By endowing El Fatmi’s theories of bars with first-order warping functions due to torsion and shear, a family of theories of bars, of various applicability ranges, is effectively constructed. The theories thus formed concern bars of arbitrary cross-sections; they are reformulations of the mentioned theories by El Fatmi and theories by Kim and Kim, Librescu and Song, Vlasov and Timoshenko. The Vlasov-like theory thus developed is capable of describing the torsional buckling and lateral buckling phenomena of bars of both solid and thin-walled cross-sections, which reflects the non-trivial correspondence, noted by Wagner and Gruttmann, between the torsional St.Venant’s warping function and the contour-wise defined warping functions proposed by Vlasov. Moreover, the present paper delivers an explicit construction of the constitutive equations of Timoshenko’s theory; the equations linking transverse forces with the measures of transverse shear turn out to be coupled for all bars of asymmetric cross-sections. The modeling is hierarchical: the warping functions are numerically constructed by solving the three underlying 2D scalar elliptic problems, providing the effective characteristics for the 1D models of bars. The 2D and 1D problems are indissolubly bonded, thus forming a unified scientific tool, deeply rooted in the hitherto existing knowledge on elasticity of elastic straight bars.

中文翻译：

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将横向剪切和扭转引起的翘曲效应纳入弹性直杆理论

通过赋予 El Fatmi 的钢筋理论由于扭转和剪切而产生的一阶翘曲函数，可以有效地构建一系列适用范围不同的钢筋理论。这样形成的理论涉及任意横截面的钢筋；它们是对上述 El Fatmi 理论以及 Kim 和 Kim、Librescu 和 Song、Vlasov 和 Timoshenko 的理论的重新表述。这样发展起来的类弗拉索夫理论能够描述实心截面和薄壁截面钢筋的扭转屈曲和侧向屈曲现象，这反映了瓦格纳和格鲁特曼指出的扭转圣路易斯截面之间的非平凡对应关系。 Vlasov 提出的 Venant 翘曲函数和轮廓定义的翘曲函数。而且，本论文给出了铁木辛哥理论的本构方程的明确构造；将横向力与横向剪切量度联系起来的方程对于所有非对称横截面的钢筋都是耦合的。建模是分层的：翘曲函数是通过求解三个基本的二维标量椭圆问题来数值构建的，为杆的一维模型提供了有效的特征。二维和一维问题密不可分，从而形成了一个统一的科学工具，深深植根于迄今为止关于弹性直杆弹性的知识。翘曲函数是通过求解三个基本的二维标量椭圆问题来数值构建的，为杆的一维模型提供了有效的特征。二维和一维问题密不可分，从而形成了一个统一的科学工具，深深植根于迄今为止关于弹性直杆弹性的知识。翘曲函数是通过求解三个基本的二维标量椭圆问题来数值构建的，为杆的一维模型提供了有效的特征。二维和一维问题密不可分，从而形成了一个统一的科学工具，深深植根于迄今为止关于弹性直杆弹性的知识。