Concentrated Force Research Articles

АНАЛИЗ УСТОЙЧИВОСТИ ПРИ СДВИГЕ СТАЛЬНЫХ БАЛОК-СТЕНОК С КРУГЛЫМИ ВЫРЕЗАМИ

In spite of half century experience of application of perforated beams the local bucking of their webposts is not finally researched till now. In manuals there is no relations for their critical loads corresponding to buckling of web-posts under shear. In the study the numerical analysis by the finite element method is performed for evaluating the local stability of simply supported cellular beam-webs with circular openings. Variant of loading of beam with concentrated force was considered, which assure constant value of shear force along the length of beam. Obtained empirical relation allow take into account not only parameters of perforation but influence also of bending moment on the shear buckling of web-posts. Analysis by FEM also confirmed permitted accuracy of analytical approach to evaluation of stability of web-posts.

Contact of the Faces of an Interface Semiinfinite Crack

We consider the equilibrium of two rigidly connected elastic half planes made of different materials, containing a semiinfinite crack on their interface, and subjected to the action of normal and tangential concentrated forces applied to the crack faces. We take into account the friction contact of the crack faces near the crack tip and at a certain distance from the tip. The solution of integral equation of the analyzed problem is obtained in the closed form with the help of the Wiener–Hopf method. We determined the boundaries of contact zones between the crack faces and the distributions of stresses in these zones and on the interface of half planes outside the crack.

Quantitative Analysis of Electromagnetic Forces by Decoupling Air-Gap Field Modulation and Force Modulation in Rotor-Permanent-Magnet Machines

This article proposes a quantitative analysis model for the electromagnetic forces in rotor-permanent-magnet (rotor-PM) machines, decoupling the air-gap field modulation and the force modulation. First, a unified analytical model of the equivalent concentrated forces (ECFs) is deduced based on the air-gap flux density (AFD) harmonics. Second, the air-gap field modulation and the force modulation are decoupled by the ECF coefficient. Then, to determine the contributions of the modulated AFD harmonics, a quantitative analytical method is proposed. On this basis, the vibration performance of a 12-slot/10-pole (12s/10p) interior PM machine is investigated considering the modulation effect. The results reveal that the influences of rotation speed, PM-excited field, and armature field on the vibration can be clearly represented by the proposed method. Finally, the modal test and the acceleration measurement are conducted to validate the analysis.

Vibration Isolation in Plates Using Flexural Cloaking

Cloaking is a novel idea, where the incident waves are guided around a space called the core. The cloaking concept can be realized by engineering the material matrix around the core. Cloaks can be used to guide waves around an obstacle with minimum reflection and scattering. On the other hand, they can be also used to isolate the middle core space, which makes cloaking an innovative potential noise control measure at challenging lower frequencies. Space isolation using cloaks can be beneficial for different applications such as architectural noise control or soundproofing the cabin in automotive and aerospace industries. However, this interesting aspect of cloaking has not been investigated in the literature yet. In this study, a flexural cloak in a thin plate is modeled using finite element method to investigate the impact of cloaking on isolation of the middle core numerically. The effect is examined using plane waves as well as concentrated force excitation. Moreover, the dependency of the isolation effect on the geometrical size of the core and the cloak is investigated with regard to the wavelength.

On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing

The present two-part paper concerns the active vibration suppression for the simplest damped continuous system, namely the transverse oscillations of an elastic string, with constant tension and mass density per unit length and friction force proportional to the velocity, described by the telegraph or wave-diffusion equation, in two complementary parts. The initial part I considers non-resonant and resonant forcing, by concentrated point forces or continuous force distributions independent of time, with phase shift between the forced and free oscillations, in the absence of damping, in which case the forced telegraph equation reduces to the forced classical wave equation. The present and final part II uses the forced wave-diffusion equation to model the effect of damping, both as amplitude decay and phase shift in time, for non-resonant and resonant forcing by a single point force, with constant magnitude or magnitude decaying exponentially in time at an arbitrary rate. Assuming a finite elastic string fixed at both ends, the free oscillations are (i) sinusoidal modes in space-time with exponential decay in time due to damping. The non-resonant forced oscillations at an applied frequency distinct from a natural frequency are also (ii) sinusoidal in space-time, with constant amplitude and a phase shift such that the work of the applied force balances the dissipation. For resonant forcing at an applied frequency equal to a natural frequency, the sinusoidal oscillations in space-time have (iii) a constant amplitude and a phase shift of π/2. In both cases, the (ii) non-resonant or (iii) resonant forcing dominates the decaying free oscillations after some time. Even by optimizing the forcing to minimize the total energy of oscillation, it remains below the energy of the free oscillation alone, but only for a short time—generally a fraction of the period. A more effective method of countering the damped free oscillations is to use forcing with amplitude decaying exponentially in time; by suitable choice of the forcing decay relative to the free damping, the total energy of oscillation over all time can be reduced to no more than 1/16th of the energy of the free oscillation.

Viscoelastic Solution for Axial Forces of Point-Anchored Rock Bolts in a Circular Tunnel

In this study, a deeply buried circular tunnel supported with point-anchored rock bolts was investigated. Assuming that the effects of each rock bolt on the surrounding rock are simplified to two identical concentrated forces acting in opposite directions, a viscoelastic analytical solution for the axial forces of the rock bolts is presented herein based on the complex variable method and correspondence principle. First, in the solution process, the surrounding rock is considered as an elastic material. A linear equation system for the axial forces was established according to the deformation coordination condition between the surrounding rock and the rock bolts. Second, the surrounding rock was considered as a viscoelastic material. Based on the correspondence principle, Laplace transformation was performed for the equation system of the axial forces to obtain its expression in the Laplace domain, and the values of the axial forces in the Laplace domain could be determined by solving the obtained expression. Finally, the viscoelastic semi-analytical solution for the axial forces was established using the numerical method of the Laplace inverse transformation. The results obtained were also compared with the numerical methods of ANSYS. Accordingly, the effects of the lateral stress coefficient, rock mass Young’s modulus, rock bolt length, viscosity coefficient, and preload on the axial forces of the rock bolts were discussed.

Flexible guided anti-collision device for bridge pier protection against ship collision: Numerical simulation and ship collision field test

The flexible guided anti-collision device (FGAD) can prolong the impact time between ship and pier, so that the ship has enough time and space to turn and deviate from the pier. In this way, the ship can retain most of the kinetic energy, and the impact force on the pier can be greatly reduced. Both of ship and pier can be protected by the FGAD. In this paper, the dynamic response and the energy-dissipation capacity of FGAD are firstly studied by the numerical simulation of ship-bridge collision process. The results show that the FGAD can effectively turn the ship's course, and the concentrated force of collision is transformed into the distributed force, which significantly reduces the energy conversion and impact force. Then, an actual ship collision field test is carried out to investigate the effectiveness of the FGAD for reducing the impact force on the pier and turning the navigation direction of ship. The test results show that the FGAD can turn the ship's course with a little change of the ship speed after the collision. Most of the kinetic energy does not participate in the energy exchange and the impact force on the pier is significantly reduced. The FGAD realizes the design idea of protecting both the bridge and the ship. The application of engineering also confirms the effectiveness and reliability of the FGAD.

Continuous and discontinuous contact problem of a functionally graded orthotropic layer indented by a rigid cylindrical punch: Analytical and finite element approaches

AbstractIn this study, the continuous and discontinuous contact problems of a functionally graded (FG) orthotropic layer lying on a homogeneous and isotropic layer were investigated. The orthotropic layer was indented by a rigid cylindrical punch which was subjected to a concentrated normal force, and a homogeneous isotropic layer was firmly attached to a rigid substrate. For the analytical solution, the general expressions for the stresses and displacements were obtained in the presence of body forces by using elasticity theory and Fourier integral transforms. The continuous and discontinuous contact problems were reduced to the singular integral equations by means of boundary conditions. The system of the singular integral equations was solved using the Gauss‐Chebyshev integration formula. Then, the finite element solution of the two cases was also performed. Analytical results and numerical results (FEM) for critical load, initial separation distance, separation region in discontinuous contact, contact lengths under the punch, and contact stress distributions between the layer‐punch and layer‐layer were obtained for the dimensionless quantities.

Calculation method for internal force and deformation of the prestressed I-beam on the elastic foundation

The elastic foundation beam theory has been widely used in civil engineering, including railway, tunnel, and building foundations. With the development of fabricated structures, more elastic foundation beams need to be prestressed. In order to explore the frame foundation beam with the fabricated anchor-cable in the slope reinforcement project, in this article, prestress is applied to both ends of the beam. Then, according to the calculation method of internal force and deformation of the beam under concentrated force and the equivalent load theory of the prestressed structure, three new methods: the finite difference method (FDM) of the Euler–Bernoulli beam on the Winkler foundation, FDM of the Euler–Bernoulli beam on the Pasternak foundation, and theoretical analytical solution of the Timoshenko beam on the Winkler foundation, are deduced to calculate internal force and deformation under prestress force and concentrated force, respectively. Typical calculation parameters are selected for design and verification via the three new methods and GEO5, respectively. The results show that the calculated values of the three new methods are basically consistent with those calculated using GEO5 software, which verifies the feasibility of the three new methods.

Experimental analysis on the interaction between underground structures and sand layer under groundwater level change

Groundwater level change stands a momentous role in affecting the geotechnical construction stability and safety of underground structures. Global warming and active underground construction cause conspicuous changes in the groundwater level, which further leaves an impact on the underground structures’ serviceability. To reveal the interaction between underground structure and soil under groundwater level change in the sand layer, model tests of circular transportation tunnel and rectangular utility tunnel were carried out. With the self-designed experimental equipment and innovative experimental methods, the changes in tunnel stress, bending moment, buoyancy, and vertical displacement of the rise and drawdown of the groundwater level in the sand layer were studied. The results revealed the developments of concentrated structural forces during the rising and falling process of the groundwater water level, indicating critical locations that should be strengthened. Meanwhile, both tunnels showed the same movement trend: settling first, floating afterwards and settling at last. And it is concluded that no reduction is required when calculating buoyancy in sands using measured pore pressure. Conclusions can provide a notable reference for future related research and engineering designs.

Method of fundamental solution using Erdogan's solution: Static and dynamic

This paper developed the method of Erdogan`s fundamental solution (MFS) for a 2D in-plane crack problem with static and dynamic loads. The analytical solutions with complex variables with a pair of static concentrated force in a cracked infinite domain were derived by Erdogan in 1965. Based on the Erdogan's fundamental solutions, the MFS is extended to 2D elasticity under static and dynamic loads. The Laplace transform technique and the Durbin inversion technique are adopted for a dynamic case to determine the time-dependent variables. The domain integrals are obtained numerically with both Constant Cell Method (CCM) and Finite Block Method (FBM). The accuracy and convergence of the MFS are demonstrated by four numerical examples. Comparisons have been implemented with both analytical solutions and the finite element method.

Dynamic analysis of an elastic half-plane soil medium with periodic wave barriers under a moving load

Most investigations on the control of ambient vibration caused by moving loads are partial to the ground buildings rather than underground structures. With the rapid development of rail transit, the effects of ambient vibration caused by moving loads on underground structures such as important underground military facilities or underground railway stations have been got wide attention in recent years. This paper focuses on the vibration attenuation of underground structures by periodic barriers under a single frequency moving concentrated force. First, under the assumption of plane strain and taking the Doppler effect into account, the closed-form solutions of an elastic half-plane subjected to a moving load on the surface are obtained, and are validated by numerical simulations. The frequency characteristics of the present solutions are comprehensively discussed. The isolation mechanism of periodic wave barriers under a moving load is revealed. Second, based on the characteristics of soil’s frequency response under a moving load, two different schemes for designing wave barriers are provided for load speed less than S-wave speed of soil mass. The effectiveness of the proposed design schemes is verified through numerical simulations for different types of traffic loads. This study shows that the periodic wave barriers can play a great role in vibration isolation of underground engineering, and the design schemes proposed in this paper work very well. The present study provides a new insight for vibration attenuation of underground structures caused by moving loads.

Experimental study on the coupling between the piezoelectric and streaming potential in wet bone

The stress-generated potential (SGP) of bone is one of the mechanisms affecting bone remodeling including piezoelectricity and streaming potential. To explore the interactions between the piezoelectric and streaming potential, an experimental setup was designed that simultaneously applied a concentrated force and liquid pressure to wet bone. Using this device, the stress-generated potential of wet bone under the two types of loads was measured. The experimental results show that under a constant liquid pressure, the measured potential curves increase over time, and its increasing rate decrease as the concentrated force increase. The measured peak amplitudes of potential decrease as the liquid pressure increase under the same concentrated force whether loading or unloading. To explain the coupling mechanism of the found phenomena, an equivalent model with two voltage sources and three equivalent resistances was established, and the equivalent electrical relationship between the piezoelectric and streaming potential was obtained by analyzing the model. The analysis discussion implies that various factors have influence on the coupling relationship between streaming and piezoelectric potentials, and the factors can be summarized as the changes of the three equivalent resistances caused by piezoelectric and streaming potentials.

The stiffness of a reinforced concrete beam with random parameters the cracks formation

The work of a reinforced concrete beam lying on an elastic foundation is considered, the deformation characteristics of which are described by a model with two Vlasov-Pasternak bed coefficients. The bed coefficients are considered as stationary random normally distributed functions of the x coordinate. Strength characteristics of concrete: cube strength and concrete tensile strength are assumed to be random variables with Gaussian probability density distributions. The concentrated forces approximating the columns of the building are applied at regular intervals and are assumed to be random normally distributed values. After the formation of cracks in the sections under concentrated forces in the beam, areas with a reduced bending stiffness are formed, and therefore the stiffness of the beam becomes variable along the length of the beam. The solution of the differential equation of bending of a beam with variable stiffness lying on an elastic foundation is proposed to be sought by a variational method minimizing the functional of the additional energy principle. The deflections of a beam with variable stiffness will be approximated by a function that satisfies the boundary conditions and the equilibrium condition. From the equality of the deflections of a beam with constant equivalent bending stiffness and deflections of a beam with variable bending stiffness due to cracking, the constant equivalent stiffness and its statistical parameters are determined. Knowing the constant equivalent bending stiffness of a concrete beam and using the known solutions of the bending equation of a beam lying on an elastic foundation with random properties, it becomes possible to find the probabilistic parameters of bending moments and shear forces and to determine the probability of the onset of limit states of the beam along the normal and inclined sections.

Support Vector Regression approach for prediction of delamination at entry and exit during drilling of GFRP Composites

The demand for Composites in the modern era have increased immensely due to its vast applications and superior properties over conventional materials. Glass Fibre Reinforced Plastic (GFRP) is one of the economic alternative to conventional engineering materials due to its high specific modulus of elasticity, high specific strength, good corrosion resistance, high fatigue strength and lightweight. Components made out from GFRP composites are usually near net shaped and require holes for assembly integration. Drilling is an important process as concentrated forces can cause major damage to the composite. Drilling of GFRP causes various damage such as thermal degradation, fibre breakage, matrix cracking and delamination. A substantial damage is caused by delamination which can occur both on the entry and exit sides of the composite, exit side delamination considered more severe. Therefore, selection of proper process parameters during drilling operation is very much essential. In the present work, a support vector regression (SVR) model is developed to predict the delamination at entry and exit during the drilling of GFRP composites. The model is developed based on the data obtained from experimentation. The model accuracy is evaluated by the three performance criteria including root mean square error (RMSE), Nash–Sutcliffe efficiency co-efficient (E) and co-efficient of determination (R2). The model provides an inexpensive and time saving alternative to study the delamination at entry and exit of the GFRP composite actual drilling operation.

Transverse vibrations of underground pipeline with pinched ends

The study of transverse oscillations of extended underground structures from the action of constant harmonic forces is of practical interest. Transverse vibrations of an underground cylindrical structure of finite length under the action of a system of concentrated forces that change according to a harmonic law are considered. The environment surrounding the structure is assumed to be elastic, while the interaction of the “environment-structure” system is studied. When considering the problem, the Fourier method and the results of solving the problem of vibrations of an unbounded cylindrical structure in an elastic medium from the action of concentrated harmonic forces were used. The problem of transverse vibrations of a cylindrical rod of finite length interacting with an elastic homogeneous medium is solved by the method of compensating loads. In this case, the structure is considered a source of waves radiated into space and satisfying the conditions of radiation to infinity. The resulting solution is approximate and reliable within the length of the rod. The analysis of the influence of soil conditions and geometric dimensions on the amplitude of oscillations of an underground structure was carried out according to the graphs of the amplitude-frequency characteristic of the object.

IN MECHANICS OF DEFORMABLE SOLIDS PRIMARY PLASTIC ON PANEL DURING POWER UP DEVELOPMENT OF DEFORMATIONS

One of the most important issues in solid mechanics is the development and application of optimal methods for estimating the resistance of dispersion materials. Currently, the theory of propagation of cracks in solid bodies is widespread due to the wide use of high-strength materials and large-sized structures and devices in various fields of technology. The article discusses the methods of studying the problems of the interaction of two factors - strengthening the panel with stiffening ribs made of unidirectional composite material and weakening it with a number of circular holes and cracks protruding from its contour. An infinite isotropic elastic riveted panel weakened by a periodic system of circular holes and two cracks along the abscissa axis segments near each hole is considered. The contours of the holes and the edges of the cracks are free from external forces. Transverse stiffness ribs made of metal composite material are riveted to the plate. In the calculation scheme, the effect of riveted reinforcing ribs is replaced by the concentrated forces applied at the points of location of the rivets. The values of the concentrated forces are found depending on the geometrical and physical parameters of the problem. Stress intensity factors and ultimate loadings were calculated depending on the geometrical and physical parameters of the problem. It has been shown that there is a stable phase of crack development under certain specified conditions. The development of initial plastic deformations (embryonic cracks) emerging from the contour of circular holes in a plate reinforced with stiffness ribs is studied. The material of the plate is elastic-perfect-plastic satisfying the Tresca-Saint-Venant condition. It is assumed that the plastic deformations are concentrated along some slip lines emanating from the contour of the hole. By satisfying the boundary conditions, the solution of the problem leads to a singular integral equation. Then the singular equation of the problem is brought to a system of algebraic equations without the intermediate step of bringing it to the Fredholm equation. The resulting system was solved by the Gaussian method with the selection of principal elements for different values of n (n is the number of Chebyshev nodes in the division of the interval). The values of concentrated forces are found depending on the geometrical and physical parameters of the problem. The dependences of the length of the plasticity bands on the applied load, the geometrical parameters of the problem and the yield point of the material were found. Using the local dissipation condition (KPT-criteria), a ratio was obtained to determine the destructive edge loading.

Deformation of a plane modelled by John's material with a rigid elliptical inclusion loaded by force and moment

An exact analytical solution is obtained for a non-linear problem of elasticity theory for a plane with a rigid elliptical inclusion. A concentrated force and a moment are applied at the center of inclusion. The elastic properties of the plane are modeled by John’s harmonic material. For this material methods of the theory of functions of a complex variable are using for solving nonlinear plane problems. Nominal stresses and displacements are expressed in terms of two analytical functions of a complex variable, which are determined from the boundary conditions on the contour of inclusion. The problems of the action of a concentrated force and moment on an elliptical core in a plane are considered separately. A comparison with a similar linear problem is made. The influence of the applied force and moment on the magnitude of stresses is studied depending on various parameters of the problem. Calculations of nominal stresses on the contour joining the plane with inclusion are performed.

Solution of contact problems of anisotropic plates bending on an elastic base using the compensating loads method

In critical structures used elements consisting of anisotropic materials with concentrated force or moment applied to them are often found. In order to equalize the internal force factors, as well as to ensure the required strength and rigidity of the structure, in places where a local load is applied, it is necessary to strengthen the plate with a rigid lining (stamp). There are several ways to solve contact problems. The way seems simpler if it is possible to determine the exact fundamental solution (FS) for the plate. The exact FS significantly reduces the amount of computational work due to the fact that the boundary conditions and the conditions of conjugation of solutions at the boundary of the contact area are met in advance. It remains only to formulate the conditions for the compatibility of movements in the contact area. With this formulation, the integral equation of the contact condition becomes a singular integral equation of the first kind. Such equations are reduced by regularization to Fredholm equations of the second kind and the problem becomes mathematically and physically correct. The paper considers contact problems without taking into account tangential interaction between objects. For such problems, the following way of simplification is possible: replacing the singular core for a half-space with a core for a layer of constant thickness modeled by an elastic base with a single base coefficient. Such a replacement for thin shells is quite justified and gives solutions that are in good agreement with the solutions of the theory of elasticity. The deflection of the plate is sought using the compensating loads method. To work out the methodology, a test contact problem was solved.

Modeling a Viscoelastic Support Considering Its Mass-Inertial Characteristics During Non-Stationary Vibrations of the Beam

Non-stationary loading of a mechanical system consisting of a hinged beam and additional support installed in the beam span was studied using a model of the beam deformation based on the Timoshenko hypothesis with considering rotatory inertia and shear. The system of partial differential equations describing the beam deformation was solved by expanding the unknown functions in the Fourier series with subsequent application of the integral Laplace transform. The additional support was assumed to be realistic rather than rigid. Thus it has linearly elastic, viscous, and inertial components. This means that the effect of a part of the support vibrating with the beam was considered such that their displacements coincide. The beam and additional support reaction were replaced by an unknown concentrated external force applied to the beam. This unknown reaction was assumed to be time-dependent. The time law was determined by solving the first kind of Volterra integral equation. The methodology of deriving the integral equation for the unknown reaction was explained. Analytic formulae and results of computations for specific numerical parameters were given. The impact of the mass value on the additional viscoelastic support reaction and the beam deflection at arbitrary points were determined. The research results of this paper can be helpful for engineers in designing multi-span bridges.