Issue 75

P. Grubits et alii, Fracture and Structural Integrity, 75 (2026) 124-156; DOI: 10.3221/IGF-ESIS.75.10

R U



K



(22)

T



This iterative scheme updates the displacement field until convergence is achieved, enabling the simulation of both elastic and elasto-plastic behavior under geometric nonlinearity. The resulting displacement vector U , representing the total nodal displacements at equilibrium, becomes a key quantity in the assessment of deformation-based criteria, such as serviceability. As such, it may also serve as the basis for imposing constraints on maximum allowable displacements in subsequent optimization formulations. Importantly, this geometrically nonlinear formulation remains fully consistent with the residual force–based framework and the evaluation of complementary plastic work introduced earlier. The use of Green–Lagrange strains and second Piola– Kirchhoff stresses ensures a coherent treatment of both material and geometrical nonlinearities, enabling accurate assessment of residual internal forces and the corresponding inelastic energy contributions throughout the deformation history. Furthermore, by operating within a total Lagrangian framework, this approach inherently allows the incorporation of initial geometric imperfections, which can significantly influence the structural response—particularly in slender systems sensitive to stability effects. xpanding on the foundational concepts discussed in the preceding sections, this part of the study presents the mathematical formulation of two optimization problems: one based on elastic limit analysis and the other on elasto plastic limit analysis. The proposed formulations are integrated into a unified computational framework, enabling flexible and accurate structural design under both material and geometrical nonlinearities. Elastic design optimization In the context of elastic limit analysis, the principal objective is to determine the elastic limit load el P , which marks the onset of yielding in a structural system characterized by a given configuration and associated structural weight s G . Beyond this threshold, plastic deformations begin to develop. Based on this theoretical foundation, the elastic design problem is formulated as an optimization task, where the aim is to minimize the structural weight s G while ensuring that the structural response remains entirely elastic and that sufficient global stability is maintained. This approach enables a design strategy that is simultaneously material-efficient and structurally robust, expressed mathematically as follows:       1 2 : s min fitness f G p m p     P (23a) where the objective function “ fitness ” quantifies the overall quality of a given design solution based on three components: the structural weight term   s f G , and two penalty terms,   1 p m P and   2 p  , which enforce the elastic limit and stability conditions, respectively. The function   s f G aims to minimize the total structural weight within prescribed design bounds, using a normalized expression that incorporates the minimum and maximum allowable cross-sectional areas min A and max A , which together define the design domain. Accordingly,   s f G is expressed as: E O PTIMIZATION

1     n n i i i i n max i Al A l    

   

   

A l

 

min i



1





f G

(23b)

s

n

A l

min i



i

i

1

1

For further clarification, the term   1 p m P is introduced to ensure that the optimized structure reaches the predefined load level 0 P , while remaining within the elastic regime and avoiding failure due to instability or yielding. This term evaluates the extent to which the structure satisfies the fundamental design requirement of attaining the desired load-carrying capacity. The formulation distinguishes between four physically meaningful scenarios that may arise during the load history:

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