Large deflection of a circular C-shaped spring

The theoretical analysis of a circular $\text{C-}\text{shaped}$ spring is presented. The analysis is based on the assumptions that the spring is perfectly elastic and that its deflection is plane under large deflections. The deformed shape of the spring is found to be a segment of a nodal elastica, and the load-deflection curve is almost linear over some range of deflection; this provides basic information for the design of the spring. Experimental verifications are performed using steel circular $\text{C-}\text{shaped}$ springs of flat rectangular cross section. There is no discrepancy between the calculated and observed results.     

Large deflection of a heated cylindrical shell

The large deflection of a heated cylindrical shell is investigated and some numerical results are presented.     

Experimental investigation on the deflection of a serpentine spring machined by WEDM

Journal of Mechanical Science and Technology - A kind of micro serpentine spring by wire electrical discharge machining (WEDM) is selected as research object, and the bending deformation mechanism...     

Large deflection analysis of a thin walled channel section cantilever beam

For a special application the large deflection behaviour of thin walled channel section beams made of thin sheet steel has been investigated. The experiments consisted of cantilever bending tests with the beam loaded through the shear centre and through the centroid. When loaded through the shear centre the beam buckling took place in the compression flange at the root of the cantilever. When loaded through the centroid however, it was noted that the compression flange buckled at a fixed distance from the fixed end. The general theory of thin walled beams developed by Vlasov was applied to the problem and indicated that the maximum compression stress at the edge of the flange would be at some distance from the fixed end. The value of the maximum compression stress obtained by the general linear theory was small and its position did not coincide with the experimental position. The Vlasov analysis has been modified to include the increase in the twisting moment due to the lateral deformation of the beam along its length. Good agreement between the modified theory and experiment both for the position of the maximum compressive stress and for the twist of the cantilever at three points along its length.

Because of the very low torsional stiffness of thin walled channel sections, the small deflection theory is only applicable for small bending loads applied through the centroid and the modified theory should be used for practical loading cases.     

联合载荷作用下悬臂梁大变形分析的打靶法

利用微积分基本知识，这里导出了集中力和任意横向分布载荷作用下悬臂梁的挠曲线微分方程。基于两点边值问题的打靶法，分别计算了两种联合载荷作用下悬臂粱的大变形。所得微分方程不仅具有普遍性，而且本文的计算方法对类似问题的解决具有一定的借鉴意义。     

Large deflection analysis of elasto-plastic beams and frames

A general method for analysis of elasto-plastic beams and frames with large displacements is described in this paper. A hybrid-type beam element with 3df at each end is used in the analysis. The element stiffness matrix is obtained by inversion of a flexibility matrix, which is computed from an assumed distribution of internal forces along the element axis. This approach with approximations of the stress fields better imitates varying stiffness along the beam than the traditional approach with assumed displacement fields.Large displacement effects are taken into account by updating the geometry (both node co-ordinates and cross-section shapes) and accurately computing the elongations of each element. Partial yielding is considered across the height of the beam as well as along the element axis.     

The elastic spring behavior of a rhombus frame constructed from non-prismatic beams under large deflection

The spring behavior due to large deflection of a flexible rhombus frame with rigid joints is studied in this paper. The system consists of two equal and opposite diagonal forces that act on two opposite corners of a rhombus frame with a given apex angle γ. The frame sides have a rectangular cross-section and are non-prismatic due to tapering in the section's dimensions. The relations between the large displacement at the corners and the applied forces are obtained using a new robust numerical technique. The technique is based on representing the angular deflection of the side beam by a polynomial on the position variable along the deflected beam axis. The polynomial coefficients are obtained through a numerical optimization procedure based on minimizing the residual error and satisfying the boundary conditions. A classical numerical integration technique is used to verify some of the results obtained by the proposed technique. The frame spring behavior is investigated for different taper ratios and apex angles. The results are transformed into regression relations to utilize its implementation in design and analysis.     

Large deflection of viscoelastic beams using fractional derivative model

This paper deals with large deflection of viscoelastic beams using a fractional derivative model. For this purpose, a nonlinear finite element formulation of viscoelastic beams in conjunction with the fractional derivative constitutive equations has been developed. The four-parameter fractional derivative model has been used to describe the constitutive equations. The deflected configuration for a uniform beam with different boundary conditions and loads is presented. The effect of the order of fractional derivative on the large deflection of the cantilever viscoelastic beam, is investigated after 10, 100, and 1000 hours. The main contribution of this paper is finite element implementation for nonlinear analysis of viscoelastic fractional model using the storage of both strain and stress histories. The validity of the present analysis is confirmed by comparing the results with those found in the literature.     

Large deflection of constant curvature cantilever beam under follower load

This paper describes a method to analyze for the large deflections of curved prismatic cantilever beams with uniform curvature subjected to a follower load at the tip. The large deflection, the deflection dependent follower load and the initial curved geometry are the important features of the beam considered in this work. Shear force formulation proposed by Lee [Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int J Non-Linear Mech 2002;37(3)] is used for deriving the governing equations. Using this approach, the resulting two point boundary value problem (TPBVP) can be reduced to an initial value problem (IVP). Fourth order Runge-Kutta method along with one parameter reverse shooting method is applied to the numerical solution to the problem. A novel approach presented in this paper of integrating from the free end to the fixed end of the cantilever beam simply replaces the two parameter shooting with a single parameter shooting yielding several advantages. This solution technique is demonstrated for various types of follower tip loads on curved and straight cantilever beams and is validated with existing solutions in the literature.     

A new approach for bending analysis of thin circular plates with large deflection

This paper is concerned with bending analysis of an axisymmetric simply supported circular plate with large deflection. Based on the linear theory of thin plates, the incremental load technique is developed for solving the bending problem of a thin circular plate with large deflection. In the proposed method, the total applied load is divided into various small load steps. In each load step, the plate stress behavior is simplified to be linear. The incremental formulations are presented for the deflection and stresses of the plate when external loads increase. A numerical example is given to show simplicity and accuracy of the present method. It is found that the proposed method can be an alternative useful tool for engineering applications.     

An exact analytical solution for the self-weight deflection of circular plates of constant thickness

An exact analytical solution, not previously noted in the literature, for the self-weight deflection of a circular plate is obtained by superposition of various elementary solutions due to Love, and the accuracy of the existing approximate procedures is examined. The expressions for stress components are obtained using the axisymmetric stress-strain relations and the validity of the derived solution is justified by satisfying the field equations and the required boundary conditions.     

圆截面卡簧的设计探讨

分析了卡簧工作时的受力状态，建立了卡簧设计中的数学方程，并给出了设计实例。     

Large deflection analysis of elastoplastic plate in steady continuous four-roll bending process

This paper presents a mathematical model to simulate the mechanics in a steady continuous bending mode for four-roll thin plate bending process. The differential equations governing the large deflection bending of elastic-perfectly plastic thin plate are derived by modelling the loading and unloading deformation of the operational process. Through a semi-analytical solution of the system equations, the physical quantities of (i) the deformation and stress resultants of bend plate, (ii) the springback, (iii) the shift of contact-angle of bend plate with rolls, (iv) the applied forces on rolls, (v) the bending torque and power, and (vi) the position of the operative side roll, etc., associated with bending of a plate to an anticipated curvature, for single-pass bending operation, are predicted and discussed.     

圆形截面与矩形截面扭簧的优化换算设计

通过对代换设计中两扭转弹簧的刚度,强度的等价关系确定,阐明了圆形截面和矩形截面扭转弹簧的换算设计方法。     

Creative approach for the deflection formula according to relative length of supporting edge in a partially supported circular plate

The purpose of this investigation is to derive the maximum deflection formula of the circular plate with nonaxisymmetric boundary condition, such as chalk valves for vessel and artificial heart valves, with respect to the length ratio of supporting edge using elastic beam theory. To evaluate the deflection characteristics of this plate, we assumed the circular plate to have a cross-section that varies in the longitudinal direction, and derived the maximum deflection formula of this circular plate with a radius of ‘r’ in four boundary conditions. Then to verify the deflection formula that is derived from elastic beam theory, five different relative lengths of the supporting edge were adopted as the design parameter, and finite element analysis was carried out for each model.     

片弹簧贮存中性能和结构变化分析

采用周期性作业法和线性回归分析方法分析了片弹簧在不同变形量贮存过程中弹性性能和残余变形随时间的变化规律和趋势,并对产品的贮存寿命进行了评估;分析了片弹簧在强化变形量条件下贮存后的组织结构变化情况.结果表明,该片弹簧在一定变形量条件下贮存3293h后,其性能依然满足设计要求,在工作负荷下预期贮存寿命可达20年.     

大圆弧弯曲模的回弹计算

通过实验数据的拟合，及数学变换得到了大圆弧弯曲模的回弹计算公式。说明了该公式的适用范围、优点等。     

Large deflexion, geometrically non-linear finite element analysis of circular arches

This paper presents the results of an investigation into the geometrically nonlinear behaviour of circular arches. A curved element is used, based on satisfying the condition that the circumferential strain and change in curvature, rather than the displacements, should be simple independent functions of the co-ordinate axes. The analysis was carried out using the linearized incremental method based on the mid-increment stiffness in conjunction with the Newton-Raphson iterative technique. A comparison is first made between the results obtained with this element and those using a simple polynomial shape function. The behaviour of a range of arches is then considered and the results are compared and found to agree well with an analytical method. The results include the behaviour of deeper but still shallow arches which exhibit “looping” of the load-deflexion curve, and bifurcation of the equilibrium path into unsymmetric deflexions of the arch.A computer program was developed to allow any of the generalized degrees of freedom which are designated for incrementation to be expressed as either forces, to avoid failure at the vertical tangent to the load deflexion curves, or as generalized displacements to avoid failure at the horizontal tangent. The program also allows the quantity subjected to incrementation to be changed as necessary to follow complex load-deflexion equilibrium paths.