## Optimum structural design: concepts, methods, and applications |

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Page 280

This problem will be discussed later in Sec. 13-5. The behavior constraints and

the analysis equations are related to the action of a single-loading condition. In

the case of several loading conditions all vectors associated with the behavior of

the structure will be transformed into matrices so that each of their columns will

correspond to a certain loading condition. The nonlinear programming

This problem will be discussed later in Sec. 13-5. The behavior constraints and

the analysis equations are related to the action of a single-loading condition. In

the case of several loading conditions all vectors associated with the behavior of

the structure will be transformed into matrices so that each of their columns will

correspond to a certain loading condition. The nonlinear programming

**problem****of Eqs**. (13-10) through (13-13) can be solved by methods described in Chap. 8.Page 282

In the latter

functions of the variables {X} and {r}. Thus, the optimal solution of the linearized

subjected to a set of different loadings, {Rop,}, defined as6 {Ropl} = ([K*] + [AK])({r

*} + {Ar}) (13-35) For the loadings computed by this equation, the equilibrium

conditions are also satisfied. Based on

expressed as ...

In the latter

**problem**all the expressions, except those of**Eq**. (13-8), are linearfunctions of the variables {X} and {r}. Thus, the optimal solution of the linearized

**problem**({X*} + {AX}, {r*} + {Ar}) can be viewed as an exact optimum of a trusssubjected to a set of different loadings, {Rop,}, defined as6 {Ropl} = ([K*] + [AK])({r

*} + {Ar}) (13-35) For the loadings computed by this equation, the equilibrium

conditions are also satisfied. Based on

**Eqs**. (13-8) and (13-30), {Rop.} can beexpressed as ...

Page 303

Using the former approach, we define the nonnegative variables {A'} and {A"} by {

A} = {A'}-{A"} (13-116) Substituting

the number of variables in the LP

advantage of this approach is that only Nqn yield stress constraints [

are required instead of 2Nq n in the original

may rewrite

Using the former approach, we define the nonnegative variables {A'} and {A"} by {

A} = {A'}-{A"} (13-116) Substituting

**Eq**. (13-116) into**Eqs**. (13-114) and (13-115),the number of variables in the LP

**problem**becomes (2Nq + l)n. However, theadvantage of this approach is that only Nqn yield stress constraints [

**Eq**. (13-115)]are required instead of 2Nq n in the original

**problem**. To see this possibility, wemay rewrite

**Eq**. (13-115) in terms of {A'} and {A"} and obtain the following n ...### What people are saying - Write a review

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### Contents

General Formulation of Optimal Design | 6 |

Approaches to Structural Design | 23 |

Structural Analysis | 31 |

Copyright | |

11 other sections not shown

### Common terms and phrases

active constraints algorithm analysis equations application approach approximate assumed beam behavior constraints buckling coefficients configuration considered constant constraints of Eq convergence convex corresponding cost cross section cross-sectional areas defined derivatives design space design variables discussed in Sec displacement constraints displacement method displacement vector dual efficient elements ELPP equality constraints expressed feasible region formulation fully stressed design geometric programming given global optimum independent inverse iteration Lagrange multipliers limits linear programming lower bound LP problem mathematical programming minimization modulus of elasticity nonlinear programming nonnegative number of variables objective function obtained optimal design problem optimal solution optimality criteria optimization problem optimum parameters penalty function plastic analysis posynomial problem of Eqs procedure recurrence relations redundants represent result satisfied shown in Fig solved stage statically statically determinate stiffness matrix straints stress constraints structural design subproblems Substituting Eq substructure Taylor series three-bar truss tion truss example value problem vector