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where S=section modulus; I=moment of inertia; and =constants that depend on the shape of the cross section; and and =positive powers. By performing regression analysis of the cross sectional data for commercially available wide-flange steel sections, nonlinear relationships between the cross-sectional properties are obtained as follows (unit:inch) (Khan 1984; Sadek 1992; Sedaghati and Esmailzadeh 2003):
Although this representation of the design optimization problem is quite restrictive, it is adopted to evaluate the alternative formulations and compare solutions with the previously published results for some example problems. After the evaluation has proven usefulness of the alternative formulations, a more realistic definition of the problem will be pursued in the future work. It is also important to note that relations in Eqs. (2)and (3) are discontinuous when A=15 and 44. These discontinuities can be troublesome for the gradient-based optimization methods.
Mathematical Statement
The design optimization problem of minimizing the volume of a frame is to determine the design variables to
where n and m=numbers of members and degrees of freedom for the frame, respectively; Ai and Li=ith member cross-sectional area and its length, respectively; and and =lower and upper bounds for the stresses nodal displacements , and design variables, respectively. To simplify the presentation, only one load case is considered; however, additional loading conditions can be treated similarly. Also, members of the structure are usually linked together into groups for symmetry and other considerations. Such linking, although not shown, can be routinely incorporated into the problem definition.
In the literature, the combined axial and bending stress equation given in the following has been used by different researchers (Saka 1980; Khan 1984; Sadek 1992). In order to compare solutions, it is adopted here as well
where Pi and Mi=axial force and bending moment, respectively and Si=section modulus of member i. The combined axial and bending stress constraints are imposed at both ends of each frame member. Note that although the shear stress and other design code constraints should be included in the formulation, they cannot be imposed due to the limitations of the design variable definitions (Saka 1980; Khan 1984; Sadek 1992)
Frame Analysis
For a general frame structure, the (m*m) equilibrium equation for a specified design variable vector A is given as
where K=global stiffness matrix, whose elements are explicit functions of A; r=vector of nodal displacements for the frame model in the global coordinate system; and R=external load vector that may also be a function of A, i.e., when the structural self-weight is considered. The equilibrium equation for member i
in the global coordinate system is given as
where ki=member stiffness matrix in the global coordinate system and Qi and qi=nodal force (including moments)and displacement vectors (6*1) in the global coordinate system. Member nodal force vectors (6*1) in the local and global coordinate systems are illustrated in Figs. 1(a and b), respectively. Note that ki= , where =member stiffness matrix in the local coordinate, system and Ti=transformation matrix between the local (member)and global coordinate systems. The member stiffness, matrix ki can be separated into axial and flexural parts as
where and =6*6 constant matrices.
Member nodal displacement vector qi can be related to the global nodal displacement vector r for the structure by a 6*m Boolean matrix Zi
Therefore from Eqs. _10_–_12_, nodal force vector for member i in the global coordinate system can be written in terms of the displacements r as
It is important to note that are 6*m transformation matrices that are independent of Ai and Ii. They are only functions of the elastic modulus E, member length and direction cosines between the local and global coordinate systems.