Pushing Resistance of Rc Slabs With Rectangular Columns

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Punching resistance of RC slabs with

rectangular columnsD. R. C. Oliveira,Ã P. E. ReganÃ and G. S. S. A. MeloÃ

University of Brası lia

Fifteen tests on high strength concrete slabs with rectangular supports and three different load patterns are

reported. The results show that current code preovisions can overestimate punching resistance in some cases. An

investigation, made using the finite element method, shows the influences of the shape of a support and the pattern

of loading on the distribution of shear. Factors are proposed to allow for these effects while using the control

perimeter and basic shear resistance of the CEB Model Code 90, and it is demonstrated that this approach provides

strength estimates better than those of MC 90, BS 8110 and ACI 318. There remains a problem of the punching

capacity of slabs almost failing in flexure and this is discussed.

Notation

c side of a rectangular column

cmax longer side of column

cmin shorter side of column

c x side of column perpendicular to span of aone-way slab

c y side of column parallel to span of a one-

way slab

d effective depth of slab ¼ (d x + d y)/2

f 9c cylinder compressive strength of concrete

f ys yield stress of steel

mflex flexural resistance per unit width

u0 perimeter of column or loaded area

u1 control perimeter

V shear strength

V ACI unfactored shear strength from ACI 318

V BS

unfactored shear strength from BS 8110

V CEB unfactored shear strength from MC90

V flex shear force corresponding to flexural failure

V MV shear strength predicted by Mowrer and

Vanderbilt’s equation

V ref reference shear strength used as calculation

parameter

V test measured shear strength

V u ultimate shear strength

íone-way shear stress for a shear failure across the

width of a one-way slab

í prop shear stress proposed

í punching shear stress for a punching failure

ö diameter of a circular columnr ratio of flexural reinforcement ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi(r x

:r y)p

Introduction

Flat plate floors supported by elongated rectangular

columns and column-supported slabs spanning predo-

minantly in one direction are fairly common forms of

construction. In both cases designers need to consider

safety in relation to punching, but most codes of prac-

tice give little if any relevant guidance.

So long as any concentrated force on a slab is con-centric with a load or support, the resulting shear

is commonly assumed to act uniformly around the

perimeter, at which shear stresses are controlled.

This assumption is contrary what might reasonably be

expected.

This paper briefly reviews available information, de-

scribes a series of tests and presents results from finite

element analyses. From this base it proposes a simple

modification of the approach to punching given in

CEB Model Code 90, and adopted in the current draft

of EC2. The method proposed is not a complete solu-

tion to all the problems, but is shown to be satisfactory

in its correlation with most available test data.

Magazine of Concrete Research, 2004, 56, No. 3, April, 123–138

123

0024-9831 # 2004 Thomas Telford Ltd

Ã University of Brasılia., Brazilia-OF, Brazil.

(MCR 1104) Paper received 2 December 2002; last revised 6 March

2003; accepted 12 March 2003



Existing literature

As early as 1946 tests by Forsell and Holmberg1

included one in which a square slab supported at four

edges was centrally loaded over a 300 3 25 mm area.

The punching strength is well predicted by code expres-

sions such as that of CEB Model Code 902

which take

no account of the ratio cmax/cmin, showing that thisratio is not itself a significant parameter so long as

cmax/d is not large. In this instance cmax/d ¼ 2·88d .

Mowrer and Vanderbilt3

made tests of shallow

(d ¼ 51 mm) lightweight aggregate slabs supported at

four sides and loaded through square central columns,

with side dimensions up to 8d . The results were com-

pared with predictions made by various methods and

the best agreement was found for a modification of

Moe’s4

equation in which the punching strength is a

function of the ratio V u/V flex.

Hawkins et al .5

tested slabs supported by rectangular

columns with aspect ratios up to 4·33. In most of the

tests the load was applied at the two edges perpendicular

to the long sides of the columns. With the length of the

column perimeter constant the punching resistance re-

duced appreciably as cmax/cmin increased. A proposal6

was made to modify the then the ACI code’s limit shear

stress of 0:332 ffiffiffiffiffi

f 9cp

, at a perimeter d /2 from the column,

by a factor (0:625 þ 0:75cmin=cmax) < 1:0. This form of

expression has been adopted in subsequent additions of

ACI 318 but in a more conservative form.

Regan and Rezai-Jorabi7

tested one-way spanning

slabs subjected to either one central load or two sym-

metric loads applied through steel plates. Some of the

failures were by punching and others by wide beamshear. In a considerable number of tests the ultimate

loads were below conventionally calculated punching

resistances and normal beam shear strengths. The ap-

proach proposed was to calculate the applied shear at

the BS 81108

control perimeter in a two-stage process.

In the first stage the concentrated force or forces are

resisted by uniform upward pressure in a distribution

zone and in the second the pressure is treated as a

downward load spanning to the supports. Summation of

the shears from the two stages gives the applied stress

which is then compared with the resistance according

to BS 8110.

Leong and Teng9 tested slabs supported on central

columns and loaded near their edges. The major vari-

able was the perforation of slabs by large openings near

the columns, but the series included a number of solid

specimens. For solid slabs Leong and Teng propose the

use of the control perimeter of Fig. 1(a) which is

similar to that of BS 8110 but with a reduction in

length at large and/or elongated columns.

Al-Yousif and Regan10

tested slabs with elongated

columns (cmax/cmin ¼ 5 and cmax/d ¼ 6·25). Loads

were applied through the columns and the slabs were

simply supported on two or four sides. The ultimate

loads varied markedly with the support conditions,

being highest for four-sided support and lowest when

the slab spanned one-way parallel to the long sides of

the columns. Available test data were used to derive a

method of reducing the BS 8110 perimeter to an ‘effec-

tive perimeter’.

ueff ¼ 2f º x(c x þ 3d ) þ º y(c y þ 3d )g (1)

with

º x One and two-way slabs: 1:09 À 0:03c x

d

< 1

º y ¼

One-way slabs: 1:09 À 0:09c y

d

< 1

Two-way slabs: 1:09 À 0:03c y

d

< 1

8>><>>:

where c y is the column dimension parallel to the

span if the slab spans predominantly in one direction.

The limit of this method is for for cy . 4:55d for

one-way slabs for which º y Á (c y þ 3d) can be taken

as 5·14d .International codes have developed different ways of

treating the effects of the column size and rectangular-

ity, but none of them has taken any account of a slab’s

general flexure (one or two-way).

ACI 318,6

in which the control perimeter for punch-

ing is at a distance of d /2 from the loaded area, origin-

ally used a single limiting shear stress with an

unfactored value of 0:332 ffiffiffiffiffi

f 9cp

. It then introduced a

reduction factor for rectangular loaded areas and later

another factor for large loaded areas. The limit stress is

now as below.

íACI ¼ 0:332Æ ffiffiffiffiffi

f 9cp

(2)

where Æ is the least of (0·5 + cmin/cmax), (0·5 + 10d /u1)

and 1·0, with u1 ¼ (2cmax + 2cmin + 4d ) as ACI 318

uses square-cornered perimeters.

The CEB-FIP Model Code of 197811

used a control

perimeter 0·5d from the load but with rounded corners

as compared to the square corners of the ACI code. For

concentric loading the distribution of characteristic

shear resistance was as shown in Fig. 1(b). For large

and/or elongated columns, the punching shear resis-

tance applied only at the corners or the shorter sides.

For the remainder of the perimeter the resistance wasas for one-way shear (í punching ¼ 1·6íone-way ).

The 1978 Model Code’s definitions of the ‘punching

part’ of the column periphery was adopted in the ‘Pre-

standard’ version of Eurocode 2.12

However the dis-

tance to the control perimeter was increased to 1·5d

and the non-punching parts of the perimeter were as-

sumed to be stress free ‘in the absence of more detailed

analysis’.

The current CEB-FIP Model Code 90 has moved the

control perimeter to 2d from the loaded area, and

makes no special provisions for large or rectangular

columns. Recent drafts of EC2 have followed this

approach.

Oliveira et al .

124 Magazine of Concrete Research, 2004, 56, No. 3



To make comparisons between Codes’ treatments of

the factors in question here the effects of other para-

meters e.g. the ratio of main steel, the absolute effec-

tive depth and the concrete strength have to be

eliminated. This can be done by relating all predicted

strengths to a reference value V ref , equal to the resis-

tance of a two-way slab at a square loaded area of side

length 2d .

Because the length of u1 is much more dependent on

the column dimensions in ACI 318 than in MC 90, if

the ACI code had not introduced the reduction coeffi-

cient Æ, the values of V ACI/V ref for large square loaded

areas would be far above those of V CEB/V ref . With Æincluded, the difference between the two Codes is

greatly reduced. Thus what the ACI 318 accomplishes

with a column-size factor is largely accomplished in

MC 90 by a different definition of the control perimeter.

Figure 2 offers comparisons between the predictions

of Codes and other methods for square loaded areas

and for ones with cmax ¼ 5cmin. The left side considers

recommendations that do not distinguish between one-

way and two-way spanning slabs and includes all the

Codes considered above. The right side of the f igure

presents results for the methods that take account of

the slab’s overall flexural behaviour and includes lines

for MC 90 to facilitate comparisons.

For square loaded areas the predictions of MC 78,

MC 90 and ACI 318 are all within 15% of one another,

and except for the largest areas Al-Yousif and Regan’s

predictions for two-way slabs are not much lower.

However for one-way slabs Al-Yousif and Regan and

Regan and Rezai-Jorabi’s methods give resistances

significantly below those of the codes. It should be

noted here that the results obtained from Regan and

(a)

One-way shear Punching shear

Column

c max

0.5a1 0.5d

0 . 5 d

0 . 5 b 1

0 . 5 b 1

0 . 5 d

c m i n

a1

c max

2

2c min

5.6dϪ b1

b1

c min

2.8d

(b)

Control perimeter

Column

x 1.5 d

c max

1 . 5 d

1 . 5 d

c m i n

c max

2

2c min

c min

25.6dϪ

x Յ

c min Յ 600 mm

d Ն 125 mm

Fig. 1. Control perimeters of (a) Leong and Teng and (b) CEB MC 78

Punching resistance of RC slabs with rectangular columns

Magazine of Concrete Research, 2004, 56, No. 3 125



Rezai-Jorabi depend upon the dimensions of the slabs,

which is not the case for other methods.

With cmax/cmin ¼ 5 the differences between the

codes are much greater, with ACI 318 predicting the

lowest strengths for small values of cmax/d and EC2

(the prestandard) giving the lowest values when cmax/d

is large. In the methods taking account of the overall

flexural behaviour of the slab there are considerable

differences between the strengths obtained for different

orientations of the loaded areas, i.e. between cases with

cmax in the direction of the span and with cmax in the

transverse direction. Fig. 2 does not include approaches

such as Mowrer and Vanderbilt’s in which V u is a func-

tion of V u/V flex but this form of equation does not allow

some account to be taken of the differences between

one-way and two-way spans.

Note:

1 – Vref ϭ Strength of two-way slab and square column with c ϭ 2d

2 – A & R – one or two way ϭ Al-Yousif and Regan’s method applied to one- or two-way slab

3 – R & R – one way ϭ Regan and Rezai-Jorabi’s method applied to one-way slab

4 – b and l are the slab width and the span length respectively for one-way slabs

5 – Except where otherwise stated predicitions by Regan and Rezai-Jorabi are for slabs with 1 ϭ b ϭ 20d

3.0

2.5

2.0

1.5

1.0

0.5

0.0

V / V

r e f

c max ϭ 5c min

0 2 4 6 8 10

c max/d

MC78

MC90Leong and Teng

ACI

EC2-92

3.0

2.5

2.0

1.5

1.0

0.5

0.0

V / V

r e f

c max ϭ 5c min

0 2 4 6 8 10

c max/d

R & R – 1 way with

c max perp. to span

MC90

A & R – 2 way A & R – 1 way withc max perp. to span

R & R – 1 way withc max parall. to span

A & R – 1 way withc max parall. to span

3.0

2.5

2.0

1.5

1.0

0.5

0.0

V / V

r e f

c max ϭ

c

min

0 2 4 6 8 10

c max/d

A & R – 2 way

MC90

R & R – 1 way

A & R – 1 way

R & R – 1 way for l ϭ 25d and b ϭ 15d

3.0

2.5

2.0

1.5

1.0

0.5

0.0

V / V

r e f

c max ϭ

c

min

0 2 4 6 8 10

c max/d

MC78

MC90

Leong and Teng

ACI

EC2-92

Fig. 2. Comparisons between the predictions of codes and other methods

Oliveira et al .




Tests

Programme

Tests were made on 15 slabs with overall dimensions

of 2280 3 1680 3 130 mm.13,14

The main reinforce-

ment was fifteen 12·5 mm bars in the long direction

with a nominal cover of 10 mm and twenty three 12·5

mm bars in the short direction, giving equal ratios of reinforcement (1·1%) both ways and a nominal mean

effective depth of 107·5 mm. At their ends these bars

were lapped with 6·3 mm hairpin bars to assist the end

anchorages. Reinforcement details are drawn in Fig. 3.

Table 1 summarises the main slab properties.

The slabs were supported at their centres through 50

mm thick steel plates, 120 mm wide and with lengths

varying from 120 to 600 mm (1·12–5·58d ). Equal

loads were applied through beams close to the slab

edges. In slabs type ‘a’ and ‘b’ the loads were at only

two opposite edges (the short and long edges respec-

tively) while in type ‘c’ all four edges were loaded.

The test arrangements for a type ‘c’ slab are shown in

Fig. 4. Those for the other types were similar but with

only pairs of loading beams. The loads were measured

independently by four load cells.

Materials

The concrete used throughout had the mix propor-

tions given in Table 2. Cement CPII – F32 is an

15 # 12·5 mm e 117 mm – 2250 mm

2 2 8 0

1 3 0

15

5 0 120

1680

1 0

400

1 0 8

Staple

2 3 # 1 2 · 5 m m e

1 0 2 m m – 1

6 5 0 m m

5

609

8

7

6

432

1

6 0

1 0 0

Fig. 3. Flexural reinforcement arrangement and positions of strain gauges

Table 1. Characteristics of the tested slabs

Slab d : r f 9c: Column: mm V test:

mm MPa kN

cmin cmax

L1a 107 0·0109 57 120 120 240·0

L1b 108 0·0108 59 120 120 322·4

L1c 107 0·0109 59 120 120 318·0

L2a 109 0·0107 58 120 240 246·0L2b 106 0·0110 58 120 240 361·0

L2c 107 0·0109 57 120 240 330·8

L3a 108 0·0108 56 120 360 240·6

L3b 107 0·0109 60 120 360 400·0

L3c 106 0·0110 54 120 360 357·6

L4a 108 0·0108 56 120 480 250·8

L4b 106 0·0110 54 120 480 395·0

L4c 107 0·0109 56 120 480 404·0

L5a 108 0·0108 57 120 600 287·4

L5b 108 0·0108 67 120 600 426·4

L5c 109 0·0107 63 120 600 446·4

Failure modes: Group ‘a’ slabs: flexural-punching

Groups ‘b’ and ‘c’ slabs: punching





ordinary Portland cement with a 6– 10% content of

filler which is primarily crushed limestone. Compres-

sion strengths were determined from tests of 100 3 200

mm cylinders, cured and stored with the slabs.

The main reinforcement was of deformed bars with a

yield stress of 749 MPa and an ultimate strength of 903

MPa. The 6·3 mm bars used for hairpins and for the

bottom steel were also deformed and had yield and ultimate stresses of 641 and 799 MPa. Stress–strain

relationships from tests of the reinforcement are given

in Fig. 5.

Instrumentation and procedure

Slab deflections were measured using dial gauges

mounted from independent frames and reading onto

targets on the top surfaces of the slabs. Strains of

reinforcement were measured at the positions shown in

Fig. 3 using pairs of gauges with 5 mm gauge lengths

so that averaged values could eliminate local bending

effects. Radial strains of the bottom surface of the con-crete were measured also using electrical resistance

gauges (gauge length 31·8 mm). Loads were applied in

increments of 40 kN of total load and, after each, the

slabs were inspected for cracking and measurements

were taken.

Results

All of the tests ended in shear failures. The type ‘c’

slabs, with loads at four sides, failed in a normal

punching mode with failure surfaces being truncated

cones. The failure surfaces for slabs type ‘a’ and ‘b’ with small reaction areas were similar, but when

cmax > 360 mm the failure surfaces in the type ‘a’ slabs

did not run around the longer sides of the reactions

(Fig. 6). In the type ‘b’ tests, the failures of slabs L3b

and L4b (cmax ¼ 360 and 480 mm) were concentrated

1680

510

9 0

7 1 0

2 2 8 0

90

SLAB

Fig. 4. Loading system (plan)

900

800

700

600

500

400

300

200

100

0

S t r e s s : M P a

0 3 6 9 12 15 18 21 24 27 30 33 36

Strain: ‰

900

800

700

600

500

400

300

200

100

0

S t r e s s : M P a

0 3 6 9 12 15 18 21 24 27 30 33 36

Strain: ‰

(a) (b)

Fig. 5. Stress–strain graphs for reinforcement: (a) 6·30 mm; (b) 12·50 mm

Table 2. Mix proportions

Materials kg/m3

Cement (CP II – F32) 600

Silica fume 60

Crushed limestone (16 mm) 1092

Sand (5mm down) 512

Water 180

Superplasticiser 7·2

Oliveira et al .




3 7 °

L3a

3 5 °

L5a

L5b

32°31°

3 0 °

3 0 °

L3b

L3c

36° 39°

L5c

30°34°

Fig. 6. Failure surfaces





to the short sides of the reaction areas although in L5b

a normal punching surface developed as in L1b and

L2b. As is confirmed by calculations of flexural capa-

cities and by the measurements discussed below all of

the type ‘a’ slabs were very close to flexural failure. In

types ‘b’ and ‘c’ the failures were purely by punching.

The deflection data showed that the displacements of

the slabs along the lines at which measurements weremade increased practically linearly with distance from

the reaction area.

The dial gauge readings from each line were used to

calculate rotations and averaged to obtain transverse

and longitudinal values. Fig. 7 shows the greater aver-

aged rotations plotted against total load for slabs L1a,

L5a, L3b and L3c. These slabs were used as there were

V test ϭ 400 kN

V test ϭ 358 kN

V test ϭ 287 kN

V test ϭ 240 kN

L1a (y)

L5a (y)

L3b (x)

L3c (y)

0 Ϫ5 Ϫ10 Ϫ15 Ϫ20 Ϫ25 Ϫ30 Ϫ35

Deflection: mm

450

400

350

300

250

200

150

100

50

0

L o a d : k N

Fig. 7. Load versus deflection

1

2

3

4

5

6

7

L5a450

400

350

300

250

200

150

100

50

0

L o a d : k N

0 2 4 6 8 10 12 14 16 18 20

Strain: ‰

8

9

1

2

3

4

5

6

7

L5b450

400

350

300

250

200

150

100

50

0

L o a d : k N

0 2 4 6 8 10 12 14 16 18 20

Strain: ‰

8

9

1

2

3

4

5

6

7

L5c450

400

350

300

250

200

150

100

50

0

L o a d : k N

0 2 4 6 8 10 12 14 16 18 20

Strain: ‰

8

9

1

2

3

4

5

6

7

L1c450

400

350

300

250

200

150

100

50

0

L o a d : k N

0 2 4 6 8 10 12 14 16 18 20

Strain: ‰

1

2

3

4

5

6

7

L1b450

400

350

300

250

200

150

100

50

0

L o a d : k N

0 2 4 6 8 10 12 14 16 18 20

Strain: ‰

1

2

3

4

5

6

7

L1a450

400

350

300

250

200

150

100

50

0

L o a d : k N

0 2 4 6 8 10 12 14 16 18 20

Strain: ‰

Fig. 8. Reinforcement strains

Oliveira et al .




readings at or close to their ultimate loads. The figure

shows the difference in behaviour between types ‘a’,

‘b’ and ‘c’ very clearly.

In type ‘a’ slabs, the strains measured on the bars in

the direction of the main span, 60 mm from the edge of

the reaction zone, showed yielding of all bars within a

width 2·2d from the centre line. At the edge the strain

reached 11·56% in slab L1a and 3·1–3·6% in the other slabs. None of the strains of the transverse bars ex-

ceeded 1·55%.

In the type ‘b’ slabs, spanning predominantly in the

shorter direction, i.e. perpendicular to the longer sides

of the reaction areas, there was no yielding of the long

(transverse) bars. The main bars did not yield in L1b,

but yielding was recorded in the instrumented bars of

the others slabs, which were however very close to the

loads (see Fig. 8).

A comparison between the experimental ultimate

loads and the unfactored resistances according to ACI

318, BS 8110 and CEB MC 90 is shown in Fig. 9. For

the type ‘a’ slabs, with the main span parallel to the

longer sides of the supports, all three codes overesti-

mate resistance and this is almost certainly because of

the partly flexural nature of the failures. For the other

two types ACI 318 is safe but not very consistent,

while BS 8110 and MC 90 resistances are close to the

actual strengths for the smaller supports but tend to

become unsafe for the larger ones.

Finite element analysis

All the slabs were modelled and analysed using thefinite element method through the program Structural

Analysis Program (SAP). The mesh was the same for all

slabs and the applied loads were the failure loads. The

elements used were rectangular shell elements 603

60 mm with four joints.

The aim of the analysis was to investigate the shear

force distribution around the columns and along the

model code control perimeter. This perimeter was

adopted to plot the results due to both its reasonable

concordance with the failure surfaces from the tests

and its giving good results for the integration of the

shear forces. The supports were modelled as follows.

For slabs type ‘b’ and ‘c’ the nodes at the boundariesof and within the support areas were pinned. For slabs

type ‘a’ only the three nodes at the short sides were

pinned since central upward displacements were ob-

served in the tests. Loads were applied uniformly along

lengths of 480 mm at short edges and/or 660 mm at

long edges.

The shear force contours around the column for slabs

L1a, L5a, L5b and L3c are presented in Fig. 10. For all

slabs with cmax=cmin . 1 is possible to note clearly the

influence of the column shape on the shear polarisation

even for one-way slabs where the applied load is paral-

lel to the long side of the column. This characteristic is

not present in the shear distributions along the model

code control perimeter shown in Fig. 11.

At the model code perimeter, the shears are greatest

near the support sides perpendicular to the spans for

slabs of types ‘a’ and ‘b’ and the variation around the

perimeter increases with the ratio cmax=cmin. For type

‘a’ the ratio of the maximum to the average shear rises

for 1·17 for cmax=cmin of 1·0–1·64 for cmax=cmin equal

to 5. For type ‘b’ the variation is much smaller and

reaches only 1·23 for slab L5b. In type ‘c’, with half

the loading applied at the short edges, the concentration

of shear is almost as great as for type ‘a’, with a maxi-

mum/average shear ratio of 1·60 for slab L5c.If the maximum shears from Fig. 11 are compared

with the unfactored model code values from equation

(3)

írk d ¼ 0:18 1 þ

ffiffiffiffiffiffiffiffi200

d

r ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi100r f 9c

3p

d (3)

calculated for the parts of the perimeters where the

shears are highest, the ratio of experimental (test + FE

analysis) shear to írk d for types ‘b’ and ‘c’ is generally

satisfactory (1·02–1·14) but rises to 1·30 for slab L5c.

The values for type ‘a’ remain low at 0·82–0·93. For

type ‘a’ the problem remains the proximity of flexural

failure.

Proposal

Is proposed here to take account of the effects of the

shear polarisation by factors to be incorporated in the

model code for cases of symmetrical punching for two-

way and one-way slabs.

For design an effective applied shear force

(Vsd,eff ¼ ºVSd ) should be calculated such that

(Vsd,eff =u1d ) can be compared with the normal shear

Slabs ‘a’

Slabs ‘c’

Slabs ‘b’

ACI

CEBBS

0 1 2 3 4 5 6

c max /d

600

500

400

300

200

100

0

V ϫ ( 6

0 / f ′ c ) 1 / 3 : k N

Fig. 9. Comparison of test results with unfactored predictions

from codes





resistance írd ¼ írk =ªm. For comparison with test data,

this implies a proposed strength equation

V Prop ¼0:18

º1 þ


d


3p

du1 (4)

Values for º have been derived from available test data

(Table 4) taking account of the principal conditions

which can occur relating to the directions in which a

slab spans and the orientations of the longer and shorter

sides of rectangular supports. The corresponding ex-

pressions for º are given in Table 3.

Mowrer and Vanderbilt’s tests are useful as they

include slabs loaded through large square columns, but

they introduce the problem of lightweight aggregate

concrete in association with a very small effective

70→140→210→280→350→420 N/mm

70→140→210→280→350→420 N/mm

200→400→600→800→1000→1200 N/mm200→300→400→500→600→700 N/mm

Slab 1a

• • ••••

• • ••••

Slab 5a

• •••••

Slab 3cSlab 5b

100

• •••••

Fig. 10. Shear contours for slabs L1a, L5a, L5b and L3c

Oliveira et al .




depth. To be able to use these data in considering the

effect of column size these predictions of equation (4)

have been multiplied by 0·9 to give an average

Vtest=V prop of 1·0.

For the other methods of calculating resistances con-

sidered here the lightweight aggregate factors of ACI

318 and BS 8110, i.e. 0·85 and 0·80 respectively have

been taken into account, while no correction has been

made for MC 90 which does not treat lightweight con-

crete. The results obtained are given in Table 5 where it

can be seen that the proposed approach reduces thecoefficient of variation of Vtest=V prop relative to those

of all three codes.

The results of comparisons for normal weight con-

crete which exclude slabs L1a–L5a are shown in Figs

12 and 13 which show that the proposal eliminates the

trend of both MC 90 and BS 8110 resistances to

become less safe as cmax/d increases. The scatter of

Vtest=CACI is larger than for the other methods princi-

pally because of ACI 318’s neglect of the influence of

the ratio of flexural reinforcement. It can be observed

that the improvements produced by the proposed meth-

od are more relevant for one-way slabs with the long

dimension of the column parallel to the span.

Figure 14 considers lightweight concrete and com-

pares Mowrer and Vanderbilt’s results with the predic-

tions of ACI 318, BS 8110 and MC 90.

The strengths of the present type ‘a’ slabs are not

predicted satisfactorily by using the º factors above.

The distinct difference between these and the other

slabs can be appreciated by considering the results for

slabs L1a and L1b. Both had square supports of the

same size and both spanned one-way. Their effective

depths, reinforcement ratios and concrete strengths

were similar and yet the ultimate loads were 240 kN

for L1a and 322 kN for L1b. The reason for the

154 N/mm126 N/mm 168 N/mm

54 N/mm

107 N/mm187 N/mm

141 N/mm200 N/mm

204 N/mm

77 N/mm

255 N/mm 38 N/mm

L1a

L3b

L3c

L5a

L5b

L5c

Fig. 11. Shear forces along the CEB control perimeter for representative slabs

Table 3. Regularisation factors

Situation º

Two-way slabs 1:03cmax

d

0:02

One-way

cmax > cmin parallel to span

cmax

d

0:17

One-way

cmax perpendicular to span 0:93cmax

d

0:14





Table 4. Application of the regularisation factors for available test data

Author Slab Type d : r: f 9c: f ys: Column: mm V test V propV test V test

mm % MPa MPa cmin 3 cmax kN kN V prop V flex

1 L1a OW1 107·0 1·09 57·0 750 120 3 120 240 – – 1·00

L1b OW1 108·0 1·08 59·0 750 120 3 120 322 331 0·97 0·49

L1c TW 107·0 1·09 59·0 750 120 3 120 318 323 0·99 0·67

L2a OW1 109·0 1·07 58·0 750 120 3 240 246 – – 0·94

L2b OW2 106·0 1·10 58·0 750 1203

240 361 354 1·02 0·56L2c TW 107·0 1·09 57·0 750 120 3 240 331 356 0·93 0·65

L3a OW1 108·0 1·08 56·0 750 120 3 360 241 – – 0·88

L3b OW2 107·0 1·09 60·0 750 120 3 360 400 382 1·05 0·62

L3c TW 106·0 1·10 54·0 750 120 3 360 358 383 0·93 0·66

L4a OW1 108·0 1·08 56·0 750 120 3 480 251 – – 0·85

L4b OW2 106·0 1·10 54·0 750 120 3 480 395 388 1·02 0·62

L4c TW 107·0 1·09 56·0 750 120 3 480 404 430 0·94 0·67

L5a OW1 108·0 1·08 57·0 750 120 3 600 287 – – 0·91

L5b OW2 108·0 1·08 67·0 750 120 3 600 426 450 0·95 0·65

L5c TW 109·0 1·07 63·0 750 120 3 600 446 495 0·90 0·66

2 1 OW1 117·3 1·12 30·9 419 305 3 305 391 364 1·08 1·08

2 OW1 117·3 1·12 26·9 419 203 3 406 358 330 1·08 0·93

3 OW1 117·3 1·12 32·6 419 152 3 457 340 346 0·98 0·85

4 OW1 117·3 1·12 31·6 419 114 3 495 337 337 1·00 0·82

5 OW2 117·3 1·12 27·4 419 152 3 457 362 363 1·00 0·746 OW2 117·3 1·12 23·1 419 152 3 457 342 343 1·00 1·06

7 TW 117·3 0·86 26·4 419 152 3 457 326 350 0·93 0·78

8 TW 120·7 0·80 26·6 422 114 3 495 321 355 0·90 0·77

9 TW 120·7 0·76 30·1 422 152 3 305 322 326 0·99 0·92

3 14R OW1 79·0 1·00 31·0 670 75 3 100 154 149 1·03 0·65

15R OW2 79·0 1·00 30·8 670 100 3 150 172 169 1·02 0·68

19R OW2 79·0 0·99 29·0 670 100 3 150 170 165 1·03 0·55

4 B1 OW2 95·5 0·76 25·1 709 ö¼ 120 181 184 0·99 0·70

B2 OW2 101·0 0·51 25·0 665 ö¼ 120 180 175 1·03 0·98

C1 OW2 201·0 0·75 23·5 711 ö¼ 240 648 643 1·01 0·54

C2 OW2 201·0 0·52 23·2 706 ö¼ 240 547 569 0·96 0·61

S1 OW2 619·0 0·57 30·6 622 ö¼ 800 4915 4822 1·02 0·61

5 OC11 TW 105·3 1·81 36·0 452 200 3 200 423 369 1·15 0·82OC13 TW 107·3 1·71 35·8 452 200 3 600 568 497 1·14 1·09

OC15 TW 102·8 1·76 40·2 452 200 3 1000 649 627 1·04 1·06

OC13a¼1:60 TW 109·8 1·67 33·0 470 200 3 600 508 493 1·03 0·73

C11F22 TW 155·0 1·72 35·4 460 250 3 250 627 664 0·94 0·49

C13F22 TW 155·0 1·66 35·6 460 250 3 750 792 861 0·92 0·50

C15F22 TW 160·0 1·64 35·4 460 250 3 1250 1056 1102 0·96 0·17

C13F11 TW 159·0 1·07 35·5 520 250 3 750 769 767 1·00 0·61

6 2 OW2 111·0 0·64 12·3 – ö¼ 140 176 177 1·00 0·46

3 OW2 106·0 0·67 12·3 – ö¼ 140 172 167 1·03 0·49

4 OW2 110·0 0·64 12·3 – ö¼ 140 177 175 1·01 0·51

10 TW 104·0 0·68 17·6 – 25 3 300 186 190 0·98 0·84

11 TW 112·0 0·63 17·6 – 140 3 540 279 273 1·02 0·99

12 TW 108·0 0·65 17·6 – 140 3 340 265 228 1·16 0·86

7 A7 OW1 114·5 2·48 28·5 321 254 3 254 400 424 0·94 0·96A8 OW1 114·5 2·48 21·9 321 356 3 356 436 428 1·02 1·03

A2a TW 114·5 2·48 13·7 321 254 3 254 334 363 0·92 0·57

A2b TW 114·5 2·48 19·5 321 254 3 254 400 409 0·98 0·61

A2c TW 114·5 2·48 37·4 321 254 3 254 467 508 0·92 0·63

A7b TW 114·5 2·48 27·9 321 254 3 254 512 460 1·11 0·72

A5 TW 114·5 2·48 27·8 321 356 3 356 534 533 1·00 0·70

8 DT1 OW1 190·0 1·28 43·6 530 150 3 150 780 823 0·95 0·92

BD2 OW1 101·0 1·28 42·2 530 100 3 100 293 277 1·06 0·98

BD8 TW 101·0 1·28 35·3 530 100 3 100 251 253 0·99 0·66

(continued overleaf )

Oliveira et al .




difference lies in the proximity of flexural failure in all

the type ‘a’ tests.

Tests of two-way slabs15–17

show that while the ratio

V u/V CEB is generally independent of V u/V flex, there is a

decrease of punching resistance by up to 20% when the

load is at or very close to the flexural capacity. The

main reason for this is probably a lowering of resis-

tance due to wide cracks and high concrete strains.

In the case of one-way slabs there is probably an

additional effect from an increasing concentration of

shear to the column faces perpendicular to the span as

transverse yield lines develop. In the extreme this could

reduce the active part of the control perimeter to that

obtained at two edge columns contacting a slab only at

their inner faces as shown in Fig. 15 for which the

shear resistance could be estimated according to equa-

tion (5)

V min ¼ 0:18 1 þ


d


3p

3 d(2c þ 3ð d) (5)

Table 4. (continued)

Author Slab Type d : r: f 9c: f ys: Column: mm V test V propV test V test

mm % MPa MPa cmin 3 cmax kN kN V prop V flex

9 1 OW1 80·0 0·98 23·6 472 100 3 500 163 171 0·95 0·71

2 TW 80·0 0·98 23·2 472 100 3 500 209 217 0·96 0·54

3 OW2 80·0 0·98 21·2 472 100 3 500 189 187 1·01 0·84

4 TW 80·0 0·98 22·0 472 300 3 300 242 216 1·12 0·62

10 1 OW2 123·0 0·88 42·6 452 ö¼ 171 367 359 1·02 0·942 OW2 123·0 0·87 44·9 452 ö¼ 171 349 364 0·96 1·20

4 OW2 123·0 1·09 45·6 452 ö¼ 171 393 394 1·00 0·81

11 L42 TW 139·0 1·46 43·2 604 200 3 400 657 613 1·07 0·51

L42a TW 164·0 1·23 36·2 604 200 3 400 693 684 1·01 0·46

L45 TW 154·0 1·31 42·0 604 200 3 600 798 753 1·06 0·56

L46 TW 164·0 1·23 39·3 604 200 3 800 911 864 1·05 0·60

L41 TW 139·0 1·46 44·7 604 150 3 250 563 540 1·04 0·45

L41a TW 164·0 1·23 38·9 604 150 3 250 600 621 0·97 0·40

L43 TW 164·0 1·23 38·7 604 150 3 450 726 698 1·04 0·49

L44 TW 164·0 1·23 40·0 604 150 3 600 761 766 0·99 0·51

12 1 TWLW 51·0 1·10 28·6 386 102 3 102 86 79 1·08 1·14

2 TWLW 51·0 2·20 24·9 386 102 3 102 102 95 1·07 0·76

3 TWLW 51·0 1·10 21·1 386 152 3 152 79 84 0·94 1·03

4 TWLW 51·0 2·20 18·0 386 152 3 152 99 100 0·99 0·76

5 TWLW 51·0 1·10 15·5 386 203 3 203 93 87 1·06 1·18

6 TWLW 51·0 2·20 27·2 386 203 3 203 133 132 1·00 0·87

7 TWLW 51·0 1·10 23·3 386 254 3 254 109 113 0·97 1·25

8 TWLW 51·0 2·20 22·9 386 254 3 254 152 141 1·07 0·97

9 TWLW 51·0 1·10 28·0 386 305 3 305 119 134 0·89 1·25

10 TWLW 51·0 2·20 26·4 386 305 3 305 158 165 0·96 0·92

11 TWLW 51·0 1·10 27·8 386 356 3 356 138 147 0·94 1·37

12 TWLW 51·0 2·20 25·0 386 356 3 356 185 179 1·03 1·02

13 TWLW 51·0 1·10 24·9 386 406 3 406 145 155 0·94 1·35

14 TWLW 51·0 2·20 24·6 386 406 3 406 185 194 0·95 0·95

Oliveira13

Hawkins et al 5

Regan and Rezai-Jorabi7

Nylander and Sundquist15

Leong and Teng9

Forssel and Holmberg1

Elstner and Hognestad 16

Regan17

Al-Yousif and Regan10

Borges18

Mahmood 19

Mowrer and Vanderbilt3

Legend:

OW1: One-way slab with cmax parallel to span

OW2: One-way slab with cmax perpendicular to span

TW: Two-way slab

TWLW: Two-way lightweight aggregate concrete





Table 6 summarises the results for the type ‘a’ slabs.

The data are too limited for any definite conclusion to

be reached but it appears that the ultimate resistances

of such slabs when very close to flexural failure might

be calculated by either reducing the normal estimate of

V CEB/ º by 30% or considering the reduced perimeter of

Fig. 15. An alternative approach would be to accept the

reduction in the ratio º(V u/V CEB) on the basis that the

partial safety factor on resistance can be allowed to

decline from 1·5 for a shear failure to 1·15 for a

flexural failure (1·15/1·5 ¼ 0·77).

ConclusionThe results of the tests reported here and others in

the literature show that the punching resistances of flat

slabs are influenced by the shapes and sizes of their

supports and by their overall flexural behaviour in ways

not properly accounted for in current Code provisions.

Elastic analyses of uncracked slabs help to explain

the results obtained by illustrating the distributions of

Table 5. Comparison of the results from ACI, BS 8110, CEB

and proposal

Codes and

proposal

V test

V calc:

Normal weight concrete Lightweight concretea

Av SD Cv: % Cv Sd Cv: %

ACI 1·37 0·22 16·23 1·70 0·30 17·41

BS 8110 1·01 0·09 8·53 1·18 0·09 7·89

CEB 0·95 0·09 9·04 0·84 0·06 7·49

Proposal 1·00 0·06 5·77 1·00 0·06 6·21

aResults obtained using the test results of Mowrer and Vanderbilt with

two-way slabs loaded through square areas.

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

V T e s t

/ V A C I

0 2 4 6 8 10 12

c max/d

ACI

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

V T e s t

/ V B S

0 2 4 6 8 10 12

c max/d

BS

2.01.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

V T e s t

/ V P r o p

0 2 4 6 8 10 12

c max/d

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

V T e s t

/ V C E B

0 2 4 6 8 10 12

c max/d

CEB

Fig. 12. Results from codes and proposal

OW1: Prop

OW2: PropTW: PropTW: CEB

OW2: CEB

OW1: CEB

0 2 4 6 8 10 12

c max

/d

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

V T e s t

/ V

Fig. 13. Trend lines for the results from CEB with and with-

out proposed modification

Oliveira et al .




shear around the control perimeter of the CEB-FIP

model code.

Most of the effects observed can be taken into ac-

count, in the general context of the model code by the

introduction of a factor º expressing the non-uniformity

of the shear distribution that can occur even at concen-

trically loaded supports. This factor is a function of the

ratio (cmax=d ) between the larger dimension of the sup-

port and the effective depth of the slab, of the slab’s

overall flexure (one or two way spanning) and of the

direction of cmax (parallel or perpendicular to a one-

way span).

With the proposed values of º included, the modified

CEB-FIP method gives predictions of ultimate

strengths which are significantly better than those of

the unmodified model code, AC I318 and BS 8110.There remains the problem of the reduction of

punching capacity, which occurs when the load is very

close to a slab’s flexural resistance. This is discussed,

and two methods of obtaining approximate ultimate

loads are given, although it may be that the difference

between the partial safety factors governing resistances

to flexure and punching makes such calculations un-

necessary.

Acknowledgements

The authors would like to acknowledge the support

of Brazilian Scientific and Technological Development

Agency (CNPq), FINATEC, CAPES and Imperial Col-

lege (London).

References

1. Forssel C. and Holmberg A. Stampellast pa plattor av betong.

Betong , 1946, 31, No. 2, 95–123.

2. CEB-FIP. Model Code 1990, Thomas Telford, London, 1993.

3. Mowrer R. D. and Vanderbilt M. D. Shear strength of light-

weight aggregate reinforced concrete. ACI Journal , 1967, 64,

No. 11, 722– 729.

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

V t e s t / V

0 2 4 6 8 10

c /d

ACI CEB BS Proposal

Fig. 14. Results for lightweight aggregate concrete

22.5°

Column

Control perimeter

Fig. 15. Possible control perimeter for type ‘a’ slabs

Table 6. Results for type ‘a’ slabs

Slab Column: mm V test: V test

V flex

º Á V test

V CEB

V test

V minkN

cmin cmax

L1a 120 120 240 1·00 0·74 1·06

L2a 120 240 246 0·94 0·73 1·06

L3a 120 360 241 0·88 0·70 1·06

L4a 120 480 251 0·85 0·69 1·11L5a 120 600 287 0·91 0·75 1·26





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slabs with elongated columns. 6th International Symposium on

Utilization of High Strength/High Performance Concrete, Leip-

zig, Germany, June, 2002, Vol. 1, pp. 445–456.

15. Nylander, H., Sundquist, H., Genomstansning av pelarunder-

stodd plattbro av betong med ospand armering. Meddelande Nr.

104, Institutionen for Byggnadsstatik, KTH Stockholm, 1972.

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concrete slabs. ACI Journal , 1956, Proceedings 53, No. 1, July,

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Polytechnic of Central London, 1978.

Discussion contributions on this paper should reach the editor by

1 October 2004

Oliveira et al .


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