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

Frontiers of Structural and Civil Engineering

2022, Vol. 16

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

本期目录

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.

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

收稿日期: 2021-11-08 出版日期: 2022-12-22

Corresponding Author(s): Hao WU

引用本文:

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

链接本文:

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

Fig.1

Fig.2

Tab.1

Fig.3

$p (ξ)$	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

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

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

Fig.4

Tab.5

Fig.5

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

Fig.6

Fig.7

Fig.8

Fig.9

Fig.10

Fig.11

Fig.12

Fig.13

Fig.14

Fig.15

Fig.16

Fig.17

Fig.18

Fig.19

Fig.20

Fig.21

Fig.22

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

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