PREDICTING THE SHEAR STRENGTH OF CONCRETE STRUCTURES

Transcription

1 PREDICTING THE SHEAR STRENGTH OF CONCRETE STRUCTURES M.P.COLLINS; E.C.BENTZ; P.T.QUACH; A.W.FISHER; G.T. PROESTOS Department of Civil Engineering, University of Toronto, Canada SUMMARY Beause many shear design tehniques rely on empirial equations derived from rather small sale experiments there is onern that these traditional tehniques may give unonservative estimates of shear strength when applied to very large reinfored onrete members with small amounts of shear reinforement. This paper will desribe the loading to failure of a strip from a four metre thik slab and the assoiated hallenge issued to engineers to predit the shear response. Also disussed is the shear response of heavily reinfored oupling beams for shear walls. INTRODUCTION Defiienies in the shear design of onrete struture are inherently more dangerous than defiienies in flexural design beause shear failures an our with no prior warning and with no possibility for redistribution of internal fores. While aurate assessment of the shear apaity of a reinfored onrete struture is ritially important for publi safety, the traditional tehniques available for this task are open to dispute. For determining flexural apaity engineers an use the simple, aurate, general and internationally aepted plane setions theory. However, for finding shear strength engineers typially rely on restrited empirial equations whose appliability and auray are sometimes very questionable. Mat or raft foundations for high rise buildings were in the past typially onstruted to be thik enough not to need shear reinforement aording to traditional shear design proedures (e.g. ACI ). In a departure from this tradition the 5.33 m thik mat foundation of the 73 storey Wilshire Grand building (Nieblas 2014) now under onstrution in Los Angeles does ontain signifiant amounts of shear reinforement, see Figure 1. This reinforement was speified partly to alleviate onerns with the so alled size effet in shear whih is based on the observation that as depth of reinfored onrete members without shear reinforement inrease the shear stress to ause failure dereases(collins and Kuhma 1999). The size effet in shear is predited by the Modified Compression Field Theory (MCFT Vehio and Collins 1986) whih forms the basis of the shear provisions of the AASHTO-LRFD Bridge Design Speifiations, the Canadian Standard Design of Conrete Strutures, CSA A and the fib Model Code These proedures predit that the shear apaity of a slab suh as that shown in Figure 1 in regions without shear reinforement an be less than one third of the shear strength predited by ACI. To verify these preditions it was deided to fabriate and test to failure a strip from a four metre thik slab, see Figure 2.

2 Figure 1. Reinforement in mat foundation of Wilshire Grand Figure 2. Strip from four metre thik slab and 300 mm deep traditional size shear speimen

3 MODIFIED COMPRESSION FIELD THEORY (MCFT) The shear design proedures developed at the University of Toronto over the last 40 years (Collins 1978) by studying reinfored onrete elements loaded in pure shear will be briefly summarized before disussing the urrent experiments. Figure 3 shows the five equilibrium equations, five geometri onditions and five stress-strain relationships of the MCFT. In the upper part of the figure the equations deal with average stresses, average strains and the relationships between average stresses and average strains. Average stresses (e.g. f sx) and average strains (e.g. ε x) orrespond to stresses and strains averaged out over lengths long enough to damp out the loal variations that our at raks and between raks. The bottom five relationships in the figure are onerned with stresses at a rak (e.g. f sxr), the rak width and the maximum shear stress that an be transmitted aross the rak. The term v i stands for shear stress transmitted aross the rak interfae. The average stress-average strain relationships for raked onrete given by Eq. 13 and Eq. 14 in Fig. 3 are the result of experiments on reinfored onrete elements tested in pure shear. For higher strength onretes more aurate results are obtained if the 170 oeffiient in Eq. 14 of Fig. 3 is replaed by 3f where f is in MPa. The relationship used for determining the ability of the rak surfaes to transmit the interfae shear stresses, Eq. 15 in Fig. 3, was derived from the aggregate interlok experiments of Walraven (1981). Note that this shear stress limit is a funtion of the rak width, w, the maximum aggregate size, a g, and the onrete ylinder strength, f. Figure 3: The modified ompression field theory (MCFT) The 15 equations of the MCFT an be onveniently solved using program Membrane available on the web (Bentz 2014).To use the MCFT to predit the load-deformation response of a reinfored onrete beam suh as that shown in Fig. 4, the beam ould be represented as a two dimensional grid of elements with the response of eah element being predited by the MCFT. This is the basis of non-linear finite element programs suh as VeTor2 (Vehio 2014).

4 If suh a program is used to analyze the beam it will be found that in zones extending for a distane of about d away from the point loads and the reations, there will be signifiant vertial ompressive stresses in the onrete. These lamping stresses will enhane the shear strength of the elements in these zones making it probable that the shear failure will our outside of these zones. For beams with short shear spans the zones with signifiant lamping stresses will overlap and the shear strength of the beam will be onsiderably inreased. In these disturbed regions the shear stress distribution over the depth of the beam is influened by the distribution of the lamping stresses and near the loads and reations, plane setions do not remain plane. Outside of the disturbed regions it is appropriate to assume that plane setions remain plane and that the lamping stresses are negligible. With these two assumptions a beam rosssetion an be modeled as a stak of biaxially stressed elements with the response of eah element being predited by the MCFT. This is the basis of program Response (Bentz 2015) whih an be used to predit the shear stress distribution over the depth of the beam and the omplete load-deformation response of onrete setions subjeted to shear, flexure and axial load, see Fig. 4. Craked reinfored onrete Tee beam VeTor2 finite element model Response-2000 model CSA model Figure 4: Levels of Approximation in MCFT Analyses If only the shear strength of a beam ross-setion is required, then the web of the beam an be approximated by just one biaxial element loated at mid-depth and the shear stress on the element an be assumed to be V/(b wd v) where b w is the web width and d v is the flexural lever arm whih an be taken as 0.9d. The longitudinal strain, ε x, at mid-depth of the beam an be found from the alulated strain in the longitudinal flexural reinforement and the assumption that plane setions remain plane. For a given value of ε x the failure shear stress an then be alulated from the MCFT as the sum of two terms, V and V s, see Fig. 4. This simplified MCFT setional design model for shear (Bentz and Collins 2006) is the method used in the Canadian ode CSA A (2014).

5 The unfatored shear strength, V n, of a setion as alulated by the Canadian ode an be determined from the following equations where the unfatored failure shear stress v n is defined as V n/(b wd v) where d v an be taken as 0.9d: v n f f ot f (1) z y The aggregate interlok parameter β primarily depends on the width of the raks and size of the aggregate a g. As rak width depends on both the average tensile strain in the raked onrete and the rak spaing it is not surprising that the expression for β derived from the MCFT has a strain term and a rak spaing term. The equation is (2) x s xe The longitudinal strain at mid-depth, ε x, an be onservatively taken as one half of the tensile strain in the flexural tensile reinforement. Allowing for the tension in the longitudinal reinforement aused by the shear and assuming that there is no axial load or prestressing gives 1 M /( V dv ) x vn (3) 2E where M/V is the ratio of bending moment to shear at the setion being onsidered and ρ l is the geometri ratio of the area of longitudinal flexural tension reinforement to the shear area. That is: ρ l =A s/(b wd v). If the member ontains more than the speified minimum amount of shear reinforement (i.e. f f ) then it an be assumed that the rak spaing will be well ontrolled and z y hene s xe an be taken as 300 mm whih redues the rak spaing term in Eq. (2) to unity. If the member ontains only onentrated longitudinal reinforement then the spaing of vertial raks near mid-depth of the member, s x, is assumed to be equal to 0.9d. To allow for the influene of aggregate size the effetive rak spaing, derived from Eq.(15) in Fig.3, is taken as: 35sx sxe 0. 85sx (4) 15 a For high strength onrete the raks will go through the aggregate rather than around the aggregate partiles leading to smoother rak surfaes with less aggregate interlok apaity. To aount for this, if f exeeds 70 MPa the term a g in Eq. (4) is taken as zero. As f goes from 60 MPa to 70 MPa, a g is linearly redued to zero. As an additional allowane for the low aggregate interlok apaity of high strength onrete the term f in Eq. (1) is not allowed to exeed 8 MPa. The MCFT predits that the angle of the prinipal ompressive stress, θ, at shear failure for members with shear reinforement depends primarily upon the longitudinal strain, ε x as: g s x (5)

6 DESIGN OF THE SLAB STRIP SPECIMEN The large slab strip speimen, alled PLS4000, was designed so it would fail first in the 12 m long east shear span not ontaining shear reinforement, see Fig. 5. With a simple span of 19 m the speimen was loaded by its own substantial self-weight, 24.4 kn/m, and an off-entre point load. The nine high strength steel bars ating as flexural tension reinforement had a total yield fore of 3610 kn giving the setion a flexural apaity of about knm. Thus the magnitude of the point load to ause flexural failure of the speimen was predited to be 2730 kn. Figure 5. Details of the 4 m thik slab strip speimen PLS4000 Figure 6 summarizes the alulations involved in prediting the values of the point load, P, whih will ause shear failure of PLS4000 and of the small ompanion speimen PLS300. Equation (6) given below is the SI unit version of the basi expression for V developed in 1962 (ACI-ASCE 326) and still used in ACI (2014). The expression was derived based on the orret assumption that the failure shear stress will derease as the stress in the flexural tension reinforement inreases and by determining the oeffiients based on empirially fitting to the failure shears from 194 relatively small beams. V Vd f 17.2 w bwd (6) M For PLS4000 three setions along the east shear span are heked in Fig. 6 (a): Setion 1, d from the fae of the support; Setion 2 half-way along the shear span; and Setion 3, d from the fae of the load. As load P is inreased the moments and shears at these three setions inrease from the self-weight values to the predited failure values. It an be seen that Setion 1 has the smallest inrement of shear to ause failure giving an ACI predition for the

7 magnitude of P at shear failure of 2530 kn whih is 93% of the value of P to ause a flexural failure. For the small speimen self-weight is negligible and the ACI predition of the value of P to ause failure is kn, see Fig. 6(b). Equation (7) given below is the diretly omparable MCFT equation to the ACI Eq. (6): V 0.4 w v s x 1300 xe f b d (7) The resulting shear-moment interation diagrams shown in Fig. 6 were alulated by hoosing values of ε x alulating V from Eq. (7) and then finding the orresponding M from Eq. (3). The point load value predited to ause a shear failure of the large slab, PLS4000, is 662 kn whih is only 26% of the ACI predited shear failure value of P while for the small speimen, PLS300, the predited failure value of P is 86.2 kn or 84% of the ACI value. (a) PLS4000 (b) PLS300 Figure 6. Predited shear failure loads of 4 m thik and 300 mm thik slab strips LOADING TO FAILURE OF EAST SHEAR SPAN PLS4000 Speimens PLS4000 and PLS300 were ast on April 27 th 2015 and loading of PLS4000 began on June 10 th 2015 using a displaement ontrolled atuator. Figure 7 is the reording from the X-Y plotter used to ontrol the experiment. The plotter shows the relationship between the magnitudes of the applied point load on the Y axis against the defletion measured by an LVDT diretly below the point load. Three University of Toronto professors made load-deformation preditions and these were plotted on the graph paper prior to starting the test. First flexural raking was observed at P = 198 kn and full load stages were taken at 250 kn, 325, 500 and 625 kn. At eah load stage, the load was redued somewhat to ensure safety of the students

8 marking raks and measuring rak widths. At the end of eah day of testing the load was redued to zero overnight. The other redutions in load on the graph were aused by the formation of major new raks. As the point load reahed its maximum value of 685 kn (a value just 4% higher than the MCFT predition) a flexural rak about 5.5 m from the east support began to spread upwards, rossed mid-depth with a slope of about 45 and as it propagated towards the point load the fore applied by the displaement ontrolled jak redued to less than 500 kn indiating a typial shear failure. At 5.5 m from the east support the shear at the peak load was x 7/19 = 350 kn. After the peak load Load Stage 5 was taken and rak widths at mid-depth were found to be up to 3 mm wide. At this stage the photograph in Fig. 2 was taken. After the weekend the damaged speimen was reloaded and reahed a maximum load of only 433 kn at whih load the raks spread and widened and the point load redued to just 13 kn. Load Stage 6 was taken at this stage, see Fig.8, with rak widths being up to 35 mm wide. Note from Fig.8 that in the east shear span there are three visible raks whih ross over middepth of the large speimen and the horizontal spaings between these three raks are 0.68 d and 0.59 d. For the ompanion small speimen the spaings between the three raks that ross over mid-depth, see Fig.6 (b), are 0.73 d and 0.50 d. Thus as the depth d goes up by a fator of 14.5, the rak spaing at mid-depth goes up by about the same ratio resulting in muh wider raks in the larger member for the same tensile strain in the longitudinal reinforement. This is the prime reason for the size effet in shear. Figure 7. Load-defletion response of PLS4000 east Figure 8. East span of PLS4000 after failure

9 LOADING TO FAILURE OF WEST SHEAR SPAN OF PLS4000 In order to determine the shear apaity of the shorter west shear span whih ontained shear reinforement the failed east end of the speimen was repaired by strapping that shear span with four pairs of 36 mm diameter post-tensioned Dywidag threadbars. The point load was then inreased and the resulting load-deformation response is shown in Fig. 9. Load stages were taken when the point load reahed 1000 kn, 1375 kn, 1750 kn and 2020 kn and failure ourred with rushing of the onrete near the loading plate when P reahed 2162 kn, see Fig. 10. Figure 9. Load-defletion response of PLS4000 west Figure 10. Speimen PLS4000 after failure of short shear span with shear reinforement Note that the shear fore required to fail the shorter west shear span with widely spaed minimum shear reinforement was x 12/19 = 1512 kn whih is 1512/350 = 4.3 times the magnitude of the failure shear of the longer east shear span with no shear reinforement. PREDICTIONS SUBMITTED BY ENGINEERS Figure 11 (a) ompares the experimental result from the east shear span with the 66 preditions made by engineers who responded to the hallenge of prediting the failure load of the very thik slab. Also shown on the plot are the preditions made by six different odes. Given the large range of values shown in the figure and the almost uniform distribution of predited values aross the entire range it is evident that prediting the shear strength of very thik slabs not ontaining shear reinforement was a hallenging task for the profession. The upper red zone in the figure identifies very unonservative preditions where the ratio of

10 predited failure load to observed failure load goes from 1.5 to 5.5. The yellow band in the figure, on the other hand, indiates the gold standard predition range of plus or minus 10% from the observed strength. It an be seen that based on this measure, eight of the preditions from industry, five from aademia and three preditions from odes were exellent. While 20% of the entries were very aurate, the onern is that 44% of the entries and two of the odes were in the red zone and thus made very unonservative preditions. Figure 11(b) ompares the experimental value of the point load required to fail the west shear span with the 44 preditions made by engineers who responded to the hallenge of prediting the failure load of the speimen if the east shear span had also ontained shear reinforement. Comparing Fig. 11(a) and (b) it an be seen that while 29 of the 66 entries were in the red zone for the east shear span without shear reinforement, for the west shear span only one of the 44 entries was in the red zone. Further for the west shear span 66% of the preditions were onservative while for the east shear span only 24% were onservative. For the west shear span ten of the preditions, five from industry, four from aademia and the ACI value were within 10% of the experimental value. There are two CSA preditions shown in Fig.11 one based on setional analysis, Eq. (1), and the seond on a strut-and-tie (S&T) analysis. For this speimen the strut-and-tie estimate of failure was only 11% higher than the setional value indiating that strut ation while signifiant is not yet as dominant as it would be for a somewhat shorter shear span. (a) East Shear Span (b) West Shear Span Figure 11. Comparisons of preditions of point load P to ause failure with test results

11 THE TORONTO SIZE EFFECT SERIES FOR MEMBERS IN SHEAR In designing the large slab strip speimen and its ompanion small speimen the shear span to depth ratio, a/d, the perentage of longitudinal reinforement and the onrete strength were hosen so that the new speimens would be ompatible with previous tests at Toronto investigating the size effet in shear (Collins and Kuhma 1999, Lubell et al 2004, Sherwood et al 2010). The eleven speimens with the similar properties are desribed in Table 1 and the experimental results are ompared with the ACI and MCFT preditions in Fig. 12. It is interesting to note how well the MCFT preditions math the experimental results over the member size range where the largest speimen is 35 times as big as the smallest speimen. Table 1. Properties of speimens in Toronto size effet series. Name f MPa a g mm ρ % a/d b w mm d mm V exp kn V exp kn/m v exp MPa v mft MPa v exp v mft BN BN PLS S-10N S-10N BN BN L-10N L-10N YB2000/ PLS Figure 12. The size effet in shear for members without shear reinforement

12 HEAVILY REINFORCED COUPLING BEAMS In the design of high-rise residential buildings minimizing storey-to-storey height typially improves the eonomy of the building. Beause of this the oupling beams of the shear walls of suh buildings are often made from high strength onrete and are very heavily reinfored so that their depth an be minimized. There is onern that for suh members the over onrete may spall as failure approahes reduing the effetive width of the beam. To investigate these onerns four full sale heavily reinfored oupling beams are to be tested to failure, with the ompletion of the first of these experiments, CBF1, being shown in Fig. 13. The properties of the four beams are given in Fig. 14. Note that three of the beams exeed the maximum amount of shear reinforement permitted by the ACI ode. Figure 13. Speimen CBF1 after failure, V max = 1918 kn, right wall pushed down 70 mm. Figure 14. Properties of the four oupling beams, the ACI ode limits A v/a v,min to 11

13 The ACI ode has traditionally limited the shear apaity of a reinfored onrete setion to 10 f b wd with an upper limit of 1000 b wd (psi units). For CBF1 this upper limit orresponds to a shear fore of 1064 kn or just 56% of the experimental failure shear. The simplified MCFT of Eq. (1) on the other hand predits a failure shear of 1840 kn or 96% of the experimental failure load. Program Membrane using the full MCFT predits that the failure shear at mid-span of the oupling beam, where M=0, will be 1940 kn while program Response predits 1949 kn. The experimental load-deformation is ompared with the preditions from these two MCFT programs in Fig. 15. Note that the Membrane predition inludes only the deformation due to shear strain. The wall displaement measuring rod, rak widths and onrete splitting and spalling near peak load are shown in Fig. 16. Post peak the beam ontinued to resist substantial shear as the deformation inreased. Figure 15. Observed and predited load-deformation response of CBF1 Figure 16. Instrumentation on bak fae, wall-to-wall displaement rod.

14 CONCLUDING REMARKS This paper has summarized the shear design proedures developed at the University of Toronto over the past 40 years. These proedures are based on the Modified Compression Field Theory (MCFT) whih is apable of prediting aurately the shear behaviour of raked reinfored onrete under general biaxial loading onditions. In addition the paper has desribed two urrent researh projets one involving testing to failure of a strip from a four metre thik slab and the other involving loading full-sale oupling beams of shear walls. The paper has summarized the 2014 Canadian shear provisions based on the MCFT and shown how they an be used to alulate the shear strength for a wide variety of members and strutures. To investigate the ability of urrent design proedures to predit the shear strength of very thik slabs engineers were invited to predit the magnitude of the point loads required to fail the two shear spans. The shear strength of the longer shear span with no shear reinforement of the very thik slab was dangerously overestimated by many engineers resulting in 44% of the 66 entries prediting failure loads more than 1.5 times the experimental values and 12% submitting preditions more than three times the experimental value. The ACI predition, whih ignores the size effet, was 3.7 times the experimental value while EC2 (whih underestimates the size effet) predited 2.0 times the experimental value. It is onluded that these two influential shear design proedures an seriously overestimate the strength of very thik slabs in long shear spans not ontaining shear reinforement. The seond and more positive onlusion is that, as shown by 20% of the entries and three of the odes, exellent estimates of failure load for suh shear spans an be made. The third and perhaps most important onlusion is that adding minimum shear reinforement essentially eliminates the size effet in shear and so greatly inreases the shear strength of very thik slabs. While predited flexural apaities given by different internationally respeted design odes agree very losely, the predited shear apaities may differ by fators as high as three. The diffiulty is that shear strength is influened by many more parameters than flexural strength and most laboratory experiments have been onduted on rather small speimens with a rather narrow range of parameters. Beause of this, the traditional empirially-based shear provisions an give unsafe preditions when applied to large, lightly-reinfored members or to members made with new materials suh as high strength onrete, self ompating onrete, FRP reinfored onrete or members reinfored with high strength steel. Given the large number of existing onrete strutures designed using shear provisions now known to be unonservative, and the limited resoures available to inrease publi safety, it is essential that the shear design proedures used to evaluate these strutures be as aurate as possible. In ranking strutures most in need of repair it is equally problemati to have design methods that for some strutures are exessively onservative, suh as the ACI estimate for the oupling beam, as it is to have methods that are for some strutures dangerously unonservative, suh as the EC2 preditions for the thik slab. Currently shear provisions based on the MCFT provide the most aurate estimates of shear apaity. The low satter demonstrated by these provisions is despite, or, perhaps, beause of the fat that these shear provisions are theoretially based, rather than being fitted to the experimental database. It is hoped that the methods presented in this paper will allow better alloation of resoures to strutures that are most in need of repair.

15 REFERENCES AASHTO, (2012), LRFD Bridge Design Speifiations and Commentary, 6 th Edition, Amerian Assoiation of State Highway Transportation Offiials, Washington, 2012, 1264 pp. ACI Committee 318, (2014), Building Code Requirements for Strutural Building (ACI ) and Commentary, Amerian Conrete Institute, Farmington Hills, MI, 2014, 519 pp. ACI-ASCE Committee 326, (1962), Shear and Diagonal Tension, ACI Journal, Proeedings, V.59, No.1, 2, and 3, Jan., Feb., and Mar., 1962, pp. 1-30, , and and disussion and losure, Ot 1962 pp Bentz, E.C. and Collins, M.P., (2006), Development of the 2004 CSA A23.3 Shear Provisions for Reinfored Conrete, Canadian Journal of Civil Engineering, V. 33(5), pp Bentz, E.C. (2014), Membrane webpage last aessed 2014/02/13 Bentz, E.C. (2014), Response webpage last aessed 2014/02/13 Collins, M.P., (1978), Towards a Rational Theory for Reinfored Conrete Members in Shear, Journal of the Strutural Division, ASCE, 104(4), pp Collins, M.P. and Kuhma, D., (1999), How Safe Are Our Large, Lightly Reinfored Conrete Beams, Slabs and Footings?, ACI Strutural Journal V. 96, No. 4, July-August, pp CSA Tehnial Committee on Reinfored Conrete Design,(2014), CSA A Design of Conrete Strutures, Canadian Standards Assoiation, Mississauga, Ontario, Canada, 290 pp. European Committee for Standardization, (2004), CEN, EN :2004 Euroode 2: Design of Conrete Strutures- Part 1-1: General rules and rules for buildings, Brussels, Belgium, 225 pp. International Federation For Strutural Conrete (fib), (2013), fib Model Code for Conrete Strutures 2010, Ernst & Sohn, Lausanne Switzerland, 402 pp. Lubell, A.S., Sherwood, E.G., Bentz, E.C. and Collins, M.P., (2004), Safe Shear Design of Large, Wide Beams, ACI Conrete International, V.26, No. 1, pp Nieblas, G.M., (2014), Reahing New Heights in Los Angeles, STRUCTURE magazine, NCSEA, Sherwood, E.G., Bentz, E.C. and Collins, M.P., (2007), Effet of Aggregate Size on Beam- Shear Strength of Thik Slabs, ACI Strutural Journal, Vol. 104, No. 2, Marh-April 2, pp Vehio, F. J. and Collins, M. P.,(1986), The Modified Compression-Field Theory for Reinfored Conrete Elements Subjeted to Shear, ACI Strutural Journal, Vol. 83, No. 2, Mar.-Apr., pp Vehio, F.J., (2014), VeTor2 webpage, last aessed 2014/02/13 Walraven, J.C., (1981), Fundamental Analysis of Aggregate Interlok, Journal of the Strutural Division, ASCE, 107(ST11), pp