Draft - Thesis March 2005 FINAL

PUNCHING SHEAR OF FLAT SLABS:

FAILURE AND CAPACITY CALCULATION

OVERVIEW OF CURRENT DESIGN PRACTICE

PROPOSED STRENGTHENING AND REMEDIAL PROCEDURES

EXPERIMENTAL TESTING OF REMEDIAL MEASURES

SCHALK WILLEM MARAIS

THESIS PRESENTED IN PARTIAL FULFILLMENT OF REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING (CIVIL)

AT THE UNIVERSITY OF STELLENBOSCH

SUPERVISOR: PROF.G.P.A.G. VAN ZIJL

STELLENBOSCH APRIL 2005

DECLARATION

I, the undersigned, declare that the work contained in this thesis is my own original work and has not been submitted in its entirety or in part for a degree at any other university.

______________ ____________

Opsomming

Moderne beton konstruksie maak meestal gebruik van plat blaaie sonder kolomkoppe of blad verdikkings, in plaas van die meer konvensionele balk-en-blad stelsels. Die gebruik van plat blaaie bied aansienlike voordele ten opsigte van koste, relatief maklike

konstruksie en meer vryheid met die argitektoniese uitleg van die gebou. Ongelukkig word die volle voordele en kapasiteit van plat blaai nie noodwendig benut nie. Heelwat

ontwerpkodes bied onakkurate metodes om ponsskuif weerstand te voorspel.

Gepaardgaande hiermee word die gebruik van plat blaaie gepenaliseer in gebiede wat onderhewig is aan matige seismiese aktiwiteit, as gevolg van moontlike moment

geïnduseerde ponsskuif swigting.

Hedendaags word al meer bestaande geboue omskep en herbenut. In baie gevalle veroorsaak die nuwe uitlegte en veranderde gebruik dat ekstra strukturele kapasiteit van die bestaande kolom en blad verbindings benodig word. Somtyds is daar reeds skade aan hierdie verbindings as gevolg van ontwerpfoute of ongewenste praktyke tydens

konstruksie.

Na aanleiding van die voorafgenoemde spreek hierdie verslag die volgende aspekte aan: Eerstens word „n aantal ontwerpkodes se voorspelling van ponsskuif swigting vergelyk. Uit die vergelyking volg dat weinig van die benaderings in lyn is met moderne

betroubaarheidsbeginsels vir die ontwerp van strukture. Op hede is daar slegs een kode wat op alle beskikbare toetsdata gekalibreer is om te voldoen aan 5% moontlikheid van swigting. Hierdie kode is die nuutste DIN 1045-1 (2001) ontwerpkode.

Tweedens word verskeie metodes voorgestel vir die herstel en versterking van bestaande kolom en blad verbindings. Gepaardgaande hiermee word „n universele klassifikasie van skade voorgestel om te sorg dat die korrekte stappe vir remediërende werk geneem kan word.

Derdens is „n kolom en blad verbinding eksperimenteel getoets. Hierdie toets het die welbekende, bros gedrag van ponsskuif swigting uitgelig, asook „n groot verskil tussen die

voorspelde skuif kapasiteit en die getoetse kapasiteit van die model. Meganistiese modellering toon dat die swiglas van die model wel in die regte ordegrootte was.

Laastens is daar gepoog om die beskadigde blad te herstel deur vertikale wapening stawe in die blad te installeer met „n hoë sterkte epoksie. Hierdie stawe is binne die oorspronklike skuifwapening omtrekke geïnstalleer, asook op „n nuwe wapeningsomtrek. Ongelukkig het die herstelde blad nie dieselfde kapasiteit as die oorspronklike blad gehad nie. Ten spyte hiervan kan die sukses van die herstel toegeskryf word aan die feit dat die uiteindelike swigting buite die vegrote bewapening sone geskied het. Die addisionele skuifstaal het die skuifkrake forseer om weg van die kolom te migreer en uiteindelik buite die versterkte sone te swig.

Daar word voorgestel dat verdere toetse gedoen word om die presiese bydrae van die addisionele skuif wapening te bepaal, sowel as die meer akkurate voorspelling van die kapasiteit van herstelde blaaie.

Synopsis

Modern concrete construction favours the use of flat slabs without drop-panels or column capitols as opposed to more conventional slab and beam systems. Flat slabs offer

numerous advantages in terms of cost, ease of construction and architectural flexibility. However, more often than not, the full advantage of using flat slabs is not harnessed. Many design codes offer inaccurate formulations to predict punching shear capacity; furthermore, they discriminate against their use in modestly active seismic regions due to potential moment induced punching failure.

Lately, more and more existing structures are being refurbished and renovated. In many cases the change in architectural layouts and altered use necessitate additional load carrying capacity from the existing slab-column connections. In some cases the existing structures already show distress due to under-designed slab-column interfaces or dubious construction methods.

Based on the aforementioned points the aims of this report are the following:

Firstly, a comparison of several current codified design approaches is performed to highlight the fact that some of the favoured codified approaches do not comply with modern reliability-based structural design philosophies. At this stage, there is only one design approach that has been calibrated with virtually all available test data, and scaled to comply with a 5% probability of failure. This is the formulation presented in the latest DIN 1045-1 (2001) design code.

Secondly, numerous methods of strengthening and repair of existing slab-column connections are presented. Accompanying these suggested methods, a universal

classification of damage is proposed to aid in the effective repair of damaged connections.

Thirdly, experimental testing shows the well-known brittle behaviour of punching shear failure and the difference between the predicted and actual failure loads measured. Mechanistic modelling of the test panel shows the failure load to be in the correct order of magnitude.

Lastly, an attempt was made to repair the damaged slab by adding vertically doweled reinforcing bars, bonded with high strength epoxy grout within the original shear reinforced zone, as well as on a new perimeter. Even though the failure load of the repaired test panel did not meet that of the original panel, the effectiveness of the repair was evident in the fact that punching failure did not take place within the extended shear reinforced zone. The additional perimeter of shear reinforcing forced the inclined shear cracks to migrate away from the column, causing failure outside the shear reinforced zone.

It is proposed that further future testing is done to quantify the added benefit of additional reinforcing, as well as the accurate prediction of the punching shear capacity of repaired slab-column connections.

Acknowledgments:

Special thanks go out to the following people for their patience, time, support, ideas and critique:

Gideon van Zijl University of Stellenbosch Billy Boshoff University of Stellenbosch Wayne Ritchie Sutherland Associates (Pty) Ltd Gerrit Bastiaanse BKS (Pty) Ltd

Ralph Kratz University of Cape Town

Without the support and enthusiasm of the engineering industry the experimental testing would not have been possible. Very special thanks go out to each of the following

companies and their representatives for the supply of construction materials and expertise:

Dave Miles

Lafarge South Africa

Grant Pistor

Contents

1. Overview of Punching Shear Failure 13

1.1. Introduction 13

1.2. Classical Punching Failure 16 1.3. Punching failure due to lateral loading of the structure 18 2. Proposed Analytical and Empirical Models 20 2.1. Synthesis of Punching Shear Failure, as proposed by Menétrey (2002) 20 2.1.1. Experimental results 20 2.1.2. Numerical simulations 21 2.1.2.1. Model description 22 2.1.2.2. Simulation of the punching failure 22 2.1.2.3. Parametric analysis 23

2.1.3. Analytical Model 26

2.1.3.1. Punching- vs. Flexural capacity 27 2.1.3.2. Tensile force in the concrete 27 2.1.3.3. Contribution of the dowel effect 29 2.1.3.4. Contribution of the shear reinforcing 29 2.1.3.5. Contribution of prestressing tendons 33 2.2. Proposed Punching Capacity Increase due to the use of Fibre Reinforced

Concrete 33

2.2.1. Experimental testing 34 2.2.2. Prediction of Punching Shear Strength 35 2.2.3. Observations and discussions based on the experiments 36

3. Current Design Practice 39

3.1. German design code – DIN 1045-1988 40 3.2. British Standard 8110-1:1997 42

3.3. ACI 318M-02 45

3.4. Eurocode 2 48

3.5. DIN 1045-1:2001 49

3.6. CSA A23.3 53

3.7. CAN/CSA-S6-00 Canadian Highway Bridge Design Code 54 3.8. Comparisons of code equations for punching shear with and without shear

reinforcing – standardized approach. 56 3.8.1. German design code DIN 1045 (88) 58

3.8.2. Eurocode 2 (EC2) 59

3.8.3. British Standard 8110-1:1997 61

3.8.4. ACI 318-95 63

3.8.5. DIN 1045-1 (2001) 65

3.8.5.1. Punching shear resistance of a slab without shear reinforcing 66 3.8.5.2. Maximum punching shear capacity 66 3.8.5.3. Punching shear strength within the shear reinforced area 67 3.8.5.4. Punching shear strength outside the shear reinforced area 67 4. Accuracy of Modelling and Codified Design Rules 69 4.1. Accuracy of Experimental Testing 69 4.1.1. Single Column Tests 69 4.1.1.1. The effect of boundary conditions 70 4.1.1.2. The effect of compressive membrane action 71

4.1.2. Slab subsystems 72

4.2. Accuracy of Code Predictions 72 4.2.1. Compilation of databank 73 4.2.2. Comparisons between Design Code Rules and Experimental Results for Flat

Slabs without Shear Reinforcing 74 4.2.3. Comparisons between Design Code Rules and Experimental Results for Flat

Slabs with Shear Reinforcing 75 4.2.4. Discussion of the Comparison of Test Data and Codified Predictions 77

5. Proposed Repair Methods for Punching Shear Failure and Preventative Measures against Punching Shear Failure 78 5.1. Strengthening of Existing Slab-column Connections 79 5.1.1. Increasing the Effective Slab Depth 79 5.1.1.1. Slab strengthened with additional concrete and vertical bolts 79 5.1.1.2. Slab strengthened with additional concrete and bonded steel plate80 5.1.2. Increasing the area of load transfer 81 5.1.3. Installation of additional shear reinforcing 82 5.1.3.1. Doweling additional bars into the existing slab 82 5.1.3.2. Slab strengthened with vertical bolts 83 5.2. Repairing slab-column connections showing distress due to punching shear

failure or near failure 84

5.2.1. Proposed Classification of Damage 84 5.2.1.1. Damage Level 1 – Minor to medium levels of damage 85 5.2.1.2. Damage Level 2 – Medium to severe levels of damage 85 5.2.1.3. Damage Level 3 – Extreme levels of damage 86 5.2.2. Proposed remedial works for the different levels of damage 86 5.2.2.1. Repair of Damage Level 1 87 5.2.2.2. Repair of Damage Level 2 87 5.2.2.3. Repair of Damage Level 3 88 5.3. Retrofitting of slab-column connections for improved behaviour under seismic

loading conditions 88

5.3.1. Fibre reinforced concrete infill panel 89 5.3.2. Demolition of part of the existing concrete slab and replacement thereof with

fibre reinforced concrete 90 6. Experimental Testing of an Undamaged Slab-column Connection 91 6.1. Experimental Test Setup 92 6.2. Proposed Test Procedure 95 6.3. Actual Test – Virgin test panel 96

6.3.1. Material Test Results 96

6.3.2. Load Application 97

6.3.3. Placement and Setting Up of the Test Panel 98 6.3.4. Original Panel – Load Application 1 99 6.3.5. Original Panel – Load Application 2 104 6.3.6. Original Panel – Load Application 3 106 6.3.7. Original Panel – Combination of results – Loads 1, 2 &3 109 6.3.8. Verification of test results with the method proposed by Menétrey (2002)

112

7. Experimental Testing of Repaired Slab-column Connection 117 7.1. Classification of Damage and Proposed Method of Repair 117 7.2. Repair of the damaged slab 118 7.3. Testing of the Repaired Panel 121 7.3.1. Material Test Results 121

7.3.2. Load Application 121

7.3.3. Repaired Panel – Load Application 1 122 7.3.4. Repaired Panel – Load Application 2 128 7.3.5. Dismantling of the failed slab panel 135 8. Conclusions and Recommendations 139

8.1. Conclusions 139

8.2. Recommendations 140

9. Appendix A – Estimation of the experimental model‟s punching shear capacity and the design of the required shear reinforcing 142 10. Appendix B – Calculations using the Mechanistic Model Proposed by Menétrey147

10.1. Virgin Test Panel 147

11. Appendix C – Construction Details 155 12. Appendix D – Prediction of Flexural Failure at Slab-column Connections – Yield

Line Approach 158

13. Appendix E – Method for Epoxy Crack Injection 160

1. Overview of Punching Shear Failure

1.1. Introduction

Building construction with flat plates has become more and more popular lately. This construction method may dominate all modern reinforced concrete

construction in conventional buildings.

Flat plates – better known as flat slabs – offer numerous advantages:  Architectural flexibility

 More clear spaces

 Reduced overall building height – equating to lower construction and maintenance cost

 Simpler and less costly formwork systems  Shorter construction times

Some of the disadvantages associated with the use of flat slabs systems are the following:

 From a serviceability point of view, designs are often governed by

deflection criteria. Accurate estimation of deflections in two-way spanning slab systems is debatable.

 Flat slab construction is penalised by certain codes in seismic regions, e.g. Eurocode and SABS(SANS).

 From an ultimate limit state point of view the greatest limiting factor in the design process is punching shear failure of the flat slab at the column-slab interfaces.

Punching shear failure can be defined for two specific cases. Firstly, flat slabs

without shear reinforcing, and secondly flat slabs with shear reinforcing.

Flat slabs without shear reinforcing typically tend to fail in a brittle manner with the telltale signs of failure being a conical concrete plug perforating the slab in

combination with a fair amount of flexural cracking evident on the top surface of the slab. The brittle behaviour of the slab-column connection at failure is clearly depicted on a load-deflection curve (Fig 1.1), showing a sudden loss in load carrying capacity of the connection.

Flat slabs with shear reinforcing commonly fail in a less spectacular fashion. The addition of shear reinforcing causes increases the toughness of the connection. The failure mode is shifted from pure punching failure towards a more ductile flexural failure mode. This intermediate failure behaviour can be seen on the load-deflection plot (Fig 1.1). Even though the connection is more resilient it still shows a rather steep decline in load carrying capacity. At failure the connection shows more warning of distress by means of a more pronounced flexural

cracking pattern originating at the column, and circumferential cracking around the loaded surface. In some cases delamination of the concrete at the level of the tension reinforcing may occur.

Fig. 1.1 Response curves for flexural- and punching failure (Menétrey 2002)

Both failures, with and without punching shear reinforcing, can be considered as brittle failure designated by the sudden reduction in the load carrying capacity of the structure. Due to their sudden nature these failures are more often than not, disastrous. However with shear reinforcement a more acceptable failure can be achieved.

A number of structural failures and collapses can be attributed to punching shear failure of slab-column connections, of these a few examples are shown in

Fig 1.2 and Fig 1.3.

Fig. 1.2 Progressive collapse of the Sampoong department store in Seoul Korea

(Gardner et al 2002)

During the years numerous ways of countering punching shear failure of flat slabs have been proposed and used, all with varying rates of success. Some of these methods are:

 Drop panels  Column capitals

 Additional flexural reinforcing  The use of pre-stress

 Pre-fabricated shear heads

Fig. 1.3 Collapse of the upper parking deck at Pipers Row Car Park,

Wolverhampton, UK (Wood et al, 1998)

Numerous codified approaches to the design of shear-reinforced slabs exist, as well as number of mechanical and numerical models to predict the punching behaviour and capacity of flat slabs. Most of these methods are based on limited number of tests and formulated in such a way that no clear comparison between the methods can be used, even though most of them are defined by the same parameters.

This report aims to compare the various approaches to predict shearing resistance of slab-column connections. One approach for punching shear

enhancement is selected and studied both analytically and experimentally. After initial testing of the undamaged plate a repair will be attempted. The capacity of the repaired slab will be compared with the original capacity and verified with a mechanistic failure model.

1.2. Classical Punching Failure

At overloading a typical slab-column connection will fail at a perimeter, proportional to the effective depth of the slab, from the column face.

Excessive loading, in combination with an unbalanced moment over the column will cause the shear stress on this critical perimeter to exceed the capacity of the structural system.

These shear stresses cause angled cracks to develop from the column face to the upper surface of the slab. In slabs without shear reinforcing the crack growth is rapid resulting in a concrete plug being pushed out of the slab. This behaviour is clearly illustrated in Fig 1.4.

Fig. 1.4 Punching failure of slab without shear reinforcing (Beutel 2002)

When the slab has been reinforced with shear reinforcing in the area around the column the reinforcing stirrups, clips or studs bridge the cracks and prevents the conical concrete plug to separate from the rest of the slab. The behaviour of a slab-column connection with shear reinforcing is illustrated in Fig 1.5. The shear reinforcing also causes cracking to migrate away from the column. If the shear capacity of the concrete outside the shear-reinforced zone is insufficient the failure will be similar to a connection without shear reinforcing.

Fig. 1.6 Punching failure zone evident on top of a failing slab (Wood et al 1998)

In the event that the moment transfer to the column is negligible, the punching failure would typically be similar to the failure pattern shown in Fig 1.6. However, when moment transfer is more significant, the circumferential degradation would only be visible on one side of the column – corresponding to the stress

distribution indicated in Fig 1.7.

1.3. Punching failure due to lateral loading of the structure

It is common practise to design multi-storey buildings with two independent structural systems. The first for resisting gravity loading and the second to resist lateral loading imposed by wind and earthquake excitations, where applicable. In most cases these two systems are designed independently.

However, due to building drift and the flexibility of the gravity resisting structure (mostly reinforced concrete or post-tensioned slabs) unbalanced moments develop over the columns. The stresses caused by the moment transfer to the columns are additional to the stresses caused by the normal gravity loading –Fig

1.7.

(a) (b)

Consequently slab-column connections may fail at gravity loads below their intended design scope if significant effects of lateral loading are present.

2. Proposed Analytical and Empirical Models

Various approaches to predict punching shear resistance have been formulated in the available literature and the numerous structural design codes. In this section two interesting approaches will be studied. Firstly the method proposed by Menétrey, a particularly rigorous method, taking different components of resistance into account. Secondly a method proposed to include the beneficial shear properties of fibre

reinforcing will be overviewed. In the next chapter the different approaches of various design codes will be compared.

2.1. Synthesis of Punching Shear Failure, as proposed by

Menétrey (2002)

Menétrey presents a general model for predicting the punching capacity of a slab. The punching resistance of the slab is obtained by integrating the vertical components of the tensile stresses around the punching crack. The contribution of flexural reinforcing, shear reinforcing, prestressing tendons and the inherent resistance of the concrete are accounted for by means of addition of the vertical components of tensile forces of each crossing the punching crack.

2.1.1. Experimental results

The tests conducted by Menétrey focused on the difference between

flexural- and punching failure of slabs. Flexural and punching failure can be distinguished with the help of a load vs. deflection plot for the test, see Fig

1.1. A steep capacity drop for increasing deflection characterizes punching

failure, while flexural failures show a rather steady decrease in load carrying capacity with increased deflection.

The experiments show that increasing the cross-sectional area of the flexural reinforcing can increase the failure loads. However, by increasing the bending steel even more, a transition is made from a flexural, fairly tough failure to brittle punching shear failure at higher loads.

Another experimental observation was that controlling the shape of the punching cone, i.e. different cone inclinations, it is possible to reveal a transition between punching and flexural failure.

The inclination of a pure punching failure crack is in the order of 30°. This is reported both by Menétrey and Mervitz (1971), who studied a series of flat slab punching experiments. However, Menétrey managed to control the inclination of the punching crack artificially. Placing a reinforcing ring concentrically around the column position and varying the ring radius achieved control of the crack inclination, since the shear crack always crossed the reinforcing ring. By increasing the crack inclination (), see Fig

2.1, the behaviour became less brittle. The most ductile failure was

achieved at  = 90°. This angle in fact implies flexural failure, as the crack is perpendicular to the flexural stresses in the slab.

Fig. 2.1 Punching cone shape enforced by steel ring reinforcement

By denoting the failure load for 30 0 

as the punching load Fpun, and

the load at flexural failure (90) as Fflex, these results could be fitted with

the following expression:









2.1.2. Numerical simulations

Finite element analyses were performed by Menétrey (2002) to study the slab column interaction. Thereby insight was gained, leading to the

eventual formulation of an analytical expression for punching shear capacity prediction.

2.1.2.1. Model description

Axi-symmetry was considered for analysing punching shear in round plates. Finite element analysis enabled the consideration of the complicated stress state in the structure. As failure criterion, the concrete constitutive law developed by Menétrey & William (1995) was considered. The dilatancy observed experimentally is matched to a specific flow rule. Concrete cracking is described using a smeared crack approach with a strain softening formulation. For this softening Hillerborg, et al‟s (1976) fictitious crack model is used for reducing the tensile stresses (t) as controlled by the crack opening (w) and the

fracture energy, which is defined as the amount of energy absorbed per unit area in opening of the crack from zero to the crack rupture opening wr.



The fracture energy is forced to be invariant with the finite element size by adoption of the crack band concept by Bazant and Oh (1983). The connection between the brittleness of failure and the state of stress is reproduced by the introduction of a fictitious number of cracks.

2.1.2.2. Simulation of the punching failure

The analyses were done on slabs similar to those of Kinnunen & Nylander (1960). These slabs were chosen because of their perfect axially symmetric geometry. The cracking phenomenon in the vicinity of the column is clearly shown by the simulation, in reasonable

agreement with experimentally observed cracking.

The punching crack is initiated by the coagulation of micro cracks at the top of the slab. As the vertical displacement increases, the inclined crack expands towards the corner of the slab-column intersection.

Simultaneously the other inclined micro cracks are closing. Failure occurs when the inclined crack reaches the corner of the slab-column connection.

The crack angle is found to be approximately 45°, as opposed to the 30° reported earlier. This phenomenon is ascribed to the effect of the upper layers of flexural reinforcing, which directs the initially 40°-45° shear crack to an eventual 30° by delamination along the longitudinal reinforcing. This observation was also made by Mervitz (1971).

2.1.2.3. Parametric analysis

Having found reasonable numerical results, a parametric analysis was performed on a circular slab reinforced with orthogonally placed

reinforcing. Partial bond between the concrete and reinforcing steel was simulated by means of rigidly fastening the reinforcing element to the concrete at the end of a fictitious fastening length. This fictitious length allows some cracks to grow, while others close. The fictitious length is related to the observed spacing of cracks in tensile tests of reinforced concrete.

In Fig 2.2 and Fig 2.3 it is shown that the load capacity increases with increased uniaxial tensile strength of the concrete, as well as with increasing fracture energy.

Fig 2.2 Influence of tensile concrete strength on response curves

(Menétrey 2002)

Fig 2.3 Influence of fracture energy on response curves (Menétrey

2002)

By varying the percentage of flexural reinforcing the following can be shown: Firstly, all the slabs show a similar cracking pattern,

regardless of the percentage of longitudinal flexural reinforcing.

Secondly, all the slabs show similar initial elastic behaviour. Lastly it is shown that the post-elastic behaviours vary considerably with varying percentages of reinforcing. The higher the reinforcing ratio is, the higher is the failure load, and with increasing reinforcing percentages the ductility of the connection decreases. This is in agreement with the experimentally observed transition from flexural, tough failure to high capacity brittle punching failure.

Fig 2.4 Influence of the percentage of flexural reinforcing on response

curves (Menétrey 2002)

The size effect was simulated using varying slab thicknesses with similar scaling factors applied to the concrete geometry and reinforcing steel area, while the boundary conditions and material characteristics remained similar. The nominal shear stresses at failure were computed as follows:





d Effective depth of the slab Pfailure Failure load

rs Column radius

n Nominal shear stress

Assuming constant fracture energy, the size-effect law by Bazant & Cao (1987) can be used. The experimental data is adjusted using the RILEM recommendations for linear regression yielding the following equation: 2 1 34 1 55 . 1            ft d n  (2.4)

d Effective depth of the slab ft Uniaxial tensile strength of the

concrete

Fig 2.5 Size-effect law obtained by numerical analysis (Menétrey

2002)

2.1.3. Analytical Model

The model is based on the assumption that the punching load is influenced by the tensile stress in the concrete along the inclined punching crack. The magnitude of the punching load is obtained by integrating the vertical stress components along the punching crack and summation of the vertical force components of the flexural reinforcing, shear reinforcing and

prestressed tendons crossing the punching crack. Thus the general formulation is: p sw dow ct pun F F F F F     (2.5)

Fct Vertical component of concrete

tensile force

Fdow Dowel force contribution by the

flexural reinforcing

Fsw Vertical components of force in

the shear reinforcing

Fp Vertical components of force in

the prestressing tendons

Even though punching failure is sudden it is due to the amalgamation of micro cracks. This formation takes place progressively and consequently the steel forces are activated gradually and can be added to the tensile concrete forces.

Fig 2.6 Typical cross-section showing relevant parameters (Menétrey 2002)

2.1.3.1. Punching- vs. Flexural capacity

The influence of the inclination of the punching crack can be expressed by equation 2.1, with30 90.

The following special cases can be highlighted:

  Ffail = Fpun

  Ffail = Fflex

Menétrey calculated Fflex using the following equation:

e s r flex r r m F     1 2  (2.6)

mr Bending moment resistance

rs Column radius

re Radius of the slab

2.1.3.2. Tensile force in the concrete

The punching crack is assumed to form the border of the punching cone. The bottom radius is defined as r1 and the top radius as r2.

 tan 10 1   r d r s (2.7)  tan 2 d r r  _s  (2.8)

Subsequently the inclined length is:



 



1

2 r 0.9 d

r

s    _(2.9)

In order to simplify the formulation, a constant stress distribution is assumed, leading to the vertical component of the concrete tensile force being:









The shear resistance is seen to be proportional to the concrete tensile strength to the power 2/3, i.e.

3 2

t ct f

F  (2.11)

From the results of the numerical simulations, Menétrey determined the influence of the percentage of longitudinal reinforcing to be approximated by the following relations:

35 . 0 46 . 0 1 . 0  2         _for 02% 87 . 0   _for  2% _(2.12)

The size effect is incorporated in the formulation with the following expressions : 2 1 1 6 . 1           a d d  with d 3da _(2.13)

In order to predict the failure load of a slab with shear reinforcing and a failure outside the shear reinforced area the parameter  is used.

25 . 1 5 . 0 1 . 0 2            h r h r_{s s}  for 5 . 2 0  h r_s 625 . 0   _for r_hs 2.5 _(2.14)

2.1.3.3. Contribution of the dowel effect

According to Menétrey the contribution of longitudinal reinforcing crossing the punching crack can be evaluated as being equal to:









s Axial tensile stress in reinforcing bar

fs Reinforcing yield strength

 Angle between punching crack and reinforcing, in the vertical plane

2.1.3.4. Contribution of the shear reinforcing

Different types of shear reinforcing are used to increase the failure load of slabs and to lessen the sudden decrease in load carrying capacity of the slab, i.e. to improved post-peak ductility. Generally systems such as studs, stirrups, bent-up bars and bolts are used.

Three different positions of the punching crack are possible at failure. 1. Punching crack between the column face and the first row of

The calculation should consider the interaction between the punching load and the flexural capacity in terms of the crack inclination  and the bending failure load.

        d r r_{swi s} arctan 1  (2.18)

Fig 2.7 Crack position 1.

2. Punching crack outside the shear reinforced area. The capacity of this scenario is calculated in a way similar to a slab without shear reinforcing. Instead of using the column radius (rs), the

radius of the outermost row of reinforcement (rsc) should be

used. The size effect is to be considered using the parameter

.

Fig 2.8 Crack position 2.

Fig 2.9 Crack position 3.

The ultimate punching load is to be the minimum value calculated from the three cases presented above.

The punching load in scenario 3 can be calculated as follows:

Firstly some distinction is to be made with regard to the bond properties of the shear reinforcing. Reinforcing made of plain bars will be denoted as studs, and those made with high bond (deformed) bars will be denoted as stirrups.

The contribution of injected strengthening bolts, installed after drilling through the slab, will be determined similar to either stirrups or studs, depending on their respective bond properties.

Interestingly Menétrey & Brühwiler (1997) found that non-injected bolts do not interact and consequently the

concrete- and shear reinforcing contributions cannot be added.

Fig 2.10 Crack formation in a stud-reinforced slab

The failure mechanism is initiated by the formation of micro cracks. Due to the crack formation the slab depth increases and resulting in the reinforcing bars to start taking load. The studs are subjected to displacement controlled loading. The displacement corresponds to the summation of the micro cracks opening.

The stud elongation at failure can be expressed as: 

cos  

l wr (2.19)

Consequently the deformation is:

l w_r sw  cos    (2.20)

The crack rupture opening (wr) is approximated as:

sw f r f G w  5 (2.21)

The maximum force can then be expressed as:



reinforcing contribution decreases at a rate inversely proportional to the stud length.

sw sw r f E w l₀  cos (2.23)

The contribution of high bond bars can be evaluated in a similar way. Due to the micro crack formation and the increased slab depth the generated tensile forces in the bars are distributed beyond the micro crack zone by means of bond stress to the concrete along the transmission length. The transmission length is defined as the length of bar along which slip between the steel bar and concrete occurs. If the necessary length is available the yield stress of the stirrup can be reached.



If the required length is not available, the force developed in the stirrup is a function of the anchorage at the stirrup‟s extremity.

2.1.3.5. Contribution of prestressing tendons

Taking the contribution of inclined prestressing tendons into account can enhance the punching shear resistance of a slab.



2.2. Proposed Punching Capacity Increase due to the use of

Fibre Reinforced Concrete

Harajli et al (1995) propose a design equation to predict the increased

resistance to punching shear failure of flat slabs by using deformed steel fibre reinforcing in the concrete. The equation is based on a number of small-scale test specimens. These tests were also compared to work done by Alexander & Simmonds (1992).

Due to the brittle nature of punching shear failure, it should be avoided at all costs. The general design philosophy of the North American codes (ACI & CSA) is to design flexural members in such a way that the structure develops a yield mechanism and therefore fails in a ductile, flexural manner.

From this point of view and the known fact that fibre reinforcing enhances the mechanical properties of concrete, by controlling crack growth, numerous researchers have investigated the influence of fibres on slab-column

connections. Fibre reinforcing leads to higher load carrying capacities, improved ductility of shear failure and better energy absorption properties.

However, experimental studies are still limited and there is no established method to predict the contribution of the fibre reinforcement as a function of the fibre parameters. Harajli et al used the following experimental setup to calibrate the capacity enhancement due to fibres:

2.2.1. Experimental testing

The panels tested by Harajli et al (1995) consisted of square slabs (650mm x 650mm) with a monolithically cast 100mm x 100mm column. Two slab thicknesses were used, i.e. 55mm and 75mm. The specimens are representative of slabs setups with span-depth ratios of 26 and 18 respectively. Two identical slabs for each different input variable were tested to minimize possible scatter.

The slabs were rather heavily reinforced ( = 1.12%) in order to ensure that they failed by means of punching prior to flexural failure. Fibre reinforcing consisted of one of the following:

 Loose 30/50 hooked steel fibres (30mm long, 0.5mm diameter)  Collated 50/50 hooked steel fibres

 12.5mm long monofilament polypropylene fibres

The slabs were reinforced with fibres at the following densities:  80kg/m3 _{– 1% 30/50 fibres}

 160kg/m3 _{– 2% 30/50 fibres}

 35kg/m3 _{– 0.45% 50/50 fibres}

 64kg/m3 _{– 0.8% 50/50 fibres}

 8.8kg/m3 _{– 1% polypropylene fibres}

Fig. 2.11 Typical specimen cross-section showing reinforcing details

(Harajli et al 1995)

2.2.2. Prediction of Punching Shear Strength

In order to obtain the design capacity of the connection, the capacity of a normal slab setup without fibres is added to the additional capacity provided by the fibres.

The best-fit equation for the prediction of the additional capacity is:





Adjusted to a zero y-intercept and a reduction factor of 0.9 a reasonably safe equation follows:





The above equations are limited to cases where fibre reinforcing is less than 2% volume fraction and where the reinforcing used is similar to those of the experiments, i.e. hooked-, crimped-, corrugated- and paddle fibres.

2.2.3. Observations and discussions based on the

experiments

Harajli et al (1995) concluded the following:

1. The addition of steel fibres increased the ultimate punching shear capacity of a slab-column connection by ±36%

2. The increased punching capacity is related to the volume fraction of fibres added and not the length or aspect ratio of the fibres

3. Steel fibres cause the failure mode to change from punching to flexural- or combined flexural-punching failure

4. Improved ductility of shear failures

5. The inclination of the shear failure plane decreased with the addition of steel fibres. This causes the failure surface to move away from the column face, resulting in an increased failure load.

6. Polypropylene fibres led to improved ductility and energy absorption in the post-failure portion of the test. However, the polypropylene fibres made an insignificant difference in the ultimate failure loads.

From the experimental results it is clear that the punching capacity increases linearly with an increased volume of steel fibres. This increase is not

significantly influenced by the span-depth ratio of the slabs.

Table 2.1 provides a summary of the behaviour of the two groups of slabs

tested, accompanying this the load vs. deflection behaviour of the panels are illustrated in Fig 2.12 and Fig 2.13.

Slab Fibre Volume Fraction (%) Aspect Ratio Failure Mode

Test ACI Test/ACI

Normalized Strength A1 - 0.0 - Punch 0.53 0.33 1.61 A2 Steel 0.45 100 Punch 0.57 0.33 1.73 A3 Steel 0.8 100 Flexural 0.64 0.33 1.94 A4 Steel 1.0 60 Flex-Punch 0.64 0.33 1.94 A5 Steel 2.0 60 Flex 0.64 0.33 1.94 A6 Polypr.* 1.0 0.5in Punch 0.53 0.33 1.61 B1 - 0.0 - Punch 0.52 0.33 1.58 B2 Steel 0.45 100 Punch 0.60 0.33 1.82 B3 Steel 0.8 100 Punch 0.61 0.33 1.85 B4 Steel 1.0 60 Punch 0.64 0.33 1.94 B5 Steel 2.0 60 Punch 0.79 0.33 2.39 B6 Polypr.* 1.0 0.5in Punch 0.60 0.33 1.82

* Polypropylene Table 2.1 Summary of test variables and results (Harajli et al 1995)

Fig. 2.12 Normalized load-deflection behaviour for Series A slabs (Harajli

Fig. 2.13 Normalized load-deflection behaviour for Series B slabs (Harajli

3. Current Design Practice

The general approach to determine the punching shear capacity of slab-column connections can be summarized as follows: The shear strength of the concrete is determined on a predetermined critical perimeter (u0 – See Fig 3.1) at a specified

distance from the loaded area (i.e. column) – this is to cater for the presence of an inclined shear crack in the assumed region. If the capacity of the system is adequate, the connection can be deemed satisfactory. If the resistance is inadequate, either the slab depth or the cross section of the column needs to be increased. If this is not desired, additional shear reinforcing in the slab depth needs to be provided.

The shear reinforcing provides resistance additional to the shear capacity of the concrete and the dowel action of the flexural reinforcing. In order to determine the amount of reinforcing needed, the required area of reinforcing steel is evaluated on consecutive perimeters (u1, u2, u3, etc.) around the loaded area. Reinforcing is needed

up to a perimeter such that the following perimeter under consideration does not need any additional shear reinforcing.

Various prescriptions of what the considered parameters, such as the critical perimeters, contribution of the shear reinforcing, as well as other requirements are given by the different codes. In this chapter the requirements of the most important codified approaches are summarized.

Due to the unavailability of original copies of certain codes a simplified presentation used by Albrecht (2002) is used for codes marked with **. The simplification uses the following notation: h c c    2 2 1  (3.1) h d 0.85 (3.2) List of symbols:

c1 First sectional dimension of the column c2 Second sectional dimension of the column

h Total height i.e. depth of the concrete slab d Effective depth i.e. distance from the centroid

of the tension reinforcement to the extreme compression fibre

fyd Design yield stress of steel

u Control perimeter

VULS Ultimate load imposed on the connection

f Common loading factor determined by the

weighted average of the load factors for imposed and permanent loads (live and dead loads)

 Percentage of longitudinal tension reinforcing in the considered cross-section

Vrc Resistance provided by the concrete

Vmax Maximum allowed punching shear resistance

with shear reinforcing

3.1. German design code – DIN 1045-1988 **

According to the 1988 formulation of the German design code the critical perimeter is calculated as a circle concentric to an equivalent circular column cross section. This circle is a distance 0,5d from the equivalent circular column

face. Moment transfer to the columns is ignored if the panel spans differ by less than 33%.

Fig 3.2 Critical perimeter and relevant parameters

The calculations are based upon a circular column cross-section. However, rectangular cross-sections are converted to equivalent circles with radius dst.

2 1 13 . 1 c c d_st    _(3.3)

From this the critical perimeter can be determined,





u1.13 1 2 _(3.4)

It should be noted that the ratio of the side lengths of a rectangular column is limited to less than 1.5.

2 5 . 1 1 c c   _(3.5)

The contribution by the concrete and the longitudinal tensile reinforcing is expressed as





The maximum shear resistance allowed for slabs with shear reinforcing is

rc f V V _{ } 4 . 1 max  (3.7)

If the shear force is higher than the capacity of the concrete, shear reinforcing is to resist 75% of the force, i.e.

ULS s

sd V

The required cross-sectional area of shear reinforcing is: yd sd sv f V A  (3.9)

The shear reinforcing is to be placed into two consecutive perimeters. The first placed at 0.5d from the column face and the second at 1.0d from the column face.

3.2. British Standard 8110-1:1997

The requirements of the British standard stipulate that the critical perimeter is a rectangle at a distance 1.5d from the column face.

Fig 3.3 Critical perimeter and relevant parameters

Accordingly the control perimeter is:





u2 1 2 12 _(3.10)

In order to allow for moment transfer to the column the total shear force needs to be factored. In the absence of detailed calculation, internal column loads in braced structures with approximately equal spans; the enhancement is done with a predetermined factor of 1.15.

t eff V

V 1.15

(3.11)

In case moment transfer is calculated in the structural analysis, the shear load enhancement is determined according to the following equation

         x V M V V t t t eff 5 . 1 1 (3.12)

At corner columns and edge columns bending about an axis parallel to the free edge an enhancement factor of 1.25 can be used.

t eff V

V 1.25 (3.13)

Alternatively the enhanced shear force for edge columns bending about an axis perpendicular to the free edge can be calculated with the following equation.

         x V M V V t t t eff 5 . 1 1 (3.14)

Alternatively the shear force should be enhanced with a factor of 1.4.

t eff V

V 1.25 (3.15)

It should be noted that Mt may be reduced by 30% if an equivalent frame

analysis with pattern loading was done.

The maximum stress at the column face is not allowed to exceed the lesser value of:





MAX

fmax  0.8 cu,5 (3.16)

The concrete contribution to the shear resistance is derived as follows:

3 1 4 1 3 1 25 1 400 100 79 . 0                          cu m v s c f d d b A v  (3.17)

The nominal shear stresses on the specific perimeter under consideration can be calculated with the following equation:

d u V v   (3.18)

If the shear stress at the control perimeter is less than vc, no additional shear

The shear stress is to be checked on consecutive perimeters, each taken at 0.75d from the former perimeter, until a perimeter is reached where no shear reinforcing is needed. For the perimeters requiring reinforcing the amount of shear steel is determined as follows:

 v1.6vc













As Cross-sectional area of the longitudinal

tensile reinforcing

d Effective slab depth

fcu Characteristic concrete cube strength fyv Characteristic strength of the shear

reinforcing

Mt Design moment transferred to the column u Control perimeter

u0 First control perimeter taken at 1.5d V Factored shear force

Veff Effective shear force

x Length of the side of the considered perimeter parallel to the axis of bending

Asv Area of shear reinforcement

 Angle between the plane of the slab and the shear reinforcing

3.3. ACI 318M-02

The ACI recommendations consider a critical perimeter taken at 0,5d from the column face. Moment transfer to the column is assumed to be due to a stress distribution as indicated in Fig 3.4.

Fig 3.4 Critical perimeter, relevant parameters and shear distribution due to moment transfer

For non-prestressed members Vc should be taken as the lesser value of the

following: 6 4 2 1 0 ' d b f V c c c           (3.21) 12 2 0 ' 0 d b f b d V s c c          (3.22)

s Critical Section with:

40 4 Sides, i.e. internal columns 30 3 Sides, i.e. edge columns 20 2 Sides, i.e. corner columns

Table 3.1 Shear enhancement factors

d b f Vc c 0 ' 3 1  (3.23)

For prestressed members Vc should be taken as





c f f bd V

V   ' 0.3 ₀  (3.24)

with

-                       12 5 . 1 ; 29 . 0 b0 d MIN s p   (3.25)

- Vp – The vertical component of prestress

- c is to be taken as the ratio of the longest overall dimension of the

effective loaded area to the largest overall perpendicular

dimension of the effective loaded area. The effective loaded area is taken as the area that totally encloses the actual loaded are, for which the perimeter is a minimum.

Shear reinforcement is allowable in slabs where the effective depth is greater than 150mm.

c s n V V

V   (3.26)

Vc should be taken as above but not greater than Vc fcb0d ' 6 1  (3.27) s d f A Vs  v y (3.27)

Vn should not exceed Vn fcb0d ' 2 1

 (3.28)

When shear reinforcing is used the yield strength of the reinforcing is limited to 420MPa. The maximum allowable yield strength of the shear reinforcing is an empirical value. The reasoning behind limiting the tensile strength of the reinforcing is that with decreasing slab depth, full yield capacity of the steel is less likely to be reached before punching shear failure takes place.

The required shear reinforcing is placed in the slab similar to shear stirrups in beams – see Fig. 3.5. The control perimeter outside the shear-reinforced zone is taken at a distance 0.5d outside the last line of shear stirrups. However, the shape of the outer control perimeter is quite different from the original control perimeter.

When using the ACI recommendations for punching shear design it should be kept in mind that the required integrity steel must be provided and that the tensile reinforcing is adequate to resist bending failure.

Integrity reinforcing is the provision of adequately anchored sagging (bottom) reinforcing that has to be provided through the core of the column.

Fig 3.5 ACI control perimeters and shear reinforcing details (ACI 318M-02) List of symbols:

Av Area of shear reinforcing b0 Critical perimeter

d Effective depth

fc’ Characteristic compressive cylinder

strength of concrete

fpc Average pre-stressing stress after losses fy Shear reinforcing yield stress

s Spacing of shear stirrups

Vc Punching shear capacity of the concrete Vp Vertical component of pre-stress force after

Vs Punching shear capacity contributed by

shear reinforcing

s Shear enhancement factor

c Ratio of column dimensions

3.4. Eurocode 2 **

The Eurocode considers a control perimeter with rounded corners at 1.5d from the column face.

Fig 3.6 Critical perimeter and relevant parameters

Moment transfer is considered as a shear force per unit of perimeter.

u VULS sd



   (3.29)

 For internal columns the enhancement factor is 1.15

The control perimeter is calculated as:





u2 1 2 3 _(3.30)

It should be noted that the ratio of the column side lengths is limited to 2

2 2 1 c c   _(3.31)







If the shear force is higher than the capacity of the concrete, shear reinforcing is used with the total resistance calculated by the addition of the concrete- and steel resistances.

3.5. DIN 1045-1:2001

The latest DIN recommendations are formulated using a critical perimeter taken at a distance equal to1.5d from the column face.

When dealing with rectangular columns or walls the critical perimeters should be taken as indicated in Fig 3.7.

Fig 3.7 Critical perimeter for a rectangular column or wall









The critical perimeter is selected to be such that it has the shortest length and at a distance 1.5d from the column face – see Fig 3.8.

Fig 3.8 Typical Columns

Penetrations in the close proximity of the column should be taken into account as shown in Fig 3.9.

Fig 3.9 Penetrations close to the column

When corner and edge columns are located closer than 3d from edge of the slabs the critical perimeter should be taken according to Fig 3.10.

Fig 3.10 Corner and Edge Columns

The geometrical parameters used in the calculation of the punching shear capacity and the required reinforcing are indicated in Fig 3.11.

In order to take moment transfer between the slab and the column into account, the enhancement factors given in Table 3.2 are used to increase the shear stress around the column.

 Type of Support: 1.05 Internal Column

1.4 Edge Column 1.5 Corner Column

Table 3.2 Shear enhancement factors

u V v Ed Ed    (3.35)

Flat slabs without shear reinforcing should conform to:

max ,

Rd Ed v

v  (3.36)

Flat slabs with shear reinforcing should conform to the following: The upper limit of the punching capacity is given by

max ,

Rd Ed v

v  (3.37)

Within the shear reinforced area

sy Rd Ed v

v  _, (3.38)

Outside the shear reinforced area

a ct Rd Ed v

v  _{, ,} (3.39)

Flat slabs without shear reinforcing:













2 , ,x cdy cd cd      [MPa] (3.44) i c i Ed i cd A N , , ,   (3.45)

Flat slabs with shear reinforcing:

ct Rd Rd v

v _,max 1.5 _, (3.46)

For the first perimeter of shear reinforcing within 0.5d from the column face

u f A v

v_Rd,sy  _Rd,cs sw yd (3.47) For the perimeters with reinforcing within spacing s_w0.75d

w yd sw s c Rd sy Rd s u d f A v v       _,  , (3.48) ct Rd c Rd v v ,  , (3.49) 0 . 1 400 400 3 . 0 7 . 0 7 . 0 _s   d  (3.50)

Bent down bars within 0.5d from the column face is considered using the following equation: u f A v v_Rd,sy  _Rd,c1.3 ssin() yd (3.51) Outside the shear reinforced area the critical perimeter is taken as 1.5d from the last row of shear reinforcing, with

ct Rd a a ct Rd v v , ,   , (3.52) 71 . 0 5 . 3 29 . 0 1   d lw a  (3.53) List of symbols:

Ac Cross-sectional area of concrete under

consideration

Asw Area of shear reinforcing in the considered

d Effective depth of the tension steel

fck Design crushing strength of a standard

cylinder

fyd Design yield strength of reinforcing steel lw Radial distance from the column face to the

last reinforcing row

N Axial force on the above mentioned cross-sectional area

sw Spacing of the shear reinforcing u Critical perimeter

VEd Imposed axial column load vEd Design shear stress

VRd Design resistance shear load vRd Design resistance shear stress

 Angle of the bent down bar measured from horizontal. 45o60o

 Factor f moment transfer

 Shear enhancement factor

1 1.0 for normal concrete – refer to DIN1045-1

for lightweight concrete

 Size effect factor

s Effectiveness factor of shear reinforcing

cd Effective pre-stress in the considered

cross-section

3.6. CSA A23.3 **

The Canadian building code considers a critical perimeter taken at 0,5d from the column face. Moment transfer between the slab and column is similar to the assumptions of the ACI-318 code – Fig 3.12.

Fig 3.12 Critical perimeter, relevant parameters and shear distribution due to

moment transfer

The control perimeter is:





u2 1 2 4 _(3.54)

The contribution of the concrete and longitudinal reinforcing is given by:



 



The upper limit of resistance is set as:

2 max 12.82 h

V  _f  (3.56)

Similar to the ACI recommendations the capacity of the concrete and the shear steel can be added together. In principle the ACI and CSA approaches are similar, but the detailing of the shear reinforcing differs. According to the ACI the shear reinforcing is fixed as beam strips, while the CSA method uses evenly arranged reinforcing on the control perimeters.

3.7. CAN/CSA-S6-00 Canadian Highway Bridge Design Code

According to the Canadian Bridge Code the shear resistance of slabs should be the more severe of the following cases:

1. Beam action, with a critical section extending in a plane across the entire width and located at a distance, d, from the face of the concentrated load or reaction area, or from any change in slab thickness.

2. Two-way action, with a critical section perpendicular to the plane of the slab and located so that its perimeter, u, is a minimum. This perimeter need not be closer than 0.5d to the perimeter of the concentrated load or reaction area. The shear resistance should also be checked at critical

sections located at a distance no closer than 0.5d from any change in slab thickness and should be located such that the perimeter, u, is a minimum.

The shear resistance for two-way action is calculated as follows

f r V V  (3.57)





d Effective depth – distance from the extreme

compression fibre to the centroid of the tensile force (mm)

fcr Cracking strength of concrete (MPa)

fpc The average of the compressive stresses in

the two directions in concrete after all prestress losses have occurred, at the centroid of the cross-section (MPa)

u Perimeter of the critical section (mm)

Vf Shear demand (factored applied load) (kN)

Vp Shear resistance provided by reinforcing (kN) Vr Shear resistance (kN)

c Material resistance factor for concrete (0.75)

p Material resistance factor for reinforcing

3.8. Comparisons of code equations for punching shear with

and without shear reinforcing – standardized approach.

In order to compare the provisions made for punching shear by the various codes the method used by the International Federation of Structural Concrete (fib) will be presented.

The fib has done extensive research on the topic of punching shear and the related performance of available codified approaches to the problem of punching shear failure. In their publication “Punching of Structural Concrete Slabs” (2001) a comparison of the available test data and commonly used design code

predictions are presented. In order to compare the different codes standardization was necessary. The results of their standardization and comparisons are overviewed in the following sections.

Nominal punching shear stress is taken as a shear force (_FV_F), divided by a control surface around the loaded area. The resistance partial shear factor to avoid punching failure is determined by comparing the nominal shear stress of tests with a strength parameter of the concrete.

To determine admissible punching shear strength, partial safety factors for the actions (imposed loading) and resistances (material characteristics) are used.

R R F F V V     (3.59)

This states that the demand is less than the capacity of the system. Standardization of the punching shear capacity or resistance is done by considering the concrete shear resistance and that of the reinforcing steel superimposed as follows: c c outside c s s c c R Rd V V V V V      max ,     (3.60)

d u f

k

V_c _c  (_l)  (3.61) and the steel shear capacity contribution is

) sin(      sw s y s A f V (3.62) List of symbols:

Asw Cross sectional area of the shear reinforcing d Effective depth

f(l) Function of the longitudinal tension

reinforcing

fy Yield strength of the shear reinforcing k Size effect factor of the effective depth

u Control perimeter

Vc Characteristic punching resistance without

shear reinforcing, or the contribution of the concrete to the punching shear capacity in the presence of shear reinforcing

Vc,outside Characteristic shear capacity outside the

shear reinforced area

Vf Characteristic value of the acting force Vmax Characteristic maximum shear capacity

VR Punching shear capacity

Vs Characteristic shear capacity of the shear

reinforcing

l,g Ratio of the longitudinal tension reinforcing

 Inclination of the shear reinforcing

c Partial material factor for the concrete

f Partial safety factor for the imposed forces

s Partial material factor for the reinforcing steel

s Efficiency of the shear reinforcing