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Literature review of punching shear in reinforced concrete slabs
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School of Civil and Environmental Engineering
Structural Engineering, Mechanics and Materials
Research Report No. 09-10
Literature Review of Punching Shear in Reinforced
Concrete Slabs
for CEE 8956, Special Research Problem
by
Eva Lantsoght
August 2009
Summary
This literature review describes the four different methods to study the problem of
punching shear in reinforced concrete bridge deck slabs and the ACI 318 and EN 1992-1-1
code provisions regarding punching shear.
The first chapter of this literature review is an introduction to the problem of
punching shear in reinforced concrete bridge deck slabs. The lay-out of bridge deck slabs in
Dutch and North-American practice is introduced, as well as the wheel loading model used in
the European and North-American design codes. Punching shear failure in slabs is described
and the difference with shear failure in beams is explained.
The second chapter introduces the shear stress theory. This theory is the basis of the currently
used design codes. The influence of the concrete strength, the flexural reinforcement, the
support and loading conditions, the side length of the loaded area, the size effect, the
restraints and the column shape are studied and compared to experimental data.
The third chapter introduces the solutions based on plate theory and finite element methods.
The fourth chapter introduces the beam analogy method. The currently used beam analogy
method as well as the method used before 1960 are briefly explained.
The fifth chapter introduces the strut and tie model as developed by Alexander and
Simmonds (1986). Consequently, the bond model developed by Alexander and Simmonds
(1992) is introduced.
In the sixth chapter, the code provisions of ACI 318-08 and EN 1992-1-1:2004 are cited and
consequently compared to experimental data.
The seventh chapter is the discussion of the literature review. Critique on the
introduced theories is given and possible future work is pointed out.
The last chapter contains the conclusions of this literature review. A distinction is made
between the design practice based on the code equations and the analysis practice which
requires a thorough understanding of the mechanics at the basis of the punching shear
problem.
ii
Acknowledgements
First and foremost I would like to thank Dr. Lawrence Kahn for his guidance during
my reading of the literature on punching shear and during the writing of this report and for
advising me during my master’s studies.
I also would like to thank Prof. Joost Walraven, Dr. Cor van der Veen and Ir. Joop den Uijl,
Technical University of Delft, for sending information and literature during my reading and
Xuying Wei and Gert Jan Bakker for answering my questions regarding their MSc theses.
Funding for my master’s studies has been provided by the Belgian American
Educational Foundation and the Fulbright program. I am very grateful for the received
sponsorship.
iii
Table of Contents
Summary ................................................................................................................................... ii
Acknowledgements.................................................................................................................. iii
Table of Contents ..................................................................................................................... iv
List of Tables ............................................................................................................................ v
List of Figures .......................................................................................................................... vi
1. Introduction........................................................................................................................... 1
1.1. Purpose and objectives................................................................................................... 1
1.2. Shear in slabs ................................................................................................................. 6
1.3. Comparison with shear in beams ................................................................................... 7
2. Shear Stress Theory ............................................................................................................ 10
2.1. Shear stress on critical perimeter ................................................................................. 10
2.2. Critical shear crack theory ........................................................................................... 20
2.3. Influence of concrete strength...................................................................................... 23
2.4. Influence of flexural reinforcement ............................................................................. 26
2.5. Influence of support and loading conditions................................................................ 31
2.6. Influence of the side length of the loaded area ............................................................ 31
2.7. Size effect..................................................................................................................... 33
2.8. Influence of restraints .................................................................................................. 37
2.9. Influence of the shape of the loaded area..................................................................... 41
2.10. Discussion of shear stress theory ............................................................................... 42
3. Plate Theory and Finite Elements ....................................................................................... 44
3.1. Material modeling........................................................................................................ 44
3.2. 3D solids model ........................................................................................................... 44
3.3. Axi-symmetric model .................................................................................................. 46
3.4. Discussion of plate theory and finite elements solutions............................................. 49
4. Beam Analogy .................................................................................................................... 50
4.1. Currently used beam analogy....................................................................................... 50
4.2. Design based on the equivalent width of a beam strip................................................. 50
4.3. Discussion of beam analogy methods.......................................................................... 51
5. Strut and Tie models ........................................................................................................... 52
5.1. Strut and tie model ....................................................................................................... 52
5.2. Bond model .................................................................................................................. 57
5.3. Discussion of strut and tie models ............................................................................... 61
6. Code Provisions .................................................................................................................. 62
6.1. ACI 318-08 .................................................................................................................. 62
6.2. EN 1992-1-1: 2004 ...................................................................................................... 62
6.3. Comparison of code provisions ................................................................................... 64
6.3.1. Comparison of ACI 318 and EN 1992-1-1 ........................................................... 64
6.3.2. Size effect.............................................................................................................. 67
6.3.3. Influence of concrete strength............................................................................... 68
6.3.4. Influence of reinforcement ratio ........................................................................... 69
7. Discussion ........................................................................................................................... 71
8. Conclusions......................................................................................................................... 73
9. References........................................................................................................................... 75
Appendix A: Notations ........................................................................................................... 78
A.1. List of notations by symbol......................................................................................... 78
A.2. Table of notations by parameter.................................................................................. 82
iv
List of Tables
Table 1: Shear stresses at the inclined cracking load, Moe (1961)......................................... 13
Table 2: Conversion factors for compressive concrete strength, Dilger, Birkle and Mitchell
(2005)...................................................................................................................................... 26
Table 3: Test values with the analytical (Pa), experimental (Pe) and predicted (Pp) solutions,
Bakker (2008). ........................................................................................................................ 39
Table 4: List of test results and calculated results, Wei (2008). ............................................. 41
Table 5: Comparison of test results reported by Taylor, Ranking, Cleland and Kirckpatrick
(2007) and finite element modeling, Bakker (2008)............................................................... 49
Table 6: Overview of the notations by parameter................................................................... 82
v
List of Figures
Fig. 1: Punching shear failure of a bridge deck, Ngo (2001).................................................... 1
Fig. 2: Traffic load according to the NEN-EN 1991-2, Bakker (2008). ................................... 2
Fig. 3: Dimension of the ZIP profile and the edge beam, Bakker (2008)................................. 3
Fig. 4: Cross-section of the bridge and one span of the compression layer zoomed into,
Bakker (2008). .......................................................................................................................... 4
Fig. 5: Characteristics of the design truck, US customary units, AASHTO (2007). ................ 5
Fig. 6: Characteristics of the design truck, SI units, AASHTO (2007). ................................... 5
Fig. 7: Typical North-American cross-section of a bridge, PCI (2003). .................................. 6
Fig. 8: Crack formation in column area of slab, ASCE-ACI Committee 426 (1974). ............. 8
Fig. 9: Horizontal forces on sections near inclined cracks, ASCE-ACI committee 426 (1974).
................................................................................................................................................... 8
Fig. 10: Testing arrangement, Moe (1961). ........................................................................... 11
Fig. 11: Inclined cracking observed by Moe (1961)............................................................... 11
Fig. 12: Response curves for flexural and punching failure, Menétry (1998), taken from
Menétry (2002). ...................................................................................................................... 12
Fig. 13: Punching cone with different inclinations: 30°, 45° and 60°, Menétry (1998), taken
from Menétry (2002)............................................................................................................... 12
Fig. 14: Stresses after inclined cracking, Moe (1961). ........................................................... 13
Fig. 15: Schematic presentation of cracking, Theodorakopoulos and Swamy (2002). .......... 14
Fig. 16: Strain and stress distribution in concrete section, Theodorakopoulos and Swamy
(2002)...................................................................................................................................... 15
Fig. 17: Interaction between shearing and flexural strength, Moe (1961).............................. 16
Fig. 18: Comparison of test results and Eq. 8, Moe (1961). ................................................... 17
Fig. 19: Elimination of the possibility of shear failure in slab, Moe (1961)........................... 18
Fig. 20: Eq. 9, Moe (1961)..................................................................................................... 19
Fig. 21: Influence of relative shearing strength (Eq. 10) Elstner and Hognestad (1956). ...... 20
Fig. 22: Test by Guandalini and Muttoni: (a) cracking pattern of slab after failure; (b)
theoretical strut developing across the critical shear crack; (c) elbow-shaped strut; and (d)
plots of measured radial strains in soffit of slab as function of applied load, Muttoni (2008).
................................................................................................................................................. 21
Fig. 23: Slab deflection during punching test: (a) measured values of w at top and bottom
face of a slab tested by Guandalini, Burdet and Muttoni (2009); and (b) inpterpretation of
measurements according to critical shear crack theory. ......................................................... 22
Fig. 24: Design procedure to check the punching strength of a slab, Muttoni (2008)............ 23
Fig. 25: Influence of concrete strength, Elstner and Hognestad (1956). ................................ 24
Fig. 26: Effect of concrete strength on shear strength (tests by Elstner and Hognestad, (1956),
figure by Mitchell, Cook and Dilger (2005). .......................................................................... 25
Fig. 27: Comparison of square root and cube root functions with test results reported by
Ghannoum (1998) and McHarg et al. (2000), figure by Mitchell, Cook and Dilger (2005). . 26
Fig. 28: Failure load vs. flexural reinforcement ratio (h ≈ 150 mm ≈ 6 in ), from Dilger,
Birkle and Mitchell (2005)...................................................................................................... 27
Fig. 29: Test results from Vanderbilt (1972), from Dilger, Birkle and Mitchell (2005). ....... 28
Fig. 30: Test results from Mardouk and Hussein (1991) from Dilger, Birkle and Mitchell
(2005)...................................................................................................................................... 28
Fig. 31: Test results from Hallgren (1996), from Dilger, Birkle and Mitchell (2005). .......... 28
Fig. 32: Test results from Richart (1948), from Dilger, Birkle and Mitchell (2005).............. 28
vi
Fig. 33: Influence of flexural reinforcement ratio on punching shear strength according to
ACI 318-08 and EN 1992-1-1 (fck = 30 MPa = 4350 psi, fyk = 414 MPa = 60 ksi, c/d=1,
l/d=25, l1=l2), Guandalini, Burdet and Muttoni (2009)........................................................... 29
Fig. 34: Normalized load-deflection curve for all 11 specimens (deflection w was measured
between center of column and reaction points at perimeter), Guandalini, Burdet and Muttoni
(2009)...................................................................................................................................... 30
Fig. 35: Results of 99 punching tests analyzed by Guandalini, Burdet and Muttoni (2009) and
88 tests taken from the literature compared with the failure criterion of the critical shear crack
theory: (b) 99 tests with identification of the reinforcement ratio; and (c) 99 tests with
identification of effective depth. ............................................................................................. 31
Fig. 36: Effect of large c/d values on shear strength, ASCE-ACI committee 426 (1974)...... 32
Fig. 37: Beam and slab effect as function of r/d, Moe (1961). ............................................... 32
Fig. 38: Effect of the ratio bo/d on the punching shear strength of slab-column connections,
from Sherif, Emara, Ibrahim and Magd (2005). ..................................................................... 33
Fig. 39: Normalized punching shear stress ( v / f c' ) vs. average effective depth for tests
reported by Li (2000), Mitchell, Cook and Dilger (2005). ..................................................... 34
Fig. 40: Normalized punching shear stress ( v / f c' ) vs. average effective depth for tests
reported by Nylander and Sundquist (1972), from Mitchell, Cook and Dilger (2005). ......... 35
Fig. 41: Normalized punching shear stress ( v / f c' ) vs. average effective depth for tests
reported by Regan (1986) and Birkle (2004), from Mitchell, Cook and Dilger (2005). ........ 35
Fig. 42: Influence of slab thickness on failure stress in slabs without shear reinforcement,
Birkle and Dilger (2008). ........................................................................................................ 36
Fig. 43: Idealized restrained slabs: forces and stress distributions, Hewitt and Batchelor
(1975)...................................................................................................................................... 37
Fig. 44: Punching shear failure model, taking compressive membrane action into account,
Bakker (2008). ........................................................................................................................ 38
Fig. 45: Modified Hallgren model considering boundary restraint, Wei (2008). ................... 40
Fig. 46: Effect of column rectangularity on shear strength, ASCE-ACI committee 426 (1974).
................................................................................................................................................. 41
Fig. 47: Influence of rectangularity of column on shear strength from tests by Hawkins et al.
(1971), Leong and Teng (2000) and Oliveira et al. (2004), figure by Mitchell, Cook and
Dilger (2005)........................................................................................................................... 42
Fig. 48: Models for behavior of concrete in tension: brittle and Hordijk tension softening,
Bakker (2008). ........................................................................................................................ 44
Fig. 49: Ideal plastic model used for concrete in compression, Bakker (2008)...................... 44
Fig. 50: Dimensions of the 3D solids model. All the edges are clamped and horizontally
restrained. Dimensions are in mm, Bakker (2008). ................................................................ 45
Fig. 51: 3D solid element of DIANA software, Bakker (2008).............................................. 45
Fig. 52: Model of a slab using 3D solid elements. The load is applied to the red zone. Figure
by Bakker, personal communication 07-15-2009................................................................... 45
Fig. 53: The load-displacement graphs of the 3D solids model, Bakker (2008). ................... 46
Fig. 54: Axi-symmetric model, Bakker (2008)....................................................................... 47
Fig. 55: CQ16A element from DIANA (2008), used in the axi-symmetric model by Bakker
(2008)...................................................................................................................................... 47
Fig. 56: Load-displacement graphs for 4 different models of the axi-symmetric model,
Bakker (2008). ........................................................................................................................ 48
Fig. 57: Crack pattern just before (loadstep 16) and after (loadstep 17) failure, Bakker (2008).
................................................................................................................................................. 48
vii
Fig. 58: Limiting strength combinations for beam analogy, Park and Gamble (1999). ......... 50
Fig. 59: In-plane or anchoring struts, Alexander and Simmonds (1986)................................ 52
Fig. 60: Comparison of corbel with out-of-plane or shear strut, Alexander and Simmonds
(1986)...................................................................................................................................... 53
Fig. 61: Calibration of α, Alexander and Simmonds (1986). ................................................. 55
Fig. 62: Effect of the reinforcement density, Alexander and Simmonds (1986). ................... 56
Fig. 63: Space truss analogy for torsion, MacGregor and Ghoneim (1995)........................... 56
Fig. 64: Layout of radial strips, Alexander and Simmonds (1992). ....................................... 58
Fig. 65: Equilibrium of radial strip, Alexander and Simmonds (1992). ................................. 58
Fig. 66: Free-body diagram of one-half radial strip, Alexander and Simmonds (1992). ....... 60
Fig. 67: Bond model results using ACI one-way shear, Alexander and Simmonds (1992). . 60
Fig. 68: Verification model for punching shear at the ultimate limit state, figure 6.12 from
EN 1992-1-1:2004. ................................................................................................................. 63
Fig. 69: Typical basic control perimeters around loaded areas, Figure 6.13 from EN 1992-11: 2004. ................................................................................................................................... 63
Fig. 70: Comparison of test/predicted using ACI 318-05 with rounded corners shear
perimeter, by Gardner (2005). ................................................................................................ 65
Fig. 71: Comparison of test/predicted using ACI 318-05 with assumption of square shear
perimeter, by Gardner (2005). ................................................................................................ 66
Fig. 72: Comparison of test/predicted using CEB-FIP MC90 and EN 1992-1-1:2003, by
Gardner (2005)........................................................................................................................ 66
Fig. 73: Comparison of code expressions with results reported by Li (2000) from Mitchell,
Cook and Dilger (2005). ......................................................................................................... 67
Fig. 74: Failure criterion: punching shear strength as function of width of critical shear crack
compared with 99 experimental results and ACI 318-05 design equation, Muttoni (2008)... 68
Fig. 75: Comparison of code expressions with test results reported by Ghannoum (1998) and
McHarg et al. (2000), figure by Mitchell, Cook and Dilger (2005). ...................................... 69
Fig. 76: Effect of analysis on perceived scatter, Alexander and Hawkins (2005).................. 70
viii
1. Introduction
1.1. Purpose and objectives
The purpose of this report is to provide a review of the literature regarding the shear
capacity and behavior of bridge decks and two-way structural concrete slab systems. The
scope is limited; influence of shear reinforcement, moment transfer and holes in the slab is
not included. The notations are kept identical to the notations used in the referenced
literature. A list of notations can be found in Appendix A.
A typical punching shear failure of a bridge deck can be seen in Fig. 1.
Fig. 1: Punching shear failure of a bridge deck, Ngo (2001).
Currently, new attention is given to the punching shear problem, as the traffic loads
are increased and older bridges might not have enough punching shear resistance. The traffic
load model according to the EN 1991-2 is shown in Fig. 2. A typical lay-out of a bridge deck
using the ZIP-girder system from the Dutch company Spanbeton is shown in Fig. 3. The area
over which the wheel load is spread is 350x600 mm (13.73x23.62 in), as can be seen in Fig.
4.
Fig. 2: Traffic load according to the NEN-EN 1991-2, Bakker (2008).
2
Fig. 3: Dimension of the ZIP profile and the edge beam, Bakker (2008).
3
Fig. 4: Cross-section of the bridge and one span of the compression layer zoomed into,
Bakker (2008).
According to AASHTO (2007), either a design truck or a design tandem is used in
North-America. The design truck is given in Fig. 5 and Fig. 6. The design tandem consists of
a pair of 25 kip axles (110 kN) spaced 4 ft (1200 mm) apart and with a transverse spacing of
6 ft (1800 mm). The tire contact area is taken as a single rectangle with width 20.0 in (510
mm) and length 10.0 in (250 mm). The tire pressure is assumed to be uniformly distributed
over the contact area. A typical North-American bridge lay-out is shown in Fig. 7. The
girders are spaced at 9 ft on center (2.74 m) and the slab is 8 in thick (20.32 cm).
4
Fig. 5: Characteristics of the design truck, US customary units, AASHTO (2007).
Fig. 6: Characteristics of the design truck, SI units, AASHTO (2007).
5
Fig. 7: Typical North-American cross-section of a bridge, PCI (2003).
1.2. Shear in slabs
Although the mechanics of punching shear are not entirely understood, many methods
have been developed over the years. The first to experimentally study the punching problem
was Talbot in 1913.
According to Park and Gamble (1999), the behavior of the failure region is extremely
complex, because of the combined flexural and diagonal tension cracking and the 3D nature
of the problem. However, Moe (1961) stated: “safe design equations apparently can be
developed without a full understanding of the fundamental laws governing the phenomenon
under consideration”.
The methods which have been developed can be divided into 4 categories, according
to Park and Gamble (1999): methods which calculate a nominal shear stress on a critical
section, beam analogies in which slab strips are calculated as beams under a combination of
moment, shear and torsion, methods based on a combination of plate theory and nonlinear
finite element methods and truss models. These four methods are studied in this literature
review.
Alexander and Simmonds (1986) cited the description of Masterson and Long of the
four basic stages in the punching failure of a slab-column connection. First, flexural and
shear cracks form in the tension zone of the slab near the face of the loaded area. Then, the
slab tension steel close to the loaded area yields. Consequently, flexural and shear cracks
extend into what was the compression zone of the concrete. Finally, failure occurs before
yielding extends beyond the vicinity of the loaded area. A possible reason for punching is
rupture of the reduced compression zone in the slab.
6
According to ASCE-ACI committee 426 (1974), most available test data come from
slab-column tests. These tests consist of a slab-column specimen with the slab piece up to the
line of contraflexure. The behavior of slab-column specimens differs from the real behavior
of a slab, since in-plane forces cannot develop.
This literature review treats the four previously mentioned categories and the ACI 318
and EN 1992-1-1 code provisions. Most attention is given to methods based on a nominal
shear stress on a critical section, as stress methods are most commonly used in design and
serve as a basis for the code provisions.
1.3. Comparison with shear in beams
The difference between the shear stress theory in beams and slabs is that the nominal
ultimate shear stress which can be developed in a slab is higher than in a beam. According to
ASCE-ACI committee 426 (1974) this increase is due to the following effects: the location of
the inclined crack, the stress conditions at the apex of the crack, the proportionally greater
dowel forces, the distribution of moments, the lack of symmetry, the inadequacy of simple
static analysis and the in-plane forces generated by restraints provided by the supports and
non-yielding portions of the slab.
The distribution of moments can be seen from Fig. 8. The radial moment Mr
decreases at a rapid rate with distance from the loaded area. It causes yielding to develop first
at the perimeter of the loaded area. However, an increasing tangential moment Mθ will
restrain any rotation at the inclined crack. Most punching failures do not occur until
aggregate interlock effects are markedly diminished by yielding in both the Mr and Mθ
direction.
When the direction of the principal moment does not match the direction of the layout of the reinforcement, in-plane compression forces are developed to balance the tension
force resulting from the reinforcement. The in-plane forces due to this lack of symmetry thus
increase the maximum loading.
7
Fig. 8: Crack formation in column area of slab, ASCE-ACI Committee 426 (1974).
Fig. 9 shows that statics does not provide a unique value for the compressive force C1
above the inclined crack for a slab. This is the limitation of static analysis. The in-plane
forces due to the elastic band around the outer boundary of the yielding portion of the slab
which restrains the outward displacement and generates axial compressive forces, increases
the flexural and shear capacities.
Fig. 9: Horizontal forces on sections near inclined cracks, ASCE-ACI committee 426 (1974).
8
The ability of a slab to resist higher unit shear stresses diminishes as the size of the
loaded area increases relative to the slab thickness and as the direction of the reinforcement
more closely parallels the direction of the maximum moment when there is essentially oneway action. A slab can then fail as a wide beam.
9
2. Shear Stress Theory
2.1. Shear stress on critical perimeter
The shear stress theory compares the shear stress on a critical section with a
maximum shear stress. According to Alexander and Simmonds (1986) the shear stress theory
is perhaps the simplest approach and is favored by most design codes.
Moe (1961) tested 43 slabs (Fig. 10) and investigated the results of 140 footings and
120 slabs tested in the literature. He reported inclined cracking at 60% of the ultimate load
(Fig. 11). This inclined cracking started from bending cracks, then rapidly extended up to the
proximity of the neutral axis and finally developed rather slowly but leaving only a very
narrow depth of compression zone unaffected.
Moe (1961) introduced three levels of shear force:
Vi
= the shear force at which inclined cracks form,
Ve
= the shear force at which failure in the compression zone above the inclined cracks
occurs,
Vflex
= the shear force at the ultimate flexural strength.
With these three forces, he classified four possible types of failure:
Vi < Ve < Vflex: shear-compression failure. This type of failure occurs when the compression
zone of the critical section (reduced in size due to inclined cracks) fails under a combined
action of compressive and shearing stresses.
Ve < Vi < Vflex: the slab fails at the moment of formation of inclined cracks, this failure mode
is called inclined tension failure.
Vi < Vflex < Ve: the slab will fail in flexure after the formation of inclined cracks.
Vflex < Vi: the slab will fail in flexure before the formation of inclined cracks.
10
Fig. 10: Testing arrangement, Moe (1961).
Fig. 11: Inclined cracking observed by Moe (1961).
Menétry (2002) claimed that the distinction between the flexural and punching failure
is controlled by the punching crack inclination angle α (Fig. 12 and Fig. 13). For a crack
inclination of 30° the failure mode is pure punching and for a crack inclination of 90° the
failure mode is pure flexure. Fig. 12 shows a typical lay-out of a slab-column test at the right
of the figure. In the test lay-out the loading is applied on the top of the slab, resulting in a
compression zone at the top. Fig. 13 displays the column on the bottom as typically seen in
flat plates, resulting in a compression zone at the bottom.
11
Fig. 12: Response curves for flexural and punching failure, Menétry (1998), taken from
Menétry (2002).
Fig. 13: Punching cone with different inclinations: 30°, 45° and 60°, Menétry (1998), taken
from Menétry (2002).
According to Moe (1961) the periphery of the loaded area is the critical one for a
shear-compression failure. For the inclined cracking load, the critical zone is not so clear. A
statistical analysis of the data in Table 1 demonstrated that the best agreement between the
individual values resulted for a critical perimeter at a distance d/2 away from the periphery of
the column or loaded area. Therefore the distance d/2 away from the face of the loaded area
was chosen as the location to compute the shear stress for the inclined cracking.
12
Table 1: Shear stresses at the inclined cracking load, Moe (1961).
After the inclined cracking, the following stresses occur in the compression zone (Fig. 14):
vertical shearing stresses (vo),
direct compressive stresses (fc),
vertical compressive stresses (f3), which make the shear strength of a slab higher than the
shear strength of a beam, and
lateral compressive stresses (f2), which increase the compressive strength of a slab as
compared to a beam.
Fig. 14: Stresses after inclined cracking, Moe (1961).
These stresses can be used in theories of failure in concrete under combined stresses.
There are two groups of theories: physical theories, based on the internal non-isotropic
13
structure of the material, and phenomenological theories, based on the external behavior
under different stress combinations. The physical theories are mathematically complicated
and the phenomenological theories, using Mohr’s theory of failure, do not give a reliable
criterion of failure. Therefore, Moe (1961) concluded that a better understanding of the stress
distribution in the critical zone of the slab and the stress conditions during failure was
needed.
The depth of the zone above the inclined cracking (Fig. 14) is important for the shear
strength, as this zone carries the load after inclined cracking. Theodorakopoulos and Swamy
(2002) developed an expression for the depth of the critical section based on the harmonic
mean of Xs (depth of the critical section for punching) and Xf (depth of the critical section for
flexure) (Fig. 15).
Fig. 15: Schematic presentation of cracking, Theodorakopoulos and Swamy (2002).
The depth of the shear critical section is found as
X s = 0.25d .
Eq. 1
The depth of the flexural critical section is found as
Xf =
ρf s − ρ ' f s'
k1 f cu
Eq. 2
d
with
k1 = 0.67
ε cu − Aε o / 3
ε cu
Eq. 3
and
εo =
f cu
4115
(SI units)
Eq. 4
14
εo =
f cu
49380
(US customary units)
in which
fs and fs’
= the steel stresses in tension and compression respectively,
ρ
= the reinforcement ratio of tensile steel,
ρ’
= the reinforcement ratio of compression steel,
εcu
= the ultimate concrete strain (Fig. 16),
εo
= the concrete strain at the level of the end of the rectangular stress block,
A
= a coefficient, =1 for normal density concrete,
d
= the effective depth of the slab,
fcu
= the ultimate concrete strength in MPa (SI units) or psi (US customary units).
Fig. 16: Strain and stress distribution in concrete section, Theodorakopoulos and Swamy
(2002).
The harmonic mean is then given by
1
1
1
=
+
.
X 2X s 2X f
Eq. 5
This method requires iterations and is not very useful for design practice.
Alexander and Simmonds (1986) questioned the importance of diagonal cracking,
arguing that, although diagonal cracks appear visually important, test observations have
shown that these cracks occur at 50-70% of the ultimate loading. The slab in that cracked
15
state is very stable since it can be unloaded and reloaded without affecting the ultimate
capacity.
As a workable solution to the difficulties encountered, Moe (1961) developed a semiempirical formula for the ultimate strength. The nominal stress is computed as
v=
V
bd
Eq. 6
with
V
= the shear force,
b
= the width of critical section in shear,
d
= the effective depth of the slab.
Vflex is introduced as a parameter governing the shear strength of slabs, but the magnitude of
Vflex has no direct physical relation to the mechanism of failure. Vo is defined as the fictitious
shear strength if bending could be eliminated (Fig. 17).
Fig. 17: Interaction between shearing and flexural strength, Moe (1961).
Considering the influence of the ratio of column size to slab thickness and φo = V/Vflex, Moe
(1961) suggested the following relationship:
16
 

r
v =  A1 − C  − Bφ o  f c' (US customary units)
d
 

 

r
v = 12 A1 − C  − Bφ o  f c'
d
 

Eq. 7
(SI units)
The constants A, B and C from Eq. 7 were obtained with a statistical analysis (Fig. 18). The
purpose of the statistical analysis was to determine the specific combination of constants A,
B and C which for the available test data yields to an average value of the ratio of tested to
calculated ultimate loads (Vtest/Vcalc) equal to one and a minimum standard deviation for
Vtest/Vcalc. This analysis led to:
r
15(1 − 0.075 )
d
=
=
'
'
f c bd f c
bd f c'
1 + 5.25
V flex
v
V
(US customary units)
Eq. 8
r
180(1 − 0.075 )
v
V
d
(SI units)
=
=
'
'
f c bd f c
bd f c'
1 + 63
V flex
Fig. 18: Comparison of test results and Eq. 8, Moe (1961).
17
A shear failure is not desirable in a slab, and therefore Moe (1961) limited the stress
such that a slab fails in flexure (Fig. 19). The shear failures in Fig. 19 are calculated by Eq. 8.
The point of balanced design is chosen as point B, where V =1.1Vflex. In order to be sure of
obtaining a flexural failure, the shear stress should be limited to:
r
v = (9.23 − 1.12 ) f c'
d
for r/d ≤ 3
(US customary units) Eq. 9
d
v = (2.5 + 10 ) f c'
r
for r/d > 3
(US customary units)
r
v = (110.76 − 13.44 ) f c'
d
for r/d ≤ 3
(SI units)
d
v = (30 + 120 ) f c'
r
for r/d > 3
(SI units)
The equation for r/d > 3 was derived theoretically based on the fact that for a value of r/d
approaching infinity, the value for the shear stress should approach the corresponding
shearing strength of a beam and because test data were only available for values of r/d up to
3.1.
Fig. 19: Elimination of the possibility of shear failure in slab, Moe (1961).
18
Fig. 20: Eq. 9, Moe (1961).
Elstner and Hognestad (1956) used a shearing stress, v2, computed at zero distance
from the column face or loaded area and used an equation based on statistical analysis:
v2
P
333 0.046
= shear = ' +
(US customary units)
'
ϕ
fc 7
fc
'
o
bdf c
8
Eq. 10
v2
P
2.296 0.046
= shear =
+
(SI units)
'
7
ϕ
fc
f c'
'
o
bdf c
8
with
fc’
= the cylinder strength (psi for US customary units, MPa for SI units),
φo = Pshear/Pflex,
Pshear
= the ultimate shear capacity, and
Pflex
= the ultimate flexural capacity of the slab computed by the yield-line theory without
regard to a shear failure (Fig. 21). No resistance factor is used.
19
Fig. 21: Influence of relative shearing strength (Eq. 10) Elstner and Hognestad (1956).
2.2. Critical shear crack theory
The critical shear crack theory describes the relationship between the punching shear
strength of a slab and its rotation at failure. Muttoni (2008) gave the following evidence
supporting the role of the shear critical crack in the punching shear strength. After reaching a
maximum level, the radial compressive strain decreases; and shortly before punching, tensile
strains may be observed. These strains can be explained by the development of an elbowshaped strut (Fig. 22) with a horizontal tensile member along the soffit due to the
development of the critical shear crack. Also, experimental results on slabs with a particular
lay-out of circular reinforcement in which only radial cracks form and in which the formation
of circular cracks is avoided, confirmed the role of the critical shear crack.
20
Fig. 22: Test by Guandalini and Muttoni: (a) cracking pattern of slab after failure; (b)
theoretical strut developing across the critical shear crack; (c) elbow-shaped strut; and (d)
plots of measured radial strains in soffit of slab as function of applied load, Muttoni (2008).
The critical shear crack theory is described in Guandalini, Burdet and Muttoni (2009).
This theory is based on the assumption that the shear strength of members without transverse
reinforcement is governed by the width and roughness of an inclined shear crack that
develops through the inclined compression strut carrying shear. In two-way slabs the width
wc of the critical shear crack is assumed proportional to the slab rotation ψ and the effective
depth d of the member (Fig. 23).
The following failure criterion was obtained:
VR
b0 d f c
VR
b0 d f c
=
3/ 4
1 + 15
1 + 15
Eq. 11
d g0 + d g
9
=
(SI units)
ψd
ψd
(US customary units)
d g0 + d g
in which
VR
= the shear strength,
b0
= a control perimeter at d/2 from the edge of the column,
d
= the effective depth of the member,
fc
= the compressive strength of the concrete,
21
dg
= the maximum size of the aggregate (accounting for the roughness of the lips of the
cracks),
dg0
= a reference aggregate size equal to 16 mm = 0.63 in.
Fig. 23: Slab deflection during punching test: (a) measured values of w at top and bottom
face of a slab tested by Guandalini, Burdet and Muttoni (2009); and (b) inpterpretation of
measurements according to critical shear crack theory.
The failure load is obtained at the intersection (Fig. 24) of the failure criterion (Eq.
11) with the load-rotation curve of the slab, which for practical purposes can be
approximated by:
r f  V 
ψ = 1.5 s . y 
d E s  V flex 
3/ 2
Eq. 12
with
rs
= the radius of the slab,
Vflex
can be estimated with the yield-line method.
22
According to Muttoni (2008), the load-rotation relationship can, in a more general
case, be obtained from a nonlinear numerical simulation of the flexural behavior of the slab,
or in the axi-symmetric case by a numerical integration of the moment-curvature relationship.
An advantage of this method is that it finds the value of the rotation capacity of the slab, and
thus of its ductility. Due to the relation between the shear carried across a crack and the depth
of a section, this method takes the size effect into account. An earlier treatment of this
method can be found in Muttoni (2003).
Fig. 24: Design procedure to check the punching strength of a slab, Muttoni (2008).
2.3. Influence of concrete strength
The shear strength is related to the concrete strength fc’. It is not clear however if this
relationship is a square root or cubic root dependence. Early research did not consider a
square or cube root dependence. Elstner and Hognestad (1956) used the concrete strength fc’
in their study and tested slabs with concrete strengths fc’ ranging from 2000 to 7000 psi.
23
Fig. 25: Influence of concrete strength, Elstner and Hognestad (1956).
Moe (1961) used the square root for the following two reasons. First, a shear failure is
of a splitting type, comparable to the type of failure observed in specimens under tension.
The tensile strength was generally assumed to be proportional to
f c' . Second,
f c' approaches zero when fc’ approaches zero. A function a + bf c' which fits test results
would not approach zero and therefore would not be satisfactory.
Mitchell, Cook and Dilger (2005) compared the influence of the square and cube root
of the concrete strength with experimental data. Fig. 26 compares test results to ( f c' ) with
n
n=1/2, 1/3 and 2/3. These figures show that the overall trend is reasonably presented by
n=1/3 as well as by n=1/2.
24
Fig. 26: Effect of concrete strength on shear strength (tests by Elstner and Hognestad, (1956),
figure by Mitchell, Cook and Dilger (2005).
Fig. 27 shows test results compared by Mitchell, Cook and Dilger (2005) to the square root
and cube root of the concrete strength. The two functions were normalized to give a value of
1.0 at a concrete strength of 30 MPa. For each of the tests, the normalized shear ratio is taken
as the failure load divided by the failure load for the case with a concrete compressive
strength of 30 MPa. Based on Fig. 27, the cube root function appears to fit the data for high
strength concrete in a more conservative manner. However, Mitchell, Cook and Dilger
(2005) concluded that it is not clear whether the punching strength is proportional to the
square or cube root of the concrete strength and that additional research is needed to enable
the development of design expressions for punching shear that are applicable to a wide range
of concrete strengths, especially high strength concrete.
25
Fig. 27: Comparison of square root and cube root functions with test results reported by
Ghannoum (1998) and McHarg et al. (2000), figure by Mitchell, Cook and Dilger (2005).
2.4. Influence of flexural reinforcement
ASCE-ACI committee 426 (1974) collected different methods to calculate the shear
strength and divided them into two categories: expressions dependent primarily on the
concrete strength and expressions dependent primarily on flexural effects. Tensile
reinforcement was advised to improve the flexural behavior of the slab in the service load
range. Compression reinforcement was advised because it acts like a suspension net,
supplying an alternate load path that holds the slab together even after a punching failure.
Elstner and Hognestad (1956) tested the influence of compression reinforcement on
the ultimate shearing strength and concluded that it did not have an effect on the ultimate
shearing strength of slabs. They did not include the influence of reinforcement in the
expression they developed (Eq.10).
Dilger, Birkle and Mitchell (2005) studied the influence of the reinforcement ratio on
the ultimate shearing strength. Fig. 28 shows the comparison of test results from a variety of
researchers. The stresses at failure were adjusted for varying concrete strength and the
reinforcement ratio was taken as the average reinforcement ratio in both directions. The
concrete strength was adjusted as
 30 

v adj = vtest 
 f' 
 c 
1/ 3
 4350 

v adj = vtest 
 f' 
 c 
(SI units)
Eq. 13
1/ 3
(US customary units)
in which fc’ is the concrete strength measured in MPa (SI units) or psi (US customary units)
and adjusted according to Table 2.
26
Table 2: Conversion factors for compressive concrete strength, Dilger, Birkle and Mitchell
(2005).
Fig. 28 indicated a tendency of increasing punching strength with increasing reinforcement
ratio, but the data showed a lot of scatter. Therefore the data were studied per researcher.
Fig. 28: Failure load vs. flexural reinforcement ratio (h ≈ 150 mm ≈ 6 in ), from Dilger,
Birkle and Mitchell (2005).
The test results per researcher are shown in Fig. 29, Fig. 30, Fig. 31 and Fig. 32.
Dilger, Birkle and Mitchell (2005) concluded from these test series that with an increase in
flexural reinforcement ratio the stresses along the punching cone and, hence, the load
carrying capacity in shear were increased. Dilger, Birkle and Mitchell (2005) cited an
explanation given by Richart (1948), who found that significant yielding of the flexural
reinforcement produced large cracks, which decreased the effective area resisting the shear.
Assuming that little or no shear can be transferred through the portion of the depth of the slab
that is cracked, it is easy to conclude that the width and, hence, the depth of the crack, which
are controlled by the amount of flexural reinforcement, have a significant influence on the
shear capacity.
27
Fig. 29: Test results from Vanderbilt (1972), from Dilger, Birkle and Mitchell (2005).
Fig. 30: Test results from Mardouk and Hussein (1991) from Dilger, Birkle and Mitchell
(2005).
Fig. 31: Test results from Hallgren (1996), from Dilger, Birkle and Mitchell (2005).
Fig. 32: Test results from Richart (1948), from Dilger, Birkle and Mitchell (2005).
28
Guandalini, Burdet and Muttoni (2009) investigated the punching strength of slabs
with low reinforcement ratios. The scope of their research was slabs with low reinforcement
ratios, because there was not much data available for slabs with low reinforcement ratios
failing in shear, as researchers tried to avoid flexural failures, and because the code
provisions differ significantly (Fig. 33). The results were recorded as load-deflection curves
(Fig. 34) which shows unexpectedly low strengths for slabs with low reinforcement ratios.
The results were compared to results form the literature (Fig. 35). Guandalini, Burdet and
Muttoni (2009) concluded that future research is needed to investigate this observation and
that special attention should be given to the cases in which the code provisions significantly
underestimate the punching shear strength.
Fig. 33: Influence of flexural reinforcement ratio on punching shear strength according to
ACI 318-08 and EN 1992-1-1 (fck = 30 MPa = 4350 psi, fyk = 414 MPa = 60 ksi, c/d=1,
l/d=25, l1=l2), Guandalini, Burdet and Muttoni (2009).
29
Fig. 34: Normalized load-deflection curve for all 11 specimens (deflection w was measured
between center of column and reaction points at perimeter), Guandalini, Burdet and Muttoni
(2009).
30
Fig. 35: Results of 99 punching tests analyzed by Guandalini, Burdet and Muttoni (2009) and
88 tests taken from the literature compared with the failure criterion of the critical shear crack
theory: (b) 99 tests with identification of the reinforcement ratio; and (c) 99 tests with
identification of effective depth.
2.5. Influence of support and loading conditions
According to ASCE-ACI committee 426 (1974), long-term loading did not have a
negative effect on the shear strength. Rapid loading resulted in an increase in strength.
Elstner and Hognestad (1956) tested slabs with different support conditions. They reported
that a two-edge support as compared to a four-edge support decreased Pflex, increased φo and
decreased the estimated maximum load.
2.6. Influence of the side length of the loaded area
Fig. 36 shows the results for the ultimate shear strength for tests with different side
length to effective depth (c/d) ratios as compared by ASCE-ACI committee 426 (1974). The
data show a decrease in the shear strength for increasing c/d ratios.
31
Fig. 36: Effect of large c/d values on shear strength, ASCE-ACI committee 426 (1974).
r
Moe (1961) assumed a linear expression (1 − 0.075 ) where r is the side length of
d
the loaded area and d is the effective depth of the slab. In the case of a continuous slab
supported on a wall, r/d becomes infinite and the shearing strength approaches the shearing
strength of a beam (Fig. 37).
Fig. 37: Beam and slab effect as function of r/d, Moe (1961).
32
Sherif, Emara, Ibrahim and Magd (2005) studied the influence of the bo/d ratio (with
bo the perimeter of the critical section) on the punching capacity. From a theoretic point of
view they stated that the punching shear strength decreased with an increase in the bo/d ratio,
because confinement was reduced. This effect can also be seen in the data gathered in Fig.
38. The critical section was taken at a distance d/2 from the face of the loaded area.
Fig. 38: Effect of the ratio bo/d on the punching shear strength of slab-column connections,
from Sherif, Emara, Ibrahim and Magd (2005).
2.7. Size effect
Elstner and Hognestad (1956) questioned the extrapolation of observations on thick
footing slabs to flat plate floors from a theoretical point of view, since lower thickness-tospan ratios and higher moment-to-shear ratios are more associated with floor slabs than with
footings.
33
According to ASCE-ACI committee 426 however, there was no effect of scale on
shear strength, provided that deformed bars are approximately scaled, concrete mixes are
scaled, and measurements of compressive strength are made on scaled cylinders.
Collins and Kuchma (1999) investigated the importance of the size effect on beams,
slabs and footings and concluded that the size effect has to be taken into account and that
high-strength concrete members display a more significant size effect. They pointed out that
the shear stress at failure decreases, both as the member depth increases and as the maximum
aggregate size decreases. According to Collins and Kuchma (1999), the size effect had to be
studied especially in slabs and footings, as these members can be both very thick and very
lightly reinforced.
Mitchell, Cook and Dilger (2005) stated that it is difficult to gather experimental data solely
on the size effect, as many reported experiments varied other parameters together with the
thickness. For example, the reinforcement ratio was changed together with the slab thickness
to keep the ratio of flexural capacity to shear capacity constant. Mitchell, Cook and Dilger
(2005) gathered information of tests where only the size was varied. It is clear from the data
in Fig. 39 that there is a size effect for slabs thicker than about 200 mm (8 in). The data in
Fig. 40 also show a size effect, even for slabs with a thickness smaller than 200 mm (8 in).
Fig. 41 also shows the size effect. Tests with varying maximum aggregate size are not
included. As can be seen in Fig. 39, Fig. 40 and Fig. 41, the shear stress at punching failure
decreases as the effective depth increases. According to Mitchell, Cook and Dilger (2005) the
size effect is significant, but the available data are scarce.
Fig. 39: Normalized punching shear stress ( v / f c' ) vs. average effective depth for tests
reported by Li (2000), Mitchell, Cook and Dilger (2005).
34
Fig. 40: Normalized punching shear stress ( v / f c' ) vs. average effective depth for tests
reported by Nylander and Sundquist (1972), from Mitchell, Cook and Dilger (2005).
Fig. 41: Normalized punching shear stress ( v / f c' ) vs. average effective depth for tests
reported by Regan (1986) and Birkle (2004), from Mitchell, Cook and Dilger (2005).
Birkle and Dilger (2008) state that the size effect has to be taken into account for
slabs with an effective depth larger than 220 mm (9 in), as can be estimated from Fig. 42.
35
Fig. 42: Influence of slab thickness on failure stress in slabs without shear reinforcement,
Birkle and Dilger (2008).
According to Sundquist (2005), no good analysis method has been presented to date
that can really explain the size effect. A model developed by Hallgren (1996) was cited,
based on fracture mechanics that incorporated the aggregate size. This led to the formula:
3.6 G F 0
1.4 x
(SI units)
90 G F 0
889 x
(US customary units)
ε cTu =
ε cTu =
Eq. 14
where
ε cTu
= the ultimate tangential strain,
x
= the depth of the compression zone in mm (SI units) or in (US customary units),
GF0
= the fracture energy equal to 0.025, 0.030, 0.038 for aggregate size da = 8 mm (0.31
in), 16 mm (0.62 in), 32 mm (1.26 in), respectively.
With the ultimate tangential strain, the stress distribution in a section at a given location in
the critical zone can be calculated. Then the forces are found and the maximum punching
force is calculated. The expression in Eq. 14 could be optimized by including information on
the concrete properties.
Hallgren and Bjerke (2002), however, stated that the influence of tensile strength and
fracture energy has been found significant for the size effect in earlier research, but in their
research based on non-linear finite element analysis of footings, no significant influence was
found.
36
2.8. Influence of restraints
Hewitt and Batchelor (1975) stated that restraining forces at the slab boundaries can
result from compressive membrane (arch) action as well as from “fixed boundary action”
(Fig. 43). The compressive membrane action gives a net in-plane force at the slab boundaries,
while fixed boundary action is due to moment restraint with no net in-plane force at the slab
boundary. Compressive membrane forces can be induced in a cracked concrete slab but,
unlike fixed boundary moments, cannot occur in a slab that is uncracked or made from a
material having the same stress-strain relationships in compression and tension. Thus,
compressive membrane action can occur in a cracked unreinforced concrete slab, whereas
fixed boundary action in a cracked slab requires the provision of tension reinforcement at the
boundary. A restrained reinforced slab loaded to its punching load goes through the
following stages: fixed boundary action, cracking, compressive membrane action
superimposed on fixed boundary action, and finally punching shear failure.
Fig. 43: Idealized restrained slabs: forces and stress distributions, Hewitt and Batchelor
(1975).
The area of the slab beyond the line of contraflexure and external frames enhances the
capacity of slabs due to the action of in-plane compression forces. The portion of the slab
beyond the line of contraflexure acts as a tension ring which reacts against compressive
37
forces induced in the inner portion of the slab. Hewitt and Batchelor (1975) wrote that the
first model which took fixed boundary action into account, was the model developed by
Kinnunen and Nylander (1960).
Tests carried out by Csagoly (1979) for the Ontaria Ministry of Transportation and
Communications also indicated the increase in punching resistance due to membrane action.
Very high factors of safety against punching of slabs designed by conventional methods were
found.
While Hewitt and Batchelor (1975) made a clear distinction between the compressive
membrane action and the fixed boundary action, Bakker (2008) and Wei (2008) name the
existence of any restraining force “compressive membrane action”.
Bakker (2008) studied the influence of the compressive membrane action on the
decks of plate-girder bridges. The influence of the compressive membrane action on the
punching shear strength is presented in the model shown in Fig. 44. In Fig. 44, the following
symbols are used:
nr
= the dimensionless radial membrane force working on the surface of the failure
cone,
na
= the dimensionless membrane force in the mid-depth of the slab,
d1
= the outer diameter of the punched cone,
a
= the radius of the slab,
r(x)
= a function of the failure surface over the height,
d0
= the length over which the concentrated load is spread,
α
= the angle between yield surface and displacement rate vector,
β
= a factor between 0 and 0.5.
Fig. 44: Punching shear failure model, taking compressive membrane action into account,
Bakker (2008).
38
In this model, the following assumptions are made: the failure mechanism consists of a solid
cone-like plug, the compressive membrane force has a constant value, the behavior is rigidly
plastic, and the energy in hoop expansion outside the plug is neglected. An upper-bound
solution is found by using virtual work theory. The value for na is derived by using the flow
theory. The method is suitable for the use with a calculation sheet. Details of the method and
an example Maple sheet can be found in Bakker (2008). This method is compared with test
results as shown in Table 3. The “analytical” solution is the solution obtained by using the
Dutch code NEN 6720. The NEN 6720 takes the critical perimeter at d/2 from the face of the
loaded area and calculates the maximum shearing stress.
Table 3: Test values with the analytical (Pa), experimental (Pe) and predicted (Pp) solutions,
Bakker (2008).
Wei (2008) developed a model which considers boundary restraint based on
Hallgren’s punching shear model. The dowel effect is ignored in Wei’s model. In Fig. 45, the
following symbols are used:
T
= the inclined compressive force in the compressed conical shell,
∆φ
= the central angle,
RcT
= the horizontal force in the concrete crossing the shear crack,
RsT
= the horizontal force in the reinforcement crossing the shear crack,
RsR
= the horizontal force in the reinforcement at right angles to the radial cracks,
P
= the external load,
Fb
= the total boundary restraint force,
Mb
= the boundary restraint moment,
B
= the diameter of the loading area,
xm
= the neutral axis depth at mid-span or plastic hinges,
c
= the diameter of the slab area with negative radial bending moment,
xb
= the neutral axis depth at the boundary.
39
Fig. 45: Modified Hallgren model considering boundary restraint, Wei (2008).
This model led to a set of three kinematic equations, eight constitutive equations in St (lateral
stiffness of surrounding structures) and Sφ (rotation stiffness depending on the boundary
restraint), and six equilibrium equations. The lateral and rotational stiffness can be found by
40
using the elasticity theory. Results obtained by using this method are compared with test
results in Table 4.
Table 4: List of test results and calculated results, Wei (2008).
2.9. Influence of the shape of the loaded area
According to ASCE-ACI committee 426 (1974), slabs supporting circular columns
have greater punching shear resistance than the same slabs supporting square columns. The
tire contact area is typically rectangular. Fig. 46 shows slab-column test results for the
ultimate shear strength as a function of the column rectangularity. The data suggest a
decrease in ultimate shear strength for an increase in rectangularity.
Fig. 46: Effect of column rectangularity on shear strength, ASCE-ACI committee 426 (1974).
Mitchell, Cook and Dilger (2005) also stated that the normalized shear stress
decreases as the column rectangularity increases. Fig. 47 shows the decrease in normalized
shear stress for increasing column rectangularity.
41
Fig. 47: Influence of rectangularity of column on shear strength from tests by Hawkins et al.
(1971), Leong and Teng (2000) and Oliveira et al. (2004), figure by Mitchell, Cook and
Dilger (2005).
Sherif, Emara, Ibrahim and Magd (2005) remarked that slab-column specimen test
results from the literature indicate an increasingly high concentration of concrete strains
towards the corners of the loaded area with an increased column aspect ratio (c1/c2). The
results from the literature also indicated that the effect of c1/c2 diminishes for ratios greater
than 3.
2.10. Discussion of shear stress theory
Alexander and Simmonds (1986) pointed out that the shear stress theory assumes that
the vertical load is carried by shear stress on some critical section. The critical section is a
vertically oriented surface at some distance from the face of the column or loaded area. The
description of a punching failure suggests that it is unlikely that vertical load on a slabcolumn connection is controlled by shear stress on some vertical plane. Shear stress on a
vertical plane creates diagonal principal tension and compression stresses which may be
regarded as a diagonal tension field. However, diagonal cracking at a relatively early load
stage should preclude the tension field. The area of concrete available to participate in the
tension field ought to be confined to the uncracked region in the compression zone of the slab
at the face of the loaded area. In spite of this, most critical sections are placed at some
distance from the column or loaded area and the area of the critical section is based on the
depth of the reinforcement rather than the thickness of the compression zone.
42
Another weakness of the shear stress model, as pointed out by Alexander and
Simmonds (1986), is the way it accounts for the flexural reinforcement. Some simplified
design techniques neglect the flexural reinforcement, and those methods which do not usually
assume a smooth distribution of reinforcement. There are three main drawbacks for this
idealization. The first reason is that the slab reinforcement is discrete. At collapse, a bar
either crosses a failure surface or not. Secondly, reinforcement is often irregularly spaced,
making smooth distributions difficult to define. Finally, with design moments and shears
based on smoothly distributed reinforcement, there is no clear indication as to where a
particular bar is best placed.
A last critique on the shear stress theory, as stated by Alexander and Simmonds
(1986) is the assumption of a critical section. The shape of the failure surface changes with
the ratio of moment to shear. It was suggested to remedy this problem by making the critical
section a variable of the model. It is also not clear how to account for slab discontinuities,
such as reinforcing bars, which are located at or very near the assumed section. Using a
distributed reinforcing ratio is the usual solution to these discontinuities, but this amounts
into covering up one inaccuracy with another.
Regardless of these three points of critique, the shear stress theory is still the most
commonly used method and serves as a basis for the code provisions. The shear stress theory
offers an easy way to design structures for shear, but it does not explain the mechanics of the
punching shear problem and is, therefore, unable to precisely assess the ultimate strength of
existing structures.
43
3. Plate Theory and Finite Elements
According to Alexander and Simmonds (1986), plate theory and finite element
methods range from simple elastic plate models to sophisticated nonlinear models which
account for cracking and plastic behavior. Most models assume that the reinforcement can be
described by a thin membrane rather than by discrete bars.
3.1. Material modeling
Bakker (2008) used finite element modeling to calculate the ultimate punching strength by
using the software DIANA. The cracking model used is the total strain rotating crack model.
The total strain crack model describes the tensile and compressive behavior of a material
with one stress-strain relationship, and the rotating model calculates the direction of the crack
in each load step. Both the brittle and Hordijk tension softening models are used to describe
the behavior of concrete in tension (Fig. 48). The ideal plastic model as shown in Fig. 49 is
used to describe the concrete in compression. The reinforcement steel is modeled as ideal
plastic.
Fig. 48: Models for behavior of concrete in tension: brittle and Hordijk tension softening,
Bakker (2008).
Fig. 49: Ideal plastic model used for concrete in compression, Bakker (2008).
3.2. 3D solids model
First a slab (Fig. 50) was modeled using 3D solid elements. The CHX60 element was
used (Fig. 51). The slab was modeled using two symmetry lines, resulting in a model of a
fourth of the slab using 108 elements with the load applied on 9 elements (Fig. 52). The size
of the elements is smaller on the loaded area and larger towards the outer corners. The
44
smallest used element was 2.77mm x 2.77mm x 50mm (0.11in x 0.11in x 1.97in), and the
largest element was 75mm x 75mm x 50mm (2.95in x 2.95in x 1.97in).
Fig. 50: Dimensions of the 3D solids model. All the edges are clamped and horizontally
restrained. Dimensions are in mm, Bakker (2008).
Fig. 51: 3D solid element of DIANA software, Bakker (2008).
Fig. 52: Model of a slab using 3D solid elements. The load is applied to the red zone. Figure
by Bakker, personal communication 07-15-2009
The predicted solution in Fig. 53 is based on the method described in Bakker (2008) and
briefly introduced in section 2.8. Model (a) was simply supported and did not take
compressive membrane action into account, while model (b) was clamped and laterally
45
restrained and hence took compressive membrane action fully into account. The results of the
3D solids model seemed not to correspond very well to the expected values.
Fig. 53: The load-displacement graphs of the 3D solids model, Bakker (2008).
3.3. Axi-symmetric model
Another way punching shear strength was studied by Bakker (2008), was by using an
axi-symmetric model (Fig. 54). In the axi-symmetric model, CQ16A elements were used
(Fig. 55). Note that Fig. 54 shows how the model was built up; the number of elements used
in the finite element calculation of the 2D slice was about 500, because the axi-symmetric
model does not have a long calculation time. This model was evaluated for the same four
cases as the 3D solids model (Fig. 56). The axi-symmetric model with brittle material
behavior gave the best results. It also led to a realistic crack pattern (Fig. 57).
46
Fig. 54: Axi-symmetric model, Bakker (2008).
Fig. 55: CQ16A element from DIANA (2008), used in the axi-symmetric model by Bakker
(2008).
47
Fig. 56: Load-displacement graphs for 4 different models of the axi-symmetric model,
Bakker (2008).
Fig. 57: Crack pattern just before (loadstep 16) and after (loadstep 17) failure, Bakker (2008).
The results of measurements on bridge decks reported by Taylor, Rankin, Cleland and
Kirckpatrick (2007) were compared with the deflection results obtained through the finite
element modeling. Bakker (2008) reported that the results for the deflections (Table 5) did
not match at all with the experimentally found values. Further improvement of the model is
needed.
48
Table 5: Comparison of test results reported by Taylor, Ranking, Cleland and Kirckpatrick
(2007) and finite element modeling, Bakker (2008).
Polak (2005) used layered shell finite elements for punching shear analysis. This
finite element type allows for the use of 3D constitutive models. Special attention was given
to the formulation of the concrete cracked shear modulus, the calculation of the cracking load
and tension stiffening formulations and modeling the dowel action.
3.4. Discussion of plate theory and finite elements solutions
Finite element solutions require a very good understanding of the material behavior
and the software. The correct material property models have to be selected and implemented.
The user needs to be aware of the limitations of the material models, element types and
calculation techniques he is using. Even though in previous sections good approximations for
the ultimate load were found, none of the consulted literature reported a finite elements
model which succeeded to realistically display the behavior from first loading until the
ultimate punching shear loading.
49
4. Beam Analogy
4.1. Currently used beam analogy
The beam analogy method is described by Park and Gamble (1999). The slab
segments adjacent to the loaded area are considered to act as beams running in two directions
at right angles as shown in Fig. 58. The slab strips making up the beams are subjected to
bending moment, torsional moment and shear force; and redistribution of these actions is
able to occur between the beams. Each beam is assumed to be able to develop its ultimate
bending moment, torsional moment, and shear force, and interaction effects can be taken into
account. The total strength is the sum of the contributions of the strength of the beams.
Failure occurs when at least three beams reach their ultimate strength. A detailed description
of the method can be found in Park and Gamble (1999). This beam analogy predicts up to
eight possible limiting strength combinations of bending, torsion and shear (Fig. 58). The
large number of possible limiting strength combinations makes the application relatively
difficult.
Fig. 58: Limiting strength combinations for beam analogy, Park and Gamble (1999).
In a simplified beam analogy model the ultimate strength is obtained as the sum of the
flexural, torsional and shear strength of all the beams. Sufficient ductility in bending, torsion
and shear to allow the simultaneous development of the ultimate capacities is assumed.
Other beam analogies of semi-empirical nature exist. According to Alexander and
Simmonds (1986), the differences between the several beam analogies lie largely in the
method by which shear and torsional strengths are calculated and in the degree of
redistribution allowed between beam elements.
4.2. Design based on the equivalent width of a beam strip
Elstner and Hognestad (1956) indicated that the shearing stress
50
v=
V
bjd
Eq. 15
can also be calculated with b the “equivalent width”, the width of a fictitious beam strip of
slab across which a concentrated load should be considered as distributed to produce
calculated shearing stresses equal to the maximum stress that actually occurs in the slab. To
evaluate this procedure, eight beam specimens representing center strips of slabs were tested.
They observed that six of the eight beams representing the slabs failed in flexure while the
corresponding fully tested slabs failed in shear. This observation led to the conclusion that
the behavior of beam strip specimens does not reflect the behavior and mode of failure of a
corresponding slab and that beam strips are unsuitable to evaluate the shearing strength of
slabs even though they are successfully used to model the flexural behavior of slabs.
4.3. Discussion of beam analogy methods
The beam analogy methods lead to an approximate ultimate punching shear load
which can be used for design purposes. However, the beam analogy methods are not based
upon the mechanics of the punching shear problem and, therefore, are not suitable for the
analysis of existing structures.
Alexander and Simmonds (1986) pointed out that beam analogy methods, just like the
shear stress methods discussed in section 2.10, assume that the vertical load is carried by
shear stress on some vertically inclined critical section. Also, the beam analogy methods do
not account correctly for the influence of the flexural reinforcement and use the assumption
of a critical section, which is the geometry of the beams for the beam analogy.
51
5. Strut and Tie models
5.1. Strut and tie model
A more plausible source of shear strength than explained through the shear stress
theory and beam analogy, is an inclined compression field in the concrete. Together with
steel tension ties, this approach is often referred to as a truss model. Not only does this
mechanism provide a load path for shear forces in the presence of diagonal cracking, it
explains the role that flexural reinforcement plays in determining shear strength. The
variables used in the strut and tie model developed by Alexander and Simmonds (1986) are:
the overall geometry of the connection, the concrete strength, the strength of the flexural
reinforcement, and the placement of the flexural reinforcement.
The strut and tie model, as developed by Alexander and Simmonds (1986), consisted
of two types of compression struts: in-plane or anchoring struts (parallel to the slab) and outof plane or shear struts (at some angle (α) to the plane of the slab).
The anchoring, in-plane struts are presented in Fig. 59, showing a plane parallel to the
plane of the slab. Four anchoring struts are shown. Each is equilibrated by two mutually
perpendicular reinforcing bars: one passing through the loaded zone and the other at some
distance from the loaded zone. This mechanism gives an explanation for the influence of the
flexural reinforcement on the shearing strength: bars at some distance from the loaded zone
are able to exert flexural moment.
Fig. 59: In-plane or anchoring struts, Alexander and Simmonds (1986).
52
An out-of-plane or shear strut can be compared to the familiar force diagram used in
corbel design (Fig. 60). However, there are two differences between the shear struts for slabs
and the struts in corbel design. First, the point of load application does not coincide with the
junction of the tensile and compressive force, and as a result the angle of inclination of the
shear strut, α, is not pre-set. The second difference is that the vertical component of the
compression strut is no longer equilibrated at the junction by the applied load. There exists a
force component out of the plane of the slab which must be balanced by some form of
tension field within the concrete, resulting in a three-dimensional truss.
Fig. 60: Comparison of corbel with out-of-plane or shear strut, Alexander and Simmonds
(1986).
In a slab, the amount of steel participating in the tension ties (called “shear steel”) is
not clearly defined. Alexander and Simmonds (1986) assumed that all steel aligned through
the loaded zone participates in the tension tie, plus some fraction of the steel within a
distance ds (the effective depth) of the face of the loaded zone. This fraction decreases
linearly from 1 at the face of the loaded zone to 0 at a distance ds from the face of the loaded
zone.
Three conditions can lead to failure in the strut and tie model of Alexander and
Simmonds (1986): failure of the tension tie, failure of the compression strut, and failure when
the out-of-plane component of the compression strut exceeds the confining strength of the
slab.
53
The primary assumption in the strut and tie model of Alexander and Simmonds
(1986) is that the shear steel will always reach yield, and, therefore, a compression failure
never governs. This statement is false for heavily reinforced slabs. The justification for this
assumption is that test data have shown that the shear steel yields before failure and that
predicting a compression failure requires many assumptions, while a compression failure is
not ductile. Therefore a compressive failure mode is precluded rather than described.
The ultimate capacity of an in-plane bar-strut unit is limited by the yield of the
reinforcing bars. To define the ultimate capacity, the bar force at yield and the angle of the
compression strut (α) need to reach a critical value. The parameters which are likely to affect
α are assembled in a non-dimensional, empirically determined, term. From geometric
considerations, tan(α) equals the ratio of the out-of-plane component (defined by the ability
of the slab to confine the bar, function of tributary width of each bar (s), cover (d’) and
concrete strength) to the in-plane component (yield force in steel). These observations led to
the expression:
tan α =
K=
Pfailure
K
top
ASV
fy
s eff .d ' . f c'
Abar . f y .(c / d s )
0.25
Eq. 16
where
Pfailure = the failure load,
ASVtop = the top mat shear steel,
fy
= the yield strength of the steel,
seff
= the maximum of s or 3d’,
d’
= the cover of the reinforcement measured to the near side of the slab,
ds
= the cover of the reinforcement measured to the far side of the slab,
c
= the column dimension perpendicular to the bar being considered,
fc’
= the concrete strength, and
Abar
= the area of a single reinforcing bar.
Based on these theoretical considerations, a design equation for α was determined from test
results. The data from Fig. 61 led to a design equation:
tan α = 1.0 − e −0.85 K
(US customary units)
tan α = 1.0 − e −2.25 K
(SI units)
Eq. 17
54
Fig. 61: Calibration of α, Alexander and Simmonds (1986).
Alexander and Simmonds (1986) compared this model to test results from the
literature. The accuracy of the predicted results depended upon the density of the
reinforcement. As can be seen in Fig. 62, for reinforcement ratios of the top shear steel
between 1 to 2.5%, an excellent prediction was obtained. In lightly reinforced specimens,
strain hardening led to an underestimation of the capacity. In heavily reinforced specimens,
the assumption of yielding bars was not always met, and some specimens had a compression
failure. In this strut and tie model,
f c' is used determine α. A better result could be attained
by using the split cylinder tensile strength. The main reason why the split cylinder tensile
strength is not used, is the absence of the value of the tensile strength in the tests from the
literature that Alexander and Simmonds (1986) used to calibrate the factor α.
55
Fig. 62: Effect of the reinforcement density, Alexander and Simmonds (1986).
Alexander and Simmonds (1986) also stated that the truss model is applicable to a
wide range of boundary conditions and in-plane forces can be taken into account. The ACI
318-08 and EN 1992-1-1:2004 based the provisions for torsion on the use of a thin-walled
tube, space-truss analogy (Fig. 63). The background to the ACI code provisions using a three
dimensional truss model is discussed in MacGregor and Ghoneim (1995).
Fig. 63: Space truss analogy for torsion, MacGregor and Ghoneim (1995).
56
5.2. Bond model
Alexander and Simmonds (1992) developed a bond model in which radial arching
action and the concept of a critical shear stress on a critical section are combined. The bond
strength of the reinforcement is the significant factor. With the bond model, a simple lower
bound estimate of the ultimate punching strength was derived. The bond model is the result
of experiments carried out by Alexander and Simmonds after the development of the truss
model discussed in the previous section. The tests showed that the radial compression struts
are actually curved and parallel to the reinforcement in plan, which changed the mechanics of
the truss model. The revised model could be called a “force gradient model for punching
shear”.
The bond model combines features of the truss model with the concept of a limiting
shear stress, thereby explaining why code procedures give good results. The basis of the
method is the following expression of the shear force:
V=
d (Tjd ) d (T )
d ( jd )
=
jd +
T
dx
dx
dx
Eq. 18
in which the first part is carried by beam action (requiring strong bond forces) and the second
part by arching action (requiring only remote anchorage of the reinforcement).
In Eq. 18 the following parameters are used:
T
= the steel tension force,
jd
= the effective moment arm.
The geometry of the curved arch (which replaced the compression strut) is not governed by
conditions at the intersection of the arch and the reinforcement tying the arch, but rather by
the interaction between the arch and the adjacent quadrants of the two-way plate. The radial
strips (Fig. 64) extend from the loaded zone, up to a “remote end”, which is a position of zero
shear. The shear carried in the radial compression arch varies from a maximum near the
loaded zone where the slope of the arch is large, to a minimum at the intersection of the arch
and the reinforcing bar, where the slope is small. The shear carried by a radial strip needs to
be dissipated some distance away from the loaded area, depending on the curvature of the
arch.
57
Fig. 64: Layout of radial strips, Alexander and Simmonds (1992).
Fig. 65 describes the radial strip as a cantilever beam. The length l is called the loaded area,
and w the uniformly distributed load. For four radial strips extending from the loaded area, a
lower bound of the shear capacity is expressed as:
P = 8 M sw
Eq. 19
Fig. 65: Equilibrium of radial strip, Alexander and Simmonds (1992).
The flexural capacity of the strip Ms and the loading term w are consequently defined to meet
two conditions: the equilibrium of the strip has to be satisfied and both the flexural capacity
and shear capacity of the strip may not be exceeded at any point in the strip. The flexural
capacity depends upon the amount of reinforcement that effectively acts within the strip and
is composed of the negative and positive moment capacity.
58
M neg = ρ neg f y jd 2 c
Eq. 20
M pos = k r ρ neg f y jd 2 c
Eq. 21
In these equations, the following symbols are used:
ρ neg =
AsT
bd
= the negative effective reinforcing ratio,
ρ pos =
AsB
bd
= the positive effective reinforcing ratio,
AsT
= the total cross-sectional area of top steel within the radial strip plus one-half
the area of the first top bar on either side of the strip,
AsB
= the total cross-sectional area of bottom steel within the radial strip plus one-
half
the area of the first top bar on either side of the strip,
b
= the total distance between the first reinforcing bars on either side of the
radial strip,
d
= the effective depth,
jd
= the internal moment arm,
c
= the width of the radial strip,
fy
= the yield stress of the reinforcement,
kr
= a factor which accounts for the proportion of the bottom steel that can be
developed by the rotational restraint at the remote end of the strip. This
is zero if the remote end is simply supported.
The loading term w represents a lower bound estimate of the maximum shear load that may
be delivered to one side of a radial strip by the adjacent quadrant of the two-way plate. Fig.
66 shows any load applied directly to the strip (q) and the internal shears and moments. The
near side face of the half-strip is loaded in shear (v), torsion (mt) and bending (mn). Two
approximations are made: the direct load (q) and the torsional shear are neglected.
The maximum value of the loading term w is based on the maximum value of beam action
shear
w ACI = 0.166d f c'
(SI units)
Eq. 22
w ACI = 2d f c' (US customary units)
59
Fig. 66: Free-body diagram of one-half radial strip, Alexander and Simmonds (1992).
Fig. 67 shows the predicted failure load compared to 115 test results reported in the literature.
The mean value is 1.29. This value is reasonably close to unity, which suggests that the
mechanics of the bond model are not unrealistic. The effect of the reinforcement is taken into
account by estimating the flexural capacity Ms of the radial strip and, thus, can be combined
with an estimate of the one-way shear strength which does not take the reinforcement ratio
into account.
Fig. 67: Bond model results using ACI one-way shear, Alexander and Simmonds (1992).
The bond model also explains how load may be carried in the presence of diagonal
cracking. Test results have shown that diagonal cracking occurs at 50 to 70% of the ultimate
load. The bond model is limited to plates with a value c/d larger than 0.66. Most practical
slabs have a value of c/d larger than 1.5, and, thus, this requirement is not considered as a
serious limitation of the bond model.
60
5.3. Discussion of strut and tie models
The strut and tie model as developed by Alexander and Simmonds (1986) does not
have the weak points which were pointed out in the discussion of the shear stress theory and
the beam analogy models. Strut and tie models are successfully used to calculate forces in
various types of reinforced concrete structures. Three dimensional strut and tie models are
currently used to evaluate torsion and seem to answer several questions regarding the basic
mechanics of the punching shear problem.
Testing of the strut and tie model developed by Alexander and Simmonds (1986),
showed that the compression strut is actually curved. This experimental observation led to
the development of the bond model as described by Alexander and Simmonds (1992). The
bond model combined elements of the shear stress theory and the truss model, and gave an
explanation why code provisions, based on shear stress theory, give results which safely
match experimental data. Further research on the bond model could lead to a powerful
method to analyze existing structures.
61
6. Code Provisions
A good overview of different code provisions is written by Gardner (2005).
6.1. ACI 318-08
The nominal shear strength Vc shall be taken as the smallest of (ACI 318-08 §11.11.2.1, in
US customary units):

4
Vc =  2 + λ f c' bo d
β

Eq. 23
α d

Vc =  s + 2 λ f c' bo d
 b0

Eq. 24
Vc = 4λ f c' bo d
Eq. 25
in which:
f c'
= the specified concrete cylinder strength, psi,
β
= the ratio of the long side to the short side of the column, concentrated load or
reaction area,
λ
= the factor to account for concrete density (1.0 for normal density concrete),
bo
= the perimeter of the critical section for shear,
αs
= 40 for interior columns, 30 for edge columns, 20 for corner columns,
d
= the distance from the extreme compression fiber to the centroid of tensile
reinforcement.
The critical section is taken at a distance of d/2 away from the periphery of the loaded area.
6.2. EN 1992-1-1: 2004
The critical section is taken at 2d from the loaded area (Fig. 68). Around rectangular loaded
areas, rounded corners are used (Fig. 69).
62
Fig. 68: Verification model for punching shear at the ultimate limit state, figure 6.12 from
EN 1992-1-1:2004.
Fig. 69: Typical basic control perimeters around loaded areas, Figure 6.13 from EN 1992-11: 2004.
63
The design punching shear capacity is calculated as follows (equation 6.47 in the EN 1992-11:2004, in SI units):
v Rd ,c = C Rd ,c k (100 ρ l f ck )
1/ 3
+ k1σ cp ≥ (v min + k 1σ cp )
Eq. 26
with
fck
= the characteristic concrete strength in MPa,
200
≤ 2,0
d
k = 1+
d
= the effective depth in mm,
ρ l = ρ ly .ρ lz ≤ 0,02
ρly, ρlz
relate to the bonded tension steel y- and z-directions respectively. The values
ρly and ρlz should be calculated as mean values taking into account a slab width
equal to the column width plus 3d each side.
σ cp = (σ cy + σ cz ) / 2
σcy, σcz
= the normal concrete stresses in the critical section in y- and z-directions
(MPa, positive if compression) σ cy =
NEd,y, NEd,z
N Ed , y
Acy
and σ cz =
N Ed , z
Acz
= the longitudinal forces across the full bay for internal columns and the
longitudinal force across the control section for edge columns. The force may
be from a load or prestressing action.
Ac
= the area of concrete according to the definition of NEd.
The values of the next parameters depend on the National Annex. The recommended values
are:
CRd,c = 0.18/γc with γc=1.5,
v min = 0.035k 3 / 2 f ck1 / 2 ,
k1 = 0.1.
6.3. Comparison of code provisions
6.3.1. Comparison of ACI 318 and EN 1992-1-1
Gardner (2005) compared experimental data with the provisions of ACI 318-05, Fig.
70 and Fig. 71, and EN 1992-1-1:2003, Fig. 72. According to Gardner, comparison of the
code provisions with experimental results is not straightforward because the code expressions
were developed to be conservative and use specified or characteristic concrete strengths,
64
depending on the code and not the mean concrete strength reported for experimental studies.
The code punching shear predictions were calculated using the reported mean concrete
cylinder strengths. A second note to the data is that the median thickness of the tested slabs
was 140 mm (5.51 in), with a maximum of 320 mm (12.6 in), which is smaller than slabs
used in practice. The data (Fig. 70, Fig. 71 and Fig. 72) show that only ACI 318-05 with a
rounded shear perimeter meets the criterion of a 5% fractile value greater than one. The
results obtained by using EN 1992-1-1:2003 seemed to be unconservative, but the coefficient
of variation was smaller than for the results obtained by using ACI 318-05.
Fig. 70: Comparison of test/predicted using ACI 318-05 with rounded corners shear
perimeter, by Gardner (2005).
65
Fig. 71: Comparison of test/predicted using ACI 318-05 with assumption of square shear
perimeter, by Gardner (2005).
Fig. 72: Comparison of test/predicted using CEB-FIP MC90 and EN 1992-1-1:2003, by
Gardner (2005).
Albrecht (2002) remarked that the different code provisions have been derived from
tests which take into account the reinforcement practices common in the respective countries.
It should be emphasized that design and construction form an integral whole. As an example,
Albrecht (2002) points out that the comparatively high punching shear resistance provided by
the ACI code should be seen together with the required integrity reinforcement. Integrity
reinforcement is continuous reinforcement required in bridge decks to prevent progressive
66
collapse. EN 1992-1-1:2004 §9.10.1 requires that structures, which are not designed to
withstand accidental actions should have a suitable tying system to prevent progressive
collapse. In practice, this requirement could also result in continuous reinforcement.
6.3.2. Size effect
Mitchell, Cook and Dilger (2005) investigated the influence of the size effect. As can
be seen in Fig. 73, the code predictions match the test data up to an average effective depth of
300 mm (11.8 in). ACI 318-05 does not have a size factor and overestimates the punching
capacity of thicker slabs. The capacity of the 500 mm (19.7 in) thick slab is overestimated by
40%. EN 1992-1-1:2003 has a size factor, but overestimates the capacity of the 500 mm
(19.7 in) thick slab by about 20%.
Fig. 73: Comparison of code expressions with results reported by Li (2000) from Mitchell,
Cook and Dilger (2005).
Muttoni (2008) claimed that a failure criterion based on the critical shear crack theory
predicts the size effect better than ACI 318, which becomes unconservative for large values
of
ψd
d g0 + d g
, as can be seen in Fig. 74. According to Muttoni (2008), there are two
explanations for the difference between the test data and the value obtained by using ACI for
slabs with a value of
ψd
d g0 + d g
larger than 0.1. When the ACI formula was developed in the
1960s, only tests with relatively small effective depths were available in which the size effect
was not apparent. Tests in which punching occurred as a second failure after reaching the
flexural capacity are considered in the comparison (empty squares in Fig. 74).
67
Fig. 74: Failure criterion: punching shear strength as function of width of critical shear crack
compared with 99 experimental results and ACI 318-05 design equation, Muttoni (2008).
6.3.3. Influence of concrete strength
Mitchell, Cook and Dilger (2005) investigated the influence of the concrete strength
and compared test results to the code equations. Note that ACI 318 uses a square root of the
concrete strength and EN 1992-1-1:2004 a cube root. The authors concluded from Fig. 75
that the EN 1992-1-1 prediction seems to be more conservative than the ACI 318 prediction.
This conclusion is only valid for high strength concrete. As seen in section 6.3.1.
experimental data indicate that EN 1992-1-1 leads to less conservative results than ACI 318.
68
Fig. 75: Comparison of code expressions with test results reported by Ghannoum (1998) and
McHarg et al. (2000), figure by Mitchell, Cook and Dilger (2005).
6.3.4. Influence of reinforcement ratio
The ACI 318 code does not take the reinforcement ratio into account, while EN 19921-1 does.
Dilger, Birkle and Mitchell (2005) concluded that the reinforcement ratio should be
used in the ACI 318 code equations, as gathering test data (Fig. 28, Fig. 29, Fig. 30, Fig. 31
and Fig. 32) showed a distinct decrease in punching shear resistance with decreasing
reinforcement ratio. The reinforcement ratio should be calculated in the region where the
punching cone occurs. The authors noted that a definition of the extent of this region needs to
be developed.
Alexander and Hawkins (2005) on the other hand advised not to include the flexural
reinforcement ratio in code equations. They pointed out that ACI 318 works as well as any
detailed analytical model available for design purposes, with a fraction of the computational
effort, provided that the slab has been correctly designed for flexure. The reason is that ACI
318 is based on the work of Moe (1961) (see section 2.1) in which the goal is to prevent
punching shear failures prior to developing full flexural strength. As can be seen from Eq. 7
and Eq. 8, the ACI 318 code equation actually comes from an analytical equation that does
account for the flexural reinforcement. Alexander and Hawkins (2005) concluded that the
scatter of the ACI 318 code equation which is criticized by several authors is, in fact,
statistically invalid (Fig. 76). The code equation is intended to handle the case where the
69
flexural and shear strengths match. Specimens designed for higher or lower loads are from
other populations. The code equation was never meant to handle these cases; the goal of the
code has never been to predict slab-column test results.
Fig. 76: Effect of analysis on perceived scatter, Alexander and Hawkins (2005).
70
7. Discussion
While the mechanics at the basis of the punching shear problem are still not very well
understood, engineers need tools to design and analyze slabs. Several methods have been
developed over the years. These methods are typically compared to the results of slabcolumn tests. The behavior of the samples used in these tests is not fully comparable with the
behavior of slabs in real buildings or bridge decks. However, due to the high cost of largescale tests, slab-column tests are used to study the effect of several variables on the shear
strength of slabs.
To design slabs, code provisions have been developed. There are significant
differences between the code provisions. Comparing and judging these code equations is
hard, as there is not a generally accepted mechanical model representing the behavior of a
slab in shear to compare with these code equations. The ACI 318 code provisions for shear in
slabs appear to be easier to use for practicing engineers than the EN 1992-1-1 provisions.
The influence of several effects has been discussed in the previous sections. The work of
Guandalini, Burdet and Muttoni (2009) indicates that ACI 318 might not be conservative for
slabs with low reinforcement ratios (ρ < 0.5%), and the work of Muttoni (2008) indicates that
ACI 318 might not be conservative for slabs thicker than 300 mm (11.81 in). As Alexander
and Hewitt (2005) argued, it might not be needed to add the reinforcement ratio to the code
expression because the code expression assures that the slab fails in flexure before punching.
To decide whether the punching shear strength needs to be calculated using the square or the
cube root of the concrete compressive strength, a better understanding of the behavior of
concrete in tension and under splitting is needed.
In spite of these drawbacks, the code provisions generally appear to lead to safe designs.
To analyze the strength of slabs in existing structures, a good mechanical model is
needed. Test results, such as those by Taylor, Rankin, Cleland and Kirckpatrick (2007) and
Csagoly (1979) indicate that the shear strength of bridge deck slabs is significantly higher
than calculated by the code provisions. This observation is typically contributed to the
influence of compressive membrane forces and fixed boundary action.
Among the methods that calculate the shear strength using the shear stress on a
critical perimeter, recent developments include the critical shear crack theory, which
accounts for the size effect, and models which take the fixed boundary action and
compressive membrane action into account. All of these models use empirical formulations,
and none of them fully explains the mechanics of the punching shear problem.
71
The combination of plate theory and finite element models could give a more realistic
assessment of the real punching capacity. In these models, the material behavior can be
selected to approach the real material behavior. Results up to now found a good
approximation of the ultimate loading as compared to test results. A good model should
however reproduce the full behavior of a specimen (such as the load-deflection curves) up to
punching failure. Current models do not meet this criterion yet.
Beam analogy methods are widely used in the design of slabs in flexure. By combining the
effects of flexure, shear and torsion, beam analogy methods can be used to find the ultimate
punching strength. However, beam analogy methods do not give a realistic mechanical model
of shear in slabs.
The truss analogy (strut and tie) method is a different concept and seems to answer some of
the fundamental problems of the shear stress and beam analogy methods. Test results
indicated that the compression struts have a curved shape, which led to the bond model. The
bond model unites the apparently conflicting methods of the truss model and the shear stress
method. It might be an interesting method for the analysis of existing structures. Further
research on the bond model is needed to verify its applicability.
It can be concluded that, even though a large number of methods exist which describe
the punching shear problem, none of these methods explains the mechanics at the basis of the
punching shear problem. Therefore, semi-experimental formulas have been developed which
lead to safe designs for commonly used structures. However, there is a need for an
understanding of the mechanics at the basis of the punching shear problem to find the real
shear capacity of existing structures.
72
8. Conclusions
In previous sections, four methods to find the ultimate punching shear strength in
reinforced concrete slabs and punching shear code provisions were discussed. None of the
methods is fully theoretical, and test results from slab-column specimens are widely used to
determine semi-empirical formulations.
The methods based on the shear strength on a critical section are most commonly
used in practice. The shear stress on a critical section at a certain distance from the face of the
loaded area is compared to a maximum shear stress. The basis for the most commonly used
method was provided by Moe (1961). The distance from the face of the loaded area was
determined through a statistical analysis of test results and determined to be d/2. Then, a
criterion was developed to ensure that slabs would fail in a ductile flexural manner rather
than in a usually brittle punching shear manner. Most of the methods based on the shear
stress take the concrete strength and/or the flexural reinforcement into account. Other factors,
such as the shape of the column or loaded area, size of the loaded area, boundary conditions,
load duration, size effect and influence of restraints can be taken into account, but are
generally considered of less importance. Muttoni (2008) however argued that the size effect
should be taken into account because the ACI code provisions are not conservative for slabs
of thickness larger than 300 mm (11.81 in). The critical shear crack theory takes the size
effect into account and correctly predicts the ultimate punching shear loading of slab-column
specimens with a large thickness.
Methods based on plate theory and non-linear finite elements are currently being
developed. The difficulty here is to select the appropriate material behavior when modeling a
specimen.
Methods based on beam analogies are not often used. Accounting for redistribution of
forces and interaction between shear, flexure and torsion can approach the behavior of a test
specimen, but it does not provide a mechanical model of the shear force in slabs.
The strut and tie, truss model uses steel ties and concrete struts and gives a different
approach to the shear problem. Additional testing led to the bond model, in which the
concrete compression strut is curved and a load gradient is carried along the strut up to a
certain point where the internal loading reaches zero. The external loading is calculated on
strips extending from the faces of the loaded area. The maximum loading on these strips is
taken as the maximum loading per area for a beam given in the ACI code.
73
The provisions of ACI 318 and EN 1992-1-1 appear very different at first sight. These
code provisions are, however, based on the same theory, namely the shear stress on a critical
perimeter. An obvious difference is the place of the critical perimeter. While ACI 318 takes
this critical perimeter at d/2 away from the face of the loaded area and straight around
rectangular areas, EN 1992-1-1 takes it at 2d away from the loaded area and uses rounded
corners for the perimeters. Another apparent difference is the use of the square root of the
concrete strength in ACI 318 and the cube root in EN 1992-1-1. There is no consensus up to
now as to which of these is better. While EN 1992-1-1 takes the flexural reinforcement into
account, ACI 318 does not use reinforcement ratio in its provisions. Using the flexural
reinforcement seems to match the experimental data better, but is not guaranteed to make the
code a better design tool.
In general, there is a significant distinction between the design practice which needs a
good tool to guarantee a safe design on one hand and the analysis practice which needs a
model which accurately describes the mechanical behavior of a slab under shear. The first
need seems to be satisfied with code provisions that work well in practice. The second need
is not fulfilled, as none of the above mentioned models fully explain the behavior of slabs in
shear.
74
9. References
AASHTO, 2007, AASHTO LRFD Bridge Design Specifications, American Association of
State Highway and Transportation Officials, Washington D.C., 4086 pp.
ACI Committee 318, 2008, Building Code Requirements for Structural Concrete (ACI 31808) and Commentary, American Concrete Institute, Farmington Hills, MI, 465 pp.
Albrecht, U., 2002, “Design of flat slabs for punching – European and North American
practices,” Cement & Concrete Composites, V. 24, No. 6, pp. 531-538.
Alexander, S.D.B. and Hawkins, N.M., 2005, “A Design Perspective on Punching Shear,”
SP-232, Ed. Polak, M.A., American Concrete Institute, Farmington Hills, MI, pp. 97-108.
Alexander, S. and Simmonds, S., 1986, “Shear-Moment Transfer in Slab-Column
connections,” Structural Engineering Report No. 141, University of Alberta, Edmonton,
Alberta, 95 pp.
Alexander, S. and Simmonds, S., 1992, “Bond Model for Concentric Punching Shear,”, ACI
Structural Journal, V. 89, No. 3, pp. 325-334.
ASCE-ACI Committee 426, 1974, “The Shear Strength of Reinforced Concrete Members –
Slabs,” Proceedings, ASCE, V.100, No. ST8, pp. 1543-1591.
Bakker, G.J., 2008, A finite element model for the deck of plate-girder bridges including
compressive membrane action, which predicts the ultimate collapse load, MSc Thesis, Delft
University of Technology, Delft, The Netherlands, 203 pp.
Birkle, G., Dilger, W., 2008, “Influence of Slab Thickness on Punching Shear Strength,” ACI
Structural Journal, V. 105, No.2, pp. 180-188.
CEN, 2004, Eurocode 2 – Design of Concrete Structures: Part 1-1 General Rules and Rules
for Buildings, EN 1992-1-1, Comité Européen de Normalisation, Brussels, Belgium, 225 pp.
Collins, M.P. and Kuchma, D., 1999, “How Safe Are Our Large, Lightly Reinforced
Concrete Beams, Slabs, and Footings?,” ACI Structural Journal, V. 96, No. 4, pp. 482-490.
Csagoly, P. F., 1979, Design of Thin Concrete Deck Slabs by the Ontario Highway Bridge
Design Code, Ministry of Transportation and Communications, Ontario, 50 pp.
DIANA, 2008, User’s Manual – Release 9.3., TNO DIANA, Delft, The Netherlands,
http://www.tnodiana.com/upload/files/DIANA/HTML/Diana.html, 5380 pp.
Dilger, W., Birkle, G. and Mitchell, D., 2005, “Effect of Flexural Reinforcement on
Punching Shear Resistance,” SP-232, Ed. Polak, M.A., American Concrete Institute,
Farmington Hills, MI, pp. 57-74.
75
Elstner, R., Hognestad, E., 1956, “Shearing Strength of Reinforced Concrete Slabs,” Journal
of the American Concrete Institute, V. 28, No. 1, pp. 29-58.
Gardner, N.J., 2005, “ACI 318-05, CSA A.23.3-04, Eurocode 2 (2003), DIN 1045-1 (2001),
BS 8110-97 and CEB-FIP MC 90 Provisions for Punching Shear of Reinforced Concrete Flat
Slabs,” SP-232, Ed. Polak, M.A., American Concrete Institute, Farmington Hills, MI, pp.122.
Guandalini, S., Burdet, O.L. and Muttoni, A., 2009, “Punching Tests of Slabs with Low
Reinforcement Ratios,” ACI Structural Journal, ACI, V. 106, No. 1, pp. 87-95.
Hallgren, M., Bjerke, M., 2002, “Non-linear finite element analyses of punching shear failure
of column footings,” Cement & Concrete Composites, V. 24, No. 6, pp. 491-496.
Hewitt, B. and Batchelor, B., 1975, “Punching Shear Strength of Restrained Slabs,” Journal
of the Structural Division, V. 101, No. 9, pp. 1837-1853.
Menétrey, Ph., 2002, “Synthesis of punching failure in reinforced concrete,” Cement &
Concrete Composites, V. 24, No. 6, pp. 497-507.
Mitchell, D., Cook, W.D. and Dilger, W., 2005, “Effects of size, Geometry and Material
Properties on Punching Shear Resistance,” SP-232, Ed. Polak, M.A., American Concrete
Institute, Farmington Hills, MI, pp. 39-56.
Moe, J., 1961, “Shearing Strength of Reinforced Concrete Slabs and Footings under
Concentrated Loads,” Bulletin D47, Portland Cement Association, Skokie, IL, 135 pp.
Muttoni, A., 2003, “Schubfestigkeit und Durchstanzen von
Querkraftbewehrung,” Beton- und Stahlbetonbau, V. 98, No.2, pp. 74-84.
Platten
ohne
Muttoni, A., 2008, “Punching Shear Strength of Reinforced Concrete Slabs without
Transverse Reinforcement,” ACI Structural Journal, V. 105, No. 4, pp. 440-450.
Ngo, D. T., 2001, “Punching shear resistance of high-strength concrete slabs,” Electronic
Journal of Structural engineering, V. 1, No. 1, pp. 2-14.
Normcommissie 351 001, 1995, Technische Grondslagen voor Bouwvoorschriften,
Voorschriften Beton TGB 1990 – Constructieve Eisen en Rekenmethoden (VBC 1995), NEN
6720, Civieltechnisch centrum uitvoering research en regelgeving, Nederlands Normalisatieinstituut, Delft, The Netherlands, 245 pp.
MacGregor, J.G. and Ghoneim, M.G, 1995, “Design for Torsion,” ACI Structural Journal, V.
92, No. 2, pp. 211-218.
Park, R. and Gamble, W., 1999, Reinforced Concrete Slabs, John Wiley & Sons, New York,
716 pp.
PCI, 2003, Precast Prestressed Concrete Bridge Design Manual, Precast/Prestressed
Concrete Institute, Chicago, IL, 1340 pp.
76
Polak, M.A., 2005, “Shell finite element analysis of RC plates supported on columns for
punching shear and flexure,” Engineering Computations, V. 22, No. 1, pp. 409-428.
Sherif, A.G., Emara, M.B, Ibrahim, A.H. and Magd, S.A., 2005, “Effect of the Column
Dimensions on the Punching Shear Strength of Edge Column-Slab Connections,” SP-232,
Ed. Polak, M.A., American Concrete Institute, Farmington Hills, MI, pp. 175-192.
Sundquist, H., 2005, “Punching Research at the Royal Institute of Technology (KTH) in
Stockholm,” SP-232, Ed. Polak, M.A., American Concrete Institute, Farmington Hills, MI,
pp. 229-256.
Taylor, S.E., Rankin, B., Cleland, D.J., Kirckpatrick, J., 2007, “Serviceability of bridge deck
slabs with arching action,” ACI Structural Journal, V. 104, No. 1, pp. 39-48.
Theodorakopoulos, D. D., Swamy, R.N., 2002, “Ultimate punching shear strength analysis of
slab-column connections,” Cement & Concrete Composites, V. 24, No. 6, pp. 509-521.
Wei, X., 2008, Assessment of Real Loading Capacity of Concrete Slabs, MSc Thesis, Delft
University of Technology, Delft, The Netherlands, 112 pp.
77
Appendix A: Notations
A.1. List of notations by symbol
a
radius of the slab (Bakker, 2008)
A
coefficient, =1 for normal density concrete (Theodorakopoulos and Swamy,
2002)
A
constant, determined by statistical analysis (Moe, 1961)
Abar
area of a single reinforcing bar (Alexander and Simmonds, 1986)
Ac
area of concrete
AsB
total cross-sectional area of bottom steel within the radial strip plus one-half
the area of the first top bar on either side of the strip (Alexander and
Simmonds, 1992)
AsT
total cross-sectional area of top steel within the radial strip plus one-half the
area of the first top bar on either side of the strip (Alexander and Simmonds,
1992)
ASVtop
top mat shear steel (Alexander and Simmonds, 1986)
b
width of critical section in shear (Moe, 1961)
b
total distance between the first reinforcing bars on either side of the radial strip
(Alexander and Simmonds, 1992)
b0
control perimeter at d/2 from the edge of the column (Guandalini, Burdet and
Muttoni, 2009)
bo
perimeter of the critical section at d/2 form the edge of the loaded area
B
constant, determined by statistical analysis (Moe, 1961)
B
diameter of the loading area (Wei, 2008)
c
length of side of loaded area
c
diameter of the slab area with negative radial bending moment (Wei, 2008)
c
column dimension perpendicular to the bar being considered (Alexander and
Simmonds, 1986)
c
width of the radial strip (Alexander and Simmonds, 1992)
C
constant, determined by statistical analysis (Moe, 1961)
C1
compressive force above inclined crack (ASCE-ACI Committee 426, 1974)
d
effective depth of the slab
78
d’
cover of the reinforcement measured to the near side of the slab (Alexander
and Simmonds, 1986)
d0
length over which the concentrated load is spread (Bakker, 2008)
d1
outer diameter of the punched cone (Bakker, 2008)
dg
maximum size of the aggregate (Guandalini, Burdet and Muttoni, 2009)
dg0
reference aggregate size equal to 16mm = 0.63 in (Guandalini, Burdet and
Muttoni, 2009)
ds
cover of the reinforcement measured to the far side of the slab (Alexander and
Simmonds, 1986)
f2
lateral compressive stresses (Moe, 1961)
f3
vertical compressive stresses (Moe, 1961)
fc
direct compressive stresses (Moe, 1961)
fc
compressive strength of the concrete (Guandalini, Burdet and Muttoni, 2009)
fc’
ultimate concrete strength, cylinder stength
fck
characteristic concrete strength
fcu
ultimate concrete strength
fs
steel stresses in tension (Theodorakopoulos and Swamy, 2002)
fs’
steel stresses in compression (Theodorakopoulos and Swamy, 2002)
fy
yield strength of the steel
Fb
total boundary restraint force (Wei, 2008)
GF0
fracture energy equal to 0.025, 0.030, 0.038 for aggregate size da = 8mm,
16mm, 32 mm respectively (Sundquist, 2005)
h
thickness of the slab
jd
moment arm between centre of compression zone and tensile reinforcement
k
size factor (EN 1992-1-1)
kr
a factor which accounts for the proportion of the bottom steel that can be
developed by the rotational restraint at the remote end of the strip (Alexander
and Simmonds, 1992)
K
constant (Alexander and Simmonds, 1986)
mn
bending (Alexander and Simmonds, 1992)
mt
torsion (Alexander and Simmonds, 1992)
Mb
boundary restraint moment (Wei, 2008)
Mneg
negative moment capacity
79
Mr
radial moment (ASCE-ACI Committee 426, 1974)
Mpos
positive moment capacity
Ms
flexural capacity of the strip (Alexander and Simmonds, 1992)
Mθ
tangential moment (ASCE-ACI Committee 426, 1974)
na
dimensionless membrane force in the mid depth of the slab (Bakker, 2008)
nr
dimensionless radial membrane force working on the surface of the failure
cone (Bakker, 2008)
NEd,y
longitudinal force across the full bay for internal columns and the longitudinal
force across the control section for edge columns (EN 1992-1-1)
NEd,z
longitudinal force across the full bay for internal columns and the longitudinal
force across the control section for edge columns (EN 1992-1-1)
P
lower bound for punching shear capacity (Alexander and Simmonds, 1992)
P
external load (Wei, 2008)
Pshear
ultimate shear capacity (Elstner and Hognestad, 1956)
Pfailure
failure load (Alexander and Simmonds (1986)
Pflex
ultimate flexural capacity of the slab computed by the yield-line theory
without regard to a shear failure (Elstner and Hognestad, 1956)
q
direct load on strip (Alexander and Simmonds, 1992)
r
side length of loaded area
r(x)
a function of the failure surface over the height (Bakker, 2008)
rs
radius of the slab (Guandalini, Burdet and Muttoni, 2009)
RcT
horizontal force in the concrete crossing the shear crack (Wei, 2008)
RsR
horizontal force in the reinforcement at right angles to the radial cracks (Wei,
2008)
RsT
horizontal force in the reinforcement crossing the shear crack (Wei, 2008)
s
spacing of bars
T
inclined compressive force in the compressed conical shell (Wei, 2008)
T
steel force (Alexander and Simmonds 1992)
vo
vertical shearing stresses (Moe, 1961)
v
nominal shear stress (Moe, 1961)
v2
shearing stress at face of loaded area (Elstner and Hognestad, 1956)
vadj
adjusted shear strength (Dilger, Birkle and Mitchell, 2005)
vtest
tested shear strength (Dilger, Birkle and Mitchell, 2005)
80
V
shear force (Moe, 1961)
Ve
shear force at which failure in the compression zone above the inclined cracks
occurs (Moe, 1961)
Vflex
shear force at the ultimate flexural strength (Moe, 1961)
Vi
shear force at which inclined cracks form (Moe, 1961)
Vo
fictitious shear strength if bending could be eliminated (Moe, 1961)
VR
shear strength (Guandalini, Burdet and Muttoni, 2009)
wc
width of critical shear crack (Guandalini, Burdet and Muttoni, 2009)
w
loading
x
depth of the compression zone (Sundquist, 2005)
xb
neutral axis depth at the boundary (Wei, 2008)
xm
neutral axis depth at mid-span or plastic hinges (Wei, 2008)
Xf
depth of critical section for straight flexural crack (Theodorakopoulos and
Swamy, 2002)
Xs
depth of critical section for inclined shear crack (Theodorakopoulos and
Swamy, 2002)
α
punching crack inclination (Menétry, 2002)
α
angle between yield surface and displacement rate vector (Bakker, 2008)
α
angle of the compression strut (Alexander and Simmonds, 1986)
αs
factor, 40 for interior columns, 30 for edge columns, 20 for corner columns
(ACI 318)
β
factor between 0 and 0,5 (Bakker, 2008)
β
ratio of the long side to the short side of the column, concentrated load or
reaction area (ACI 318)
∆φ
central angle (Wei, 2008)
ε cTu
ultimate tangential strain (Sundquist, 2005)
εcu
ultimate concrete strain (Theodorakopoulos and Swamy, 2002)
εo
concrete strain at the level of the end of the rectangular stress block
(Theodorakopoulos and Swamy, 2002)
φo
ratio of ultimate shear capacity to ultimate flexural capacity (Moe, 1961)
λ
factor to account for concrete density (ACI 318)
ψ
rotation of slab (Guandalini, Burdet and Muttoni, 2009)
ρ
reinforcement ratio of tensile steel
81
ρ’
reinforcement ratio of compression steel (Theodorakopoulos and Swamy,
2002)
ρly
bonded tension steel in y-direction (EN 1992-1-1)
ρlz
bonded tension steel in z-direction (EN 1992-1-1)
ρneg
negative effective reinforcing ratio
ρpos
positive effective reinforcing ratio
σcy
normal concrete stress in the critical section in y-direction
σcz
normal concrete stress in the critical section in y-direction
A.2. Table of notations by parameter
Table 6: Overview of the notations by parameter
Parameter
ACI 318 EN 1992-1-1 Other authors
Adjusted shear strength
vadj (Dilger, Birkle
and Mitchell, 2005)
Angle between yield surface and
α (Bakker, 2008)
displacement rate vector
Angle of the compression strut
α (Alexander and
Simmonds, 1986)
Area of concrete
Ac
Ac
Area of single reinforcing bar
Abar
Bending
mn (Alexander and
Simmonds, 1992)
Bonded tension steel in y-direction
ρly
Bonded tension steel in z-direction
ρlz
Boundary restraint moment
Mb (Wei, 2008)
Central angle
∆φ (Wei, 2008)
Characteristic/Nominal compressive
fck
strength
Compressive force above inclined crack
C1 (ASCE-ACI 426,
1974)
Compressive strength
fc (Guandalini,
Burdet and Muttoni,
2009)
Considered steel in bond model
AsB, AsT (Alexander
and Simmonds,
1992)
Crack width
wk
wc (Guandalini,
Burdet and Muttoni,
2009)
Critical perimeter
bo
u1
b0
Depth of compression reinforcement
d’
d’
from extreme compression fibre
82
Depth of compression zone
Depth of critical section for inclined
shear crack
Depth of critical section for straight
flexural crack
Depth of tensile reinforcement from
extreme tension fibre
Diameter of critical perimeter
c
x
ds
h-d
b1, b2
by, bz
Diameter of slab with negative radial
bending moment
Dimensionless membrane force
Dimensionless radial membrane force
Direct compressive stress
Direct load on strip (bond model)
Distance between reinforcing bars
s
w
Effective depth
External shear force
d
d
Factor to account for concrete density
Failure surface over height
Fictitious shear strength if bending could
be eliminated
Flexural capacity of strip (bond model)
λ
Fracture energy
Horizontal force in the concrete crossing
the shear crack
Horizontal force in the reinforcement at
right angles to the radial cracks
Horizontal force in the reinforcement
crossing the shear crack
Inclined compressive force in the
compressed conical shell
Lateral compressive stress
Load
Longitudinal force
Maximum size of aggregate
Moment arm between centre of
compression zone and tensile
reinforcement
Negative effective reinforcing ratio
Negative moment capacity
Neutral axis depth at boundary
Neutral axis depth at mid-span
x (Sundquist, 2005)
Xs (Theodorakopoulos and Swamy,
2002)
Xf (Theodorakopoulos and Swamy, 2002)
d1 (Bakker, 2008),
B (Wei, 2008)
c (Wei, 2008)
na (Bakker, 2008)
nr (Bakker, 2008)
fc (Moe, 1961)
q (Alexander and
Simmonds, 1992)
b (Alexander and
Simmonds, 1992)
P (Wei, 2008), V
(Moe, 1961)
r(x) (Bakker, 2008)
Vo (Moe, 1961)
MS (Alexander and
Simmonds, 1992)
GF0 (Wei, 2008)
RcT (Wei, 2008)
RsR (Wei, 2008)
RsT (Wei, 2008)
T (Wei, 2008)
f2 (Moe, 1961)
w
P
jd
w
NEd,y, NEd,z
dg
z
ρneg
Mneg
xb (Wei, 2008)
xm (Wei, 2008)
83
Nominal shear stress
Normal concrete stress in the critical
section in y-direction
Normal concrete stress in the critical
section in y-direction
Positive effective reinforcing ratio
Positive moment capacity
Proportion of bottom steel which can be
developed at the end of a strip (bond
model)
Punching crack inclination
Punching shear capacity
vn
vEd
σcy
σcz
ρpos
Mpos
kr (Alexander and
Simmonds, 1992)
Vc
θ
VEd
Radius of the slab
r
r
Ratio of the long side to the short side of
the column, concentrated load or reaction
area
Ratio of ultimate shear capacity to
ultimate flexural capacity
Reference aggregate size
Reinforcement ratio of compression steel
Reinforcement ratio of tensile steel
Rotation of slab
β
Radial moment
Shear force at the ultimate flexural
strength
Shear force at which failure in the
compression zone above the inclined
cracks occurs
Shear force at which inclined cracks form
Shear steel (truss model)
v (Moe, 1961)
v2 (Elstner and
Hognestad, 1956)
α (Menétry, 2002)
P (Alexander and
Simmonds, 1992),
Pshear (Elstner and
Hognestad, 1956),
Pfailure (Alexander
and Simmonds,
1986), VR
(Guandalini, Burdet
and Muttoni, 2009)
Mr (ASCE-ACI 426,
1974)
a (Bakker, 2008)
rs (Guandalini,
Burdet and Muttoni,
2009)
φo (Moe, 1961)
dg0
ρ’
ρ
ρl’
ρl
ψ (Guandalini,
Burdet and Muttoni,
2009)
Vflex (Moe, 1961)
Ve (Moe, 1961)
Vi (Moe, 1961)
Asvtop, ASVbottom
(Alexander and
Simmonds, 1992)
84
Side length of loaded area (column)
c1, c2
Size factor
Steel force
Steel stress in compression
Steel stress in tension
Tangential moment
c
k
T (Alexander and
Simmonds, 1992)
fs’
fs
fs2
fs1
Mθ (ASCE-ACI 426,
1974)
vtest (Dilger, Birkle
and Mitchell, 2005)
Tested shear strength
Thickness of slab
Torsion
Total boundary restraint force
Ultimate compressive strength
Ultimate concrete strain
d0 (Bakker, 2008)
r
h
h
mt (Alexander and
Simmonds, 1992)
Fb (Wei, 2008)
fc’
fcu
εcu2, εcu3
Ultimate flexural capacity calculated with
yield line theory
Ultimate tangential strain
Vertical compressive stress
Vertical shearing stress
Width of critical section
Width of radial strip (bond model)
vc
b
vRd,c
ui
Yield strength of steel
fy
fy
εcu (Theodorakopoulos and Swamy,
2002)
Pflex (Elstner and
Hognestad, 1956)
ε cTu (Sundquist,
2005)
f3 (Moe, 1961)
vo (Moe, 1961)
b (Moe, 1961)
c (Alexander and
Simmonds, 1992)
85
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