PSI - Issue 78

Salvatore Mottola et al. / Procedia Structural Integrity 78 (2026) 623–630

625

To simulate the behavior of the retrofitted system, a simplified analytical model is developed comprising two nonlinear springs – one representing the primary structure and the other representing the exoskeleton – each associated with its mass. An additional nonlinear spring can be used to model the connection between the two components (Fig. 2a). In practical applications, the exoskeleton is typically joined to the existing structure using rigid connections to facilitate efficient load transfer. As such, the system can be approximated as two nonlinear springs arranged in parallel (Fig. 2b). The equivalent single degree of freedom (SDOF) representation of the structure is derived from the base shear roof displacement relationship obtained through pushover analysis, specifically using the curve with the lowest base shear demand (Fig. 3a). This capacity curve is then idealized using the Takeda hysteresis model (Takeda et al. , 1970). For this purpose, the initial stiffness, K 0 (S) , corresponds to the tangent stiffness of the pushover curve at the origin (Fig. 3b), and the ratio of cracking to yielding strength is set to 0.30 (i.e., V c (S) / V y (S) =0.30).

K (S) ,μ (S)

K (S) ,μ (S)

K (C)

K (E) ,μ (E)

K (E) ,μ (E)

(a)

(b)

Fig. 2. (a) Two-degree-of-freedom model (flexible connection); (b) Single-degree-of-freedom model (rigid connection).

V

r (S) K

(S)

0

V y

(S)

K 0

(S)

Shear Force

V c

(S)

Pushover curve Trilinear Idealization

Displacement

d c

d

µ (S) d

d (S)

(S)

(S)

(S)

y

y

(a)

(b)

Fig. 3. (a) Pushover curve and trilinear approximation; (b) Parameters of the Takeda hysteresis model.

2.2. Performance spectra The Performance spectra, or P-Spectra, are valuable tools in seismic design, enabling engineers to achieve desired performance goals without iterative recalculations while controlling various response factors under seismic loads. First introduced by Guo and Christopoulos (2013), they were originally developed for structures utilizing hysteretic, viscous, and viscoelastic damping devices. The main structure ( S ) is represented by an idealized SDOF model characterized by parameters including the initial stiffness K 0 (S) , the fundamental period T 0 (S) , and the equivalent mass m e * =Σ( m i  i ) 2 / Σ( m i  i 2 ) where  i denotes the displacement of the i th floor in the fundamental vibration mode. The relative stiffness of the structure compared to the total system stiffness is quantified by the stiffness ratio, expressed as:

( ) S K K K + 0 ( ) S

(1)

 =

( ) E

0

0

The initial natural period of the combined system - comprising the main structure ( S ) and the exoskeleton ( E ) – is given by the following expression:

m

*

(2)

( ) SE

( ) S

2

T

T

=



=



e

0

0

( ) S K K +

( ) E

0

0

The normalized maximum response factors for displacement, R d , and base shear, R a , are defined as the ratios of the