In higher-order beam theories, cross-sectional deformations causing complex responses of thin-walled beams are considered as additional degrees of freedom. To fully capture their bending responses, enriched sectional modes departing from Vlasov’s assumptions have been utilized in recent studies. However, due to these bending-related modes, no available higher-order beam bending theory has established explicit stress-generalized force relations that are fully consistent with those by the classical beam theories and earlier studies based on Vlasov’s assumptions. If they are available, physical significance of the bending-related generalized forces can be readily understood. In addition, equilibrium conditions at a joint of multiple thin-walled beams can be explicitly derived. Here, we newly propose a higher-order beam bending theory that not only includes as many bending-related sectional modes as desired, but also provides the desired explicit stress-generalized force relations. To this end, we establish a recursive analysis method that derives hierarchical bending-related sectional modes. We show that this method can yield certain relations among the sectional mode shapes, which are critical in establishing the desired explicit relations. The validity of the present theory is confirmed by calculating the static, free vibration, and buckling responses of several thin-walled rectangular hollow section beams.

在高阶梁理论中，将导致薄壁梁复杂响应的横截面变形视为额外的自由度。为了充分捕捉其弯曲响应，最近的研究中使用了偏离弗拉索夫假设的丰富截面模式。然而，由于这些与弯曲有关的模式，目前尚无可用的高阶梁弯曲理论建立与传统梁理论和基于Vlasov假设的早期研究完全一致的显式应力-广义力关系。如果有的话，可以很容易地理解与弯曲有关的广义力的物理意义。此外，可以明确导出多个薄壁梁的节点处的平衡条件。这里，我们新提出了一种高阶梁弯曲理论，该理论不仅包括所需的与弯曲有关的截面模式，而且还提供了所需的显式应力-广义力关系。为此，我们建立了一种递归分析方法，该方法可得出与弯曲相关的分层截面模式。我们表明，该方法可以在截面模式形状之间产生某些关系，这对于建立所需的显式关系至关重要。通过计算几个薄壁矩形空心截面梁的静，自由振动和屈曲响应，可以证实本理论的有效性。我们建立了一种递归分析方法，该方法可得出与弯曲相关的分层截面模式。我们表明，该方法可以在截面模式形状之间产生某些关系，这对于建立所需的显式关系至关重要。通过计算几个薄壁矩形空心截面梁的静，自由振动和屈曲响应，可以证实本理论的有效性。我们建立了一种递归分析方法，该方法可得出与弯曲相关的分层截面模式。我们表明，该方法可以在截面模式形状之间产生某些关系，这对于建立所需的显式关系至关重要。通过计算几个薄壁矩形空心截面梁的静，自由振动和屈曲响应，可以证实本理论的有效性。