Proposal for a new shear design method

Transcription

1 Chapter 6 Proposal for a new shear design method The behaviour of beams failing in shear has been studied in the previous hapters, with speial attention paid to high-strength onrete beams. Some aspets not usually onsidered have been highlighted, suh as the dependene of the size effet on the onrete ompressive strength or the non-linear proportionality between stirrups and shear strength. Furthermore, proedures used for alulating the ultimate shear strength from five different odes of pratie have been evaluated. In this hapter, a new shear design method based on the observed behaviour is proposed, and an effort is made to keep it simple enough to make it suitable for implementation in a ode of pratie. 6.1 Beams without web reinforement Summary of the observed behaviour The main onlusions drawn from the observed behaviour of beams failing in shear, that form the basis for our two proposals are that: 133

2 Chapter 6 - The EHE proedure generally shows a satisfatory orrelation for normalstrength onrete beams. However, the orrelation an be improved with minor hanges. - The preditions made by the EHE shear proedure are unonservative for members with an effetive depth under 100 mm (Table 5.4). The benefit of the size effet should be limited for small members. - The size effet is related to the onrete ompressive strength. For deep beams, the benefit of a higher onrete ompressive strength is outweighed by the loss aused by the size effet (Figure 5.20). - The size effet is also related to the maximum spaing between layers of longitudinal reinforement. - The influene of the longitudinal reinforement is greater than that proposed by the EHE proedure. It would not be neessary to limit its value to 2% for high-strength onrete beams (Figure 5.21). - The shear span-to-depth ratio, a/d, influenes the failure shear strength even for beams with an a/d greater than 2.5 (Figure 5.22). The AASHTO LRFD Speifiation takes the bending moment into aount to alulate the failure shear strength. Nevertheless, the equations proposed will predit a onservative result, sine they do not take the shear span-to-depth ratio into onsideration Shear design method For members without web reinforement, the following equation is proposed, diretly derived from the analyses arried out in Chapter 5: where V 2 3 [ 0.13ξ ( 100 ) f ] 1 / 1 / b d = (6.1) ρ s w 134

3 Proposal for a new shear design method ξ = sx f 1,1 f , is the size effet with f 25 MPa, ξ f = 25 MPa f = 50 MPa f = 75 MPa f = 100 MPa sx Figure 6.1: Proposed size effet in funtion of the onrete ompressive strength. s x is whihever is smaller, d v or the vertial distane between longitudinal distributed reinforement as indiated in Figure 6.2, d is the effetive depth in mm, d v is the mehanial depth taken to be 0.9 d, A = sl f + ρ s is the amount of longitudinal reinforement, b d 100 w f 100 MPa and b w is the smallest width of the ross-setion area in mm. flexural ompression zone As > 0,003 bw sx sx = dv sx Figure 6.2: Value of s x for members without web reinforement. 135

4 Chapter 6 Equation 6.1 does not take the onrete safety fator into aount. To onsider it, the first onstant should be modified, resulting in the following expression: V 2 3 [ 0.10ξ ( 100 ) f ] 1 / 1 / b d = (6.2) ρ s w Simplified shear design method Inluding the size effet in equations 6.1 and 6.2, whih was linearly derived in from equations developed by Fujita et al. (2002) results in equation that are probably too omplex to be implemented in a ode of pratie. For this reason, a simplified shear design method is proposed. The simplified shear design method adopts a size effet term similar to the EHE one, and it limits the onrete ompressive strength to 60 MPa to keep from being unonservative for deep high-strength onrete beams. where V [ 0.225ξ ( 100 ) f ] 1 / b d = (6.3) ρ s 200 ξ = is the size effet, s x s x is whihever is smaller, d v or the vertial distane between longitudinal distributed reinforement as indiated in Figure 6.2, d v is the mehanial depth, taken to be 0.9 d, d is the effetive depth in mm, A = sl f + ρ s is the amount of longitudinal reinforement, bwd 100 f 60 MPa and b w is the smallest width of the ross-setion area in mm. w Equation 6.3 does not take the onrete safety fator into aount. If we fator it in, the resulting equation is: 136

5 Proposal for a new shear design method V [ 0.18ξ ( 100 ) f ] 1 / b d = (6.4) ρ s w Verifiation of the proposed equation using the experimental database In order to ompare the proposed equations with the ode proedures desribed in Chapter 2, the statistial analyses presented in Chapter 5 are here ompared with the preditions made by equations 6.1 and 6.3. Table 6.1 ompares the predited values with the empirial results for the entire database of members without web reinforement. Proposed equations 6.1 and 6.3 offer very similar results, although the first one gives a slightly better oeffiient of variation than the seond proedure. Nevertheless, both equations orrelate better with the empirial results than do the other proedures. 5.2 explains the meaning of eah parameter in Table 6.1. Proedure EHE EC-2 AASHTO ACI 11-5 ACI 11-3 Eq. 6.1 Eq. 6.3 Average Median Standard deviation COV (%) COV LOW 50% (%) COV HIGH 50% (%) Minimum (V test / V pred ) 1% Maximum (V test / V pred ) 99% Table 6.1: Verifiation of proposed shear proedures using the entire database for beams without web reinforement. The Total Demerit Point lassifiation is given in Table 6.2. It an be seen that the simplified shear design proedure (equation 6.3) obtains a vaguely better sore, as the general design proedure is slightly more onservative. 137

6 Chapter 6 V V test pred Classifiation DP EHE EC-2 AASHTO ACI 11-5 ACI 11-3 Eq. 6.1 Eq. 6.3 < 0.50 Extremely dangerous Dangerous Low safety Appropriate safety Conservative > 2.00 Extremely onservative Total Demerit Points Table 6.2: Comparison of Demerit Points lassifiations for beams without web reinforement. The results of the partial set analyses are given in Table 6.3. The proposed methods represent an improvement in terms of the oeffiient of variation over all the other ode proedures and all partial datasets exept for the AASHTO proedure. This ode presents the best orrelation for two sets: beams with a low amount of longitudinal reinforement, ρ l 1%, and for normal-strength onrete beams. Proposed equation 6.1 gives an almost idential orrelation for normal-strength and high-strength onrete beams, with average V test /V pred ratios of 1.14 and 1.16, and oeffiients of variation of 15.96% and 15.53% respetively. The distribution of V test /V pred ratio perentages is plotted in Figure 6.3, and an be ompared to the graphs given in Chapter 5 for the different ode proedures. Both proposed odes show a higher perentage of preditions in the band than any other ode proedure. It an also been seen that, although equation 6.3 gets a better Demerit Point sore, the proposed equation 6.1 shows a better distribution, with 60% of the beams falling in the band. 138

7 Proposal for a new shear design method Beam speimens nº beams EHE Average V test / V pred ACI ACI EC-2 AASHTO Eq. 6.1 Eq. 6.3 COV V test / V pred EHE EC-2 AASHTO ACI ACI Eq. 6.1 Eq. 6.3 All d 900 mm d 100 mm ρ l 1% f > 50 MPa f 50 MPa ρ l > 2% f > 50 MPa ρ l > 2% f 50 MPa Table 6.3: Verifiation of different ode proedures using partial sets of the database for beams without web reinforement. a) b) % of values of Vtest/Vpred < > < > 2.50 Figure 6.3: Correlation of the proposed equations with empirial tests for beams without web reinforement. a) Equation 6.1. b) Equation 6.3. % of values of Vtest/Vpred 139

8 Chapter Verifiation of the proposed equation for elements with longitudinal distributed reinforement As was stated in Chapter 2, Collins and Kuhma (1999) arried out an experimental ampaign to evaluate the parameters influening on the size effet. They onluded that it was related to the maximum spaing between the layers of longitudinal reinforement rather than the overall member depth. Test beams with longitudinal reinforement were not inluded in the database presented in 5.1, as only the AASHTO LRFD Speifiations take this effet into aount, and therefore the performane of the other odes proedures would have been poorer. Table 6.4 gives the geometrial parameters of the beams with longitudinal distributed reinforement, test results, and preditions given by the EHE Code proedure, the AASHTO LRFD Speifiations, and the proposed equations. All beams in Table 6.4 had a greater amount of longitudinal reinforement distributed in the web than the minimum amount given in Figure 6.2. All were tested by Collins and Kuhma (1999) exept for the last two beams, tested as part of this thesis. The EHE shear proedure does not take the effet of distributed longitudinal reinforement into onsideration, and is exessively onservative for the 17 beams ontaining it, with an average V test /V pred ratio of The preditions made by the AASHTO proedure improve the orrelation, resulting in an average of The standard deviation for both odes is Equations 6.1 and 6.3 do take this effet into aount and they improve the performane observed for the EHE shear proedure for members without web reinforement. The average V test /V pred ratio is 1.09 for equation 6.1 and 1.20 for equation 6.3, and their standard deviations are 0.16 and 0.11 respetively. 140

9 Proposal for a new shear design method Beam f MPa b mm d mm a/d ρ l s x V fai V predited V test / V predited (KN) EHE LRFD Eq.6.1 Eq.6.3 EHE LRFD Eq.6.1 Eq.6.3 B100D BND BND BND BHD BHD100R BHD BHD50R BH25D SE100B SE100B-45-R SE50B SE100B SE100B-83-R SE50B H50/ H100/ Distane between layers of long. reinforement Average Standard deviation Coeffiient of var Table 6.4: Summary of preditions by EHE, AASHTO LRFD, equation 6.1 and equation 6.3 for elements with longitudinal distributed reinforement. 6.2 Beams with web reinforement Summary of the observed behaviour A general shear design method is proposed in this setion, as well as two simplified shear proedures based on the following observations for members with web reinforement: - The EHE, EC-2, ACI 11-5 and ACI 11-3 shear proedures do not orrelate satisfatorily for members with web reinforement, as was disussed in Chapter

10 Chapter 6 - The AASHTO LRFD shear proedure, based on the modified ompression field theory, performs muh better than the other urrent odes do. In this proedure, the angle between the ompression struts and the longitudinal axis of the beam, θ, is obtained by ompatibility, and it depends on the shear stress and the longitudinal strain of the web. - The onrete ontribution to the shear strength is the vertial omponent of the shear stress transferred aross the rak and therefore depends on the rak width. The greater the amount of shear reinforement, the lesser rak width, and the larger the onrete ontribution will be ( 2.3.5). - The influene of the amount of web reinforement is not linearly proportional to the failure shear strength ( 5.5.3). Truss models, like EC-2, ould be unonservative for highly reinfored onrete members, as an be seen in Table The use of high-strength onrete tends to prevent shear-ompression failure and to ensure a diagonal tension failure instead, thus inreasing the effetiveness of the shear reinforement. - For members with low shear reinforement, the size effet redues the shear stress at failure, although most odes do not take it into aount for members with stirrups (Figure 5.24). - An inrease in the amount of longitudinal reinforement produes an inrease in shear strength. It would not be neessary to limit its value to 2% as is required by the EHE proedure (Figure 5.26) General shear design method: proedure and justifiation Proedure For members with web reinforement, the failure shear strength is given by: V = V + V s (6.5) 142

11 Proposal for a new shear design method V [ 0.17ξ ( 100ρ ) f ] 1 / τ 1 / b d = (6.6) s w where 200 ξ = is the size effet, s x s x is whihever is smaller, d v or the vertial distane between longitudinal distributed reinforement as indiated in Figure 6.2, d v is the mehanial depth whih an be taken as 0.9 d, d is the effetive depth in mm, Asl ρ s = 0.04, is the amount of longitudinal reinforement, b d w f 100 MPa, τ = Vd b d w v 3 MPa, and b w, the smallest width of the ross-setion area in mm. And, where Asw Vs = dv f ywd otθ (6.7) s A sw is the ross-setional area of the shear reinforement. s is the spaing of the stirrups f ywd is the design yielding strength of the shear reinforement, and θ is the angle of the ompression struts, derived as follows: τ θ = ε x (6.8) f k where ε x is the longitudinal strain in the web, expressed in 1/1000, alulated by the following expression: 143

12 Chapter 6 τ 0.05 f k M d + Vd dv ε x (6.9) E A s sl The expression of the longitudinal strain in the web is a onservative simplifiation of the real strain. It assumes that in the web the strain is equal to one half the strain in the tension reinforement, and that the maximum longitudinal strain of the reinforement is Figure 6.4: Longitudinal strain in the web (from Collins 2001) Equation 6.6 does not take the onrete safety fator into aount. To onsider it, the first onstant should be modified, resulting in the following expression:: V [ 0.14ξ ( 100ρ ) f ] 1 / τ 1 / b d = (6.10) s w Justifiation Equation 6.8 was diretly derived from the AASHTO LRFD Table given in Figure 2.29, in an attempt to find the simplest equation that still followed the general trend. The proposed value for θ is always onservative ompared with the AASHTO preditions. One the angle was obtained, the steel ontribution was able to be determined. Equation 6.3, from the simplified shear design method for beams without stirrups, was taken to be a good proedure for evaluating the onrete ontribution for a beam with only longitudinal reinforement. An extra term was added to take into aount the 144

13 Proposal for a new shear design method stirrups influene on the shear frition. As the amount of transversal reinforement is unknown during the design proess, the new term is a funtion of the designing shear stress, τ to the power of 1/3. This value of the power and the onstant 0.17, used to multiply V, were derived empirially to adjust the test beam results Simplified shear design method To apply the proedure presented in it is neessary to evaluate the shear strength in different setions of the beam, due to the interation between the bending moment and the shear strength. The simplified shear design method assumes that the longitudinal strain in the web, ε x, is equal to 1, and therefore that the longitudinal reinforement yields; this is the worst ondition under whih to alulate the shear strength. Hene: where τ 0.05 f k τ θ = (6.11) f k With the value of the angle of the ompression struts given by 6.11, the failure shear strength an be alulated using equations 6.5, 6.6 and Simplified shear verifiation method To verify the ultimate shear strength of a given setion it would be possible to use the expressions given in 6.2.2, although it would be neessary to iterate to find the solution, as V d is an input to obtain both the onrete and the steel ontributions. Moreover, the ultimate shear strength would depend not only on the ross-setion of the beam, but also on the bending moment in that setion. The simplified shear verifiation method estimates τ, so the ultimate shear strength an be alulated from equations 6.5, 6.6 and 6.7, assuming that ε x = 1: 145

14 Chapter 6 est f ywd Asw 3.5 sx bw s τ = (6.12) The above estimation is equivalent to a truss model using a variable angle of inlination for the trusses. For small members, suh as s x = 200 mm, the estimated shear failure would be given by a truss model in whih ot θ = 3.5. For a bigger beam, for example s x = 1000 mm, the inlination of the truss would be given by ot θ = Verifiation of the proposed equation with the experimental database The proposed equations are ompared in Table 6.5 with the database s 123 test beams with shear reinforement. The three proposed proedures orrelate muh better with empirial tests than do the EHE, EC-2, ACI 11-5, or ACI 11-3 proedures. For example, the V test /V pred ratio for the urrent EHE ode is 1.64, with a standard deviation of 0.43, while, it is 1.11, with a standard deviation of 0.21 for the general shear design method ( 6.2.2). Nevertheless, the AASHTO LRFD shear proedure performs very similarly to the proposed equations. It an also be seen in Table 6.5, that the two simplified methods are slightly more onservative than the general design method, as they do not take into aount the influene of the bending moment, and they assume that the longitudinal rebars yield. Proedure EHE EC-2 AASHTO ACI 11-5 ACI Average Median Standard deviation COV (%) COV LOW 50% (%) COV HIGH 50% (%) Minimum (V test / V pred ) 1% Maximum (V test / V pred ) 99% Table 6.5: Verifiation of proposed shear proedures for beams with web reinforement using the entire database. 146

15 Proposal for a new shear design method The Total Demerit Point lassifiation for beams with web reinforement is given in Table 6.6. The AASHTO proedure obtains the best sore, 36 points, followed by the general shear design method whih gets 37 points. The two simplified proposed methods sore 41 and 39 Demerit Points. V V test pred Classifiation DP EHE EC-2 AASHTO ACI 11-5 ACI < 0.50 Extremely dangerous Dangerous Low safety Appropriate safety Conservative > 2.00 Extremely onservative Total Demerit Points Table 6.6: Comparison of Demerit Points lassifiation for beams with web reinforement. Beam speimens nº beams EHE Average V test / V pred ACI ACI EC-2 LRFD COV V test / V pred ACI ACI EHE EC-2 LRFD All d 750 mm ρ w 1MPa ρ w > 1MPa ρ w 2MPa ρ w > 2MPa f 50 MPa f > 50 MPa ρ l 2 % Table 6.7: Verifiation of different ode proedures using partial sets of the database for beams with web reinforement. 147

16 Chapter 6 To hek the ability of the proposed methods to predit the shear strength for different types of beams, the results of the partial set analyses are given in Table 6.7. The proposed proedures represent an improvement over the performane of the EHE, EC- 2, ACI 11-5, and ACI 11-3 proedures for all the groups of beams studied. For the biggest members, where d 750 mm, most odes do not take into aount the size effet when stirrups are provided. It was shown in Figure 5.24 that, for members with low shear reinforement, the size effet auses a redution in shear strength. The proposed equations orretly reprodue this behaviour, as very little redution in the overall safety fator (V test /V pred ) is observed. Another set of beams that requires speial attention is the group of seven beams with high shear reinforement (ρ w > 2 MPa). The EC-2 proedure is absolutely unonservative with an average V test /V pred ratio of The EHE ode, with a ratio of 0.99, is also somewhat unonservative, signifying an approximate 40% redution in the safety fator with respet to the average oeffiient for all the 123 beams. The general shear design method presents the same ratio as the EHE ode, but the derease in the safety fator is only by about 11%. The simplified shear design methods are not unonservative for this set of beams, although the best performane is ahieved by the AASHTO proedure. For beams with a low amount of longitudinal reinforement, ρ l 2%, the proposed methods perform satisfatorily, while other odes present slightly unonservative results. Finally, the distribution in perentages of the V test /V pred ratio is plotted in Figure 6.5, and it an be ompared with the Figures given in Chapter 5 for the different ode proedures. The three proposed methods present the highest predition perentage in the band ompared with other odes. 148

17 Proposal for a new shear design method a) % of values of Vtest/Vpred < b) ) 70 > % of values of Vtest/Vpred < > 2.50 % of values of Vtest/Vpred < > 2.50 Figure 6.5: Correlation of the proposed equations with empirial tests for beams with web reinforement. a) Equation b) Equation ) Equation Equivalene between the simplified shear design method and the simplified shear verifiation method The simplified shear proedures given in and are respetively intended for design and verifiation. Both proedures assume the longitudinal strain in the web to be equal to 1, and, that the longitudinal reinforement will yield. In this setion it will be shown that the results obtained by the two methods are pratially idential, the verifiation proedure being slightly (1%) more onservative than the design method. 149

18 Chapter 6 % of values of V 6.2.3/V < >1.15 Figure 6.6: Correlation between the simplified shear design method ( 6.2.3) and the simplified shear verifiation method ( 6.2.4). The average V /V ratio for the database s 123 test beams with web reinforement, where V is the shear strength predited by the simplified shear design method and V is the shear strength predited by the simplified shear verifiation method, is equal to 1.01, and its oeffiient of variation is 3.85%. The distribution of V /V ratio perentages is given in Figure 6.6. For 82% of the test beams, the value of the ratio falls in the band. 6.3 Comparison of the proposed method with beams tested at the Strutural Tehnology Laboratory. Table 6.8 summarises the preditions made by the EHE proedure, the 2002 Final Draft of the Euroode-2, the AASHTO LRFD, ACI , and proposed general shear design proedures, in addition to the simplified proedures for beams with and without web reinforement. For beams with stirrups the verifiation proedure was used. It an be seen that the proposed equations orrelate satisfatorily with the beams tested at the Strutural Tehnology Laboratory of the Tehnial University of Catalonia, with a oeffiient of variation lower than 10%. 150

19 Proposal for a new shear design method Beam f MPa b mm d mm a/d ρ w ρ l V fai V predited V test / V predited KN EHE EC LRFD ACI Gen* Sim+ EHE EC LRFD ACI Gen Sim+ H50/ H50/ H50/ H50/ H50/ H60/ H60/ H60/ H60/ H75/ H75/ H75/ H75/ H100/ H100/ H100/ H100/ H100/ * General proposed proedure Average Simplified shear proedure (verifiation) Stand. Deviation COV (%) Table 6.8: Comparison of the proposed general and simplified shear proedures and urrent odes with test results of the experimental ampaign. 151

20 Chapter 6 152