Article publié par le Laboratoire de Construction en Béton de l'epfl. Paper published by the Structural Concrete Laboratory of EPFL

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1 Article publié par le Laboratoire de Contruction en Béton de l'epfl Paper publihed by the Structural Concrete Laboratory of EPFL Title: The level-of-approximation approach in MC 2010: application to punching hear proviion Author: Muttoni A., Fernández Ruiz M. Publihed in: Volume: Page: Structural Concrete Vol. 13 pp Year of publication: 2012 Type of publication: EPFL InfoScience link: Peer reviewed journal article Pleae quote a: Muttoni A., Fernández Ruiz M., The level-of-approximation approach in MC 2010: application to punching hear proviion, Structural Concrete, Vol. 13, 2012, pp [Muttoni12] Downloaded by infocience ( on :03

2 Article Aurelio Muttoni Miguel Fernández Ruiz* DOI: /uco The level-of-approximation approach in MC 2010: application to punching hear proviion In order to addre how new knowledge influence deign expreion, deign code have in mot cae become ignificantly more complex over the lat decade. However, thi tendency i leading to code that are too complicated for preliminary deign but till not ufficiently accurate for aeing exiting tructure (where even more realitic model of behaviour are ometime required). An alternative code trategy i that propoed by code baed on a level-of-approximation (LoA) approach. Thi approach i baed on the ue of theorie baed on phyical parameter where the hypothee for their application can be refined a the accuracy required increae. The approach propoe adopting afe hypothee during the firt tage of deign, leading to relatively quick and imple analye. In cae where uch a degree of accuracy i not ufficient (e.g. deign of complex tructure, aement of exiting tructure, ignificant potential economic aving), the hypothee can be refined in a number of tep, leading to better etimate of the behaviour and trength of member. Thi approach, recently adopted in the firt complete draft of Model Code 2010 for a number of deign iue, i dicued within thi paper with reference to punching hear proviion. Keyword: level-of-approximation approach, deign code, Model Code 2010, aement of tructural afety, critical hear crack theory 1 Introduction * Correponding author: miguel.fernandezruiz@epfl.ch Submitted for review: 12 July 2011 Revied: 20 September 2011 Accepted for publication: 20 September 2011 Mot experienced engineer have alway tackled the problem of deigning new tructure or aeing the trength of exiting one by following a level-of-approximation (LoA) approach. Thi i rather intuitive a the limit to the trength of a tructure can uually be calculated with fairly imple model. Provided that uch model are baed on ound theorie, ome of their phyical parameter (e.g. angle of compreion trut or effectivene factor) can be better etimated by devoting more time to their analyi, leading to improved (typically higher) etimate of the trength of a member. However, thi deign trategy ha not alway been reflected in code of practice, where in many intance it i not poible to refine the parameter ued in their deign expreion. Thi i typically the cae with empirical formula baed on geometrical dimenion and material propertie, but not on phyical parameter. A a conequence, code eldom overrule mot apect of deign (which i uually time-conuming and leave little opportunity for deigner to ue advanced tate-of-the art deign method) or are exceively open (which might be dangerou in the hand of inexperienced deigner). In order to addre the influence of new knowledge on deign expreion uitably, code are alo increaing in complexity (particularly when empirical model are ued). Thi tendency i leading to code that are ometime too complicated for preliminary deign but till not ufficiently accurate for aeing exiting tructure (where even more realitic model of behaviour are ometime required). In many countrie thi i giving rie to a debate on the need for concie code for deigning imple tructure and aement code for exiting tructure. However, their deign model are not alway conitent and thi lead to confuion for deigner. An alternative code trategy i that propoed by the LoA approach [1, 2], ee Fig. 1. Thi approach propoe uing theorie baed on phyical model. When preliminary etimate of the trength of a member are required, the mechanical parameter of the deign expreion can be aeed in a imple (yet afe) manner. Thi allow the limit of trength to be determined even though very little time need to be devoted to the analye, which i normally ufficient for preliminary deign purpoe and even for many tructural member without a given governing failure mode. However, in cae where uch a degree of accuracy i not ufficient (e.g. critical element, detailed deign), the value of the mechanical parameter can be refined in a number of tep. Thi mean devoting more time to analye, ee Fig. 1, but lead to better etimate of accuracy IV III II I time devoted to analyi level of approximation Fig. 1. Level-of-approximation approach: accuracy of the etimate a a function of the time devoted to analye (adapted from [1]) Ernt & Sohn Verlag für Architektur und techniche Wienchaften GmbH & Co. KG, Berlin

3 the behaviour and trength of member. The LoA approach thu allow the preliminary deign phae a well a advanced deign and aement to be covered with the ame et of expreion. With repect to the deign of new tructure, thi approach allow for a gradual increae in the accuracy of the analye a the project evolve from preliminary deign tudie to a contruction project. Thi help to expend the neceary time at each deign tage. It alo allow a refined deign to be carried out for unuual element with pecial ignificance regarding the afety of the tructure (e.g. dicontinuity region or coupling member). And the incremental approach i alo very convenient when aeing exiting tructure which, even if they were correctly deigned according to code of practice at the time they were built, might not comply with current code recommendation due to change in load or more tringent code proviion. Thi doe not mean, however, that uch tructure are unafe. Deign rule are provided to cover a erie of uncertaintie and to be applied to a wide number of cae, although they might be exceively conervative in ome ituation. In thee cae, the ue of more refined analyi method to ae the tructural afety i fully jutified (even if they are more timeconuming) a expenive trengthening can be avoided. The LoA approach wa formally preented [1] and implemented for the calculation of econd-order effect in the Swi Code for tructural concrete [3] in Recently, it ha alo been conidered in the firt complete draft of Model Code 2010 [4, 5] for hear, punching hear and buckling deign. In thi paper, the fundamental of thi approach are preented with reference to the punching hear proviion of Model Code A practical example of the ue of thi approach i alo introduced, helping the reader to undertand the increae in accuracy expected a higher level of approximation are employed. (a) (b) 2 The LoA approach for punching hear in Model Code 2010 Punching hear ha been a topic of reearch in tructural concrete ince the The firt rational approach to punching hear deign wa developed in Sweden by Kinnunen and Nylander [6]. Thi approach uccefully explained the behaviour and trength of punching hear in flat lab without tranvere reinforcement. Although the approach of Kinnunen and Nylander wa rather atifying, it reulted in omewhat complicated deign expreion. A a conequence, it implementation in code of practice wa difficult and currently mot code of practice both in Europe [7] and in America [8] are till baed on empirical expreion without a phyical bai for the punching hear deign of member without tranvere reinforcement. In order to provide a better undertanding of the phenomenon, intenive reearch ha been performed in recent decade. A detailed tate of the art report and comparion of approache can be found in pecialized publication [9, 10] and reearch work [11, 12]. Following thee invetigation, and contrary to previou edition of the Model Code, the punching hear proviion in Model Code 2010 are baed on a phyical theory rather than on empirical formula. The theory behind MC (c) Fig. 2. Critical hear crack developing through the compreion trut: (a) location of trut and critical hear crack [12], (b) failure envelope for reinforced concrete lab a a function of lab rotation (reult for pecimen with effective depth of mm, flexural reinforcement ratio of %, concrete trength of MPa, aggregate ize of 8 32 mm and column diameter of mm), and (c) comparion of failure band and reult from 99 punching hear tet [12] 2010 proviion i critical hear crack theory (CSCT). The baic principle of CSCT with repect to punching hear deign were developed by Muttoni and Schwartz in 1991 [13] and were later refined and extended to hear deign of one-way member by Muttoni [14]. A erie of recent experimental and theoretical work have provided jutifica- 33

4 tion for it mechanical model [12, 15, 16] and have alo extended it ue to member failing in hear after developing platic train in the flexural reinforcement [17, 18], hearreinforced lab [19, 20] and other topic. An extended ummary of recent development and application can be found elewhere [21]. 2.1 Mechanical model of CSCT Critical hear crack theory i baed on the aumption that the hear trength in member without tranvere reinforcement i governed by the width and roughne of a hear crack that develop due to the inclined compreion trut carrying hear [12, 15], ee Fig. 2a. The hear trength reulting from thi aumption can be calculated by auming two rigid bodie with kinematic at failure characterized by the rotation of the lab (developed in agreement with tet meaurement [16]). Auming uch kinematic, both tenile tree and tree due to aggregate interlock (relative lip between the lip of the crack) develop along the critical hear crack. The hear trength can thu be calculated by integrating both contribution (concrete in tenion and aggregate interlock) along the failure urface (dowel action i neglected due to palling of the concrete cover to the flexural reinforcement, Fig. 2a). Fig. 2b how the reult obtained by performing uch integration, uing the numerical approach detailed in [16] and for ignificant variation of the mechanical parameter implied. The plot i normalized on both axe to account for upport region ize, concrete compreive trength, member depth and aggregate ize. It can be een that the punching hear trength decreae a rotation and effective depth increae (leading to greater opening of the critical hear crack). Thi i logical becaue wider crack reduce both the concrete in tenion and aggregate interlock contribution. It i alo intereting to note that failure occur in a well-defined and rather narrow band for all cae (Fig. 2b). Comparing the failure region with the reult of 99 punching hear tet (whoe data i detailed elewhere [12]) i hown in Fig. 2c, howing a very atifactory agreement. For deign purpoe, and taking into account the narrow width of the failure band, detailed calculation of the failure envelope by integrating concrete in tenion and aggregate interlock contribution i not uually neceary. For thee cae, a implified failure criterion wa propoed by Muttoni [14, 12]. It aume that the punching hear trength (traditionally correlated to the quare root of the concrete compreive trength after the work of Moody et al. [22]) i a function of the width and roughne of a hear crack a jutified by the previou mechanical model: VR b d 0 where: V R b 0 v = f f w, d (1) c ( g) hear trength hear-reiting control perimeter (et at d v /2 of the edge of the upport region auming a uniform ditribution of hear force) Fig. 3. Comparion of failure region (Fig. 2c) and average and characteritic CSCT failure criteria d v f c w d g hear-reiting effective depth of member (ditance between centroid of flexural reinforcement and the urface at which the lab i upported) compreive trength of concrete width of critical hear crack maximum ize of aggregate (accounting for the roughne of the lip of the crack) In order to evaluate the width of the critical hear crack w, Muttoni and Schwartz [13] aumed it to be proportional to the lab rotation ψ multiplied by the effective depth of the member (ee Fig. 2a): w Baed on thee aumption, the following failure criterion wa propoed by Muttoni [14, 12] for member without hear link and auming average value for the trength: where: d g0 d d VR 3/4 = b0 d f d v c d + d g0 reference aggregate ize equal to 16 mm i to be introduced in [mm]. g In Fig. 3 thi equation i compared with the failure band calculated on the bai of the mechanical model, howing good agreement. For deign purpoe, a characteritic failure criterion ha to be adopted (target 5 % fractile, refer to Muttoni [12] and Model Code 2010 [5]), ee Fig. 3: 0 V v Rd b d f ck c 1 = dk 0.6 where k dg i a coefficient accounting for the maximum aggregate ize d g, whoe value can be calculated a k dg = 48 [mm]/(16+d g ). dg (2) (3) (4) 34

5 (a) Fig. 4. Calculation of failure point according to CSCT: interection between failure criterion and load rotation curve 2.2 Calculation of failure load The punching hear trength of a lab without hear reinforcement can be directly calculated uing the CSCT failure criterion. To do o, the interection between the failure criterion and the actual behaviour of the lab (characterized by it load-rotation curve) ha to be calculated, ee Fig. 4. It hould be noted that thi procedure allow not only the calculation of the punching trength but alo the etimation of the deformation capacity (rotation) at failure. Thi provide the deigner with valuable information on the behaviour of the tructure (e.g. ductility, brittlene). Moreover, the rotation (a an etimate of the hear crack opening) can be ued to calculate the activation of the tranvere reinforcement for hear-reinforced lab [19] or to etimate how fibre contribute to punching hear trength [21], accounting for their oftening behaviour. 2.3 Application to hear-reinforced lab The theory can alo be conitently applied to hear-reinforced lab. A number of potential failure mode can develop [19, 20] uch a: punching within the hear-reinforced area, punching outide the hear-reinforced area, cruhing of concrete trut, delamination of concrete core, hear reinforcement pull-out, flexural failure. Detail of the way thee failure mode can be treated within the frame of CSCT are invetigated in depth elewhere [19, 20, 21]. Of particular ignificance i the failure mode by punching within the hear-reinforced zone, ee Fig. 5a. The trength in thi cae depend on the contribution of the concrete and the tranvere reinforcement: V = V + V Rd Rd, c Rd, Thi fact ha been acknowledged by mot deign model. However, mot code of practice till propoe empirical formulation for etimating the contribution of the two term. For intance, a contant reduction in the concrete contribution with repect to the trength of member without hear reinforcement i provided for in EC-2 (25 %) [7] and ACI [8] (50 %). Thee code alo give empirical formula or contant value for the tre in the hear reinforcement. The CSCT approach i, however, rather different and take advantage of the phyical hypothee of the theory. Thi approach can be undertood with the help of Fig. 5a, (5) Fig. 5. Slab with tranvere reinforcement: (a) activation of hear reinforcement by critical hear crack, (b) concrete and hear reinforcement contribution, and (c) um of concrete and hear reinforcement contribution a a function of lab rotation which how that the tranvere reinforcement i activated a the critical hear crack open (Fig. 5b). Thi mean that the tree in the reinforcement increae until they eventually reach their yield trength. On the other hand, the concrete contribution to the trength decreae with the opening of the hear crack (Fig. 3). Thi i conitent with the aumption of empirical code but allow the calculation of a uitable reduction in the contribution of the concrete (V Rd,c /V Rd,c0 ratio in Fig. 5c) for each pecific cae on the bai of the rotation (deformation capacity) at failure. With repect to the activation of the hear reinforcement, uitable analytical law have been derived elewhere [19, 20] a a function of the lab rotation (correlated to the critical hear crack opening) and bond condition of the reinforcement. A code-like formulation of thee model [19, 26], accounting for bond and inclined reinforcement, ha recently been introduced in the firt complete draft of Model Code [5]. In it general formulation, thi equation i a follow: where: α f bd wd f ywd φ w E f = (in + co ) in + 6 f bd ywd d f w ywd angle between lab axi and hear reinforcement value of bond trength (which for deign purpoe can be adopted a 3 MPa for ribbed bar) yield trength of hear reinforcement diameter of hear reinforcement (b) (c) (6) 35

6 2.4 Calculating the load-<rotation behaviour uing an LoA approach Variou method can be ued to etimate the load rotation behaviour neceary to calculate the punching hear trength (Fig. 4 and 5c). Model Code 2010 propoe both implified formula derived on the bai of the analytical formula [12] and numerical procedure. With repect to the analytical formula, the general form of the load rotation relationhip propoed by MC 2010 [5] i: r f 1.5 yd m = d d E m (7) where: r ditance from column axi to line of contraflexure of radial bending moment f yd yield trength of flexural reinforcement E modulu of elaticity of flexural teel m d average moment per unit length for calculating flexural reinforcement in upport trip m Rd average flexural trength per unit length in upport trip The value of the variou mechanical parameter in the formula can be aeed with different degree of accuracy, leading to the level-of-approximation (LoA) approach LoA I For preliminary deign purpoe, a afe hypothei can be adopted by auming m d = m Rd. Thi implie that, at failure, bending reinforcement yield over the entire width of the upport trip, thu leading to large crack opening (which decreae the punching hear trength). Thi therefore lead to a afe etimate of the trength becaue if punching trength atifie thi condition, the trength of the lab will be governed by it bending capacity. Furthermore, lab atifying thi condition exhibit a very ductile behaviour, avoiding brittle failure problem. The load-rotation equation i therefore: 1.5 r f = d E where for regular flat lab the value of r can be calculated on the bai of the pan length a r 0.22 (calculated baed on the geometry of the tructure) LoA II yd Rd 1.5 Approximation level II i a implified etimate of the moment acting in the upport trip m d. Thi i carried out with an analytical expreion relating the moment in the upport trip to the hear force acting V Ed and the moment tranferred from the lab to the upport region (characterized by it eccentricity e u ). For intance, for inner column of flat lab: (8) m d where: 1 e = V u Ed + 8 2b V Ed /8 average moment for calculating flexural reinforcement acting in upport trip without moment tranfer V Ed e u moment tranferred to column 2b width where thi tranferred moment act (half the moment acting on each ide of the column) In pite of it implicity, thi expreion provide excellent etimate of the load-rotation behaviour of a lab a hown elewhere [12, 21]. For intance, the ratio between the meaured and calculated punching hear trength for the tet in Fig. 2c i 1.07, with a coefficient of variation of 9 % (ignificantly better than mot deign code [12]) LoA III If a linear-elatic analyi i performed for deigning the flexural reinforcement in a flat lab, the reulting moment field can be ued to improve the etimate of the mechanical parameter of Eq. (7). Thi i, for intance, imple for the value of r and m d, where bending and torion moment can be integrated directly for the latter. In thi cae, and due to the better etimate of the variou parameter, coefficient 1.5 in Eq. (7) can be replaced by 1.2 (leading to tiffer behaviour and thu to higher trength for equal mechanical parameter): r f 1.2 yd m = d d E m LoA IV Rd 1.5 (10) In ome pecial cae, the load-rotation behaviour of a flat lab can be invetigated by integrating the moment-curvature diagram of the tructure directly. Thi i jutified, for intance, if expenive trengthening can be avoided by performing a more refined analyi. However, it hould be borne in mind that uch analye are very time-conuming and that the accuracy of LoA III i already very atifactory. Significant improvement in the trength by uing thi level hould only be expected for lab with fairly low reinforcement ratio over column (with ignificant teniontiffening effect) or when large reditribution of bending moment between column and mid-pan region are expected. Procedure for the numerical integration have been preented elewhere [23, 24, 26]. Many commercial oftware package alo provide tool for uch analye. Thee method are, however, generally enitive to the choice of the parameter involved and hould be applied by experienced uer who have previouly checked the reult of the numerical imulation againt actual tet. (9) 36

7 (a) (c) (b) Fig. 6. Example of application: (a) geometry and region invetigated, (b) cro-ection over column, and (c) governing load cae 3 Choice of a uitable LoA The choice of a uitable level of approximation depend motly on the context of the analyi performed (preliminary or detailed calculation) and on the potential aving that can be achieved if a more refined level of approximation i performed. In general, thi choice i a deciion that belong to the deigner, but ome objective criteria to guide thi choice are provided below. LoA I provide imple and afe hypothee for evaluating the phyical parameter of deign equation. It i quick and imple and thu i in mot cae ufficient for preliminary deign purpoe. Another ignificant ue of LoA I i to check whether a given failure mode cannot govern. Thi i the cae for tructure where ufficient trength i provided even under the afe aumption of LoA I. In uch cae, it i unneceary to perform further analye uing more accurate level of approximation. For more accurate level of approximation, the phyical parameter of deign equation are evaluated through implified analytical formula. Thee level are again quick and are uually ufficient to cover mot deign cae. Their ue i advied for tender and detailed deign of mot new tructure a well a for aeing exiting tructure. A the mot refined level of approximation, numerical method can be ued to etimate the value of the phyical parameter conidered by the deign equation (numerical integration of the moment-curvature diagram of the tructure). The ue of uch level i typically very time-conuming and only advied for the detailed deign of very complex tructure or for the aement of critical exiting tructure. Thi i jutified when a more accurate etimate can lead to ignificant aving for the client (avoiding or limiting the trengthening of tructure). 4 Example of application Thi ection explain the ue of the level-of-approximation approach with the help of a practical example. It conit on the aement of the trength of an exiting flat lab built in the 1970 in Switzerland. The lab ha a contant thickne of 0.52 m and i upported on a erie of wall around it periphery and by two inner column (with a diameter of 1.20 m at the lab upport point), ee Fig. 6a. Concrete compreive trength wa updated by tet to a value of f ck = 59.4 MPa (accounting for long-term effect) and the maximum aggregate ize i 32 mm. The lab ha large amount of flexural reinforcement in the column region, with average ratio in the upport trip of ρ = 1.7 % in the x direction and ρ = 1.4 % in the y direction. The reinforcement yield trength i f yd = 390 MPa. The lab alo ha a number of bent-up bar (16 ection of bar 16 mm diameter inclined at 45 and interected by the conical punching failure urface). It i ubjected to elf-weight, earth cover and traffic load. The geometry and governing load cae for punching hear trength are hown in Fig. 6a and 6b. The tructure wa analyed and eventually trengthened following the guideline of MC In thi paper, only the evaluation of it punching hear trength will be dicued. 37

8 (a) (b) Fig. 7. Shear field analyi: (a) hear field of flat lab for governing load cae, and (b) ditribution of hear force along control perimeter 4.1. LoA I A preliminary etimate of the punching hear trength can be obtained uing LoA I. The total hear acting on the control perimeter can be obtained from a linear analyi of the tructure (column reaction minu force acting within control perimeter) a V Ed = 5.54 MN. The minimum trength correpond in thi cae to the direction of maximum rotation (maximum pan length), which can be calculated from Eq. (8): r f yd max = = 1.5 = d E % For etimating the trength, the hear-reiting control perimeter b 0 ha to be reduced with repect to the total perimeter available at a ditance d v /2 from the border of the upport region (deignated the baic control perimeter b 1 in Model Code 2010) to account for concentration on the hear field. Thi can be done by applying the following relationhip [5]: b = k b 0 e 1 (12) Approximated value for the coefficient of eccentricity k e are provided in Model Code 2010 [5]. However, a tated in the code, due to the preence of ignificant concentrated load in the vicinity of the upport region, larger hear concentration than thoe conidered by the value of Model Code 2010 can occur. Thu, intead of a value k e = 0.90 (correponding to inner column of flat lab ubjected to ditributed load according to Model Code 2010) a afer value k e = 0.80 will be adopted [25]. Thi choice will be dicued and refined below. The concrete and reinforcement contribution can thu be directly calculated for the governing rotation: b0 dv fck / c VRd = VRd, c + VRd, = d k max + A k in = = 3.50 w e wd (13) where the hear reinforcement contribution can be calculated on the bai of Eq. (6), leading to σ wd = 390 MPa (yield trength). In thi cae, other potential failure mode dg + (11) (cruhing of concrete trut, punching at other perimeter) are not governing. The compliance factor (n = V Rd /V Ed ) i then 0.63, which can be conidered a very low, adviing urgent meaurement. 4.2 LoA II A better etimate of the punching hear trength can be obtained by uing the reult of a linear-elatic analyi. Coefficient k e can be etimated on the bai of the hear field of the tructure around the column region, ee Fig. 7. Thi figure how concentration of the hear field near the concentrated load to the left of the column (where heavier concentrated load are applied). In uch a cae, according to Model Code 2010, the coefficient of eccentricity can be calculated a: V 1 ke = d = 0.78 v b perp, d,max 1 (14) where v perp,d,max correpond to the maximum value of the hear force component per unit length perpendicular to the baic control perimeter, ee Fig. 7b. Thi value how that the previou etimate (level I) for the coefficient of eccentricity wa already rather good. In LoA II, the load rotation curve are calculated baed on Eq. (7) and etimating m d according to the expreion i provided by level II of MC 2010, m d = V Ed (1/8 + e u /(2b )). The parameter required, r x (= 0.22 x = 2.51) and r y (= 0.22 y = 2.22), can be obtained by conidering the geometry. The reult are plotted in Fig. 8a. In thi figure, the concrete trength (V Rd,c, etimated according to Eq. (4)), the contribution of the hear link (V Rd,, etimated according to Eq. (6)) and the load-rotation behaviour of the lab (etimated according to Eq. (7) and (9)) are plotted a a function of the rotation in the x and y direction (maximum punching hear trength i not governing). The reult yield V Rd = 4.22 MN, with the y direction governing. The compliance factor (n = V Rd /V Ed ) i 0.76, till rather low but not o critical a in LoA I. It can be een that in thi cae the tre in the tranvere reinforcement i σ wd = 271 MPa, which i below the yield trength (contrary to the outcome of LoA I). Thi i 38

9 be een that the tructure will mot probably not have ufficient trength (V Rd < V Ed ) even if more refined level of approximation are ued (the trength of the lab i only higher than the action for very limited rotation). However, the analyi with higher level of approximation can till be intereting for the optimization of the potential trengthening [20]. Thi wa the cae in thi example, which jutified more in-depth analye by uing higher LoA. 4.3 LoA III ψ ψ (a) (b) Taking advantage of the linear analyi performed, value of m d and r can be etimated in a more accurate manner from the moment field a decribed in MC The value of r are 2.48 and 2.61 m for the x and y direction repectively (fairly in agreement with the etimate of LoA II). Integrating the bending moment lead to the reult hown in Fig. 8b. The reult provide a lightly higher trength prediction, although with a limited gain. The punching hear trength i V Rd = 4.35 MN, leading to a compliance factor of n = LoA IV Finally, a refined analyi of the load-rotation behaviour wa performed uing a non-linear finite element model. The model wa local (lab between mid-pan axe) and conited of 32 region where the different reinforcement near the column region, the upport trip and the field where conidered. The reult are hown in Fig. 8c, where it can be een that the reult i only a limited increae in the trength, with a value of V Rd = 4.46 MN leading to a compliance factor of n = Thi reult i conitent with other work [12], howing that the MC 2010 expreion for LoA II and III are quite accurate for the flexural behaviour of lab with large amount of bending reinforcement and afer for lab with low amount of flexural reinforcement (where tenion-tiffening effect can play a ignificant role). 4.5 Comment on the reult ψ Fig. 8. Reult for the variou level of approximation: (a) LoA II, (b) LoA III, and (c) LoA IV (c) due to the fact that the rotation at failure (ψ = 0.64 %) i maller than that etimated in LoA I. However, although the contribution of the hear reinforcement decreae, the total trength increae becaue the concrete contribution i larger for maller rotation. From the plot of Fig. 8a, it can alo The reult obtained confirm that the etimate of the accuracy of the punching hear trength can be refined progreively by uing the level-of-approximation approach. Thi i poible becaue a better etimate of the phyical parameter of the deign model i provided in each ubequent level of approximation (LoA). LoA I and II can be performed in jut a couple of minute. LoA III required more than twice that time. LoA IV, a performed in thi example, took ome day. The gain in the etimated trength i, however, limited, and LoA II or III are uually ufficient for detailed deign and aement. In thi cae (due to the complexity of the location of the flat lab) all level were performed in order to optimize the amount of hear reinforcement and the method to be ued during retrofitting. 5 Concluion Thi paper ha dicued the main idea of the level-of-approximation (LoA) approach for deigning and aeing 39

10 tructure. It ue in the new Model Code 2010 i alo explained with reference to the punching hear chapter and an example of an application to a real tructure i preented. The main concluion of the paper are: 1. The LoA approach i baed on the idea that conitent (phyically ound) theorie hould be ued for deign. The variou mechanical parameter ued within thee theorie can be etimated uing different degree of accuracy. 2. When little work i devoted to the analyi (firt level of approximation), afe (yet realitic) value of the trength and behaviour of tructural member hould be provided by thi approach. 3. The accuracy can be increaed thereafter by performing additional analye. Thi allow a better etimation of the phyical parameter required by the deign equation. 4. Such an approach i convenient for deigning or aeing tructure: a. With repect to deign, it allow an increae in the accuracy of the analye a the project evolve from a conceptual deign to a contruction project. b. With repect to the aement of exiting tructure, it allow the refinement of ome hypothee adopted for deign (typically afe and introduced to cover a broad range of cae). In thee cae, devoting a ignificant amount of time to the analye i certainly jutified if expenive (and unneceary) trengthening can be avoided. 5. The firt complete draft of Model Code 2010 propoe following the LoA approach with repect to hear, punching hear and buckling. Conitent deign model have been obtained, encouraging the extenion of the LoA approach to other domain. Acknowledgement The author have implemented the idea of the LoA approach into Model Code 2010 within the cope of fib Tak Group 4.2. The author would like to thank the other member of the editorial group of TG 4.2 (Evan Bentz, Stephen Foter and Viktor Sigrit) a well a Joot Walraven. The author are alo appreciative of the comment and uggetion of Karel Thoma and Bruno Zimmerli (i-beratung GmbH, Switzerland) for the example preented within thi paper. Notation A w E V V Ed V R V Rd V Rd,c V Rd,max V Rd, b 0 cro-ectional area of hear reinforcement modulu of elaticity of reinforcement hear force deign value of hear force acting punching hear trength deign punching hear trength deign concrete contribution to punching hear trength maximum punching hear trength deign hear reinforcement contribution to punching hear trength hear-reiting control perimeter b 1 b d v baic control perimeter trip width hear-reiting effective depth d effective depth d g maximum diameter of aggregate d g0 reference aggregate ize (16 mm) e u load eccentricity with repect to centroid of baic control perimeter f bd deign bond trength f c average compreive trength of concrete (cylinder) f ck characteritic compreive trength of concrete (cylinder) f yd deign yield trength of flexural reinforcement f ywd deign yield trength of hear reinforcement k dg coefficient for aggregate ize k e coefficient of eccentricity pan length m d average moment per unit length (deign of flexural reinforcement) in trip m Rd average flexural trength per unit length in upport trip n compliance factor (= V Rd /V Ed ) r ditance between column and line of contraflexure of moment v perp,d,max maximum hear force perpendicular to baic control perimeter w critical hear crack opening α angle between lab axi and hear reinforcement γ c partial afety factor for concrete φ w diameter of hear reinforcement ρ flexural reinforcement ratio σ wd deign tre in hear reinforcement ψ rotation of lab outide column region ψ max maximum rotation of lab outide column region Reference 1. Muttoni, A.: Introduction to SIA 262 code (in French: Introduction à la norme SIA 262), Documentation SIA, D 0182, Zürich, Switzerland, 2003, pp Muttoni, A., Fernández Ruiz, M.: Deign through an incremental approach: the Swi experience, 2010 Joint IABSE-fib Conference, Dubrovnik, Croatia, 2010, p SIA. Code 262 for Concrete Structure, Swi Society of Engineer and Architect, Zürich, Switzerland, 2003, p Fédération Internationale du Béton (fib), Model Code 2010 Firt complete draft, fédération internationale du béton, Bulletin 55, Lauanne, Switzerland, 2010, vol. 1, p Fédération Internationale du Béton (fib), Model Code 2010 Firt complete draft, fédération internationale du béton, Bulletin 56, Lauanne, Switzerland, 2010, vol. 2, p Kinnunen, S., Nylander, H.: Punching of Concrete Slab Without Shear Reinforcement, Tranaction of the Royal Intitute of Technology, No. 158, Stockholm, Sweden, 1960, p Eurocode 2, Deign of concrete tructure Part 1-1: General rule and rule for building, CEN, EN , Bruel, Belgium, 2004, p ACI, Building Code Requirement for Structural Concrete (ACI ) and Commentary (ACI 318R-08), American Concrete Intitute, Farmington Hill, Mich., USA, 2008, p

11 9. fib TG 4.3, Punching of tructural concrete lab, Bulletin 12, fédération internationale du béton, Lauanne, Switzerland, 2001, p Polak, M. A.: Punching Shear in Reinforced Concrete Slab, American Concrete Intitute, Special Publication SP-232, Farmington Hill, Mich., 2005, p Hegger, J., Häuler, F., Ricker, M.: Critical review of the punching hear proviion according to Eurocode 2 (in German: Zur Durchtanzbemeung von Flachdecken nach Eurocode 2), Beton- und Stahlbetonbau, vol. 103, No. 2, 2008, pp Muttoni, A.: Punching hear trength of reinforced concrete lab without tranvere reinforcement, ACI Structural Journal, vol. 105, No. 4, 2008, pp Muttoni, A., Schwartz, J.: Behaviour of Beam and Punching in Slab without Shear Reinforcement, IABSE Colloquium Stuttgart, vol. 62, IABSE, Zurich, Switzerland, 1991, pp Muttoni, A.: Shear and punching trength of lab without hear reinforcement, (in German, Schubfetigkeit und Durchtanzen von Platten ohne Querkraftbewehrung ), Beton- und Stahlbetonbau, vol. 98, 2003, pp Muttoni A., Fernández Ruiz, M.: Shear trength of member without tranvere reinforcement a function of critical hear crack width, ACI Structural Journal, vol. 105, No. 2, 2008, pp Guidotti, R.: Punching of flat lab ubjected to very large column loading (in French: Poinçonnement de plancherdalle avec colonne uperpoée fortement ollicitée), PhD thei, École Polytechnique Fédérale de Lauanne, Switzerland, 2010, p Guandalini, S., Burdet, O., Muttoni, A.: Punching tet of lab with low reinforcement ratio, ACI Structural Journal, vol. 106, No. 1, 2009, pp Vaz Rodrigue, R., Muttoni, A., Fernández Ruiz, M.: Influence of hear on the rotation capacity of R/C platic hinge, American Concrete Intitute, Structural Journal, vol. 107, No. 5, 2010, pp Fernández Ruiz, M., Muttoni, A.: Application of the critical hear crack theory to punching of R/C lab with tranvere reinforcement, ACI Structural Journal, vol. 106, No. 4, 2009, pp Fernández Ruiz, M., Muttoni, A., Kunz, J.: Strengthening of flat lab againt punching hear uing pot-intalled hear reinforcement, ACI Structural Journal, vol. 107, No. 4, 2010, pp Muttoni, A., Fernández Ruiz, M.: MC2010: The Critical Shear Crack Theory a a mechanical model for punching hear deign and it application to code proviion, fédération internationale du béton, Bulletin No. 57, 2010, pp Moody, K. G., Viet, M., Eltner, R. C., Hognetad, E.: Shear Strength of Reinforced Concrete Beam Part 1: Tet of Simple Beam, ACI Journal, Proceeding vol. 51, No. 4, 1954, pp Muttoni, A. (ed.), Fernández Ruiz, M., Fürt, A., Guandalini, S., Hunkeler, F., Moer, K., Seiler, H.: Structural afety of parking garage (in French: Sécurité tructurale de parking couvert), Doc. D 0226 SIA, Société Suie de ingénieur et de architecte, Zurich, Switzerland, 2008, p Vaz Rodrigue, R.: Shear Strength of Reinforced Concrete Bridge Deck Slab, Thèe EPFL, No. 3739, Lauanne, Switzerland, 2007, p Muttoni, A., Fernández Ruiz, M., Guandalini, S.: Punching of lab bridge (in French: Poinçonnement de pont-dalle), 4. FBH / ASTRA tudy conference Neue au der Brü - ckenforchung, Doc. D0223 SIA, Societé uie de ingénieur et architect, Zurich, Switzerland, 2007, pp Tainari, L.: Aymmetric punching of R/C lab with hear reinforcement (in French : Poinçonnement aymétrique de dalle en béton armé avec armature de poinçonnement), Thèe EPFL No. 5030, Lauanne, Switzerland, 2011, p Prof. Dr. Aurelio Muttoni Ecole Polytechnique Fédérale de Lauanne ENAC Station 18 Lauanne CH-1024 Switzerland Dr. Miguel Fernández Ruiz Ecole Polytechnique Fédérale de Lauanne ENAC Station 18 Lauanne CH-1024 Switzerland 41