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The jth nodal force component Qji in the global coordinate system for member i is
where ej=6*1 Boolean vector that is independent of Ai and Ii to determine the position of the corresponding nodal force component j. From Eq. (13), the axial force and the nodal bending moment of a frame member (in the local coordinate system)are given as
where and =direction cosines of, member i with respect to the global x and y directions; li and pi=6*1 transformation vectors that are independent of Ai and Ii. Therefore from Eq. (8), the combined axial and bending stress for a frame member is
where ti=height of the cross section for member i, related to the variables as ti=2Ii /Si.
From Eqs. (15_)and (16), Eq.(8) can also be expressed directly in terms of the member force vector Qi, as follows:
Optimal Design Formulations
Conventional Formulation „CF…—Only Areas as Optimization Variables
Only the cross-sectional areas of the frame members are considered as design variables. The design problem is to minimize the cost funct ion of Eq. (4), subject to the constraints of Eqs. (5)–(7). This formulation uses the smallest number of optimization variables. However, the problem functions are implicit in terms of the variables, since the displacement vector r is an implicit function of the member cross-sectional properties. Explicit expressions for r and, thus, for the constraint functions in terms of the design variables cannot be obtained. The equilibrium equations need to be solved for evaluation of constraint functions. Also, gradients of the functions need to be evaluated using special sensitivity analysis procedures (Arora 1995)
Alternate Formulation 1 „AF1…—Areas and Nodal
Displacements as Optimization Variables
If the nodal displacements r are also treated as independent variables in the optimization formulation, the implicit problem functions are transferred to explicit ones. In addition, the global equilibrium equations need not be solved explicitly as they are treated as equality constraints in the optimization process. Thus decomposition of the global stiffness matrix is avoided. The optimum design problem is to determine A and r to minimize the cost function of Eq. (4), subject to the constraints as.
where Qjk=nodal force of member k which has same direction as the displacement j. Member k is connected to the node which has displacement j as one of the degrees of freedom _in the global coordinate system_. NEj=number of members connected to the same node and Rj=resultant external load acting at node p in the direction of displacement j; note here that Qjk and Rj can be either forces or moments. Eq. (19) in fact includes the equilibrium equation for the degree of freedom j. Displacement constraints in Eq. (6) become simple bound constraints. The lower bound constraint on stress can be treated similar to that in Eq. (20).
Differentiating Eqs. (19) and (20) with respect to all the variables directly, explicit expressions for derivatives are obtained. For the equality constraints in Eq. (19)
where =0 if member s is not connected to displacement j, and =1 if member s is connected to displacement j. The subscript q represents the qth component of the vector.
For the inequality behavior constraints in Eq. (20)
where =0, if and =1, if i=s. The constraint functions and their derivatives in Eqs. (19)–(24) can be easily calculated using the current nodal displacement vector r and cross-sectional areas A.
Alternate Formulation 2 „AF2…—Areas, Nodal Displacements and Member Forces as Optimization Variables
In AF1, the constraint functions in Eqs. (19) and (20)need to be expressed explicitly in terms of areas and displacements. It is seen that evaluation of Eqs. (19)–(24) for the functions and their derivatives is a little tedious. However, if the member forces are also treated as variables, the constraints in Eqs. (19) and (20) become very simple. The stress in Eq. (20)can now be replaced by Eq. (18). This leads to simpler expressions and computer implementation. To evaluate this and compare efficiency, member nodal forces are also treated as variables in AF2. This formulation has not been presented previously in the literature and its success can open doors for other alternative formulations for optimal design of framed structures. The formulation is to determine A, r, and Q to minimize the cost in Eq. (4), subject to