Non-prismatic Cantilever Beam Research Articles

A Finite Difference Scheme for the Large-Deflection Analysis of Non-Prismatic Cantilever Beams

A Finite Difference Scheme for the Large-Deflection Analysis of Non-Prismatic Cantilever Beams

Semi-Analytical Solution for Elastoplastic Deflection of Non-Prismatic Cantilever Beams with Circular Cross-Section

A solution for the elastoplastic deflection of cantilever beams with linearly variable circular cross-section loaded by shear force at the free end, which is suitable for practical use, has not yet been developed. A semi-analytical solution for such a problem is proposed in this paper. The solution involves beams made of homogenous and isotropic materials with bilinear elastoplastic strain hardening behavior. The Bernoulli–Euler formula is used for determining the elastic deflection. However, for the plastic domain of material behavior, the differential equation of beam bending does not have a solution in closed form. Therefore, an incremental procedure for determining the curvature of the plastified region of the beam is suggested. Deflection of the cantilever beam is calculated via integration of the approximated function of the beam curvature. The proposed semi-analytical solution is validated using experimental results of the seismic energy dissipation device components which have been selected as a sample of a real engineering system. Also, validation is done via finite element analysis of six different cantilever beam models with varying geometric and material characteristics. A satisfying agreement between the proposed semi-analytical results and the subsequent experimental and numerical results is herein achieved, confirming its reliability.

Modified bar simulation method for shear lag analysis of non-prismatic composite box girders with corrugated steel webs

In this study, a modified bar simulation method is proposed for analyzing the shear lag effect of non-prismatic composite box girders with corrugated steel webs (CBGCSWs) during the elastic stage. In this theoretical method, formulas for the equivalent area of stiffening bar and the shear flow in corrugated steel webs (CSWs) are derived by properly considering the mechanical properties of non-prismatic CBGCSWs. The governing differential equations for shear lag are formulated and solved with given boundary conditions. The feasibility of the proposed method is validated by the finite element (FE) method based on a non-prismatic cantilever beam under different loading conditions. In addition, the shear lag behavior of non-prismatic cantilever CBGCSWs is compared with that of corresponding concrete box girders, and the effects of width-span ratio, girder height ratio, changing curve of girder height and loading form on the shear lag effect of non-prismatic cantilever CBGCSWs are examined based on the modified bar simulation method. Results indicate that non-prismatic cantilever CBGCSWs exhibit more significant shear lag behavior than the corresponding concrete box girders under gravity. The distribution rule of shear flow in concrete flanges, which first increase and then decrease from the cantilever end to the fixed end, causes the transformation between negative and positive shear lag effects in non-prismatic cantilever beams. Increasing the width-span ratio or girder height ratio leads to more serious shear lag effect. The effect of loading form on the shear lag behavior is significant, while the changing curve of girder height has minimal influence.

Large Deflection Analysis of Prismatic Cantilever Beam Comparatively by Using Combing Method and Iterative DQM

There is no exactly analytical solution for the large deflection problem of prismatic cantilever beams under general loading conditions. In the case of considering a non-prismatic cantilever beam, the difficulty of the larger deflection problem is increased. In this study, the comparison of the Iterative Differential Quadrature Method (I-DQM) and the Combining Method (CM) was performed. Numerical solution of the large deflection problem was separately performed with both the I-DQM and the CM for prismatic cantilever beams. The obtaining results show that both of these methods gave more accurate solutions compared with a reliable semi-analytic method which was introduced by Dado and Sadder (2005). Besides, it was demonstrated that the I-DQM provided a more wide-range solution than the CM.

Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading

This study presents the elastica of non-prismatic cantilever beams with rectangular cross-sections that are subjected to combined loading. The considered beams are nonlinearly elastic and obey Ludwick’s constitutive law. The combined loading system used in this study provides uniform loading, tip point loading, and tip couple loading individually or in combination. This loading system can create a total of seven loading cases that have not been covered in the literature. Ordinary differential equations governing the large deformed shapes of the elastica of the beams are derived and solved numerically. The effects of beam parameters on elastica behavior, including tip responses and strains and stresses loaded onto the cross-sections, were studied. It was observed that the exponential constant of the mechanical properties is very sensitive to elastica behavior. The results also demonstrate that the stresses near the neutral axis are larger with a larger exponential constant.

Design of a Novel Carbon-Fiber Ankle-Foot Prosthetic using Finite Element Modeling

Recently, researchers worldwide have made considerable efforts to enhance amputees’ quality of life by designing improved prosthetic feet. The ideal prosthetic for this job is that one which seamlessly mimics the functions of the human foot. To achieve this objective, accurate design is thus required. In this paper, a novel design for a carbon-fibre ankle-foot prosthetic is considered. The geometry of the designed ankle-foot prosthetic was created to satisfy a wide range of anthropometric parameters, taking into consideration the available previous designs in the published literature. AutoCAD version 15 was used in this work for modeling and creating the geometry, and ANSYS Workbench software version 16.1 was used for the finite element modelling based on geometry retrieved from the AutoCAD software. The complexity of the suggested design was simplified by partitioning the design into parts according to function. The roll-over shape concept was considered in the current design, which acts as an important tool in the design, evaluation, and alignment of lower limb prostheses and orthoses. Carbon fibre-epoxy composite material was utilised to manufacture a light and high strength prosthetic foot. The current study revealed that the proposed design offered a smooth roll-over shape and good response to energy return requirements in ankle-foot prosthetics, with the keel and heel design behaving as a non-prismatic cantilever beam. The keel and heel thickness require optimisation based on the specific materials used. FE analysis showed successful heel and keel deflections, although inadequate deflection was demonstrated in the vertical loading test. The heel deflected by 29.18 mm under a load of 300 N, while the keel deflected more than 25 mm under 1,230 N, and the deflection under a vertical load of 1,230 N was about 8.1 mm. The strain and stress seen in the three tests were within safe limits, while most of the energy was absorbed in the ankle component of the prosthesis.

Static deformation modeling and analysis of flexure hinges made of a shape memory alloy

The flexure hinge is a key element in compliant mechanisms to achieve continuous motion; however the motion range of a flexure hinge is severely restricted by the material’s allowable strain. Due to the superelasticity effect, shape memory alloys (SMAs) can undergo much larger strain than other metals; this means that they are excellent candidates for the fabrication of flexure hinges with a large motion range. In this paper, a simple static deformation modeling approach is proposed for a flexure hinge made of a SMA. The superelastic behavior of the SMA is described by Brinson’s constitutive model. The flexure hinge is considered as a non-prismatic cantilever beam associated with geometrical and material nonlinearities. Govern equations of the flexure hinge are derived and solved numerically by applying the nonlinear bending theory of the Euler–Bernoulli beam. Experimental tests show that the proposed modeling approach can predict the deformation of the flexure hinge precisely; the maximum relative error is less than 6.5%. Based on the static deformation model, the motion capacity, the stiffness characteristic and the rotational error of the flexure hinge are also investigated. The results reveal that the flexure hinge made of a SMA has great potential to construct compliant mechanisms with a large motion range.

Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end

This work studies large deflections of slen- der, non-prismatic cantilever beams subjected to a combined loading which consists of a non-uniformly distributed con- tinuous load and a concentrated load at the free end of the beam. The material of the cantilever is assumed to be non- linearly elastic. Different nonlinear relations between stress and strain in tensile and compressive domain are considered. The accuracy of numerical solutions is evaluated by com- paring them with results from previous studies and with a laboratory experiment.

Some Comments on ‘Exact Large Deflection Analysis of Nonprismatic Cantilever Beams of Nonlinear Bimodulus Material Subjected to Tip Moment’

Some Comments on ‘Exact Large Deflection Analysis of Nonprismatic Cantilever Beams of Nonlinear Bimodulus Material Subjected to Tip Moment’

Large deflections of nonlinearly elastic non-prismatic cantilever beams made from materials obeying the generalized Ludwick constitutive law

This paper studies large deflections of nonlinearly elastic cantilever beams made from materials obeying the generalized Ludwick constitutive law. An exact moment-curvature formula which can be applied to study arbitrarily loaded and supported beams of rectangular cross-sections is developed. Several advantages of the generalized Ludwick’s model are illustrated. Numerical examples considered in this materially and geometrically nonlinear analysis clearly indicate rich nonlinear behavior of the beams.

Exact Large Deflection Analysis of Nonprismatic Cantilever Beams of Nonlinear Bimodulus Material Subjected to Tip Moment

This paper presents an exact solution for large deflection behavior of nonprismatic cantilever beams of nonlinear bimodulus type material and subjected to tip concentrated moment. The highly nonlinear simultaneous first-order differential equations were solved analytically using the power series approach. Several numerical examples are carried out to investigate the effect of bimodulus properties, material parameter, n, and the applied tip moment on the large deflection behavior of nonprismatic cantilever beams. The numerical results show that the interaction (i.e., coupling) between the bimodulus material properties, the material constant, n, and the applied tip moment, plays a significant role on the large deflection behavior of cantilever beams. A comparative study with ADINA has been made to verify the accuracy of the presented analytical solution, and excellent agreement has been obtained.

Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings

An efficient scheme, called quasi-linearization finite differences, is developed for large-deflection analysis of prismatic and non-prismatic slender cantilever beams subjected to various types of continuous and discontinuous external variable distributed and concentrated loads in horizontal and vertical global directions. Simultaneous equations of highly nonlinear and linear terms are obtained when casting the derived exact highly nonlinear governing differential equation using central finite differences on the nodes along the beam. A quasi-linearization scheme is used to solve these equations based on successive corrections of the nonlinear terms in the simultaneous equations. The nonlinear terms in the simultaneous equations are assumed constant during each correction (iteration). Several representative numerical examples of prismatic and non-prismatic slender cantilever beams with different loading conditions are analyzed to illustrate the merits of the adopted numerical scheme as well as its validity, accuracy and efficiency. The results of the present scheme are checked using large-displacement finite element analysis by the MSC/NASTRAN program. A comparison between the present secheme, MSC/NASTRAN and available results from the literature reveals excellent agreement. The advantage of the new scheme is that the load can be applied in one step with few iterations (3–6 iterations).

A new technique for large deflection analysis of non-prismatic cantilever beams

This paper studies the very large deflection behavior of prismatic and non-prismatic cantilever beams subjected to various types of loadings. The formulation is based on representing the angle of rotation of the beam by a polynomial on the position variable along the deflected beam axis. The coefficients of the polynomial are obtained by minimizing the integral of the residual error of the governing differential equation and by applying the beam’s boundary conditions. Several numerical examples are presented covering prismatic and non-prismatic cantilever beams subjected to uniform, non-uniform distributed loads and tip concentrated loadings in vertical and horizontal directions. The loads considered in this study are restricted to the non-follower type loads. Cases with different loadings and geometries are compared with MSC/NASTRAN computer package. However, for some very large deflection case, the MSC/NASTRAN failed to predict the deflected shape due to divergence problems.