Abstract The Heavy Elastica problem is a classical issue and has been addressed by many authors. The only analytical approach applied is the perturbative method, and the expansion through Taylor's series is the most recurrent technique. This transforms the nonlinear differential problem into a system of nonlinear (trigonometric) algebraic equations to compute the expansion coefficients. The main disadvantage is that the size of the obtained nonlinear systems is equal to the degree of the series expansion of the solution; therefore, to gain a good accuracy, it is necessary to manage the series expansion up to a considerable degree. Possibly for this reason, the use of perturbative techniques has gradually been abandoned and many authors have addressed the problem of Heavy Elastica with numerical methods. In this paper, a new analytical method based on the parabolic curvilinear abscissa mapping is presented. The result is that the problem of the Heavy Elastica takes on the same complexity as the well-known Elastica problem (concentrated forces and moment applied) and is therefore solved. The proposed method is compared with the Runge-Kutta integration approach and finite element results, showing the correctness of the solution suggested and its remarkable computational advantage. A further comparison is carried out by means of a number of experimental tests that agree with the analytical expected values. By expressing the proposed solution in a dimensionless form, Design Charts are given; they provide the results in the whole field of the domain, so that both kinematic and static variables can be deduced without computer aid.

中文翻译：

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重型弹性土支撑，在端部施加提升载荷和弯矩：一种用于超大位移和实验验证的新分析方法

摘要 重弹性问题是一个经典问题，已被许多作者解决。唯一应用的分析方法是微扰方法，通过泰勒级数进行扩展是最常见的技术。这将非线性微分问题转换为非线性（三角）代数方程系统，以计算展开系数。主要缺点是得到的非线性系统的大小等于解的级数展开程度；因此，为了获得良好的精度，必须在相当程度上管理系列扩展。可能由于这个原因，微扰技术的使用逐渐被放弃，许多作者已经用数值方法解决了重弹性的问题。在本文中，提出了一种基于抛物曲线横坐标映射的新解析方法。结果是，Heavy Elastica 的问题与众所周知的 Elastica 问题具有相同的复杂性（集中力和施加的力矩），因此得到解决。将所提出的方法与Runge-Kutta积分方法和有限元结果进行比较，表明所提出的解决方案的正确性及其显着的计算优势。通过与分析预期值一致的大量实验测试进行进一步比较。通过以无量纲的形式表达所提出的解决方案，给出了设计图表；它们提供整个域中的结果，因此无需计算机辅助即可推导出运动学和静态变量。