# Shear and Punching Capacity Predictions for One-Way Slabs under Concentrated Loads Considering the Transition between Failure Mechanisms

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## Abstract

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## 1. Introduction

_{eff}), contributes to the one-way shear capacity (Figure 1b). In practice, this means that the load effect from each loading axle can be distributed over a certain length (the effective shear width) to calculate the total shear stress at the control section (Figure 1b). In assessing existing structures, the actual shear demand v

_{E}is compared to the nominal shear resistance v

_{R}to define if the structure satisfies the requirements of one-way shear resistance. Figure 1b shows the approach frequently named the French load-spreading method resulting in the French effective shear width [3,4], which assumes that the load is spread horizontally from the load back sides with a fixed angle of 45 degrees.

_{test}for the achieved sectional shear in the test and V

_{R}for the predicted sectional one-way shear capacity). In the last approach, the effective shear width b

_{eff}is frequently multiplied by the unitary shear resistance (v

_{R}) in comparing tested and predicted resistances.

_{v}/d

_{l}≤ 2) or when the slabs fail in one-way shear as wide beams [13]. However, this approach presents the shortcoming that the effective shear increases excessively as the distance from the load to the support a

_{v}increases. In practice, this causes the predicted one-way shear capacity to frequently overestimate the tested resistance when the loads are placed far from the support [7,10,13,14]. Until the last years, this has not been considered a significant deficiency because the most critical position of the design tandem for one-way shear was always considered close to the support (a

_{v}= 2d

_{l}). In practice, this occurs because arching action improves the shear capacity for a distance a

_{v}≤ 2d

_{l}and because placing the load close to the support increases the shear demand at the support v

_{E}(Figure 2a,b). However, with the advancing understanding of the one-way shear behavior [15,16,17], this argument is subject to discussion. In fact, increasing the load distance from the support decreases the load effect v

_{E}close to the load (action side). On the other hand, the unitary shear capacity v

_{R}also decreases by increasing a

_{v}due to the higher bending moments around the load (Figure 2c). In summary, the most critical position for one-way shear in Figure 2a can be at the mid-span and not at the support, depending on how the load effect v

_{E}and the shear resistance v

_{R}vary by increasing a

_{v}. Additionally, the approach of checking only one position of the design tandem for one-way shear comes from the use of hand calculations in the past. Nowadays, with the aid of computational tools, it is possible to calculate the load effect for several load positions and for each control section in such a way as to search for the most critical position (resulting in the highest ratio between shear demand v

_{E}and shear capacity v

_{R}). Therefore, a correction in the predicted effective shear width may be needed to evaluate the one-way shear resistance at different load positions.

## 2. Literature Review

#### 2.1. Background Calculations for One-Way Shear

_{R,shear}according to EN 1992-1-1:2004 [25] are presented below (the legend of each parameter appears in the Notations Section; these expressions use average values for material properties to compare tested and predicted resistances):

_{R,shear}according to fib Model Code 2010 [21] are presented below:

_{R,shear,MC}is a function of the applied concentrated load F and, consequently, the unitary bending moments in the control section m

_{E}. In the design or assessment of existing structures, the loads F are generally known, and the solution of v

_{R,shear,MC}becomes direct (closed-form solution). However, in the comparison between tested and calculated resistances, the load that causes the failure or the predicted one-way shear capacity is determined iteratively by varying the applied load F(i) until the calculated unitary shear capacity v

_{R(i)}is equal to the unitary shear demand v

_{E(i)}(Figure 3a). Figure 4 shows a summary of the main calculations in the iterative process to calculate the unitary shear capacity and punching shear capacity with the fib Model Code expressions (the punching capacity calculations are discussed in more detail in the next sections).

_{E}becomes straightforward, for slabs, most designers recur to using finite element analyses to determine the relation between F(i) and v

_{E(i)}and between F(i) and m

_{E(i)}in the case of one-way slabs under concentrated loads [10].

_{R,shear,MC}and the related concentrated load F(i) for a 1 m slab width, the total shear capacity and concentrated loads are multiplied by the respective effective shear width b

_{eff}(Figure 5c). According to the fib Model Code 2010, the control section to evaluate the shear force v

_{E}and bending moments m

_{E}is placed at the distance of d

_{l}≤ a

_{v}/2 from the load edge for simply supported slabs and at d

_{l}≤ a

_{v}/2 from the support edge for cantilever slabs. For continuous members, the control section is placed closer to the load or the support, depending on which one leads to the lower shear resistance (typically the section with the higher ratio m

_{E}/v

_{E}∙d

_{l}).

_{Ed}for loads placed at a

_{v}≤ 2d

_{l}by considering that a portion of the load is taken directly to the support by strut or arching action. Figure 6 shows the cracking pattern of slender members failing in flexure-shear (here assumed with a

_{v}/d

_{l}> 2, Figure 6a) and non-slender members failing in compression-shear (Figure 6b). It can be noted that the flexure-shear crack in Figure 6a disturbs the load transfer in the fictitious strut between the load and the support. On the other hand, when the concentrated loads are placed closer to the support (Figure 6b), the members fail in shear-compression, and direct load transfer between the load and the support can occur by a strut, which is named herein as arching action.

_{Fu}caused by the concentrated load F

_{u}(Table 1).

_{E}) can be calculated as (considering the effect of the self-weight v

_{g}over the effective shear width assumed):

_{E}[11]. In this way, the factor μ = 1/β can be multiplied by the unitary shear resistance v

_{R}, and V

_{E}and V

_{R}become:

_{eff}[3,4,5,12], also referred to as the French approach, is based on the horizontal load spreading toward the supports from the back sides of the load (Figure 7a). In the French approach, the angle of this horizontal spreading is fixed at 45 degrees. In the fib Model Code 2010 [21], the angle of spreading varies as a function of the support conditions (Figure 7b), and the reference line to calculate the effective shear width is placed at min{d

_{l};a

_{v}/2} from the support edge.

#### 2.2. Insights from the Literature for One-Way Shear

_{Fu}was normalized by the effective depth d

_{l}and the root of the compressive strength of concrete f

_{c}. Figure 8a shows the normalized shear resistance as a function of the shear slenderness a

_{v}/d

_{l}for 75 test results and Figure 8b shows the cracking pattern of a set of tests from Reiβen et al. [6] that vary only the ratio a

_{v}/d

_{l}.

_{v}/d

_{l}, as also demonstrated by other authors [5,7,14]. In fact, the tested shear load decreases markedly until a certain value of a

_{v}/d

_{l}= 2 or 3, and after this, it keeps almost the same level. Added to that, it was observed that punching failures become the critical failure mechanism for a

_{v}/d

_{l}> 4. In practice, this means that the failure restricts to a narrower length around the load, which could support the definition of a reduced effective shear width when the shear slenderness a

_{v}/d

_{l}increases.

_{v}/d

_{l}(these figures show the cracking pattern by cut views at the symmetry axis of a set of slabs tested by Reiβen et al. [6]). When the load is placed relatively close to the support, arching action and shear compression failure are most likely to occur along a larger slab strip. In such cases, it is frequent that a one-way shear crack is also visible at the slab sides. Increasing the ratio a

_{v}/d

_{l}, the cracking pattern from such tests indicates a punching failure around the load, which naturally is a failure mechanism concentrated around the load. Therefore, the corresponding effective shear width should be decreased compared to that with the load closer to the support.

#### 2.3. Background Calculations for Punching Capacity Predictions

_{R,punch}is the unitary punching capacity (punching capacity per unit length) and b

_{0}is the shear-resisting control perimeter.

_{E}, some adaptations are needed. The first one is to replace the relation m

_{E}/m

_{R}by P

_{E(i)}/P

_{flex}in the expression of ψ. In this study, P

_{E(i)}is the actual applied concentrated load and P

_{flex}is the slab flexural capacity calculated according to yield line analyses. In this way:

_{E(i)}equals the calculated punching capacity P

_{R,predicted}(see Figure 3b).

_{s,ij}is generally defined as the length of the loading center to the point of contra-flexure (zero bending moment) in the evaluated direction. In this study, the following values are adopted as a simplification in the absence of results from numerical analyses:

_{flex}was assumed as the load F that causes M

_{span}or M

_{sup}to be equal to M

_{r}. The yielding moment M

_{r}was calculated according to the equation (where ρ

_{l}is the reinforcement ratio and b

_{slab}the slab width) [17]:

_{0}is set at the distance α∙d

_{avg}from the loaded area according to the studied code. The value of α is 2 for the Eurocode [25] and 0.5 for the fib Model Code 2010 [21].

_{k}of the control perimeter (see Figure 10d). For slab–column connections, the length b

_{k}has the following values for both Eurocode [25] and fib Model Code 2010 [21]:

_{k}(used in the following calculations):

#### 2.4. Insights from the Literature for Punching Capacity Predictions

_{v}/d

_{l}) and slab width (or the ratio b

_{slab}/l

_{load}). At this point, it is important to note that the shear flow through the sides of the control perimeter close to the free edges is higher or lower as a function of the slab width (Figure 11a). Therefore, for punching capacity predictions, the effective contribution of these sides should be considered as a function of the slab width.

_{slab}/l

_{load}. The reader can realize that by increasing the ratio b

_{slab}/l

_{load}, the normalized failure load increases almost linearly for each series of tests until reaching a plateau on which the governing failure mechanism starts to be punching. Before reaching such a plateau, the governing failure mechanism of such slabs is most likely one-way shear as a wide beam.

## 3. Proposed Approach

#### 3.1. Proposed Approach for One-Way Shear

_{EC}is suggested for both shear codes (Eurocode and fib Model Code) since it correlated better with test results for loads close to the support.

_{v}/d

_{l}[6,11] (see Figure 8a) and to avoid unsafe predictions of one-way shear capacity for a

_{v}/d

_{l}> 3 [10] (as shown in the following sections), a correction in the predicted effective shear width is proposed. In practice, this correction allows considering that the failure occurs along a narrower region in front of the load, mainly by punching, by increasing the shear slenderness a

_{v}/d

_{l}(Figure 12b). To this effect, a factor μ

_{shear,2}is multiplied by the effective shear width calculated according to the French approach, b

_{eff,french}. Consequently, the predicted effective shear width decreases as a

_{v}/d

_{l}increases (see Figure 12a).

_{shear,1}; (ii) the French effective shear width was used for both codes to estimated V

_{R,predicted}; (iii) a linear fitting function was assumed for simplicity. In this study, the calibration of the factors according to the support condition was based on the different coefficients observed for the fitting functions. In practice, this method can be justified mainly due to the different shear flow that occurs for cantilever members (one line support) and simply supported or continuous members (two line supports).

_{shear,2}calibrated for each shear code provision. These factors were derived based on linear regression analyses observing the functions that fitted the ratio V

_{test}/V

_{R,predicted}for each code (see Figure 13). For each code (V

_{R,predicted}), the French effective shear width model and the factor μ

_{shear,1}were applied.

_{R,predicted}:

#### 3.2. Proposed Approach for Punching Shear Capacity Predictions

_{0x,1}as the length covered by the dashed blue lines (Figure 14b) or the straight length that intercepts the support (Figure 14c). However, such approaches would lead to very small values of b

_{0x,1}when the load is placed at a

_{v}= 0 or when a

_{v}= 2d

_{avg}, respectively. Because of this, we concluded that the most consistent approach would consider b

_{0,x1}as the length between the middle points of the rounded corners (Figure 14d). Alternatively, and most simply for the Eurocode control perimeter, one can also use a squared control perimeter at 1.5d

_{avg}(Figure 14e), which simplifies the calculations for the partitions of the control perimeter, as suggested by Regan [19]. In this way, the length b

_{0,x1}would not vary as a function of a

_{v}(Figure 14f), and the total control perimeter b

_{0}would be very similar to that using rounded corners.

_{avg}provides similar results (this is demonstrated in the next sections).

_{0,x1}) when a

_{v}/d

_{l}≤ 2; (ii) the reductions in the capacity for the lateral sides b

_{0,y1}and b

_{0,y2}as a function of the slab width.

_{0x,1}is achieved by multiplying the unitary shear capacity on this side by the factor μ

_{punch,1}, which is equal to that for one-way shear.

_{0y,1}and b

_{0y,2}to the punching capacity of slabs with reduced width b

_{slab}is considered by multiplying these sides by a correction factor μ

_{punch,2}(Table 3). In this study, the same factors were proposed for both codes.

_{punch,2}was calibrated. A comparison between tested and predicted resistances with the punching capacity expressions from Eurocode was performed (Figure 15a). This comparison demonstrated sensitivity in the predictions with the value of t and, consequently, the slab width. In this way, the function of μ

_{punch,2}(assumed linear) was modified until removing this sensitivity (Figure 15b). As only the simply supported and continuous slabs varied the parameter t more extensively in the database than the respective cantilever slabs, the derivation of such parameters was based only on the results of simply supported and continuous slabs (97 test results).

_{v}/d

_{l}≈ 2 (see Figure 16a). Since the effect of arching action by μ

_{punch,1}was considered only until a

_{v}/d

_{l}= 2, this could indicate that a different factor should be used for cantilever slabs. However, even extending the length of influence until a

_{v}/d

_{l}= 3 did not significantly change the results. Because of this, another explanation was needed for such results. A possible explanation for these results was that observing the cracking pattern of the tests from Henze et al. [7], the failure mechanism for loads closer to the support of cantilever slabs would be more influenced by the longitudinal reinforcement at the top side of the slabs (Figure 16c,d). In other words, considering the predicted resistance as a function of the bottom reinforcement of cantilever slabs would lead to overly conservative predictions since such reinforcement ratios are significantly lower than the ones used at the top side of the slab. At this point, it is important to note that the bending moments for cantilever slabs are slightly different from cantilever beams since we always have sagging bending moments and hogging bending moments for cantilever slabs (see Figure 16e,f). In order to keep the core of the expressions for the punching capacity of cantilever slabs by considering the bottom reinforcement in the expressions, an alternative is to consider another load position parameter in the expressions, such as the ratio a/l

_{span}(Figure 16b). Consequently, it would be possible to consider some enhancements in the punching capacity as a function of the load position on the cantilever slab not related to arching action.

_{R,punch}) with the fib Model Code 2010 [21] expressions already lead to enhanced resistances for loads close to the support by considering that the flexural capacity of the slabs P

_{flex}is enhanced in such regions and, hence, smaller slab rotations ψ around the load occur for these regions. In the end, this effect is combined with the one expected from arching action, which leads to significant enhancements in resistance. In the expressions from Eurocode [25], on the other hand, the unitary shear capacity predicted is not influenced by the load position and can be enhanced only as a function of the arching action.

_{punch,3}(Table 4).

_{0,x2}= 0):

## 4. Database

## 5. Results

#### 5.1. Results with the European Code Expressions

_{test}/V

_{R,predicted}) and punching shear (P

_{test}/P

_{R,predicted}) considering two cases: (i) without the use of any correction factor μ

_{shear}or μ

_{punching}; (ii) with the use of the correction factors μ

_{shear}or μ

_{punching}.

_{v}/d

_{l}< 2, even using β

_{EC}to decrease the sectional shear V

_{test}(V

_{test,red}/V

_{R,predicted}> 1). On the other hand, the predictions of one-way shear capacity for the tests with a

_{v}/d

_{l}> 5 may become quite unsafe (V

_{test}/V

_{R,predicted}< 0.6). In Figure 17a, this occurs because the effective shear width predicted with the French approach increases excessively by increasing the shear slenderness a

_{v}/d

_{l}. Figure 17d shows that the accuracy and precision of the predictions improved notably by using the proposed correction factors to consider the arching action and the reduced effective shear width for large shear slenderness a

_{v}/d

_{l}. The average ratio V

_{test}/V

_{R,predicted}changed from 1.22 to 1.25 and the coefficient of variation decreased from 33.1% to 17.2%.

_{slab}/l

_{load}and without the use of any correction factor (μ

_{punching,1}and μ

_{punching,2}). Figure 17b shows that the predictions may become quite unsafe for the tests that failed as wide beams in one-way shear, mainly for ratios b

_{slab}/l

_{load}< 7.5 (see Detail 1 in Figure 17b). This occurs because the contribution of the sides of the control perimeter parallel to the slab-free edges was overestimated with the traditional approach (which is rarely discussed in most publications). In the same way, the predictions of punching capacity with the European code expressions may become overly conservative if arching action is not considered in the portion of the control perimeter close to the line support when a

_{v}/d

_{l}< 2 (see Detail 2 in Figure 17b). On the other hand, Figure 17e shows that the proposed recommendations significantly improve the predictions of punching capacity. Comparing Figure 17b,e, the average ratio between tested and predicted resistances changes from 1.44 to 1.17 and the coefficient of variation decreases from 40.1% to 22.1%.

_{test}/P

_{R,predicted}, with and without the corrections factors for punching, respectively, but using a square control perimeter such as suggested by Regan [19] and sketched in Figure 14e,f. As can be seen, the results using a square control perimeter is similar to that achieved with the round control perimeter but slightly more conservative (see Figure 17c,f).

#### 5.2. Results with the Fib Model Code 2010 Code Expressions

_{shear,1}or μ

_{shear,2}. At this point, however, it is important to note that the fib Model Code 2010 adopts the factor β

_{MC}, to consider the arching action (having a similar effect to the proposed μ

_{shear,1}). Figure 18d shows the results, including the proposed correction factors μ

_{shear,1}or μ

_{shear,2}.

_{v}/d

_{l}< 2. On the other hand, these predictions were quite unsafe for a

_{v}/d

_{l}> 5 (Figure 18a). Using the correction factors μ

_{shear,1}and μ

_{shear,2}, these shortcomings are corrected (Figure 18d). The average ratio V

_{test}/V

_{Rpredicted}reduced from 1.44 to 1.26 and the coefficient of variation decreased from 31.9% to 18.2%, comparing Figure 18a,d.

_{punching}already reached enhanced levels of accuracy (coefficient of variation lower than 20%, for instance, and the average ratio P

_{test}/P

_{R,predicted}between 1.0 and 1.20). Using the proposed correction factors to consider the disturbed contribution as a function of the slab width, the predictions on this range were slightly enhanced (Figure 18e). Comparing Figure 18b,e, the average ratio P

_{test}/P

_{R,predicted}decreased slightly from 1.14 to 1.09 and the coefficient of variation varied from 17.0% to 15.6%.

_{avg}makes the total length of the control perimeter more similar, even with the different shapes.

## 6. Discussion

_{v}/d

_{l}< 2) (Figure 17a, for instance). Additionally, it can be seen that the ultimate loads were enhanced for a

_{v}/d

_{l}< 2 regardless of the governing failure mechanism being one-way shear or punching shear (Figure 17a). Therefore, these results indicate a close relation between one-way shear mechanisms and two-way shear mechanisms for loads close to the support in one-way slabs.

_{v}/d

_{l}≤ 2 and unsafe predictions for large values of a

_{v}/d

_{l}. Since the shear expressions of the fib Model Code 2010 already take into account the beneficial effect of short shear spans and the detrimental effect of the large shear spans through the calculations of m

_{E}and v

_{E}, this result draws attention. In practice, this occurs because the influence of the arching action is more significant than the effect of smaller m

_{E}for a

_{v}/d

_{l}< 2 [15]. In the same way, the effect of larger bending moments m

_{E}for a

_{v}/d

_{l}> 5 (which decreases the unitary shear capacity v

_{R,shear,MC}) does not compensate for the increase in the effective shear width when the load is placed so far from the support with the current French approach b

_{eff,French}or fib Model Code rules to determine the effective shear width. Therefore, the predictions of one-way shear capacity for large shear slenderness can be safe only if the effect of a reduced effective shear width is considered in the expressions, as proposed in this paper.

_{avg}from the loaded area, and (ii) because the expressions of the fib Model Code 2010 already take into account the influence of a reduced shear span in the expressions by calculating the slab rotations. Moreover, the unitary punching capacity is greater than the corresponding unitary shear capacity in one-way shear [43], as the failure for the first always occurs around the load, which is enhanced by significant confinement stresses [44]. In this way, the enhanced punching capacity for loads close to the support is already partially considered without a factor μ

_{punching,1}through the (i) enhanced unitary punching capacity compared to the one-way shear capacity and (ii) the calculations of the slab rotations around the load.

_{E}by the factor β or improved shear capacity V

_{R}by μ

_{shear,1}leads to similar results for the tests with a

_{v}/d

_{l}< 2. Concerning the effective shear width, this study demonstrates with different one-way code expressions that a simple correction in the French effective shear width by the factor μ

_{shear,2}allows for improving the accuracy of the one-way shear expressions for loads far from the support. In this way, the factor μ

_{shear,2}allows consideration that a possible one-way shear failure or punching failure will develop under a narrower slab strip.

_{span}plays a key role in enhancing the predictions of punching capacity, mainly when the slabs may fail as wide beams and for cantilever slabs. On the other hand, the fib Model Code expressions are less dependent on such factors to reach enhanced levels of accuracy. In practice, this occurs mainly because the arching action is indirectly considered for the punching shear expressions through the calculations of the slab rotations, which decrease for loads close to the support, and also because the unitary punching strength in the fib Model Code is larger than the corresponding unitary one-way shear strength.

## 7. Conclusions

- The ultimate capacity of one-way slabs under concentrated loads increases significantly when the loads are placed close to the support at distances a
_{v}/d_{l}≤ 2, due to arching action, regardless of the slabs failing in one-way shear as wide-beam (WB) or punching shear (P). In this study, the enhancement factor μ_{shear,1}and μ_{punch,1}are applied for both one-way shear and punching shear expressions to consider this mechanism. Comparatively, the ultimate resistance of the slabs decreases significantly when the loads are placed at distances a_{v}/d_{l}≥ 3. At such positions, most slabs from the database failed by punching, which is a failure mechanism concentrated around the load. Therefore, a reduced effective shear width should be employed if the one-way shear resistance needs to be checked at such positions. In this study, the factor μ_{shear,2}allows for decreasing the effective shear width for larger distances from the load to the support. - In the punching shear resistance predictions, however, the slab width may also play a significant role. For slabs with a reduced slab width compared to the effective depth, for instance, t < 5 with t = (b
_{slab}− l_{load}− 4d_{avg})/d_{avg}, the contribution of the sides of the control perimeter parallel to the free edges is reduced due to the small shear flow going through these sides. In this study, it is proposed to multiply the contribution of these sides by μ_{punch,2}to consider this effect. In the case of cantilever slabs, and particularly with the Eurocode expressions, another aspect considerably influences the predictions of punching capacity. The punching capacity expressions use the bottom reinforcement of the slab in the calculations, and many of these slabs fail in one-way shear, presenting higher demand on the top reinforcement of the slabs. Consequently, the predictions of punching capacity can become overly conservative for loads placed at distances a_{v}/d_{l}close to 2. Because of this, a third factor was needed to reach enhanced predictions for cantilever slabs using the Eurocode expressions. - The one-way shear capacity predictions are significantly enhanced by considering the arching action for loads close to the support by a factor μ
_{shear,1}. Furthermore, the transition from one-way shear failures as wide-beam (WB) to punching failures (P) by increasing the shear slenderness a_{v}/d_{l}can be considered in a simplified way by multiplying the predicted effective shear width b_{eff,french}by the factor μ_{shear,2}. In this way, enhanced predictions of one-way shear capacity can be achieved for the tests, even when they fail by punching. In practice, these observations were valid for both codes (Eurocode and fib Model Code). - The predictions of punching capacity with the Eurocode expressions are significantly enhanced considering the factors related to arching action and to the slab width in the proposed approach. In the case of the fib Model Code 2010 expression, these enhancements were less pronounced since the results without the proposed factors already led to relatively enhanced predictions. In other words, the proposed recommendations to calculate the slab rotations and respective shear capacity on each portion of the control perimeter (without the use of numerical models) have already led to good levels of accuracy.
- In general, the one-way shear and punching shear predictions led to similar levels of accuracy when using the proposed recommendations. In the case of the current Eurocode, the average ratio V
_{test}/V_{R,predicted}was 1.25 with a coefficient of variation of 17.2%, while the average ratio P_{test}/P_{R,predicted}was 1.17 with a coefficient of variation of 22.1%. In the case of the fib Model Code 2010, the average ratio V_{test}/V_{R,predicted}was 1.26 with a coefficient of variation of 18.2% and the average ratio P_{test}/P_{R,predicted}was 1.09 with a coefficient of variation of 15.6%. Therefore, both one-way shear and punching shear predictions may lead to close estimations of the ultimate capacity, regardless of the governing failure mechanism of the slabs, when the parameters that influence the transition from one failure mechanism to another are embedded in the calculations.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notation

a | shear span: distance between the center of the support and the center of the load |

a_{v} | clear shear span: distance between face of support and face of load |

b_{0} | total length of the shear-resisting control perimeter |

b_{0,x1} | side of the control perimeter in the spanning direction between the load and the closer support |

b_{0,x2} | side of the control perimeter in the spanning direction between the load and the far support |

b_{0,y1} | side of the control perimeter in the transverse direction close to the free edge 1 |

b_{0,y2} | side of the control perimeter in the transverse direction close to the free edge 2 |

b_{eff} | effective shear width |

b_{eff,proposed} | proposed effective shear width |

b_{eff,french} | French effective shear width |

b_{slab} | slab width |

b_{load} | size of the concentrated load in the slab width direction (transverse direction) |

d_{avg} | average effective depth of the flexural reinforcement |

d_{l} | effective depth towards longitudinal steel |

d_{t} | effective depth towards transverse steel |

d_{g} | maximum aggregate size |

d_{g0} | reference aggregate size (=16 mm in fib Model Code 2010) |

d_{dg} | parameter that considerers the crack roughness |

f_{c} | average compressive strength measured on cylinder specimens |

f_{yi} | steel yielding stress in the evaluated direction (x = longitudinal direction and y = transverse direction) |

h_{slab} | slab thickness |

k | constant accounting for size effect in one-way shear for EN 1992-1-1:2004 |

k_{1} | factor accounting for axial forces in one-way shear for EN 1992-1-1:2004 |

k_{dg} | coefficient for aggregate size (=32/(16 mm + dg) in fib Model Code 2010) |

k_{v} | factor accounting for strain effect and member size in fib Model Code 2010 |

k_{ψ} | factor accounting for effect of crack widths and roughness of cracks on shear strength in fib Model Code 2010 |

l_{span} | span length |

l_{load} | size of the concentrated load in the span direction |

l_{s} | is the length of the sides with one-way shear behavior |

m_{R,i} | yielding moment per unit length in the evaluated direction |

m_{max} | maximum bending moment at the control section for a given applied load |

m_{s,ij} | averaged acting bending moment at the loading plate edge ij within the width b_{s} |

m_{Ed} | design bending moment at the control section |

r_{s} | distance between the center of the concentrated load and the line of contraflexure of moments (subscripts x, y refers to the direction considered) |

r_{s,ij} | distance between the center of the concentrated load and the point of contraflexure in the evaluated direction |

v | shear force per unit length (nominal shear force) |

v_{E} | shear force at the control section |

v_{Ed} | design shear force at the control section |

v_{min} | minimum one-way shear resistance per unit length in EN 1992-1-1:2004 |

v_{R,shear} | unitary one-way shear resistance |

v_{g} | shear forcer per unit length in the control section placed at a/2 due to the self-weight |

w_{cr} | width of the critical shear crack |

z | effective shear depth in fib Model Code 2010 |

A_{s} | cross-sectional area of flexural reinforcement |

C_{R,c} | calibration factor in the shear and punching expressions of EN 1992-1-1:2004 |

E_{c} | modulus of elasticity of concrete |

E_{s} | steel modulus of elasticity |

F | applied concentrated load |

F_{Ed} | design concentrated load |

F_{predicted} | predicted load that causes a one-way shear failure or two-way shear failure |

F_{u} | applied concentrated load at failure |

F_{hyp} | arbitrary concentrated load |

L | span length |

V_{control} | total shear force going through the evaluated direction along the slab width |

V_{Ed} | design shear action |

V_{Fu} | shear force due to the concentrated load F_{u} |

V_{test} | measured one-way shear force at failure in the tests for a section at a/2. |

V_{R} | one-way shear capacity |

V_{R,predicted} | predicted one-way shear resistance |

V_{Rd} | design one-way shear capacity |

V_{R,CSCT} | predicted one-way shear resistance with the CSCT expressions |

V_{R,ij} | punching shear strength corresponding to b_{0,ij} |

P_{test} | maximum applied concentrated load at failure |

P_{Ed} | design concentrated loads |

P_{Rd} | design punching capacities |

P_{predicted} | predicted punching resistance |

P_{flex} | concentrated load associated with the slab flexural capacity |

P_{R,punching} | total shear force resisted by punching |

β_{shear} | enhancement factor to account for arching action |

ρ_{avg} | average flexural reinforcement ratio considering both directions |

ρ_{l} | flexural reinforcement ratios in longitudinal direction |

ρ_{t} | flexural reinforcement ratio in transverse direction |

ψ | rotations around the loaded area |

ψ_{ij} | rotations in each side of the control perimeter |

ε_{x} | strain in the control depth for one-way shear analyses |

ε_{y} | is the flexural reinforcement yield strain |

γ | concrete specific weight (assumed = 24 kN/m^{3} in this study) |

γ_{c} | partial safety factor of concrete |

μ_{shear,1} | factor accounting for arching action in one-way shear analyses |

μ_{shear,2} | factor accounting for reduced b_{eff} for loads far away from the support |

μ_{punch,1} | factor accounting for arching action in punching shear analyses |

μ_{punching,2} | factor accounting for the influence of the slab width in the effective contribution of the sides b_{0,y1} and b_{0,y2} to the punching capacity |

μ_{punching,3} | correction factor related to the load position in cantilever slabs for punching capacity predictions |

AVG | average |

COV | coefficient of variation |

P | observed failure mode is punching failure |

SS | test was performed with the load closer to the simple support |

CS | tests was performed with the concentrated loads close to a continuous support |

CT | test was performed with the concentrated load applied on a cantilever slab |

WB | observed failure mode is wide-beam shear failure |

WB + P | the observed failure mode combines characteristics of WB and P |

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**Figure 1.**(

**a**) Example of bridge deck slab under the concentrated loads of the European design tandem; (

**b**) example of calculation of the design shear stress v

_{E}at the support combining the actions from the self-weight and the ones from the load axes using the effective shear width definition; (

**c**) example of laboratory test.

**Figure 2.**(

**a**) Sketch of the test S35B1-1 from Reiβen [14] and position of the control section on which the acting bending moments and shear demand are evaluated with fib Model Code 2010 [21]; (

**b**) example of possible distribution in shear demand v

_{E}and shear resistance v

_{R}using one-way shear models that do not consider the influence of the ratio m

_{E}/v

_{E}∙ d

_{l}in the unitary shear capacity (note that the most critical position for one-way shear will always be close to the support in this approach); (

**c**) example of possible distribution in shear demand v

_{E}and shear resistance v

_{R}at the control section with the models that consider the lower shear resistance increasing the ratio m

_{E}/v

_{E}d

_{l}.

**Figure 3.**Determination of the (

**a**) unitary shear resistance v

_{R,shear}and (

**b**) punching capacity P

_{R,predicted}iteratively for the fib Model Code expressions.

**Figure 4.**Flowchart of calculations for (

**a**) one-way shear resistance predictions and (

**b**) punching shear capacity with the fib Model Code 2010 expressions in the comparisons between tested and predicted resistances.

**Figure 5.**Proposed approach to calculate the relation between the applied concentrated load and the acting shear v

_{E}and bending moments m

_{E}when using the fib Model Code 2010 expressions for one-way shear: (

**a**) Sketch of a test to be evaluated; (

**b**) assumption of fictitious a test with 1 m width with equal support conditions to calculate v

_{E}, m

_{E}and, subsequently, the unitary shear capacity v

_{R}; (

**c**) multiplying of the test sketched in (

**b**) by the respective effective shear width calculated in (

**a**).

**Figure 6.**Influence of the shear slenderness (here defined as a

_{v}/d

_{l}) in the cracking pattern and failure mechanism of members loaded over the entire width: (

**a**) slender beams failing in flexure-shear; (

**b**) non-slender beams failing in shear-compression (adapted from Muttoni and Fernandez Ruiz [17]).

**Figure 8.**(

**a**) Normalized sectional shear achieved in the tests as a function of (

**a**) shear slenderness a

_{v}/d

_{l}and (

**b**) change in the cracking pattern and corresponding failure mechanism as a function of the ratio a

_{v}/d

_{l}(cut views of a set of tests from Reiβen et al. [6]). Source: Adapted from Sousa, Lantsoght and El Debs [24]. Note: WB = wide-beam shear failure (one-way shear); P = punching failure; WB + P = mixed failure mode between one-way shear and punching shear.

**Figure 9.**Yield line pattern for flexural assessment of simply supported, continuous and cantilever slabs.

**Figure 10.**Different layouts of the control perimeter for simply supported slabs: (

**a**) four sides, (

**b**) two sides and (

**c**) three sides; and for cantilever slabs: (

**d**) three sides, (

**e**) one side and (

**f**) two sides.

**Figure 11.**(

**a**) Shear flow and possible shear failure mechanisms for the slab: one-way shear as wide beams, punching shear around the load and a mixed-failure mode; (

**b**) sides enhanced or disturbed due to the ratio a

_{v}/d

_{l}and b

_{slab}/l

_{load}for a simply supported slab; (

**c**) influence of the slab width to load size ratio b

_{slab}/l

_{load}on the failure load of slabs under concentrated loads (adapted from [24]); (

**d**) influence of the slab width on the cracking pattern and consequently on the failure mechanism of the tests (adapted from Reiβen et al. [6]). Note: WB = wide-beam shear failure (one-way shear); P = punching failure; WB + P = mixed failure mode between one-way shear and punching shear.

**Figure 12.**(

**a**) Effective shear width corrected as a function of the shear slenderness a

_{v}/d

_{l}(the proposed effective shear width decreases as the shear slenderness increases) and (

**b**) failure mechanism varying from one-way shear as wide-beam (WB) to punching shear (P) increasing a

_{v}/d

_{l}for the tests from Reiβen et al. [6].

**Figure 13.**Calibration of μ

_{shear,2}for (

**a**) simply supported and continuous slabs using the current Eurocode expressions; (

**b**) cantilever slabs using the current Eurocode expressions; (

**c**) simply supported and continuous slabs using the fib Model Code 2010 expressions; (

**d**) cantilever slabs using the fib Model Code 2010 expressions. Note: SS = simple support (hinged support); CS = continuous support; CT = cantilever slab.

**Figure 14.**Possible partitions of the shear-resisting control perimeter for simply supported slabs and EN 1992-1-1:2004: (

**a**–

**d**) using the rounded corners; (

**e**,

**f**) using the squared control perimeter.

**Figure 15.**Comparison between tested and predicted resistances using the punching capacity expressions from Eurocode according to the parameter t (related to the slab width) for the dataset of simply supported and continuous slabs (97 tests results): (

**a**) using only the factor μ

_{punch,1}and (

**b**) using both factors μ

_{punch,1}and μ

_{punch,2}.

**Figure 16.**(

**a**) Comparison between tested and predicted resistances with the punching capacity expressions from the Eurocode for cantilever slabs as function of the shear slenderness a

_{v}/d

_{l}; and (

**b**) as function of the ratio a/l

_{span}; (

**c**) sketch of the tests performed by Henze et al. [7] on cantilever slabs; (

**d**) cracking pattern of the tests performed by Henze et al. [7] (cut views); (

**e**) distribution in the bending moments in the axis of symmetry of the slabs (axis x); (

**f**) distribution in the bending moments in the transverse direction of the slabs (axis y).

**Figure 17.**Comparison between tested and predicted resistances the European code expressions for: (

**a**) one-way shear and without the correction factors μ; (

**b**) punching shear, without the correction factors μ, and using round corners for the control perimeter; (

**c**) punching shear, without the correction factors μ, and using square corners for the control perimeter; (

**d**) one-way shear with the correction factor μ; (

**e**) punching shear expressions, with the correction factors, and using round corners for the control perimeter; (

**f**) punching shear expressions, with the correction factors μs, and using square corners for the control perimeter. Note: P = punching failure; WB = wide-beam shear failure in one-way shear; WB + P = mixed mode between one-way shear and punching.

**Figure 18.**Comparison between tested and predicted resistances the fib Model Code 2010 expressions for: (

**a**) one-way shear and without the correction factors μ; (

**b**) punching shear, without the correction factors μ, and using round corners for the control perimeter; (

**c**) punching shear, without the correction factors μ, and using square corners for the control perimeter; (

**d**) one-way shear, with the correction factors μ; (

**e**) punching shear expressions, with the correction factors, and using round corners for the control perimeter; (

**f**) punching shear expressions, with the correction factors, and using square corners for the control perimeter. Note: P = punching failure; WB = wide-beam shear failure in one-way shear; WB + P = mixed mode between one-way shear and punching.

Code | |
---|---|

EN 1992-1-1:2004 [25] | ${\beta}_{EC}=\frac{{a}_{v}}{2\cdot {d}_{l}}\left\{\begin{array}{l}\le 1.00\\ \ge 0.25\end{array}\right.$ (11) |

fib Model Code 2010 [21] | ${\beta}_{MC}=\frac{{a}_{v}}{2\cdot {d}_{l}}\left\{\begin{array}{l}\le 1.00\\ \ge 0.50\end{array}\right.$ (12) |

**Table 2.**Factors μ

_{shear,2}to correct the predicted effective shear width with the French approach when using different codes and support conditions. Note: Sup. Cond = support condition close to the load; SS = simple support (hinged support); CS = continuous support; CT = cantilever slab.

Code | Sup. Cond. | Factors |
---|---|---|

EN 1992-1-1:2004 [25] | SS, CS | ${\mu}_{shear,2}=-0.144\cdot {a}_{v}/{d}_{l}+1.456\left\{\begin{array}{l}\ge 0\\ \le 1.50\end{array}\right.$ (34) |

EN 1992-1-1:2004 [25] | CT | ${\mu}_{shear,2}=0-0.184\cdot {a}_{v}/{d}_{l}+1.400\left\{\begin{array}{l}\ge 0\\ \le 1.50\end{array}\right.$ (35) |

fib Model Code 2010 [21] | SS, CS | ${\mu}_{shear,2}=-0.104\cdot {a}_{v}/{d}_{l}+1.392\left\{\begin{array}{l}\ge 0\\ \le 1.50\end{array}\right.$ (36) |

fib Model Code 2010 [21] | CT | ${\mu}_{shear,2}=-0.184\cdot {a}_{v}/{d}_{l}+1.624\left\{\begin{array}{l}\ge 0\\ \le 1.50\end{array}\right.$ (37) |

Code | Parameter | Factor μ_{punch,2} |
---|---|---|

Eurocode EN 1992-1-1:2004 | $t=\left({b}_{slab}-{l}_{load}-4\cdot {d}_{avg}\right)/{d}_{avg}$ | ${\mu}_{punch,2}=0.2\cdot t\left\{\begin{array}{l}\ge 0\\ \le 1\end{array}\right.$ (40) |

fib Model Code 2010 | $t=\left({b}_{slab}-{l}_{load}-{d}_{avg}\right)/{d}_{avg}$ | ${\mu}_{punch,2}=0.2\cdot t\left\{\begin{array}{l}\ge 0\\ \le 1\end{array}\right.$ (41) |

Code | Factor μ_{punch,2} |
---|---|

Eurocode EN 1992-1-1:2004 | ${\mu}_{punch,3}=0.672\cdot {\left(a/{l}_{span}\right)}^{-0.76}\left\{\begin{array}{l}\ge 0.80\\ \le 2.15\end{array}\right.$ (42) |

fib Model Code 2010 | ${\mu}_{punch,3}=1$ (43) |

Parameter | Minimum | Maximum |
---|---|---|

h (m) | 0.10 | 0.30 |

b_{slab} (m) | 0.60 | 4.50 |

t = (b_{slab} − l_{load} − 4d_{avg})/d_{avg} | 0.45 | 27.21 |

l_{span} (m) | 0.90 | 4.00 |

b_{slab}/l_{load} (-) | 1.67 | 23.08 |

b_{slab}/d_{l} (-) | 5.66 | 29.41 |

a_{v}/d_{l} (-) | 0.24 | 7.66 |

f_{c} (MPa) | 19.20 | 77.74 |

ρ_{l} (%) | 0.602 | 2.150 |

ρ_{t} (%) | 0.132 | 1.526 |

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## Share and Cite

**MDPI and ACS Style**

de Sousa, A.M.D.; Lantsoght, E.O.L.; El Debs, M.K.
Shear and Punching Capacity Predictions for One-Way Slabs under Concentrated Loads Considering the Transition between Failure Mechanisms. *Buildings* **2023**, *13*, 434.
https://doi.org/10.3390/buildings13020434

**AMA Style**

de Sousa AMD, Lantsoght EOL, El Debs MK.
Shear and Punching Capacity Predictions for One-Way Slabs under Concentrated Loads Considering the Transition between Failure Mechanisms. *Buildings*. 2023; 13(2):434.
https://doi.org/10.3390/buildings13020434

**Chicago/Turabian Style**

de Sousa, Alex Micael Dantas, Eva Olivia Leontien Lantsoght, and Mounir Khalil El Debs.
2023. "Shear and Punching Capacity Predictions for One-Way Slabs under Concentrated Loads Considering the Transition between Failure Mechanisms" *Buildings* 13, no. 2: 434.
https://doi.org/10.3390/buildings13020434