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Ultimate limit state of punching inthe (fib) FIP recommendationsforthe design of post-tensioned slabsand foundations
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Structural Concrete
2000,1, No.3
Sept., 143-149
Ultimate limit state of punching inthe (fib) FIP recommendationsforthe design of post-tensioned slabsand foundations
v. J. G.Lúcio New University of Lisbon, Portugal
J. A.5. Appleton Technical University of Lisbon, Portugal
J. F. Almelda Technical University of Lisbon, Portugal
Thls paper descrlbes the formulatlon of the Ultlmate LlmltState of punchlng reslstance of prestressed slabs proposed In the new FIP Recomo
mendatlons for the deslgn of post-tensloned slabs and foundatlon rafts. The prestress effects were consldered on the actlon slde deflnlng an ef-
fectlve applled punchlng load. For the evaluatlon of the punchlng reslstance, CEB-FIP Model Code 90 was followed. An example of the
appllcatlon of thls method to an Interior column of a prestressed flat slab 15presented.
Introductlon
After the publication of CEB-FIP Model Code 90 (MC90) 1 itbecame necessary to update the previous FIP recommendations
for the design of flat slabs in post-tensioned concrete, 2 published
in 1980. The new document, 'Recommendations for the design of
post-tensioned slabs and foundation rafts', 3 was prepared by aworking group of FIP Commission 3 on practical design, and wasfinally published in 1999 by SETO(for the fib).
The ultimate limit state of punching resistance of prestressedslabs is one of the subjects revised in the documento In that revi -sion the concept of prestress as an action was considered in the
quantification ofthe applied punching load. The effects ofthe pre -stress (isostatic and hyperstatic) on the punching load are sepa -rated into the effects of the equivalent prestress forces
perpendicular to the slab plan and the effects of equivalent pre -stress forces in the plane of the slab (compression due to
prestress).
The equivalent prestress forces perpendicular to the slab planeinclude the moments due to eccentricities at the anchorages andthe transversal deviation forces of the strands. For the evaluation
of the effects of these forces on the punching force, the deviation
forces within a perimeter at 0.5 h from the periphery of the column
1464-4177 @ 2000 Thomas Telford Ltd and fib
- - -- .-----
may be considered to be transferred directly to the column re
ducing the value of the effective punching load.The eccentricity of the punching force is taken into account
using the formulation presented in MC90. 1
This new formulation considers a rectangular distribution, over
the control perimeter, of the shear stresses that equilibrate partof the moment transferred between the column and the slab. This
method is much less conservative than the one previouslyadopted in MC78, which used a triangular distribution ofthe shearstresses.4
For the quantification of the punching resistance, with or
without transversal reinforcement, the formulation proposed by
MC90 is adopted. The quantification ofthe punching resistance ismade considering a control perimeter at a distance from the
column equal to 2 d, allowing resistant shear stresses for
punching to equal the ones used for the shear resistance of slabs,
and avoiding the previous difficulties in the case of large columns.The quantification of the maximum punching resistance is more
rational than before since the crushing resistance of the concrete
in the column perimeter governs it. In this way the value of the
maximum punching resistance is again less conservative than
those proposed by the previous recommendations.
143
Lúcio et aI.
Rg. 1 Transversal effect of the prestress for a typified prestressed flatslab
An example of the application of the formulation described in
this paper is presented.
Effective value of the punchlng force
Theeffective designvalueofthe punchingforce ( PSd.efl) is definedas the value of a concentric punching force that produces uniform
shear stresses, over the control perimeter equal to the maximum
shear stress caused by the eccentric punching force. The effective value of the punching force may be obtained multiplying the
ITIllJ
Plana
eccentric punching force PSd(p),due the applied load p, by a factor(3greater than the unity, to take into consideration the eccen
tricity effect on the stresses around the control perimeter:
PSd.ett= (3PSd(p) (1)
The effective punching force must also account for the prestress
effects: the effect the equivalent prestressing forces transversal
to the slab PSd(P),and the in plane prestress force effect PPo:
(2)
Transversal prestress effect
The punching force is determined taking ínto consideration the
loads applied to the slab and the equivalent prestressing forces
transversal to the slab ( PSd(p,P)), including its hyperstatic effect.For this quantification ali the transversal equivalent prestress
forces (Figure 1) are considered, except those which are transferred directly to the column and do not influence the shear
stresses around the column. For this purpose it is considered thatthe prestress equivalent forces acting inside a perimeter 0.5 h
from the column periphery, together with the applied loads in that
area, are transferred directly to the column (Figure 2(a)). This isequivalent to reducing the column reaction by the value of Ptan aof the tendons which cross that perimeter (Figure 2(b)), where a
is the angle between the tendons and the plane of the slab at adistance 0.5 h from the column.
Rg. 2 Punching load
O.5h
(a) t P~(p, P) (b)
O.5h
Plana
tO.5h
PSd(p, P)
Rg. 3 Moments transferred between the slaband the columns
H
~MI
144
Rg. 4 Contrai perimeter u,
Structural Concrete , 2000, 1, No. 2
Eccentricity of the punching force
Usually the forces transferred between the slab and the columnsare not centred with the column centroid. Due to horizontal forces
applied to the structure, unequal spans or unequalload values on
two adjacent spans, or at the columns of an end span (Figure 3),there are moments transferred between the slab and the columns.
The maximum shear force per unit width on the control perim -eter (vmax) may be evaluated as follows, 4 where MSd = MSd(P, P) isthe moment transferred between the slab and the column, due to
the applied loads and the effects of the prestress:
PSd kMSdVmax=-+-
u1 w1
The control perimeter ( u1) is taken at a distance 2 d from theperiphery of the applied force or column and must be constructed
50 as to minimize its length (Figure 4).
The parameter w1 is defined as
ULS 01punching in the ( fib) FIP recommendations
c,
The lesser 01
1.5d and O.5c1
H
I~
(a) (b)
(3) Flg. 5 Contrai perimeters at edge columns
(4)I The lesser 01
1.5d and O.5c__u1
where di is an elementary length of the perimeter and e is the dis-
tance of d I to the axis about which the moment MSdacts. I (a) (b)
The coefficient k represents the proportion of applied moment
(MSd)transferred to the slab by shear stresses along the control Flg. 6 Contrai perimetersat comer columnsperimeter. This coefficient depends on the ratio between the
columndimensions c1(parallelto the eccentricity MSdI PSd) and c2(perpendicular to the eccentricity):
In accordance with MC90, 1 the effective punching force due to
the eccentricity effect may be estimated as
where 13is given by
13= 1 + k MSd u1PSdw1
In the case of double eccentricity of the punching force the foi
lowing expression may be used:
'
(ke
)2
(ke
)2
13=1+u1./ w1.+ w1 y
where e. and ey represent the eccentricities MSdlPSdalong x andy, respectively.
In the new recommendations, simpler expressions are proposed
for interior rectangular and circular columns:
13=1+1.8.{::J +(~Jfor rectangular columns, and
le 2 +e 2
13 = 1 + 0.61r ". yD+4d
for circular columns, where b. and by(Figure 4) are the dimensionsof the control perimeter and D is the diameter of the circular
column. Expression (8) was obtained using the 'Ieast squares'
method on the values of expression (7). The values obtained with
Structural Concrete , 2000,1, No.3
(5)
(6)
(7)
Flg. 7 Compression due to prestress
(8) expression (8) are very close to the ones of expression (7) forcommon slab and column dimensions.
(9)
Slab-edge column connectlons. At slab-edge column connec-
tions, where the eccentricity perpendicular to the slab edge is
towards the interior, it is considered that the punching force is
uniformly distributed along the control perimeter u~, as defined inFigure 5. Thevalue of 13maythen be determined as
(10)
145
c.1C2 k
0.5 0.451.0 0.602.0 0.703.0 0.80
Lúcio et ai.
Flg. 8 Decompression punching force p'o due to compression effect
. . .. ___--e, .,
.. A- -. - -8 --/..- '~\/ / . .. \.,11'8 .TT.
.+~. . ..T.I I. . . . . . . .\' ... //\
j/
8-_ S,__. -.- - 8
. . .Flg. 9 Punching-shear reinforcement
Flg. 10 Control perimeters for maximum resistance
where e is the component of the eccentricity parallel to the slab
edge.If the eccentricity perpendicular to the slab edge is not towards
the interior, expression (6) applies.
In any case, in the definition of the control perimeter, the spaceneeded for the anchorage recess must not be considered
(Rgure 5).
SlalH:orner column connectlons. At slab-corner column connec -
tions, where the eccentricity is towards the interior of the slab, it
is assumed that the punching force is uniformly distributed along
the control perimeter u~, as defined in Rgure 6. The {3value maythen be considered as
If the eccentricity is towards the exterior, expression (6) applies.
146
Campressian effects af the prestress
The compression effects of the prestress influence the punchingbehaviour of the slab.
The value of this compression depends on the position of the an -chorages in relation to the punching area and the restriction tothe in plane deformation of the slab caused by the vertical struc
tural elements, like shear walls or large columns (Figure 7).For foundation rafts, where the friction between the foundation
and the subgrade may be significant, a detailed analysis must beperformed in order to quantify the compression stresses.
Unless the prestress force is very high, the influence of thecompression stresses on the punching behaviour is usually very
small. For doubtful situations the compression effect is
neglected.
The compression effect of the prestress may be considered on
the action side of the ultimate limit check (expression (2», and
different compression stresses in two directions ( x and y) may beconsidered. The compression stresses due to prestress delay theopening of the shear cracks, reduce their widths and increase the
depth of the concrete compressed area at the slab cross-sectionnear the column faces.
A decompression punching force Ppo is defined as the forceneeded to compensate the compression stresses due to the com -
pression effect of the prestress. To take into account differentprestress forces in two orthogonal directions x and y the decom-
pression punching force may be evaluated as follows:
(12)
where b. and by are the dimensions of the control perimeter along
x and y, respectively, and p'o and Pyoare the decompression forcescorresponding to the prestress forces in those directions (Figure8).
It is assumed that the decompression punching force is the
punching force which corresponds to a bending moment thatcauses tension stresses on the slab top surface equal to the com -
pression stresses due to the compression effect of the prestress,as is shown in the following paragraphs.
The decompression forces in each direction may be easily eval -uated as a proportion to the actual punching force PSd(p,P) and
bending moments M.Sd(P,P) and MySd(p,P):
(13)
The moments M.Sd(p, P) and MySd(p, P) are the total bending
momentsat the columnface in the widths b x and by'respectively,and M.o and Myo stand for the decompression moments in thewidths b. and by' respectively, defined as
(14)
where I7cP' and I7cpy are the mean concrete stresses, due to the pre -
stress axial compression, in the by and b. widths, respectively.
Punchlng reslstance
(11)
In order to check the ultimate limit state of punching resistance,
the effective punching force must not be greater than the
punching resistance:
PSd. ell ~ PRd (15)
Structural Concrete , 2000, 1, No. 2
Flg. 11 Slab geometry (dimensions in mm)
For the quantification of the punching resistance the CEB-FIPModel Code is followed.
Punching resistance without shear reinforcement
The punching resistance. along the control perimeter u1. may betaken as
PRd,= 0.12((100p,"k)1/3u1d (16)
where (= 1 + -../(200/ d) expresses the size effect, d (mm) beingthe effective depth of the slab. If more than one layer of bonded re-
inforcement exists. the mechanical centre of resistance must be
considered in the definition of d. The ratio of reinforcement may be
determined as (= -../(PXpy),where Px and Py are the ratios of alibonded reinforcement (reinforcing bars and bonded tendons) inthe two orthogonal directions. These reinforcement ratios are cal
culated as the average along the widths by and bx. respectively.The characteristic value ofthe concrete cylinder's compression
strength fCk(MPa), in this expression, is limited to 50 MPa.
Punching resistance with shear reinforcement
The punching resistance with shear reinforcement may be evaluated as
PR~ = tPRd + 1.5:1 A.w f jd sina, 5,
where Aswis the total area of shear reinforcement in a layer around
the column, 5, is the radial spacing between layers (Figure 9) anda is the angle between the shear reinforcement and the plane of
the slab. The design strength of the reinforcement fydshall not betaken greater than 300 MPa.
Maximum punching resistance
The maximum load /3PSd(P),not considering the prestress effects,must not be greater than
where Uois the length of the periphery of the supporting column
and ,"d2the design resistance. of the concrete under compressionin a cracked zone, and is given by
Structural Concrete .2000, 1.No. 3
ULS of punching in the ( fib) FIP recommendations
I I-I ~
T-I II II CenlreUnes01 I
lhe slabpanel
A- A0.80.~
-r +-I
Ix= 9.0
Section A-A --------
---------
Flg. 12 Prestress tendons
Seclion A-A016/10.15
0.225
------
Flg. 13 Ordinary reinforcement (dimensions in m, diameter in mm)
(17)fcd2 = 0.60 (1- fck )fcd
250
The segments of perimeters Uonormal to the slab edge at comerand edge columns are limited to 1.5 d as are shown in Figure 10.
(19)
Example
In this section an example of the application of these recommen
dations is presented.
Design data
(18)The example refers to the punching check of an interior panel of a
prestressed solid slab, shown in Rgure 11. The slab is 0.225 m
thick, with spans I. = 9.0 m and Iy= 7.0 m. The prestress con-sists of unbonded tendons with an effective prestress, after
losses, of P = 150 kN per tendon. There are 16 and 12 tendons on
147
Lúcio et ai.
I------------------------I
I
I ___06
I
I
I
I
I
I
I
I
I
I
I
I
-, 1-
I
Flg.14 Punching reinforcement (dimensions inm, diameter inm)
the column lines in the x and y directions, respectively (Figure
12).
The amount of flexural reinforcement is As = 13.4 cm2/m in
both x and y directions (Figure 13). The materiais used in the
design are concrete grade C30 and steelgrade forordinaryrein-
forcement 5500, and the prestress reinforcement isdefined by
fptk = 1800 MPa.
For the geometry of this column and a slab effective depth
d= (dx+ dy)/2 = 0.19 m:
bx= 0.8 + 4 x 0.19 = 1.56 m
by= 0.6 + 4 x 0.19 = 1.36 m
u1= 2 x (0.80 + 0.60) + 1\"x 4 x 0.19 =5.19 m
p = Px'" Py =7.05 X 10-3
Acting forces
The forces transferred between the slab and the column are
shown inTable 1.The firstloadcombination referstogravityloads
plus the transversal prestress action, and the other two load com -
binations refer to the quasi-permanent value of the gravity loads
and the earthquake actions along x and y directions plus the pre -
stress action.
Effective punching force
From the eccentricitiesin Table 1 the value of the effective
punching force may be quantified as shown in Table 2.
Punching resistance neglecting the prestress compressioneffect
PRd,1 =0.12[1 + --J(200/190 mm)](100 x0.00705 x 30 MN/mm 2)1/3x 103 x5.19 m x 0.19 m
= 663 kN < PSd.ell (combination 2)
148
Punching reinforcementisthus needed forload combination 2.
Considering vertical stirrups, with a radial spacing
s, = 0.125 m < O.75d (Figure 14), then
p. _:lp,A. = Sd,eff 4 Rd,l
w (1.5d/ s,)xfyd
then
817 - i x 663 X104 =4.67 cm2A.w= -
Ten6 mmdia. stirrups with two legs were used in each layer(5.66 em 2 per layer), as shown inFigure14.
Punching resistance considering the prestress compressioneffect
Assuming thatalithe compression due to prestressis transmittedto the slab and the compression stresses are uniform along the
panel width, the favourable compression effects may be consid
ered in the following calculations:
Px= 16 x 150 kN = 2400 kN
.Py= 12 x 150 kN= 1800 kN
O"cpx= 2400 kN/(0.22 5 m x 7.0 m) = 1524 kPa
O"cpy= 1800 kN/(0.22 5 m x 9. Om)=889 kPa
M O =15241.36 mx(0.225 m)2 -17.5 kNmy 6
M O =8891.56 m x (0.225 m)2 =11.7 kNmx 6
Comblnatlon 1. Considering for the totalbending moments on
the slab widths bxand b" respectively:
MySd =110 kN m
M xSd= 80 kN m
p = 17.5 kNm x 550 kN=88 kNXo 110~0 kNm
p = 11.7 kNm x 550 kN=80 kNyo 80.0 kNm
p = 80 x 1.56 + 88 x 1.36 = 84 kNPo 1.56 + 1.36
psd.ett= I3PSd(p,P) - PpO= 644 - 84 = 560 kN
Table 3 presents these results for alithe load combinations.
These values are slightlylowerthan those presented inTable 2,
where the compression effectof prestresswas neglected.
Comparing the values of PSd, ellwith the value of PRd.1 = 663 kN
it can be seen that less punching reinforcement is required for
load combination 2 than in the previous case: Asw= 3.45 cm2 perlayer, with s, = 0.125 m.
Checking the maximum punching force
uo=2 x (0.80 x 0.60) = 2.80 m
,"d2=0.6[1- (30/250)] x 20 =10.56 MPa
PRd,max=0.5 x 10.56 MN/m2x103 x 2.80 m x 0.19 m
= 2809 kN > I3PSd(p)
Structural Concrete , 2000, i, No. 3
Table 1 Transferred forces between the slab and the column
ULS of punching in the ( fib) FIP recommendations
PSd (kN)
MSd' (kN m)
(ey (~))
Msdy (kN m)
(e, (m))
Load combination 1:
1.5(g + q) + P
550
40
(0.073)
60
(0.109)
Load combination 2:
g + 'l'2q + P + 1.5E,
180
30
(0.167)
480
(2.667)
Load combination 3:
g + 'l'2Q + P + 1.5Ey
180
380
(2.11)
40
(0.222)
Table 2 Effective punching load neglecting the compression due to prestress
Load combination 1 Load combination 3Load combination 2
/3PSd.eff = /3PSd(kN)
1.17644
Table 3 Effective punching load including the compression effect due to prestress
4.54817
3.45621
Load combination 1 Load combination 3Load combination 2
MySd (kN m)P 'o (kN)
MySd (kN m)P'o (kN)
P""PSd.ett
11088
8080
84560
3688
2680
84733
3688
2680
84537
Table 4 Maximum punching force check
PSd(P) (kN)
MSd' (kN m)
(ey(m))
MSdY(kN m)
(e, (m))
(3
(3PSd(p) (kN)
Load combination 1:
1.5(g + Q)
857
40
(0.047)
60
(0.070)
1.11
951
Load combination 2:
g + .'l'2Q+ 1.5E,
487
30
(0.062)
480
(0.986)
2.31
1125
Load combination 3:
g + 'l'2Q+ 1.5Ey
487
380
(0.780)
40
(0.082)
1.91
930
The values of /3PSd(P)are obtained from Table 4, where the pre -stress effects were not considered.
References
1. CEB-FIP. CEB-FIP Model Code 1990, Design Code . Thomas Telford,
London, 1993.
2. FIP. RP Recommendations for the Design of Flat Slabs in Post-tensioned Concrete (Using Unbonded and Bonded Tendons) . Cementand Concrete Association, Wexham Springs, 1980. .
3. fib. Design of post-tensioned slabs and foundation rafts. FIP
recommendations, SETO (for fib), London, 1999.
4. fib. Structural Concrete, Textbook on Behaviour, Design andPerformance. fib, Lausanne, 1999.
. Prepared by a Working Group with the following members: João Almeida(POR), Thomas Friedrich (CH), M. Jartoux (F, FIP Commission 2) ManfredMiehlbradt (CH), K. Schütt (D, FIP Commission 4), Júlio Appleton (POR),H.Ganz (CH), Válter Lúcio (POR), L Schúbert (D), Paul Regan (UK, CEBCommission).
Structural Concrete , 2000, 1, No. 3
--
V.J. G.Lúcio,PhD
Researcher of ICIST (UTL), Associated Professor atUNL, Dep. Eng. Civil, Universidade Nova de Lisboa,
Quinta da Torre, 2825-114 Caparica, Portugal
J. A. S. Appleton, PhD
Head of ICIST (UTL), Fuli Professor at 1ST,Dep. Eng.Civil, Instituto Superior Técnico, Av. Rovisco Pais,1049-001 Lisboa, Portugal
J. F.Almelda,PhD
Researcher of ICIST (UTL), Associated Professor at1ST,Dep. Eng. Civil, Instituto Superior Técnico, Av.
Rovisco Pais, 1049-001 Lisboa, Portugal
149