A multi-objective design method for seismic retrofitting of existing reinforced concrete frames using pin-supported rocking walls

doi:10.1007/s11709-022-0851-z

Front. Struct. Civ. Eng.

2022, Vol. 16

Issue (9) : 1089-1103 https://doi.org/10.1007/s11709-022-0851-z

RESEARCH ARTICLE

A multi-objective design method for seismic retrofitting of existing reinforced concrete frames using pin-supported rocking walls

Yue CHEN^1,^2,³, Rong XU², Hao WU^1,³(

), Tao SHENG⁴

¹. Institute of Engineering Mechanics, China Earthquake Administration; Key Laboratory of Earthquake Engineering and Engineering Vibration of China Earthquake Administration, Harbin 150088, China
². Research Center of Industrialization Construction Technology of Zhejiang Province, Ningbo University of Technology, Ningbo 315106, China
³. State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China
⁴. College of Civil and Environmental Engineering, Ningbo University, Ningbo 315211, China

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Abstract

Over the past several decades, a variety of technical ways have been developed in seismic retrofitting of existing reinforced concrete frames (RFs). Among them, pin-supported rocking walls (PWs) have received much attentions to researchers recently. However, it is still a challenge that how to determine the stiffness demand of PWs and assign the value of the drift concentration factor (DCF) for entire systems rationally and efficiently. In this paper, a design method has been exploited for seismic retrofitting of existing RFs using PWs (RF-PWs) via a multi-objective evolutionary algorithm. Then, the method has been investigated and verified through a practical project. Finally, a parametric analysis was executed to exhibit the strengths and working mechanism of the multi-objective design method. To sum up, the findings of this investigation show that the method furnished in this paper is feasible, functional and can provide adequate information for determining the stiffness demand and the value of the DCFfor PWs. Furthermore, it can be applied for the preliminary design of these kinds of structures.

Keywords pin-supported rocking wall reinforced concrete frame seismic retrofit stiffness demand drift concentration factor multi-objective design genetic algorithm Pareto optimal solution

Corresponding Author(s): Hao WU

Just Accepted Date: 26 July 2022 Online First Date: 17 November 2022 Issue Date: 22 December 2022

Cite this article:

Yue CHEN,Rong XU,Hao WU, et al. A multi-objective design method for seismic retrofitting of existing reinforced concrete frames using pin-supported rocking walls[J]. Front. Struct. Civ. Eng., 2022, 16(9): 1089-1103.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-022-0851-z
https://academic.hep.com.cn/fsce/EN/Y2022/V16/I9/1089

Fig.1 Reinforced concrete rocking walls: (a) pin-supported rocking wall; (b) stepping rocking wall.

Fig.2 Schematic of the EMO using GA.

Tab.1 Parameters adopted by EMO using GA

Fig.3 Model for RF-PWs: (a) lateral load; (b) PW with a rotational spring; (c) RF; (d) RF-PW; (e) parametric model of RF-PWs.

$p (ξ)$	differential equations	solutions of differential equations	boundary conditions
$q$	$? λ 2 d 2 y d ξ 2 = q H 4 E w I w$	$y (ξ) = C 1 + C 2 ξ + A sinh ? λ ξ + B cosh ? λ ξ ? q H 2 2 K ξ 2$	$(1) When x = H, (ξ = 1), Shearblance, (d 3 y d ξ 3 ? λ 2 d y d ξ) ξ = 1 = 0 .$
$q ? ξ$	$? λ 2 d 2 y d ξ 2 = q ξ H 4 E w I w$	$y (ξ) = C 1 + C 2 ξ + A sinh ? λ ξ + B cosh ? λ ξ + q H 2 6 K ξ 3$	$(2) When x = 0, (ξ = 0), Momentbalance, (R f d y d ξ ? d 2 y d ξ 2) ξ = 0 = 0.$
$q ? ξ$	$? λ 2 d 2 y d ξ 2 = q ξ H 4 E w I w$	$y (ξ) = C 1 + C 2 ξ + A sinh ? λ ξ + B cosh ? λ ξ + 4 q H 2 15 K ξ 52$	$(3) When x = 0, (ξ = 0), y = 0, y ξ = 0 = 0.$
F	$d 4 y d ξ 4 = λ 2 d 2 y d ξ 2$	$y (ξ) = C 1 + C 2 ξ + A sinh ? λ ξ + B cosh ? λ ξ$	$(4) When x = H, (ξ = 1), (d 2 y d ξ 2) ξ = 1 = 0.$

Tab.2 Expressions of

y (ξ)

corresponding to the four different

p (ξ)

coefficients	uniformly distributed load	inverted triangular distributed load	parabolic distributed load	concentrated load at the top of the structure
$φ$	$1 ? cosh ? λ ? R f cosh ? λ sinh ? λ + R f λ cosh ? λ$	$1 ? R f (12 ? 1 λ 2) cosh ? λ sinh ? λ + R f λ cosh ? λ$	$1 ? R f (23 ? 1 2 λ 2) cosh ? λ sinh ? λ + R f λ cosh ? λ$	$R f cosh ? λ sinh ? λ + R f λ cosh ? λ$
C₁	$? q H 2 K λ 2 (1 + R f + R f λ φ)$	$? q H 2 K λ 2 [R f λ 2 (12 ? 1 λ 2) + R f λ φ]$	$? q H 2 K λ 2 [R f λ 2 (23 ? 1 2 λ 2) + R a λ φ]$	$? F H K λ 2 (R f ? R f λ φ)$
C₂	$q H 2 K$	$q H 2 K (12 ? 1 λ 2)$	$q H 2 K (23 ? 1 2 λ 2)$	$F H K$
A	$q H 2 K λ 2 φ$	$q H 2 K λ 2 φ$	$q H 2 K λ 2 φ$	$? F H 2 K λ 2 φ$
B	$q H 2 K λ 2 (1 + R f + R f λ φ)$	$q H 2 K λ 2 [R f λ 2 (12 ? 1 λ 2) + R f λ φ]$	$q H 2 K λ 2 [R f λ 2 (23 ? 1 2 λ 2) + R f λ φ]$	$F H K λ 2 (R f ? R f λ φ)$

Tab.3 Solutions of C₁, C₂, A and B

position	lateral displacement (mm)	$I D R RF$
roof	7.16577	1/5709
4th floor	6.64028	1/3045
3rd floor	5.65518	1/2083
2nd floor	4.21509	1/1588
1st floor	2.32695	1/1289
ground	0	?

Tab.4 Lateral displacements and IDRs of the RF

Fig.4 Two-dimensional five-story RF before retrofitted.

Tab.5 Material and geometrical dimensions of the RF

Fig.5 (a) Lateral displacement and (b) IDR of the RF.

No.	$η$	$D C F PW$	$E w I w × 10 ? 7$ (kN·m²)	$b w$ (m)	$λ$	$I D R max PW$	$ζ$	$V wb$ (kN)	$δ$
No.	$η$	$D C F PW$	$E w I w × 10 ? 7$ (kN·m²)	$t w$ = 0.6 m	$λ$	$I D R max PW$	$ζ$	$V wb$ (kN)	$δ$
1	0.311	1.321	4.058	3.0	3.211	1/1537	83.8%	430.5	25.9%
2	1.266	1.037	67.1	7.6	0.790	1/1958	65.8%	712.6	42.8%
3	1.514	1.026	96.0	8.6	0.660	1/1978	65.2%	723.2	43.4%
4	0.258	1.380	2.784	2.6	3.877	1/1472	87.6%	370.8	22.3%
5	0.188	1.469	1.472	2.1	5.332	1/1382	93.3%	278.4	16.7%
6	0.392	1.249	6.439	3.5	2.549	1/1626	79.3%	502.6	30.2%
7	0.331	1.301	4.592	3.1	3.019	1/1560	82.6%	450.1	27.0%
8	0.223	1.422	2.09	2.4	4.475	1/1428	90.3%	327.3	19.7%
9	0.758	1.094	24.0	5.4	1.319	1/1857	69.4%	656.6	39.4%
10	0.723	1.101	21.9	5.3	1.383	1/1844	69.9%	649.0	39.0%
11	1.197	1.041	60.0	7.4	0.835	1/1950	66.1%	708.5	42.6%
12	0.249	1.390	2.603	2.6	4.010	1/1461	88.2%	360.4	21.6%
13	0.531	1.165	11.8	4.3	1.883	1/1742	74.0%	585.6	35.2%
14	0.269	1.367	3.033	2.7	3.714	1/1486	86.7%	384.2	23.1%
15	0.169	1.494	1.2	2.0	5.905	1/1359	94.8%	252.4	15.2%
16	0.358	1.277	5.373	3.3	2.791	1/1591	81.0%	474.8	28.5%
17	1.120	1.047	52.5	7.0	0.892	1/1940	66.4%	703.2	42.2%
18	1.514	1.026	96.0	8.6	0.660	1/1978	65.2%	723.2	43.4%
19	0.657	1.119	18.1	4.9	1.522	1/1815	71.0%	631.7	37.9%
20	0.172	1.490	1.24	2.0	5.809	1/1363	94.6%	256.4	15.4%
21	0.484	1.189	9.812	4.0	2.065	1/1708	75.5%	562.3	33.8%
22	0.806	1.084	27.2	5.7	1.241	1/1873	68.8%	665.9	40.0%
23	1.040	1.054	45.3	6.7	0.962	1/1927	66.9%	696.4	41.8%
24	1.404	1.031	82.5	8.2	0.712	1/1970	65.4%	719.1	43.2%
25	0.293	1.340	3.599	2.9	3.410	1/1515	85.0%	411.3	24.7%
26	0.245	1.396	2.504	2.6	4.089	1/1455	88.6%	354.4	21.3%
27	0.196	1.458	1.61	2.2	5.098	1/1393	92.5%	290.4	17.4%
28	1.346	1.033	75.8	8.0	0.743	1/1966	65.6%	716.6	43.0%
29	1.346	1.033	75.8	8.0	0.743	1/1966	65.6%	716.6	43.0%
30	0.238	1.404	2.366	2.5	4.205	1/1446	89.1%	345.9	20.8%
31	1.467	1.028	90.1	8.4	0.681	1/1975	65.3%	721.5	43.3%
32	0.574	1.147	13.8	4.5	1.741	1/1770	72.8%	603.9	36.3%
33	0.475	1.194	9.423	4.0	2.107	1/1700	75.8%	556.9	33.4%
34	0.874	1.073	31.9	6.0	1.145	1/1892	68.1%	676.9	40.7%
35	0.806	1.084	27.2	5.7	1.241	1/1873	68.8%	665.9	40.0%
36	0.268	1.368	3.004	2.7	3.733	1/1484	86.8%	382.7	23.0%
37	0.289	1.345	3.488	2.9	3.464	1/1510	85.4%	406.3	24.4%
38	0.551	1.157	12.7	4.4	1.815	1/1756	73.4%	594.3	35.7%
39	0.180	1.479	1.361	2.1	5.545	1/1373	93.9%	268.2	16.1%
40	1.169	1.043	57.2	7.2	0.856	1/1947	66.2%	706.6	42.4%
41	0.429	1.222	7.715	3.7	2.329	1/1661	77.6%	529.2	31.8%
42	0.311	1.321	4.058	3.0	3.211	1/1537	83.8%	430.5	25.9%
43	0.612	1.133	15.7	4.7	1.634	1/1792	71.9%	617.6	37.1%
44	0.228	1.416	2.181	2.4	4.381	1/1434	89.9%	333.6	20.0%
45	1.273	1.037	67.8	7.7	0.786	1/1959	65.8%	713.0	42.8%
46	0.948	1.063	37.6	6.3	1.055	1/1910	67.5%	686.7	41.2%
47	0.408	1.237	6.952	3.6	2.454	1/1641	78.5%	514.0	30.9%
48	0.210	1.439	1.854	2.3	4.751	1/1411	91.3%	310.0	18.6%
49	0.169	1.494	1.2	2.0	5.905	1/1359	94.8%	252.4	15.2%
50	0.331	1.301	4.592	3.1	3.019	1/1560	82.6%	450.1	27.0%

Tab.6 50 POSs at Generation 50

Fig.6

η

of 50 POSs at Generation 50.

Fig.7

D C F PW

of 50 POSs at Generation 50.

Fig.8 Trade-off correlation between

η

and

D C F PW

at Generation 50.

Fig.9 3D diagram of the quantity distribution of

η

and

D C F PW

at Generation 50.

Fig.10 Plan diagram of the quantity distribution of

η

and

D C F PW

at Generation 50.

Fig.11 Scatter diagram of the distribution of

E w I w

at Generation 50.

Fig.12 Bar diagram of the distribution of

E w I w

at Generation 50.

Fig.13 Correlation between ζ and

D C F PW

(when

t w

= 0.6 m).

Fig.14 Bar diagram of the distribution of

b w

at Generation 50 (when

t w

= 0.6 m).

Fig.15 Bar diagram of the distribution of

λ

at Generation 50.

Fig.16 Bar diagram of the distribution of

I D R max PW

at Generation 50.

Fig.17 Bar diagram of the distribution of

ζ

at Generation 50.

Fig.18 Correlation between

η

and

ζ

Fig.19 Correlation between

ζ

and

I D R max PW

Fig.20 Bar diagram of the distribution of

V wb

at Generation 50.

Fig.21 Bar diagram of the distribution of

δ

at Generation 50.

Fig.22 Correlation between

η

and

δ

No.	$η$	$D C F PW$	$E w I w × 10 ? 7$ (kN·m²)	$b w$ (m)	$λ$	$I D R max PW$	$ζ$	$V wb$ (kN)	$δ$
No.	$η$	$D C F PW$	$E w I w × 10 ? 7$ (kN·m²)	$t w$ = 0.6 m	$λ$	$I D R max PW$	$ζ$	$V wb$ (kN)	$δ$
3	1.514	1.026	96.0	8.6	0.660	1/1978	65.2%	723.2	43.4%
21	0.484	1.189	9.812	4.0	2.065	1/1708	75.5%	562.3	33.8%
42	0.311	1.321	4.058	3.0	3.211	1/1537	83.8%	430.5	25.9%

Tab.7 Performance indexes of the 3 chosen individuals

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