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Showing papers on "Bending moment published in 2010"


Journal ArticleDOI
TL;DR: In this article, the effects of large deflection, material distribution, velocity of the moving load and excitation frequency on the beam displacements, bending moments and stresses have been examined in detail.

201 citations


Journal ArticleDOI
TL;DR: In this paper, the authors apply the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach to derive the new equilibrium conditions, domain governing differential equation and boundary conditions for bending of nanobeams.
Abstract: 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, domain 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 conditions 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.

148 citations


Journal ArticleDOI
TL;DR: In this paper, an experimental and analytical program was designed and conducted to evaluate the magnitude and distribution of seismically induced lateral earth pressures on cantilever retaining structures with dry medium dense sand backfill.
Abstract: An experimental and analytical program was designed and conducted to evaluate the magnitude and distribution of seismically induced lateral earth pressures on cantilever retaining structures with dry medium dense sand backfill. Results from two sets of dynamic centrifuge experiments and two-dimensional nonlinear finite-element analyses show that maximum dynamic earth pressures monotonically increase with depth and can be reasonably approximated by a triangular distribution. Moreover, dynamic earth pressures and inertia forces do not act simultaneously on the cantilever retaining walls. As a result, designing cantilever retaining walls for maximum dynamic earth pressure increment and maximum wall inertia, as is the current practice, is overly conservative and does not reflect the true seismic response of the wall-backfill system. The relationship between the seismic earth pressure increment coefficient (ΔK AE ) at the time of maximum overall wall moment and peak ground acceleration obtained from our experiments suggests that seismic earth pressures on cantilever retaining walls can be neglected at accelerations below 0.4 g. This finding is consistent with the observed good seismic performance of conventionally designed cantilever retaining structures.

145 citations


Journal ArticleDOI
TL;DR: In this article, a simplified method for evaluating the moment carrying of a segmental tunnel liner was proposed using a result from a FEM analysis in which parameters were obtained by calibration against a true scale model test.

133 citations


Journal ArticleDOI
TL;DR: In this article, three-dimensional finite element analyses were performed to study the behavior of piles in sloping ground under undrained lateral loading conditions, and analytical formulations were derived for the ultimate load per unit length and the initial stiffness of hyperbolic p-y curves.
Abstract: Three-dimensional finite element analyses were performed to study the behavior of piles in sloping ground under undrained lateral loading conditions. Piles of different diameter and length in sloping cohesive soils of different undrained shear strength and several ground slopes were considered. Based on the results of the finite element analyses, analytical formulations are derived for the ultimate load per unit length and the initial stiffness of hyperbolic p-y curves. New p-y criteria for static loading of piles in clay are proposed, which take into account the inclination of the slope and the adhesion of the pile-slope interface. These curves are used through a commercial subgrade reaction computer code to parametrically analyze the effect of slope inclination and pile adhesion on lateral displacements and bending moments. To validate the proposed p-y curves, a number of well documented lateral load tests are analyzed. Remarkable agreement is obtained between predicted and measured responses for a wide range of soil undrained shear strength and pile diameter, length, and stiffness.

120 citations


DOI
30 Nov 2010
TL;DR: In this paper, an evolutionary spread plasticity macro-element has been developed and static pushover procedures for individual masonry walls, as well as entire buildings, are presented and discussed in detail.
Abstract: Non-linear analysis is the most viable tool to get accurate predictions of the actual response of masonry structures under earthquake loading. Analytical methods based on the idealisation of masonry walls with openings as systems of macro-elements allow not only to capture the main failure modes observed after past earthquakes, but also to ensure a limited computational demand in engineering practice. The present thesis deals with non-linear incremental static (pushover) analysis on masonry buildings modelled through evolutionary spread plasticity macro-elements. In the first part of the work, fundamentals of structural analysis, seismic risk, and performance-based seismic design are reviewed along with non-linear static procedures and macro-element methods for seismic analysis of masonry buildings. The second part of the work deals with theoretical advances in non-linear seismic analysis of masonry buildings. In particular, an evolutionary spread plasticity macro-element has been developed and static pushover procedures for individual masonry walls, as well as entire buildings, are presented and discussed in detail. The proposed macro-element has been defined as ‘evolutionary’ because its inner reacting domain changes as the lateral drift demand increases. Such an evolution is caused by the spreading of cracking and yielding within the masonry. Namely, tensile cracking of masonry induces significant reductions in the effective width of cross-sections, which then depends on the magnitude of the given lateral drift (geometrical non-linearity). Yielding of masonry develops near the extreme parts of the macro-element, which are subjected to maximum bending moment. Mechanical non-linearity of masonry in compression is also taken into account through a deformation-based approach. In fact, the mechanical behaviour of the macro-element is characterised for different constitutive laws of masonry, by means of (1) two- and three-dimensional flexural strength domains (to be coupled with classical strength domains), (2) moment-curvature relationships, and (3) force-displacement diagrams. Two-dimensional strength domains derived from strength degrading constitutive models allow to account for more real characteristics of masonry. The comparison between flexural strength domains corresponding to different constitutive models have shown that current simplified formulas lead to higher values of ultimate shear force and bending moment, if the given axial force is not significantly higher than one-half of the allowable axial force. The implementation of full non-linear stress-strain relationships (which have been obtained by recent uniaxial compression tests in the direction orthogonal to mortar bed joints of masonry) has been found to provide more conservative estimations of ultimate shear force corresponding to flexural failure of macro-elements (i.e., toe crushing). Flexural strength domains have been defined at cracking, elastic, and ultimate limit states of masonry cross-sections and panels, in both cracked and uncracked conditions. The explicit consideration of strain ductility of masonry has allowed to assess the evolution in strength domains. In this regard, less significant variations in both ultimate shear force and bending moment have been detected for a given axial force lower than one-fourth of the allowable axial force. Finally, flexural strength of macro-elements has been also investigated for any boundary condition through the development of three-dimensional strength domains corresponding to elastic and ultimate limit states, in both cracked and uncracked conditions. Moment-curvature relationships have been defined for rectangular unreinforced masonry cross-sections by means of an incremental iterative procedure. Their development has allowed to assess key parameters of sectional behaviour, such as flexural overstrength, strength degradation due to strain softening of masonry, yielding and ultimate curvatures, and curvature ductility. Such parameters have been estimated for a number of constitutive laws and the relationship between curvature and strain ductilities has been also investigated. The implementation of empirical stress-strain relationships presented in the first appendix of this thesis has let to derive moment-curvature relationships where both yielding and ultimate strains of masonry, as well as strain softening, are explicitly considered. It has been found that, if the applied axial force does not exceed one-half of the allowable axial force, the ratio between curvature ductility and strain ductility is greater than unity and does not depend on the magnitude of the applied axial force. In order to define force-displacement diagrams of macro-elements, a specific incremental iterative procedure has been developed; it is based on the monitoring of the maximum axial strain over the cross-section(s) subjected to the maximum bending moment. It has been shown that force-controlled procedures can lead to significant underestimations of displacement capacity of masonry panels and to underestimations of both lateral stiffness and maximum resisting shear force. Force-based pushover procedures in response control have been developed for individual walls with openings and entire masonry buildings, separately, in order to predict also their non-linear softened response. Such procedures have been implemented in a novel computer program named RAN CODE, which is specifically devoted to structural analysis of masonry buildings. Amongst several numerical applications aimed at validating the developed procedures, the outcomes of a series of global pushover analyses on a masonry building designed in compliance with Eurocode 8 and Italian building code are discussed. The pushover procedure developed for single masonry walls with openings could be employed in the case of existing buildings (either single buildings or building units within aggregates), which have often flexible floor diaphragms and lacking, or poor, connections between diaphragms and walls, as well as between orthogonal walls. The procedure developed for global pushover analysis of masonry buildings accounts for torsional effects due to both inherent (i.e., structural) and accidental eccentricities between centres of mass and centres of stiffness. The use of spread plasticity macro-elements which change with the given deformation state allows to relate the ‘local’ response of masonry panels to the ‘global’ response of the structure. Finally, three appendices include results and empirical models obtained through experimental programs aimed at supporting non-linear modelling and analysis of masonry buildings. Appendix A deals with mechanical characterisation of masonry under uniaxial compression along directions parallel and orthogonal to mortar bed joints. Such a characterisation is consisted in the definition of mechanical parameters and constitutive models able to simulate non-linear behaviour of masonry up to large inelastic strains. Appendix B deals with mechanical characterisation of masonry in sliding shear along mortar bed joints. Also in this case, both classical and advanced mechanical parameters have been defined and empirical models have been derived. Such models include shear stress versus shear strain relationships and a shear response surface, which allows to simulate non-linear shear behaviour of unit-mortar interfaces over the whole range of allowable strains (that is, from elastic to inelastic range). Appendix C summarises the main results of three quasi-static lateral loading tests on a full-scale masonry wall with a opening and no tensile-resistant elements (e.g., reinforced concrete bond beams, steel ties), which is the typical case of existing masonry buildings. Namely, the first monotonic test allowed to investigate non-linear behaviour of the wall up to the first significant damage to the spandrel panel above the opening. The second test was carried out on the pre-damaged wall under cyclic displacements, in order to assess residual properties and to reach some hints on seismic performance of masonry walls subjected to earthquake sequences. The last test was performed under cyclic displacements on the wall after repairing and upgrading of the spandrel panel with an inorganic matrix-grid composite system. The aim of that test was to assess the effectiveness of the strengthening system for seismic retrofit of masonry structures and rapid remedial works during seismic emergency scenarios. Data processing for all lateral loading tests has shown that the damage to the spandrel panel affected both load-carrying capacity and strength degradation of the wall, whereas rocking behaviour of piers produced large displacement capacity and low residual drifts (that is, high re-centring capacity).

117 citations


Journal ArticleDOI
TL;DR: In this paper, an analytic model is developed to investigate the wave propagation and sound transmission characteristics of an infinite sandwich structure reinforced by two sets of orthogonal rib-stiffeners when subjected to convective fluid-loaded pressure.
Abstract: An analytic model is developed to investigate the wave propagation and sound transmission characteristics of an infinite sandwich structure reinforced by two sets of orthogonal rib-stiffeners when subjected to convective fluid-loaded pressure. The rib-stiffeners are assumed to be identical and uniformly spaced, which can exert not only tensional forces and bending moments but also torsional moments on the facesheets. Inertial terms of the tensional forces, bending moments and torsional moments are introduced to account for inertial effects arising from the mass of the rib-stiffeners. With the surrounding acoustic fluids restricted by the acoustic wave equation, fluid–structure coupling is considered by imposing velocity continuity condition at fluid–panel interfaces. By applying the Bloch theorem for periodic structures, the structural and acoustic responses are expressed in a superposition form of space harmonics for a given wavenumber. The application of the virtual work principle for one periodic element yields two infinite sets of simultaneous algebraic coupled equations, which are numerically solved by truncating them in a finite range insofar as the solution converges. The validity and feasibility of the analytic model is qualified by comparing model predictions with existing results, in which the necessity and advantage of the exact modeling of rib-stiffener motions are also demonstrated. Specifically, the influences of inertial effects arising from rib-stiffener mass, the periodicity spacing of rib-stiffeners, and the airborne as well as structure-borne paths on the transmission of sound across the sandwich structure are quantified and conclusions of significant practical implications are drawn.

113 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of tunneling beneath buried pipelines and the relationship between soil strains and pipeline bending behavior was examined. But, the authors focused on the tunnel deformation rather than the pipe bending behavior.
Abstract: The paper examines the problem of tunneling beneath buried pipelines and the relationship between soil strains and pipeline bending behavior. Data are presented from centrifuge tests in which tunnel volume loss was induced in sand beneath pipelines of varying stiffness properties. The model tunnel and pipelines were all placed at a Perspex wall of the centrifuge strong box such that image-based deformation analyses could be performed. The method provided detailed data of subsurface soil and pipe displacements and illustrated the soil-pipe interaction mechanisms that occurred during tunnel volume loss, including the formation of a gap beneath the pipes. The relationship between tunnel volume loss, soil strain, and pipe bending behavior is illustrated. Experimental results of pipe bending moments are compared against predictions: (1) assuming the pipe simply follows greenfield displacements; (2) using an elastic continuum solution; and (3) using a new method in which an "out-of-plane" shear argument, due to soil-pipe interaction, is introduced into the elastic continuum solution. It is shown that the new method gives the best prediction of experimental pipe bending moments.

108 citations


Journal ArticleDOI
TL;DR: In this paper, the higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties, and a general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions.
Abstract: The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.

104 citations


Journal ArticleDOI
TL;DR: In this article, a model of a thin elastic plate subjected to bending and tension is used to describe the bending moment and the distribution of membrane forces, and the stability of the plate is investigated with the help of an analytical approach.

92 citations


Journal ArticleDOI
TL;DR: In this paper, the structural behavior of an innovative hybrid fiber reinforced polymers (FRP) beam consisting of carbon/glass fibers and vinyl-ester resin was investigated under four-point bending varying ratio of flange to web width and volume content of carbon and glass fiber in the flanges.

Journal ArticleDOI
TL;DR: In this paper, a new roll-based process and machine for three-dimensional bending of profiles with symmetrical and asymmetrical cross-sections have been developed, where a torque is superposed to the bending moment to define the spatial geometry of the workpiece.

Journal ArticleDOI
15 Aug 2010-Spine
TL;DR: This study confirms the importance of immediate in-brace correction to predict long-term outcome of the treatment and provides insights in the understanding of brace biomechanics.
Abstract: Study design Multiple brace designs were simulated using a finite element model and their biomechanical effect was evaluated. Objective To study correlations between immediate in-brace correction of coronal curves and bending moments acting on the apical vertebrae. Summary of background data Immediate in-brace correction has often been deemed as fundamental to long-term brace effect but the biomechanical explanation is unclear. Methods Three-dimensional geometry of 3 patients was acquired using multiview radiographs and surface topography techniques. A finite element model of the patients' trunk including gravitational forces and a parametric brace model were created. Two sets of mechanical properties of the spine (stiff and flexible) were tested. Installation of the brace on the patients was simulated. Using an experimental design framework including fourteen design factors, 1024 different virtual braces were tested for each patient. For each brace, immediate in-brace correction of the coronal Cobb angles and the bending moment acting on the apical vertebrae were computed and their correlation was studied. Results Immediate correction of coronal curves and corresponding impact on the apical vertebrae bending moments were linearly correlated (mean R = 0.88). The amount of immediate correction necessary to nullify the bending moment ranged between 19% and 61% with average 48% (flexible spine model) and 27% (stiff spine model). The braces corrected the apical vertebrae bending moment more in the flexible spine model. In the framework of the Hueter-Volkmann principle, the correlation between coronal immediate in-brace correction and corresponding apical bending moment can be interpreted as a correlation between immediate in-brace correction and long-term treatment outcome. The amount of immediate correction necessary to invert the bending moments, and in theory counteract the progression of the scoliotic deformity, depends on spine stiffness and spine segment. Conclusion This study confirms the importance of immediate in-brace correction to predict long-term outcome of the treatment and provides insights in the understanding of brace biomechanics.

Journal ArticleDOI
TL;DR: In this paper, the dynamics of a single-walled carbon nanotubes (SWCNT) subjected to a moving nanoparticle is examined in the framework of the nonlocal continuum theory of Eringen.
Abstract: Single-walled carbon nanotubes (SWCNTs) can be promising delivery nanodevices for a diverse range of applications, however, little is known about their dynamical interactions with moving nanoscale particles. In this paper, dynamic response of a SWCNT subjected to a moving nanoparticle is examined in the framework of the nonlocal continuum theory of Eringen. The inertial effects of the moving nanoparticle and the existing friction between the nanoparticle surface and the inner surface of the SWCNT are incorporated in the formulation of the problem. The equivalent continuum structure associated with the SWCNT is considered and modeled using nonlocal Rayleigh beam theory under simply supported boundary conditions. The governing equations are then established both in the strong and weak forms. The set of linear equations are solved in the time domain using generalized Newmark-β method. The effects of mass weight of the moving nanoparticle, its velocity, and small scale effect parameter on the dynamic amplitude factors of longitudinal and transverse displacements as well as those of axial force and bending moment are studied in some detail. Additionally, the possibility of moving nanoparticle separation from the inner surface of the SWCNT is investigated. The role of influential parameters on the possibility of this phenomenon is also addressed and discussed.

Journal ArticleDOI
TL;DR: In this article, the effect of an increase in aspect ratio with respect to several behaviors was analyzed, including the significance of the level of shear on the failure modes and its effect on the torsional and bending strength under combined loadings.

Journal ArticleDOI
TL;DR: In this paper, a theoretical solution is obtained to predict the dynamic response of peripherally clamped square metallic sandwich panels with either honeycomb core or aluminium foam core under blast loading, where the deformation of sandwich structures is separated into three phases, corresponding to the transfer of impulse to the front face velocity, core crushing and overall structural bending/stretching, respectively.

Journal ArticleDOI
TL;DR: In this article, the kinematic seismic interaction of single piles embedded in soil deposits is evaluated by focusing the attention on the bending moments induced by the transient motion, which is performed by modeling the pile like an Euler-Bernoulli beam embedded in a layered Winkler type medium.

Journal ArticleDOI
TL;DR: In this paper, the one-to-one correspondence between the R-curve and the softening curve is revisited and adapted for any kind of specimen geometry and for the bilinear approximation of the softness function, well-known to successfully describe the failure of a wide group of quasibrittle materials used in Civil Engineering such as concrete or wood.

Journal ArticleDOI
TL;DR: In this article, the effect of sheet pile wall construction type for varying soil conditions and wall heights were investigated using finite element modeling and analysis, and the results indicated that walls constructed by backfill method yield significantly higher bending moments and wall deformations.

Journal ArticleDOI
TL;DR: In this paper, a new method of determining bending moments at the overlap ends of single lap joints is proposed, based on the assumption that the overlap region does not deform under load.

Journal ArticleDOI
TL;DR: In this paper, the authors reported the results of an investigation into the characteristics of catenary action and their effect on structures and proposed novel retrofitting schemes to enhance the survival capability of the steel framed structures subjected to a terrorist blast.


Journal ArticleDOI
TL;DR: In this paper, the authors presented the response of piles in liquefiable soil under seismic loads and the effects of soil, pile, and earthquake parameters on the two potential pile failure mechanisms, bending and buckling, are examined.
Abstract: This paper presents the response of piles in liquefiable soil under seismic loads. The effects of soil, pile, and earthquake parameters on the two potential pile failure mechanisms, bending and buckling, are examined. The analysis is conducted using a two-dimensional plain strain finite difference program considering a nonlinear constitutive model for soil liquefaction, strength reduction, and pile-soil interaction. The depths of liquefaction, maximum lateral displacement, and maximum pile bending moment are obtained for concrete and steel piles for different soil relative densities, pile diameters, earthquake predominant frequencies, and peak accelerations. The potential failure mechanisms of piles identified from the parametric analysis are discussed.

Journal ArticleDOI
TL;DR: In this paper, a kinematic rigid-plastic homogenization model for the limit analysis of masonry walls arranged in random texture and out-of-plane loaded is proposed.

Journal ArticleDOI
TL;DR: In this paper, a finite element analysis with the nonlinear isotropic/kinematic (combined) hardening model has been used to evaluate ratchetting behavior of the cylinder under mentioned loading condition.

Journal ArticleDOI
TL;DR: In this article, an experimental apparatus was developed to investigate the behavior of vertically loaded free-head piles in sand undergoing lateral soil movement (wf), and a large number of tests have been conducted.
Abstract: An experimental apparatus was developed to investigate the behaviour of vertically loaded free-head piles in sand undergoing lateral soil movement (wf). A large number of tests have been conducted ...

Journal ArticleDOI
TL;DR: In this article, the deformation of a photodeformable film material is studied based as regards the constitutive relations and a model for the deflection of the laminated sample upon the irradiation of ultraviolet (UV) light is established and the bending moment is deduced.
Abstract: The deformation of a photodeformable film material is studied based as regards the constitutive relations. In the theoretical analysis, a model for the deflection of the laminated sample upon the irradiation of ultraviolet (UV) light is established and the bending moment is deduced. Furthermore, the deflection of the film can be calculated using finite-element analysis (FEA) software. Then the attempt to utilize the superior characteristics including shape memory and large internal force on the application of a micropump is undertaken in our study. The force produced in the reciprocating deformation of the film sample is of potential to activate the pump membrane. A set of experimental devices are designed to test the performance of the membrane in the experiment. The flow volume in a stroke is close to simulation result obtained by FEA.

Journal ArticleDOI
TL;DR: In this article, the structural response of post-tensioned prestressed beams with unbonded tendons, including both the deflections under service loading, before or after cracking, and the ultimate bearing capacity are considered.

Journal ArticleDOI
TL;DR: In this paper, the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series, were derived.
Abstract: This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.

Journal ArticleDOI
TL;DR: In this article, the bending angle suffered by the block is specified instead of the usual applied moment, and the general moment-bending angle relationship is then obtained and is shown to be dependent on only one nondimensional parameter: the product of the aspect ratio of the block and bending angle.
Abstract: The classical flexure problem of nonlinear incompressible elasticity is revisited assuming that the bending angle suffered by the block is specified instead of the usual applied moment. The general moment-bending angle relationship is then obtained and is shown to be dependent on only one nondimensional parameter: the product of the aspect ratio of the block and the bending angle. A Maclaurin series expansion in this parameter is then found. The first-order term is proportional to , the shear modulus of linear elasticity; the second-order term is identically zero because the moment is an odd function of the angle; and the third-order term is proportional to 41, where is the nonlinear shear coefficient, involving third-order and fourth-order elasticity constants. It follows that bending experiments provide an alternative way of estimating this coefficient and the results of one such experiment are presented. In passing, the coefficients of Rivlin’s expansion in exact nonlinear elasticity are connected to those of Landau in weakly (fourth-order) nonlinear elasticity. DOI: 10.1115/1.4001282