The Effects of the Longitudinal Axis of Loading upon Bending, Shear and Torsion of a Thin-Walled Cantilever Channel Beam

Three aluminium channel sections of US standard extruded dimension are mounted as cantilevers with x-axis symmetry. The flexural bending and shear that arise with applied axial torsion are each considered theoretically and numerically in terms of two longitudinal axes of loading not coincident with the shear centre. In particular, the warping displacements, stiffness and stress distributions are calculated for torsion applied to longitudinal axes passing through the section’s centroid and its web centre. The stress conversions derived from each action are superimposed to reveal a net sectional stress distribution. Therein, the influence of the axis position upon the net axial and shear stress distributions is established compared to previous results for each beam when loading is referred to a flexural axis through the shear centre. Within the net stress analysis is, it is shown how the constraint to free warping presented by the end fixing modifies the axial stress. The latter can be identified with the action of a ‘bimoment’ upon each thin-walled section.

Exploring the Limits of a Deflection-Based Method for the Estimation of Dynamic Stress in Beams

Estimation of the dynamic stress in structures,such as beams and plates,has previously been made using the relationship between stress and velocity spatial maxima based on far-field assumptions.This paper presents a method for the estimation of dynamic stress in a beam using Euler–Bernoulli beam theory,where deflection data from a grid of measurement points on the surface of the beam is used to estimate the dynamic bending stress in the structure.The limitations of the method are investigated via response data provided by a numerical model of a freefree beam.A nondimensional wavenumber analysis is used to determine the number of points required for an accurate estimate of stress.Beams with a range of material and geometric parameters are modeled in order to explore the limits of the estimation method,and parameters representative of several real-world materials are used to assess the suitability of the method for practical applications.

Flapwise Bending Vibration Analysis of Rotating Tapered Rayleigh Beams for the Application of Offshore Wind Turbine Blades

The flapwise bending vibrational equations of tapered Rayleigh beam are derived based on Hamilton’s principle.The corresponding vibrational characteristics of rotating tapered Rayleigh beams are investigated via variational iteration method(VIM).Natural frequencies and corresponding mode shapes are examined under various rotation speed,taper ratio and slenderness ratio focusing on two types of tapered beam.The convergence of VIM is examined as part of the paper.Validation of VIM solution is made by referring to results available in other literature and corresponding results show that VIM is capable of yielding precise results in a very efficient way.

Behavior of interfacial stresses between RC beams and GFRP sheets

Seven reinforced concrete （RC） beams with epoxy-bonded glass fiber reinforced plastic （GFRP） sheets and two control RC beams were experimentally tested to investigate the bond behavior of the interfaces between RC beams and GFRP sheets. The variable parameters considered in test beams are the layers of GFRP sheets, the bond lengths and the reinforcement ratios. The results indicate that the flexural strength of the repaired beams is increased, but the ultimate load of beams with GFRP sheets debonding failure is reduced relatively. The bond length is the main factor that results in bonding failure of the strengthened beams. An experimental method of interfacial shear stress is proposed to analyze the distribution of shear stress according to experimental results. The analytical method of shear and normal stresses and a simple equation are proposed to predict the peeling loads. The proposed model is applied to experimental beams. The analytical results show a good agreement with the experimental results.

Comparison study on calculation methods of bending behavior of Chinese reinforced concrete beams from 1912 to 1949

In order to study the calculation methods of bending behavior of Chinese reinforced concrete beams from 1912 to 1949, tests on the mechanical performance of 66 rebars from different modem Chinese concrete buildings, the concrete compressive strength of 12 modem Chinese concrete buildings, and the concrete cover thickness of 9 modem Chinese concrete buildings are carried out; and the actual material properties and structural conformations of modem Chinese concrete buildings are obtained. Then, the comparison on calculation methods of bending behavior including the original Chinese calculation method, the present Chinese calculation method, the present American calculation method and the present European calculation method is studied. The results show that the original Chinese calculation method of bending behavior is based on the allowable stress calculation method, and the design safety factors are 3.55 to 4. 0. In term of the calculation area of longitudinal rebars of reinforced concrete beams, without considering earthquake action, the original Chinese structural calculation method is safer than the present Chinese structural calculation method, the present European structural calculation method, and the present American structural calculation method. The results can provide support for the structural safety assessments of modem Chinese reinforced concrete buildings.

Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model

The bending responses of functionally graded （FG） nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index, and surface stress on dimensionless deflection are discussed in detail.

Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section

The bending and free vibrational behaviors of functionally graded(FG)cylindrical beams with radially and axially varying material inhomogeneities are investigated.Based on a high-order cylindrical beam model,where the shear deformation and rotary inertia are both considered,the two coupled governing differential motion equations for the deflection and rotation are established.The analytical bending solutions for various boundary conditions are derived.In the vibrational analysis of FG cylindrical beams,the two governing equations are firstly changed to a single equation by means of an auxiliary function,and then the vibration mode is expanded into shifted Chebyshev polynomials.Numerical examples are given to investigate the effects of the material gradient indices on the deflections,the stress distributions,and the eigenfrequencies of the cylindrical beams,respectively.By comparing the obtained numerical results with those obtained by the three-dimensional(3D)elasticity theory and the Timoshenko beam theory,the effectiveness of the present approach is verified.

Theoretical Prediction of the Bending Stiffness of Reinforced Thermoplastic Pipes Using a Homogenization Method

The accurate prediction of bending stiffness is important to analyze the buckling and vibration behavior of reinforced thermoplastic pipes(RTPs)in practical ocean engineering.In this study,a theoretical method in which the constitutive relationships between orthotropic and isotropic materials are unified under the global cylindrical coordinate system is proposed to predict the bending stiffness of RTPs.Then,the homogenization assumption is used to replace the multilayered cross-sections of RTPs with homogenized ones.Different from present studies,the pure bending case of homogenized RTPs is analyzed,considering homogenized RTPs as hollow cylindrical beams instead of using the stress functions proposed by Lekhnitskii.Therefore,the bending stiffness of RTPs can be determined by solving the homogenized axial elastic moduli and moment of inertia of cross sections.Compared with the existing theoretical method,the homogenization method is more practical,universal,and computationally stable.Meanwhile,the pure bending case of RTPs was simulated to verify the homogenization method via conducting ABAQUS Explicit quasi-static analyses.Compared with the numerical and existing theoretical methods,the homogenization method more accurately predicts the bending stiffness and stress field.The stress field of RTPs and the effect of winding angles are also discussed.

Analysis of bending and buckling of pre-twisted beams:A bioinspired study

Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.

Analysis of Stresses and Deflection of Sun Gear by Theoretical and ANSYS Method

Gearing is one of the most critical components in mechanical power transmission systems. This article examines the various stresses and deflection developed in sun gear tooth of planetary gearbox which is used in Grabbing Crane. Article includes checking sun gear wear stresses and bending stresses using IS 4460 equations. Also calculate various forces acting on gear tooth. In this study, perform the calculation for sun gear tooth to calculate bending, shear, wear & deflection using theoretical method. 3D model is created of circular root fillet & trochoidal root fillet of gear tooth for simulation using ProE Wildfire 3. In Pro-E, the geometry is saved as a file and then it is transferred from Pro-E to ANSYS 10 in IGES format. The results of the 3 D analyses from ANSYS are compared with the theoretical values. Comparison of ANSYS results in circular root fillet & trochoidal root fillet also carry out.

On Applications of Generalized Functions in the Discontinuous Beam Bending Differential Equations

This paper discusses the mathematical modeling for the mechanics of solid using the distribution theory of Schwartz to the beam bending differential Equations. This problem is solved by the use of generalized functions, among which is the well known Dirac delta function. The governing differential Equation is Euler-Bernoulli beams with jump discontinuities on displacements and rotations. Also, the governing differential Equations of a Timoshenko beam with jump discontinuities in slope, deflection, flexural stiffness, and shear stiffness are obtained in the space of generalized functions. The operator of one of the governing differential Equations changes so that for both Equations the Dirac Delta function and its first distributional derivative appear in the new force terms as we present the same in a Euler-Bernoulli beam. Examples are provided to illustrate the abstract theory. This research is useful to Mechanical Engineering, Ocean Engineering, Civil Engineering, and Aerospace Engineering.

Effect of ion-beam assisted deposition on the film stresses of TiO_2 and SiO_2 and stress control

Based on Hartmann-Shack sensor technique, an online thin film stress measuring system was introduced to measure the film stresses of TiO2 and SiO2, and comparison was made between the film stresses prepared respectively by the conventional process and the ion-beam assisted deposition. The effect of ion-beam assisted deposition on the film stresses of TiO2 and SiO2 was investigated in details, and the stress control methodologies using on-line adjustment and film doping were put forward. The results show that the film stress value of TiO2 prepared by ion-beam assisted deposition is 40 MPa lower than that prepared by conventional process, and the stress of TiO2 film changes gradually from tensile stress into compressive stress with increasing ion energy; while the film stress of SiO2 is a tensile stress under ion-beam assisted deposition because of the ion-beam sputtering effect, and the film refractive index decreases with increasing ion energy. A dynamic film stress control can be achieved through in-situ adjustment of the processing parameters based on the online film stress measuring technique, and the intrinsic stress of film can be effectively changed through film doping.

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory:equilibrium,governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.

Effect of Grinding and Polishing on the Residual Stress and Bending Strength of a Silicon Nitride Ceramic

The residual stresses on the surface of the d ifferently ground and polished silicon nitride ceramics were measured using X-r ay diffraction and identified by SEM.The effect of the residual stress on the be nding strength was investigated.The investigations show that the grinding proces s can introduce subatantial tensile residual stresses up to 290MPa on the surfac e of silicon nitride ceramics,which has a significant effect on reducing the ben ding strength of the ceramics after grinding.Thus,in comparison with the ceramic s with a rough surface,the ceramics with a mirror image surface may have a lower strength.Polishing can smooth the residual stresses.When we evaluate the qualit y of the ceramic components after grinding,we must take residual stress into con sideration. The grinding methods and grinding conditions must be carefully selec ted in order to get the favorite residual stress as well as the surface smoothne ss.

BENDING THEORIES FOR BEAMS AND PLATES WITH SINGLE GENERALIZED DISPLACEMENT

Classical bending theories for beams and plates can not be used for short, stubby beams and thick plates since transverse shearing effect is excluded, and ordinary theories with multiple generalized displacements can not be used for long, slender beams and thin plates since the innate relation between rotation angle and deflection is ignored. These two types of theories are not consistent due to the contradiction of dependence and independence of the rotation angle. Based on several basic assumptions, a new type of theories which not only include the transverse shearing effect is presented, but also the relation between potation angle and deflection is obtained. Analytical solutions of several simple beams are given. It has been testified by numerical examples that the new theories can be used for either long, slender beams and thin plates or short, stubby beams and thick plates.

Fugue Effect on Bending of Nonlinear Viscoelastic Beams

Based on the Leaderman constitutive relations in nonlinear viscoelasticity and the linear geometrical assumption, a mathematical model for the bending of nonlinear viscoelastic beams was established in this paper. The Laplace transformation method and the Titchmarsh theorem were used to prove that some relations exist between solutions to bending problems of visco- and elastic beams, which reveals the fugue effect of viscoelastic materials. The high-order Galerkin approximate solution to the quasi-static response of nonlinear viscoelastic beams under a step load was obtained by using the new method suggested in this paper as well as the Mathematica software and the Newton iteration technique.

Fatigue tests of composite beam by steel fiber reinforced self-stressing concrete in the hogging bending

Through the experiments of 7 T-section composite beams, steel fiber reinforced self-stressing concrete （SFRSC） as the composite beam in the composite layer was studied under the hogging bending. The tests simulated composite layer tensile strain under the hogging bending of inverted loading composite beams, giving the relationship under the different fatigue stress ratios between fatigue cycles and steel bar’s stress range, crack width, stiffness loss and damage, etc., in composite layer. This article established fatigue life equation, and analyzed SFRSC reinforced mechanism to crack width and stiffness loss. The results show that SFRSC as the composite beam concrete has excellent properties of crack resistance and tensile, can reinforce the fatigue crack width and stiffness loss of composite beams, and improve the durability and in normal use of composite beams in the hogging bending zone.

BENDING OF CANTILEVER RECTANGULAR PLATES WITH THE EFFECT OF TRANSEVERSE SHEAR DEFORMATION

On the basis for Reissner's theory, the exact solutions of the bending of cantilever rectangular plates are obtained by means of the concept of generalized simply-supported boundary. From the results obtained, it can be found that the method is valid.

Bending of simply supported incompressible saturated poroelastic beams with axial diffusion

Based on the mathematical model of the bending of the incompressible saturated poroelastic beam with axial diffusion, the qUasi-static bendings of the simply supported poroelastic beam subjected to a suddenly applied constant load were investigated, and the analytical solutions were obtained for different diffusion conditions of the pore fluid at the beam ends. The deflections, the bending moments of the solid skeleton and the equivalent couples of the pore pressures were presented in figures. It is also shown that the behavior of the saturated poroelastic beams depends closely on the diffusion conditions at the beam ends, especially for the equivalent couples of the pore pressures. It is found that the Mandel-Cryer effect also exists in the bending of the saturated poroelastic beams under specific diffusion conditions at the beam ends.

Numerical Investigation on the Ultimate Strength of Box Beams with Impact Damage

The objective of this paper is to study the residual ultimate strength of box beams with impact-induced damage,as a model of what may occur in ship hulls.The bottom and side plates of ship hulls can suffer denting or fracture damage due to grounding,collision and other contacts during the ship’s service life and these impact-induced damages could result in considerable strength degradation.Box beams are firstly subjected to impact loading and then four-point bending loading is imposed on the damaged structures to assess the residual strength using ANSYS/LS_DYNA.The ultimate moment and collapse modes are discussed considering the effect of impact location.The impact-induced deformation is introduced in the four-point bending simulation,and the impact-induced stress is included or not to determine the effect of residual stress and distortion after impact.It is shown that impact location has significant influence on the residual ultimate bending moment of the damaged box beam providing that the impact energy is kept constant.The collapse modes also change when the impactor strikes on different locations.Damaged hard corner and inclined neutral axes might explain the reduction of ultimate strength and diverse collapse modes.The residual stress in the box beam after impact may increase or decrease the ultimate strength depending on impact location.