Vertical Concentrated Force Research Articles

Modeling and design of a SMA self-actuating bistable beam

Bistable beams with implanted actuators show the self-actuating ability that further improve performance of the system. In this paper, a novel self-actuating bistable mechanism coupled by elastic straight beam and SMA curved beam was designed and tested. Snap-back behavior of the presented system was triggered by SMA curved beam actuator, which generated actuation moment and rotation angle at the end of flexible beam. The bistable mechanism was modeled through nonlinear governing equation of large deflection beam. A solution strategy that considered multiple inflection points was developed to solve the governing equation. Loading path, deflection curves, and energy landscape of bistable beam were illustrated and further analyzed. Moreover, parameters of SMA curved beam actuator were determined by a simplified model. A prototype of the self-actuating bistable beam was fabricated and the snap-back phenomenon was observed on the test bench. Accompanied by the snap-back, a 44% decrease of axial load could be reached. It was also found that the bistable beam could be reset by applying vertical concentrated force. Reciprocating snap-back tests were conducted to confirm that point. Possible applications of the presented bistable beam could be variable load mechanisms and novel flexible switches.

Enhancement of the behaviour of reinforced concrete dapped end beams including single-pocket loaded by a vertical concentrated force

In precast building construction, some beams are designed to support one or several concentrated loads resulting from the reactions of the attached cross beams. Consequently, the pocket beams without or with dapped ends may represent one of the innovative solutions to constitute the joints between the two attached members. In the present research work, the behaviour of dapped end beams that included rectangular opening loaded with in-plane force, have been investigated. Several detailing have been proposed, in addition the vertical stirrups, to improve the strength of the opening region including the use of steel fibre concrete (SFC),Configuration of the inclined crossed bars, jacketing with steel plates and the composite section technique with two arrangements of the reinforcement of the dapped end . Ten specimens have been tested under gradually increased vertical static loading. The tested specimens are categorized into two sets based on the configuration of dapped end reinforcement. Two variable have been considered which are the strengthening configuration of the opening region and the configuration of the dapped end reinforcement. The response has been discussed in terms several indicators including, the cracking and failure loads, maximum deflection, mode of failure, load-deflection curves, crack patterns, crack width, to recognize the optimum strengthening proposal of the opening. Results revealed that using the inclined steel bars (modified arrangement) yield better response than the conventional (standard) reinforcement (vertical stirrups) within the dapped end. For beams with pockets strengthened with crossed inclined bars, failure load enhanced in range of (8.5-11%) whereas the enhancement was in ranged in (8-10%) for the steel SFC method. Moreover, an improvement by about (11-13%) in load capacity increased when the jacketing with steel plates was applied. The maximum improvement was obtained when using the embedded rolled section within the opening with values of (21-23%).

To Clarify Decision on Action of Concentrated Force to Elastic Quarter-Space for Arbitrary Poisson’s Ratio

Previously, the author obtained an analytical solution for determining the vertical displacements of the homogeneous isotropic quarter-space face, which is affected by a vertical concentrated force. This expression gave an exact solution for determining the vertical displacements of a quarter-space face made of incompressible material and an approximate solution for a Poisson ratio different from 0.5. Later, in the published paper by S. V. Bosakov and P. D. Skachek “Action of Concentrated Force on 1/8 of Homogeneous Isotropic Space”, it was shown that by combining solutions for determining the vertical displacements of a quarter-space and a half-space from the action of concentrated forces, one can find vertical displacements for one-eighth face of a homogeneous isotropic space. The resulting expressions allow solving contact problems for non-classical domains in the form of a quarter of a space and one eighth of a space. Below, the author obtains the first approximation for displacements of a quarter-space face from the action of a vertical concentrated force. In this paper, the author gives the first approximation for determining the vertical displacements of a quarter-space face from the action of a vertical concentrated force, widely using the special approximation method developed in the works of V. M. Alexandrov and allowing to calculate successfully improper integrals. The constructed graphs show good results when determining displacements with a Poisson’s ratio different from 0.5 and equal to 0.5. It should be noted that the approach indicated in the paper by S. V. Bosakov and P. D. Skachek can be successfully used in determining all displacements of the faces of one-eighth face of a homogeneous isotropic sharpened forces tangent to the edge of a quarter-space and this will allow to solve contact problems taking into account friction forces in the contact zone of the beam or plate.

Steady-State Response Analysis of the Incompressible Nonlocal Saturated Poroelastic Beam under a Vertical Harmonic Load

Based on the nonlocal theory and the theory of saturated porous media, the mathematical and physical model and governing equations of the steady-state response of the incompressible nonlocal saturated poroelastic beam under vertical harmonic loading are established with assumption of the movement of the liquid-phase fluid only in the axial direction of the beam and considering the nonlocal effects such as particle size, pore size, and pore dynamic stress. The dynamic response of a saturated poroelastic cantilever beam with permeability at both ends under a vertical harmonic concentrated force at the free end is studied. In the frequency domain, the analytical expressions of deflection amplification factor and equivalent couple amplification factor of liquid fluid pressure are given. The effects of nonlocal coefficient τ, mechanical parameter α, and geometric parameter β on the deflection amplification factor and equivalent couple amplification factor at the midpoint of the nonlocal saturated poroelastic cantilever beam are studied. The results show that the steady-state vibration of the incompressible nonlocal saturated poroelastic cantilever beam has resonance. When the nonlocal effect is considered, the deflection amplification factor and the equivalent couple amplification factor are larger, so the influence of the nonlocal effect on the steady-state response of the beam should not be ignored. The geometric parameter β has significant effect on the peak positions of the curves of the deflection amplification factor and the equivalent couple amplification factor varying with frequency.

Girder Longitudinal Movement and Its Factors of Suspension Bridge under Vehicle Load

Vehicle load may not only cause vertical deformation and vibration of suspension bridge but also lead to longitudinal deformation and vibration. And the longitudinal behavior is closely related to the durability of the girder end devices and the bending fatigue failure of suspenders. In this study, the longitudinal deformation behavior and longitudinal vibration of suspension bridge under vehicles, as well as the related influencing factors, are investigated. The underlying mechanism of girder longitudinal movement under the moving vehicles is revealed. Based on the simplified vehicle model of vertical concentrated force, the characteristics of main cable deformation and girder longitudinal displacement under vertical loads are analyzed first. Then, the longitudinal motion equation of the girder under vertical moving loads is derived. Finally, a single long‐span suspension bridge is employed in the case study, and the girder longitudinal response and influencing factors are investigated based on both numerical simulation and field monitoring. Results indicate that the asymmetric vertical load leads to cable longitudinal deflection owing to the geometrically nonlinear characteristic of the main cable, leading to longitudinal movement of the girder. The results of field monitoring and numerical simulation indicate that the girder moves quasi‐statically and reciprocates longitudinally with centimeter amplitude under normal operational loads.

Effort and bearing capacity testing of laterally restrained asymmetrical stepped steel I-columns

This paper presents extensive experimental investigations of models of I-shaped steel columns with segmentally prismatically variable stiffness. The models are discretely braced with lateral restraints along their length. With regard to the geometry of the actual columns, the models were made on the semi-industrial scale of 1:2. An external load, in the form of a vertical concentrated force, was applied to the header (in the Ch test series) or to the bracket (in the Cb and Chb test series). Selected static equilibrium paths of the linear displacements and the angles of rotation of column cross sections, determined in each of the test series, are presented and their behaviour is described in detail. Moreover, in the case of the Ch3 and Cb3 models, stress intensity in selected cross sections is examined by means of electrical resistance strain gauges, and the increase in normal stress with regards to the external load is shown. In addition, the experimental investigations included relevant material studies to determine the strength and chemical composition of the steel of which the columns were made.

Long-Term Deformation Analysis for a Vertical Concentrated Force Acting in the Interior of Fractional Derivative Viscoelastic Soils

Abstract This paper gives a set of time-dependent displacement solutions for the case of a vertical concentrated force acting in the interior of fractional derivative viscoelastic soils. First, a t...

Experimental analysis of the effect of discrete aggregate on the bending stability of thin-walled cylindrical shells

The effect of a discrete aggregate on the bending stability of thin-walled cylindrical shells was studied. The samples for the study were made of aluminum alloy 3004 (temper H19). The samples were cantilevered; a vertical concentrated cross force was applied to its free end. Ten empty samples and ten samples 90% filled with iron powder were tested. The samples were loaded in stages by 10N, and when approaching the moment of buckling the samples were loaded by 1N or less. The force, deflection of the free end of the sample, and axial strain were recorded at each stage of loading. Displacement of the free end of the sample versus cross force diagrams is plotted. The buckling force was determined by the inflection point of the diagrams. The samples lost the stability in elasticity. The diagrams for the empty and filled samples for each series almost coincide until the loss of stability of the empty sample. This means that when the empty sample loses its stability, the buckle directed towards the center line is formed; and for the filled samples, the formation of buckles is prevented by the discrete material, therefore the critical force increased by 18.8%. The influence of discrete aggregate is considered on the basis of approximation [24] for tank trucks manufactured at the enterprise. The critical stress is calculated using the superposition principle, since the stability is lost in the elasticity. The critical stress for the filled tank is determined by the sum of the critical stress for an empty shell with the stresses created by the weight load and the hydrostatic pressure of the discrete filler. The calculation of critical stresses showed that for the pattern samples the influence of discrete filler is 8.3%, and for the full-sized tanks the influence of discrete filler on the value of the critical voltage is significant and amounts to 62%. We studied the effect of various discrete fillers on stability, such as river sand, iron and copper powders at different filling ratios of the samples. Due to an increase in powder density and filling volume, the value of the critical force increased. For the samples filled to 90% with the river sand had the critical force increase by 13.3%, while those filled with the iron powder had their critical force increase by 40.5%, the copper powder samples had an increase of the critical force by 43.1%.

Study on the Static Performance of H-shaped Steel Beam to Square Tubular Column Semi-Rigid Connections Steel Frame

In order to study the overall mechanical behavior of the new H-shaped steel beam to square tubular column semi-rigid connections in steel frame, the static performance, failure characteristics and bearing mechanism of the steel frame are studied by the medium-span static load test of the full-size steel frame and finite element simulation. The results show: when the vertical concentrated force is applied to the beam span, the buckling occurs in the mid-upper flange of the beam spans, and the plastic hinge is formed at the position of the beam span, while the deformation of the joint and the column is small, so the safety factor of the frame column and the joint region is high. Due to the semi-rigidity of the joint, the force and deformation of the joint region are smaller than the span of the beam, and the rotation angle of the joint meets the requirements for normal use of the structure. Therefore, the carrying capacity mechanism of the steel frame is reasonable.

On the Kelvin, Boussinesq, and Mindlin problems

Three fundamental three-dimensional axisymmetric problems of elasticity are revisited: the Kelvin problem of a concentrated force in an infinite space, the Boussinesq problem of a vertical concentrated force at the boundary of a half-space, and the Mindlin problem of a vertical concentrated force in the interior of a half-space. New elements of the derivation of the solution to each of these problems are included to make the presentation more appealing for the coverage of the topic in graduate courses of solid mechanics. Two approaches are employed and compared, one based on the Galerkin method and Love’s potential function, and the other based on the Papkovich–Neuber displacement representation and Boussinesq’s potential functions. A historical perspective and a referral to more recent contributions in the field are also given.

Analysis of a beam with a crack under irregular cyclic load

This paper considers a beam with a technological longitudinal section on the level of the gravity center of the cross section that is loaded by a vertical concentrated force. We study the structure durability depending on the load amplitude and the initial crack length.

Derivation Method for the Foundation Boundaries of Hydraulic Numerical Simulation Models Based on the Elastic Boussinesq Solution

The foundation boundaries of numerical simulation models of hydraulic structures dominated by a vertical load are investigated. The method used is based on the stress formula for fundamental solutions to semi-infinite space body elastic mechanics under a vertical concentrated force. The limit method is introduced into the original formula, which is then partitioned and analyzed according to the direction of the depth extension of the foundation. The point load will be changed to a linear load with a length of2a. Inverse proportion function assumptions are proposed at parameteraand depthlof the calculation points to solve the singularity questions of elastic stress in a semi-infinite space near the ground. Compared with the original formula, changing the point load to a linear load with a length of2ais more reasonable. Finally, the boundary depth criterion of a hydraulic numerical simulation model is derived and applied to determine the depth boundary formula for gravity dam numerical simulations.

Semi-exact solutions for large deflections of cantilever beams of non-linear elastic behaviour

The problem of large deflections of cantilever beams made of materials obeying a Ludwick type stress–strain law under the combined action of one vertical concentrated force at the free end and a uniformly distributed load were investigated in the so-called bending moment formulation. A corrected bending moment is introduced and it is shown that the resulting governing differential equation for the slopes can be solved semi-analytically for some few selected cases of interest in which materials like acetal plastic and glass fibre enter. A discovered coupling between the material and geometrical non-linearities involved in the problem divides the general solution into two regions separated by one in which there is no coupling at all and where it is expected to exist an elastic critical behaviour.

The cantilevered beam: an analytical solution for general deflections of linear-elastic materials

The deflections of a cantilevered beam made of a linear-elastic material under the influence of an external vertical concentrated force at the free end are analysed in detail and it is found that a factor that is always ignored in the theory commonly developed in the literature concerned permits the solution of the elastic curve to be obtained analytically. I introduce this analytical solution and show that the widely known simple solution for small deflections is a limiting case, which holds when forces applied are much smaller than the force needed for breaking the beam. A simple relation for an upper bound to this latter force is obtained which may be of practical importance for engineering. Finally, the most simple results obtained theoretically are compared with experiments in the laboratory.

Experimental investigation of local damage in high strength concrete columns using a shaking table

In this paper the accumulation of local damage during the cyclic loading in reinforced high-strength concrete columns is experimentally investigated. Two identical column specimens with annular cross-section and spiral reinforcement were designed and two tests, up to failure, under the action of a constant vertical concentrated force and a time-dependent concentrated horizontal force, were carried out at the LNEC shaking tables facility. Sine type signals, controlled in amplitude, frequency and time duration were used for these experiments. The concept of local damage based on local stiffness degradation is considered in detail and illustrated by experimental results. The specimens were designed and reinforced in such a way that the accumulation of damage was predicted by dominating deformations (cracking and crushing of the concrete) while the increasing of the loading values was a dominating factor of damage. It was observed that the local damage of HSC columns has exposed their anisotropic local behaviour. The damage accumulation was slightly different from the expected in accordance with the continuum damage concept, and a partial random character was observed.

DAMAGE OF HIGH-STRENGTH REINFORCED CONCRETE COLUMNS UNDER HORIZONTAL AND VERTICAL LOADING

The global damage of the high-strength reinforced concrete columns is investigated experimentally. Two identical specimens with annular cross-section and spiral containment were designed and two tests up to failure for the both specimens were carried out at the LNEC shaking table facilities. The columns worked as plane cantilever beams under the action of the constant vertical concentrated force and time-dependent concentrated horizontal force. The global damage concept is based on stiffness degradation. A variation of the observed horizontal displacements and forces during experiments as well as evolution of damage are presented and considered in details. The results show that increase of loading values is dominating factor of the global damage.

A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part I, development of governing equations

This two-part contribution presents a novel and efficient method to analyze the two-dimensional (2-D) electromechanical fields of a piezoelectric layer bonded to an elastic substrate, which takes into account the fully coupled electromechanical behavior. In Part I, Hellinger–Reissner variational principle for elasticity is extended to electromechanical problems of the bimaterial, and is utilized to obtain the governing equations for the problems concerned. The 2-D electromechanical field quantities in the piezoelectric layer are expanded in the thickness-coordinate with seven one-dimensional (1-D) unknown functions. Such an expansion satisfies exactly the mechanical equilibrium equations, Gauss law, the constitutive equations, two of the three displacement–strain relations as well as one of the two electric field-electric potential relations. For the substrate the fundamental solutions of a half-plane subjected to a vertical or horizontal concentrated force on the surface are used. Two differential equations and two singular integro-differential equations of four unknown functions, the axial force, N, the moment, M, the average and the first moment of electric displacement, D 0 and D 1, as well as the associated boundary conditions have been derived rigorously from the stationary conditions of Hellinger–Reissner variational functional. In contrast to the thin film/substrate theory that ignores the interfacial normal stress the present one can predict both the interfacial shear and normal stresses, the latter one is believed to control the delamination initiation.

Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space

We present the solutions for displacements and stresses subjected to a vertical point load in a continuously inhomogeneous transversely isotropic half-space with Young’s and shear moduli varying exponentially with depth. Planes of transverse isotropy are assumed to be parallel to the horizontal surface. The solutions for the half-space are obtained by superposing the solutions of two full spaces, one with a point load in its interior and the other with opposite traction of the first full space along the z ¼ 0 plane. The Hankel transform in a cylindrical co-ordinate system is employed for deriving the solutions. However, the resulting integrals for displacements and stresses involve polynomial, exponential function, and Bessel function that cannot be given in closed form; hence, numerical techniques are adopted in this work. In order to check the accuracy of numerical procedures, the comparisons are carried out with the homogeneous solutions of Liao and Wang, and the calculated results agree with those to nine decimal places. Furthermore, two illustrative examples are presented to elucidate the effect of inhomogeneity, and the type and degree of rock anisotropy on the vertical surface displacement and vertical normal stress in the inhomogeneous isotropic/ transversely isotropic rocks subjected to a vertical concentrated force acting on the surface. The calculated results show that the induced displacement and stress are decisively influenced by the inhomogeneity, and the degree and type of material anisotropy. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice. r 2003 Elsevier Ltd. All rights reserved.

DAMAGE OF HIGH-STRENGTH REINFORCED CONCRETE COLUMNS UNDER HORIZONTAL AND VERTICAL LOADING

The global damage of the high-strength reinforced concrete columns is investigated experimentally. Two identical specimens with annular cross-section and spiral containment were designed and two tests up to failure for the both specimens were carried out at the LNEC shaking table facilities. The columns worked as plane cantilever beams under the action of the constant vertical concentrated force and time-dependent concentrated horizontal force. The global damage concept is based on stiffness degradation. A variation of the observed horizontal displacements and forces during experiments as well as evolution of damage are presented and considered in details. The results show that increase of loading values is dominating factor of the global damage. First Published Online: 26 Jul 2012

Elastic solutions for a transversely isotropic half-space subjected to a point load

We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the effect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite different for the displacement and stress calculated from isotropic solutions. © 1998 John Wiley & Sons, Ltd.