BASKARAN and MORLEY - Strength Assessment of Flat Slabs on Nonrectangular Column Grid

8/21/2019 BASKARAN and MORLEY - Strength Assessment of Flat Slabs on Nonrectangular Column Grid

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Strength assessment of flat slabs on non-

rectangular column grid K. Baskaran* and C. T. Morley†

University of Moratuwa; University of Cambridge

Flat slab construction has proved efficient and large numbers have been built. Often the layout of columns is

limited to rectangular grid owing to the scarcity of structural analysis tools validated by experimental evidence.

Recently, to break through this barrier, two irregular flat slabs were tested at Nanyang Technological University,Singapore. However, they failed at almost twice the design ultimate load, showing the need for more work. Also,

there are efforts to apply yield line analysis to flat slabs on irregular column grid. However, such efforts need

validation through comparison with experimental results. To fill this vacuum, nine flat slab panels supported on

non-rectangular column grid were designed and tested at Cambridge University Engineering Department (CUED).

The current paper presents experimental details and analytical results of five flat slab specimens tested at CUED:

three interior panel specimens and one corner panel specimen of parallelogram column layout and an irregular

seven-column specimen. Failure loads are predicted using yield line analyses based on energy approach and

nonlinear finite element analyses using a commercially available finite element package (DIANA). Further

confirmation of predictions is made by comparisons with experimental observations.

Notation

a, b, c, d some dimensions of length used in

calculating collapse load

f t tensile strength of concrete obtained from

splitting cylinder tests

f cu compressive strength of concrete obtained

from testing cubes

m total yield moment capacity across a yield

line

p load per unit area

shear retention factor

maximum deflection in the slab when a

collapse mechanism has formed

Ł relative rotation across a yield line

Introduction

With limited land resources, demand for optimum

use of available floor area in urban areas is increasing.

By avoiding downstand beams, flat slab construction is

cheaper and permits one or more additional floors with-in the allowable height in multi-storey buildings. Also

flat slabs are quite flexible in accommodating modifi-

cation to partition locations in case of change in use.

However, in contrast to two-way slabs on beams, in flat

slabs the span that influences the behaviour is the long-

est. Therefore it is better to reduce the distance between

the columns. Architects, however, demand fewer col-

umns or for them to be hidden inside walls. This

inevitably leads to inefficient designs, if columns are

restricted to a rectangular grid. To overcome this, flat

slabs supported on non-rectangular column grid were

suggested 1 and attempts have been made to study de-

sign and assessment methods for flat slabs supported

on such column layouts.2

Despite the long history of flat slab construction,

experiments on flat slabs supported on irregular column

grid are scarce. A seven-column and a fourteen-column

specimen were designed and tested at Nanyang Techno-

logical University, Singapore.3–5 However, the experi-

mental failure loads in both slabs were almost twice the

design ultimate loads. In addition, deflections reached

critical values almost at the design ultimate load, con-

firming how conservative the adopted design approach

was. Recently, suggestions were made6 to extend the

application of yield line analysis to flat slabs on an

*Department of Civil Engineering, University of Moratuwa, Sri

Lanka

†Department of Engineering, University of Cambridge, UK

(MCR 61497) Paper received 3 January 2006; last revised

25 November 2006; accepted 4 January 2007

Magazine of Concrete Research, 2007, 59, No. 4, May, 273–286

doi: 10.1680/macr.2007.59.4.273

273

www.concrete-research.com 1751-763X (Online) 0024-9831 (Print)# 2007 Thomas Telford Ltd


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irregular column grid. However, they are not validated

by comparison with experimental evidence. To fill this

vacuum of experimental results, a series of flat slabs

supported on non-rectangular column grid were de-

signed and tested at Cambridge University Engineering

Department (CUED).2 Yield line and nonlinear finite

element (FE) analysis have been used to assess those

slabs. In the present paper, an attempt is made, toencourage the use of yield line and nonlinear FE meth-

ods to analyse flat slabs on an irregular column grid by

presenting predictions compared with experimental re-

sults. Also, it is hoped that the experimental results will

help to validate future approaches to assess flat slabs

supported on a non-rectangular column grid.

Details of five model specimens relevant to this

paper and some observations made on them during the

tests are described in the next section. This is followed

by application of yield line analysis to assess flexural

capacity of these slabs and the NTU seven-column

specimen.5 Some yield line patterns and corresponding

predictions are illustrated to encourage the use of yield

line analysis in the industry. In the next section, details

of FE models and their predictions are presented. Com-

parisons between experimental observations and analy-

tical predictions are made prior to conclusions.

Experimental study

Specimen detailsSlabs 1, 4 and 5 were interior panels and slab 7 was

a corner panel of flat slabs supported on 308 skew

column layout with 1200 mm column centre-to-centre

spacing. The difference between slabs 1 and 4 was the

reinforcement direction (see Fig. 1). Details of the

specimen are given in Table 1. Slab 9 was a flat slab,

with 50 mm thickness and supported on seven columns

in an irregular grid. Reinforcement layouts of parallelo-

gram slab specimens and CUED seven-column speci-

men are shown in Figs 1 and 2 respectively. Mild steel

bars of 4 mm diameter having yield strength 395 MPa

were used as reinforcement. To improve the bond

characteristics, prior to use, bars were allowed to form

rust on their surface. Concrete mix used in all speci-

mens had ingredients in the following ratios

Slab 1 Slab 4

Slab 5 Slab 7

Bottom Top Bottom Top

Fig. 1. Reinforcement layouts for parallelogram slab specimens

Table 1. Details of experimental slabs

Specimen Description Thickness:

mm

Reinforcement

direction

f t: MPa f cu: MPa Experimental

failure load: kPa

Yield line

prediction: kPa

Finite element

prediction: kPa

Slab 1 Interior panel of a

parallelogram column

layout

50 Parallel to column

lines

2.84 34 19.64 20.96 19.7



layout

50 Parallel to

diagonals

3.22 51.3 18.92 19.41 19.51



layout


lines

3.18 51.5 25.31 24.84 23.02

Slab 7 Corner panel of a


layout


lines

3.4 48.1 23.81 26.34 23.96

Slab 9 Irregular 50 Orthogonal 3.47 45.2 25.86 25.34 30.1

Baskaran and Morley

274 Magazine of Concrete Research, 2007, 59, No. 4


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water : cement : fine aggregate : 6 mm

aggregate ¼ 0:5:1:2:35:2:87

To measure compressive and tensile strengths,

100 mm cubes and cylinders measuring 100 mm in dia-

meter and 200 mm in height were cast from the same

batch of concrete as control specimens and tested on

the same day as the experimental specimens. Obtained

strength values are averaged and given in Table 1.

After casting, specimens were allowed to cure for a

fortnight under the cover of polythene before being

tested in the vacuum rig. The vacuum rig is a closed

chamber, having steel sheets as bottom surface, alumi-

nium ring beam as edges and clear polythene as a top

cover. Fig. 3 shows a view of vacuum rig. Details of

the vacuum rig are discussed elsewhere.7 Positions of

the deflection gauges for parallelogram slabs and the

seven-column specimen are shown in Fig. 4.

Observations during tests (refer to Fig. 4 to identifylocations)

In slab 1, cracks were first observed on the top

surface above acute corner columns A, C at 10 kPa. At

13.1 kPa bottom surface cracks along middle strips

formed. This was followed by top surface cracks along

column lines between 14.2 and 15.4 kPa. With further

increase in load, slab 1 failed by punching of column D

after reaching 19.64 kPa load with central deflection

36.6 mm (0.73 times slab thickness). The observed top

surface crack pattern in slab 1 is shown in Fig. 5.

As the experimental specimens are fairly small (de-

signed for flexure considering one fourth of common

span, say 5 m) punching shear failure is prominent. To

reduce the scale effects bar diameter (4 mm) and maxi-

mum size of aggregates (6 mm) were selected small.

However, the observations made on slab 1 showed theneed to postpone punching shear failure further, with-

out affecting flexural capacity. Therefore from slab 2

onwards shear reinforcement in the form of spirals was

added above the columns.

In slab 4, the first cracks appeared above the acute

corner columns on the top surface at 9.7 kPa. Bottom

surface cracks were first seen at 12.3 kPa along diag-

onal AC. With the formation of diagonal cracks, deflec-

tion gauges TR 7, TR 8 and TR 9 measured almost

equal readings. At 12.5 kPa load, a diagonal crack

along column line BD was observed on the top surface.

With more load, cracks along other column lines also

formed before column B failed by punching at

18.92 kPa with maximum deflection equal to 67% of

the slab thickness.

In slab 5, bottom surface cracks along diagonal AC

were observed at 13.4 kPa. Cracks along column lines

on the top surface formed around 23.6 kPa. Finally slab

5 failed by fracture of bottom reinforcement at

25.31 kPa. The central deflection at that load was

43 mm (0.72 times the slab thickness). In the corner

panel specimen, first cracks were seen above column B

at 18.9 kPa. With further increase in load, top surface

cracks in acute corners and cracks along column lines

AB and BC formed at 21.3 kPa and 22 kPa respectively.At 23.8 kPa, the interior column B failed by punching.

In the seven-column specimen, first cracks were ob-

served above column G on the top surface around

14.8 kPa. With increasing load, hogging cracks appeared

above column F and along column line BG at 16.5 kPa.

Top surface cracks along column lines EG and FG

formed between 18 and 19 kPa load. Close to failure,

deflection gauge TR 17 showed decreasing deflection

owing to rotation of column C about its inner edge.

Finally, bottom reinforcement bars in panel BCDG frac-

tured at 25.86 kPa in flexure. Crack patterns observed in

seven-column specimen are shown in Fig. 6.

Assessment based on yield line analysis

To assess the strength of slabs, Ingerslev8 proposed

the yield line analysis in 1923. Johansen9 extended the

application by tests, confirming the reality of yield

lines and validity of the theory. Later, several research-

ers considered yield line analysis to assess the strength

of slabs having rectangular layout and reported its

advantages in assessing slabs with low steel ratios.

Recently Kennedy and Goodchild 6 proposed inserting

the largest possible rectangle through columns and ap-

(a) (b)

A B

C

D

G

E

F

A

F

E

G

B

C

D

Fig. 2. Reinforcement layouts for CUED seven-column

specimen (slab 9): (a) bottom reinforcement; (b) top

reinforcement

Fig. 3. A view of the vacuum rig after testing slab 5

Strength assessment of flat slabs on non-rectangular column grid

Magazine of Concrete Research, 2007, 59, No. 4 275


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(a)

(b)

A B

C

D

G

E

F

1233

1133

1700

2100

966

100

634

1135 333633

792

958

157

633

TR 2

TR 5

TR 4

TR 7

TR 6

TR 3

TR 1

R 1

R 3

R 5

R 2

R 8

TR 20 TR 21

TR 22

TR 9

TR 10

TR 8

TR 17

TR 10

TR 6 TR 7

TR 9

TR 8

TR 5

TR 2

A B

CD

1800

6 0 °

60

Fig. 4. Position of deflection gauges in experimental specimens: (a) deflection gauge locations in parallelogram slabs;

(b) deflection gauge locations in seven-column specimen. Note: All dimensions are in millimetres

Baskaran and Morley



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plying yield line patterns for rectangular slabs sup-

ported on simple or continuous supports to assess flat

slabs on irregular column grid. In the current paper,

instead of following such tactics, the actual probable

yield line patterns (without worrying about optimisa-

tion) are considered to analyse the experimental slabs.

Yield line method

The ultimate load in a yield line analysis is calcu-

lated by postulating a failure mechanism, involving

rigid slab portions rotating about axes of rotation with

plastic deformation along their boundaries (yield lines),

compatible with the boundary conditions. This is based

on the assumptions that final failure is attributable to

flexure and the structure has enough ductility. As thestructure has turned into a mechanism, the predicted

load by each mechanism is an upper bound value and

unsafe. Therefore an engineer needs to find the lowest

prediction, from several possible collapse mechanisms

for the structure being analysed. However in slabs,

membrane action owing to restraints and strain hard-

ening of steel have often made predictions based on the

yield line theory safe.10,11

Yield line mechanisms for flat slabs can be classified

into two categories: local and global failure mechan-

isms. Local failure mechanisms involving fans are com-

mon with uniform steel layouts and point loads.12

However, flat slab design in general considers uniform

load and concentrates steel in the column region lead-

ing to global (fold type) failure mechanisms. The slabs

analysed in the present paper fall into this second category and only global failure mechanisms are con-

sidered in their analysis.

A yield line mechanism must be compatible for its

existence. In the current paper hodographs (rotations

drawn as a vector plot) are used to ensure compatibility.

Calculations are performed using a work equation, that

is, equating the work done by the external forces to the

internal energy dissipated for the mechanism consid-

ered. From this work equation the load capacity, the

only unknown in the equation, can be found. To calcu-

late the work done by the uniform load the following

procedure can be used.

(a) Divide the rigid slab bodies into triangles.

(b) Find the deflections at the corners of the rigid slab

bodies.

(c) Calculate the centroidal deflection of each triangu-

lar slab portion and multiply by the force acting on

them to find the work done. Better use a computer

aided design (CAD) program to calculate the area

in a multipanel slab.

(d ) Sum up work done in all triangles to find the total

work done by the uniform external forces.

In the case of point or line loads, the equivalent load

multiplied by the displacement along the force direc-

tion gives the work done by these forces. To calculate

the energy dissipated across yield lines, moment capa-

city across each yield line should be multiplied by the

corresponding relative rotation and summed. This pro-

cedure is applied to the experimental slabs in the fol-

lowing section.

Fig. 5. Crack pattern observed on top surface of slab 1

Fig. 6. Crack patterns observed in CUED seven-column specimen: (a) bottom surface; (b) top surface




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Application

Four yield line mechanisms, as shown in Fig. 7 were

considered to assess interior panels of parallelogram

slabs.13. To calculate the relative rotations across yield

lines and deflections at corners of rigid slab bodies,

hodographs (diagrams of rotations regarded as vectors)

given in the same figure were used. For each slab two

calculations were made: one considering only the inter-

ior panel alone as a typical panel in a continuous floor,

15

A

(a)

1,4,13,16

(b)

(c)

(d)

2

3

4

1

Sagging yield line

Hogging yield line

Continuous edge

Column

321

5 6

4

78

9 10

13 14

1211

15 16

7,10,19,22

2,5,14,17

3,6,15,189,12,21,24

8,11,20,23

00

1,3,9,11

2,4,10,126,8,14,16

5,7,13,15

Yield line pattern

Hodograph

21

31 526

13

4

7

12

16

22

17

24

89

10

11

14

19

20

15

18

23

Yield line pattern

Hodograph

0 1,32,4

Yield line pattern

Hodograph

Yield line pattern

0

6,16

1,11

7,13

4,10

5,15

2,12

8,14

3,9

6 7

1110

1 32 4

8

12

1614

9

13

5

B

CD

X

Y

ZPQ S

Fig. 7. Yield line patterns considered for interior panel of parallelogram column layout: (a) mechanism one; (b) mechanism two;

(c) mechansim three; (d) mechanism four

Baskaran and Morley



7/14

and the other including the overhang slab portions with

edge shear also contributing to the work done by ex-

ternal forces. Predicted failure loads for slabs 1, 4 and

5 are tabulated in Table 2. Detailed calculation for slab

1 considering mechanism one that includes edge shear

forces is given in the Appendix. Similar yield line

patterns, but slightly modified to include two free

edges, were used to assess the corner panel specimen,with no effort made to find the critical location of yield

lines. The yield line patterns considered are shown in

Fig 8 and predictions are given in Table 2.

Part of slab 9 (shown in Fig. 9(a)) was analysed for a

valley type mechanism (mechanism PI) prior to the

experiment. Hogging yield line along column line BGE

and sagging yield line along MN are the only yield

lines considered. This mechanism predicted 26.4 kPa as

the failure load. However, in the experiment the ob-

served crack pattern (shown in Fig. 6) suggested a dif-

ferent failure mechanism. Based on this, a more

complicated failure mechanism (mechanism W), shown

in Fig. 9(c), was considered in which rigid slab bodies

a, b, c, d, e, f, g, h, i, j, k and l rotate about axes of

rotations along column line GE, column D, column D,

column G, column C, column B, column B, column A,

column G, column F, column line EG and column F

respectively. This mechanism predicted the collapse

load as 25.34 kPa.

To assess the flexural capacity of part of the full-

scale NTU seven-column specimen3,5 two yield line

mechanisms were considered: valley type mechanism

PI and one similar to that observed in the experiments

PII (see Fig. 9). Few details of the NTU seven-column

specimen are shown in Fig. 10. The predicted failureload 20.34 kPa, agreed well with the experimental fail-

ure load 20 kPa. Yield line predictions and experimen-

tal results will be compared later, after the FE analysis

has been described.

Assessment based on nonlinear FE

analysis

In the FE analysis, a continuum is replaced by a

finite number of elements connected at their nodes.

Material and geometrical properties are attached to the

elements before the assembly is analysed using stan-

dard structural theories. Ngo and Scordelis14 pioneered

the application of FEs to analyse reinforced concrete

beams. FE analysis provides more information than

yield line analysis. However, a nonlinear analysis must

be performed to exploit the benefits of ductility and

predict the deflections in the post-cracking region. Un-fortunately what makes the FE analysis of reinforced

concrete difficult is the modelling of cracks and the

deterioration of bond between steel and concrete.

Model

In slab analysis, to include the post-cracking changes

in stiffness, two approaches have been followed. In the

first approach the stiffness is modified 15 based on the

state of the section, that is, cracked or uncracked. How-

ever deformation of a two-way slab is attributable to

stretching, bending and shear. Also the material proper-

ties vary through the thickness. These are not repre-sented in the modified stiffness approach. Therefore

another approach based on multiple layers was intro-

duced to analyse reinforced concrete slabs16–18 and is

adopted here.

To model the slabs, eight-noded, curved, layered

shell elements based on Mindlin formulations divided

into three layers were used. To avoid locking, the only

available integration scheme over the area is 2 3 2

instead of 3 3 3. Each node had three translational and

two rotational degrees of freedom. Columns were mod-

elled as point supports with no vertical translation.

Horizontal translations of few columns were restricted as in the experiment to avoid the whole specimen mov-

ing together as a rigid body. Reinforcement was em-

bedded inside the layers using bar elements. To obtain

the stress versus strain characteristics in compression

for concrete, the relationships given by Shah et al .19

were used. The peak strength was assumed at a strain

of 0.003.

To model cracking, the smeared crack approach20

was used. The initiation of the first crack at an integra-

tion point was based on the principal stress exceeding

the tensile strength of concrete. If multiple cracks are

allowed to form with strength criterion alone then with

Table 2. Predicted failure load from different yield line mechanisms

Specimen Yield line mechanism

Mechanism one Mechanism two Mechanism three Mechanism four Mechanism PI Mechanism W Mechanism PII

Slab 1 20.96 21.04 21.06 21.59 — — —

Slab 4 19.98 21.07 20.65 19.41 — — —

Slab 5 25.04 26.36 24.84 25.58 — — —

Slab 7 26.84 27.24 26.34 26.62 — — —

Slab 9 — — — — 26.4 25.34 —

NTU 7-column — — — — 21.31 — 20.34




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small rotations in principal tensile stress, new cracks

will form and close the existing cracks. To avoid this,

the program allowed successive cracks to form only

when the tensile stress exceeded the tensile strength

and the angle between the two cracks is above a thresh-

old angle (default value 608). In the post-cracking state,

concrete between cracks contributes to carry tensile

forces. This tension stiffening effect was included in

the model by having a descending branch in the stress–

strain curve for concrete in tension. Massicotte et al .21

reported similarities between tension stiffening and ten-

sion softening curves. For the crack to propagate

further and to transform into a macro crack, energy is

required. This fracture energy, which was included in

8

(a) (b)

(c)

(d)

0

4

8

5

3,7

1,9

6

2

0 1,32

4 5

8

1 2 3

6

97

Yield line pattern

Hodograph

Yield line pattern

Hodograph

Yield line pattern

Hodograph

Yield line pattern

Symbol additional to those in Fig. 7

0

2,8

1,3,7,94,6

5

31 5264

10

1314

78

9

11

1215

16

1,4,13

2,5,11,14

3,6,12,157,10,16

9

0

2

3

1

Free edge

21

4

3

56

78

9

Fig. 8. Yield line patterns considered for corner panel of parallelogram column layout: (a) mechanism one; (b) mechanism two;

(c) mechansim three; (d) mechanism four

Baskaran and Morley



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the model via tension stiffening, was found from em-

pirical equations22 based on direct tension tests. In the

post-cracking region, contributions to shear resistance

by various sources such as aggregate interlock, dowel

action, etc. were represented by a constant shear reten-

tion factor ( ¼ 0:2).From a parametric study on various tension stiffen-

ing models, a linear tension softening model was se-

lected for the present analysis.2 In compression, a

loading surface based on von Mises yield criterion

was used. Steel was modelled as elastic, strain hard-

ening plastic with similar behaviour in tension and

compression. A perfect bond between steel and con-

crete was assumed to avoid complexity in modelling.

For the analysis, the Newton Raphson method was

used with force norm as convergence criteria. To re-

duce the storage requirements, an incremental iterative

approach was implemented with maximum iterations

within a load increment limited to 750. Also to cali-

brate the non-linear finite element (NLFE) tool, slab 1

was used as a calibration model. For the parallelogram

specimens, edge shear was applied as distributed load

on the elements along the edge.

The mesh used to analyse slabs 1 and 4 is shown in

Fig. 11(a). Similar but slightly modified meshes were

used in slabs 5 and 7. The mesh used for the CUED

seven-column specimen is shown in Fig. 11( b). In the

same figure, the division of layers along the thickness

direction for all slabs is also shown. For the specimens

with parallelogram layout the bottom reinforcement in

(a) (b)

(c)

o

a

b

cd

e

f

g

h

i

j

k

l

a

b

d e

c

f

g

i

h

j

l

k

A

E

G

B

C

D

F

A B

G

E

F

M

N

1

2 0

1

2

Hogging yield lines

Sagging yield lines

Free edge

Axes of rotations

o

1

2

3

1

2

3

P

O

A B

E

F

R

Q

G

Fig. 9. Yield line patterns considered to analyse CUED seven-column specimen: (a) valley-type mechanism (PI); (b) yield line

mechanism (PII); (c) yield line pattern (W) and corresponding hodograph




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(a) (b)

(c) (d)

200 200 mm wire meshT 10 mm diameter

bars at 170 mm c/c

T 10 mm diameter bars at 200 mm c/c

200 200 mm wire mesh

R- ar 10 mm diameter 400 c/c along strong bandb

A AB B

C C

G G

D D

E E

F F

T 10 mm diameter

bars at 140 mm c/c

Fig. 10. Details of NTU seven-column specimen: (a) top surface cracks observed; (b) bottom surface cracks observed; (c) top

reinforcement layout; (d) bottom reinforcement layout

(a) (b)

461 462 463 464

465 466 467 468

469470

471472

473474

475476

477478

479480

481482

483484

485 486 487 488

489 490 491 492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

0·3h

0·3h

0·4h

Fig. 11. Mesh used in the analyses of experimental slabs: (a) for parallelogram slabs; (b) for CUED seven-column specimen

Baskaran and Morley



11/14

the central panel was notably low. Therefore no tension

stiffening was provided for concrete in central ele-

ments. Also, concrete in the middle layer in all speci-

mens was modelled as brittle.

Finite element predictions

In slab 1, the first predicted cracks appeared above

acute corner columns on the top surface between 9 and

10 kPa. With further loading, bottom surface cracks

formed around 13.25 kPa. Bottom surface cracks ex-

tended along the middle strips and top surface cracks

penetrated in the thickness direction. Finally the pro-

gram stopped with a zero stiffness message for an

integration point close to a column. Subsequent checks

on the previous converged step showed that hogging

cracks had penetrated the layers except for the extreme

bottom layer integration point. Similar behaviour was

observed in slab 4. In slab 5, the ‘no stiffness message’

was for an element in the central panel. In the previous

converged step, bottom surface cracks had progressed up to the top layer except for two integration points in

that element.

In the corner panel specimen, initial cracks on the

top surface were predicted to form between 13 and

14 kPa. With further increase in loading, bottom sur-

face cracks formed between 14 and 15 kPa. With the

opening of bottom surface cracks the deflection con-

tours suggested a valley-type failure mechanism. Pre-

dicted crack patterns for slabs 1 ,4, 5 and 7 are shown

in Fig. 12. In the analysis of CUED seven-column

specimen, cracks formed at different locations with in-

creasing loads. These are shown in Fig. 13. The pre-

dicted failure was a result of the bottom surface cracks

penetrating across the thickness in panel BCDG, as

seen in the experiment. Predicted failure loads for all

specimens are given in Table1.

Comparison between experiment and yield

line predictions

The predicted failure loads based on yield line analy-

sis (lowest value out of several values from different

mechanisms) for the experimental slabs are tabulated

along with the measured experimental values in Table

Slab 4 at 19·5 kPa

Slab 7 at 24 kPa

Slab 1 at 19·7 kPa

Slab 5 at 23 kPa

Fig. 12. Predicted crack pattern for parallelogram slabs.

Note: both top and bottom surface cracks are shown together

at 16·5 kPaat 15·5 kPa

at 18·9 kPaat 18·6 kPa at 18·8 kPa

at 30·1 kPaat 26·2kPaat 21·6 kPaat 20·1 kPa

at 15 kPa at 16·8 kPa

at 17·5 kPa

Fig. 13. Predicted crack pattern for slab 9 at various stages in the NLFE analysis. Note: both top and bottom surface cracks are

shown together




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1. Only the prediction for the corner panel differs by

more than 10 % (10.6% to be exact). This may be a

result of either punching shear intervening in the ex-

periment prior to reaching full flexural capacity (the

strength of the spirals used as shear reinforcement in

slab 7 and CUED seven-column specimen were low

compared with other slabs; 400 MPa compared with

700 MPa) or the location of yield lines may need somemodifications to improve the predictions.

With increasing load, tensile stresses at highly

stressed regions exceed the tensile strength of concrete

and form cracks. Reinforcement bars crossing these

cracks carry the tensile forces previously carried by

concrete. With further increase in load, steel stresses

reach yield strength. If the steel is ductile enough then

the heavily stressed steel bars deform while carrying

the same stresses, to bring lesser-stressed neighbouring

bars into the resisting pattern. This will result in more

cracks and more steel yielding at more locations until

the structure converts itself into a mechanism. There-fore, the crack pattern at failure on the top and bottom

surfaces is a good indicator of yield line locations.

According to mechanism one in Fig. 7, top surface

cracks caused by hogging moments along column lines,

Table 3. Expected and observed crack pattern

Specimen Expected crack pattern according to yield

line

Observed crack pattern

predictions Top surface Bottom surface

Slab 1

Slab 4

Slab 7

Slab 9

A

F

f B

G

C

E

D

Baskaran and Morley



13/14

and bottom surface cracks along panel centre lines

owing to sagging moments, are expected. Observed

crack patterns for slab 1 agree with that expected for

mechanism one and confirm the prediction. Expected

and observed crack patterns for the slabs 1, 4, 7 and 9

are tabulated in Table 3.

In a yield line mechanism, slab portions separated by

yield lines rotate as rigid bodies about their axes of rotation. By considering this fact and requirements for

compatibility, change in deflection readings at final

stages of tests can be effectively used to identify the

yield line mechanism. To support this claim, by taking

deflections at 90% of the total experimental failure load

as reference, changes in deflections were calculated for

the CUED seven-column specimen at various linear

variable differential transducer (LVDT) locations. The

average values of these changes in deflection were

divided by that at LVDT location TR 10 to find the

normalised change in deflections. Based on the collapse

mechanism (W) predicted changes in deflections were

calculated. Expected and predicted changes in deflec-

tions are compared in Table 4, and agree quite well.

Comparison between experiment and

finite element predictions

Except for the seven-column specimen (slab 9) the

FE-predicted failure loads for the specimens are within

10% of the experimental values. The observed crack

pattern in the seven-column specimen suggests tensile

membrane action may be the cause for the increased

strength prediction. One drawback observed in the pre-

sent analyses was that the slopes of the load against

deflection diagram did not change with crack initiation

but changed only after the cracks were fully opened.

Also the type of element used is not capable of predict-

ing punching shear failures.

Conclusions

Results of experiments on flat slabs supported on a

non-rectangular column layout were presented. Ulti-mate failure loads were predicted based on yield line

theory and nonlinear FE analyses. These values are

compared in Fig. 14. Agreement between prediction

and experimental results is good. However, the non-

linear FE analysis needs a good calibration model.

Considering the accuracy of the predictions and the

resources used, the yield line theory is better compared

with NLFE to assess flat slabs (with low steel ratios)

on non-rectangular column grid.

Appendix

Example: Consider calculating failure load ( p kPa)

for slab 1 according to mechanism one including the

edge shear forces. The yield line pattern consists of 16

rigid bodies as shown in Fig. 7. If the deflection at X is

then those at P, Q, S, Y and Z will be 0:25, 0:5,0.5, 0.5 and 0.75 respectively

AB ¼ BC ¼ CD ¼ DA ¼ 2a ¼ 1:2m and

YZ ¼ PQ ¼ b ¼ 0:3m, DÂB ¼ 608

Table 4. Comparison between observed and predicted nor-

malised deflection ratios in CUED seven-column specimen

LVDT location Observed normalised

value

Predicted normalised

value

TR 2 1.0 0.99

TR 4 1.08 0.90

TR 5 0.82 0.96

TR 6 0.93 0.95

TR 7 0.78 0.82TR 8 0.26 0.28

R 1 0.28 0.49

TR 9 0.69 0.67

TR 10 1.0 1.0

TR 17 -0.24 -0.24

TR 20 0.83 0.68

TR 21 0.98 1.0

TR 22 0.83 0.73

R 2 0.29 0.28

R 3 0.06 0.01

R 5 0.23 0.19

R 8 0.4 0.29

TR 1 0.15 0.33

TR 3 0.05 0.5

Experimental failure

load

Yield line

prediction

Finite element

prediction

Seven-columnSlab 7Slab 5Slab 40

5

10

15

20

25

30

35

Load:kPa

Slab 1

Fig. 14. Comparison between experimental and analytical

failure loads




14/14

Work done by the uniform load

¼ 4 p(volume swept by rigid bodies 1, 2, 5, and 6)

¼ ffiffiffi

3p

4 p 4a2 þ 6ab þ 2b2ð Þ

Lengths of the simply supported edge shear steel plates

along column and middle strips are 2c ¼ 0.9 m, 2d ¼0.3 m respectively.

Work done by the edge shear forces

¼ 8 p(ca sin 60 3 0:375 ¼ db sin 60 3 0:625)

Therefore

Total work done by the external forces ¼ 2:065 pThis is equal to the energy dissipated across the yield

lines. If the total hogging moment resistances across

rigid bodies 1, 2 and 2, 6 are m3, and m1 respectively

and total sagging moment resistances across rigid bodies 2, 3 and 6,7 are m4 and m2 respectively

Then

Energy dissipated across yield lines ¼(8m1 þ 4m2 þ 8m3 þ 4m4) 3 Ł

where Ł ¼ 2= ffiffiffi 3p a is the relative rotation betweenrigid bodies obtained from the hodograph.

From calculations considering the number of bars

crossing the yield line, width, effective depth ¼40 mm, and strength values for concrete and steel

m1 ¼1:15kNm, m2 ¼ 0:75kNm,m3 ¼1:03kN and m4 ¼ 0:5kNm

Substitution of these values gives the total energy dis-

sipated across yield lines ¼ 22:44 3 2= ffiffiffi 3p a ¼ 43:19From the work equation

Work done by the external forces

¼ Energy dissipated across the yield lines

suggesting 20.9 kPa as the failure load.

Note: The values given in Tables 1 and 2 were

calculated using an Excel spreadsheet in which exacteffective depths were used instead of the average

40 mm in this example.

References

1. Wiesinger F. P. Design of flat slabs with irregular column

layout. ACI Journal, 1973, 70, 117–123.

2. Baskaran K . Flexural Behaviour of Reinforced Concrete Flat

Slabs Supported on Non-Rectangular Column Grid . PhD thesis,

Cambridge University, 2004.

3. Tan H. N. and Teng S. Test of a 7-column Irregular Flat-plate

Floor . Research report, phase 2A, NTU-Singapore, August 2000.

4. Jianzeng G., Teng S. and Kiat C. H. Test of a 14-column

Irregular Flat-plate Floor . Research report, phase 2B, NTU-

Singapore, August 2000.

5. Teng S., Tan H. N., Irawan P. and Kiat C. H. Test of a 7-

column irregular flat-plate floor. Seminar on Flat Plate for

Residential Projects. NTU-Singapore, September 2000.

6. Kennedy G. and Goodchild C. Practical Yield line design.

Proposed RCC publication draft 18. British Cement Association,

London, 2002.

7. Baskaran K. and Morley C. T. A new approach to testing

reinforced concrete flat slabs. Magazine of Concrete Research,

2004, 56, No. 6, 367–374.

8. Ingerslev A. The strength of rectangular slabs. The Institution

of Structural Engineers Journal , 1923, 1, 3–14.

9. Johansen K. W. Yield-Line Theory. Cement and Concrete Asso-

ciation, London, 1962.

10. Ockleston A. J. Load tests on a three storey reinforced con-

crete building in Johannesburg. The Structural Engineer , 1955,

33, No. 10, 304–322.

11. Park R. Ultimate strength of rectangular concrete slabs under

short-term uniform loading with edges restrained against lateral

movement. Proceedings of the Institution of Civil Engineers,

1964, 28, June, 125–150.

12. Park R. T. and Gamble W. L. Reinforced Concrete Slabs, 2nd

edn. John Wiley & Sons, London, 2000.

13. Baskaran K. and Morley C.T. Using yield line theory for

interior panels of slabs. Proceedings of the Institution of Civil

Engineers, Structures and Buildings, 2004 , 157, No. 6, 395–

404.

14. Ngo D. and Scordelis A. C. Finite element analysis of rein-

forced concrete beams. ACI Journal , 1967, 64, No. 3, 152–163.

15. Jofriet J. C. and McNeice G. M. Finite element analysis of

reinforced concrete slabs. Journal of Structural Division, Pro-

ceedings of ASCE , 1971, 97, No. ST3, 785–806.

16. Hand

F. R., Pecknold

D. A. and Schnobrich

W. C. Nonlinear layered analysis of RC plates and shells. Proceedings of ASCE ,

1973, 99, No. ST7, 1491–1505.

17. Lin C. S. and Scordelis A. C. Nonlinear analysis of RC shells

of general form. Journal of the structural division. Proceedings

of ASCE , 1975, 101, No. ST3, 523–538.

18. Barzegar. F. Layering of RC membrane and plate elements in

nonlinear analysis. Journal of Structural Engineering, ASCE ,

1988, 114, No. 11, 2474–2492.

19. Wang P. T., Shah S.P. and Naaman A.E. Stress–strain curves

of normal and lightweight concrete in compression. ACI Journal

Proceedings, 1978, 75, No. 11, 603–611.

20. Rashid Y. R. Ultimate strength analysis of prestressed concrete

pressure vessels. Nuclear Engineering and Design, 1968, 7, No.

4, 334–344.

21. Massicotte

A., Elwi

A. E. and MacGregor

J. G. Tension– stiffening model for planar reinforced concrete members. Jour-

nal of Structural Engineering, ASCE , 1990, 116, No.11, 3039–

3057.

22. Phillips D. V. and Binsheng Z. Direct tension tests on notched

and un-notched plain concrete specimens. Magazine of Concrete

Research, 1993, 45, No. 162, 25–35.

Discussion contributions on this paper should reach the editor by

1 November 2007

Baskaran and Morley


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BASKARAN and MORLEY - Strength Assessment of Flat Slabs on Nonrectangular Column Grid