## 2 Influence of Material and Mechanical Parameters on Structural Response and Toughness
In such a case, the total elongation of a tie will result in the one corresponding to the uncracked region plus the opening of the crack:  
\[\delta=\delta_{el}+w=\epsilon_{el}\cdot\ell+w \tag{2}\]  
where both \(\epsilon_{el}\) and \(w\) refer to a given level of stress \(\sigma\). As it can be noted (Figure 1c), when the stress is completely released, the elongation of the tie is equal to the opening of the crack at which tensile stresses vanish (\(w_{\mathrm{res}}\)). This parameter is thus instrumental for the post-peak response of the system and characterizes its structural toughness (area under the force-displacement curve). As it can be noted, the slope characterizing the structural response depends on both material and geometrical parameters (with longer ties associated with more brittle responses, as the released energy in the uncracked regions is higher):  
\[K_{sof}=\frac{F_{cr}}{w_{res}-\delta_{cr}}=\frac{F_{cr}}{w_{res}-\epsilon_{cr}\cdot\ell} \tag{3}\]  
This is a significant observation, as the expected structural response depends not only on the selected material but also on the structural configuration. On the contrary, when a strain hardening response is expected (Figure 1d), multiple cracks develop (when the stresses increase for increasing levels of strain). In this case, the smeared nature of cracking allows considering an equivalent (average) strain in the tension member (refer to \(\epsilon_{w}\) in Figure 1d). Thus, the application of limit analysis for design, assuming smeared deformations, remains fully valid. The responses for isolated elements previously discussed have strong implications on the structural response. Two cases can again be distinguished depending upon the material response. The former refers to a plastic response (Figure 2a), where no strain localization is expected at any element (as for Figure 1c). In this case, redistributions of internal forces happen without restriction. The structural resistance can, in this case, be determined as the load equilibrating the internal forces for a state of stresses where the yield conditions are respected and where the increment of plastic strains are compatible with an admissible failure mechanism (corresponding to an exact solution for associate plasticity conditions [26]). For the simple case shown in Figure 2a, such resistance corresponds to the sum of the individual resistances (Figure 2b), independently of the load path followed (disregarding, for instance, if self-stresses were present before loading). Figure 1: Influence of material properties on the structural response: (**a**) Concrete in compression and stress–strain response in the failure region; (**b**) Uncracked response in tension; (**c**) Cracked region response in case of softening response in tension and localization of strains; (**d**) Cracked response in case of perfectly-plastic or hardening response in tension and smeared cracking. The latter case corresponds to members where strain localization occurs (Figure 2c). In this case, the structural resistance depends on the structural toughness of its elements. For relatively tough members (Figure 2d), the load after the first event of cracking can still be increased. However, the failure load is normally reduced with respect to the theoretical plastic resistance (sum of individual resistances). When the structural toughness of its components is reduced, the structural resistance is also reduced and may be governed by the first event of cracking (Figure 2e,f). In addition, such cases are sensitive to potential self-stresses or imposed strains. Figure 2: Structural response as a function of material and mechanical parameters: (**a**) Investigated elements and system in case of plastic material response; (**b**) Structural plastic response; (**c**) Investigated elements and system in case of softening material response; (**d–f**) Response for reducing levels of material toughness. The numbers with the ring refer to “F_1” and “F_2” in the right-hand diagrams.