Shear strength prediction for steel reinforced concrete deep beams

Journal of Constructional Steel Research 62 (2006) 933–942 www.elsevier.com/locate/jcsr

Shear strength prediction for steel reinforced concrete deep beams Wen-Yao Lu Department of Civil Engineering, China University of Technology, No. 56, Section 3, Xinglong Rd, 116 Taipei, Taiwan, ROC Received 1 October 2005; accepted 8 February 2006

Abstract This study proposes an analytical method for determining the shear strengths of steel reinforced concrete deep beams under the failure mode of concrete crushing originally based on the softened strut-and-tie model. The proposed method is a good physical model that can correlate well with the observed failure phenomenon of steel reinforced concrete deep beams. By comparing the predictions of the proposed method with the available test results from the literature, it was found that the proposed method is capable of predicting the shear strengths for steel reinforced concrete deep beams with sufficient accuracy. The shear-carrying behavior of steel reinforced concrete deep beams is highly influenced by the ratios of flange width to gross width, the shear span-to-depth ratios, and the concrete strengths. When the ratio of flange width to gross width is low, the shear-carrying capacities of steel reinforced concrete deep beams increase with the increasing ratio. However, if the ratio of the flange width to gross width is higher than a critical value, then the failure mode of steel reinforced concrete deep beams will be converted from diagonal compression failure into bearing failure. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Steel reinforced concrete; Deep beam; Shear strength; Diagonal compression failure; Bearing failure

1. Introduction Concrete deep beams are those having a shear span-to-depth ratio (a/d) less than 2.5 [1], and they generally occur as transfer girders in tall building construction. Since the strength of steel reinforced concrete (SRC) deep beams is usually controlled by shear, an understanding of the shear behavior of SRC deep beams is very important. The shear action in the web of SRC deep beams leads to diagonal compression and tension in a direction perpendicular to it. The possible failure modes of SRC deep beams could be diagonal compression failure and bearing failure, as shown in Fig. 1. The AIJ Standards [2] use the concept of strength superposition to calculate the ultimate shear strengths of SRC members, i.e. the shear strengths of SRC members are the sum of the shear strengths of the reinforced concrete portion and the steel portion. The shear strengths of reinforced concrete deep beams have been accurately predicted by Hwang et al. [3]. Very few, if any, theoretical model have been developed that include

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2006.02.007

composite action for predicting the shear strengths of SRC deep beams. Based on the softened strut-and-tie (SST) model [3,4], this paper proposes an analytical method for determining the shear strengths of SRC deep beams under the failure mode of concrete crushing. Beginning with the pioneering work of Ritter [5] and M¨orsch [6] about a century ago, numerous researchers have examined the application of strutand-tie model concepts to structural design problems [7,8]. So far, the strut-and-tie model has been incorporated into many design codes [9–12]. In the conventional strut-and-tie model [7–12], the stresses are usually determined by the equilibrium condition alone, while the strain compatibility conditions are neglected. However, the SST model, which satisfies equilibrium, compatibility and constitutive laws of cracked reinforced concrete, has been proposed for determining the shear strengths of reinforced concrete deep beams [3]. Based on the available experimental data, the applicability of the proposed SST model to SRC deep beams for predicting the shear strength is examined. Also, a parametric study was performed to demonstrate the variation in the shear-carrying capacities of SRC deep beams caused by various parameters.

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W.-Y. Lu / Journal of Constructional Steel Research 62 (2006) 933–942

Notation a a/d AB Ads Adu Af A0f as As A0s Asc A0sc Atv Av b b f /b C Cd Cds Cdu d d0

dc

D dt f c0 Fv Fv fy f yc fy f f yv f yw h hs k jdc Kv Kv `n `n / h `b

shear span calculated per the SST model [3,4] shear span-to-depth ratio bearing area of the nodal zone concrete area of the softened concrete contributing to the diagonal strut area of the un-softened concrete contributing to the diagonal strut area of the tension flange of the steel beams area of the compression flange of the steel beams depth of the diagonal strut area of flexural tension bars area of flexural compression bars area of the tension cover plate area of the compression cover plate area of the vertical tie area of the vertical hoops gross width of SRC deep beams ratio of flange width to gross width resultant compression force diagonal compression strength of SRC deep beam diagonal compression strength provided by the softened concrete diagonal compression strength provided by the un-softened concrete distance from the extreme compression fiber to the centroid of the flexural tension bars distance from the extreme compression fiber to the centroid of resultant compression force of SRC deep beams distance from the extreme compression fiber to the centroid of resultant tension force of SRC deep beams compression force in the diagonal strut (negative for compression) distance between the centroid of resultant tension force and the bottom of SRC deep beams compressive strength of concrete tension force in the vertical tie (positive for tension) balanced amount of vertical tie force yield strength of flexural bars yield strength of cover plates yield strength of the steel beams flange yield strength of vertical hoops yield strength of the steel beams web overall depth of SRC deep beam overall depth of the steel beam coefficient distance of the lever arm vertical tie index vertical tie index with sufficient vertical hoops clear span clear span-to-depth ratio width of bearing plate

n s T tc tf tw Vbh Vbv Vbv,calc Vbv,test VB VDC θ γv ρc ρt ρv ζ

modular ratio of elasticity spacing of the vertical hoops within the shear span resultant tension force thickness of the cover plates flange thickness of the steel beams web thickness of the steel beams horizontal shear force vertical shear force predicted shear strengths of SRC deep beams measured shear strengths shear strength due to the bearing failure shear strength due to the diagonal compression failure angle of inclination of the diagonal strut fraction of vertical shear transferred by the vertical tie in the absence of the horizontal tie ratio of the total compression reinforcement of SRC deep beams ratio of the total tension reinforcement of SRC deep beams ratio of the vertical hoops = Av /b/s softening coefficient of concrete

2. Proposed method Consider a typical SRC deep beam loaded on the top and simply supported at the bottom, as shown in Fig. 1. The failure modes of beams could be the diagonal compression failure and bearing failure (Fig. 1). The macromodel and the associated solution procedure for predicting the shear strengths of SRC deep beams are as follows. 2.1. Macromodel Fig. 2 shows the external shear forces acting on the SRC deep beam and the proposed force transferring mechanisms in view of the SST model [3,4]. By considering the distances between couples [Fig. 2(a)], it will be sufficiently accurate to express the following relationship between vertical and horizontal shears: jdc Vbv ≈ Vbh a

(1)

where Vbh is the horizontal shear force; Vbv is the vertical shear force; a is the shear span calculated per the SST model [3,4]; and jdc is the distance of the lever arm. According to experimental observations [13], the flexural tension forces of SRC deep beams come from the tension bars, the tension flange and the tension cover plate. The distance between the centroid of the resultant tension force and the bottom of the SRC deep beam (dt ) can be estimated as h−h +t

As f y (h − d) + A f f y f ( 2s f ) + Asc f yc ( h−h2s −tc ) dt = As f y + A f f y f + Asc f yc (2)

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Fig. 1. Failure modes of SRC deep beams.

(a) Shear element.

(b) Nodal zone.

(c) Force transferring mechanisms. Fig. 2. Shear resisting mechanisms of SRC deep beams.

where As is the area of the flexural tension bars; f y is the yield strength of the flexural bars; h is the overall depth of the SRC deep beam; d is the distance from the extreme compression fiber to the centroid of the flexural tension bars; A f is the area of the tension flange of the steel beams; f y f is the yield strength of the steel beams flange; h s is the overall depth of the steel beam; t f is the flange thickness of the steel beams; Asc is the area of the tension cover plate; f yc is the yield strength of the cover plates; and tc is the thickness of the cover plates. As shown in Fig. 2, the distance from the extreme compression fiber to the centroid of the resultant tension force

of the SRC deep beams (dc ) is equal to h − dt . The lever arm jdc can be approximated as dc − kdc /3, where the coefficient k can be estimated as follows: q k = [nρt + (n − 1)ρc ]2 + 2[nρt + (n − 1)ρc d 0 /dc ] − [nρt + (n − 1)ρc ]

(3)

where ρt =

As + A f + Asc bdc

(4)

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ρc =

W.-Y. Lu / Journal of Constructional Steel Research 62 (2006) 933–942

A0s + A0f + A0sc bdc

(5)

in which n is the modular ratio of elasticity; ρt is the ratio of the total tension reinforcement of the SRC deep beams; ρc is the ratio of the total compression reinforcement of the SRC deep beams; A0s is the area of the flexural compression bars; A0f is the area of the compression flange of the steel beams; A0sc is the area of the compression cover plate; and d 0 is the distance from the extreme compression fiber to the centroid of the resultant compression force of the SRC deep beams. As shown in Fig. 2(a), the cross-section of SRC deep beams can be divided into a nodal zone and a non-nodal zone. The area above the top flange of the steel beam or below the bottom flange of the steel beam can be classified as the nodal zone, while the middle section can be classified as the nonnodal zone. The concrete in the nodal zone is simultaneously subjected to vertical, horizontal and diagonal compression [Fig. 2(b)]; it can be recognized as un-softened concrete due to the confining effect. The word softened emphasizes the importance of the compression phenomenon, which means that cracked reinforced concrete in compression exhibits lower strength and stiffness than uniaxially compressed concrete [14]. According to experimental observations [15], both a softened area and an un-softened area could exist in the middle section of SRC deep beams [Fig. 2(c)]. The concrete in the outer shell of the middle section cracked [15] before the ultimate state was reached, so it should be recognized as softened concrete. In contrast, the concrete in the core of the middle section did not crack until failure; therefore, it could be recognized as unsoftened concrete. The SST model [3,4] is based on the concept of struts and ties and is derived to satisfy equilibrium, compatibility and the constitutive law of cracked reinforced concrete. In practice, the horizontal hoops are not detailed in SRC deep beams. As shown in Fig. 2(c), the shear forces are then transferred by the reduced mechanisms, namely, diagonal plus vertical mechanisms within SRC deep beams. The diagonal mechanism is a diagonal strut. The depth of the diagonal strut depends on its end condition [3], which can be estimated as s  2 `b 2 as = (kdc ) + (6) 2 where as is the depth of the diagonal strut, and `b is the width of the bearing plate. The bearing area of the nodal zone concrete can be estimated as A B = as sin θ × b

(7)

where θ = tan−1



jdc a

 (8)

in which A B is the bearing area of the nodal zone concrete, b is the gross width of the SRC deep beams; and θ is the angle of inclination of the diagonal strut.

The area of the diagonal strut can be estimated as follows: Ads = as × (b − b f )

(9)

Adu = as × (b f + (n − 1)tw )

(10)

where Ads is the area of the softened concrete contributing to the diagonal strut; b f is the flange width of the steel beams; Adu is the area of the un-softened concrete contributing to the diagonal strut; and tw is the web thickness of the steel beams. The vertical mechanism consists of one vertical tie and two steep struts [3,4]. The vertical tie is made up of vertical hoops. When computing the area of the vertical tie (Atv ), it is roughly assumed that the vertical hoops within the center half are fully effective, and the other vertical hoops are included as 50% effectiveness [3]. If the vertical hoops are uniformly distributed within the shear span, then Atv = 0.75Av , where Av is the area of the vertical hoops within the shear span. 2.2. Evaluation of shear strength The predicted shear strengths of the SRC deep beams can be estimated as Vbv,calc = min(VDC , VB )

(11)

where Vbv,calc is the predicted shear strengths of the SRC deep beams, VDC is the shear strength due to diagonal compression failure, and VB is the shear strength due to bearing failure. The shear strength due to diagonal compression failure can be estimated as VDC = Cd sin θ

(12)

where Cd is the diagonal compression strength of the SRC deep beam. The shear strength due to bearing failure can be estimated as VB = f c0 A B

(13)

where f c0 is the compressive strength of concrete. The diagonal compression strengths of SRC deep beams can be estimated as (14)

Cd = Cds + Cdu

where Cds is the diagonal compression strength provided by the softened concrete, and Cdu is the diagonal compression strength provided by the un-softened concrete. The diagonal compression strength provided by the softened concrete can be estimated as [4] Cds = K v ζ f c0 Ads

(15)

where K v is the vertical tie index, and ζ is the softening coefficient of concrete. According to Hwang and Lee [4], the vertical tie index K v can be estimated as K v = 1 + (K v − 1)

Atv f yv Fv

≤ Kv

(16)

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3. Experimental verification A total of 16 test specimens and their results in the available literature are used to verify the proposed method. These are the test results of Chen et al. [13], Weng et al. [16], and Weng and Wang [15], which are listed in Table 1 in chronological order for easy reference. In selecting these test data, the following screens were applied: (1) All of the test specimens were SRC deep beams. The reinforced concrete deep beams were omitted. (2) The test specimens have a shear span-to-depth ratio (a/d) less than 2.5 and clear span-to-depth ratio (`n / h) less than 4.

Fig. 3. Solution procedure for the diagonal compression strength provided by the softened concrete.

where Kv = γv =

1 1 − 0.2(γv + γv2 ) 2 cot θ − 1 , 3

but 0 ≤ γv ≤ 1

F v = γv K v ζ f c0 Ads sin θ

(17) (18) (19)

where K v is the vertical tie index with sufficient vertical hoops, γv is the fraction of vertical shear transferred by the vertical tie in the absence of the horizontal tie, f yv is the yield strength of vertical hoops, and F v is the balanced amount of the vertical tie force. According to Hwang and Lee [4], the softening coefficient of concrete ζ can be estimated as 3.35 ζ = p ≤ 0.52. f c0

(20)

The solution procedure for the diagonal compression strength provided by the softened concrete is shown in Fig. 3. The diagonal compression strength provided by the unsoftened concrete can be estimated as Cdu = f c0 Adu .

(21)

In Table 1, the seven SRC deep beams tested by Chen et al. [13] were composed of a 350 mm×600 mm rectangular section, three of which had vertical hoops and four of which were without vertical hoops. Both the top flange and the bottom flange were covered with a 130 mm × 20 mm cover plate [13]. The nine SRC deep beams tested by Weng et al. [16] and Weng and Wang [15] were composed of a 200 mm × 350 mm rectangular section, seven of which had vertical hoops and two of which were without vertical hoops. In Table 1, ρv is the ratio of the vertical hoops, ρv = Av /b/s, s is the spacing of the vertical hoops within the shear span, and Vbv,test are the measured shear strengths. The shear strengths of 16 SRC deep beams were predicted by the proposed method, as shown in Table 2. The predicted shear strengths are taken as the smaller of the shear strengths due to diagonal compression failure and the shear strengths due to bearing failure. The failure modes are all predicted as diagonal compression failure (Table 2). However, the deviations between the predicted shear strengths due to diagonal compression failure and the predicted shear strengths due to bearing failure are very small (Table 2). The bearing failure of the SRC deep beams is likely to occur in practice. Table 3 compares the measured shear strength (Vbv,test ) with the predicted shear strength (Vbv,calc ) of the proposed method. Accuracy for the proposed method is gauged in terms of a strength ratio, which is defined as the ratio of the measured to the calculated strength. The mean of the test-to-calculated strength ratio was 1.06, with a coefficient of variation of 0.09 for the proposed method (Table 3). It is concluded that the proposed method can accurately predict the shear strengths of SRC deep beams. Fig. 4 shows the effect of the shear span-to-depth ratios on the shear strength predictions of the proposed method. The predictions of the proposed method are consistent for the shear span-to-depth ratios of 1.11 and 1.92 (Fig. 4). Fig. 5 shows the effect of the vertical hoops on the shear strength predictions. No matter whether SRC deep beams have vertical hoops or no vertical hoops, the proposed method can accurately predict the shear strength. 4. Parametric study The parametric study was performed to demonstrate the variation in shear-carrying behavior of SRC deep beams caused

1.71 1.71 1.71

1.71 1.71 1.71 1.71 1.71 1.71

Weng et al. [16] D1-N 600 D2-FS 600 D3-WS 600

Weng and Wang [15] DB1-15-NS 600 DB2-15-NS 600 DB3-NT-NS 600 DB4-15-FS 600 DB5-15-WS 600 DB6-NT-WS 600

`n / h

3.17 3.17 3.17 3.17 3.17 3.17 3.17

`n (mm)

1900 1900 1900 1900 1900 1900 1900

Chen et al. [13] SRC1-00 SRC1-00-T SRC1-00-E SRC1-00-D SRC1-50 SRC1-25 SRC1-17

Beam

Table 1 Specimen details

337.5 337.5 337.5 337.5 337.5 337.5

337.5 337.5 337.5

975 975 975 975 975 975 975

a (mm)

1.11 1.11 1.11 1.11 1.11 1.11

1.11 1.11 1.11

1.92 1.92 1.92 1.92 1.92 1.92 1.92

a/d

200 × 350 200 × 350 200 × 350 200 × 350 200 × 350 200 × 350

200 × 350 200 × 350 200 × 350

350 × 600 350 × 600 350 × 600 350 × 600 350 × 600 350 × 600 350 × 600

b×h (mm)

198 × 99 × 4.5 × 7 198 × 99 × 4.5 × 7 198 × 99 × 4.5 × 7 198 × 99 × 4.5 × 7 198 × 99 × 4.5 × 7 198 × 99 × 4.5 × 7

198 × 99 × 4.5 × 7 198 × 99 × 4.5 × 7 198 × 99 × 4.5 × 7

300 × 150 × 6.5 × 9 300 × 150 × 6.5 × 9 300 × 150 × 6.5 × 9 300 × 150 × 6.5 × 9 300 × 150 × 6.5 × 9 300 × 150 × 6.5 × 9 300 × 150 × 6.5 × 9

Steel beam (mm)

0 0 0 0 0 0

0 0 0

2600 2600 2600 2600 2600 2600 2600

Asc (A0sc ) (mm2 )

254 254 254 254 254 254

254 254 254

3042 3042 3042 3042 3042 3042 3042

As (mm2 )

254 254 254 254 254 254

254 254 254

1014 1014 1014 1014 1014 1014 1014

A0s (mm2 )

#3@150 mm #3@150 mm – #3@150 mm #3@150 mm –

#3@150 mm #3@150 mm #3@150 mm

– – – – #3@500 mm #3@250 mm #3@170 mm

Vertical hoops

325 325 325 325 325 325

325 325 325

332 332 332 332 332 332 332

fy f (MPa)

325 325 325 325 325 325

325 325 325

326 326 326 326 326 326 326

f yw (MPa)

0 0 0 0 0 0

0 0 0

370 370 370 370 370 370 370

f yc (MPa)

387 387 387 387 387 387

387 387 387

427 427 427 427 427 427 427

fy (MPa)

407 407 0 407 407 0

407 407 407

0 0 0 0 380 380 380

f yv (MPa)

0.48 0.48 0 0.48 0.48 0

0.48 0.48 0.48

0 0 0 0 0.08 0.16 0.24

ρv (%)

23.3 24.5 23.7 23.9 23.9 24.2

24.5 23.9 23.9

28.9 28.9 28.9 28.9 27.7 32.2 28.9

f c0 (MPa)

391 409 396 414 398 430

408 415 395

772 751 799 695 861 877 923

Vbv,test (kN)

938 W.-Y. Lu / Journal of Constructional Steel Research 62 (2006) 933–942

0.0169 0.0169 0.0169 0.0169 0.0169 0.0169

0.0169 0.0169 0.0169 0.0169 0.0169 0.0169

0.385 0.382 0.384 0.384 0.384 0.383

150 150 150 150 150 150

279.4 279.4 279.4 279.4 279.4 279.4

100 100 100 100 100 100 100

Weng and Wang [15] DB1-15-NS 70.6 DB2-15-NS 70.6 DB3-NT-NS 70.6 DB4-15-FS 70.6 DB5-15-WS 70.6 DB6-NT-WS 70.6

0.492 0.492 0.492 0.492 0.494 0.486 0.492

48 260 48 260 48 260 48 260 48 480 47 690 48 260

47 035 47 035 47 035 47 035 47 516 45 833 47 035

32 183 32 183 32 183 32 183 32 183 31 797 32 183

401 401 401 401 401 402 401

131.1 130.5 130.9 130.8 130.8 130.7

13 246 13 183 13 224 13 214 13 214 13 198

17 596 17 384 17 524 17 488 17 488 17 436

15 343 15 270 15 318 15 306 15 306 15 288

244 244 244 244 244 244

130.5 13 183 17 384 15 270 244 130.8 13 214 17 488 15 306 244 130.8 13 214 17 488 15 306 244

241.3 241.3 241.3 241.3 242.4 238.4 241.3

35.8 35.8 35.8 35.8 35.8 35.8

35.8 35.8 35.8

22.4 22.4 22.4 22.4 22.4 22.4 22.4

23.3 24.5 23.7 23.9 23.9 24.2

24.5 23.9 23.9

28.9 28.9 28.9 28.9 27.7 32.2 28.9

1.00 1.00 1.00 1.00 1.12 1.21 1.34

Kv

726 726 726 726 783 966 975

0.52 0.52 0.52 0.52 0.52 0.52

1.23 1.23 1.00 1.23 1.23 1.00

198 207 163 202 202 166

410 426 415 418 418 422

426 418 418

1361 1361 1361 1361 1318 1478 1361

Cds Cdu (kN) (kN)

0.52 1.23 207 0.52 1.23 202 0.52 1.23 202

0.52 0.52 0.52 0.52 0.52 0.52 0.52

`b as Ads Adu AB jdc θ f c0 ζ (mm) (mm) (mm2 ) (mm2 ) (mm2 ) (mm) (degree) (MPa)

279.4 0.0169 0.0169 0.382 150 279.4 0.0169 0.0169 0.384 150 279.4 0.0169 0.0169 0.384 150

0.0295 0.0295 0.0295 0.0295 0.0295 0.0295 0.0295

k

Weng et al. [16] D1-N 70.6 D2-FS 70.6 D3-WS 70.6

0.0416 0.0416 0.0416 0.0416 0.0416 0.0416 0.0416

ρc

480 480 480 480 480 480 480

dt dc ρt (mm) (mm)

Chen et al. [13] SRC1-00 120 SRC1-00-T 120 SRC1-00-E 120 SRC1-00-D 120 SRC1-50 120 SRC1-25 120 SRC1-17 120

Beam

Table 2 Shear strength predicted by the proposed method

795 795 795 795 801 931 890

608 633 578 620 620 588

356 370 338 363 363 344

795 795 795 795 801 931 890

Vbv,cal (kN)

357 374 363 366 366 370

356 370 338 363 363 344

374 370 366 363 366 363

930 930 930 930 896 1024 930

VDC V B (kN) (kN)

633 370 620 363 620 363

2087 2087 2087 2087 2101 2444 2336

Cd (kN)

Diagonal compression Diagonal compression Diagonal compression Diagonal compression Diagonal compression Diagonal compression

Diagonal compression Diagonal compression Diagonal compression

Diagonal compression Diagonal compression Diagonal compression Diagonal compression Diagonal compression Diagonal compression Diagonal compression

The predicted failure mode

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W.-Y. Lu / Journal of Constructional Steel Research 62 (2006) 933–942

Table 3 Experimental verification Author

Beam

a/d

ρv f yv

ρt

ρc

(MPa)

Chen et al. [13]

Weng et al. [16]

Weng and Wang [15]

Total

SRC1-00 SRC1-00-T SRC1-00-E SRC1-00-D SRC1-50 SRC1-25 SRC1-17

D1-N D2-FS D3-WS

DB1-15-NS DB2-15-NS DB3-NT-NS DB4-15-FS DB5-15-WS DB6-NT-WS

1.92 1.92 1.92 1.92 1.92 1.92 1.92

1.11 1.11 1.11

1.11 1.11 1.11 1.11 1.11 1.11

0 0 0 0 0.30 0.61 0.89

1.95 1.95 1.95

1.95 1.95 0 1.95 1.95 0

16

Fig. 4. Effect of shear span-to-depth ratio on shear strength predictions.

by various parameters. The effects of the ratio of flange width to gross width (b f /b), shear span-to-depth ratio (a/d), and the compressive strength of concrete ( f c0 ) on the shear-carrying capacities of SRC deep beams are shown in Figs. 6 and 7. The studied beams have b f /b values varying from 0.286 to 0.857; a/d values of 0.5, 1.25 and 2.0 (Fig. 6); and f c0 values of 20.67, 41.36 and 62.05 MPa (Fig. 7). When b f /b is low, the more softened concrete will exist in the outer shell of the middle section. Therefore, the diagonal

0.0416 0.0416 0.0416 0.0416 0.0416 0.0416 0.0416

0.0169 0.0169 0.0169

0.0169 0.0169 0.0169 0.0169 0.0169 0.0169

0.0295 0.0295 0.0295 0.0295 0.0295 0.0295 0.0295

0.0169 0.0169 0.0169

0.0169 0.0169 0.0169 0.0169 0.0169 0.0169

Vbv,test Vbv,calc

Vbv,test

Vbv,calc

(kN)

(kN)

772 751 799 695 861 877 923

795 795 795 795 801 931 890

0.97 0.94 1.01 0.87 1.07 0.94 1.04

AVG COV

0.98 0.07

370 363 363

1.10 1.14 1.09

AVG COV

1.11 0.03

356 370 338 363 363 344

1.10 1.10 1.17 1.14 1.10 1.25

AVG COV

1.14 0.05

AVG COV

1.06 0.09

408 415 395

391 409 396 414 398 430

Fig. 5. Effect of vertical hoops on shear strength predictions.

compression failure is likely to occur in beams with low b f /b. If the shear-carrying capacity of the beam is dominated by the diagonal compression strength, then the shear-carrying capacities increase with increasing b f /b (Figs. 6 and 7). These tendencies are because, with higher b f /b, there is higher b f , more un-softened concrete in the core of middle section, and higher diagonal compression strength. When the b f /b value is higher than a critical value, then the failure mode of steel reinforced concrete deep beams will be converted from

W.-Y. Lu / Journal of Constructional Steel Research 62 (2006) 933–942

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Fig. 7. Effect of b f /b and f c0 on shear-carrying capacities. Fig. 6. Effect of b f /b and a/d on shear-carrying capacities.

diagonal compression failure into bearing failure (Figs. 6 and 7). If the failure mode of SRC deep beams is bearing failure, then the shear-carrying capacities increase with the increasing b f /b at a much lower rate (Figs. 6 and 7). With higher a/d values, the vertical mechanism is more effective in the SST model [3], and the beam can gain higher diagonal compression strength. Thus the shear-carrying capacity of the beam is dominated by the bearing strength. That is why the critical value of b f /b for beams with high a/d is lower than that for beams with low a/d (Fig. 6). This leads to the bearing failure being likely to occur in beams with high a/d, whereas diagonal compression failure is likely to occur in beams with low a/d (Fig. 6). The softening effect on high-strength concrete is more pronounced than that on normal-strength concrete, which leads to the shear-carrying capacities being dominated by the

diagonal compression strengths for beams with high f c0 . With higher values of f c0 , there are higher critical values of b f /b (Fig. 7). Bearing failure is likely to occur in beams with low f c0 , whereas diagonal compression failure is likely to occur in beams with high f c0 (Fig. 7). 5. Conclusions An analytical method for determining the shear strengths of SRC deep beams under the failure mode of concrete crushing based on the SST model [3,4] is proposed. Based on the available test results in the literature [13,15,16] and their comparisons with the proposed method, the following conclusions can be made: 1. The proposed method can accurately predict the shear strengths of SRC deep beams with vertical hoops or without vertical hoops.

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2. The major factors influencing the shear-carrying behavior of SRC deep beams have been found to be the ratio of the flange width to the gross width (b f /b), shear span-to-depth ratio (a/d), and the compressive strength of concrete ( f c0 ). 3. When b f /b is low, the shear-carrying capacities of SRC deep beams increase with increasing b f /b. However, if b f /b is higher than a critical value, then the failure mode of SRC deep beams will be converted from diagonal compression failure into bearing failure. If the failure mode of SRC deep beams is bearing failure, then the shear-carrying capacities increase with increasing b f /b at a much lower rate. 4. Bearing failure is likely to occur in SRC deep beams with high a/d and low f c0 . On the contrary, diagonal compression failure is likely to occur in SRC deep beams with low a/d and high f c0 . 5. Currently, the available experimental data on the shear strengths of SRC deep beams is very limited; it is hoped that further experimental work should be performed in the near future. Acknowledgments This research study was partially sponsored by the National Science Council of the Republic of China under Project NSC 92-2218-E-163-001. The author would like to express his gratitude for the support. References [1] ACI-ASCE Committee 426. Shear strength of reinforced concrete members. Proceedings ASCE 1973;99(ST6):1091–187 [Reaffirmed in 1980 and published by ACI as Publication No. 426R-74]. [2] Architectural Institute of Japan. AIJ Standards for structural calculation

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