Ludwick Cantilever Beam in Large Deflection Under Vertical Constant Load

The aim of this paper is to calculate the horizontal and vertical displacements of a cantilever beam in large deflections. The proposed structure is composed with Ludwick material exhibiting a different behavior to tensile and compressive actions. The geometry of the cross-section is constant and rectangular, while the external action is a vertical constant load applied at the free end. The problem is nonlinear due to the constitutive model and to the large deflections. The associated computational problem is related to the solution of a set of equation in conjunction with an ODE. An approximated approach is proposed here based on the application Newton-Raphson approach on a custom mesh and in cascade with an Eulerian method for the differential equation.

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Large Deflection of a Non-Linear, Elastic, Asymmetric Ludwick Cantilever Beam

ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis, Volume 2 ◽

10.1115/esda2010-24257 ◽

2010 ◽

Cited By ~ 12

Author(s):

Alberto Borboni ◽

Diego De Santis ◽

Rodolfo Faglia

Keyword(s):

Cantilever Beam ◽

Cross Section ◽

Mechanical Model ◽

Elastic Moduli ◽

Large Deflection ◽

Neutral Surface ◽

Linear Elastic ◽

Non Linear ◽

Vertical Displacements ◽

Tension And Compression

The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated vertical constant load at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler-Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper. Precisely, the proposed problem is a generalization of similar cases in literature, consequently numerical comparisons are performed with these previous works, i.e. assuming linear elastic materials or assuming symmetric Ludwick type material with same behavior in tension and compression like aluminum alloy and annealed copper. After verifying a proper agreeing with the literature, in order to investigate the effect of the different material behavior on the horizontal and vertical displacements of the free end cross-section, numerical results are obtained for different values of elastic moduli and strain exponential coefficients. The arising conclusions are coherent with the assumed hypotheses and with similar works in literature.

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Large deflections of a spring-hinged tapered cantilever beam with a rotational distributed loading

The Aeronautical Journal ◽

10.1017/s0001924000021667 ◽

1987 ◽

Vol 91 (909) ◽

pp. 429-437 ◽

Cited By ~ 3

Author(s):

B. Nageswara Rao ◽

G. Venkateswara Rao

Keyword(s):

Cantilever Beam ◽

Cross Section ◽

Large Deflection ◽

Rectangular Cross Section ◽

Large Deflections ◽

Elastic Curves ◽

Distributed Load ◽

Tapered Cantilever Beam ◽

Large Deflection Problem ◽

Special Case

SummaryLarge deflection problem of a spring loaded hinged nonuniform cantilever beam subjected to a rotational distributed loading is formulated by means of a second-order non-linear integro-differential equation. The problem is examined by considering the beam of rectangular cross-section with linear depth taper subjected to a uniform rotational distributed load. The elastic curves of a beam for this special case are presented.

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A Seminalytical Approach to Large Deflections in Compliant Beams under Point Load

Mathematical Problems in Engineering ◽

10.1155/2009/910896 ◽

2009 ◽

Vol 2009 ◽

pp. 1-13 ◽

Cited By ~ 14

Author(s):

N. Tolou ◽

J. L. Herder

Keyword(s):

Cantilever Beam ◽

Large Deflection ◽

Compliant Mechanisms ◽

Compliant Mechanism ◽

Geometrical Nonlinearity ◽

Adomian Decomposition Method ◽

Analytical Form ◽

Point Load ◽

Large Deflections ◽

Adomian Decomposition

The deflection of compliant mechanism (CM) which involves geometrical nonlinearity due to large deflection of members continues to be an interesting problem in mechanical systems. This paper deals with an analytical investigation of large deflections in compliant mechanisms. The main objective is to propose a convenient method of solution for the large deflection problem in CMs in order to overcome the difficulty and inaccuracy of conventional methods, as well as for the purpose of mathematical modeling and optimization. For simplicity, an element is considered which is a cantilever beam out of linear elastic material under vertical end point load. This can further be used as a building block in more complex compliant mechanisms. First, the governing equation has been obtained for the cantilever beam; subsequently, the Adomian decomposition method (ADM) has been utilized to obtain a semianalytical solution. The vertical and horizontal displacements of a cantilever beam can conveniently be obtained in an explicit analytical form. In addition, variations of the parameters that affect the characteristics of the deflection have been examined. The results reveal that the proposed procedure is very accurate, efficient, and convenient for cantilever beams, and can probably be applied to a large class of practical problems for the purpose of analysis and optimization.

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Large deflection analysis of clamped annular sector plates

The Journal of Strain Analysis for Engineering Design ◽

10.1243/03093247v191001 ◽

1984 ◽

Vol 19 (1) ◽

pp. 1-8 ◽

Cited By ~ 5

Author(s):

R S Srinivasan ◽

V Thiruvenkatachari

Keyword(s):

Integral Equation ◽

Parametric Study ◽

Large Deflection ◽

Large Deflections ◽

Lateral Loads ◽

Bending Stresses ◽

Deflection Analysis ◽

Annular Sector ◽

Newton Raphson ◽

Sector Angle

Thin annular sector plates undergoing large deflections due to lateral loads are considered in this paper. For such plates exact solutions are not available. A matrix method using integral equation of beams and the Newton Raphson procedure has been adopted for the analysis of clamped annular sector plates. Numerical values for the deflection, the membrane and the bending stresses at the interior of the plate, and the bending stresses at the edges of the plate are obtained. A parametric study has been carried out by varying the sector angle from 30 to 90 degrees in steps of 30 degrees, and the ratio of the inner and outer radii from 0 to 0.6 in steps of 0.2. The results are presented in non-dimensional graphical format.

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Large Deflections and Buckling Loads of Cantilever Columns with Constant Volume

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455417500912 ◽

2017 ◽

Vol 17 (08) ◽

pp. 1750091 ◽

Cited By ~ 6

Author(s):

Joon Kyu Lee ◽

Byoung Koo Lee

Keyword(s):

Differential Equations ◽

Cross Section ◽

Large Deflection ◽

Nonlinear Differential Equations ◽

Constant Volume ◽

Large Deflections ◽

Buckling Loads ◽

Geometrical Nonlinear ◽

Deflection Theory ◽

Large Deflection Theory

This paper deals with the large deflections and buckling loads of tapered cantilever columns with a constant volume. The column member has a solid regular polygonal cross-section. The depth of this cross-section is functionally varied along the column axis. Geometrical nonlinear differential equations, which govern the buckled shape of the column, are derived using the large deflection theory, considering the effect of shear deformation. The buckling load of the column is approximately equivalent to the load under which a very small tip deflection occurs. In regard to the numerical results, both the elastica and buckling loads with varying column parameters are discussed. The configurations of the strongest column are also presented.

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Proposing a Pretwisted Bending Actuator

Aerospace ◽

10.1115/imece2006-13281 ◽

2006 ◽

Author(s):

Frank Dienerowitz ◽

Nicole Gaus ◽

Wolfgang Seemann

Keyword(s):

Magnetic Fields ◽

Cross Section ◽

Large Deflection ◽

Mechanical Energy ◽

Electric Energy ◽

Small Strain ◽

Large Deflections ◽

Direct Transformation ◽

Active Elements ◽

Bending Actuator

Piezoelectric bending actuators take advantage of both piezoelectricity and kinematics of beams, i.e. (1) direct transformation of electric energy into mechanical energy without causing significant magnetic fields and (2) to be capable of turning small strain modifications into large deflections, provided the cross-section is rather flat. Unfortunately the latter usually implies that bending actuators provide only one axis of large deflection. Herein a pretwisted bending actuator is investigated, similar to a helicoid. The active elements along the beam axis are subdivided and controlled separately, hence allowing independent control of the curvature of each section. Due to the pretwist, this bending actuator can provide not only one but two axes of deflection. For a slender pretwisted bending actuator the problems emerging are presented and discussed, covering the work space of the actuator, optimization of electrode connecting patterns and experimental results.

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Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

Shock and Vibration ◽

10.1155/2018/1056397 ◽

2018 ◽

Vol 2018 ◽

pp. 1-14 ◽

Cited By ~ 4

Author(s):

Zhongmin Wang ◽

Rongrong Li

Keyword(s):

Differential Equation ◽

Boundary Conditions ◽

Cantilever Beam ◽

Cross Section ◽

Transverse Vibration ◽

Slenderness Ratio ◽

Circular Cross Section ◽

Dimensionless Parameters ◽

Inner Radius ◽

Tapered Cantilever Beam

Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

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The Nonlinear Static Formulation of Variable Cross-Section Beam Element Based on Positional Description

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.557-559.2367 ◽

2012 ◽

Vol 557-559 ◽

pp. 2367-2370

Author(s):

Lv Zhou Ma ◽

Jian Liu ◽

Xun Lin Diao ◽

Yu Qin Yan

Keyword(s):

Cross Section ◽

Large Deflection ◽

Variable Cross Section ◽

Beam Element ◽

Solution Procedure ◽

Variable Cross ◽

Hyper Elastic ◽

Nonlinear Static ◽

Newton Raphson ◽

Raphson Iteration

Based on positional FEM (finite element method), the nonlinear static formulation to treat large deflection of variable cross-section beam element is created by using the lowest potential energy theory. Adopting linear constitutive relation for hyper-elastic materials, the formulation and the solution procedure by Newton-Raphson iteration method are very simple.

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