Seismic responses of reinforced concrete intermediate short columns failed in different modes

Engineering Structures 206 (2020) 110173

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Seismic responses of reinforced concrete intermediate short columns failed in different modes

T

⁎

XiangLin Gua,b, , JingJing Huaa,b, Mao Caic a

Key Laboratory of Performance Evolution and Control for Engineering Structures of Ministry of Education, Tongji University, 1239 Siping Rd, Shanghai 200092, PR China Department of Structural Engineering, Tongji University, 1239 Siping Rd, Shanghai 200092, PR China c China Shipbuilding NDRI Engineering Co., LTD, 303 Wuning Rd, Shanghai 200063, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Reinforced concrete intermediate short column Seismic response Failure mode Shear spring

The main objective of this paper is to observe the seismic responses of intermediate short columns failed in different modes and develop an efficient analytical model reproducing their behavior based on the experimental observation, which will eventually be used to execute assessments of seismic behavior for existing RC frame structures. Five RC columns designed to fail in different modes were tested under constant axial load plus unidirectional cyclic lateral loading, the effect of height-to-depth ratio, axial load ratio and transverse reinforcement ratio on their seismic response was investigated. Interaction of flexural and shear responses was studied by comparing their component deformations. Test results shown that for a column whose shear-resisting mechanism was significantly degraded, flexural responses would not develop while shear responses increased rapidly. However, for a column possessing a shear-resisting mechanism without significant deterioration, shear responses would stably increase as flexural responses continued to develop over the entire range of imposed displacement ductility. In the proposed model, flexural behavior was estimated by the multi-springs model employing the hysteretic behavior of steel and concrete materials. Pre-peak monotonic shear behavior was modeled by applying the truss model, and the degrading slope after shear failure was estimated by the shear-friction model. The shear hysteretic model proposed by Ozcebe and Saatcioglu was expanded for seismic response evaluation before and after failure. The column model was validated with results of test columns both from this paper and the literature.

1. Introduction Generally, an RC column with a lower height-to-depth ratio (in between 2 and 4) are called intermediate short columns. Post-earthquake reconnaissance has shown that in-service structures with RC intermediate short columns may fail in different modes. The ultimate failure of a column in brittle shear occurs at a relatively low displacement accompanied by a large drop in lateral load resistance. Columns that first yield in flexure may do so under cyclic loading and then ultimately fail in a mode showing prominent shear phenomena, such as increasing diagonal crack widths, a yielding of transverse steel, and a drop of peak force resistance with cycling). This failure mode is termed “flexure-shear” failure [1]. By contrast, columns that fail in flexure show a prominent flexure phenomena (e.g., single wide cracks perpendicular to a member axis at the section of maximum moment, as

well as concrete crushing or longitudinal buckling next to that section). Meanwhile, any diagonal cracks that may have formed initially decrease in width [2]. The intermediate short columns usually fail at an early stage during an earthquake due to their relatively high stiffness; however, a complete understanding of the seismic performance that led to their failure in different modes is lacking. Moreover, a reliable model to predict the failure processes of columns is necessary for the seismic assessment and retrofitting of RC buildings [3,4]. Hence, investigating the seismic behavior and providing the analytical model for intermediate short columns failed in different modes are primary research objectives of this paper. A comprehensive test designed to compare the seismic behavior of RC intermediate short columns that failed in different modes is scarcely seen. The post-crack stiffness of an RC intermediate short column should be accurately estimated because it always sustains larger shear

⁎ Corresponding author at: Key Laboratory of Performance Evolution and Control for Engineering Structures of Ministry of Education, Tongji University, 1239 Siping Rd, Shanghai 200092, PR China. E-mail addresses: [email protected] (X. Gu), [email protected] (J. Hua), [email protected] (M. Cai).

https://doi.org/10.1016/j.engstruct.2020.110173 Received 7 June 2019; Received in revised form 2 January 2020; Accepted 2 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

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strength for RC columns with rectangular sections by combing and modifying a shear-strength model for truss-tension failures [25] with softened strut-and-tie model for strut-compression failures [6], furthermore, the authors added two new steps to consider truss-tension and truss-compression mechanisms of failure. Comparison of the measured and calculated results prove the accuracy of the shear strength estimation, as well as the accuracy of the identification of the observed shear-resistance mechanism for a database of 62 shear-critical columns. Although these strength models are useful for predicting the force transfer mechanism and shear capacity for shear-critical RC components when they reach the shear strength point, it is unclear whether they can accurately predict the failure point for columns experiencing shear failure after flexural yielding [23]. In reality, the degradation of shear response can be initiated when the column lateral deformation exceeds the estimated ultimate deformation, and several models have been developed to predict the ultimate displacement for RC columns failed in flexure-shear [23,32,33]. In this paper, five columns under a constant axial load and cyclic lateral loading were tested up to axial failure or severe loss in lateral strength to obtain a clear understanding of their seismic performances. Experimental results yielded crack patterns, strength, stiffness, and hysteretic behavior. Total lateral deformations of column specimens were decoupled into shear and flexural deformation to study their interactions. Great attention was paid to the distribution and magnitude of shear deformation when a column was loaded in the inelastic range. This helped us determine whether the shear deformation was important enough to consider in our structural analysis. Strength and stiffness degradation of column specimens that failed in shear under cyclic loadings were discussed in detail. Based on experimental observations, a shear model was developed incorporating shear-flexure interaction and a modified shear hysteretic model accounting for shear strength and stiffness deterioration in the post-peak range. The modified shear model was implemented as a shear spring accompanied by a multisprings model to effectively reproduce the cyclic response of RC intermediate short columns. A modified shear strength model considering different shear resisting mechanisms [31], accompanied by an ultimate displacement model [23] was adopted to predict the point of shear failure. Tests verified that the proposed model is computationally effective and reliable in predicting seismic behavior for shear-, flexureshear- and flexure-dominated columns.

force and can affect the distribution of lateral forces among other members. Lynn et al. [5] demonstrated that for RC columns with inadequate transverse reinforcement and a height-to-depth ratio larger than 5, the contribution of shear deformation to total lateral displacement at peak strength is relatively small; furthermore, inaccuracy in calculating shear deformation has had a minor impact on the total deformation calculation. In this century, Li and Hwang [6] believed the shear deformation could not be ignored because for short columns with a height-to-depth ratio of less than 2, the average percentage of shear deformation at peak strength reached as high as 60%. Because of the limited number of Li and Hwang’s [6] tests, more tests are needed to study the inelastic shear responses of intermediate short columns failed in different modes. Current modeling approaches used for simulating the seismic performance of an RC structure can accurately predict the flexure-critical RC columns, while the incorporation of cyclic shear effects and the coupling of shear-flexure interaction need further development. Two methods are proposed to predict the column shear response. The first approach is based on the fiber section discretization. In this case, the shear response is modeled for each concrete fiber of the selected section, and the strain field of the section incorporates the classical plane section hypothesis for the longitudinal strain and assumes a constant or parabolic shear strain [7,8]. Additionally, an accurate sectional analysis has also been proposed for the estimation of shear strain distribution [9]. Bidimensional or tridimensional concrete constitutive models with cyclic capabilities were adopted in this approach, which led to several theories, such as the smeared crack theory [10,11], the microplane theory [7,8] and damage mechanics [12].The above models can model the principle mechanism of shear deformation and failure directly, accounting for the axial-shear-flexure interaction in a more physical and rational way. Practical limitations of this approach are the lack of quantity information about the response of severely cracked concrete under post-yield load reversals. Besides, it is not economical to describe shear behavior in its full complexity when the model is to be used in the dynamic response analysis of large-scale structures. In the second approach, inelastic hysteretic shear deformation is modeled independently from the flexural response. This separation of shear and flexure response allows freedom in combining shear and flexural responses. On the basis of the shear modeling approach, this model is further divided into three categories: the elastic model [13–15], the continuous inelastic model [16], and the macro model [17–20]. Previous research has revealed that computing shear deformation based on elastic properties significantly underestimates the shear deformation of members with shear cracking [21]. Few models attempt to generate continuous shear responses of concrete columns under seismic loadings, due to its deficiency in efficiency and stability, as well as the difficulty in applying the appropriate hysteretic constitutive model. Nowadays, a macro model including a predefined shear force-shear deformation primary curve and a set of loading/unloading rules has been most-widely used. The hysteretic shear model proposed by Ozebe and Saatcioglu [22] can be used to accurately capture the pinching, stiffness softening, and strength deterioration of a column; however, their model is limited to the deformation range where strength decay is not significant. An important aspect of modelling inelastic shear behavior is to accurately predict the point of “shear failure”. Several models for column shear strength have been proposed and used for design of new buildings and/or assessment of existing structures [23–30], however, few of them are easily to be adopted in detecting the shear failure point of columns in which the critical shear-resistance mechanism is unknown. Li et al. [29] divided the shear capacity for shear-critical intermediate short columns into shear compressive failure and shear tensile failure based on experimental observation, shear compressive strength and shear tensile strength were estimated by the softened strut-and-tie model [30] and ASCE/SEI 41–13 formula [27], respectively. Hua et al. [31] proposed a versatile, multi-mechanism methodology to estimate the shear

2. Experimental investigation 2.1. Specimen Statistical data in previous study revealed that the parameters affecting columns’ seismic response include height-to-depth ratio, axial load ratio, longitudinal and transverse reinforcement ratio [31,34]. Therefore, in this study, five column specimens were designed and tested quasi-statically to further understand how different parameters affect the seismic behavior and failure modes of intermediate short columns [35,36]. The design parameters and measured material properties are listed in Table 1 and Fig. 1. All specimens in this study had a 350 mm × 350 mm square cross-section with a clear cover of 30 mm. Each specimen consisted of heavily reinforced 1400 × 500 × 700 mm beams at the top and bottom, simulating a stiff foundation or floor elements. The top beam was connected to the loading frame and the bottom beam was connected to the strong floor. For the columns designed to fail in brittle shear (C-S1, C-S2, and CS3), 14 deformed bars with nominal diameters of 18 mm were employed as the longitudinal reinforcement, resulting in a longitudinal reinforcement ratio (total reinforcement area divided by gross area) of 3.28%. This large amount of longitudinal reinforcement kept the flexural behavior of the column in an elastic range, which enabled shear failure and excluded flexure-shear failure. Longitudinal reinforcements were bent at the top and bottom inside the beams to ensure a good 2

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Table 1 Test parameters and material properties of column specimens. Specimen

b × h, mm

L/h

ρt, %

ρl, %

N/Ag f c' ,

Material Properties, MPa

Failuremode

f c'

fyl

ful

fyt

This study C-S1 C-S2 C-S3 C-FS C-F

350 350 350 350 350

350 350 350 350 350

3.7 3.1 3.7 3.7 3.7

0.14 0.14 0.14 0.28 0.57

3.28 3.28 3.28 2.59 1.70

0.13 0.13 0.30 0.13 0.13

32 38 32 38 38

398 398 398 354 498

572 572 572 526 563

303 303 303 303 303

S S S FS F

Gill et al. [37] No. 1 No. 2 No. 3

550 × 550 550 × 550 550 × 550

4.4 4.4 4.4

0.71 1.10 0.76

1.79 1.79 1.79

0.26 0.21 0.42

23 41 21

375 375 375

636 636 636

297 316 297

F F F

Tanaka and Park [38] No. 5 550 × 550 No. 7 550 × 550

6 6

0.75 0.91

1.25 1.25

0.1 0.3

32 32

511 511

675 675

325 325

F F

Matamoros et al. [39] C5-00N 203 × 203 C5-00S 203 × 203

6 6

0.92 0.90

1.93 1.93

0 0

38 38

572 572

729 729

514 515

F F

Ramirez and Jirsa [40] 00-U 305 × 305 120C-U 305 × 305

3 3

0.32 0.32

2.5 2.5

0 0.19

35 31

374 450

– –

455 455

FS S

Umehara and Jirsa [41] OUS 229 CUS 229 2CUS 229 OUW 406 CUW 229

2.3 2.3 2.3 4 4

0.28 0.28 0.28 0.28 0.28

3.1 3.1 3.1 3.1 3.1

0 0.16 0.27 0 0.16

40 35 42 40 35

441 441 441 441 441

745 745 745 745 745

414 414 414 414 414

S S S S S

Yoshimura et al. [42] CE 175 × 175 BE 175 × 175 LE 175 × 175

2 2 2

0.29 0.29 0.29

5.2 5.2 5.2

0.1 0.1 0.1

26 33 42

388 344 344

569 498 498

312 312 322

S S S

Aboutaha et al. [43] SC9 457 × 914

2.7

0.08

2.0

0

16

434

690

400

S

Yoshimura and Nakamura [44] N27C 300 × 300 2C 300 × 300 3C 300 × 300

3 2 2

0.19 0.19 0.19

2.7 2.7 2.7

0.27 0.2 0.3

27 25 25

380 396 396

– – –

375 392 392

FS S S

Yoshimura et al. [45] No.6 300 × 300 No.7 300 × 300

4 4

0.19 0.13

1.8 1.8

0.2 0.2

31 31

409 409

– –

392 392

FS FS

Ousalem et al. [46] No.1 No.11 No.12 No.14

300 300 300 300

2 3 3 3

0.38 0.13 0.13 0.38

1.8 2.4 2.4 2.4

0.2 0.2 0.2 0.2

28 28 28 26

447 447 447 447

– – – –

398 398 398 398

S S S FS

Moretti and Tassios [47] Specimen 1 250 × 250 Specimen 2 250 × 250 Specimen 3 250 × 250

2 2 2

1.21 1.21 1.21

2.0 2.0 2.0

0.3 0.3 0.3

36 39 35

480 415 415

740 630 630

300 300 305

FS S S

Tran [48] SC-1.7-0.05 SC-1.7-0.20 SC-1.7-0.35 RC-1.7-0.05 RC-1.7-0.20 RC-1.7-0.35

3.4 3.4 3.4 3.5 3.5 3.5

0.13 0.13 0.13 0.18 0.18 0.18

2.1 2.1 2.1 2.1 2.1 2.1

0.05 0.2 0.35 0.05 0.2 0.35

30 28 26 33 25 27

408 408 408 408 408 408

607 607 607 607 607 607

393 393 393 393 393 393

FS S FS FS FS FS

300 300 300 300

350 350 350 250 250 250

× × × × ×

× × × × ×

× × × ×

× × × × × ×

406 406 406 229 406

350 350 350 490 490 490

*Note: 1. b = column section width perpendicular to the applied shear; h = column section height parallel to the applied shear; L = length of column; ρt = is the area ratio of transverse reinforcement to concrete (ρt = Ast/bh0, Ast = transverse reinforcement area within the spacing, s, in the loading direction; h0 = distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement); ρl = total longitudinal reinforcement ratio; N = axial load; Ag = gross area of section; f c' = cylinder compressive strength of concrete ( f c' = 1.18 × fck MPa, fck = prism compressive strength of column)); fyl = yield strength of longitudinal steel bars; ful = ultimate strength of longitudinal steel bars; fyt = yield strength of transverse steel bars. 2. Failure mode: S-shear failure; FS-flexure-shear failure; F-flexural failure.

anchorage. This reinforcement consisted of closed square perimeter hoops made of an 8 mm diameter rebar with 135-degree hooks. The transverse reinforcement spaced at 200 mm intervals, resulted in a

transverse reinforcement ratio equal to 0.14% (area of transverse reinforcement in one horizontal direction divided by the product of section width and hoop spacing). Specimen C-S1, considered as the 3

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700 mm 1300 mm [1100 mm]

1400 mm

D8@200 δD8@100ε

C-S1ȽC-S2 C-S3ȽC-FS

12D14

350 mm

350 mm

14D18 (14D16, 14D14)

350 mm

700 mm

1300 mm [1100 mm]

350 mm

700 mm

2700 mm [2500 mm]

350 mm

500 mm

700 mm

1400 mm

500 mm

D8@100

350 mm C-F

Note: [ ] shows the elevation detail of specimen C-S2 Fig. 1. Specimen elevation and cross-section details.

minimum displacement increment of δ (δ is the displacement in the last cycle controlled by force), up to axial failure or to the point where the measured shear strength degraded to 10% of the maximum lateral strength. Lateral loading was applied slowly in three-cycle increments at each displacement level.

reference specimen, had a clear height of 1300 mm, resulting in heightto-depth ratio equal to 3.7, and was tested under a constant axial load ratio of 0.13 (applied axial load divided by the product of cylinder concrete compressive strength and section gross area). For other specimens, only one parameter was changed with respect to the reference specimen C-S1. C-S2 had a height-to-depth ratio of 3.1, and C-S3 had an axial load ratio of 0.30. Specimen C-FS was designed to fail in flexure-shear. In this case, 14 deformed bars with nominal diameters of 16 mm were employed as the longitudinal reinforcement. Specimen C-FS had hoop sets with 135degree hooks at 100 mm spacing, resulting in a transverse reinforcement ratio of 0.28%. Other test parameters were the same as specimen C-S1. Specimen C-F was designed to fail in flexure. In this case, 14 deformed bars with nominal diameters of 14 mm were employed as the longitudinal reinforcement. Additionally, Specimen C-F had two identical rectangular stirrups, which were stacked in a cross-shaped arrangement and were used in addition to the perimeter hoops at 100 mm spacing, resulting in a transverse reinforcement ratio of 0.56%. The remaining test parameters were the same as specimen C-S1. Concrete compressive strength were obtained from standard compression tests on the 150 mm × 150 mm × 300 mm prisms and ranged from 27 to 32 MPa. The corresponding cylinder concrete strength can be found in Column 7 of Table 1. Yield strengths of the longitudinal bars and transverse bars can also be found in Column 8 to Column 10 of Table 1. The applied axial load ratios are listed in Column 6 of Table 1.

2.3. Determining shear and flexural deformations The lateral deformation of column consisted of deformations due to flexure (Δf) and shear (Δs). External measuring instruments were used in this test measuring different deformation components to investigate their contribution. As depicted in Fig. 3(a), the region surrounded by a series of LVDTs was considered as a unit measuring element. Average curvature in the element was calculated as the difference between two vertical LVDTs divided by the distance between them and the vertical height between the top and bottom horizontal LVDTs. The total flexural deformation Δf, including both the flexural deformation and deformation due to the fixed-end rotation, was obtained by integrating the average curvature over the height of the column and can be calculated by Eq. (1), where, θfi is the average rotation angle for the ith measuring element. As shown in Fig. 3(b)–(f), average shear deformation Δsi in the ith measuring element was calculated as a change in the diagonal length Δdi (minus the length change due to the vertical and horizontal elongation, as shown in Eqs. (3) and (4)) divided by the sine of the angle between the diagonal LVDT and the column longitudinal axis. The total shear deformation Δs was calculated by summing the average shear deformation of each measured element (Eq. (2)).

2.2. Test program and instrumentation As shown in Fig. 2, all columns were tested in an anti-symmetric double curvature configuration, and the specified axial load was applied on the specimen by the vertical hydraulic actuator acting on the loading frame through the axis of the top section of each specimen. The uniaxial lateral loading was applied by the horizontal hydraulic actuator through the mid-height of the column. Any rotation of the top beam was prohibited by the four-bar linkage. Cyclic lateral loading was applied as specified in JGJ/T 101-2015 [49]; thus, the lateral load was first applied in a force control mode. The minimum force increment was 10% of the estimated maximum shear strength. When the prominent inclined cracks appeared or the longitudinal reinforcement yield, the lateral load was applied in a displacement control mode with the

Δf =

∫0

L

i

φxdx ≈

1

i

Δs =

i

∑ 1

4

i

∑ θfi ∙di = ∑

Δsi =

∑ 1

Δdsi = sin ϕi

1 i

∑ 1

Δli − Δri ∙di H

Δdi − Δdhi − Δdvi sin ϕi

(1)

(2)

Δdvi =

1 Δ + Δri Δvicosϕi , Δvi = li 2 2

(3)

Δdhi =

1 Δ + Δti Δhisinϕi , Δhi = bi 2 2

(4)

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vertical load

hydraulic actuator lateral load direction

support area

support area rigid steel beam

four-bar linkage

hydraulic actuator

130mm

specimen hydraulic jack

hydraulic jack

stopper

Fig. 2. Test setup and measurement instrumentation.

inclination of the critical shear failure plane θ (defined as the angle with vertical axis, as shown in Fig. 4(d)). In addition to above parameters, the stiffness of the line connecting the ultimate displacement point (Δu, 0.8Vmax) and the point when applied shear reduced to 20% of Vmax, Kdeg, is used here to describe the rate of lateral resistance degradation with increasing lateral displacement at the post-peak range. Test results illustrated in Table 2 and Fig. 5 were compared according to each of the main experimental parameters. Specimens with a high axial load exhibited a higher shear resistance than specimens under a low axial load. At the post-peak range, severe deterioration of concrete occurred because of the high axial load, resulting in a greater lateral strength degradation rate; thus, the inclination of the shear failure plane was found to increase with increasing axial load. Axial failure occurred with the occurrence of apparent concrete crushing at the top of specimen C-S3 (Fig. 4(e)). When comparing specimens with different height to depth ratios (=L/h), the maximum shear strength of specimens shows a tendency to increase with decreasing L/h. Shear failure occurred suddenly with the occurrence of a steep crack along the diagonal direction of the column body. Strength degradation was more severe, and the gradient of shear failure plane was slightly steeper than specimens with larger L/h. According to previous study, the shear strength of columns is sensitive to the amount of transverse reinforcement, and it can be

3. Experimental results and discussion 3.1. Test observations All specimens failed in the designed failure modes. Fig. 4 shows the failure crack patterns developed in column specimens. The cracks in specimens that failed in shear and flexure-shear were primarily diagonal cracks, along with the spalling of cover concrete at the intersection of diagonal cracks. Some of the inclined cracks propagated widely at the end of test (shown by red line in Fig. 4). As for specimen C-F, flexure failure occurred with the occurrence of horizontal cracks and apparent concrete crushing near the top and bottom of the column. No apparent diagonal cracks were observed at the end of the test. Hysteretic loops for each specimen are shown in Fig. 5. Drift ratio instead of top displacements are plotted on the x-axis to facilitate the comparison between columns of different heights. Table 2 summarizes the key test results including the maximum shear strength Vmax and horizontal displacement at yield point Δy (indicated by strain gauges). For specimens failed in shear without flexural yielding, Δy was not recorded. Table 2 also compares horizontal displacement at maximum strength Δmax and ultimate displacement Δu (determined based on the backbone curve from the test data and was taken as the displacement when the applied shear dropped to 80% of Vmax), as well as the

Fig. 3. Instrumentation Arrangement. 5

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Fig. 4. Crack pattern at failure point for specimens: (a) C-S1, (b) C-S2, (c) C-S3, (d) C-FS, and (e) C-F.

approximation; however, a slight discrepancy exists because of experimental errors, such as the assumption of curvature distribution and not noting the differences in the lengths of the diagonals due to flexural deformations [50]. Since this study is primarily concerned with the relative ratio of the shear and flexural deformation component to total deformations, these errors will not be discussed here. Fig. 6 plots the contributions of different deformation components to the total lateral displacement at the maximum displacement of each cycle. The relative contributions of shear and flexural deformations vary differently with the lateral drifts for columns failed in different modes. For shear-critical columns, such as specimen C-S1, shear deformations were relatively small and flexural deformation dominated the response initially, while shear deformation grew dramatically to about 50% of the total displacement at peak strength. Shear deformation increased in a larger proportion than flexural deformation, reaching 80% at a drift ratio of 1.7%, which corresponded to diagonal concrete crushing at shear failure; however, flexural deformation remained approximately constant once the columns reached their maximum strength. For columns forming flexural hinges, the shear-resisting mechanism did not significantly deteriorate, and both flexure and shear

reconfirmed that specimen C-FS has a higher strength than specimen CS1; however, when comparing specimens C-FS and C-F, it seems that the strength is insensitive to the amount of transverse reinforcement. With increasing transverse reinforcement ratio, the failure mechanism is transferred from the concrete crushing in diagonal cracks or stirrup fracture to the disintegration of the compression zone at the top/bottom section or at the buckling of longitudinal bars next to that section. Any diagonal cracks forming at this time will initially decrease in width and the transverse reinforcement contribute little to the lateral resistance.

3.2. Contributions of flexure and shear to total deformations Using external measuring instruments with various functions placed at different positions, flexural and shear deformations could be calculated approximately. Table 3 compares the calculated lateral deformation Δcalc (sum of Δf and Δs) and the measured lateral deformation Δtest (measured from the LVDTs installed on the top and bottom of the specimen at peak and failure stages). This table shows that the average ratio of the observed to the calculated total displacements is approximately 0.9, and the corresponding coefficient of variation is 0.05 for deformation at peak strength and 0.10 for the ultimate deformation. Comparative results show the reliability of the deformation Yield point Maximum point

(a)

(b)

(c)

(d)

(e)

Fig. 5. Hysteretic response for specimens: (a) C-S1, (b) C-S2, (c) C-S3, (d) C-FS, and (e) C-F. 6

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Table 2 Test results of specimens. Specimen

Δy, mm

Vmax, KN

C-S1 C-S2 C-S3 C-FS C-F

– – – 7.6 9.6

260 372 336 372 337

Maximum point

Failure point

Δmax, mm

Δmax/L, %

Δu, mm

Δu/L, %

8.4 9.6 8.6 19.8 31.1

0.6 0.9 0.7 1.5 2.4

8.8 10.9 11.0 30.3 62.2

0.7 1.0 0.9 2.3 4.8

C-S1 C-S2 C-S3 C-FS C-F

Peak, V = Vmax

Failure, V = 0.8Vmax

Dtest, mm

Dcalc, mm

Dtest/Dcalc

Dtest, mm

Dcalc, mm

Dtest/Dcalc

8.4 9.6 8.6 19.8 31.1

6.6 8.0 7.6 17.6 27.0

0.79 0.83 0.88 0.89 0.87

8.8 10.9 11.0 30.3 62.2

6.6 9.4 9.0 29.0 58.0

0.75 0.86 0.81 0.96 0.93

0.90 0.05

Average Coefficient of variation

Average Coefficient of variation

μ

θ, °

12.79 31.16 18.02 9.36 1.80

– – – 4.0 6.5

30 24 16 24 –

Shear deformations were obtained at peak displacement in the first cycle of each displacement amplitude. Similar results were obtained for other specimens. Although shear force in a column subjected to a concentrated lateral load at the top is constant over the height, the corresponding shear deformations are not evenly distributed when the column cracks or when the longitudinal reinforcement yields. For shear-critical columns (Fig. 7(a), (b) and (c)), shear deformations were relatively small until shear strength significantly deteriorated. Comparison of the shear deformation distributions and crack patterns (Fig. 4) shows large shear deformation concentrate in regions where inclined cracks crossing. For flexure-critical columns whose shear-resisting mechanism did not significantly deteriorate (Fig. 7(e)), shear deformation mainly concentrated in the plastic zone where tensile strains in the longitudinal steel were large and additional elongations due to flexural response were obvious. This result indicates that the magnitude of shear deformation can be linked to the magnitude of tensile strains in the longitudinal steel or axial extensions due to flexural response. For a column forming plastic hinges first but failing in shear (Fig. 7(e)), shear deformation initially develops in the plastic zone, while it increases substantially while concentrating on the extended diagonal cracks where shear failure occurred. Here, the deformation increasing was accompanied by the dominant response transform from flexure to shear.

Table 3 Comparison of measured and calculated displacement. Specimen

Kdeg

0.90 0.10

deformations increased with displacement demand. Flexure response dominated the total response as it held the highest percentage of the total deformation. This is illustrated in Fig. 6(e) where the shear deformation of specimen C-F accounts for more than 20% of the total deformation at peak strength, and the ratio of shear to total deformation is approximately constant over the entire subsequent ductility range. In Fig. 6(d), specimen C-FS also exhibits this type of behavior during the cycles before peak strength. Therefore, shear deformation should not be ignored when RC intermediate short columns fail in shear, flexure-shear, and flexure.

3.4. Verification of the plan-section-assumption 3.3. Distribution of shear deformations A series of LVDTs were installed near the top of the column specimen to investigate the average deformation within 130 mm from the

Fig. 7 shows the variation of shear deformations for all specimens.

(a)

(b)

(c)

(d)

(e)

Fig. 6. Contribution of displacement components to total lateral displacement for specimen: (a) C-S1, (b) C-S2, (c) C-S3, (d) C-FS, and (e) C-F. 7

Engineering Structures 206 (2020) 110173 DS4

15

DS3 DS2 DS1

10 5 0 0.2

1.2

1.8

30 25 20 15 10 5 0

15 10

DS3

DS2 DS1

5 0

(b) +

+

DS4

2.4

Drift Ratio (%)

Shear Deformation (mm)

(a)

0.6

20

Shear Deformation (mm)

+

0.5 1.0 1.5 2.0 2.5 3.0

DS4 DS3 DS2 DS1

+

+

DS4 DS3

10

DS2 DS1

5 0 0.3

0.5

0.9

1.3

1.7

Drift Ratio (%)

DS4

15

10

DS3 DS2 DS1

5 0 0.6 0.8 1.5 2.3 3.0 4.0 4.7

0.6 0.8 1.5 2.4 3.0 4.0

(d)

20

15

(c)

Drift Ratio (%)

Shear Deformation (mm)

20

Shear Deformation (mm)

Shear Deformation (mm)

X. Gu, et al.

(e)

Drift Ratio (%)

Drift Ratio (%)

Fig. 7. Shear deformation distribution for specimen: (a) C-S1, (b) C-S2, (c) C-S3, (d) C-FS, and (e) C-F.

Fig. 8. Average deformation distribution over the section of a column.

buckling/fracture of the flexural reinforcement in the critical section. Flexural deformation grows rapidly with increasing displacement demand and will become dominant. Shear deformation increases stably over the entire loading range, showing little difference before and after reaching the nominal peak strength. A simplified analytical model consisting of two sub-elements representing inelastic flexure and shear deformation mechanisms was developed to describe the hysteretic behavior of RC intermediate short columns failed in different modes. Fig. 9 shows both the flexural and shear sub-elements connected in a series to sustain the external forces. Deformation in each sub-element is then superimposed for estimating the total deformation. In this paper, the modified multi-springs model [51] is adopted to demonstrate the flexural sub-element, and a

4. Analytical model

Flexure subelement

According to this study’s experimental results, it can be assumed that the load-carrying mechanism for an RC intermediate short column includes flexure and shear mode. The flexural and shear responses developed simultaneously before any one of the resistance mechanisms failed, and both the flexural and shear responses engage in a series activities to undertake the applied lateral load. If one of the mechanisms disintegrates, for example, the concrete in the diagonal cracks crushes or transverse reinforcements crossing the diagonal cracks’ ruptured. The shear-resisting mechanism then deteriorates, and the shear deformation increases rapidly until the shear response becomes dominant. In other cases, flexural hinges form in a column, and a stable shear-resisting mechanism initiates, along with concrete crushing or

Inelastic element Elastic element

Shear sub-element

L

top section at each load step (Fig. 2), and the results for specimen C-S1, C-FS and C-F are shown in Fig. 8, where the average deformation was plotted in each direction (pull is positive and push is negative). Measurements from LVDTs revealed that when columns failed in shear or flexure-shear, the average deformation distribution was almost linear over the section before shear failure. However, as the damage accumulated, a critical shear crack formed and intersected with the end section, and the deformation linear distribution assumption may no longer be considered. However, for specimen C-F, the distribution remained linear throughout the test. Therefore, it could be concluded that the plane section assumption is always available for the flexure-critical column, while for columns that ultimately failed in shear, the application of the assumption needs further verification.

Uniaxial concrete steel spring element Fig. 9. Multi-springs model considering shear. 8

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once the rotation and the deformation of the neutral axis of the section in the inelastic element are decided, the total deformation of the column could be calculated using Eq. (6). The analytical procedure is schematically summarized in the red dotted box in Fig. 21.

Reinforcement springs

Shell concrete springs

Δ = 2(Δp + Δe) = θp L + ML2 /(6Ec Ic )

where Δp and Δe are the calculated lateral displacement based on the inelastic and elastic model, respectively; θp is the rotation of section in the inelastic model; Ec is the elastic modulus of normal strength con-

Core concrete springs

crete (=4700 f c' ) [55], which may not be suitable for higher strength concrete or fillers [56]; Ic is the moment of inertia of section about centroid axis.

Fig. 10. Subdivision of the column section.

simplified model is proposed for the shear sub-element. The components of these models, as well as their interaction, are illustrated in the following sections.

4.1.2. Constitutive model of the concrete springs The restoring force and the deformation of the concrete springs are defined as:

4.1. Flexural sub-element

⎧ pc = Ac σc ⎨ ⎩ d c = l p εc

4.1.1. Element formulation Multi-springs model [52–54] consists of an elastic element representing elastic behavior and two inelastic elements of zero-length at both ends representing an inelastic response in the plastic region. Each inelastic element includes a series of concrete and longitudinal reinforcement springs, as shown in Fig. 9. The sites of the reinforcement springs are the real location of the longitudinal springs and the representative area is the corresponding steel section area. Based on the stirrup confinement effect, the concrete cross section of an element is divided into two parts: the covering layer (or the shell) and the core section. Each part consists of several lattices, as shown in Fig. 10. The sites of the concrete springs are the centroids of the corresponding lattices, and the representative area is the area of the corresponding lattices. The real length of the concrete and the reinforcement springs is zero and their representative length lp, also known as the length of the plastic hinge region, which is defined in Eq. (5), where a = shear span (distance from the point of maximum moment to the point of zero moment).

1.5ha/ h 0 ≥ 3.0 lp = 0.5ha/ h 0 2.0 < a/ h 0 < 3.0 ⎨ 1.0ha/ h 0 ≤ 2.0 ⎩

(6)

(7)

where Ac, dc, pc are the representative area, deformation and the restoring force of a concrete spring, respectively; εc and σc are the mean strain and mean stress of a concrete lattice. The force-deformation relationship of the concrete springs was proposed by Li [53,54]. A linear model with five segments was used here as the primary curve defining the monotonically increasing deformation response of concrete springs, as shown in Fig. 12. It includes five characteristic points: tensile failure point (2dct,0), maximum tensile strength point (dct, pct), cracking point (0.3dcy,0.5pcy), maximum compressive strength point (dcy,pcy) and ultimate displacement point (dcu,0). The superscript “shell” and “core” in Fig. 12 represents concrete springs in the shell section and core section, respectively. A concrete spring fails when its compressive deformation reaches dcu or the tensile deformation reaches 2dct. The spring constants can be calculated as follows:

⎧

(5)

dct = εt 0 lp, pct = ft Ac

(8)

dcty = εc 0 lp, dcu = εcu lp, pcy = fck Ac

(9)

where εt0 and εc0 are strains corresponding to the concrete tensile strength and concrete compressive strength, respectively. εcu is the ultimate concrete compressive strain; dct, dcy and dcu are deformations corresponding to concrete strains εt0, εc0 and εcu, respectively. pct and pcy are the restoring forces of the concrete springs corresponding to the concrete tensile and compressive strengths, respectively. For the core section concrete, considering the confinement condition of the stirrups, the primary curve [57] is adopted, and springs

Fig. 11 shows the free body diagram for the bottom portion of the column model from Fig. 9. According to the plane section assumption,

Fig. 12. Primary curve adjustment of concrete springs considering stirrup confinement.

Fig. 11. Idealized column model. 9

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and deformation of the concrete springs can be calculated as:

constants are defined as follows.

dcyCore

= (1 +

5K ) dcyshell,

Core dcu

=

6.67dcCore 85

−

5.67dcyCore

pcyCore = pcyshell + k1 fle Ac , pcyshell = K coshell ∙dct , pctcore = K cocore ∙dct

K=

(10)

D pcmax , i = (1 − ξ Pc Dc, i ) Pcmax , i

(17)

(11)

pcyD = K cy (dcy − dcmax , i ) + pcmax , i

(18)

D dcu = dcy + |pcyD / K c|

(19)

k1 fle f c'

(12)

Core dcCore + dcshell 85 = 260ρdcy 85 , ρ =

nsl Ast1 s (bcor + hcor )

b b 1 k1 = 6.7(fle )−0.17 , k2 = 0.26 ⎛ cor ⎞ ⎛ cor ⎞ ⎜⎛ ⎟⎞ ≤ 1.0 ⎝ s ⎠ ⎝ s l ⎠ ⎝ fl ⎠ ⎜

fle = k2 ∙fl , fl =

where dcmax,i is the peak displacement at cycle i, and Pcmax,i is the restoring force on the primary curve corresponding to peak displacement D at cycle i. pcmax , i is the deteriorated restoring force corresponding to peak displacement after cycle i; pcyD is the deteriorated maximum strength after cycle i; dcy is the displacement corresponding to the maximum strength; Kcy is the initial stiffness of the concrete springs, Kcy = 0.5Pcy/(0.7dcy) for shell concrete springs, and Kcy = 0.5Pcy/dcy for core concrete springs, respectively. Kc is the degrading stiffness at the post-peak stage, Kc = − Pcy/(dcu-dcy); ξPc is the parameter defining the rate of strength deterioration, which is 0.15 in this study [51]. When reload occurs at the post-peak stage (Fig. 13b), both the strength and stiffness deterioration should be considered. Hence, their updated value can be estimated as:

(13)

⎟

(14)

∑ Ast1 f yt sinα sbcor

(15)

where K is a factor representing the effect of confinement and could be estimated using Eq. (12); k1 is a coefficient related to Poisson's ratio; k2 is a coefficient reflecting the reduction of average pressure for a rectangular section column; fl is the uniform confining pressure. It should be noted that the uniform lateral pressure fl is computed when the hoop steels yield, this assumption may result in underestimation of the actual effect of lateral pressure. The confined model adopted here may not be appropriate for columns for whom the test specimens are not representative. It is recommended that the hoop strain equation should be included in the future modification to the model, such equation has been proposed for the calculation of the concrete-filled-steel-tube columns [58–64]; fle is the equivalent uniform confining pressure; dcCore 85 is the deformation at the 85% strength level beyond the peak stress of the confined (core) concrete springs. Ast1 is sectional area of a single stirrup; nsl is the number of the stirrup legs; bcor and hcor are the width and height of the core section, respectively; sl is the spacing of laterally supported longitudinal reinforcement; α is the angle between the transverse reinforcement and bcor, α = 90° if the transverse reinforcement is perpendicular to bcor. An example of the hysteretic loop segments of the concrete model is shown in Fig. 13. The cyclic deterioration in cycle i is defined by the parameter Dc, which is given by the following expression:

Dc, i =

(dmax )α + (∑ di ) β (du )α + (∑ di ) β

D Pcmax , i = (1 − ξPc ΔDc, i ) Pcmax , i

(20)

K ced, i = (1 − ξ K c Dc, i ) K co

(21)

ΔDc, i = Dc, i − Dc, i − 1

(22)

D D dcu , i = dcmax , i + |pcmax , i / K c|

(23)

where Kco is the stiffness at the pre-cracking stage. Kco = 0.5Pcy/ (0.3dcy) for shell concrete, and Kco = 2Pcy/dcy for core concrete, respectively. Dc,i-1 and Dc,i are the calculated deteriorated parameters before and after cycle i; ξKc is the parameter defining the rate of concrete strength deterioration, which is 0.27 in this study [51]; Kced,i is D the deteriorated unload stiffness of cycle i, and dcu , i is the updated ultimate deformation after cycle i. 4.1.3. Constitutive model of the reinforcement springs The restoring force and the deformation of the reinforcement springs are defined as follows:

⎧ ps = As σs ⎨ ⎩ ds = lp εs

(16)

where dmax is the maximum displacement before cycle i; du is the ultimate displacement; ∑ di is the displacement in all previous cycles through loading in both positive and negative directions; the parameters, α and β, are the exponent describing the rate of deterioration and were calibrated from experimental results, Gu et al. [51] suggested that α = 3 and β = 0.5 are appropriate for concrete, and α = 0.5 and β = 1.0 are appropriate for the reinforcement used in this study. When reload occurs at the pre-peak stage (Fig. 13a), only the reloading strength deterioration is considered; thus, the updated force

(24)

where ɛs, σs, As, ps and ds are the mean strain, mean stress, representative area, restoring force and deformation of a reinforcement spring, respectively. As illustrated in Fig. 14, their force-deformation relationship is based on the Takeda model [65] and previous findings [66,67]. A steel spring fails when the compressive or tensile deformation is greater than the ultimate value dsu.

dsy = εsy lp, Psy = f y As

Fig. 13. Hysteretic curve of concrete springs. 10

(25)

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V

Enter shear failure stage B

Vu Vcr A

Kdeg

Kcr

Ki

C

O ǻs,cr

ǻs,u

ǻs,f,

ǻs

Fig. 14. Force-deformation relationship of reinforcement springs. Fig. 15. Primary curve for shear force-shear displacement relationship.

dsu = εsu lp, Psu = fu As

(26)

where G = shear modulus = Ec/2(1 + μ), and μ = Poisson’s ratio, assumed as 0.30 for normal strength concrete. f = non-uniform distributed factor of shear stress, approximated as 1.2 for the rectangular section [69]. The second stage indicates the portion from shear cracking till shear failure. The truss model is adopted here to describe the shear behavior within the range before the onset of significant lateral strength deterioration. As shown in Fig. 16 (a), the truss model consists of diagonal concrete struts DC, steel ties in longitudinal directions (LT in tension and LC in compression, respectively) and transversal directions (T). The compatibility conditions of the truss models are depicted in Fig. 16. Compression deformations in the diagonal concrete strut, DC, is caused by column longitudinal elongation, tensile deformation of the transverse reinforcement, and shear deformation of the analytical element; therefore, the shear deformation is calculated using Eq. (35). The first term represents axial strain, εv, which is caused by flexural moments. Because no obvious axial length change was captured during the prepeak stage during testing, the influence of this term is ignored. The second and third term represent the contribution of the horizontal strain of transverse reinforcement (εh) and the compressive strain in the diagonal concrete strut to the shear strain (εd). Both are caused by the external lateral load Vs. Equilibrium of the internal and external forces of the truss model is depicted in Fig. 17, assuming linear elastic behavior of the strut and tie. εd and εh can be approximated as shown in Eqs. (36) and (37). The stiffness predicting the post-crack behavior before shear failure, Kcr, is estimated using Eq. (38).

where εsy and εsu are strains corresponding to the yield and tensile strength of corresponding longitudinal steel. Then, dsy and dsu are deformations corresponding to steel strains εsy and εsu, respectively. Psy and Psu are the restoring forces of the steel springs corresponding to the reinforcement tensile and compressive strengths, respectively. Damage-accumulation effect is considered here, and spring constants are defined as follows: D Psmax , i = (1 − ξPs ΔDs ) Psmax , i

(27)

K ced, i = (1 − ξKs Ds, i ) Kso

(28)

Psmax , i =

D Psmax ,i−1

+ Ksy (dsmax , i − dsmax , i − 1)

ΔDs, i = Ds, i − Ds, i − 1

(29) (30)

where dsmax,i is the peak displacement of reinforcement springs at cycle i, Psmax,i is the restoring force on the primary curve corresponding to peak displacement at cycle i; Ds,i-1 and Ds,i are the calculated deterioD rated parameters before and after cycle i; Psmax , i is the deteriorated restoring force corresponding to peak displacement after cycle i; ξPs and ξKs are parameters defining the deterioration rate of reinforcement strength and stiffness. Thus, the deterioration rate of reinforcement strength and stiffness values are taken as 0.15 and 0.27, respectively [51]. 4.2. Shear sub-element

γs =

The proposed shear model includes the primary shear force-shear displacement relationship under monotonic loading and unloading/reloading branches of hysteresis loops under cyclic loading. The primary curve is used as the backbone curve to describe the cyclic lateral loadshear deformation responses. 4.2.1. Primary curve As shown in Fig. 15, the suggested primary curve is trilinear, and the representative range identified in the proposed model includes the initial (O-A), pre-peak (A-B) and post-peak (B-C). The first stage represents elastic behavior up to the shear cracking point, which indicates the initiation of diagonal cracks. The initial shear stiffness is calculated using the principle of elasticity from Eq. (31), and the shear cracking strength is predicted by using Eq. (32) and Eq. (33) [68].

Ki =

Vcr = 0.8νb Ag + 0.167

(33)

Δs, cr = Vcr / Ki

(34)

VDC Vs /sinθ = ≤ Ac Ec bh 0cosθ∙Ec Ec

(36)

εh =

f yt Vh Vs = ≤ Ast Es ρt bh 0cotθEs Es

(37)

ρ bh 0 Es cot2 θ 1 Vs ≈ t ∙ γs L 1 + αE ρt csc 4θ L

(38)

ρ bd

⎛ αE ρt + ξ1 ρt Ag ⎞ l θ = tan−1 ⎜ ⎟ ⎜ 1 + αE ρt ⎟ ⎝ ⎠

(32)

νb = (0.067 + 10ρl ) f c' ≤ 0.2 f c'

εd =

where Es is the elastic modulus of the reinforcing bar; αE = Es/Ec; θ is the inclined crack angle with respect to the longitudinal axis of the column. The crack angle proposed by Kim and Mander [70,71] is based on minimizing the external work caused by the unit shear force and was applied here; thus, θ could be estimated as follows:

(31)

hN a

(35)

f c'

K cr =

GAg fL

εv εd εd + εhtanθ − ≈ εhtanθ − tanθ sinθcosθ sinθcosθ

0.25

(39)

where ξ1 is the boundary condition factor of the column: ξ1 = 0.57 for the fixed-fixed ends and ξ1 = 1.57 for the fixed-pinned ends. The modified truss model is not appropriate for describing the shear 11

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Fig. 16. Compatibility of shear deformations in the truss model.

shear, the shear failure point could be better estimated by a limiting lateral displacement. In this study, a modified shear strength model considering different shear failure mechanisms [31], accompanied by Elwood and Moehle’s ultimate displacement model [23] is applied to determine the shear failure. If any of the above failure principles were satisfied, the degrading shear behavior was triggered. A brief description of the shear strength model and the ultimate deformation model are given herein, for further details, refer to Hua et al. [31] and Elwood and Moehle [23]. Ultimate shear strength model:

response when lateral strength degrades rapidly due to the deficiency of the hysteretic stress-strain relationship for the diagonal strut at the post-peak range. Experimental studies have shown that axial failure tends to occur when column shear strength deteriorates to about zero [72]. Once the shear strength degradation is triggered by the limiting force or displacement, as shown in Fig. 18, a linear degrading line is defined to capture the strength deterioration behavior of shear-critical columns. The stiffness of the degrading line (Kdeg) in the shear subelement is determined based on the slope of the line connecting the shear failure point and loss of the axial-load-carrying capacity point t (Eq. (40)). The degrading slope for the total response, K deg , can be calculated in Eq. (41). The shear failure point (Δu, Vu) is determined based on the proposed shear failure principle, as introduced in the next section. The total deformation at axial failure (Δalf) can be estimated in Eq. (42) by the Elwood and Moehle [73], where θalf is the critical angle and is assumed to be 65°. Δu, Δs,u and Δf,u are the total deformation, shear deformation and flexure deformation when shear failure occurs. −1

1 1 ⎞ K deg = ⎜⎛ t − ⎟ K K unload ⎝ deg ⎠ t K deg

(40)

Vu = Δalf − Δu

Δalf =

4 100

(41)

1 + (tanθalf

)2

L

⎧ifV ≥ 0.1V ⎧ Vtruss − tension truss − tension min ⎪ s ⎨ ⎩Vtruss − compression Vu = max ⎨ Vdiagonal − tension ⎪ Vstrut − compression ⎩

(43)

⎛ 0.5 f c' Vdiag − tension = ⎜ ⎜ a/h 0 ⎝

1+

(44)

⎛ 0.5 f c' Vtruss − tension = ⎜ ⎜ a/h 0 ⎝

1+

⎞ ⎟ Ag 0.5 f c' Ag ⎟ ⎠ N

N 0.5

f c' Ag

⎞ ⎟ Ag + 0.9Ast f yt h 0 / s ⎟ ⎠

(45)

Vtruss − compression = 0.3ζf c' Ag

(46)

Vstrut − compression = 1.2Kζf c' (kd × b)cosθstr

(47)

s

tanθalf + N ⎛ A f h tanθ ⎞ ⎝ st yt cor alf ⎠

(42)

where Vdiag-tension and Vstrut-compression are shear resisted by the diagonal tension mechanism and strut compression mechanism; Vtruss-tension and Vtruss-compression are the shear resisted by the truss mechanism controlled by either steel tension of by concrete compression; ζ is the softening

4.2.2. Shear failure criterion Shear failure is usually initiated through a limiting shear force. However, for columns that first yield in flexure and ultimately fail in

Fig. 17. Equilibrium of shear forces in the truss model. 12

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Fig. 18. Definition of shear strength degradation stiffness Kdeg.

Fig. 19. Hysteretic model for the shear force-shear displacement relationship.

load (e.g., A → B), the unloading stiffness is given in Eq. (50), and below cracking load (e.g., C → D, the unloading stiffness is equal to k1.

coefficient of the cracked concrete in compression, ζ = 3.35/ f c' ≤ 0.52(f c' , MPa) ; K is the strut-and-tie index. Hwang and Lee [74] describe the procedure for calculating this parameter; kd is the depth of the flexural compression zone of the elastic column, the value is approximated following the suggestion of Paulay and Priestley [75]; θstr is the inclination angle of the diagonal compressive strut with respect to horizontal axis. Ultimate displacement model:

Δu A V = 0.033 + 5 st − 0.047 ≥ 0.01 L bs f c' bh 0

k = k1 −

A ⎛when st > 0. 004⎞ bs ⎠ ⎝ (48)

A ⎛when st ≤ 0. 004⎞ bs ⎝ ⎠

(50)

(3) The unloading stiffness of post-peak unloading branches above the cracking load (e.g., G → H, M → N, R → S) and below the cracking load (e.g., H → I, N → O, S → T) are given by Eq. (51) and Eq. (52).

A V N Δu = 0.03 + 4 st − 0.024 − 0.025 ' ' L bs A g fc f c bh 0 ≥ 0.01

|Δs| − Δs, cr (k1 − k2) Δs, u − Δs, cr

k = k2 (1 − 0.05 |Δs|/Δs, u) ≤ k3

(51)

k = 0.6k2 (1 − 0.07 |Δs|/Δs, u) ≤ k 4

(52)

(4) In the pre-peak range, if Vcr in the direction of the reloading has not been previously exceeded, reloading aims at its cracking point (e.g., B → C, I → J). If the preceding unloading branch is completed before it reaches the zero shear force level, reloading goes back to the onset point of unloading branch (e.g., T → R). Otherwise, reloading goes up straight to the ‘pinching point (Δs,p, Vs,p)’(e.g., O → P, V → W), and then goes up toward the ‘maximum point (Δs,m, Vs,m)’ beyond the pinching load until reaching the primary curve (e.g., P → Q, W → X). Hysteretic parameters are defined in Eq. (53)–Eq. (58), where Vm is the shear force on the primary curve corresponding to the historic maximum displacement (Δs,m) on the primary curve. Moreover, i is number of cycles in one direction within its maximum displacement range. Δs,m ± Δs,cr. Thus, Δs,r is the shear displacement when reloading in one direction begins, E is the shear unloading stiffness and can be calculated using Eq. (51) or Eq. (52). Px and Py are pinching coefficients for displacement and force during reloading, assuming Px = 0.5 and Py = 0.2 in this study.

(49)

Where Δu/L is the drift ratio at shear failure, and V is the maximum applied shear force. 4.2.3. Shear hysteretic model Ozcebe and Saatcioglu’s hysteretic shear model [22] was revised and recalibrated in this study to expand its application to the post-peak range where shear strength decay is significant. The modified shear hysteretic model is illustrated in Fig. 19(a) and summarized by the following rules. Four reference stiffness parameters, k1, k2, k3, and k4, as shown in Fig. 19(b), are defined to prevent the extraordinary flat unloading slopes [76]. (1) Initial loading and uncracked unloading/reloading follow the primary curve; (2) For pre-peak cracked unloading from points above the cracking 13

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Fig. 20. Development of flexure and shear response for (a) the shear-critical column, (b) the flexure-shear-critical column, and (c) the flexure-critical column.

Vm' = Vm e[−0.014i

|Δs, m|/Δs, u − 0.01 i (|Δs, m|/Δs, u)]

Δ's, m = Δs, m

(54)

V p' = py Vm'

(55)

Δ's, p = Δ1 + px (Δ2 − Δ1)

(56)

Δ1 = Δs, r + py (Δ's, m − Δs, r )

(57)

Δ2 = Δ's, m − (1 − py )

Vm' E

the sum of flexural and shear deformation calculated by the flexure and shear sub-element, respectively. Assume the column behavior is initially controlled by flexure. Once shear failure is detected, the degradation behavior is dominated by the shear response and the flexural response keeps the same force with shear response through unloading (Fig. 20a and b). On the other hand, if flexural failure is detected, the shear response will unload to keep the same force with flexural response (Fig. 20c). The analysis program for RC column, as shown in Fig. 22 was developed using the Visual C++ 6.0 program language, the Microsoft Foundation Class library (MFC) and the Matlab. The framework of the analysis program is illustrated in Fig. 23. The program has a pre-processor module, which includes submodules with tasks such as geometric modelling, definition of material properties for the component springs, loads and boundary conditions. In the numerical analysis module, a pre-analysis is first carried out to obtain the model parameters, including the primary curve, the initial and post-cracking shear stiffness, afterwards, the full load-deformation responses of the column can be simulated quantitatively. After reading

(53)

(58)

4.3. Combination of flexural sub-element and shear sub-element The multi springs model and the uniaxial shear spring are connected based on the following principles, and the analytical procedure is schematically summarized in Fig. 21. For each step, both elements have the same force equal to the external lateral load. Lateral deformation is 14

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Fig. 21. Solution procedure for the analytical model.

Pre-processor module

Numerical analysis module

Fig. 22. Interface of the analysis program.

15

Post-processor module

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Analysis program Pre-processor module (geometric modelling, defining materials, loads and boundary conditions)

Numerical analysis (pre-analysis for model parameters, cyclic loading analysis)

Post-processor module (generate data tables, lateral loaddisplacement curves, and characteristic values including the strength, ductility, and so on)

Fig. 23. Framework of the analysis program.

C-S1

C-S3

C-S2

C-F

C-FS

Fig. 24. Comparison of experimental and analytical results in this study.

displacement, Δu. Their values are presented in Table 4. Note that all specimens represent an equivalent column in a single curvature. For specimens tested in a double curvature, the displacements in Table 4 are half of the experimental record. It could be illustrated that the calculation of peak strength, ultimate displacement, and the strength degradation are in good agreement with the experimental results, the pinching effect and stiffness deterioration in the hysteretic relationship are reflected well, and the failure mode is best estimated by a limiting shear force or ultimate deformation. The average value of the measured to estimated peak strength ratio is 1.0 and the coefficient of variation is 0.10. The average value of the measured to estimated peak and ultimate displacement ratio are respectively 1.4 and 1.2, and the coefficient of variation are 0.39 and 0.49. It should be noted that for columns failed in flexure, the estimated peak and ultimate deformation varied a lot with the experimental results even if they showed similar force-displacement responses; therefore, only ultimate strengths were examined for flexure-critical columns.

the data of the numerical results, a post-processor module operated by Matlab can generate full and/or envelop curve of shear force-displacement response, and characteristic values such as the failure mode, shear force and displacement corresponding to flexural yielding, peak strength and the loss of lateral strength, respectively. 5. Comparison of analytical model with experimental results The proposed analytical model was applied to 41 columns, including columns tested by the authors, which have already been discussed in this paper, and columns tested by Gill et al. [37], Tanka and Park [38], Matamoros et al. [39], Ramirez and Jirsa [40], Umehara and Jirsa [41], Yoshimura et al. [42], Aboutaha [43], Yoshimura and Nakamura [44], Yoshimura et al. [45], Ousalem et al. [46], Moretti and Tassios [47], Tran [48]. Experimental data were obtained from the PEER column database compiled by Berry et al. [34] and other sources cited in this paper. Column specimens were selected based on the following criteria: height-to-depth ratio 2 ≤ L/h ≤ 6; concrete strength, 25 < f c' < 42 MPa; steel yield strength ranging from 344 to 573 MPa; a longitudinal reinforcement ratio of 0.013 < ρl < 0.052; a transverse reinforcement index of 0.07 < ρt f yt / f c' < 0.97; an axial load ratio of

6. Conclusions Five intermediate short columns with the height to depth ratio of 3.1 and 3.7 failed in different modes were tested to analyze their seismic behavior. Based on the experimental results, an analytical model including two individual sub-elements connected in series representing member flexural and shear response was developed to simulate the nonlinear cyclic responses for shear critical, flexure-shear critical, and flexure critical intermediate short columns with height-todepth ratio of 2–4. According to the test observation, specimens with intermediate height-to-depth ratio and higher axial load ratio tended to have a higher shear strength and a more severe strength degradation. Failure

(N / f c' Ag ) ≤ 0.42, and an apparent failure phenomena dominated by shear or flexure. Table 1 summarizes the geometry, reinforcement, material properties, axial load ratio and failure mode of the examined columns. A comparison of the computed cyclic shear force during total displacement and experimental loops of column specimens are shown in Figs. 24 and 25, where test results are consistently plotted in solid black lines and simulated results are plotted in dotted red lines for all comparisons. Selected response quantities include maximum shear force, Vmax and the corresponding displacement, Δmax, as well as the ultimate 16

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Gill No.1

Tanaka No.5

Gill No.2

Tanaka No.7

Gill No.3

C5-00N

C5-00S

00-U

120-U

OUS

CUS

2CUS

OUW

CUW

CE

BE

LE

SC9

Fig. 25. Comparison of experimental and analytical results from other sources.

accompanied by increasing flexure deformations. Moreover, the ratio of shear-to-flexure deformation remained approximately constant, which was also applicable for the cycles after flexural yielding and before the onset of diagonal concrete crushing due to shear failure. Shear

modes of column specimens might change from brittle shear failure to ductile flexure failure with an increasing hoop/longitudinal reinforcement ratio. For columns forming a flexural hinge and a stable sheartransfer mechanism, shear deformation increased gradually 17

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N27C

2C

3C

No.6

No.7

No.1

No.11

No.12

Specimen 1

SC-1.7-0.05

RC-1.7-0.05

No.14

Specimen 3

Specimen 4

SC-1.7-0.20

SC-1.7-0.35

RC-1.7-0.35

RC-1.7-0.20

Fig. 25. (continued)

mechanism was significantly degraded, shear deformations increased in a larger proportion than flexural deformations. Measured results showed that the shear deformations concentrated in regions where prominent inclined cracks developed.

deformation concentrated in the limited zones where shear-flexural interaction was more pronounced (such as the plastic hinge regions in double-curvature columns), although shear effects actually spread throughout the elements. For columns whose shear-resisting 18

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the initial shear behavior, pre-peak shear behavior, point of shear failure and post-peak shear behavior. The pre-peak response was described by the truss model, and the post-peak response was described based on shear-friction concepts. Shear failure was determined based on the applied shear exceeding the shear strength estimated from a versatile, multi-mechanism methodology and the total lateral deformation exceeding the ultimate deformation predicted by the drift capacity model. Ozcebe and Saatcioglu’s shear hysteretic model was simplified and expanded to apply in the deformation range where shear strength decay was significant. Comparative results indicated that the numerical model adequately captured the failure modes, peak strengths, ultimate displacements, and strength degradation responses of 41 RC intermediate short columns failed in shear, flexure-shear or flexure. The simplicity, accuracy and computational efficiency of the developed model make it a valuable tool for the assessment of the seismic behavior for existing RC frame structures.

Table 4 Comparison of test and analytical results (cantilever). Specimen

Test data

Vmax , kN

Analytical results

Δmax , mm

Δu , mm

Vmax Vprop.

Δmax Δprop.

Δu Δprop.

4.2 4.8 4.3 9.9 15.6

4.4 5.5 5.5 15.2 31.2

0.9 1.1 0.9 1.2 1.1

1.3 1.9 1.1 0.6 –

0.8 1.1 1.0 0.7 –

33.8 25.9 14.1

– – –

1.1 1.0 1.0

– – –

– – –

74.1 62.3

– –

1.1 1.1

– –

– –

Matamoros et al. [39] C5-00N 59 C5-00S 58

18.7 18.1

35.1 35.1

0.9 0.9

– –

– –

Ramirez and Jirsa [40] 00-U 259 120C-U 312

8.4 6.7

– 8.0

1.2 1.0

1.3 1.9

– 1.3

Umehara and Jirsa [41] OUS 266 CUS 330 2CUS 422 OUW 251 CUW 272

7.7 5.0 4.2 9.9 6.3

7.9 7.3 5.7 14.9 10.1

1.0 1.0 1.1 1.2 1.0

3.3 2.2 2.1 2.4 1.4

0.9 1.7 1.5 1.5 1.5

Yoshimura et al. [42] CE 97 BE 96 LE 165

0.7 0.5 1.1

– 0.9 2.0

1.3 1.1 0.9

1.3 0.9 1.0

– – 1.0

Aboutaha et al. [43] SC9 605

8.7

13.5

1.1

1.5

0.6

Yoshimura and Nakamura [44] N27C 266 2.2 2C 224 0.8 3C 267 1.4

4.2 3.2 5.3

1.0 0.9 1.0

0.8 0.7 1.2

1.1 0.9 2.7

Yoshimura et al. [45] No.6 226 No.7 213

12.2 10.0

– –

1.1 1.0

1.3 1.6

– –

Ousalem et al. [46] No.1 326 No.11 248 No.12 253 No.14 301

2.1 2.3 2.1 4.5

4.8 3.9 4.4 10.0

1.0 1.0 1.1 1.0

1.0 0.9 1.0 1.0

1.3 1.1 1.2 1.0

Moretti and Tassios [47] Specimen 1 339 Specimen 2 361 Specimen 3 363

2.4 2.0 2.2

5.4 3.7 4.2

1.0 1.0 1.0

1.3 1.4 1.5

1.5 1.1 1.1

Tran [48] SC-1.7-0.05 SC-1.7-0.20 SC-1.7-0.35 RC-1.7-0.05 RC-1.7-0.20 RC-1.7-0.35

7.4 7.7 7.6 16.5 13.3 10.6

9.3 8.4 8.1 23.5 19.9 14.4

1.1 0.9 0.9 1.0 0.9 0.9

1.9 1.6 1.1 1.7 1.3 1.2

0.8 1.2 0.9 0.8 1.1 1.0

1.0 0.10

1.4 0.39

1.2 0.49

This study C-S1 260 C-S2 372 C-S3 336 C-FS 372 C-F 337 Gill et al. [37] No. 1 698 No. 2 832 No. 3 671 Tanaka and Park [38] No. 5 427 No. 7 666

272 296 340 288 316 350

Average Coefficient of variation

CRediT authorship contribution statement Xiang-Lin Gu: Conceptualization, Methodology, Writing - review & editing, Supervision, Project administration, Funding acquisition. JingJing Hua: Methodology, Data curation, Software, Validation, Writing original draft. Mao Cai: Methodology, Investigation, Resources. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported in part by the National Natural Science Foundation of China [Grant No. 51938013, 50978191] and the National Key R&D Program of China [Grant No. 2017YFC1500700]. References [1] Kowalsky MJ, Priestley MJN. Improved analytical model for shear strength of circular reinforced concrete columns in seismic regions. ACI Struct J 2000;97:388–96. [2] Fardis, MN. Seismic design, assessment and retrofitting of concrete buildings: based on EN-Eurocode 8 (Vol. 8). Springer; 2009. [3] Fardis MN. Seismic design, assessment and retrofitting of concrete buildings: based on EN-Eurocode 8. Springer Science & Business Media; 2009. [4] Foraboschi P. Versatility of steel in correcting construction deficiencies and in seismic retrofitting of RC buildings. J Build Eng 2016;8:107–22. [5] Lynn AC. Seismic evaluation of existing reinforced concrete building columns. Ph.D. thesis. Department of Civil and Environmental Engineering, University of California, Berkeley, California, 2001. [6] Li YA, Hwang SJ. Prediction of lateral load displacement curves for reinforced concrete short columns failed in shear. J Struct Eng 2016;143:04016164. [7] Petrangeli M, Pinto PE, Ciampi V. Fiber element for cyclic bending and shear of RC structures. I: Theory. J Eng Mech 1999;125:994–1001. [8] Petrangeli M. Fiber element for cyclic bending and shear of RC structures. II: Verification. J Eng Mech 1999;125:1002–9. [9] Bentz EC. Sectional analysis of reinforced concrete members PhD thesis Toronto, Canada: University of Toronto; 2000. [10] Vecchio FJ, Collins MP. Predicting the response of reinforced concrete beams subjected to shear using modified compression field theory. ACI Struct J 1988;85:258–68. [11] Ceresa P, Petrini L, Pinho R. Flexure-shear fiber beam-column elements for modeling frame structures under seismic loading – state of the art. J Earthquake Eng 2007;111:46–88. [12] Kotronis P, Mazars J. Simplified modelling strategies to simulate the dynamic behavior of R/C walls. J Earthqu Eng 2005;9:285–306. [13] Ricles JM, Yang Y, Priestley MJN. Modeling nonductile R/C columns for seismic analysis of bridges. J Struct Eng 1998;124:415–24. [14] Elwood KJ. Modelling failures in existing reinforced concrete columns. Can J Civ Eng 2004;31:846–59. [15] Mergos PE, Kappos AJ. A gradual spread inelasticity model for R/C beam–columns, accounting for flexure, shear and anchorage slip. Eng Struct 2012;44:94–106. [16] Mansour M, Hsu TT. Behavior of reinforced concrete elements under cyclic shear. II: Theoretical model. J Struct Eng 2005;131:54–65.

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