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Long span roofs are widely used in stadium, shopping mall, public hall, warehouse, airport and so on as they can provide large internal spaces. It is a challenging task to design a long span roof for the structural engineers using traditional linear design method due to uncertainty in the assumption of effective length. Second-order direct analysis (SODA) shows many advantages over the traditional linear design approach in terms of structural safety and cost efficiency. In this paper, the design considerations and procedures of using SODA for design of long span roofs are presented. Both the P- and P- effects as well as initial imperfections have been taken into account. More importantly, this innovative method is used to trace the structural behaviour of long span structures during offload process so that the structures can be safely and progressively deformed to their designed position with minimize lock-in stress. A nonlinear spring element is proposed to capture the behaviour of the mechanical jack which is the device for implementation of offload process. Thus, a comprehensive solution is introduced here for offload analysis of long span structures. Finally, a practical long-span single layer roof structure in Macau is used to demonstrate the validity and accuracy of the proposed method.

In the past decades, the stiffness-based elements with plastic hinge method and the flexibility-based elements with plastic zone method are widely used for second-order inelastic analysis. The former emphasizes the computational efficiency with relatively less accurate structural responses while the later aims to precisely simulate the structural behaviour but normally needs more computer times. For the member under complex loads, several stiffness-based elements are generally required to capture its plastic behaviour. In such case, one-element-per-member for member initial imperfection required in second-order direct analysis is violated. On the contrary, a flexibility-based element with several integration points is sufficient for material yielding. However, the more integration points associated with fibre section the more computational efforts are needed. It is also noted that the conventional flexibility-based type elements cannot meet the codified requirements of direct analysis due to lack of consideration of member initial imperfection. In this paper, a new flexibility-based element equipped with plastic hinge method without huge computational time is proposed for second-order inelastic analysis. This element is also able to take the member initial bowing into account. Thus, an innovative solution is proposed for routine design. Finally, several numerical examples are used to demonstrate the validity and accuracy of the proposed method.

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