ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Title No. 113-S09

Tangential Strain Theory for Punching Failure of Flat Slabs

by Carl Erik Broms

A novel mechanical model for the punching failure at interior

columns of flat slabs without shear reinforcement is presented. It
is based on fundamental structural mechanics and the stress-strain
relation of compressed concrete. The shear force is assumed to be
transferred to the column by an inclined circumferential compres-
sion strut that squeezes the concrete within the column perimeter,
and when the compression stress in this area approaches the yield
level, an increasing part of the squeezing pressure is anchored
back to the surrounding concrete. Ultimately, the compression zone
outside the column is assumed to collapse due to the generation of
radial tension strain. Validation against test results in the literature
demonstrates that the proposed model can accurately predict the
punching capacity and the concurrent ultimate slab rotation. Both
concentric and eccentric punching can therefore be analyzed.

Keywords: flat slab; mechanical model; punching failure.

INTRODUCTION
The punching failure of flat slabs resembles shear failure
Fig. 1—Radial tensile stress at concentrated force.
of beams in the sense that it is characterized by a “shear
crack” from the supporting column up to the top surface of
the slab. Consequently, the majority of researchers and most
building codes define punching capacity in terms of nominal
shear strength on a control perimeter at a certain distance
from the column. But the methods do not give the designer
any indication of the limited rotation capacity of the slab at
the column support. This shortcoming was first studied by
Kinnunen and Nylander,1 but their model is complicated and
cannot predict punching capacity with the same accuracy as
Fig. 2—Truss model for support region of flat slab.
current statistical methods.
Broms2,3 proposed improvement of the Kinnunen and below the column is squeezed by the inclined compression
Nylander1 model, in which a solution for the ultimate rota- struts and the resulting volume reduction of the concrete
tion was derived and consideration was given to the size is restrained by the surrounding concrete. This restraining
effect (decreasing punching strength with increasing height force creates radial tensile stress and tangential compression
of the compression zone in the slab). stress outside the column.
Hallgren4 tested high-strength concrete specimens and A similar truss model for an interior column of a flat slab
presented a theory in which the height of the compres- is depicted in Fig. 2, where the shear force is transferred to
sion zone was again used to derive the size effect based on the column by a circumferential inclined compression strut
nonlinear fracture mechanics. that resembles the struts in Fig. 1. The horizontal component
Muttoni5 and Muttoni et al.6 presented the Critical Shear of the strut force will thus squeeze the column and, when the
Crack Theory (CSCT). Here, the width of a critical shear stress inside the column perimeter reaches the “yield level,”
crack is assumed to be proportional to the rotation (slope) the stiffness for squeezing pressure will rapidly decrease.
of the slab. The failure criterion is based on a theory for the An increasing portion of the horizontal component of the
influence of crack width and aggregate interlock on shear inclined strut force will then instead be anchored back to
force transfer across a shear crack. the surrounding concrete by means of a radial tensile force.
In these models, the basic concept advanced by Kinnunen When the resulting radial tensile strain at the column edge
and Nylander1 has been adopted, in which a circle sector of
the area around an interior column is studied. However, it ACI Structural Journal, V. 113, No. 1, January-February 2016.
appears that a well-known effect similar to force transfer MS No. S-2014-347.R3, doi: 10.14359/51687942, received March 5, 2015, and
reviewed under Institute publication policies. Copyright © 2016, American Concrete
from a column to a larger structure has been overlooked Institute. All rights reserved, including the making of copies unless permission is
(refer to Fig. 1). The indicated strut-and-tie model simu- obtained from the copyright proprietors. Pertinent discussion including author’s
closure, if any, will be published ten months from this journal’s date if the discussion
lates the internal force flow. The concrete situated locally is received within four months of the paper’s print publication.

ACI Structural Journal/January-February 2016 95

Fig. 3—Fan-type yield lines. Fig. 4—Bending moments according to theory of elasticity
for circular slab supported on edge of circular column.
exceeds the radial compression strain due to the bending
moment, a tensile crack will develop around the column. B V  a a 3π
∆M = V =  + →B= a
This crack is assumed to initiate the punching failure. 2π 4  2 4  8 (1)
The scenario described takes place when the tangential
compression strain due to the bending moment at the column The following expressions for a circular slab with a diameter
edge reaches a critical value. Consequently, the model is, in C are derived from Eq. (84) and (85) by Timoshenko and
this context, called the “Tangential Strain Theory” (TST). The Woinowsky-Krieger7 (with Poisson’s ratio ν = 0). Most of
analytical treatment is very simple; no advanced algorithms these equations are not used in the normal design situation,
are involved—only basic structural mechanics equations. but are given herein for the sake of completeness.
Tangential moment
RESEARCH SIGNIFICANCE
The failure mechanism of the model is conceivable and will V  C B2 B2 
hopefully contribute toward a greater understanding of the mt =  2 ln + 2 − −  (2)
8π  2r 4r 2 C 2 
punching failure at interior columns of flat slabs. The model
predicts the punching capacity and the concurrent slab rota-
tion. Both concentric punching and eccentric punching due Radial moment
to imposed rotation of the slab strip in relation to the column
(or vice versa at interstory drift) can therefore be analyzed. V  C B2 B2 
mr =  2ln + 2 − 2  (3)
8π  2r 4r C 
BENDING MOMENTS
The moment distribution at the columns of a flat-slab
structure is nearly polar-symmetric, and a rational calcula- Tangential and radial moments at the column edge
tion model is shown in Fig. 3. A common arrangement for
the testing of flat slabs is therefore a circular or square spec- V  B B2 
m1 = 1 − 2ln − (4)
imen loaded along its perimeter and supported on a column 8π  C C 2 
at its center (refer to Fig. 4). The perimeter of the specimen
is intended to reflect the circular line of contraflexure for the
Tangential moment at the slab edge
bending moment in a radial direction in a continuous flat
slab. According to the theory of elasticity, this circle has a
radius of ≈0.22L in a flat slab with square panels, where L V  B2 
m2 = 1− (5)
is the span width. The following equations assume either 4π  C 2 
circular or square specimen arrangement. In the latter case,
the diameter of the equivalent circular slab is assumed to be The bending moment at overall yield becomes
equal to the width of the square slab if the corners are free to
lift in the square slab. V  B
Up to the load level, when the flexural reinforcement starts my = 1 −  (6)
to yield near the column, the theory of elasticity is assumed 2π  C 
to be valid for the bending moment distribution. The support The deflection of a slab specimen consists of bending defor-
reaction falls close to the column edge due to the slab rota- mation and shear deformation, where the latter is not insig-
tion (slope). A square column with a side length a is replaced nificant near the column (refer to Fig. 5). When these effects
by a fictitious circular column with a diameter B that gives are superimposed, the resulting deformation configuration
the same reduction ΔM of the total bending moment resembles a truncated cone with the radial rotation ψ.

96 ACI Structural Journal/January-February 2016

Fig. 5—Elastic slab deflection due to bending moment and Fig. 6—Transfer of load from flat slab to column.
shear force.

m2 C V  B 2  C
y= = 1− (7)
EI 2 4π  C 2  2 EI

The Kinnunen and Nylander1 model presupposes that the

sector elements rotate in the form of rigid bodies around the
column edge. The tangential bending moments at the column
edge m1 and specimen edge m2, respectively, become

 B Fig. 7—Radial concrete stress distribution.

1−
B V  C  (8)
m2 = m1 = “shear failure” governed by the diagonal tensile strength of
C 2π B
1 − ln concrete. Instead, punching failure appears to occur when
C
the compression zone adjacent to the column collapses. The
model depicted in Fig. 6 may simulate this zone. Inclined
compression struts transfer the shear force V to an internal
 B column capital that in turn transfers the load further to the
1−
B V  C  (9) column. Finally, a flat shear crack with an inclination of
m2 = m1 =
C 2π B approximately 1:2 develops and runs down to the neutral
1 − ln
C axis of the slab. Any shear force transfer by aggregate
interlock across this crack is conservatively neglected. A
The slope of the moment curve at rigid body rotation is single compression strut running below the shear crack will
much steeper than the slope of the moment curve according therefore ultimately transfer the entire shear force V to the
to the theory of elasticity. For the common relation B/C = column capital.
1/8, the ratio m1/m2 becomes
Squeezing effect
according to the theory of elasticity: The stress situation in the compression zone of the slab
m1/m2 = 0.2046/0.0783 = 2.61 near the column edge is shown in Fig. 7. The inclined
compression strut will cause additional horizontal pressure
and for rigid body rotation: m1/m2 = 0.3618/0.0452 = 8 p according to Eq. (10), where s is the absolute value of the
concrete stress in the slab due to the bending moment m near
The substantial difference between the two moment distri- the column. It is assumed that the compression stress s has
butions is the main reason why the theory of elasticity should a triangular distribution over the compression zone height x
be applied up to the load level when the reinforcement starts because punching occurs at such low strains that concrete
to yield at the column. Consequently, the Kinnunen and “yielding” does not take place.
Nylander1 model, which assumes rigid body moment distri- With a strut inclination of 1:2, m = 0.9V/(2π), and
bution right from the start, will not provide accurate infor- s = 2m/(0.9dx), the pressure p becomes
mation on the bending moment distribution in the slab.
V 2 × 2 0.9V 2 × 4 d 4d
p= ≈ =s (10)
FAILURE MECHANISM πB � x 2π 0.9d x B B
Shear force transfer to column
Inclined “shear cracks” near the column usually develop The pressure p will squeeze the column, whereby the
well below the ultimate load. Hallgren,4 for instance, found surrounding compression zone will restrain the volume
that shear cracking started already at approximately 25% of decrease within the column. This effect is illustrated in
the ultimate load for high-strength concrete slabs. Although Fig. 8, which describes how a uniform pressure p is distrib-
the column may be surrounded by shear cracks, flat slabs uted if the compression zone within the column has the same
are nevertheless stable and can be unloaded and reloaded stiffness as the surrounding compression zone. It is evident
without any decrease in the ultimate load.8 It is therefore that the stress transferred to the surrounding area decreases
evident that the punching failure mechanism is not a pure rapidly with increasing distance from the column edge.

ACI Structural Journal/January-February 2016 97

 If the concrete stiffness within and outside the column is S1
and S2, respectively, the stress transferred to the surrounding
area becomes pS2/(S1 + S2). With a triangular distribution of
the pressure p over the compression zone height x, the ratio
S1/S2 is approximately 5 because the peak stress is applied on
the free edge of the slab but strikes the column at a consider-
able distance from a free edge.

Example
A simple example may clarify the failure mechanism.
Study a flat slab with B/d = 1.25 (refer to Fig. 7). The
squeezing pressure according to Eq. (10) becomes p =
–s × 4/1.25 = –3.2s. The resulting radial stress at the column
edge becomes σr = –s + 3.2s/(1 + 5) = –0.47s and the
tangential stress at the column edge becomes σt = –s – 3.2s/
(1 + 5) = –1.53s.
It thus appears that a tensile radial stress will not develop
Fig. 8—Effect of squeezing pressure.
at the column edge. However, the stress within the column
perimeter becomes σr = σt = s(–1 – 5 × 3.2/6) = –3.67s. This
stress will ultimately reach the “yield level”, whereby the
stiffness S1 rapidly decreases. Punching occurs when the
stiffness S1 has decreased to 2.2S2, whereby the resulting
tangential stress at the column edge becomes σt = –2.0s and
the radial stress approaches σr = 0. A tensile crack at the
column edge then opens up and initiates the punching failure.

Recorded strains
Concrete strains were recorded for test specimens of
normal-strength concrete by Tolf 9 and with high-strength
concrete by Hallgren4 (refer to Fig. 9). Detailed information
about the test specimens is given in Table A1 in the Appen-
dix.* The ratio B/d = 250/200 = 1.25 was identical to the
aforementioned example.
In full conformity with the described failure mechanism,
the radial compression strain started to approach zero before
the punching failure in the case of all specimens.
The distribution of radial concrete strain over the height
of the compression zone outside the column that is shown
in Fig. 10 was recorded for the high-strength concrete slab
in Fig. 9. The distribution was triangular, as illustrated in
Fig. 7, until the reinforcement at the column started to yield
at V ≈ 850 kN. The height of the compression zone within the
column will then decrease to approximately 35 mm (1.4 in.),
Fig. 9—Strain recordings. but the height of the compression zone in the radial direction
outside the column remains at 45 mm (1.8 in.) due to elastic
conditions. This is the probable explanation for the shape of
the strain distribution for loads above 850 kN (191 kip).
The abrupt decrease in radial strain for the high-strength
concrete slab in Fig. 9 is evident from Fig. 11. The curve for
90 MPa (13,000 psi) concrete is almost linearly elastic all
the way up to the compression strength level, at which point
the concrete fails without any “yield plateau.” The sudden
loss of stiffness when the stress within the column reaches
90 MPa (13,000 psi) explains the sudden large decompres-
sion of the radial compression strain outside the column.

*
The Appendix is available at www.concrete.org/publications in PDF format,
Fig. 10—Distribution of radial concrete strain over compres- appended to the online version of the published paper. It is also available in hard copy
sion zone height. (Note: 1 mm = 0.04 in.) from ACI headquarters for a fee equal to the cost of reproduction plus handling at the
time of the request.

98 ACI Structural Journal/January-February 2016

 A sudden decrease in the stiffness of the concrete within
the column area would not occur in the case of normal-
strength concretes. The curved stress-strain relation
according to Fig. 11 means a gradual loss of stiffness for the
concrete within the column, which in turn means a gradual
decompression of the radial strain outside the column with
increasing column reaction (refer to Fig. 9). However, the
failure tends to be very brittle once the radial compression
strain at the column edge reaches zero.

Failure criterion Fig. 11—Stress-strain diagram for various concrete grades.

The behavior described previously would appear to be
so forceful that a tangential compression strain of 1.0‰ at
the column due to the bending moment has been chosen as
the failure criterion. This is because the total compression
stress inside the column edge will then reach the “yield
level” for normal-strength concrete (refer to Eq. (10) and
Fig. 11). The accompanying loss of stiffness is a precondi-
tion for the radial compression stress at the column edge to
decrease to zero. The critical tangential strain is assumed to
be valid for a compression zone height of 150 mm (6 in.). It
is further assumed that the critical strain level decreases with
increasing concrete strength because high-strength concretes
are more brittle. The size effect—decreasing ultimate mate-
rial strength with increasing structural size—and the varying
degree of concrete brittleness are taken into account by the
failure criterion

1 0.1
 x  3  10 
ε cpu = 0.001  0    [MPa]
 x   f ck 
1 (11)
 x  3  1450 
0.1
Fig. 12—Structural calculation principle.
= 0.001  0  
 x   f ck 
[psi]
should be lower than 0.5. Theoretically, the more nonlinear
stress distribution a structure displays, the lower the abso-
lute value of the exponent becomes—as low as zero with a
where εcpu is the critical tangential compression strain at the
plastic stress distribution (= no size effect).
column edge due to the bending moment; x0 is the reference
The chosen exponent 1/3 in Eq. (11) therefore appears to
size = 0.15 m (6 in.); and x is the height of compression zone
be reasonable and is assumed to be valid for at least slab
at linear elastic stress conditions
sizes covered by the validation of the theory in Table A1 of
the Appendix—that is, slabs with an effective depth varying
SIZE EFFECT
from 100 to 600 mm (4 to 24 in.). The upper limit can prob-
In the case of very brittle failures characterized by a
ably be increased because the theory presented presupposes
linear stress distribution, the size effect would be described
an elastic behavior of the concrete in flexure, which is more
by the linear elastic fracture mechanics equation for the
realistic the larger the structure becomes. However, thick
failure strength
slabs may display a more pronounced apparent size effect
−0.5
due to possible induced cracks in the compression zone by
d uneven temperature distribution over the slab depth during
f = k  (12)
 d0  the concrete hydration.
The choice of the compression zone height as a reference
dimension for the size effect is a natural consequence of the
where d is the actual size of the structure and d0 is a refer- hypothesis that punching occurs when the compression zone
ence size. The equation is, for instance, applied in the case of near the column collapses. The effect of maximum aggre-
the very brittle pullout strength for expansion bolts anchored gate size can be taken into account by adjusting the reference
in plain concrete. size x0 in Eq. (11). However, there are currently no system-
Most concrete structures display a nonlinear stress distri- atic tests on flat slabs to support such refinement of the size
bution for brittle fractures, which means that the absolute effect factor.
value of the exponent in the fracture strength equation

ACI Structural Journal/January-February 2016 99

 PUNCHING CAPACITY  
2
The punching capacity is determined as indicated in x = d αρ  1 + − 1 (16)
Fig. 12. The theory of elasticity is assumed to be valid for  αρ 
the bending moment distribution until the reinforcement
starts to yield at the column. Then the sector elements start to Punching capacity
rotate as rigid bodies around the column edge. The ultimate Failure criterion
load is reached when the tangential compression strain at the
column edge reaches the critical value εcpu. The calculation 1 0.1
 0.15  3  10 
procedure that follows is radically simplified in relation to ε cpu = 0.001 
 x   f ck 
[MPa, m]
the original equations shown in the Appendix. The equations
are arranged herein to facilitate the use of a spreadsheet.
1 0.1
(17)
 6  3  1450 
Geometric data = 0.001   
 x   f ck 
[psi, in.]
h is slab thickness
d is effective depth for reinforcement
a is side length of square column Yield strain of reinforcement εsy = fyk/Es
C is diameter of circular test specimen Reinforcement strain at the column edge when punching
occurs
Materials
fck is cylinder compression strength of concrete
d−x
fyk is yield strength of reinforcement ε s1 = ε cpu (18)
ρ is reinforcement ratio x

Basic slab properties If εs1 turns out to be greater than the yield strain εsy, then
Diameter of equivalent circular column the reinforcement yields before punching occurs, and a ficti-
tious strain εs2 is applied
3π
B= a (13) 1
8  ε sy  3
Average compression strength according to Eurocode 2 εs2 = ε s1   (19)
ε s1
(EC2)10

f cm = f ck + 8 [ MPa ] = f ck + 1160[psi] (14) (

ε s = if ε sy < ε s1 ; ε s 2 ; ε s1 ) (20)

Young’s modulus of elasticity for reinforcement Es = 200,000 Fictitious bending moment at the column edge when
[MPa] = 29,000 [ksi] punching occurs
Young’s modulus of elasticity for concrete at low strains
according to EC210  x
mε = ρd 2 Es ε s 1 −  (21)
 3d 
0.3
 f 
Ec 0 = 22, 000  cm  [MPa]
 10  Finally, the column reaction at punching failure Vε is
determined
0.3
 f  8π
= 3.2 × 106  cm  [psi] Vε = mε (22)
 1450   B B
2
1 − 2ln   − 2
 C C
The secant modulus to the strain 0.001 is taken as
The column reaction Vy2 at overall reinforcement yield
  f  
4
becomes
Ec10 = 1 − 0.6 1 − ck   Ec 0 [MPa]
  150  
 x
(15) m y = ρd 2 f yk 1 −  (23)
 3d 
  f ck  
4

= 1 − 0.6 1 −  E [psi]
  22, 000   c 0
2π
The relation between the modulus of elasticity for reinforce- Vy 2 = m y (24)
B
ment and concrete α = Es/Ec10 1−
C
Depth of the compression zone in the slab at linear elastic
stress distribution

100 ACI Structural Journal/January-February 2016

 The reaction capacity VR is the lesser of Vε and Vy2

(
VR = if Vε < Vy 2 ;Vε ;Vy 2 ) (25a)

V Vy 2 Vε Vy 2 
VRd = if  ε < ; ; (25b)
 1.5 1.15 1.5 1.15 

The capacity VR is preferably plotted as a function of

the reinforcement ratio for comparison with other design
methods (refer to Fig. 13).

ROTATION CAPACITY
Ultimate radial rotation (slope) y
Flexural stiffness after cracking is denoted EI. The stiff-
ness in the diagonal direction (with orthogonal reinforce-
ment) is only 50% of the stiffness parallel with the reinforce- Fig. 13—Punching capacity versus reinforcement ratio.
ment directions; refer to the derivation in the Appendix. The
0.3 (30)
average stiffness is thus 75% of the maximum stiffness. A E 2 6 0.0013  1450 
factor of 0.75 is therefore applied when calculating the effect = c10
f yk2 4d 2 ρ2  f ck 
[psi, in.]
of symmetrical loads

 x  x Curvature at column edge at start of reinforcement yield

EI = 0.75ρd 3 Es 1 −  1 −  (26)
 d   3d 
χy1 = εsy/(d − x) (31)
With tensile strength in flexure according to EC2, 3.1.8, Sector element rotation if no reinforcement yields when
fctm,fl = (1.6 – h)0.3fck2/3, the bending moment at start of flex- punching occurs
ural cracking becomes

2 Vε − Vcr  B 2  C
h2 y0 = 1− (32)
mcr = 0.3 (1.6 − h ) f [MPa, m]
3
ck 4π  C 2  2 EI
6

2
(27) Elastic sector element rotation for the column reaction Vy1
h2
= 0.04 (63 − h ) f 3
ck
6
[psi, in.]
Vy1 − Vcr  B 2  C
y y1 = 1− (33)
The corresponding column reaction becomes 4π  C 2  2 EI

8π
Vcr = mcr 2
(28) If all or part of the reinforcement yields when punching
 B B occurs, then the rotation is the sum of elastic rotation and
1 − 2ln   − 2
 C C rigid body rotation

Column reaction at start of reinforcement yield at the

column (
y u = y y1 + χu1 − χ y1 ) B2 (34)

Decide which of ψ0 and ψu is the governing value

8π
Vy1 = m y (29)
B B2
1 − 2ln − 2
C C
(
y = if ε s1 < ε sy ; y 0 ; y u ) (35)

Ultimate curvature at the column edge if the reinforce- Imposed rotation capacity qu
ment yields (as derived in the Appendix) The provisions for eccentric punching in EC2 presuppose
that the unbalanced moment is known. However, the unbal-
Ec210 0.150 0.0013  10 
0.3
anced moment is usually a statically indeterminate quantity
χu1 =
f yk2 4d 2 ρ2  f ck 
[MPa, m] that is difficult to estimate correctly. The TST method there-
fore offers a safer and more efficient way of dealing with
eccentric punching because it is the rotation capacity that is

ACI Structural Journal/January-February 2016 101

checked rather than the unbalanced moment capacity. The Table 1—Summary of test result comparison in
bending moments in a flat-slab structure are usually calcu- Table A1
lated by means of the strip method, in which the imposed Vtest/Vcalc TST CSCT EC2 ACI 318
rotation is achieved by assuming that the strip is pin-
Average 1.08 1.18 1.03 1.45
supported on fictitious beams that connect the columns. The
imposed rotation θ is then equal to the slope of the strip at Coefficient of variation 0.090 0.104 0.065 0.238
the support studied. If the slope is zero (horizontal tangent), Maximum 1.27 1.38 1.22 2.28
then there is no imposed rotation at the vertical column. Minimum 0.93 0.95 0.92 0.97
If the factored column reaction V is lower than the
punching capacity VRd, the slab can resist an imposed rota-
tion. A conservative solution is achieved if the imposed rota-
tion θ plus the radial rotation (slope) due to the concentric
column reaction is limited to the rotation capacity ψ (refer to
Eq. (36) and (37)). The column is assumed to be much stiffer
than the slab.

 Vy1 
Vy1  V− 
V≤
1.5
: qu = y −  y y1 +
 Vy1
(
1.5 y − y
y1 ) 
 (36)
 VRd − 
1.5

Vy1 1.5V
V≤ : qu = y − y y1 (37)
1.5 Vy1

VALIDATION
Fig. 14—Punching capacity according to TST, CSCT, and
Punching capacity—comparison with test results
EC2. (Note: a = 300 mm [12 in.]; d = 200 mm [8 in.]; fyk =
The punching expression in Eurocode 2,10 which is
500 MPa [72.5 ksi].)
entirely empirical and based on a vast quantity of published
test results, would appear to be the best method available for 32
determining the punching capacity of slender flat slabs. The kdg =
16 + d g
[mm] ≥ 0.75
capacity is expressed as formal shear strength at a control
perimeter with rounded corners at a distance of 2d from
the column 1.2
=
0.6 + d g
[in.] ≥ 0.75 (42)
1
vR , c = 0.18ξ (100ρf ck )3 [MPa] Flexibility equation
1 (38)
= 5ξ (100ρf ck )3 [psi,in.] 3

C f yk  VR , c  2
y = 1.2 (43)
2d Es  V flex 
200 8 where b0 is the control section with rounded corners at
ξ = 1+
d
[ mm] = 1 +
d
[in.] (39)
a distance of 0.5d from the column perimeter; dg is the
maximum aggregate size; fck is the cylinder concrete
where d shall not be taken less than 200 mm (8 in.) and ρ compression strength; and Vflex is the column reaction at
is limited to 2%. However, when checking test results, the overall yield of flexural reinforcement.
limit for d should be disregarded. ACI 318-0811 expresses the formal punching shear
In the half-empirical CSCT method, the punching capacity capacity on a control section at a distance of 0.5d from the
is determined by solving the equation system. column perimeter. A square control section is accepted for
Failure criterion square columns. No consideration is given to either the influ-
ence of the reinforcement ratio or the size effect.
VR , c = ky b0 d f ck [ MPa ]
1
νR ,c = f ck [MPa] = 4 f ck [psi] (44)
(40) 3
= 12ky b0 d f ck [ psi] where fck is limited to 69 MPa (10,000 psi).
In Table A1 in the Appendix, the predicted punching
capacity Vcalc is compared to various test results Vtest found
1 in the literature. The results are summarized in Table 1.
ky = ≤ 0.6 (41)
1.5 + 0.9kdg yd

102 ACI Structural Journal/January-February 2016

Fig. 15—Slenderness effect on punching shear strength Fig. 16—Punching shear strength versus ultimate rotation.
according to EC2 and TST. (Note: d = 200 mm [8 in.]; fck = (Note: d = 200 mm [8 in.].)
35 MPa [5075 psi]; ρ = 0.8%)
Comparison of TST and CSCT failure criteria
The TST method gives as accurate a prediction level as The CSCT gives the relation between the punching shear
the two methods CSCT and EC2, where EC2 displays the strength and the slab rotation at failure in the form of a single
lowest coefficient of variation for the database studied. The equation: the failure criterion Eq. (40). Although the equa-
somewhat higher coefficient for the CSCT method is mainly tion is based on the hypothesis that failure occurs when the
due to the calculation principle, in which the intersection width of a critical shear crack becomes too large, some of the
point for two approximate curves determines the assessed observations made in this paper would appear to contradict
punching shear strength. this hypothesis.
The large coefficient of variation for ACI 318 demonstrates The CSCT failure scenario, with a shear crack that prop-
the disadvantage of a design code that gives no consider- agates down through the compression zone, would appear
ation to the size effect or the flexural reinforcement ratio. to be inconsistent with the concrete strain variation that
Figure 14 demonstrates more clearly how the punching precedes the failure as was described previously. Those
capacity varies with the flexural reinforcement ratio. The observations indicate that the failure instead starts at the
shape of the TST curves differs slightly from EC2 and CSCT bottom of the slab at the column edge.
because the latter methods do not identify the reinforcement Furthermore, the CSCT assumes that the width of the crit-
stress variation over the slab width. The line for Vy1 defines ical shear crack is proportional to the slope of the slab, which
the column reaction when the reinforcement at the column means that the punching capacity is predicted to decrease
starts to yield, which can only be determined by means of with time because the slope of the slab, and thus the width of
the TST method. the shear crack, will increase with time due to concrete creep
EC2 is valid for comparatively slender slabs with a C/2d and drying shrinkage.
of approximately 7. The increased shear strength with In reality, creep seems instead to be favorable with respect
decreasing slab slenderness is captured by the TST and to the punching capacity. Moe,8 for instance, tested Spec-
CSCT methods, as demonstrated in Fig. 15. imen H15 with sustained load over a period of 3 months. The
sustained load was 70% of the final ultimate load and the
Ultimate rotation deflection increased by 80% from the initial 0.25 to 0.45 in.
The ultimate rotation determined by the TST method for (6.4 to 11.4 mm) during the course of sustained loading. The
a slab with effective depth of 200 mm (8 in.) is compared to width of the flexural cracks also increased by 80%. The ulti-
the ultimate rotation curves of the CSCT method in Fig. 16. mate deflection at the punching failure was 0.6 in. (15 mm)
Ultimate rotation curves for slabs with 100 and 400 mm and the punching capacity tested was estimated to be some
(4 and 16 in.) effective depth are shown in the Appendix. 4% higher than the short-term capacity. Moe8 concluded that
The TST curves indicate a somewhat larger ultimate rotation the creep deformation had no adverse effect on the punching
than the CSCT curves, which corresponds to the difference capacity, but rather the opposite.
in predicted punching capacity in Table 1. It is evident that The TST model confirms Moe’s8 conclusion; punching
the theoretically derived TST model can accurately predict capacity is assumed to increase due to concrete creep, which
not only the punching capacity but also the concurrent ulti- is taken into account by multiplying the critical strain εcpu by
mate rotation that is represented by the CSCT curves, which the creep factor (1 + φ) and dividing Ec0 by (1 + φ). A slab
are calibrated against test results in the literature. However, with an effective depth of 200 mm (8 in.), column diam-
the TST method appears to overestimate the ultimate rotation eter of 250 mm (10 in.), and a 0.8% reinforcement ratio
for specimens with such low reinforcement ratio as 0.25%. was checked by applying the creep factor (1 + φ) = 3. The

ACI Structural Journal/January-February 2016 103

punching capacity was found to increase by 14% and the AUTHOR BIOS
rotation by 100% in relation to the short-term results. Carl Erik Broms is a Senior Consultant with WSP Sweden AB, Stock-
holm, Sweden. He received his MS and PhD from the Royal Institute of
The aforementioned observations thus cast some doubt Technology (KTH), Stockholm, Sweden. His research interests include
on the shear crack interpretation of the CSCT failure crite- the design of flat-plate structures with an emphasis on punching capacity,
rion. Despite this fact, the failure equation Eq. (40) summa- ductility, and safety against accidental loads.
rizes very elegantly the short-term rotation of test speci-
NOTATION
mens as a function of the punching shear strength and the a = side length of square column
effective depth. B = diameter of circular column
C = diameter of circular slab specimen
d = average effective depth for flexural reinforcement
CONCLUSIONS Ec10 = secant modulus of elasticity for concrete to the strain 1.0‰
A novel mechanical model for the concentric punching EI = flexural stiffness of slab after concrete cracking
failure at interior columns of flat slabs without shear rein- h = slab thickness
L = span width
forcement is presented. The shear force is assumed to be m = bending moment per unit width
transferred to the column by a circumferential inclined mε = fictitious bending moment at column edge when punching
compression strut that squeezes the compression zone of occurs
my = bending moment at reinforcement yield
the slab within the column perimeter. The resulting volume p = squeezing pressure
decrease of the concrete within the column is restrained s = absolute value of concrete compression stress at column edge
by the surrounding concrete. The restraining force creates due to bending moment
V = concentric column reaction
radial tensile stress and tangential compression stress Vcr = column reaction at start of flexural cracking
outside the column. With increasing column reaction, the Vy1 = column reaction when reinforcement starts to yield at column
compression stress within the column will ultimately reach Vy2 = column reaction when all flexural reinforcement yields
Vε = column reaction when punching occurs
the “yield level” and the stiffness for squeezing pressure will x = height of compression zone in slab (at linear stress conditions)
rapidly decrease. An increasing portion of the horizontal χ = curvature of slab = m/EI =εc/x = εs/(d – x)
component of the inclined strut force will then be anchored εc = compression strain of concrete
εcpu = tangential compression strain of concrete at column edge when
back to the surrounding concrete by means of an increasing punching occurs
radial tensile force. When the resulting tensile stress exceeds εs = tensile strain of reinforcement
the radial compression stress due to the bending moment, θ = imposed rotation of slab in relation to column (or vice versa)
θu = rotation capacity of slab
a tensile crack develops around the column. This crack is ρ = flexural reinforcement ratio
assumed to initiate the punching failure. ψ = ultimate radial rotation (slope) of sector element
Concrete “yielding” within the column, which is caused by
the horizontal component of the inclined shear strut, is thus REFERENCES
assumed to be the source of the punching failure. However, 1. Kinnunen, S., and Nylander, H., “Punching of Concrete Slabs without
Shear Reinforcement,” Transactions of the Royal Institute of Technology,
this component is shown to be a function of the concrete No. 158, Stockholm, Sweden, 1960, 112 pp.
stress at the column edge due to the bending moment, and 2. Broms, C. E., “Punching of Flat Plates—A Question of Concrete Prop-
this is the reason why the punching failure can be coupled to erties in Biaxial Compression and Size Effect,” ACI Structural Journal,
V. 87, No. 3, May-June 1990, pp. 292-304.
a critical concrete strain at the column edge, which greatly 3. Broms, C. E., “Concrete Flat Slabs and Footings—Design Method for
facilitates the analytical treatment. Punching and Detailing for Ductility,” PhD thesis, Structural Design and
The failure mechanism described gives a plausible expla- Bridges, Royal Institute of Technology, Stockholm, Sweden, 2005, 114 pp.
4. Hallgren, M., “Punching Shear Capacity of Reinforced High Strength
nation for why the radial concrete compression strain near Concrete Slabs,” PhD thesis, Bulletin 23, Department of Structural Engi-
the column is lower than the tangential strain, despite the neering, Royal Institute of Technology, Stockholm, Sweden, 1996, 206 pp.
fact that they should be equal according to the theory of 5. Muttoni, A., “Punching Shear Strength of Reinforced Concrete Slabs
without Transverse Reinforcement,” ACI Structural Journal, V. 105, No. 4,
elasticity. The model also explains why the radial concrete July-Aug. 2008, pp. 440-450.
strain at the column announces impending punching failure 6. Muttoni, A.; Ruiz, M. F.; Bentz, E.; Foster, S.; and Sigrist, V., “Back-
by beginning to decrease down to zero. The final zero strain ground to fib Model Code 2010 Shear Provisions—Part II: Punching
Shear,” Structural Concrete, V. 14, No. 3, 2013, pp. 204-214. doi: 10.1002/
is not a consequence of the failure, but rather the cause of suco.201200064
the failure. 7. Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and
Contrary to current building codes, the TST model predicts Shells, International Student Edition, McGraw-Hill, New York, 1959,
580 pp.
both the punching capacity and the concurrent ultimate rota- 8. Moe, J., “Shearing Strength of Reinforced Concrete Slabs and Foot-
tion, which means that the model can also treat eccentric ings under Concentrated Loads,” Development Department Bulletin d47,
punching. The eccentricity of the column reaction is usually Portland Cement Association (PCA), Skokie, IL, 1961, 130 pp.
9. Tolf, P., “Influence of the Slab Thickness on the Punching Strength of
caused by a so-called unbalanced moment, which is in turn Concrete Slabs—Tests with Circular Slabs,” Bulletin No. 146, Department
caused by an imposed rotation of the slab in relation to the of Structural Mechanics and Engineering, Royal Institute of Technology,
column. So instead of checking the shear stress at the column Stockholm, Sweden, 1988, 64 pp. (in Swedish with a summary in English)
10. Eurocode 2, “Design of Concrete Structures—Part 1-1: General
caused by a statically indeterminate unbalanced moment, the Rules and Rules for Buildings (EN 1992-1-1),” Comité Européen de
TST model provides an efficient shortcut by checking the Normalisation, Brussels, Belgium, 2004, 225 pp.
imposed rotation against the rotation capacity of the slab. 11. ACI Committee 318, “Building Code Requirements for Structural
Concrete (ACI 318-08) and Commentary,” American Concrete Institute,
Finally, the squeezing effect plays an important role in Farmington Hills, MI, 2008, 473 pp.
the punching of column footings and pile caps. But that is
a different story.

104 ACI Structural Journal/January-February 2016

 APPENDIX

1. Orthogonal reinforcement mesh

Elastic reinforcement stress and flexural stiffness

x
s
y


F

Fig. A1 – Elastic stresses in orthogonal reinforcement mesh

Force F from reinforcement stress in x-direction

𝐹𝑥 = 𝜎𝑥 𝜌 𝐿 sin2 𝛼

Displacement in F-direction due to reinforcement stress in x-direction

𝜎𝑥 𝑠 𝐹𝑥 𝑠
𝛿= =
𝐸sin 𝛼 𝐸 𝜌 𝐿 sin4 𝛼
2

𝐹 𝐹𝑥 𝐹y 𝐸 𝜌 𝐿
Stiffness: = + = ( sin4 𝛼 + cos4 𝛼)
𝛿 𝛿 𝛿 𝑠

𝐹 sin4 𝛼 𝐹 sin2 𝛼
Reinforcement stress: 𝜎x = ∙ = ∙
𝜌 𝐿 sin2 𝛼 sin4 𝛼 + cos4 𝛼 𝜌 𝐿 sin4 𝛼 + cos 4 𝛼

𝐹 cos 2 𝛼
Reinforcement stress: 𝜎y = ∙
𝜌 𝐿 sin4 𝛼 + cos 4 𝛼

𝐹
𝛼 = 60° → 𝜎x = 1.2 !
𝜌𝐿
 𝐹 𝐸𝜌𝐿 1 1
𝛼 = 45° → = ( + )
𝛿 𝑠 4 4

2. Ultimate curvature at column edge if the reinforcement yields

Failure criterion
1
0.15 3 10 0.1
𝜀cpu = 0.001 ( ) ∙( ) (A1)
𝑥pu 𝑓ck
Force equilibrium
𝑥pu
𝜌 𝑑 𝑓yk = 𝐸c10 ∙ 𝜀cpu ∙ (A2)
2

Tangential curvature at the column edge

10 0.3
𝜀cpu 3
𝜀cpu ∙ 𝑥pu 0.0013 ∙ 0.15 ∙ ( ) 2
𝐸c10 0.15 0.0013 10 0.3
𝑓ck
𝜒u1 = = 2 2
= 2 = 2 ∙ ∙ ∙( ) (A3)
𝑥pu 𝜀cpu ∙ 𝑥pu (2𝜌 𝑑 𝑓yk ) 𝑓yk 4𝑑 2 𝜌2 𝑓ck
⁄ 2
𝐸c10

3. Exact solution for punching capacity

Failure criterion, compression strain at the column edge at punching

1
𝑥0 3 10 0.1
𝜀cpu = 0.001 ( ) ∙ ( ) (A4)
𝑥 𝑓ck

Yield strain of reinforcement

𝑓𝑦𝑘
𝜀sy = (A5)
𝐸s

Reinforcement strain at the column edge at punching

𝑑−𝑥
𝜀s1 = 𝜀cpu (A6)
𝑥

Bending moment at the column edge at punching

𝑥
𝑚ε = 𝜌 𝑑 2 𝐸𝑠 𝜀s1 (1 − ) (A7)
3𝑑
The punching capacity V if no reinforcement yields becomes

8π
𝑉ε0 = 𝑚ε ∙ (A8)
𝐵 𝐵2
1 − 2ln (𝐶 ) − 2
𝐶

Column reaction at start of reinforcement yield at the column

8π
𝑉y1 = 𝑚y (A9)
𝐵 𝐵2
1 − 2ln 𝐶 − 2
𝐶

Column reaction for overall reinforcement yield

2π
𝑉y2 = 𝑚y (A10)
𝐵
1−𝐶
Ultimate curvature at column edge

2
𝐸c10 0.150 0.0013 10 0.3
𝜒u1 = 2 ∙ ∙ ∙( ) (A11)
𝑓yk 4𝑑 2 𝜌2 𝑓ck

Yield curvature at column edge

𝜀sy
𝜒y1 = (A12)
𝑑−𝑥

Determine distance ry to the point where reinforcement no longer yields

𝑉y1 𝐶 𝐵2 𝐵2 𝐵
𝑚y = (2ln + 2 − 2 − 2 ) + (𝜒u1 − 𝜒y1 ) 𝐸𝐼 (A13)
8π 2𝑟y 4𝑟y 𝐶 2𝑟y

Determine punching capacity Vy for partial reinforcement yield over the slab width

4π
𝑉εy = [𝑚 ∙ 𝑟
𝐶−𝐵 y y
0.5𝐶 𝑉y1 𝐶 𝐵2 𝐵2 𝐵
+∫ [ (2ln + 2 − 2 − 2 ) + (𝜒u1 − 𝜒y1 ) 𝐸𝐼] d𝑟] (A14)
𝑟y 8π 2𝑟 4𝑟 𝐶 2𝑟

Determine punching capacity V

𝑉ε = if(𝜀s1 < 𝜀sy ; 𝑉ε0 ; 𝑉εy ) (A15)

Check whether overall yield is governing

 𝑉R = if(𝑟𝑦 < 0.5𝐶 ; 𝑉ε ; 𝑉y2 ) (A16)

References
12. Elstner, R. C., and Hognestad, E., “Shearing Strength of Reinforced Concrete Slabs,” ACI
Journal Proceedings, V. 53, No. 7, July 1956, pp. 29-58.
13. Kinnunen, S.; Nylander, H.; and Tolf, P., “Influence of the Slab Thickness on the Punching
Strength of Concrete Slabs—Tests with Rectangular Slabs,” Bulletin No. 137, Department of
Structural Mechanics and Engineering, Royal Institute of Technology, Stockholm, Sweden, 1980,
73 pp. (in Swedish with a summary in English)
14. Tomaszewicz, A., “Punching Shear Capacity of Reinforced Concrete Slabs,” High Strength
Concrete. SP2–Plates and Shells, Report 2.3, Report No. STF70 A93082, SINTEF Structures and
Concrete, Trondheim, Norway, 1993, 36 pp.
 0.6 fck
(MPa) 7 fck
(psi)
0.5 6
VR CSCT TST VR
b0 d b0 d
5
0.4

4
0.3
3
0.2
dg = dg = 2
16 m 27 m
m m
0.1
1

0
0 0.020 0.040 0.060 0.080 0.100 

Fig. A1 – Punching shear strength versus ultimate rotation, d = 100 mm [4 in.]

0.6 fck
(MPa) 7 fck
(psi)
0.5 6
VR VR
CSCT TST
b0 d b0 d
5
0.4

4
0.3
3
0.2
2

0.1 dg =
16 m
m
1
dg = 27 mm
0
0 0.010 0.020 0.030 0.040 0.050 

Fig. A2 – Punching shear strength versus ultimate rotation, d = 400 mm [16 in.]
Table A1 – Observed ultimate loads of flat plate specimens compared to
predictions according to TST, CSCT, EC 2 and ACI 318-08

Authors Test fcc fsy  d C Column Vtest Vtest /Vcalc

slab MPa MPa % mm mm size dg
No mm mm kN TST CSCT EC 2 ACI 318

Elstner, A-1b 25.2 332 1.16 118 1780 254 25.4 365 1.134 1.138 1.030 1.333
Hognestad A-1c 29.0 " " " " " 356 1.064 1.061 1.000 1.212
A1-d 36.8 " " " " " 351 0.983 0.978 0.978 1.061
(1956) 11 A-1e 20.3 " " " " " 356 1.174 1.190 1.017 1.449
A2-b 19.5 321 2.50 114 " " 400 1.149 1.196 1.010 1.735
A-2c 37.4 " " " " " 467 1.019 1.108 0.948 1.462
A-7b 27.9 " " " " " 512 1.262 1.345 1.147 1.856
A-3b 22.6 " 3.74 " " " 445 1.134 1.139 1.070 1.793
A-3c 26.5 " " " " " 534 1.268 1.285 1.217 1.987
A-3d 34.5 " " " " " 547 1.158 1.190 1.142 1.783
A-4 26.1 332 1.18 118 " 356 400 1.032 1.019 1.019 1.109
A-5 27.8 321 2.50 114 " " 534 1.115 1.084 1.027 1.496
A-6 25.0 " 3.74 " " " 498 0.999 0.945 0.992 1.394
A-13 26.2 294 0.554 121 " " 236 1.0 1.0 1.0 1.0
B-1 14.2 324 0.476 114 " 254 178 1.0 1.0 1.0 1.0
B-2 47.6 321 " " " " 200 1.0 1.0 1.0 1.0
B-4 47.7 303 1.01 " " " 334 1.0 1.0 1.0 1.0
B-9 43.9 341 2.00 " " " 505 1.113 1.140 0.973 1.460
B -14 50.5 325 3.02 " " " 578 1.085 1.101 1.063 1.558

Kinnunen, 5 26.8 441 0.80 117 1710 150 32 255 1.101 1.210 0.973 1.506
Nylander1 6 26.2 454 0.79 118 " " 275 1.172 1.298 1.048 1.622
24 26.4 455 1.01 128 " 300 430 1.073 1.191 1.088 1.459
(1960) 25 25.1 451 1.04 124 " " 408 1.081 1.196 1.085 1.479
32 26.3 448 0.49 123 " " 258 1.0 1.0 1.0 1.0
33 26.6 462 0.48 125 " " 258 1.0 1.0 1.0 1.0


Moe 12 R2 26.5 328 1.38 114 1780 152 25.4 311 1.158 1.216 0.963 1.646
(1961) M1A 20.8 481 1.50 " " 305 433 1.207 1.199 1.087 1.583


Kinnunen, S1 30.6 621 0.574 619 4680  32 4915 0.932 1.101 1.051 0.966
Nylander,
Tolf 13
(1980)

Tolf 8 S1.1 28.6 706 0.80 100 1190 125 16 216 0.991 1.281 1.062 1.714
(1988) S1.2 22.9 701 0.81 99 " " “ 194 0.984 1.260 1.038 1.746
S2.1 24.2 657 0.80 200 2380 250 32 603 0.930 1.048 0.946 1.300
S2.2 22.9 670 0.80 199 " " “ 600 0.953 1.068 0.966 1.340
S1.3 26.6 720 0.35 98 1190 125 16 145 1.018 1.230 0.991 1.228
S1.4 25.1 712 0.34 99 " " “ 148 1.054 1.273 1.026 1.272
S2.3 25.4 668 0.34 200 2380 250 32 489 1.016 1.172 1.004 1.029
S2.4 24.2 664 0.35 197 " " “ 444 0.948 1.094 0.938 0.979
Tomaszewicz 65-1-1 64 500 1.49 275 2500 200 16 2050 1.185 1.373 1.150 1.680
(1993)14 95-1-1 84 " " " " " 2250 1.211 1.381 1.152 1.776
115-1-1 112 " " " " " 2450 1.250 1.375 1.140 1.933
95-1-3 90 " 2.55 " " " 2400 1.137 1.203 1.089 1.894
65-2-1 70 500 1.75 200 2200 150 1200 1.173 1.349 1.078 1.764
95-2-1 87 " " " " " 1300 1.204 1.360 1.086 1.911
115-2-1 119 " " " " " 1400 1.226 1.322 1.054 2.058
95-2-3 90 " 2.62 200 2200 150 1450 1.226 1.320 1.146 2.131
115-2-3 108 " " " " 1550 1.266 1.324 1.152 2.278
95-3-1 85 " 1.84 88 1100 100 330 1.098 1.292 1.024 1.623
 
Hallgren4 HSC0 90 643 0.80 200 2400 250 18 965 1.019 1.225 0.977 1.233
(1996) HSC1 91 627 0.80 200 " " 1021 1.086 1.297 1.030 1.304
HSC2 86 620 0.82 194 " " 889 0.998 1.197 0.949 1.186
HSC4 92 596 1.19 200 " " 1041 0.949 1.123 0.916 1.330
HSC6 104 633 0.60 201 " " 960 1.144 1.327 1.012 1.217
HSC9 84 634 0.33 202 " " 565 1.0 1.093 1.0 1.0
N/HSC8 95 631 0.80 198 " " 944 1.014 1.204 0.953 1.223

 Average 1.08 1.18 1.03 1.45

COV 0.090 0.104 0.065 0.238
Max 1.27 1.38 1.22 2.28
Min 0.93 0.95 0.92 0.97