PSI - Issue 78

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

626

peak responses of the combined system to those of an elastic system with the same natural period T 0 (S) , and damping  0 (S) , as expressed below:

d

*(SE)

(3)

R

=

(

)

d

0 , 

S T

( ) S

( ) S

0

d

V

A

*

*

(SE)

(SE)

(4)

R

=

=

(

)

(

)

a

,

,

* m S T 

S T

0 

0 

( ) S

( ) S

( ) S

( ) S

0

0

e

a

a

The parameters d *(SE) , V *(SE) , and A *(SE) are the peak displacement, force, and acceleration of a nonlinear single degree of freedom (SDOF) model that represents the integrated Structure ( S ) and Exoskeleton ( E ) system. The values S d ( T 0 (S) , ξ 0 (S) ) and S a ( T 0 (S) , ξ 0 (S) ) correspond to spectral displacement and acceleration at the fundamental period T 0 (S) of the structural component. These P-Spectra are derived using nonlinear time-history analysis (NLTHA) on a simplified SDOF system, consisting of two parallel nonlinear springs - one with trilinear Takeda characteristics modeling the structure, and another with elastoplastic kinematic properties for the exoskeleton (see Fig. 4a). Fig. 4b presents the performance spectra (P-spectra) for a structure characterized by an initial period T 0 (S) , damping ratio ξ 0 (S) , and normalized strength ν y (S) - defined as the ratio of peak strength V y *(S) to elastic shear m e * ⸱ S a ( T 0 (S) , ξ 0 (S) ) - under a specified seismic load. Bold black curves represent systems sharing the same stiffness ratio α , while the gray curves correspond to systems with equal ductility ratio µ (E) . The point R a = R d = 1.0 indicates the elastic response of the main structure, whereas the lower-right corner ( µ (E) → ∞ ) reflects its inelastic response. Accordingly, the ratios R d and R a express how acceleration and displacement differ between inelastic and elastic spectral responses. Notably, R d is primarily influenced by stiffness ratio α , while R a is governed mainly by ductility µ (E) . As such, P-Spectra offers an efficient framework for evaluating seismic performance, enabling the achievement of stability and energy dissipation requirements for resilient, earthquake-resistant structures. Additionally, it simplifies structural design by mapping the required damping features of an MDOF system onto the response of an equivalent SDOF system. Using a transformation procedure, the supplemental damping characteristics of the MDOF system are adapted to achieve a targeted response that mimics the idealized SDOF behavior. P-Spectra offer a visual framework that links the responses of idealized inelastic single degree of freedom (SDOF) systems – such as peak and residual displacements, maximum base shear, and acceleration – to essential structural and damping parameters. 2.3. Design Procedure Based on the aforementioned P-spectra, a design procedure is proposed for seismic retrofit of RC buildings using dissipative exoskeletons based on hysteretic dampers. Nonlinear static (pushover) analyses are first performed using multiple vertical load patterns to simulate lateral forces. From these, the capacity curve reflecting the lowest resistance, typically aligned with the fundamental modal response, is selected for the direction of interest (i.e., X/Y). Key structural properties such as the initial period T 0 (S) and strength V y (S) are then defined.

2

T 0

(S) ; r (S) ; v y

(S)

μ (E) =1; α =0.2

( =

Elastic Base Frame Responce

0.4

(

(

1

R a

2

(

0.6

3

Inelastic Base Frame Responce

4

r ( (

P 2

R a,2 R a,1

5

10

P 1

(

0

(

R d,t

0

1

2

d (

d (

d

R d

(a)

(b) Fig. 4. (a) Nonlinear static behavior; (b) Example of P-Spectra.