarized. Indeed, PRs based on experimentally obtained 2-D strains under distinct creep and/or relaxation processes exhibit radically different time responses for identical material specimen. These results confirm the PR’s implicit path dependence in addition to their separate intrinsic time reliance. Such non-uniqueness of viscoelastic PRs renders them unsuitable as universal material descriptors. Analytical formulations and experimental measurements also examine the physical impossibility of instantaneously achieving time independent loads or strains or their rates thus making certain PR definitions based on constant state variables, while mathematically valid, physically unrealistic and unachievable. A newly developed theoretical/experimental protocol for the determination of the time when loading patterns reach stead-state conditions based on strain accelerations demonstrates the capability to measure this time from experimental data. Due to the process dependent PRs, i.e. stress and stress history paths, the non-existence of a unique viscoelastic PR and of a universal elastic-viscoelastic correspondence principle or analogy (EVCP) in terms of PRs is demonstrated. Additionally and independently, the required double convolution integral construction of linear viscoelastic constitutive relations with the inclusion of PRs is cumbersome analytically and computationally needlessly highly CPU intensive. Furthermore, there is no theoretical fundamental hint as to what loading path is required to produce a unique universal viscoelastic PR definition necessary for formulating a PR based constitutive relation or an EVCP protocol. The analysis associated with an additional Class VII viscoelastic PR establishes it as a universal representation which is loading path and strain independent while still remaining time dependent. This Class PR can be the one used if it is desired to express constitutive relations in terms of PRs, subject to the caveat applying to all PR Classes regarding the CPU intensiveness in the time space due to triple product and double convolution integral constitutive relations. However, the use PRs is unnecessary since any set of material behavior can be uniquely and completely defined in terms of only moduli and/or compliances. The mathematical model of instantaneous initial loading paths, based on Heavi-side functions, is examined in detail and shown to lead to infinite velocities and accelerations. Additionally, even if non-instantaneous gradual loading functions are employed the resulting PRs are still load and load history dependent. Consequently, they represent specialized PR responses applicable and limited to those particular load and history combinations. Although the analyses contained herein are generalized to non-homogeneous linear viscoelastic materials, the main focus is on PR time and process dependence. The non-homogeneous material results and conclusions presented herein apply equally to homogeneous viscoelasticity and per se do not influence the results or conclusions of the analytical development regarding viscoelastic PRs. In short, these PR analyses apply to all linear viscoelastic material characterization.

Thermo Elastohydrodynamic Lubrication with Liquid-Solid Lubricant

This paper presents the performance characteristics of thermo-elastohydrodynamic lubrication (TEHL) in line contact with non-Newtonian liquid–solid lubricant. The time independent Reynolds equation, energy equation, elastic equation and load carrying with solid particle equation were formulated for compressible fluid. Newton-Raphson method and multigrid technique were implemented to obtain film thickness, film pressure, film temperature, friction coefficient and load carrying with solid particle equation in the contact region at various concentrations of solid lubricant and applied loads. The simulation results showed that film thickness and film temperature increase but film pressure decreases when solid particles are added into liquid lubricant. The maximum film temperature and load carrying of solid particle increased but friction coefficient decreased when concentration of solid particle increased. For increasing applied loads, the minimum film thickness decreases but maximum film temperature and friction coefficient increase for all liquid lubricant and liquid-solid lubricants.

Initial Fracture of a Viscoelastic Isotropic Plate with Two Collinear Cracks of Equal Length

The stress–strain state of an unbounded linear viscoelastic isotropic body weakened by collinear cracks of equal length and subjected to a time-independent load normal to the cracks is studied. The expression for the crack opening displacement in the nonlinear range that is derived based on the Leonov–Panasyuk–Dugdale model is used to deduce the equations of subcritical crack growth. A numerical algorithm for solving them is presented. The solutions of the equations describing subcritical crack growth are analyzed in determining the duration of the initial crack growth period. The numerical results are presented as graphs and tables

Nonlinear dynamic Interactions between flow-induced galloping and shell-like buckling

For an elastic system that is non-conservative but autonomous, subjected for example to time-independent loading by a steadily flowing fluid (air or water), a dangerous bifurcation, such as a sub-critical bifurcation, or a cyclic fold, will trigger a dynamic jump to one or more remote stable attractors. When there is more than one candidate attractor, the one onto which the structure settles can then be indeterminate, being sensitive to infinitesimally small variations in starting conditions or parameters.In this paper we develop and study an archetypal model to explore the nonlinear dynamic interactions between galloping at an incipient sub-critical Hopf bifurcation of a structure with shell-like buckling behaviour that is gravity-loaded to approach a sub-critical pitch-fork bifurcation. For the fluid forces, we draw on the aerodynamic coefficients determined experimentally by Novak for the flow around a bluff body of rectangular cross-section. Meanwhile, for the structural component, we consider a variant of the propped-cantilever model that is widely used to illustrate the sub-critical pitch-fork: within this model a symmetry-breaking imperfection makes the behaviour generic.The compound bifurcation corresponding to simultaneous galloping and buckling is the so-called Takens-Bodganov Cusp. We make a full unfolding of this codimension-3 bifurcation for our archetypal model to explore the adjacent phase-space topologies and their indeterminacies.

Load capacity and rupture displacement in viscoelastic fiber bundles

We study the creep rupture of viscoelastic fiber bundles under uniaxial constant tensile loading, assuming global load sharing (GLS) for the redistribution of load following fiber failure. We consider loading paths such that the stress raises to a value sigma0 under a time-independent loading (negligible creep strain) and remains fixed thereafter. Motivated by experimental observations, we introduce an "effective" strain controlled failure criterion to incorporate damage into the system, thus damage is distributed over time. In addition, when a "limit" value for the effective stress is reached, failure of the remaining alive fibers is instantaneous. This enables us to show both analytically and numerically that creep rupture occurs for external loads above a critical value that is less than the static fracture bundle's strength in accordance with experimental observations. An analytical expression for this critical load is given. For stress levels below the critical value, the system suffers only partial failure since the deformation tends to a stationary solution for which the effective stress is below the limit value giving rise to an infinite lifetime. On the other hand, if the time-independent loading process ends in the softening regime, the deformation of the system monotonically increases in time resulting in global failure at finite time irrespectively of the applied load. Moreover, the applied model is found to be consistent with the experimentally observed increase of the creep rupture displacement with decreasing steady external load (above the critical value).

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.

LOAD OPTIMIZATION OF ELASTIC-PLASTIC FRAMES AT SHAKEDOWN/PRISITAIKANČIŲ TAMPRIAI PLASTIŠKŲ RĖMŲ APKROVOS OPTIMIZACIJA

In this article the theory of mathematical programming is used, composing improved mathematical models of nonlinear problems of frame loading optimization at shakedown and performing its numerical experiment. An elastic perfectly-plastic frame is considered. Frame geometry, material, load application places are considered known. Time independent load variation bounds are variable (history of loading is unknown). Mathematical model of load variation bounds optimization problem includes strength and stiffness constrains. The mentioned optimization load combines two problems. First problem is connected with the distribution of statically admissible moments at shakedown. This is a problem of residual bending moments analysis which is presented in two ways. In the first case it is formulated as a quadratic programming problem, where the objective function is non-linear, but the objective function of load optimization problem remains linear. The problem is solved by iterations, influential matrixes of residual displacements, and stresses are used. In next case, the equations of problem analysis and dependences are presented according to complete equation system of plasticity theory. Then the objective function of optimization problem becomes non-linear and it is solved in single stage. Solving the second problem, we check if it is possible to satisfy frame rigidity constrains, which are inferior or superior limits of residual displacement. This is considered as a linear programming problem. Mathematical model of frame load optimization problem at shakedown was made with the help of non-linear mathematical programming theory. Numerical experiment was realized with Rozen's gradients projecting method and using the penalty function techniques. Mathematical programming complementarity conditions prohibit taking into account the dechargable phenomena in some cross-sections, therefore analysis of residual deformation compatibility equations are performed, using linear mathematical programming.

LOAD OPTIMIZATION OF ELASTIC-PLASTIC FRAMES AT SHAKEDOWN

Summary In this article the theory of mathematical programming is used, composing improved mathematical models of nonlinear problems of frame loading optimization at shakedown and performing its numerical experiment. An elastic perfectly-plastic frame is considered. Frame geometry, material, load application places are considered known. Time independent load variation bounds are variable (history of loading is unknown). Mathematical model of load variation bounds optimization problem includes strength and stiffness constrains. The mentioned optimization load combines two problems. First problem is connected with the distribution of statically admissible moments at shakedown. This is a problem of residual bending moments analysis which is presented in two ways. In the first case it is formulated as a quadratic programming problem, where the objective function is non-linear, but the objective function of load optimization problem remains linear. The problem is solved by iterations, influential matrixes of r...

Optimal design under time-variant reliability constraints

While reliability-oriented structural optimization under time-independent loading and resistances is sufficiently well known, the same problem under time-dependent loads and resistances has found at most, grossly simplified solutions. The main reason is that reliability calculations are far more complicated than for time-invariant loading. In this paper, a first attempt is made to use the out-crossing approach for the reliability part in structural optimization. In particular, reliabilities will be determined by the out-crossing approach in the context of FORM. Two types of load models, stationary rectangular wave renewal processes and differentiable processes, respectively, will be dealt with. As in time-invariant optimal design, a one-level approach is pursued.

Mechanics of dynamic debonding

Singular fields around a crack running dynamically along the interface between two anisotropic substrates are examined. Emphasis is placed on extending an established frame work for interface fracture mechanics to include rapidly applied loads, fast crack propagation and strain rate dependent material response. For a crack running at non-uniform speed, the crack tip behaviour is governed by an instantaneous steady-state, two-dimensional singularity. This simplifies the problem, rendering the Stroh techniques applicable. In general, the singularity oscillates, similar to quasi-static cracks. The oscillation index is infinite when the crack runs at the Rayleigh wave speed of the more compliant material, suggesting a large contact zone may exist behind the crack tip at high speeds. In contrast to a crack in homogeneous materials, an interface crack has a finite energy factor at the lower Rayleigh wave speed. Singular fields are presented for isotropic bimaterials, so are the key quantities for orthotropic bimaterials. Implications on crack branching and substrate cracking are discussed. Dynamic stress intensity factors for anisotropic bimaterials are solved for several basic steady state configurations, including the Yoffe, Gol’dshtein and Dugdale problems. Under time-independent loading, the dynamic stress intensity factor can be factorized into its equilibrium counterpart and the universal functions of crack speed.

Age‐Adjusted Effective Modulus Method for Time‐Dependent Loads

A modification is proposed for the widely used Baz˘ant age-adjusted effective modulus method (AEMM) based on the generalized Baz˘ant theorem. The modified method is a numerical approximation procedure for calculating stresses and deformations in structures subject to creep, which accounts for the loading history of the structure. By applying this method to the case of time-dependent loads, it is possible to obtain exact solutions for homogeneous structures and approximate solutins for nonhomogeneous structures. The modified AEMM reduces to the usual AEMM for the case of instantaneously applied time-independent loads.

The stress intensity factor history for an advancing crack under three-dimensional loading

The stress intensity factor history due to the uniform extension of a planar crack in an unbounded elastic body under three-dimensional time-independent loading is considered. First, a fundamental (point force) solution is obtained, and this is used to write down the form of the stress intensity factor history for general loading in terms of a superposition integral. A particular traction distribution is also considered.

The viablility of the KIa procedure for predicting the arrest of a crack in a nuclear reactor steel pressure vessel

The current ASME Code procedure for predicting crack arrest in a nuclear reactor steel pressure vessel is based on a static linear elastic fracture mechanics analysis: a crack is presumed to arrest when the crack tip stress intensity factor K I ST falls below K Ia, which is assumed to be a material property and is referred to as the arrest toughness. The viability of this procedure has been questioned since the theoretical justification, in the strictest sense, for this very simple K Ia approach is based on the behaviour of a semi-infinite crack propagating in an unbounded solid due to the application of time-independent loads. Against this background, the present paper examines the effects of initial crack size and crack jump length on the viability of the K Ia procedure. A theoretical analysis shows that the procedure should give accurate predictions of the crack length at arrest certainly if the crack jump length is less than twice the initial crack size.

Consolidation by Sand Drains in Initial Gradient Soils

Phenomenon of water movement through cohesive soils is important to several fields of engineering. Many investigations on flow through such soils indicate that seepage starts only when the hydraulic gradient exceeds a certain value called the initial or threshold gradient and the velocity gradient. In the absence of the initial gradient, theoretical solutions for radial flow towards sand drains exits. In this note an analytical solution is presented that incorporates the concept of initial gradient to the problem of radial flow to sand drains. The solution is based on the assumptions of equal strain, no smear, constant compressibility and permeability, and time independent external load.

Combined distributed loads on rigid-plastic circular plates with large deflections

A theoretical investigation into the influence of geometry changes on the behavior of a simply supported circular rigid, perfectly plastic plate subjected to two independent distributed pressures is presented herein. The results indicate, as might be anticipated, that when finite deflections are considered such plates could support loads greater than the corresponding collapse pressures obtained recently by Flügge and Gerdeen [4] and Hodge and Sun [6]. The general procedure reported could be used to study the reserve strength of circular plates having different boundary conditions and other kinds of external loads, and could be developed further in order to examine the influence of geometry changes on other rigid-plastic structures subjected to time-dependent or time-independent loads.