An Alternative Procedure for the Description of Seismic Intensity Parameter-Based Damage Potential

Abstract

This study presents an alternative statistical approach for describing the damage potential of R/C structures using various seismic intensity parameters. By employing a comprehensive set of 34 Intensity Measures (IMs) and three well-known Global Damage Indices (DIs), a correlation study was initially conducted to assess the predictive capacity of the selected IMs in estimating a structure’s damage grade. Multiple regression analyses were performed to determine the most suitable IMs for damage prediction, utilizing only the highest correlated IMs with the selected DIs, employing both conventional regression models and a novel alternative approach by transforming the IMs for each predicted DI. The IMs were modified through exponentiation, using powers directly dependent on each IM’s rank correlation coefficient with the respective DI. Employing the rank correlation coefficients to modify the IMs effectively amplifies the influence of those that present the highest agreement with the observed damage. The results demonstrate that energy-related and spectral-based IMs correlate highly with structural damage. The generated models exhibit high accuracy in predicting the observed damage grade, with the models based on the proposed approach showing improved performance in estimating the sustained damage grade while maintaining computational efficiency in terms of their computational time and results’ accuracy.

1. Introduction

Modern building codes provide a framework for analyzing, designing, and assessing structures, focusing on protecting human life. This human life protection is achieved by allowing structural deterioration in sufficiently strong seismic events and preventing partial or total collapse [1,2].

Given the uniqueness and the stochastic nature of strong ground motions, the cause–effect relationship between earthquake excitation and structural response is highly complex [3]. Numerous studies have been conducted in the past decades to identify and quantify the characteristics that best describe the damaging potential of earthquake events, resulting in the realization of several Intensity Measures (IMs), as described in the works of [3,4]. Each IM represents the damage potential of a given strong motion as a single numerical value. It becomes immediately apparent that a single IM is not able to capture the characteristics of an excitation [3] completely.

On the other hand, this uniqueness of seismic events equally impacts the structural response, with significant implications for structural design and assessment [5]. Damage Indices (DIs) are employed to aid decision-makers (engineers and state authorities) both in working proactively for the design of new structures and in quantifying any damage sustained post-earthquake [6], providing damage grade approximations in the form of a single number, usually in the range [0, 1], where the former represents a healthy and the latter a collapsed structure, respectively [7,8]. Various DIs have been proposed in the literature [5,6], each derived from a different scientific standpoint and, therefore, measuring different response quantities. Several methods for damage identification, localization, and assessment have been proposed in the past decades, each catering to specific needs. Such needs arise, e.g., in Structural Health Monitoring (SHM), which deals with the compelling issue of real-time damage detection and localization [9,10,11,12,13]. In SHM, eigen perturbation techniques are commonly utilized to perform damage localization in real-time. Furthermore, Damage Indices based on eigen perturbation techniques are tasked to report sustained damage at a specific point of a structure (local damage). On the other hand, the present study does not focus on the time histories of local damages and their locations. The focus is on the interrelation between global Damage Indices, which lump the structural post-seismic status in a single numerical value, and the single-value seismic Intensity Measures. Furthermore, utilizing IMs in the form defined in the literature requires that the complete time-history record is already known.

The interdependency between seismic excitation and structural response has been verified by previous academic work from recorded post-seismic data and computer simulations [4,14,15]. Regression models have been generated in the past [16] to assess the damage grade of a structure as a function of the IMs of a specific seismic event. In the present study, an alternative procedure is proposed to strengthen the relationship between highly correlated IMs with various DIs, in which the IMs are transformed, directly considering their rank correlation coefficient with their respective DI, to provide an accurate estimation of several Damage Indices. The proposed methodology aims to aid practitioners in performing rapid post-earthquake in situ assessments with only a limited amount of readily available data. This close relationship between IMs and DIs forms the basis for fragility analysis, primarily generating a probabilistic seismic demand model that minimizes the inherent epistemic uncertainty [17]. Statistical procedures and Machine Learning algorithms are usually employed to determine the resulting damage grade of a structure given a specific seismic event, as well as the optimal IM combination for the prediction of the sustained damage, e.g., [16,18,19,20,21,22,23,24]. However, these studies do not consider one of the following key aspects: the utilization of multi-degree-of-freedom oscillators (MDOFs) [18,19], the use of modified IMs to retrieve more accurate estimations of the structural damage [16,20,21], the use of more than one response quantity (DI) to validate the method’s reliability [22], and the establishment of DI prediction equations [23,24]. In this study, IMs and DIs are related through a novel alternative approach, using well-established statistical procedures [25], by assigning an importance factor to the specified IMs to predict the corresponding DIs.

2. Methods

2.1. Seismic Intensity Measures

Characterizing seismic excitations is a challenging and complex task, given the random nature of the phenomenon and the limited understanding surrounding earthquake generation and propagation. Therefore, it is obvious that no single Intensity Measure (IM) perfectly describes and quantifies the destructive potential of a seismic event. For this reason, a series of IMs have been proposed in the past decades and the field remains open to research. IMs are typically divided into four categories: peak, duration, spectral, and energy-related [3]. Peak IMs, such as the Peak Ground Acceleration (PGA) [3], refer to the absolute maximum amplitude of the measured variable observed in a time-history record.

On the other hand, duration-related IMs identify the portions of a strong motion relevant to seismic engineering. For example, Significant Duration (or Strong Motion Duration) [26] pinpoints the significant phase of an earthquake event. Both of these IM types are often easily calculated, yet they provide little information about the overall characteristics of the strong motion. This problem is tackled by using the more sophisticated Spectral and Energy IMs. The former reveals essential information on the frequency content of the excitation and, simultaneously, allows engineers to relate strong motion records with the response of a structural system. One such IM is the Spectral Acceleration (S_a) of an equivalent single-degree-of-freedom oscillator [3]. The latter, i.e., the Energy IMs, characterize an earthquake’s energy output in a lumped manner. One of the most used energy IMs is the Arias Intensity [27].

In this study, a selection of 34 Intensity Measures is utilized to characterize each seismic event, as shown in Table 1. The selected 34 IMs are well-known representatives of peak, spectral-based, and energy-based seismic parameter categories. Initially, no IM is excluded, which could be at the expense of the model’s reliability. Thus, at the start of the proposed procedure, all the numerical values of the 34 IMs are evaluated and included in the present study. After that, a selection is performed based on the rank correlation of each IM with the considered DIs to provide an appropriate technique to rule out IMs that are not highly interrelated with the considered DIs. The numerical computation of the listed Intensity Measures is conducted using the proprietary software SeismoSignal v2024 [28]. As an extensive presentation for each employed intensity parameter is a vast topic thoroughly discussed in the literature [3,4] and is beyond the scope of the current study, interested readers are encouraged to refer to the cited bibliography.

2.2. Seismic Acceleration Time Histories

The assembled dataset must account for a wide range of damage induced to the examined structure. This task cannot be achieved by utilizing only local/national events, given the limited availability of strong Greek motion data and the much more limited availability of sufficiently large events worldwide. A selection of 90 orthogonal pairs of worldwide acceleration records (total 180) was performed to overcome this challenge. All records were collected from the PEER NGA-West2 database [29]. The maximum epicentral distance of the seismic events is about 70 km, including the damaging effects of pulse-like and non-pulse-like excitations for structures at almost any distance from the event’s location, containing near-fault and far-field earthquakes. Table 2 lists information regarding the 90 orthogonal pairs of the selected seismic events.

Since all available databases contain a vast amount of weaker seismic events and a much smaller number of excitations with a medium-to-high damage potential, the strong motion records have been scaled to provide a uniform dataset of minor, moderate, and severe damage to the examined structure while still maintaining the general characteristics that natural excitations exhibit. Details about the scaling procedure will be provided in Section 2.4. Table 1 presents the numerical ranges of all selected IMs for the chosen dataset.

Table 1. Selected Intensity Measures and the corresponding ranges for the examined seismic events.

Intensity Measure	Symbol	Units	Minimum	Maximum	Average	Standard Deviation	References
Housner Intensity	SIH	cm	15.410	673.467	161.324	135.671	[30]
Sa,avg	Sa,avg	g	0.030	1.282	0.372	0.299	[31]
Specific Energy Density	SED	cm2/s	11.378	68,792.353	5602.030	10,271.634	[32]
Velocity Spectrum Intensity	SIv	cm	21.460	635.346	180.210	134.699	[33]
PGV	PGV	cm/s	5.929	222.942	49.955	39.445	[3]
Sv,T1(5%)	Sv,T1(5%)	cm/s	9.811	364.923	84.838	73.964	[3]
Sa,T1(5%)	Sa,T1(5%)	g	0.015	1.991	0.385	0.382	[3]
Sd,T1(5%)	Sd,T1(5%)	cm	0.565	76.975	14.855	14.775	[3]
Sustained Maximum Velocity	SMV	cm/s	1.765	135.907	33.653	27.657	[34]
Velocity RMS	vRMS	cm/s	0.291	39.986	8.058	7.404	[4]
Max Incremental Velocity	MIV	cm/s	7.478	308.843	72.813	55.261	[35,36]
PGD	PGD	cm	0.451	111.511	17.283	22.265	[3]
Displacement RMS	dRMS	cm	0.030	28.531	4.163	5.876	[4]
vmax/amax	vmax/amax	s	0.017	0.436	0.093	0.069	[37]
Mean Period	Tm	s	0.102	1.479	0.498	0.288	[38]
Standardized CAV	CAVstd	g∙s	0.000	4.776	1.217	1.019	[39]
Cumulative Absolute Velocity	CAV	cm	135.287	5062.353	1382.785	1063.057	[40]
Arias Intensity	IA	m/s	0.109	18.247	3.649	3.641	[27]
Characteristic Intensity	IC	g1.5∙s0.5	0.007	0.431	0.120	0.090	[3]
Predominant Period	Tp	s	0.060	1.120	0.330	0.235	[3]
Acceleration RMS	aRMS	g	0.007	0.221	0.068	0.039	[41]
Sustained Maximum Acceleration	SMA	g	0.042	1.190	0.370	0.202	[34]
Acceleration Spectrum Intensity	SIa	g∙s	0.078	1.183	0.436	0.212	[33]
Uniform Duration	tuniform	s	1.340	61.975	12.551	9.899	[26,42]
Bracketed Duration	tbracketed	s	2.780	159.970	27.943	23.404	[43]
Significant Duration	tsignificant	s	0.760	51.200	10.625	8.906	[26]
Effective Design Acceleration	EDA	g	0.120	1.644	0.548	0.240	[40]
PGA	PGA	g	0.121	1.647	0.566	0.257	[3]
Power 0.90	P0.90	m/s2	0.013	2.199	0.404	0.412	[44]
Damage Index	D	-	0.166	51.713	6.466	8.568	[45,46]
A95 parameter	A95	g	0.084	1.332	0.409	0.188	[32]
Number of Effective Cycles	Ncy	-	0.679	17.025	3.277	2.873	[45,46]
Total Duration	ttotal	s	1.375	299.990	54.191	46.780	[3]
IP Index	IP	-	7.194	265.740	33.462	26.118	[47]

Table 1. Selected Intensity Measures and the corresponding ranges for the examined seismic events.

Intensity Measure	Symbol	Units	Minimum	Maximum	Average	Standard Deviation	References
Housner Intensity	SIH	cm	15.410	673.467	161.324	135.671	[30]
Sa,avg	Sa,avg	g	0.030	1.282	0.372	0.299	[31]
Specific Energy Density	SED	cm2/s	11.378	68,792.353	5602.030	10,271.634	[32]
Velocity Spectrum Intensity	SIv	cm	21.460	635.346	180.210	134.699	[33]
PGV	PGV	cm/s	5.929	222.942	49.955	39.445	[3]
Sv,T1(5%)	Sv,T1(5%)	cm/s	9.811	364.923	84.838	73.964	[3]
Sa,T1(5%)	Sa,T1(5%)	g	0.015	1.991	0.385	0.382	[3]
Sd,T1(5%)	Sd,T1(5%)	cm	0.565	76.975	14.855	14.775	[3]
Sustained Maximum Velocity	SMV	cm/s	1.765	135.907	33.653	27.657	[34]
Velocity RMS	vRMS	cm/s	0.291	39.986	8.058	7.404	[4]
Max Incremental Velocity	MIV	cm/s	7.478	308.843	72.813	55.261	[35,36]
PGD	PGD	cm	0.451	111.511	17.283	22.265	[3]
Displacement RMS	dRMS	cm	0.030	28.531	4.163	5.876	[4]
vmax/amax	vmax/amax	s	0.017	0.436	0.093	0.069	[37]
Mean Period	Tm	s	0.102	1.479	0.498	0.288	[38]
Standardized CAV	CAVstd	g∙s	0.000	4.776	1.217	1.019	[39]
Cumulative Absolute Velocity	CAV	cm	135.287	5062.353	1382.785	1063.057	[40]
Arias Intensity	IA	m/s	0.109	18.247	3.649	3.641	[27]
Characteristic Intensity	IC	g1.5∙s0.5	0.007	0.431	0.120	0.090	[3]
Predominant Period	Tp	s	0.060	1.120	0.330	0.235	[3]
Acceleration RMS	aRMS	g	0.007	0.221	0.068	0.039	[41]
Sustained Maximum Acceleration	SMA	g	0.042	1.190	0.370	0.202	[34]
Acceleration Spectrum Intensity	SIa	g∙s	0.078	1.183	0.436	0.212	[33]
Uniform Duration	tuniform	s	1.340	61.975	12.551	9.899	[26,42]
Bracketed Duration	tbracketed	s	2.780	159.970	27.943	23.404	[43]
Significant Duration	tsignificant	s	0.760	51.200	10.625	8.906	[26]
Effective Design Acceleration	EDA	g	0.120	1.644	0.548	0.240	[40]
PGA	PGA	g	0.121	1.647	0.566	0.257	[3]
Power 0.90	P0.90	m/s2	0.013	2.199	0.404	0.412	[44]
Damage Index	D	-	0.166	51.713	6.466	8.568	[45,46]
A95 parameter	A95	g	0.084	1.332	0.409	0.188	[32]
Number of Effective Cycles	Ncy	-	0.679	17.025	3.277	2.873	[45,46]
Total Duration	ttotal	s	1.375	299.990	54.191	46.780	[3]
IP Index	IP	-	7.194	265.740	33.462	26.118	[47]

Table 2. Data for the utilized seismic events.

RSN	Seismic Event	Station	Magnitude	Hypocenter Depth (km)	Epicentral Distance (km)	Vs30 (m/s)	Components
1	Helena, Montana-01	Carroll College	6.00	6.00	6.31	593.35	180, 270
15	Kern County	Taft Lincoln School	7.36	15.63	43.49	385.43	21, 111
33	Parkfield	Temblor pre-1969	6.19	10.00	40.25	527.92	205, 295
50	Lytle Creek	Wrightwood—6074 Park Dr	5.33	8.00	13.01	486.00	115, 205
57	San Fernando	Castaic—Old Ridge Route	6.61	13.00	25.36	450.28	21, 291
125	Friuli, Italy-01	Tolmezzo	6.50	5.10	20.24	505.23	0, 270
134	Izmir, Turkey	Izmir	5.30	5.00	2.29	535.24	L, T
136	Santa Barbara	Santa Barbara Courthouse	5.92	12.70	3.20	514.99	132, 222
139	Tabas, Iran	Dayhook	7.35	5.75	20.63	471.53	190, 280
143	Tabas, Iran	Tabas	7.35	5.75	55.24	766.77	74, 344
150	Coyote Lake	Gilroy Array #6	5.74	8.00	4.37	663.31	230, 320
182	Imperial Valley-06	El Centro Array #7	6.53	9.96	27.64	210.51	140, 230
214	Livermore-01	San Ramon—Eastman Kodak	5.80	12.00	17.13	377.51	180, 270
222	Livermore-02	Livermore—Morgan Terr Park	5.42	14.50	10.33	550.88	265, 355
225	Anza (Horse Canyon)-01	Anza—Pinyon Flat	5.19	13.60	12.68	724.89	45, 135
230	Mammoth Lakes-01	Convict Creek	6.06	9.00	1.43	382.12	90, 180
233	Mammoth Lakes-02	Convict Creek	5.69	14.00	8.60	382.12	90, 180
244	Mammoth Lakes-05	Convict Creek	5.70	4.70	9.36	382.12	90, 180
246	Mammoth Lakes-06	Benton	5.94	14.00	46.49	370.94	0, 270
256	Mammoth Lakes-07	USC McGee Creek Inn	4.73	6.00	1.33	364.84	0, 270
265	Victoria, Mexico	Cerro Prieto	6.33	11.00	33.73	471.53	45, 315
292	Irpinia, Italy-01	Sturno (STN)	6.90	9.50	30.35	382.00	0, 270
313	Corinth, Greece	Corinth	6.60	7.15	19.92	361.40	L, T
359	Coalinga-01	Parkfield—Vineyard Cany 1E	6.36	4.60	34.35	381.27	0, 90
372	Coalinga-02	Anticline Ridge Free-Field	5.09	12.00	3.38	478.63	0, 270
415	Coalinga-05	Transmitter Hill	5.77	7.40	5.99	477.25	0, 270
419	Coalinga-07	Sulphur Baths (temp)	5.21	8.40	12.02	617.43	0, 90
495	Nahanni, Canada	Site 1	6.76	8.00	6.80	605.04	10, 280
534	N. Palm Springs	San Jacinto—Soboba	6.06	11.00	33.53	447.22	0, 90
548	Chalfant Valley-02	Benton	6.19	10.00	31.25	370.94	0, 270
550	Chalfant Valley-02	Bishop—Paradise Lodge	6.19	10.00	15.42	585.12	70, 160
564	Kalamata, Greece-01	Kalamata (bsmt)	6.20	5.00	9.97	382.21	0, 270
568	San Salvador	Geotech Investig Center	5.80	10.90	7.93	489.34	90, 180
572	Taiwan SMART1(45)	SMART1 E02	7.30	15.00	71.35	671.52	0, 90
585	Baja California	Cerro Prieto	5.50	6.00	3.69	471.53	161, 251
619	Whittier Narrows-01	Garvey Res.—Control Bldg	5.99	14.60	2.86	468.18	60, 330
708	Whittier Narrows-02	Altadena—Eaton Canyon	5.27	13.30	13.04	375.16	0, 90
727	Superstition Hills-02	Superstition Mtn Camera	6.54	9.00	7.50	362.38	45, 135
779	Loma Prieta	LGPC	6.93	17.48	18.46	594.83	0, 90
802	Loma Prieta	Saratoga—Aloha Ave	6.93	17.48	27.23	380.89	0, 90
818	Georgia, USSR	Iri	6.20	6.00	40.23	437.72	X, Y
821	Erzican, Turkey	Erzincan	6.69	9.00	8.97	352.05	0, 90
825	Cape Mendocino	Cape Mendocino	7.01	9.50	10.36	567.78	0, 90
830	Cape Mendocino	Shelter Cove Airport	7.01	9.50	36.28	518.98	0, 90
864	Landers	Joshua Tree	7.28	7.00	13.67	379.32	0, 90
881	Landers	Morongo Valley Fire Station	7.28	7.00	21.34	396.41	45, 135
901	Big Bear-01	Big Bear Lake—Civic Center	6.46	13.00	10.15	430.36	0, 270
963	Northridge-01	Castaic—Old Ridge Route	6.69	17.50	40.68	450.28	0, 90
982	Northridge-01	Jensen Filter Plant Administrative Building	6.69	17.50	12.97	373.07	22, 292
1111	Kobe, Japan	Nishi-Akashi	6.90	17.90	8.70	609.00	0, 90
1114	Kobe, Japan	Port Island (0 m)	6.90	17.90	19.25	198.00	0, 90
1231	Chi-Chi, Taiwan	CHY080	7.62	8.00	31.65	496.21	0, 90
1504	Chi-Chi, Taiwan	TCU067	7.62	8.00	28.70	433.63	0, 90
1618	Duzce, Turkey	Lamont 531	7.14	14.00	27.74	638.39	0, 90
1623	Stone Canyon	Melendy Ranch	4.81	8.00	10.19	425.11	61, 331
1632	Upland	Rancho Cucamonga—Law and Justice Center FF, Foothill and Haven	5.63	4.49	12.19	390.18	0, 90
1633	Manjil, Iran	Abbar	7.37	16.00	40.43	723.95	L, T
1676	Northridge-04	Castaic—Old Ridge Route	5.93	9.83	26.95	450.28	0, 90
1734	Northridge-06	Sun Valley—Sunland	5.28	13.09	10.04	393.67	230, 320
1787	Hector Mine	Hector	7.13	14.80	26.53	726.00	0, 90
1948	Anza-02	La Quinta—Bermudas and Durango	4.92	15.20	26.88	360.32	0, 90
2383	Chi-Chi, Taiwan-02	TCU067	5.90	8.00	33.94	433.63	0, 90
2495	Chi-Chi, Taiwan-03	CHY080	6.20	7.80	29.48	496.21	0, 90
3217	Chi-Chi, Taiwan-05	TCU129	6.20	10.00	40.98	511.18	0, 90
3507	Chi-Chi, Taiwan-06	TCU129	6.30	16.00	33.15	511.18	0, 90
3689	Whittier Narrows-02	Big Tujunga, Angeles Nat F	5.27	13.30	27.52	550.11	262, 352
3733	Whittier Narrows-02	Pasadena—Old House Rd	5.27	13.30	12.25	397.27	0, 90
3748	Cape Mendocino	Ferndale Fire Station	7.01	9.50	27.85	387.95	0, 270
3764	Northridge-02	LA—UCLA Grounds	6.05	6.00	23.53	398.42	0, 90
3779	Northridge-06	Glendale—Las Palmas	5.28	13.09	22.72	371.07	177, 267
3845	Chi-Chi (aftershock 2), Taiwan	CHY006	6.20	7.80	39.60	438.19	0, 270
3865	Chi-Chi (aftershock 5), Taiwan	CHY006	6.30	16.00	56.64	438.19	0, 270
4031	San Simeon, CA	Templeton—1-story Hospital	6.5	8.50	36.63	410.66	0, 90
4040	Bam, Iran	Bam	6.6	6.00	12.59	487.40	8, 278
4097	Parkfield-02, CA	Slack Canyon	6.00	8.10	31.53	648.09	0, 90
4114	Parkfield-02, CA	Parkfield—Fault Zone 11	6.00	8.10	9.28	541.73	0, 90
4284	Basso Tirreno, Italy	Naso	6	15.00	18.04	620.56	0, 90
4456	Montenegro, Yugo.	Petrovac—Hotel Olivia	7.10	7.00	28.30	543.26	0, 90
4861	Chuetsu-oki	Nakanoshima Nagaoka	6.80	9.00	23.28	319.00	0, 90
5663	Iwate	MYG004	6.90	6.50	35.40	479.37	0, 90
5819	Iwate	Ichinoseki Maikawa	6.90	6.50	30.72	640.14	0, 90
5823	El Mayor-Cucapah	Chihuahua	7.20	5.45	20.63	242.05	0, 90
6059	Big Bear-01	Morongo Valley Fire Station	6.46	13.00	29.33	396.41	45, 135
6952	Darfield, New Zealand	Papanui High School	7.00	10.90	46.94	263.20	123, 213
8157	Christchurch, New Zealand	Heathcote Valley Primary School	6.2	6.00	1.11	422.00	116, 206
8169	San Juan Bautista	San Andreas Geophysical Obs., Hollister, CA, USA	5.17	9.13	1.79	643.80	0, 90
9071	14151344	Pinon Flats Observatory, CA, USA	5.20	15.48	13.47	763.00	0, 90
11133	21530368	Sonoma Mountain	4.50	6.99	2.19	664.57	0, 90
12261	40199209	Oakland—Hwys 13 and 24	4.20	4.13	6.02	511.12	0, 90
20125	40204628	Coe Ranch	5.45	7.49	21.25	400.00	0, 90

Table 2. Data for the utilized seismic events.

RSN	Seismic Event	Station	Magnitude	Hypocenter Depth (km)	Epicentral Distance (km)	Vs30 (m/s)	Components
1	Helena, Montana-01	Carroll College	6.00	6.00	6.31	593.35	180, 270
15	Kern County	Taft Lincoln School	7.36	15.63	43.49	385.43	21, 111
33	Parkfield	Temblor pre-1969	6.19	10.00	40.25	527.92	205, 295
50	Lytle Creek	Wrightwood—6074 Park Dr	5.33	8.00	13.01	486.00	115, 205
57	San Fernando	Castaic—Old Ridge Route	6.61	13.00	25.36	450.28	21, 291
125	Friuli, Italy-01	Tolmezzo	6.50	5.10	20.24	505.23	0, 270
134	Izmir, Turkey	Izmir	5.30	5.00	2.29	535.24	L, T
136	Santa Barbara	Santa Barbara Courthouse	5.92	12.70	3.20	514.99	132, 222
139	Tabas, Iran	Dayhook	7.35	5.75	20.63	471.53	190, 280
143	Tabas, Iran	Tabas	7.35	5.75	55.24	766.77	74, 344
150	Coyote Lake	Gilroy Array #6	5.74	8.00	4.37	663.31	230, 320
182	Imperial Valley-06	El Centro Array #7	6.53	9.96	27.64	210.51	140, 230
214	Livermore-01	San Ramon—Eastman Kodak	5.80	12.00	17.13	377.51	180, 270
222	Livermore-02	Livermore—Morgan Terr Park	5.42	14.50	10.33	550.88	265, 355
225	Anza (Horse Canyon)-01	Anza—Pinyon Flat	5.19	13.60	12.68	724.89	45, 135
230	Mammoth Lakes-01	Convict Creek	6.06	9.00	1.43	382.12	90, 180
233	Mammoth Lakes-02	Convict Creek	5.69	14.00	8.60	382.12	90, 180
244	Mammoth Lakes-05	Convict Creek	5.70	4.70	9.36	382.12	90, 180
246	Mammoth Lakes-06	Benton	5.94	14.00	46.49	370.94	0, 270
256	Mammoth Lakes-07	USC McGee Creek Inn	4.73	6.00	1.33	364.84	0, 270
265	Victoria, Mexico	Cerro Prieto	6.33	11.00	33.73	471.53	45, 315
292	Irpinia, Italy-01	Sturno (STN)	6.90	9.50	30.35	382.00	0, 270
313	Corinth, Greece	Corinth	6.60	7.15	19.92	361.40	L, T
359	Coalinga-01	Parkfield—Vineyard Cany 1E	6.36	4.60	34.35	381.27	0, 90
372	Coalinga-02	Anticline Ridge Free-Field	5.09	12.00	3.38	478.63	0, 270
415	Coalinga-05	Transmitter Hill	5.77	7.40	5.99	477.25	0, 270
419	Coalinga-07	Sulphur Baths (temp)	5.21	8.40	12.02	617.43	0, 90
495	Nahanni, Canada	Site 1	6.76	8.00	6.80	605.04	10, 280
534	N. Palm Springs	San Jacinto—Soboba	6.06	11.00	33.53	447.22	0, 90
548	Chalfant Valley-02	Benton	6.19	10.00	31.25	370.94	0, 270
550	Chalfant Valley-02	Bishop—Paradise Lodge	6.19	10.00	15.42	585.12	70, 160
564	Kalamata, Greece-01	Kalamata (bsmt)	6.20	5.00	9.97	382.21	0, 270
568	San Salvador	Geotech Investig Center	5.80	10.90	7.93	489.34	90, 180
572	Taiwan SMART1(45)	SMART1 E02	7.30	15.00	71.35	671.52	0, 90
585	Baja California	Cerro Prieto	5.50	6.00	3.69	471.53	161, 251
619	Whittier Narrows-01	Garvey Res.—Control Bldg	5.99	14.60	2.86	468.18	60, 330
708	Whittier Narrows-02	Altadena—Eaton Canyon	5.27	13.30	13.04	375.16	0, 90
727	Superstition Hills-02	Superstition Mtn Camera	6.54	9.00	7.50	362.38	45, 135
779	Loma Prieta	LGPC	6.93	17.48	18.46	594.83	0, 90
802	Loma Prieta	Saratoga—Aloha Ave	6.93	17.48	27.23	380.89	0, 90
818	Georgia, USSR	Iri	6.20	6.00	40.23	437.72	X, Y
821	Erzican, Turkey	Erzincan	6.69	9.00	8.97	352.05	0, 90
825	Cape Mendocino	Cape Mendocino	7.01	9.50	10.36	567.78	0, 90
830	Cape Mendocino	Shelter Cove Airport	7.01	9.50	36.28	518.98	0, 90
864	Landers	Joshua Tree	7.28	7.00	13.67	379.32	0, 90
881	Landers	Morongo Valley Fire Station	7.28	7.00	21.34	396.41	45, 135
901	Big Bear-01	Big Bear Lake—Civic Center	6.46	13.00	10.15	430.36	0, 270
963	Northridge-01	Castaic—Old Ridge Route	6.69	17.50	40.68	450.28	0, 90
982	Northridge-01	Jensen Filter Plant Administrative Building	6.69	17.50	12.97	373.07	22, 292
1111	Kobe, Japan	Nishi-Akashi	6.90	17.90	8.70	609.00	0, 90
1114	Kobe, Japan	Port Island (0 m)	6.90	17.90	19.25	198.00	0, 90
1231	Chi-Chi, Taiwan	CHY080	7.62	8.00	31.65	496.21	0, 90
1504	Chi-Chi, Taiwan	TCU067	7.62	8.00	28.70	433.63	0, 90
1618	Duzce, Turkey	Lamont 531	7.14	14.00	27.74	638.39	0, 90
1623	Stone Canyon	Melendy Ranch	4.81	8.00	10.19	425.11	61, 331
1632	Upland	Rancho Cucamonga—Law and Justice Center FF, Foothill and Haven	5.63	4.49	12.19	390.18	0, 90
1633	Manjil, Iran	Abbar	7.37	16.00	40.43	723.95	L, T
1676	Northridge-04	Castaic—Old Ridge Route	5.93	9.83	26.95	450.28	0, 90
1734	Northridge-06	Sun Valley—Sunland	5.28	13.09	10.04	393.67	230, 320
1787	Hector Mine	Hector	7.13	14.80	26.53	726.00	0, 90
1948	Anza-02	La Quinta—Bermudas and Durango	4.92	15.20	26.88	360.32	0, 90
2383	Chi-Chi, Taiwan-02	TCU067	5.90	8.00	33.94	433.63	0, 90
2495	Chi-Chi, Taiwan-03	CHY080	6.20	7.80	29.48	496.21	0, 90
3217	Chi-Chi, Taiwan-05	TCU129	6.20	10.00	40.98	511.18	0, 90
3507	Chi-Chi, Taiwan-06	TCU129	6.30	16.00	33.15	511.18	0, 90
3689	Whittier Narrows-02	Big Tujunga, Angeles Nat F	5.27	13.30	27.52	550.11	262, 352
3733	Whittier Narrows-02	Pasadena—Old House Rd	5.27	13.30	12.25	397.27	0, 90
3748	Cape Mendocino	Ferndale Fire Station	7.01	9.50	27.85	387.95	0, 270
3764	Northridge-02	LA—UCLA Grounds	6.05	6.00	23.53	398.42	0, 90
3779	Northridge-06	Glendale—Las Palmas	5.28	13.09	22.72	371.07	177, 267
3845	Chi-Chi (aftershock 2), Taiwan	CHY006	6.20	7.80	39.60	438.19	0, 270
3865	Chi-Chi (aftershock 5), Taiwan	CHY006	6.30	16.00	56.64	438.19	0, 270
4031	San Simeon, CA	Templeton—1-story Hospital	6.5	8.50	36.63	410.66	0, 90
4040	Bam, Iran	Bam	6.6	6.00	12.59	487.40	8, 278
4097	Parkfield-02, CA	Slack Canyon	6.00	8.10	31.53	648.09	0, 90
4114	Parkfield-02, CA	Parkfield—Fault Zone 11	6.00	8.10	9.28	541.73	0, 90
4284	Basso Tirreno, Italy	Naso	6	15.00	18.04	620.56	0, 90
4456	Montenegro, Yugo.	Petrovac—Hotel Olivia	7.10	7.00	28.30	543.26	0, 90
4861	Chuetsu-oki	Nakanoshima Nagaoka	6.80	9.00	23.28	319.00	0, 90
5663	Iwate	MYG004	6.90	6.50	35.40	479.37	0, 90
5819	Iwate	Ichinoseki Maikawa	6.90	6.50	30.72	640.14	0, 90
5823	El Mayor-Cucapah	Chihuahua	7.20	5.45	20.63	242.05	0, 90
6059	Big Bear-01	Morongo Valley Fire Station	6.46	13.00	29.33	396.41	45, 135
6952	Darfield, New Zealand	Papanui High School	7.00	10.90	46.94	263.20	123, 213
8157	Christchurch, New Zealand	Heathcote Valley Primary School	6.2	6.00	1.11	422.00	116, 206
8169	San Juan Bautista	San Andreas Geophysical Obs., Hollister, CA, USA	5.17	9.13	1.79	643.80	0, 90
9071	14151344	Pinon Flats Observatory, CA, USA	5.20	15.48	13.47	763.00	0, 90
11133	21530368	Sonoma Mountain	4.50	6.99	2.19	664.57	0, 90
12261	40199209	Oakland—Hwys 13 and 24	4.20	4.13	6.02	511.12	0, 90
20125	40204628	Coe Ranch	5.45	7.49	21.25	400.00	0, 90

2.3. Damage Indices

In contrast to Intensity Measures, which describe a seismic event’s ability to excite a structure and the degree to which this takes place, Damage Indices (DIs) provide a way to quantify the sustained damage of a single structural element or the structure as a whole. A local DI is used when a single structural element is to be assessed. In contrast, when a whole structural system is considered, a global DI is utilized to describe the sustained structural damage in a lumped manner. Furthermore, it means that the global DI does not inform us about the location of the damage, only about the overall state of the structure. Global Damage Indices result in one of two ways: as an immediate result of a specific methodology or from some form of averaging the local Damage Indices of the structure’s elements. DIs typically lie within the range [0, 1], with 0 indicating a healthy/unharmed structural state and 1 indicating total collapse of the element/structure. Detailed discussions on all the significant Damage Indices in the literature can be found in [5].

This study focuses on utilizing Global Damage Indices, which, for simplicity’s sake, will be referenced as DIs for the remainder of the article. Three well-known DIs have been investigated, each for different reasons, which will be explained in the following.

The Normalized Maximum Inter-story Drift Ratio (NMISDR) is the value of the MISDR [16], i.e., is the ratio of lateral displacement relative to story height divided by the threshold value of inter-story drift at collapse:

$$\mathrm{N}\mathrm{MISDR}={\displaystyle \frac{{\left|\mathrm{u}\right|}_{\max }}{\mathrm{h}}}·{\displaystyle \frac{1}{{\mathrm{MISDR}}_{\mathrm{threshold}}}}$$

(1)

where h is the story height.

Since it is closely related to the maximum attained member rotations, inter-story drift is an easily calculated quantity that many building codes and guidelines use as the preferred indicator to limit damage formation on structures. Reference [16] presents inter-story drift limits for various structural damage degrees, with a 2.5% threshold inter-story drift corresponding to collapse. Dividing the MISDR with the threshold value allows direct comparison with other DIs.

The Modified Park–Ang Damage Index [7] is one of the most widely used Damage Indices that incorporates ductility and hysteretic energy dissipation in a single expression, effectively capturing damage induced by both monotonic and cyclic loading effects:

$${\mathrm{D}\mathrm{I}}_{\mathrm{P}\mathrm{A},\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}={\displaystyle \frac{{\mathsf{\theta }}_{\mathrm{m}}-{\mathsf{\theta }}_{\mathrm{r}}}{{\mathsf{\theta }}_{\mathrm{u}}-{\mathsf{\theta }}_{\mathrm{r}}}}+{\displaystyle \frac{\mathsf{\beta }}{{\mathrm{M}}_{\mathrm{y}}·{\mathsf{\theta }}_{\mathrm{u}}}}·{\mathrm{E}}_{\mathrm{T}}$$

(2)

where θ_m is the maximum attained rotation during load history, θ_u is the cross-sectional ultimate rotation capacity, θ_r is the recoverable rotation at unloading, M_y is the section yield moment, E_T is the dissipated hysteretic energy, and β is a strength deterioration control parameter that is related to the degree of section detailing, with a higher value indicating a poorly detailed cross-section.

The global value of the Modified Park–Ang DI (DI_PA) is calculated with a weighted average procedure, with the dissipated hysteretic energies acting as the weights of their respective sections:

$${\mathrm{D}\mathrm{I}}_{\mathrm{P}\mathrm{A}}={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{D}\mathrm{I}}_{\mathrm{P}\mathrm{A},\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l},\mathrm{i}}·{\mathrm{E}}_{\mathrm{i}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{E}}_{\mathrm{i}}}}$$

(3)

where n is the number of locations of the computed local Damage Indices.

The last DI is the Maximum Softening (δ_Μ) [8,48], which uses the first-mode natural period of the structure to produce a lumped approximation of the accumulated damage sustained during an earthquake:

$${\mathsf{\delta }}_{\mathrm{M}}=1-{\displaystyle \frac{{\mathrm{T}}_0}{{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}$$

(4)

where Τ₀ is the first-mode natural period of the system at its unharmed state, and T_max is the maximum value of the first-mode period observed during the analysis.

A moving average smoothing procedure, described in [48], must be employed to compute the Maximum Softening in a non-linear analysis using the instantaneous natural period. This procedure attenuates the influence of all the sudden, momentary increases in the natural period during yield zone formations that would otherwise result in a significant overestimation of the actual sustained damage.

The examined DIs are well-known and extensively studied in numerical and experimental research. Many researchers have verified the modified Park–Ang DI experimentally, specifically for R/C frames [7,49,50]. The Maximum Softening DI has a sound theoretical background and agrees well with the Park–Ang DI in estimating the observed structural damage [8]. The MISDR can be found in most building codes as the defining criterion for damage limitation of structural and non-structural members [1,51]. In addition, the MISDR can be evaluated numerically and experimentally without enormous effort. Utilizing more than one DI allows for cross-checking the validity of one’s result to another’s. Thus, the results’ validity increases when all DIs provide similar damage grades for a specific seismic event.

A classification of the different damage grades based on these Damage Indices is shown in

Table 3. Classification limits for various Damage Indices.

Damage Grade	MISDR	Modified Park–Ang	Maximum Softening
Minor	≤0.5%	≤0.30	≤0.30
Moderate	0.5–1.5%	0.30–0.60	0.30–0.60
Severe	1.5–2.5%	0.60–0.80	0.60–0.80
Partial or Total Collapse	≥2.5%	≥0.80	≥0.80

, as described in [16]. The minor category corresponds to a healthy structure or one that has sustained minor damage, usually requiring no further actions. Moderate refers to repairable damage and Severe refers to either irreparable damage or collapse of the structure, partially or totally. Since an irreparable damage grade leads to the demolition of said structure, the distinction between demolition and collapse is deemed insignificant and the two can be merged into one category.

2.4. Numerical Modelling and Investigation on a Reinforced Concrete Frame Structure

The proposed methodology is applied to a 10-story, 3-bay reinforced concrete frame structure that complies with EC2 and EC8 standards [1,2]. The designed structure is doubly symmetric and regular, both in-plan and in-elevation, to eliminate the impact of torsional effects that could potentially skew the results. This building type is widespread in regions with high seismicity (e.g., in Greece). Using a two-dimensional model allows for accurate investigation of the ductile behavior of regular R/C buildings without the performance overhead of three-dimensional analyses. On the other hand, a 2D structural model neglects torsional and directionality effects. However, the present study focuses not on the effects previously mentioned but on the interrelation between seismic intensity parameters and overall structural Damage Indices. The loads considered in the analysis were self-weight, live, snow, wind, seismic excitation, and geometric imperfections. Regarding environment, soil, and seismicity conditions, exposure class XC3 and soil type B were considered, and a design acceleration of 0.24 g was achieved. The structure was designed for the medium ductility class (DCM) and importance class II. The beam and column dimensions (in cm) are illustrated in Figure 1. The slab thickness was 0.20 m, and the T-beam effective flange widths were calculated according to Eurocode standards. The fundamental period of the test frame was computed as T₁ = 1.25 s.

Upon design and detailing completion, the test frame was modeled in the computer program IDARC-2D [7] and was subjected to a series of non-linear time history analyses for all the selected ground motions to obtain all the necessary data (MISDR, DI_PA, Fundamental Period). The test frame was initially subjected to all 180 unscaled accelerograms. Then, a scale factor was adopted for each accelerogram. It was determined in such a way that, using the MISDR as the target parameter, the scale factor of each accelerogram was adjusted until a uniform set of the damage grades was achieved, i.e., 60 cases of minor, 60 cases of moderate, and 60 cases of severe damage sustained to the structure, respectively. IDARC-2D features distributed and concentrated plasticity elements with uniform and linear flexibility distributions for their rigid ends. The program incorporates a polygonal hysteretic model with stiffness degradation, strength deterioration, and pinching behavior to simulate the hysteretic response of reinforced concrete members. In this study, distributed plasticity elements with linear flexibility distribution were selected, as the calibration of the hysteretic response parameter set with experimental data has been conducted on this type of element, given that prior releases of the program did not contain all of the above options. Nominal values without slip effects corresponding to structures designed according to the Eurocodes mentioned above were adopted in the current study. The hysteretic parameters have been calibrated with experimental data, as found in the case studies section of [7] for various reinforced concrete structures. Furthermore, a Rayleigh damping model was assigned to the structure, with 5% critical damping. IDARD-2D beam and column elements utilize a trilinear backbone curve to calculate the updated tangent stiffness at each analysis step, using the cross-sections’ moment-curvature characteristics. Following the dynamic non-linear analyses, the Fundamental Period, the MISDR, and the DI_PA were directly extracted at various time steps. The Maximum Softening Damage Index was calculated according to the procedure described previously [48], using the instantaneous Fundamental Period of the structure.

2.5. Multiple Regression Analysis

Properly selecting the most appropriate IMs for characterizing a seismic event is crucial in relating IMs with the sustained damage grade in a structure. A random selection of 30 records out of the 180 representing all degrees of estimated damage were selected as a test set and the remaining 150 were used as a training set. As there is no strict methodology for dividing the dataset into training and testing subsets and given the fact that all three damage grades (minor, moderate, or severe) should be equally represented in both the training and the testing sets, the data were split into 50–10 (training–test) instances for each damage grade subset. Considering that the usual testing set is around 20% of the initial data and the small number of available acceleration records, it was deemed appropriate that the training subset should contain enough cases to avoid jeopardizing the models’ generalization capabilities. After some trial-and-error tests, it was found that the 50–10 cases split is the ideal ratio, as it provides enough instances at both training and testing procedures. Each set of 10 and 50 instances was randomly selected to ensure that the models were able to generalize and were not biased by any sequential cases. A correlation study [25] between IMs and the previously computed DIs was conducted on the training set to assess the selected Intensity Measures’ predictive capacity and eliminate any irrelevant predictors. All statistical analyses were performed using the computer program STATGRAPHICS [52]. The remaining IMs, i.e., those with the highest correlation with structural damage, should all be able to differentiate between different intensity levels when different scaling factors are applied to the same input signal. Thus, when a scale factor is applied to the accelerogram, the IMs should change values accordingly. A threshold limit ρ_lim of 90% rank correlation for the selected IMs was adopted, and only those that exhibited at least 90% rank correlation with all three DIs were used in the subsequent analysis steps.

With the remaining 10 IMs and the studied 3 DIs, a series of multiple regression analyses [25] were performed. In a multiple regression analysis, the dependent variable (y) is expressed as a linear combination of a constant term (a), (n)-independent explanatory variables (x_i, where i = 1, …, n) with the corresponding coefficients (b_i), and an error term (e), via the least squares method. The mathematical expression and its form in the current application are shown in Equation (5).

$$\mathrm{D}\mathrm{I}={\mathrm{b}}_1·{\mathrm{I}\mathrm{M}}_1+{\mathrm{b}}_2·{\mathrm{I}\mathrm{M}}_2+\cdots +{\mathrm{b}}_{\mathrm{n}}·{\mathrm{I}\mathrm{M}}_{\mathrm{n}}+\mathrm{e}$$

(5)

Here, the Damage Index (DI) is the dependent variable, and the Intensity Measures (IMs) are the (n)-independent explanatory variables. Note that the constant term (a) was omitted (a = 0), as the models should not indicate any level of induced damage when a sufficiently weak (or no) excitation is provided to predict the corresponding Damage Index.

Three multiple regression methods were tested to determine the best results: Ordinary Least Squares, Forward Stepwise Selection, and Backward Stepwise Selection [25]. The first is a direct method that incorporates all independent variables to generate the fitted model, thus resulting in models with many explanatory variables. Forward and Backward Stepwise Selection methods are also utilized to overcome this drawback. The former begins with an empty model and iteratively adds all relevant predictors to the model. This method usually produces models with the lowest possible number of predictors, generating simpler models. The latter achieves the same goal by starting with a model containing all the available independent variables and iteratively eliminating all irrelevant ones from the model. In both methods, the selection criterion is the p-value, which is defined in [25] as “the smallest level of significance that would lead to rejection of the null hypothesis H₀ with the given data” and was assigned the value of 5% in this study.

Following the statistical analysis of the initial set of models (Model Set 1: contains models 1.1, 1.2, and 1.3), a novel alternative approach is introduced (Model Set 2: includes models 2.1, 2.2, and 2.3) to highlight further the significance of each explanatory variable (IM) in predicting a specified Damage Index. This is achieved through the application of an importance factor to each of the selected Intensity Measures by raising each IM to a power related to the previously calculated rank correlation coefficient (${\mathsf{\rho }}_{{\mathrm{IM}}_{\mathrm{i}}}$) between the IM and the respective DI. By exponentiating all IMs relating to their respective rank correlation with a specific DI, a transformation of the selected IMs can be obtained, leading to a new set of independent variables for the regression model. The general form of the proposed expression and the associated exponent (c_i, where i = 1, …, n) of each IM is shown in Equations (6a) and (6b), respectively.

$$\mathrm{D}\mathrm{I}={\mathrm{b}}_1·{{\mathrm{I}\mathrm{M}}_1}^{{\mathrm{c}}_1}+{\mathrm{b}}_2·{{\mathrm{I}\mathrm{M}}_2}^{{\mathrm{c}}_2}+\cdots +{\mathrm{b}}_{\mathrm{n}}·{{\mathrm{I}\mathrm{M}}_{\mathrm{n}}}^{{\mathrm{c}}_{\mathrm{n}}}+\mathrm{e}$$

(6a)

$${\mathrm{c}}_{\mathrm{i}}={\displaystyle \frac{{\mathsf{\rho }}_{{\mathrm{I}\mathrm{M}}_{\mathrm{i}}}-{\mathsf{\rho }}_{\mathrm{l}\mathrm{i}\mathrm{m}}}{1-{\mathsf{\rho }}_{\mathrm{l}\mathrm{i}\mathrm{m}}}}$$

(6b)

Equation (6b) presents the general expression of the exponent for each IM that has been previously selected purely based on its rank correlation with the dependent variable (DI). Here, ρ_lim denotes the lower limit (threshold) of the rank correlation coefficient that all selected IMs must satisfy, which in this study was assigned the value of 90%. Any IMs that did not exceed this threshold were discarded in any of the three studied DIs.

The exponent results in values within the range [0, 1], with a value of 1 indicating perfect correlation of the corresponding IM with the predicted DI. The higher the rank correlation between an IM and a DI, the closer the modified IM is to its original value.

Table 4. Examined regression expressions and methods.

Model	Expression	Regression Method
Model 1.1	$\mathrm{D}\mathrm{I}=\sum _{\mathrm{i}}\left({\mathrm{c}}_{\mathrm{i}}\cdot \mathrm{I}{\mathrm{M}}_{\mathrm{i}}\right)$	Ordinary Least Squares
Model 1.2		Forward Stepwise Selection
Model 1.3		Backward Stepwise Selection
Model 2.1 1	$\mathrm{D}\mathrm{I}={\displaystyle \sum _{\mathrm{i}}}\left({\mathrm{c}}_{\mathrm{i}}\cdot {\mathrm{I}\mathrm{M}}_{\mathrm{i}}^{{\displaystyle \frac{{\mathsf{\rho }}_{{\mathrm{I}\mathrm{M}}_{\mathrm{i}}}-{\mathsf{\rho }}_{\mathrm{l}\mathrm{i}\mathrm{m}}}{1-{\mathsf{\rho }}_{\mathrm{l}\mathrm{i}\mathrm{m}}}}}\right)$	Ordinary Least Squares
Model 2.2 1		Forward Stepwise Selection
Model 2.3 1		Backward Stepwise Selection

¹ ρ_lim = 90%: Lower limit for the rank correlation of the selected explanatory variables.

lists all examined regression methods and the general form of the expressions produced by each method.

3. Results and Discussion

3.1. Training the Multiple Regression Models

As previously stated, a correlation study was initially conducted to determine the intensity parameters that correlate the most with the associated Damage Indices. Figure 2 illustrates the rank correlation coefficients of all IMs for a given DI in descending order.

A lower limit (threshold) of ρ_lim = 90% was adopted to select the IMs used in the multiple regression analyses. The selected IMs and their respective rank correlation coefficients are presented in

Table 5. Selected Intensity Measures and rank correlation coefficients with the examined Damage Indices.

Intensity Measure	Normalized MISDR	Maximum Softening	Modified Park–Ang
Housner Intensity	96.79%	95.32%	94.71%
Sa,avg	95.98%	95.12%	94.12%
Specific Energy Density	93.75%	95.31%	95.79%
Velocity Spectrum Intensity	96.47%	92.81%	92.84%
PGV	94.85%	92.00%	92.20%
Sv,T1(5%)	95.54%	91.88%	90.49%
Sa,T1(5%)	93.76%	93.38%	90.09%
Sd,T1(5%)	93.57%	93.32%	90.07%
Sustained Maximum Velocity	91.43%	91.01%	93.03%
Velocity RMS	91.72%	91.73%	91.69%

. The most highly correlated IMs come from the Spectral and Energy categories, with only two out of the ten selected belonging to the Peak category (PGV and Sustained Maximum Velocity). An important observation regarding the Peak-based IMs is that they are velocity-related. Velocity parameters often characterize a pulse-like strong motion, in which the pulse is usually found within the velocity time history. Specifically, PGV has been identified as a key parameter in pulse-like excitations with forward directivity effects [47].

Using the selected 10 IMs, a series of Multiple Regression Analyses were performed. Ordinary Least Squares, Forward Stepwise Selection, and Backward Stepwise Selection iterative procedures [25] were employed both on the conventional regression models (Model Set 1) and the proposed models (Model Set 2) by raising each IM to an exponent dependent upon its rank correlation coefficient with the corresponding DI. The coefficients of the fitted models, per the Damage Index and model, are presented in

Table 6. Regression model coefficients for the Maximum Softening Damage Index.

Intensity Measure	Model 1.1	Model 1.2	Model 1.3	Model 2.1	Model 2.2	Model 2.3
Housner Intensity	−0.00315	−0.00396	−0.00327	−0.03584	−0.03541	−0.03617
Sa,avg	2.14631	2.27422	2.09444	1.61169	2.01052	1.60807
Specific Energy Density	−3.55 × 10−7	-	-	0.00082	0.00079	0.00088
Velocity Spectrum Intensity	−0.00046	-	-	−0.02279	−0.06588	-
PGV	0.00101	0.00147	-	0.06651	-	-
Sv,T1(5%)	0.00207	-	0.00172	−0.03221	-	-
Sa,T1(5%)	1.28091	-	-	−9.87011	-	−9.96924
Sd,T1(5%)	−0.04399	-	−0.00987	3.02685	-	3.03894
Sustained Maximum Velocity	0.00210	0.00317	0.00214	−0.36318	-	−0.36436
Velocity RMS	0.00644	-	0.00881	0.18342	-	0.19688

,

Table 7. Regression model coefficients for the NMISDR Damage Index.

Intensity Measure	Model 1.1	Model 1.2	Model 1.3	Model 2.1	Model 2.2	Model 2.3
Housner Intensity	−0.00233	−0.00388	−0.00277	−0.01671	-	-
Sa,avg	2.56422	2.43042	2.45724	1.66802	1.18412	1.18412
Specific Energy Density	0.00000	4.30 × 10−6	3.91× 10−6	0.00925	0.00826	0.00826
Velocity Spectrum Intensity	−0.00133	0.00119	-	0.00560	-	-
PGV	0.00100	-	-	0.01069	-	-
Sv,T1(5%)	0.00361	-	0.00226	0.00840	-	-
Sa,T1(5%)	11.08590	-	-	2.94016	-	-
Sd,T1(5%)	−0.30221	-	−0.01104	−0.81300	-	-
Sustained Maximum Velocity	0.00011	-	-	−0.26576	−0.14961	−0.14961
Velocity RMS	0.00282	-	-	0.12614	-	-

and

Table 8. Regression model coefficients for the DI_PA Damage Index.

Intensity Measure	Model 1.1	Model 1.2	Model 1.3	Model 2.1	Model 2.2	Model 2.3
Housner Intensity	0.00084	-	-	0.06351	0.04842	0.07174
Sa,avg	0.84262	0.48282	1.09999	0.29184	-	-
Specific Energy Density	0.00001	0.00001	0.00001	0.00119	0.00136	0.00129
Velocity Spectrum Intensity	−0.00196	-	−0.00180	−0.28076	−0.08295	−0.20725
PGV	0.00014	-	-	0.08743	-	-
Sv,T1(5%)	0.00330	-	0.00277	3.16888	-	2.70311
Sa,T1(5%)	−11.95520	-	−0.36320	−51.95980	-	−45.04080
Sd,T1(5%)	0.29622	-	-	46.97820	-	40.80560
Sustained Maximum Velocity	0.00199	0.00207	0.00212	0.02463	-	-
Velocity RMS	0.00099	-	-	−0.04007	-	-

. Coincidentally, in the case of Model Set 2 of the Normalized MISDR, both Forward (Model 2.2) and Backward (Model 2.3) Stepwise Selection produce the same fitted model, which results from the selection criterion’s values for each IM.

As an example, the application of Equations (5), (6a) and (6b) for the prediction of the Maximum Softening DI for Model 1.2 and Model 2.2 is provided below in Equations (7) and (8), respectively:

$${\mathsf{\delta }}_{\mathsf{{\rm M}}}=-0.00396·{\mathrm{S}\mathrm{I}}_{\mathrm{H}}+2.27422·{\mathrm{S}}_{\mathrm{a},\mathrm{a}\mathrm{v}\mathrm{g}}+0.00147·\mathrm{P}\mathrm{G}\mathrm{V}+0.00317·\mathrm{S}\mathrm{M}\mathrm{V}$$

(7)

$$\begin{gathered} {\mathsf{\delta }}_{\mathsf{{\rm M}}}=-0.03541·{{\mathrm{S}\mathrm{I}}_{\mathrm{H}}}^{{\displaystyle \frac{0.9532-0.9}{1-0.9}}}+2.01052·{{\mathrm{S}}_{\mathrm{a},\mathrm{a}\mathrm{v}\mathrm{g}}}^{{\displaystyle \frac{0.9512-0.9}{1-0.9}}} \\ +0.00079·{\mathrm{S}\mathrm{E}\mathrm{D}}^{{\displaystyle \frac{0.9531-0.9}{1-0.9}}}-0.06588·{{\mathrm{S}\mathrm{I}}_{\mathrm{v}}}^{{\displaystyle \frac{0.9281-0.9}{1-0.9}}} \end{gathered}$$

(8)

The calculation time is nearly instant in both Model Sets; hence, the computational efficiency of the proposed Model Set 2 is considered on par with that of the conventional Model Set 1. While both Model Sets present comparable computational costs in terms of computation time, the proposed models provide increased accuracy in the numerical value of the predicted DI (accuracy of prediction) and comparable results in the predicted value being within the limits of the correct damage grade (success rate in damage grade identification). All models of Model Set 2 present reduced the Root Mean Square Error (RMSE) and Adjusted-R² compared to their Model Set 1 counterparts, as shown in

Table 9. Performance metrics of the statistical models.

Model	RMSE Normalized MISDR	RMSE Maximum Softening	RMSE Modified Park–Ang	Adjusted–R2 Normalized MISDR	Adjusted–R2 Maximum Softening	Adjusted–R2 Modified Park–Ang
Model 1.1	0.1429	0.1191	0.1080	94.97%	94.04%	94.00%
Model 2.1	0.1223	0.0771	0.1018	96.31%	97.50%	94.66%
Model 1.2	0.1433	0.1197	0.1121	94.94%	93.97%	93.52%
Model 2.2	0.1225	0.0804	0.1027	96.30%	97.28%	94.57%
Model 1.3	0.1419	0.1181	0.1077	95.03%	94.13%	94.03%
Model 2.3	0.1225	0.0765	0.1011	96.30%	97.54%	94.74%

.

3.2. Testing the Multiple Regression Models

As previously stated, to verify both models, a set of 30 records out of the 180 was used to test the prediction quality of the three Damage Indices. Considering that the utilized DIs have been thoroughly studied and numerically verified by various researchers [8,16,21,49], the results of the non-linear time-history analyses are treated as a reference solution that is used in comparison with the values obtained by the generated predictive models. The test set comprises ground motion excitations that have induced all degrees of damage to the test frame. The accuracy of the predicted relative to the observed Damage Indices was tested, and the success rate in predicting the correct damage grade (minor, moderate, or severe) that the structure lies in for a given record. The accuracy and success rate of the two Model Sets for both the test and the training sets are shown in Figure 3 and Figure 4, respectively. All cases presented an improvement in prediction accuracy and similar degrees of success in identifying the correct damage grade of the test structure.

Model Set 1 presents a minimum of 80.572%, 86.834%, and 80.858% agreement in the Test Set with the Maximum Softening, Normalized MISDR, and Modified Park–Ang DIs computed from the non-linear dynamic analyses, respectively. Model Set 2 had a minimum of 89.803%, 87.564%, and 86.540% agreement with the Maximum Softening, Normalized MISDR, and Modified Park–Ang DIs, respectively. The Maximum Softening and Modified Park–Ang DIs were found to benefit the most concerning the degree of correctness in predicting the observed value.

The minimum percentage of correctly identified damage grade in the Test Set were found to be 90.000%, 96.667%, and 93.333% for the Maximum Softening, Normalized MISDR, and Modified Park–Ang, respectively, for the Model Set 1, and 90.000%, 96.667%, 96.667%, respectively, for the Model Set 2. Both Model Sets provide similar levels of reliability in predicting the correct damage grade (minor, moderate, or severe). However, Model Set 2 presents the added benefit of improved accuracy with the same number of explanatory variables. An important observation regarding the case of utilizing the Forward Stepwise Selection is that it provides the same degree of accuracy in predicting both the Damage Index value and the damage grade in which the structure has entered, requiring only three or four predictors, depending on the Damage Index being predicted, allowing for damage potential predictions to be made with only a limited number of Intensity Measures.

Figure 5 presents the standard deviation of prediction accuracy that Model Set 1 and Model Set 2 exhibit separately for the test and training subsets. In both the testing and training subsets, the proposed models of Model Set 2 present, on average, an overall improvement over their Model Set 1 counterparts regarding the prediction accuracy of the observed damage grade (reference solution), as well as in correctly identifying the observed damage grade (minor, moderate, or severe).

Table 10. Averages of prediction accuracy and successful categorization in the correct damage grade.

Model	Average Prediction Accuracy: Test Set	Average Prediction Accuracy: Training Set	Average of Success Rate of Correctly Identified Damage Grade: Test Set	Average of Success Rate of Correctly Identified Damage Grade: Training Set	Average Prediction Accuracy: Test and Training Sets	Average of Success Rate of Correctly Identified Damage Grade: Test and Training Sets
Model 1.1	83.05%	78.73%	94.44%	92.22%	80.89%	93.33%
Model 2.1	88.59%	82.46%	94.44%	93.56%	85.53%	94.00%
Model 1.2	84.06%	78.39%	94.44%	91.56%	81.22%	93.00%
Model 2.2	87.97%	81.58%	96.67%	91.56%	84.77%	94.11%
Model 1.3	83.01%	78.81%	94.44%	92.44%	80.91%	93.44%
Model 2.3	88.26%	82.37%	94.44%	93.33%	85.32%	93.89%

presents the average prediction accuracy and average success in determining the correct damage grade of the three predicted DIs for each generated predictive model. Model Set 2 provides, on average, more accurate estimates of the observed damage grade (numerical value of the DI), with an improvement ranging from 3.91% to 5.55% for the test set and from 3.19% to 3.73% for the training set. Regarding the success in the correctly identified damage grades, both Model Sets present similar average success rates, with an improvement of up to 2.23% for the test set and up to 1.34% for the training set.

3.3. Comparison of the Proposed Methodology with Predictions Obtained Through Compound IM

The validity verification of the proposed methodology is carried out by comparing Model Set 2 with models that utilize Compound IMs (CIMs) generated using Canonical Correlation Analysis (CCA) [19]. An analytical discussion on the method of CCA-generated CIMs can be found in the cited literature [19]. The CIM for the 10 selected IMs is calculated through CCA for each DI (Normalized MISDR, Maximum Softening, and Modified Park–Ang). Since the CIM is related to only one DI at a time, to enable direct comparison with the proposed methodology, it is evident that the compound Engineering Demand Parameter (EDP) coincides with the DI of choice, i.e., the Normalized MISDR, the Maximum Softening, and the Modified Park–Ang DIs. After the Canonical Correlation Analysis in the logarithmic scale, the CIM and the DI are reverted to the original scale. A conventional regression analysis is carried out to obtain a relationship between the CIM and the DI in the same form as the proposed methodology, i.e., “DI = a ∙ CIM” (hereby referenced as the CCA Model). The calculated prediction equations for the Normalized MISDR, Maximum Softening, and Modified Park–Ang DIs are shown in Equations (9)–(11).

$$\mathrm{NMISDR}=0.00760519·{\mathrm{IM}}_{\mathrm{CCA}}$$

(9)

$${\mathsf{\delta }}_{\mathsf{{\rm M}}}=1.39787\times {10}^{10}· {\mathrm{IM}}_{\mathrm{CCA}}$$

(10)

$${\mathrm{DI}}_{\mathrm{P}\mathrm{A}}=1.26378 · {\mathrm{IM}}_{\mathrm{CCA}}$$

(11)

A comparison of the proposed methodology and the CCA Model Set regarding the prediction accuracy (numerical value) and the success rate in correctly identifying the observed damage grade (correct classification) for the test and training sets is shown in Figure 6, Figure 7 and Figure 8, respectively.

It can be observed that Model Set 2 provides better structural damage grade prediction. Especially in the case of Maximum Softening, the difference in prediction accuracy and success rate is significant. This difference can be attributed to various factors, such as the nature of the specific DI and the differences in the two approaches. Thus, Model Set 2 utilizes a linear combination of modified IMs to predict the damage grade directly; in contrast, the CCA Model utilizes a single Compound IM to relate to the observed damage.

4. Conclusions

The present study investigated the relationship between seismic intensity parameters and structural damage potential using a novel correlation-based statistical approach, resulting in the following findings:

• Spectral-based and energy-related IMs correlated strongly with observed damage indices, reinforcing their relevance in damage assessment models;
• Velocity-based IMs, particularly Peak Ground Velocity (PGV) and Sustained Maximum Velocity, were also significant predictors;
• The multiple regression analysis confirmed that a limited set of IMs could effectively predict the structural damage grade, reducing the complexity of damage estimation without compromising accuracy and computational efficiency;
• The introduction of correlation-based weighting, in the form of exponentiation, enhanced prediction accuracy and damage grade classification;
• Both Forward Stepwise Selection-generated models presented the best overall performance regarding prediction accuracy and damage grade classification compared to their Ordinary Least Squares and Backward Stepwise Selection counterparts.

The proposed methodology enhances seismic damage assessment and provides a practical statistical framework for damage grade approximations.