Issue 75

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

pl pl m  P P

(14)

At this point, excessive plastic deformations prevent the internal stresses from equilibrating the externally applied load 0 P . Consequently, for load multipliers in the range el pl m m m   a residual internal force vector R N , representing the remaining forces after unloading, can be defined as the difference between the plastic internal force pl N and the elastic component el N :

R el   N N N pl

(15)

s G , the complementary strain energy associated with the residual

In this context, for a truss structure of structural weight

forces can be expressed as:

n

l

1

  2 R i



i



N W 

W

(16)

p

p

0

1 E A  i

2

i

where E is the elasticity modulus of the material, R 0 p W represents the allowable limit of plastic deformation in the structure, beyond which the internal stresses can no longer maintain equilibrium, leading to structural collapse. During elastic limit analysis, the primary objective is to determine the elastic load multiplier el m for a structure with a given volume V and corresponding structural weight s G , and to compute the associated elastic limit load el P , provided that the condition 0 p W  remains satisfied. In contrast, the aim of elasto-plastic limit analysis is to determine the maximum load multiplier m such that the complementary plastic work satisfies the inequality 0 p p W W  . To improve safety and enable precise control over inelastic deformations, a more restrictive threshold , 0 0 p max p W W   may be specified, corresponding to the elasto-plastic limit load ep P and its associated multiplier ep m : i N denote the residual force of the i -th bar, and The response curves in Fig. 2 schematically illustrate the identification of the three characteristic loads over the loading history of a representative truss (Fig. 2(a)). In Fig. 2(b), the load–plastic deformation diagram starts at 0 p W  ; the elastic limit el P is read at the last equilibrium state that still satisfies 0 p W  . The elasto-plastic limit ep P is obtained at the prescribed threshold , p max W (with , 0 0 p max p W W   ), such that , p p max W W  and ep ep P m P  . Finally, the plastic limit pl P corresponds to the maximum sustained load on the path. For clarity, the ordinate is marked at { el P , ep P , pl P }, highlighting the progression from the purely elastic regime to controlled inelasticity and, ultimately, to the plastic limit. Fig. 2(c) reports the associated displacement history, which is consistent with the transitions observed on the load–plastic deformation curve and confirms that ep P is reached well before plastic collapse. This implies that, according to structural design criteria, a limited and controlled level of inelastic deformation may be permitted for a structure of given weight s G . Alternatively, the elastic and elasto-plastic limit analysis problem can be reformulated as an optimization task, where the primary objective is to attain a prescribed load multiplier—corresponding to the target load 0 P —while minimizing the structural weight s G . Within this framework, the cross-sectional areas i A of the individual members serve as design variables, framing the task as a size optimization problem, which is one of the three principal categories of structural optimization [21]. ep ep m  P P (17)

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