Enhanced empirical models for predicting the drift capacity of less ductile RC columns with flexural, shear, or axial failure modes

doi:10.1007/s11709-019-0554-2

Front. Struct. Civ. Eng.

2019, Vol. 13

Issue (5) : 1251-1270 https://doi.org/10.1007/s11709-019-0554-2

RESEARCH ARTICLE

Enhanced empirical models for predicting the drift capacity of less ductile RC columns with flexural, shear, or axial failure modes

Mohammad Reza AZADI KAKAVAND¹(

), Reza ALLAHVIRDIZADEH²

¹. Unit of Strength of Materials and Structural Analysis, Institute of Basic Sciences in Engineering Sciences, University of Innsbruck, Innsbruck 6020, Austria
². ISISE, Department of Civil Engineering, University of Minho, Guimar?es 4800-058, Portugal

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Abstract

Capacity of components subjected to earthquake actions is still a widely interesting research topic. Hence, developing precise tools for predicting drift capacities of reinforced concrete (RC) columns is of great interest. RC columns are not only frequently constructed, but also their composite behavior makes the capacity prediction a task faced with many uncertainties. In the current article, novel empirical approaches are presented for predicting flexural, shear and axial failure modes in RC columns. To this aim, an extensive experimental database was created by collecting outcomes of previously conducted experimental tests since 1964, which are available in the literature. It serves as the basis for deriving the equations for predicting the drift capacity of RC columns by different regression analyses (both linear with different orders and nonlinear). Furthermore, fragility curves are determined for comparing the obtained results with the experimental results and with previously proposed models, like the ones of ASCE/SEI 41-13. It is demonstrated that the proposed equations predict drift capacities, which are in better agreement with experimental results than those computed by previously published models. In addition, the reliability of the proposed equations is higher from a probabilistic point of view.

Keywords flexural-shear-axial failure drift capacity reinforced concrete columns statistical analysis fragility curves

Corresponding Author(s): Mohammad Reza AZADI KAKAVAND

Online First Date: 14 August 2019 Issue Date: 11 September 2019

Cite this article:

Mohammad Reza AZADI KAKAVAND,Reza ALLAHVIRDIZADEH. Enhanced empirical models for predicting the drift capacity of less ductile RC columns with flexural, shear, or axial failure modes[J]. Front. Struct. Civ. Eng., 2019, 13(5): 1251-1270.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-019-0554-2
https://academic.hep.com.cn/fsce/EN/Y2019/V13/I5/1251

researcher	model (drift) for shear failure	STD	COV
Pujol et al.	$(1 / 100) · (ρ ′ f y t / v) · (a / d)$	1.71	0.42
Pujol	$max ? (3100 · a d (1 − 7 k 1), 1100)$	1.12	0.55
Kato and Ohnishi	$(Δ y + Δ p) / L$	0.84	0.44
Elwood (flexure-shear)	$max ? (3100 + 4 ρ ′ − 140 υ f c' − 140 P A g f c', 1100)$ in MPa	0.97	0.34
Zhu et al.	shear: $2.02 ρ ′ − 0.025 s d + 0.013 a d − 0.031 P A g f c'$ flexure: $0.049 − 0.042 s d + 0.716 ρ 1 − 0.12 ρ ′ f y t f c' − 0.07 P A g f c'$	1.03	0.35 0.27

Tab.1 Proposed models in past research activities [²⁰,²²,²³,²⁵,²⁶]

parameter (unit)	flexural failure			shear failure			axial failure
parameter (unit)	range	mean	STD	range	mean	STD	range	mean	STD
$ρ "$ (%)	[0.18–2.55]	0.77	0.51	[0.07–0.93]	0.31	0.22	[0.07–0.57]	0.16	0.11
$ρ l$ (%)	[1.27–3.28]	2.03	0.50	[0.56–4.00]	2.17	0.60	[0.56–3.00]	2.11	0.59
a/d (–)	[1.18–6.64]	4.10	1.69	[1.18–5.40]	2.84	0.99	[1.18–4.44]	2.48	1.05
s/d (–)	[0.11–0.73]	0.36	0.14	[0.11–1.20]	0.49	0.30	[0.20–1.20]	0.72	0.33
$f c'$ (MPa)	[21.1–48.3]	33.0	6.44	[13.1–46.5]	26.8	7.5	[13.5–33.1]	23.8	5.6
$f y t$ (MPa)	[308–621]	494	106	[249–648]	410	78	[289–587]	405	64
$f y l$ (MPa)	[341–579]	439	80	[318–538]	424	61	[331–538]	410	66
V_p/V_n (–)	[0.32–0.60]	0.49	0.07	[0.60–2.05]	1.03	0.29	[0.52–1.57]	0.99	0.23
$P / A g · f c'$ (–)	[0–0.80]	0.30	0.22	[0–0.62]	0.19	0.14	[0.04–0.62]	0.23	0.14
Δ_max/L (%)	[0.27–7.40]	3.35	2.06	[0.38–9.00]	2.80	2.10	[0.40–8.70]	3.20	2.00

Tab.2 Statistical properties of the extracted parameters from the database [²⁹–⁷²]

Fig.1 Distribution of collected experimental results, classified by the respective failure mode. (a) Flexural and shear failure; (b) axial failure

Tab.3 Categorization of experimental results contained in the database

Fig.2 Scatter plots of the drift ratio at shear failure versus the employed parameters

method	coefficient
linear regression	$α 0 = − 0.044, α 1 = 0.031, α 2 = − 0.04, α 3 = 0.096, α 4 = 0.007$
	$R 2 = 0.490$
	test set error= 1.81%
quadratic equation (without mixed terms)	$α 0 = − 0.237, α 1 = 0.152, α 2 = − 0.06, α 3 = 0.846, α 4 = 0.02,$
	$α 11 = − 0.11, α 33 = − 0.85, α 44 = − 0.002$
	$R 2 = 0.828$
	test set error= 1.28%
full quadratic equation	$α 0 = − 2.294, α 1 = 1.005, α 2 = 0.613, α 3 = 3.255, α 4 = 0.239,$
	$α 5 = 4.181, α 11 = − 0.29, α 33 = − 2.076, α 44 = − 0.007, α 12 = − 0.336,$
	$α 13 = 0.493, α 14 = − 0.010, α 15 = − 2.80, α 23 = − 0.373, α 24 = − 0.071,$
	$α 25 = − 0.248, α 34 = − 0.057, α 35 = − 2.81, α 45 = − 0.334$
	$R 2 = 0.996$
	test set error= 24.86%
Kriging regression	$R 2 = 0.99$
	$t e s t ? s e t ? e r r o r = 0.32 %$

Tab.4 Statistical data of the parameters for the flexural failure predicted by Eq. (3)

method	coefficient
linear regression	$α 0 = 0.0199, α 1 = 0.05, α 2 = − 0.034, α 3 = − 0.025,$
	$α 4 = 0.011, α 5 = − 0.013$
	$R 2 = 0.817$
	test set error= 1.45%
quadratic equation (without mixed terms)	$α 0 = 0.15, α 1 = 0.15, α 2 = − 0.029, α 3 = − 0.354,$ $α 11 = − 0.16, α 33 = 0.202, α 44 = 0.002$
	$R 2 = 0.833$ test set error= 1.53%

Tab.5 Statistical data of the parameters for flexural-shear failure mode predicted by Eq. (3)

method	coefficient
linear regression	$α 0 = − 0.0087, α 1 = 0.202, α 4 = 0.0103, α 5 = − 0.0186$
	$R 2 = 0.745$
	test set error= 1.35%
quadratic equation (without mixed terms)	$α 0 = − 0.012, α 4 = 0.0348, α 5 = − 0.078,$ $α 11 = 0.588, α 44 = − 0.0042, α 55 = 0.0385$
	$R 2 = 0.777$
	test set error= 1.46%

Tab.6 Statistical data of the parameters for shear failure mode predicted by Eq. (3)

Tab.7 Evaluation of different functional relationships for predicting the effective coefficient of friction

Fig.3 Distributions of the ratio of calculated to measured drift capacity of RC columns according to (a) the proposed model for flexural, flexure-shear and shear failure modes and (b) Elwood model for flexure-shear failure mode [²⁵] and Zhu et al. models for flexure and shear failure modes [²⁶]

Fig.4 Distributions of the ratio of calculated to measured drift capacity of RC columns with axial failure. (a) Proposed model; (b) Elwood model

Fig.5 Probability paper to validate log-normal probability distribution of experimentally determined drift capacities with flexural and shear failure

Fig.6 Fragility curves corresponding to different failure modes. (a) Flexure-shear failure mode; (b) shear failure mode; (c) axial failure mode

Fig.7 Drift ratios corresponding to different probabilities of failure. (a) 15% probability of failure; (b) 50% probability of failure

Fig.8 Fragility curves for computing the probability of overestimating the drift capacities for different failure models by the proposed model, Elwood [²⁵] and Zhu et al. [²⁶] model and the ASCE/SEI 41-13 regulation. (a) Flexure-shear failure mode; (b) shear failure mode; (c) axial failure mode

Fig.9 Cracking patterns of non-ductile column C1. (a) top; (b) bottom; (c) axial failure at the top of C1 column. Cracking patterns of ductile columns C4; (d) top; (e) bottom and (f) at the lateral drift ratio of 9% [⁹⁰]

column	$(ρ h f y t ρ 1 f y l)$	$P A g f c$	$a d$	$s d$
C1	0.0918	0.10	3.759	0.751
C4	1.587	0.10	3.759	0.248

Tab.8 Mechanical and geometrical parameters of the tested columns [⁹⁰]

Fig.10 Comparison of the predicted results by the proposed equations, Elwood and Zhu et al. models with experimental data by Wu et al. [⁹⁰]. (a) Flexural and shear failure; (b) axial failure

frame-column	$(ρ h f y t ρ 1 f y l)$	$P A g f c$	$a d$	$s d$
MCFS-B1	0.0657	0.200	3.825	0.656
HCFS-B1	0.0657	0.350	3.825	0.656
MUFS-B2	0.0626	0.200	3.825	0.656

Tab.9 Important mechanical and geometrical parameters of the selected columns of the tested frames [⁹¹]

Fig.11 Flexural and shear cracks in columns of the tested frames. (a) MCFS; (b) HCFS; (c) MUFS and failure of columns in the frames; (d) MCFS; (e) HCFS and (f) MUFS, according to Yavari [⁹¹]

Fig.12 Comparison of the predicted results by the proposed equations, Elwood and Zhu et al. models normalized by the experimental data by Yavari et al. [⁹¹]. (a) Shear failure (b) axial failure

Fig.13 Elevation view and structural details of the RC frames, selected for the case study. (a) 3-story frame; (b) 5-story frame

Fig.14 Schematic view of the employed nonlinear structural models

Fig.15 5% damped elastic spectrum of assigned ground motion records

Fig.16 IDA curves, obtained on the basis of the modeling approaches by Elwood and the ASCE/SEI 41-13 regulation, with the drift ratio capacities corresponding to different failure modes. (a) 3-story-ASCE method; (b) 3-story-Elwood method; (c) 5-story-ASCE method; (d) 5-story-Elwood method

Fig.17 Predicted base shear in terms of the roof drift obtained from pushover analyses and drift ratios corresponding to different failure modes, predicted on the basis of the ASCE regulation [²⁷] and Elwood modeling approach [²⁵]. (a) Shear failure; (b) axial failure

Fig.18 Drift ratios predicted by the pushover analyses, based on the proposed equations for both shear and axial failure types and the modeling approaches by Elwood [²⁵] and the ASCE regulation [²⁷]. (a) Shear failure; (b) axial failure

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