Higher-order beam bending theory for static, free vibration, and buckling analysis of thin-walled rectangular hollow section beams

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.