Local OC search operator
The Optimality Criteria (OC) approach has long been recognised as a highly efficient method for element sizing optimization of large-scale structures. The rigorously derived OC method is shown to be particular suitable for lateral stiffness optimization problems associated with tall buildings subject wind-induced serviceability design constraints (Chan, 2001; Chan, 2004). Generally speaking, the design of a tall building with a height-to-width aspect ratio larger than 5 is likely to be governed by wind-induced drift and motion perception serviceability design criteria. The rigorously derived OC method is herein adopted as a local search operator for the lateral drift design of tall skeletal steelworks.
d/A)=si a; + і * dUj
To commence the rigorously derived OC method, the lateral drift constraints Eqs.(1c and 1d) must be formulated explicitly in terms of the design variables A. Using the principle of virtual work, the collective set of lateral top and interstory drift constraints can be expressed explicitly in terms of A,-:
where e, j and e’,j are the respective virtual strain energy coefficient and its correction factor of the, th steel element of the structure associated with the jth drift constraint (Chan, 1997). Once a finite element analysis is carried out for a randomly selected child design with a given topology under the actual and virtual loading conditions, the internal element forces and moments are obtained and the element virtual stain energy coefficients are then readily calculated.
Upon establishing the design constraints into explicit functions, the constrained optimization problem can then be transformed into an unconstrained Lagrangian function which involves the objective function Eq.(1a) and the explicit drift constraints Eq.(2) associated with corresponding Lagrangian multipliers. Based on the stationary conditions derived from the Lagrangian function, the following recursive linear relations can be used to resize the active sizing variables A, (Chan, 1997) : where Xj denotes the Lagrangian multiplier for the corresponding j411 drift constraint, v represents the current iteration number; and tf is a relaxation parameter. During the recursive resizing iteration process, any element found to reach its size bounds is deemed an inactive element having its size set at its corresponding size limit. Before Eq. (3) can be used to resize A, the Lagrangian multipliers Xj must first be determined. Considering the sensitivity of the drift constraints due to the changes in the design variables, one can derive a set of M simultaneous equations to solve for M number of Xj. Having the current design variables A, v, the corresponding Xj values are readily determined by solving the simultaneous equations. Having the current values of Xj, the new set of design variables AJ+1 can then be obtained by the respective recursive relations Eq. (3). Therefore, the recursive applications of the
simultaneous equations to find the If and the resizing formula Eq. (3) to find the design variables constitute the OC algorithm (Chan, 1997). By successively applying the recursive OC algorithm until convergence, a local optimal solution for the design optimization problem is then found. The local optimum with improved element sizes will then complete for survival with all parent designs and other child designs generated from crossover and mutation in the selection process.