Development of Phenomenological Model for RC Circular Columns

Abstract

The Modified Ibarra–Medina–Krawinkler (ModIMK) model, based on phenomenology, is widely used by earthquake engineering researchers due to its fast computation and effectiveness in simulating the performance deterioration of structural components. In the past, the main study objects were reinforced concrete (RC) square columns. However, RC circular columns have seldom been studied, and their ModIMK model still needs to be developed. For this reason, we collected the pseudo-static test data of 80 circular columns and calibrated the model parameters of each component; through the regression method, empirical equations between the design parameters and the hysteretic model parameters were established, i.e., the ModIMK model was determined. Using the hysteretic model in this work, the hysteretic responses of eight circular columns were predicted and found to be well fitted to the test hysteretic responses, verifying the model’s accuracy. This work enabled a more comprehensive and accurate application of the ModIMK model in earthquake engineering studies involving RC circular columns.

1. Introduction

A performance-based earthquake engineering study needs a relatively reliable hysteretic model. The model must be able to capture the structural seismic responses in different damage states, especially from the severe damage stage to the collapse stage; the model is also required to accurately capture the behaviors of the components’ continuous deterioration in strength and stiffness under earthquake motions. The Ibarra–Medina–Krawinkler (IMK) model [1] and the Modified IMK (ModIMK) model [2], which are founded on the hysteretic energy dissipation principle, could fulfill both of the previous requirements very well.

The hysteretic models for reinforced concrete (RC) components can be divided into “mechanical” and “phenomenological” models [3,4,5]. The mechanical model needs to be based on the accurate determination of the material’s mechanical properties, i.e., it needs accurate constitutive laws of the material, and the typical mechanical model is the fiber model; it is often difficult to accurately describe the phenomena, such as concrete crushing and reinforcement slipping. The empirical phenomenological hysteretic model, however, can better describe these phenomena. The ModIMK model is based on phenomenology, which not only describes complex experimental phenomena but also speeds up the computation of structural analyses; therefore, it is recommended by the U.S. Code ASCE 41, which has led to the wide application of this model in earthquake engineering studies.

The hysteretic model of a component can be determined by calibrating its individual model parameters, while empirical prediction equations for the model parameters can be obtained through statistical analyses after the calibration of the existing pseudo-static tests [6,7]. In order to ascertain the IMK model of RC square columns, Haselton [8] established a database, calibrated the model parameters of each component, and created empirical equations for each IMK model parameter. For the purpose of developing the IMK model for asymmetric components, Lignos [9,10] modified it and obtained the ModIMK model; he collected RC and steel components, calibrated each ModIMK model parameter, and then obtained its prediction equation. By contrasting the corroded and non-corroded RC square columns, Dai [11] developed the corroded columns’ empirical equations for the ModIMK model. Huang [12] established a database, calibrated the ModIMK model’s cyclic deterioration parameters for all columns, and created empirical prediction equations.

The studies mentioned above are on RC square columns, while RC circular columns have seldom been studied. Liu [13] developed the ModIMK hysteretic model for corroded circular columns, but this model relies on the ModIMK model for uncorroded columns, which is still incomplete. Dai [14] created empirical prediction equations for some parameters of the ModIMK hysteretic model, but the equations for parameters ${\theta }_{\mathrm{p}c}$ and $M_{\mathrm{c}}/{\mathrm{M}}_{\mathrm{y}}$ have not been established. Additionally, in a database of 76 components, the P-Δ effect pattern was incorrectly chosen for 27 components, leading to inaccurate calibration results for the model parameters. Therefore, the ModIMK hysteretic model for circular columns requires further improvement and development. We collected 80 sets of pseudo-static test data for RC circular columns and conducted a calibration and prediction study of the model parameters. The peak-orientated ModIMK model was adopted, and the OpenSees (3.3.0) software was utilized to numerically simulate the test curves of each column to calibrate the hysteretic model parameters. Then, a statistical regression was performed between the components’ design parameters and each model parameter, and empirical equations were established. The accuracy of these empirical prediction equations was verified through the analysis of eight columns in the database and out of the database.

2. Database

2.1. Collection of Test Data

Eighty RC columns from the Pacific Earthquake Engineering Research Center (PEER) structural performance database and published papers [15,16,17,18,19,20,21,22,23,24,25] were collected for this study, and the details of each component are listed in

Table A1. Circular columns’ calibrated model parameters.

No.	Reference	Component No.	${\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{y}}$	${\mathit{M}}_{\mathit{c}}/{\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{p}}$	${\mathit{\theta }}_{\mathit{p}\mathit{c}}$	$\mathit{\Lambda }$
1	Ma [15]	c0-15	63.46	0.0085	1.260	0.0500	—	3.10
2		c0-25	71.41	0.0094	1.254	0.0496	0.0760	3.80
3		c0-40	85.00	0.0065	1.173	0.0410	—	1.70
4	Zhu [16]	C0	79.20	0.0092	1.094	0.0130	0.0938	2.00
5	Zhou [17]	LL	36.13	0.0111	1.701	0.0545	—	1.95
6		MM	73.91	0.0176	1.532	0.0935	—	2.70
7		MH	106.51	0.0218	1.463	0.0810	—	4.15
8	Li [18,19]	C80	96.21	0.0119	1.183	0.0307	—	0.75
9		C140	86.40	0.0077	1.213	0.0085	—	0.45
10	Feng [20,21]	L0-N3-C0	104.44	0.0096	1.409	0.0225	—	0.90
11	Aquino [22]	no.6	297.17	0.0090	1.269	0.0412	0.1134	1.15
12	Lao [23]	C-0-L0-N02	127.18	0.0144	1.106	0.0240	—	0.40
13		C-0-L0-N04	135.75	0.0114	1.067	0.0180	—	0.80
14	Xie [24]	BW-1	44.90	0.0101	1.176	0.0286	—	0.75
15	PEER [25]	267	233.50	0.0213	1.087	0.0260	—	0.70
16		268	172.50	0.0079	1.052	0.0380	—	2.00
17		269	258.00	0.0236	1.039	0.0150	—	0.50
18		271	254.00	0.0225	1.080	0.0180	—	0.55
19		274	358.00	0.0195	1.226	0.0320	0.0946	0.75
20		275	355.00	0.0175	1.339	0.0550	—	1.90
21		276	360.50	0.0203	1.163	0.0360	—	0.60
22		277	323.00	0.0193	1.033	0.0135	—	0.22
23		278	320.00	0.0210	1.074	0.0370	—	1.10
24		279	337.00	0.0169	1.267	0.0630	—	0.70
25		280	245.00	0.0120	1.219	0.0340	0.1873	0.60
26		281	156.00	0.0078	1.257	0.0480	—	1.40
27		289	256.00	0.0118	1.118	0.0300	—	0.85
28		290	254.50	0.0126	1.189	0.0725	—	1.00
29		297	361.50	0.0067	1.109	0.0080	0.0536	0.50
30		298	434.00	0.0095	1.205	0.0270	0.0505	1.10
31		303	19.65	0.0247	1.047	0.0395	0.1445	1.90
32		304	22.40	0.0189	1.222	0.0700	—	2.10
33		305	23.50	0.0182	1.061	0.0650	—	3.00
34		308	47.20	0.0120	1.331	0.1540	—	1.60
35		309	56.00	0.0115	1.291	0.1200	—	1.40
36		310	41.60	0.0083	1.594	0.0700	—	2.80
37		311	47.00	0.0103	1.244	0.0960	—	1.80
38		313	48.20	0.0115	1.187	0.1000	—	2.30
39		346	100.00	0.0139	1.294	0.0470	—	2.40
40		347	98.50	0.0114	1.159	0.0320	—	5.30
41		349	109.50	0.0162	1.229	0.0400	—	2.80
42		350	101.00	0.0122	1.190	0.0565	—	1.00
43		357	332.00	0.0059	1.265	0.0315	—	0.90
44		358	456.50	0.0058	1.266	0.0140	0.0320	0.40
45		359	1192.50	0.0160	1.186	0.0800	—	3.80
46		360	502.50	0.0122	1.326	0.0420	—	1.10
47		361	198.50	0.0082	1.366	0.0180	—	1.10
48		363	775.00	0.0240	1.372	0.0850	—	1.60
49		364	302.50	0.0154	1.386	0.0750	—	1.75
50		365	827.50	0.0201	1.403	0.0850	—	3.50
51		366	579.00	0.0247	1.192	0.0765	0.3986	2.10
52		367	600.00	0.0207	1.144	0.0380	0.0574	1.40
53		368	629.00	0.0218	1.157	0.0810	—	3.20
54		369	655.50	0.0107	1.133	0.0710	—	1.10
55		370	695.00	0.0194	1.096	0.0840	—	1.60
56		371	597.00	0.0215	1.107	0.1000	—	1.80
57		372	415.00	0.0097	1.089	0.0450	—	2.00
58		373	1021.50	0.0147	1.311	0.0725	—	1.00
59		375	927.50	0.0187	1.214	0.1300	—	4.50
60		376	1080.00	0.0216	1.215	0.1450	—	3.70
61		377	651.00	0.0142	1.422	0.0710	—	1.80
62		378	649.00	0.0153	1.484	0.0630	—	1.50
63		379	721.50	0.0129	1.416	0.0915	—	1.90
64		380	121.00	0.0164	1.276	0.0810	—	7.80
65		381	112.00	0.0127	1.321	0.0630	—	6.00
66		382	135.50	0.0085	1.322	0.0600	—	2.50
67		383	126.00	0.0110	1.304	0.0190	0.0475	1.50
68		384	123.50	0.0100	1.335	0.0270	—	2.30
69		385	125.50	0.0173	1.366	0.0800	—	2.90
70		386	136.50	0.0121	1.291	0.0350	—	1.30
71		387	161.00	0.0129	1.338	0.0550	—	3.00
72		388	427.50	0.0117	1.096	0.0165	—	0.60
73		389	432.00	0.0079	1.064	0.0140	—	0.75
74		390	407.50	0.0086	1.130	0.0335	0.0269	1.00
75		391	368.00	0.0067	1.134	0.0175	—	0.70
76		392	685.00	0.0116	1.374	0.0680	0.4156	1.10
77		393	596.50	0.0123	1.456	0.0865	—	0.75
78		394	847.50	0.0116	1.160	0.0300	—	0.55
79		408	115.50	0.0108	1.228	0.0435	—	2.80
80		412	165.50	0.0099	1.186	0.0330	—	1.40

Note: ‘—’ in the θ_pc column means ‘no data’, which denotes that θ_pc cannot be calibrated.

and

Table A2. Design parameters of components.

No.	Component No.	P-Δ Type	d	c	${\mathit{f}}_{\mathit{c}}$	${\mathit{d}}_{\mathit{l}}$	${\mathit{\rho }}_{\mathit{l}}$	${\mathit{\lambda }}_{\mathit{s}\mathit{p}}$	$\mathit{\upsilon }$ + 1	${\mathit{s}}_{\mathit{n}}$	s/d	${\mathit{V}}_{\mathit{p}}/{\mathit{V}}_{\mathit{n}}$	${\mathit{f}}_{\mathit{l}\mathit{y}}$	${\mathit{f}}_{\mathit{l}\mathit{s}}$	${\mathit{d}}_{\mathit{s}}$	${\mathit{\rho }}_{\mathit{t}}$	${\mathit{f}}_{\mathit{t}\mathit{y}}$
1	c0-15	shear	260	30	25.92	16.0	0.0227	3.15	1.150	12.074	6.250	0.526	373.2	572.3	8.0	0.0100	327.0
2	c0-25	shear	260	30	25.92	16.0	0.0227	3.15	1.250	12.074	6.250	0.519	373.2	572.3	8.0	0.0100	327.0
3	c0-40	shear	260	30	25.92	16.0	0.0227	3.15	1.400	12.074	6.250	0.578	373.2	572.3	8.0	0.0100	327.0
4	C0	shear	300	30	24.34	16.0	0.0171	3.67	1.200	8.840	4.688	0.389	355.6	538.3	8.0	0.0112	230.7
5	LL	shear	240	20	32.00	12.0	0.0150	5.83	1.057	8.333	4.167	0.184	400.0	540.0	6.0	0.0113	400.0
6	MM	shear	240	20	32.00	16.0	0.0355	7.50	1.071	6.250	3.125	0.288	400.0	540.0	6.0	0.0113	400.0
7	MH	shear	240	20	32.00	20.0	0.0563	7.50	1.071	5.000	2.500	0.412	400.0	540.0	6.0	0.0113	400.0
8	C80	shear	300	30	26.48	14.0	0.0218	4.67	1.147	16.840	8.571	0.475	386.0	628.0	6.0	0.0039	367.0
9	C140	shear	300	30	26.48	14.0	0.0218	4.67	1.410	29.470	15.000	0.485	386.0	628.0	6.0	0.0022	367.0
10	L0-N3-C0	shear	300	30	35.44	16.0	0.0171	5.00	1.300	7.632	3.125	0.203	596.4	734.9	8.0	0.0167	404.9
11	no.6	shear	500	25	32.00	25.4	0.0310	4.20	1.000	16.137	7.874	0.457	420.0	620.0	9.5	0.0031	420.0
12	C-0-L0-N02	shear	250	25	26.16	16.0	0.0246	4.54	1.200	13.376	6.250	0.679	458.0	620.0	8.0	0.0100	454.0
13	C-0-L0-N04	shear	250	25	26.16	16.0	0.0246	4.54	1.400	13.376	6.250	0.652	458.0	620.0	8.0	0.0100	454.0
14	BW-1	shear	240	25	34.24	10.0	0.0174	5.83	1.100	22.334	10.000	0.237	498.8	693.6	8.0	0.0106	358.6
15	267	shear	400	18	37.50	16.0	0.0320	2.00	1.000	7.830	3.750	1.405	436.0	674.0	6.0	0.0051	328.0
16	268	shear	400	18	37.20	16.0	0.0320	2.00	1.000	6.452	3.750	0.950	296.0	457.0	6.0	0.0051	328.0
17	269	shear	400	18	36.00	16.0	0.0320	2.50	1.000	7.830	3.750	1.055	436.0	674.0	6.0	0.0051	328.0
18	271	shear	400	18	31.10	16.0	0.0320	2.00	1.000	5.220	2.500	1.223	436.0	674.0	6.0	0.0076	328.0
19	274	shear	400	18	28.70	16.0	0.0320	2.00	1.200	3.969	1.875	0.937	448.0	693.0	6.0	0.0102	372.0
20	275	shear	400	18	29.90	16.0	0.0320	2.50	1.200	3.969	1.875	0.924	448.0	693.0	6.0	0.0102	372.0
21	276	shear	400	21	31.20	16.0	0.0320	2.00	1.200	15.875	7.500	1.022	448.0	693.0	12.0	0.0102	332.0
22	277	shear	400	18	29.90	16.0	0.0320	2.00	1.200	7.937	3.750	1.184	448.0	693.0	6.0	0.0051	372.0
23	278	shear	400	18	28.60	16.0	0.0320	1.50	1.100	3.915	1.875	1.463	436.0	674.0	6.0	0.0102	328.0
24	279	shear	400	18	36.20	16.0	0.0320	2.00	1.100	3.915	1.875	1.045	436.0	679.0	6.0	0.0102	326.0
25	280	shear	400	18	33.70	24.0	0.0324	2.00	1.000	5.148	2.500	1.352	424.0	671.0	6.0	0.0051	326.0
26	281	shear	400	18	34.80	16.0	0.0192	2.00	1.000	7.830	3.750	0.932	436.0	679.0	6.0	0.0051	326.0
27	289	shear	400	21	32.30	16.0	0.0320	2.00	1.000	20.881	10.000	1.178	436.0	679.0	12.0	0.0076	332.0
28	290	shear	400	20	33.10	16.0	0.0320	2.00	1.000	14.355	6.875	1.265	436.0	679.0	10.0	0.0077	310.0
29	297	shear	400	18	37.00	16.0	0.0320	2.00	1.390	8.854	4.063	0.809	475.0	625.0	6.0	0.0047	340.0
30	298	shear	400	20	37.00	16.0	0.0320	2.00	1.390	8.173	3.750	0.962	475.0	625.0	10.0	0.0142	300.0
31	303	Feff	152	10.2	34.50	12.7	0.0558	7.50	1.241	3.667	1.732	0.228	448.0	679.6	3.7	0.0145	620.0
32	304	Feff	152	10.2	34.50	12.7	0.0558	3.75	1.241	3.667	1.732	0.448	448.0	679.6	3.7	0.0145	620.0
33	305	Feff	152	10.2	34.50	12.7	0.0558	3.75	1.351	3.667	1.732	0.475	448.0	679.6	3.7	0.0145	620.0
34	308	shear	250	9.9	24.10	7.0	0.0196	3.00	1.101	2.715	1.286	0.308	446.0	676.6	3.1	0.0141	441.0
35	309	shear	250	9.9	23.10	7.0	0.0196	3.00	1.211	2.715	1.286	0.364	446.0	676.6	3.1	0.0141	441.0
36	310	shear	250	9.7	25.40	7.0	0.0196	6.00	1.096	4.224	2.000	0.220	446.0	676.6	2.7	0.0068	476.0
37	311	shear	250	9.9	24.40	7.0	0.0196	3.00	1.100	2.715	1.286	0.322	446.0	676.6	3.1	0.0141	441.0
38	313	shear	250	9.7	23.30	7.0	0.0196	6.00	1.105	4.224	2.000	0.619	446.0	676.6	2.7	0.0068	476.0
39	346	shear	305	14.5	29.00	9.5	0.0204	4.50	1.094	4.233	2.000	0.323	448.0	690.0	4.0	0.0094	434.0
40	347	shear	305	14.5	35.50	9.5	0.0204	4.50	1.086	4.233	2.000	0.307	448.0	690.0	4.0	0.0094	434.0
41	349	shear	305	14.5	35.50	9.5	0.0204	4.50	1.086	4.233	2.000	0.341	448.0	690.0	4.0	0.0094	434.0
42	350	shear	305	14.5	35.50	9.5	0.0204	4.50	1.094	4.233	2.000	0.330	448.0	690.0	4.0	0.0094	434.0
43	357	Feff	610	15.9	30.00	12.7	0.0052	1.50	1.057	12.897	6.000	0.164	462.0	700.9	6.4	0.0028	361.0
44	358	Feff	610	15.9	30.00	12.7	0.0104	1.50	1.057	21.494	10.000	0.187	462.0	700.9	6.4	0.0017	361.0
45	359	Feff	610	27.8	41.10	22.2	0.0266	6.00	1.148	5.477	2.568	0.076	455.0	746.0	9.5	0.0089	414.0
46	360	Feff	457	24.8	38.30	15.9	0.0241	1.99	1.307	7.802	3.774	0.672	427.5	648.5	9.5	0.0114	430.2
47	361	Feff	457	24.8	39.20	15.9	0.0241	1.99	0.901	7.802	3.774	0.519	427.5	648.5	9.5	0.0114	430.2
48	363	Feff	457	26.4	35.00	19.0	0.0521	1.99	1.148	5.125	2.368	0.716	468.2	710.3	12.7	0.0270	434.4
49	364	Feff	457	24.8	35.20	15.9	0.0241	1.99	0.915	11.335	5.031	0.875	507.5	769.9	9.5	0.0085	448.2
50	365	Feff	457	26.4	35.00	19.0	0.0521	1.99	1.333	4.642	2.105	0.225	486.2	737.6	12.7	0.0304	434.4
51	366	Feff	457	30.2	36.60	15.9	0.0362	8.00	1.296	10.439	4.780	0.269	477.0	723.6	9.5	0.0092	445.0
52	367	Feff	457	30.2	40.00	15.9	0.0362	8.00	1.271	7.005	3.208	0.325	477.0	723.6	6.4	0.0060	437.0
53	368	Feff	457	30.2	38.60	15.9	0.0362	8.00	1.281	10.439	4.780	0.286	477.0	723.6	9.5	0.0092	445.0
54	369	prrdv	609.6	22.2	31.00	15.9	0.0149	4.00	1.072	4.299	2.000	0.368	462.0	630.0	6.4	0.0070	606.8
55	370	prrdv	609.6	22.2	31.00	15.9	0.0149	8.00	1.072	4.299	2.000	0.369	462.0	630.0	6.4	0.0070	606.8
56	371	shear	609.6	22.2	31.00	15.9	0.0149	10.00	1.072	4.299	2.000	0.370	462.0	630.0	6.4	0.0070	606.8
57	372	prrdv	609.6	22.2	31.00	15.9	0.0075	4.00	1.072	4.299	2.000	0.237	462.0	630.0	6.4	0.0070	606.8
58	373	prrdv	609.6	22.2	31.00	15.9	0.0298	4.00	1.072	4.299	2.000	0.595	462.0	630.0	6.4	0.0070	606.8
59	375	prrdv	609.6	28.6	34.50	19.0	0.0273	8.00	1.091	2.808	1.337	0.225	441.3	602.0	6.4	0.0089	606.8
60	376	prrdv	609.6	28.6	34.50	19.0	0.0273	10.00	1.091	2.808	1.337	0.143	441.3	602.0	6.4	0.0089	606.8
61	377	Feff	600	30.2	31.40	22.2	0.0192	3.00	1.045	9.248	4.369	0.556	448.0	739.0	9.5	0.0054	431.0
62	378	Feff	600	30.2	34.60	22.2	0.0192	3.00	1.041	9.248	4.369	0.545	448.0	739.0	9.5	0.0054	431.0
63	379	Feff	600	30.2	33.00	22.2	0.0192	3.00	1.043	6.190	2.883	0.448	461.0	775.0	9.5	0.0081	434.0
64	380	prrdv	250	13.8	65.00	16.0	0.0328	6.58	1.313	6.397	3.125	1.116	419.0	635.6	7.5	0.0015	1000.0
65	381	prrdv	250	15.6	65.00	16.0	0.0328	6.58	1.313	6.397	3.125	0.706	419.0	635.6	11.3	0.0035	420.0
66	382	prrdv	250	13.8	90.00	16.0	0.0328	6.58	1.419	6.397	3.125	0.410	419.0	635.6	7.5	0.0154	1000.0
67	383	prrdv	250	14	90.00	16.0	0.0328	6.58	1.419	6.397	3.125	0.284	419.0	635.6	8.0	0.0175	580.0
68	384	prrdv	250	15.6	90.00	16.0	0.0328	6.58	1.419	12.793	6.250	0.316	419.0	635.6	11.3	0.0174	420.0
69	385	prrdv	250	13.8	90.00	16.0	0.0328	6.58	1.209	6.397	3.125	0.338	419.0	635.6	7.5	0.0015	1000.0
70	386	prrdv	250	13.8	90.00	16.0	0.0328	6.58	1.419	6.397	3.125	0.342	419.0	635.6	7.5	0.0015	1000.0
71	387	prrdv	250	13.8	90.00	16.0	0.0328	6.58	1.419	6.397	3.125	0.219	419.0	635.6	11.3	0.0034	420.0
72	388	prrdv	508	21.3	56.20	16.0	0.0099	3.00	1.127	13.598	6.375	0.795	455.0	723.5	4.5	0.0013	455.0
73	389	prrdv	508	21.3	56.30	16.0	0.0099	3.00	1.109	13.598	6.375	0.975	455.0	723.5	4.5	0.0013	455.0
74	390	prrdv	508	21.3	57.00	16.0	0.0099	3.00	1.099	13.598	6.375	0.975	455.0	723.5	4.5	0.0013	455.0
75	391	prrdv	508	21.3	52.70	16.0	0.0099	3.00	1.107	13.598	6.375	0.926	455.0	723.5	4.5	0.0013	455.0
76	392	prrdv	609.6	22.2	37.20	15.9	0.0149	4.00	1.120	4.299	2.000	0.351	462.0	700.9	6.4	0.0070	606.8
77	393	prrdv	609.6	22.2	37.20	15.9	0.0149	4.00	1.060	8.584	3.994	0.448	462.0	700.9	6.4	0.0035	606.8
78	394	prrdv	609.6	20	32.60	19.0	0.0254	6.00	1.187	11.865	6.684	1.176	315.1	497.8	6.4	0.0017	351.6
79	408	Feff	406.4	15	36.50	12.7	0.0117	4.56	1.000	5.362	2.504	0.237	458.5	646.0	4.5	0.0053	691.5
80	412	Feff	406.4	10.4	35.40	12.7	0.0117	2.58	1.000	10.706	5.000	0.812	458.5	646.0	4.5	0.0026	691.5

Notes: In the “P-Δ Type” column, each type is abbreviated according to the meaning on the official website of PEER, where “Feff” stands for “Feff (=Mbase/L) Provided”; “shear” stands for “Shear Provided”; “prrdv” stands for “P Ram rotation decreases V”.

. Some of the following factors were considered in the collection of the test data: (1) the cross-section has to be circular; square and octagonal columns were not collected; (2) the forms of damage were flexural and flexural-shear; those with a shear damage form were not collected; (3) according to the type of construction and loading boundary conditions, the method of reference [25] was used to convert all the test data uniformly into the force–displacement form of cantilevered columns; (4) the method of reference [25] was used to differentiate the type of P-Δ effect correction, and those with an unclear type of P-Δ effect were not collected; after considering the P-Δ effect, the force–displacement form of the test data was converted into the form of moment rotation.

2.2. Design Parameters and Their Range of Values

The physical significance and the range of values of the design parameters are shown in

Table 1. Design parameters of circular components.

Parameter	Description	Parameter	Description
${\rho }_t$	volume stirrup ratio	d	diameter of circular column, mm
$\lambda$	ratio of stirrup spacing to longitudinal bar diameter	$f_{ty}$	yield strength of stirrup, MPa
c	thickness of concrete cover, mm	${\rho }_l$	longitudinal reinforcement ratio
$f_{ls}$	ultimate strength of longitudinal reinforcement	$f_{ly}$	yield strength of longitudinal reinforcement, MPa
L	shear span, mm	$V_p/V_n$	shear capacity ratio, Calculated according to the U.S. Code ACI318
$s_n$	$\mathrm{rebar} \mathrm{buckling} \mathrm{coefficient} s_n=\left(s/d_l\right){\left(f_y/100\right)}^{0.5}$	$v$	axial load ratio
s	stirrup spacing, mm	$d_s$	stirrup diameter, mm
${\lambda }_{sp}$	shear span ratio	$f_c$	cylinder compressive strength of concrete, MPa
$d_l$	longitudinal reinforcement diameter, mm

and

Table 2. Value ranges of design parameters.

Parameter	Value Range
d (mm)	152–609.6
$f_c$ (MPa)	23.1–90
v	−0.01–0.419
${\lambda }_{sp}$	1.5–10
$f_{\mathrm{l}y}$ (MPa)	296–496.4
$f_{t\mathrm{y}}$ (MPa)	230.68–1000
${\rho }_l$	0.0052–0.0563
${\rho }_t$	0.0013–0.0304

. Based on the analyses in references [8,11,26], the variables that have a more significant effect on the model parameters were selected as the design parameters. Table 2 lists the range of values of the main design parameters of the components; the axial compression ratio ranges from −0.01 to 0.419, so it can be seen that the empirical prediction equations in this work are more suitable for low and medium axial pressures.

3. The ModIMK Model and Parameters’ Calibration

3.1. The ModIMK Hysteretic Model

The ModIMK model consists of two parts, the backbone curve and the hysteresis rule, and the backbone curve of the model is shown in Figure 1. The ModIMK hysteretic model has a critical difference compared with other models; its backbone curve is not acquired by linking the peak moment point of every hysteresis loop directly, but it is obtained based on the monotonic loading curve (see Figure 1). The backbone curve connecting the peak moment points varies greatly depending on the loading history, whereas the monotonic loading curve is stable.

Because most test groups did not have monotonically loaded components, the backbone curve is generally approximated by calibrating the hysteresis curve.

The ModIMK hysteresis model has three types: bilinear, peak-orientated, and pinching; the circular columns collected in this work are more compatible with the peak-orientated type. The main model parameters are listed in

Table 3. Meaning of model parameters.

Parameter	Description	Parameter	Description
$K_0$	elastic stiffness	$M_y$	yield moment
${\theta }_p$	chord rotation of strain-hardening stage in backbone curve	$K_s$	hardening stiffness
$M_r$	residual moment in backbone curve	$K_c$	post-capping stiffness
${\theta }_y$	chord rotation of elastic stage	${\theta }_u$	ultimate rotation capacity
$M_c$	capping moment	${\theta }_{pc}$	chord rotation of softening stage in backbone curve
c	the deterioration exponent	$\Lambda$	cyclic deterioration parameter

, and other ones can be viewed on the official website of OpenSees.

3.2. The Model Parameters’ Calibration

The backbone curve reflects the component’s monotonic loading process, so the model parameters cannot be obtained directly by analyzing the test hysteresis curve. The component’s deterioration behavior is determined by the energy dissipation throughout the loading process, and the deterioration coefficients cannot be obtained through simple calculations. Only by the numerical simulation of the test process and by analyzing the fitting situation between the test curve and the simulation curve can the model parameters of the component be finally determined. Therefore, a finite element model needs to be created for each component for the numerical analysis.

The peak-orientated ModIMK model is suitable for RC circular columns where flexural deformation is dominant. According to the analysis of reference [26], six model parameters, i.e., $M_{\mathrm{y}},{\theta }_y,M_{\mathrm{c}}/{\mathrm{M}}_{\mathrm{y}},{\theta }_{\mathrm{p}},{\theta }_{\mathrm{p}c},\Lambda$, can determine the hysteretic model of a component.

According to reference [26], the analytical model was developed, and the model parameters were calibrated using the OpenSees platform. Due to the fact that the plastic hinge can only appear after a component yields, the analytical model differs from the real test, and in order to simulate this more accurately, the analytical model has to be stiffness-corrected [27,28,29].

The model parameters were calibrated in the positive and negative loading directions, and then their average values were obtained, which are listed in Table A1. The empirical prediction equations were created on the basis of the statistical regression of the average values.

4. Empirical Equations for Model Parameters

4.1. Regression Analysis Method for Model Parameters

The empirical prediction equations linking the design parameters and the calibrated model parameters were developed through the regression method. The regression process is as follows:

(1)

The form of the prediction equations

According to reference [10], each empirical prediction equation’s form was selected as follows:

$$RP=a_0\cdot {\left(X_1\right)}^{a_1}\cdot {\left(X_2\right)}^{a_2}\cdots \cdot {\left(X_n\right)}^{a_n}$$

(1)

where $RP$ represent a model parameter, $X_i$ represents the component design parameters, and $a_i$ represents the regression parameters. The 15 design parameters are listed in Table 1. Since some components do not have an axial pressure, in order to carry out the regression analyses when taking logarithms, (v + 1) is used as the input parameter for the axial pressure ratio.

(2)

The linear regression analysis method

After taking natural logarithms on both sides of Equation (1) simultaneously, Equation (2) was obtained:

$$\ln \left(RP\right)=\ln \left(a_0\right)+a_1\cdot \ln \left(X_1\right)+a_2\cdot \ln \left(X_2\right)+\cdots \cdot +a_n\cdot \ln \left(X_n\right)$$

(2)

The natural logarithm was taken for the calibrated model parameters and the design parameters of each component. Then, linear regression was carried out to determine Equation (2). The selected design parameters require a t-test significance level of less than 0.05. In order to control the multicollinearity of the selected design parameters, the VIF (Variance Inflation Factor) of each parameter is required to be less than 10.

(3)

The establishment of the empirical prediction equations

The regression equation for each model parameter can be obtained by taking the natural constant e as the bottom index for both sides of Equation (2).

4.2. Empirical Equations

(1)

The empirical prediction equations

Through a regression analysis, the empirical prediction equations were obtained as follows:

$$M_y=\exp (-14.709)\cdot d^{3.02}\cdot {\left(v+1\right)}^{1.677}\cdot {\rho }_l^{0.569}\cdot f_c^{0.274}\cdot f_{ly}^{0.513}$$

(3)

$${\theta }_y=\exp (-11.569)\cdot {\rho }_l^{0.456}\cdot {\left(s/d\right)}^{\left(-0.218\right)}\cdot f_{ls}^{1.292}\cdot c^{0.218}\cdot {\lambda }_{sp}^{0.277}\cdot {\left(v+1\right)}^{\left(-0.788\right)}\cdot {\left(V_p/V_n\right)}^{0.13}$$

(4)

$$M_c/M_y=\exp \left(-0.303\right)\cdot d_s^{0.156}\cdot {\left(s/d\right)}^{\left(-0.034\right)}\cdot d_l^{\left(-0.102\right)}\cdot f_{ty}^{\left(0.093\right)}\cdot {\lambda }_{sp}^{\left(-0.057\right)}\cdot {\left(V_p/V_n\right)}^{\left(-0.035\right)}$$

(5)

$${\theta }_p=\exp (-1.786)\cdot s_n^{\left(-0.77\right)}\cdot {\lambda }_{sp}^{0.431}\cdot d_s^{0.916}\cdot {\left(v+1\right)}^{\left(-1.782\right)}\cdot d_l^{\left(-0.688\right)}$$

(6)

$$\Lambda =\exp \left(2.028\right)\cdot {\lambda }_{sp}^{0.495}\cdot s_n^{\left(-0.745\right)}\cdot d_s^{0.898}\cdot d^{\left(-0.653\right)}\cdot {\rho }_l^{\left(-0.354\right)}$$

(7)

Most of the circular columns in the database were not loaded to softening stage, and ${\theta }_{\mathrm{p}c}$ could not be measured, so the regression method was not used to obtain the empirical equation for ${\theta }_{\mathrm{p}c}$. According to reference [8], it has been shown that there is an empirical relationship between the initial stiffness and the softening stiffness, i.e., $K_c=-0.08K_0$, and after determining $M_y,{\theta }_y,M_c$, the estimation Equation (8) can be obtained. In addition, according to the test results of 15 adequately loaded components, 0.1 is set as the upper limit value of ${\theta }_{\mathrm{p}c}$ [8]. The larger value of ${\theta }_{\mathrm{p}c}$ is generally due to the fact that the loading is not sufficiently adequate, and the deterioration in the components is not yet apparent.

$${\theta }_{pc}=\left({\theta }_y\cdot M_c/M_y\right)/0.08\le 0.1$$

(8)

(2)

Evaluation of the prediction equations

The coefficients of determination (R²) of the regression equations for model parameters $M_{\mathrm{y}},{\theta }_y,{\theta }_{\mathrm{p}},\mathrm{and} \Lambda$ are 0.98, 0.65, 0.54, and 0.50, respectively, which indicates that the regression equations are well fitted; the coefficient of determination of $M_{\mathrm{c}}/{\mathrm{M}}_{\mathrm{y}}$ is 0.36, which indicates that this parameter has a poorer correlation with the design parameters.

For each model parameter of each component, the ratio between the calibrated and predicted values was calculated. The statistics of this ratio are presented in

Table 4. Statistical information for ratios of calibrated to predicted model parameters.

	${\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{y}}$	${\mathit{M}}_{\mathit{c}}/{\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{p}}$	$\mathit{\Lambda }$	${\mathit{\theta }}_{\mathit{p}\mathit{c}}$
$\mu$	1.01	1.02	1.02	1.09	1.13	1.20
$\tilde{\mu }$	0.99	1.01	1.02	1.02	0.99	1.00
$\sigma$	0.18	0.22	0.11	0.44	0.57	—

Note: ‘—’ in the θpc column means ‘no data’.

; $\mu$, $\tilde{\mu }$, and $\sigma$ are the mean, median, and standard deviation, respectively.

Table 4 shows that the ratios’ mean values corresponding to the model parameters $M_{\mathrm{y}},{\theta }_y,M_{\mathrm{c}}/{\mathrm{M}}_{\mathrm{y}},{\theta }_{\mathrm{p}},\Lambda$ are 1.01, 1.02, 1.02, 1.09, and 1.13, and the median values are 0.99, 1.01, 1.02, 0.99, and 1.0, respectively, which indicates that the prediction equations can predict the hysteretic model parameters relatively well. The standard deviations of the ratios corresponding to the model parameters $M_{\mathrm{y}},{\theta }_y,M_{\mathrm{c}}/{\mathrm{M}}_{\mathrm{y}},{\theta }_{\mathrm{p}},\Lambda$ are 0.18, 0.22, 0.11, 0.44, and 0.57, respectively, so it can be seen that the prediction accuracies of $M_{\mathrm{y}},{\theta }_y,M_{\mathrm{c}}/{\mathrm{M}}_{\mathrm{y}}$ are high, and the dispersion is small. In contrast, the dispersion of ${\theta }_{\mathrm{p}},\Lambda$ is relatively large.

It should be noted that the statistical data of ${\theta }_{\mathrm{p}c}$ is from reference [8], and its corresponding mean and median values are 1.2 and 1.0, while the logarithmic standard deviation is 0.72. Since, after the peak point of the backbone curve, the cracking of the components, the slipping of the reinforcement, and the shear deformation are more serious, the standard deviation and dispersion are relatively large, and the accuracy of the prediction result of ${\theta }_{\mathrm{p}c}$ will be smaller than that of the other parameters.

The model parameters of the 80 columns in the database were predicted using the aforementioned empirical prediction equations. Figure 2 illustrates the comparison between the predicted and calibrated model parameters, showing that the two are generally well matched.

Generally speaking, the hysteretic model parameters can be reliably predicted using the proposed empirical prediction equations.

5. Validation of the Empirical Prediction Equations

In order to validate the accuracy of the proposed empirical equations, eight components, four inside and four outside the database, were selected for analysis. The predicted model parameters of the components were derived using the empirical prediction equations, and the predicted hysteretic curves were obtained through a numerical simulation; then, the test hysteretic curves and the predicted hysteretic curves were compared.

5.1. Validation of the Components in the Database

Based on the collected data information, the axial pressure ratios were categorized into three cases: $\upsilon \le 0.15$, low axial pressure ratio; $0.15<\upsilon <0.35$, medium axial pressure ratio; and $\upsilon \ge 0.35$, higher axial pressure ratio. In each range of axial pressure ratios, representative components were randomly selected for validation. The four components numbered 298, 360, 384, and 392 in the database have axial pressure ratios of 0.39, 0.307, 0.419, and 0.12, respectively. Their design parameters are presented in Table A2, and their predicted model parameters are listed in

Table 5. Predicted model parameters of intra-database columns.

Test No	${\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{y}}$	${\mathit{M}}_{\mathit{c}}/{\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{p}}$	$\mathit{\Lambda }$	${\mathit{\theta }}_{\mathit{p}\mathit{c}}$
298	459.50	0.0108	1.247	0.0305	1.197	0.1
360	504.36	0.0104	1.296	0.0337	1.196	0.1
384	139.57	0.0112	1.252	0.0389	2.324	0.1
392	729.74	0.0131	1.264	0.0661	1.815	0.1

. Figure 3 shows the predicted hysteresis curves and the test hysteresis curves. It is clear that the hysteresis behavior of each test can be predicted well, and the yield moment, yield rotation, hardening stiffness, and capping moment are all well fitted; through the last hysteresis loop of 298, it is also evident that the model in this work is quite accurate in predicting the softening stiffness after the capping point.

Through the hysteresis curves of the three components of 298, 360, and 392, it can be found that the model in this work has an outstanding capability for capturing the performance deterioration in the components.

The analyses of the above four components in the database prove that the model proposed in this work can predict the hysteretic responses well.

5.2. Applications Outside the Database

Since relatively few components with higher axial compression ratios were collected, the equations in this work are more applicable to the case of medium and low axial compression ratios. Four components, Wong-No.1, NiST-N5, Vu-NH3, and Calderone-328, were selected to apply the hysteretic model in this work for a hysteretic response analysis [25]. Their design parameters are listed in

Table 6. Design parameters of 4 columns for applied analyses.

Component No.	P-Δ Type	d	c	${\mathit{f}}_{\mathit{c}}$	${\mathit{d}}_{\mathit{l}}$	${\mathit{\rho }}_{\mathit{l}}$	${\mathit{\lambda }}_{\mathit{s}\mathit{p}}$	$\mathit{\upsilon }$ + 1	${\mathit{s}}_{\mathit{n}}$	s/d	${\mathit{V}}_{\mathit{p}}/{\mathit{V}}_{\mathit{n}}$	${\mathit{f}}_{\mathit{l}\mathit{y}}$	${\mathit{f}}_{\mathit{l}\mathit{s}}$	${\mathit{d}}_{\mathit{s}}$	${\mathit{\rho }}_{\mathit{t}}$	${\mathit{f}}_{\mathit{t}\mathit{y}}$
Wong-No.1	shear	400	20	38.00	16.0	0.0320	2.00	1.190	7.713	3.750	0.830	423.0	577.0	10.0	0.0142	300.0
NiST-N5	shear	250	9.9	24.30	7.0	0.0196	3.00	1.200	2.715	1.286	0.357	446.0	676.6	3.1	0.0141	441.0
Vu-NH3	Feff	457	24.8	39.40	15.9	0.0241	1.99	1.150	7.802	3.774	0.673	427.5	648.5	9.5	0.0114	430.2
Calderone-328	prrdv	609.6	28.6	34.50	19.0	0.0273	3.00	1.091	2.808	1.337	0.502	441.3	602.0	6.4	0.0089	606.8

, while their predicted model parameters are listed in

Table 7. Predicted parameters.

Component No.	${\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{y}}$	${\mathit{M}}_{\mathit{c}}/{\mathit{M}}_{\mathit{y}}$	${\mathit{\theta }}_{\mathit{p}}$	$\mathit{\Lambda }$	${\mathit{\theta }}_{\mathit{p}\mathit{c}}$
Wong-No.1	336.11	0.0108	1.253	0.0421	1.250	0.1
NiST-N5	56.70	0.0114	1.229	0.0666	1.878	0.1
Vu-NH3	410.11	0.0115	1.296	0.0423	1.196	0.1
Calderone-328	943.01	0.0161	1.263	0.0751	1.745	0.1

. The axial pressure ratios of the four components are 0.19, 0.20, 0.15, and 0.091, which represent the medium or low axial pressure ratios, respectively. From Figure 4, it is evident that the presented hysteretic model can predict the hysteretic responses well. Similar to the analysis within the database, the model parameters are predicted relatively accurately. The analyses of the last few hysteretic loops of the hysteresis curves shows that the deterioration in the components can be simulated well.

The analyses of the above four components shows that the model proposed in this work is also very applicable in applications outside the database.

6. Conclusions

The ModIMK hysteretic model for RC circular columns was created by a regression method, and the summaries and conclusions are shown below:

(1)

A database of 80 RC circular columns undergoing flexural or flexural-shear damage had been established, containing 15 design parameters and the calibrated ModIMK model parameters. For further research on various characteristics of RC circular columns, the database is an invaluable resource.

(2)

The empirical prediction equations for the hysteretic model parameters based on the design parameters were established, i.e., the ModIMK model was determined. The predicted and test hysteretic curves were compared to confirm the proposed model can predict hysteretic behavior accurately. This work enabled a more comprehensive and accurate application of the ModIMK model in earthquake engineering studies involving RC circular columns.

Of course, due to the limited numbers of the database, there exists a specific range of applicability of the design parameters; therefore, when using the prediction equations, taking note of the design parameters’ values is essential. In the future, with the continuous increase in test data, the proposed ModIMK model will be continuously improved to gradually increase its applicability and accuracy.