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