shear strength analysis and prediction for nsc and hsc beams with or without stirrups

Shear Strength Analysis and Prediction for NSC and HSC Beams with or without Stirrups

Authors: Giuliana Somma, University of Udine, Via delle Scienze 208, 33100, Udine, Italy, [email protected] INTRODUCTION The shear failure of reinforced concrete (RC) beams is due to the combined action of shear and flexure. Because of the presence of shear, beam flexural strength may result greatly reduced with respect to the pure flexure case, and failure may happen in a brittle way and without premonitory signs. Due to the complexity of the phenomenon, much research has been addressed to evaluate the resisting mechanisms of only longitudinally reinforced beams, currently called concrete mechanisms. The analysis of shear strength of reinforced Normal Strength Concrete (NSC) and High Strength Concrete (HSC) beams with only longitudinal reinforcement, in which a constant shear force acts throughout the shear span, has led the author to recently propose a shear strength formula [Russo et al., 66-74] predicting the experimental behavior in a more accurate and uniform way than other authors and Codes expressions. The shear mechanism has found to be governed by a second order differential equation, whose solution provides the internal lever arm in function of the distance from the support. The strength provided by any transverse reinforcement, which is supposed to be constituted by stirrups in almost every study, is hence taken into account by adding the resisting contribution of the truss mechanism to that due to the concrete mechanisms. In the transverse reinforcement contribution the stirrup effectiveness is taken into account. CONCRETE CONTRIBUTION TO SHEAR STRENGTH Although it has to be noticed that in HSC beams the fracture planes are smoother than in NSC beams, with consequent lower shear strength contribution of the aggregate interlock mechanism, the crack patterns are almost analogous, and the failure modes of HSC and NSC beams are similar: shear-compression failure for beams with prevalent arch action, and diagonal-tension failure for beams with prevalent beam action. The recently proposed expression for computing the shear strength of only longitudinally reinforced concrete beams [Russo et al., 66-74] has found to more accurately and uniformly predict the experimental behavior than most unknown expressions. Moreover this reliability has been noticed for both NSC and HSC beams, for a total number of 917 tested specimens. The shear strength formula for beams with no transverse reinforcement is

2007 Structures Congress: New Horizons and Better Practices 2007 ASCE

Copyright ASCE 2007 Structures Congress 2007

New Horizons and Better Practices

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d.

+= da

ylcuc daffv

45.02.189.083.039.04.0 5.013.1 (1)

where is the geometric percentage of longitudinal reinforcement, fc is the cylindrical concrete compressive strength, fyl is the yielding strength of longitudinal reinforcement, a/d is the shear span to effective beam depth ratio, and is a size effect function given by the following expression [Bazant and Sun, 259-271]

( )aa

ddd

25/1/08.51

+

+= (2)

where dais the maximum aggregate size. TRANSVERSE REINFORCEMENT CONTRIBUTION TO SHEAR STRENGTH The resisting mechanism provided by transverse reinforcement is generally assumed independent of concrete resisting contributions. Hence the shear strength of a beam with transverse reinforcement is obtained by adding the resisting contribution provided by stirrups to that provided by concrete

sucu vvv += (3) where vuc is the shear strength provided by concrete, and vs is the shear strength provided by stirrups. It was shown [Park and Paulay] that the truss mechanism is related to the beam action by increasing the dowel action, the capacity of the concrete tooth, and the aggregate interlock. It follows that the stirrups included in a beam with prevalent beam action can result very highly stressed. Consequently the transverse reinforcement contribution to shear strength, vs, takes into account not only the truss mechanism, but also some interaction between the truss action and the beam action. Therefore vs must be lower than v fyv when the arch mechanism is prevalent, and greater than v fyv when the beam mechanism is prevalent (where v and fyv are the geometric percentage and the yielding strength of the transverse reinforcement respectively). It is well known that for beams with a/d lower than the critical value of this ratio, (a/d)c arch mechanism is prevalent, and for beams with a/d greater beam mechanism is prevalent. A lot of experimental studies have been conducted to demonstrate that in correspondence of the (a/d)c value there is the maximum reduction of the beam moment capacity. This value is statistically posed in a range between 2.5 and 3, and also a little greater for HSC beams. To take into account the dependence of the transverse reinforcement shear strength contribution to the prevalence of the arch or beam actions, vs is expressed as

yvvs fv = (4) being a stirrup effectiveness function, that is chosen as:

da

3= (5)

in which the average value of (a/d)c is taken equal to 3.




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SHEAR STRENGTH FOR NSC AND HSC BEAMS WITH STIRRUPS With all the considerations done in the previous paragraphs, the here proposed shear strength expression for NSC and HSC beams is

yvv

da

ylcu fdaffv +

+= 45.02.1

89.083.039.04.0 5.013.1 (6)

where and are given by (2) and (5) respectively. This expression evidently leads to that previously proposed for only longitudinally reinforced concrete beams [(1)] if the stirrup are not present (v fyv = 0). Reliability of the Proposed Shear Strength Model The reliability of the proposed model in predicting the shear strength of NSC and HSC beams with or without stirrups has been compared with the following expressions provided by Codes [and by authors, with reference to the experimental outcomes from 917 beams with only longitudinal reinforcements and 409 beams with longitudinal and transverse reinforcements. This collection of 1326 specimens is the most numerous ever collected for shear, in the literature. ACI 318M-05

yvvu

ucu fM

dVfv ++= 1716.0 (7)

where the first addendum must be cf 29.0 , and Vu and Mu are the shear force and bending moment at ultimate state respectively. Eurocode

( ) yvvRdu fkv 78.0402.1 ++= (8) where 16.1 = dk (d in meters), = 1 for a/d 2.5 or 55.2 = ad for a/d < 2.5,

( )( )02.0;bdAmin s= , and cctkRd f 05.025.0= , with 5.1=c , ctmctk ff 7.005.0 = , and 323.0 cctm ff = .

Rebeiz

( ) yvvdcu fAadfv +

+= 310 4.0 (9)

where Ad = a/d for a/d < 2.5, and Ad = 2.5 for a/d 2.5. Kim and Park

+=

adfv cuc 4.05.3

833 (10)

where ( ) 18.0008.01 21 ++= d , = 1 for a/d 3, or ( )da 32 = for a/d < 3.




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To check the validity of the introduction of the function in the stirrup shear strength contribution, the following expression has been taken into account too, and its outcomes have been compared with the 1326 experimental results.

yvv

da

ylcu fdaffv +

+= 45.02.1

89.083.039.04.0 5.013.1 (11)

In Figures 1-6 the ratio between the experimental shear strength value and the computed one by means of the proposed formula [(6)], and each of the five above mentioned expressions [(7)-(11)] is plotted versus the shear span to effective depth ratio a/d, for the 1326 beams. The horizontal bold lines represent the perfect correspondence between experimental and computed shear strength values. So, the closer to this line the points are, the more accurate the shear strength prediction is. The thinner the width of the strip including the points is, the greater the prediction uniformity. The AVG and COV values are also reported in Figures 1-6. AVG is the index of the mean accuracy of the considered formula, and the closer to one its value is, the more accurate the expression. COV is the index of the uniformity in the prediction, with variation in beam geometrical and mechanical characteristics or the loading position, and the lower it is, the better the prediction capability of the experimental results. From Figures 1-6 it is apparent that the proposed formula for computing shear strength of beams with and without stirrups is better than the other ones, in terms of accuracy and uniformity of prediction.

FIGURE 1: EXPERIMENTAL TO COMPUTED (WITH EQUATION 6) SHEAR

STRENGTH RATIO VERSUS a/d FOR 1326 TESTED BEAMS.

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9a/d

Proposed expr. AVG = 1.01 COV = 0.20

v u,e

xp/v

u,co

mp




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9

0 1 2 3 4 5 6 7 8 9



0

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0 1 2 3 4 5 6 7 8 9



ACI 318M-05 AVG = 1.89 COV = 0.65

v u,e

xp/v

u,co

mp

a/d

Eurocode AVG = 1.69 COV = 0.37

v u,e

xp/v

u,co

mp

a/d




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0 1 2 3 4 5 6 7 8 9



0

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Rebeiz AVG = 0.90 COV = 0.25

v u,e

xp/v

u,co

mp

a/d

Kim and Park AVG = 0.98 COV = 0.22

v u,e

xp/v

u,co

mp

a/d




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STRENGTH RATIO VERSUS a/d FOR 1326 TESTED BEAMS. By observing Figures 1 and 6 it is evident that the proposed introduction of the function in the transverse reinforcement contribution to shear strength leads to an improvement of the 10% in the uniformity in the prediction. CONCLUSIONS From several experimental and theoretical investigation carried out over the years by the researchers, some interaction between the truss action and the beam action has been observed in beams subjected to shear stress. Consequently the transverse reinforcement contribution to shear strength has been taken function not only of the truss mechanism (v fyv) but also of the beam or arch action prevalence (a/d). The here proposed expression has been found to accurately predict the shear strength of Normal Strength Concrete (NSC) and High Strength Concrete (HSC) beams with or without transverse reinforcement. This accuracy and uniformity in the strength evaluation has been verified by comparing the outcomes of the expression with 1326 experimental results found in the literature. It has found that this expression exhibits a much more uniform prediction of experimental results not only than Eurocode and ACI Code, but also than two expressions provided by two different research groups. An important conclusion is also that this improvement in the uniformity prediction ability is prevalently due to the introduction of the function , which takes into account the interaction between concrete and stirrup resisting mechanisms.

Proposed expr. with =1 AVG = 0.99 COV = 0.22

v u,e

xp/v

u,co

mp

a/d




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REFERENCES [1] Russo, G, Somma, G, Mitri, D, "Shear Strength Analysis and Prediction for Reinforced Concrete

Beams without Stirrups", Journal of Structural Engineering ASCE, Vol.131, No.1, January 2005, 66-74.

[2] Bazant, Z, B, Sun, H, H, "Size effect in diagonal shear failure: Influence of aggregate size and stirrups", ACI Material Journal, Vol.84, No.4, 1987, 259-271.

[3] Park, R, Paulay, T, "Reinforced Concrete Structures", John Wiley & Sons, New York, 1975, p.769. [4] Russo, G, Puleri, G, "Stirrup Effectiveness in RC Beams under Flexure and Shear", ACI Structural

Journal, Vol.94, No.3, 1997, 576-584. [5] American Concrete Institute (ACI), "Building Code Requirements for Structural Concrete and

Commentary (ACI 318M-05)", Farmington Hills, MI. [6] ENV 1992-1-1 Comit Europen de Normalization CEN, "Eurocode 2 - Design of Concrete Structures

- Part 1-1: General Rules and Rules for Buildings Building". [7] Kim, JK, Park, YD, "Prediction of Shear Strength of Reinforced Concrete Beams without Web

Reinforcement", ACI Material Journal, Vol.93, No.3, 1996, 213-222. [8] Rebeiz, KS, "Shear Strength Prediction for Concrete Members", Journal of Structural Engineering

ASCE, Vol.125, No.3, 1999, 301-308. [9] Kani, GNJ, "A Rational Theory for the Function of Web Reinforcement", ACI Journal, Vol.66, No.3,

1969, 185-197. [10] Russo, G, Somma, G, Angeli, P, "Design shear strength formula for high strength concrete beams",

Materials and Structures, Vol.37, No.274, 2004, 680-688. [11] Russo, G, Venir, R, Pauletta, M, Somma, G, "Reinforced Concrete Corbels - Shear Strength Model and

Design Formula", ACI Structural Journal, Vol.103, No.1, 2006, 3-10.




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shear strength analysis and prediction for nsc and hsc beams with or without stirrups

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