Optimum design of nonlinear elastic framed domes
Introduction
Framed domes are used to cover large areas such as sport arenas and exhibition centers. They are one of the most spectacular type of civil engineering structures. Framed domes are either formed by using curved members forming a surface of revolution or by straight members meeting at joints which lie on the surface. They are given different names depending upon the way their surface is formed. Although the pin joint assumption is made for their analysis, it is more realistic to consider these domes as rigidly connected three-dimensional structures. This results in having bending moments in the members in addition to the axial forces. Furthermore, due to the slenderness of the members, axial forces cause lateral deflection in members which in turn generates additional bending moments. The presence of bending moments affects the axial stiffness of members due to their apparent shortening caused by the bending deformations. The interaction between bending and axial forces in members renders the overall stiffness matrix of these structures nonlinearly.
In recent years, framed domes are expected to resist ever increasing loads which come from large scale illumination, visio and audio equipments that are particularly concentrated at the apex. These concentrated loads increase to a greater extend the possibility of overall collapse of the structure[1]. Hence, it becomes necessary to consider the instability effects in the optimum design of such structures. While some of the earlier optimum design algorithms were based on linear elastic behavior of the structures2, 3, 4, others accommodated the nonlinear elastic behavior5, 6, 7, 8.
The optimum design algorithm presented in this study not only takes into account the nonlinear response of framed domes, but also considers the incremental load factor analysis to ensure that loss of stability does not take place during the optimum design cycles. It initiates the design process by carrying out the nonlinear analysis at the initial load factor. After the response of framed dome is obtained at this load factor, it is increased by the incremental amount and nonlinear analysis is performed at this load factor. The incremental load analysis is continued until the predetermined load factor is reached or the displacements of restricted joints are more than their upper bounds. At every iteration of the analysis the overall stability of the dome is checked. When the loss of stability is observed, the design process is terminated. If the nonlinear response of the dome is successfully obtained, the new values of design variables are computed by using the recursive relationships. The changes in the values of design variables necessitate updating the nonlinear response of the dome. This process of reanalysis and resizing is repeated until the convergence is obtained in the objective function.
Section snippets
Mathematical model
The optimum design of nonlinear framed domes can be described mathematically as follows:subject to:where w is the objective function which is taken as overall weight of the frame, Ak is the design variable representing the area of members belonging to group k, while mk is the total number of members in group k. Akl is the lower bound for design variable Ak. ρi and ℓi are the density and length of member i,
Design procedure
Solution of the design problem stated in (1) is obtained in an iterative process. Values of area variables are changed in every iteration in order to improve the design. It is apparent that the new values of variables are decided by the most severe constraints in the design problem. Hence, it becomes necessary to obtain expressions for updating design variables depending upon whether displacement or stress constraints are dominant. If neither of these constraints is dominant, values of design
Design examples
The algorithm presented is employed in the optimum design of three nonlinear elastic framed domes. In these examples the yield stress of material was taken as 275 N/mm2. The value of t=5 for the step size in the recursive relationship of Eq. (14)for area variables has provided reasonable speed of convergence. The step size of m=0.5 was found suitable for the iterative relationship of Eq. (15)for Lagrange multipliers. The initial value of 1000 was used for these parameters.
Conclusion
A general optimum design algorithm has been developed for three-dimensional rigidly jointed nonlinear elastic frames. It considers displacements as well as combined stress constraints. It takes into account the nonlinear response of the space frame due to the effect of axial forces in its members. The experience obtained from the design examples considered shows that in framed domes without diagonal members, the effect of nonlinearity is important. Its consideration certainly leads to an
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