Issue 75

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

As in the previous example, the FE model of the 25-bar truss is developed using B31 beam elements with a uniform 100mm mesh size and the same quad-linear constitutive model for steel. The primary material properties are as follows: elastic modulus 2 207,000N / mm E  , yield strength . 2 240N/mm y f  ., and ultimate tensile strength 2 360N/mm u f  . The load and boundary conditions are prescribed as shown in Fig. 23(a), with nodal forces specified in Tab. 8. As in the previous example, the LBA is first carried out to obtain the critical mode shape used to define the initial geometric imperfection. The analysis yields a critical buckling load factor 1 0.388     , and the corresponding mode shape is shown in Fig. 23(b). It should be noted that this configuration does not satisfy the predefined stability criterion of 1.000   , introduced in the previous benchmark optimization and adopted herein as well, as further detailed in the next section. Consistent with the first example, the length L of the most critical member is used to scale the imperfection, with amplitude 1 /1000 L   .

(a) (b) Figure 23 : The 25-bar truss: (a) the developed FE model; (b) buckling mode shape associated with the first positive eigenvalue, used to define the initial geometric imperfection.

Force ( kN )

Node

y P

z P

x P

J1 J2 J4 J5

30.0  30.0 

80.0 60.0 30.0 30.0

120.0

100.0

0.0 0.0

0.0

0.0 Table 8: The nodal loading configuration of the 25-bar space truss.

To precisely determine the complementary plastic work p W , the GAMNA is also conducted for this benchmark using the Newton–Raphson scheme. The results are summarized in Fig. 23, showing the load–plastic deformation response and the displacement in the Y-direction of node J2 at the top of the structure. It can be observed that, following the buckling of the group 2 members—consistent with the prescribed geometric imperfection—no further increase in load-carrying capacity occurs. Consequently, only approximately 61.5% of the predefined load level 0 P is achieved during the loading process. In the post-buckling regime, prior to the complete loss of stability, only moderate plastic deformation develops; therefore, the complementary plastic work is evaluated as , 43.40Nmm p init W  . As in the previous benchmark, this value is adopted as the reference 0 p W for the optimization, and the elasto-plastic limit is fixed at , 0 0.25 p max p W W  . Comparison between optimization results of the 25-bar truss After the principal characteristics of the 25-bar truss have been presented and the initial configuration evaluated, two optimization setups are introduced in this section. To characterize the stochastic behavior of the GA and assess methodological reliability, 5 independent runs are performed for each setup. The configurations are summarized in Tab. 9,

149