compressive membrane action in circular reinforced slabs

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International Journal of Mechanical Sciences 47 (2005) 1629–1647

0020-7403/$ -

doi:10.1016/j.

�Correspon

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Compressive membrane action in circular reinforced slabs

S.K. Dasa,�, C.T. Morleya,b

aParsons Brinckerhoff Ltd, Bristol BS6 6US, UKbDepartment of Engineering, University of Cambridge, Cambridge, UK

Received 30 January 2004; received in revised form 16 March 2005; accepted 6 April 2005

Available online 27 July 2005

Abstract

When a concrete slab with low steel proportion is laterally restrained, the load carrying capacity isenhanced due to the phenomenon of compressive membrane action. Despite the known advantages, theabsence of a formula, that could explicitly calculate the enhanced capacity, poses a drawback in design andassessment of laterally restrained structures. It encourages engineers to regard the additional capacity as anadded factor of safety. This paper attempts to overcome this shortcoming for slabs of circular geometry.

The elastic–plastic Deformation Theory solution for circular concrete slabs has been worked out. Theresulting graph never touches the curve for rigid-plastic strain-rate solution as it does in the strip case, Eyre(Journal of Structuring Engineering 1990;116(12):3251; 1997;123(10):1331). However, the maximummembrane force at the centre may still be used to derive an explicit load capacity as a function of materialproperties and restraint stiffness only.r 2005 Elsevier Ltd. All rights reserved.

Keywords: Compressive membrane action; Laterally restrained structures; Elastic plastic theory

1. Introduction

Compressive membrane action in concrete slabs requires low steel proportions and lateral edgerestraint, the latter of which is neither easy to provide nor straightforward to quantify in many ofthe usual slab geometries. However, surrounding panels in multi-panel floor slabs, earth pressurein underground construction, bridge girders restraining the deck slabs etc. are some of the

see front matter r 2005 Elsevier Ltd. All rights reserved.

ijmecsci.2005.04.007

ding author.

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Nomenclature

Asb total cross-sectional area of the steel restraintEc modulus of elasticity of concreteEs modulus of elasticity of steel reinforcementMr bending moment per unit width in radial directionMy bending moment per unit width in circumferential directionMo ultimate bending moment per unit width corresponding to zero normal forceNo compressive membrane force corresponding to rotation about mid-depthNa local compressive membrane force at edgeNb local compressive membrane force at centreNr local compressive membrane force in radial directionNy local compressive membrane force in circumferential directionR slab radiusRi radial distance of the individual restraining steel hoop from the slab centreS stiffnessas cross-sectional area of a steel restraining hoop of the restrainte radial extension of the elastic restraint due to membrane forcef c compressive strength of concreteh overall depth of slab sectionmo non-dimensionalised bending moment due to compression reinforcementm0

o non-dimensionalised bending moment due to tension reinforcementno non-dimensionalised compressive membrane force due to compression reinforcementn0o non-dimensionalised compressive membrane force due to tension reinforcementna non-dimensionalised compressive membrane force at edgenb non-dimensionalised compressive membrane force at centreq uniformly distributed transverse load on slabr radial distance from the slab centreu in-plane lateral displacement of slabw transverse displacement of slab at any radial pointwo transverse displacement of slab at centreda middle surface extension at edgedb middle surface extension at centreba angle made to the horizontal plane by radial span at edge after the deformationbb angle made to the horizontal plane by radial span at centre after the deformationxa neutral axis distance factor at the edge (ratio of distance of neutral axis from the mid

plane to the total slab depth at edge)xb neutral axis distance factor at the centre (ratio of distance of neutral axis from the mid

plane to the total slab depth at centre)O mechanical reinforcement degreee strain in radial directioney strain in circumferential direction

S.K. Das, C.T. Morley / International Journal of Mechanical Sciences 47 (2005) 1629–16471630

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kr curvature in radial directionky curvature in circumferential directionf flexibility parameter of the restraint

S.K. Das, C.T. Morley / International Journal of Mechanical Sciences 47 (2005) 1629–1647 1631

situations where the phenomenon may be exploited [16,17]. Low (perfectly zero) steel percentagealso ensures lower cost in maintenance and higher durability compared to the ordinaryunrestrained slabs. Punching shear capacity is also enhanced, Chana et al. [1]. The OntarioHighway Code [2] appreciates the advantage while some steel-free deck slabs have already beenconstructed in Canada [3]. Old bridges failing capacity checks may still be capable if themembrane action effect be taken into account. Wasteful replacements may hence be avoided.In most of the slab geometries, the degree of restraint of the ‘‘surround’’ and in-plane stiffness is

difficult to assess, when modelling the slabs as rigid plastic elements surrounded by linear-elasticin-plane restraints at the edge. The task is compounded by the fact that, if the slabs are notaxisymmetric, the edge restraint varies in magnitude along the perimeter. Researchers also differover the application of Deformation Plastic Theory (in which stress depends upon total plasticstrain) and Strain-Rate Plastic Theory or Flow Theory (in which stress depends upon currentincrement of plastic strain) during the loading history.In the long list of researchers, there seems to be three distinct groups of thought. Most of the

early researchers like Wood [4] and Christiansen [5] used Deformation Theory, usually implicitly.Later researchers like Morley [6,18], Al-Hassani [7], Braestrup [8], Janas [9,20] opted for Strain-Rate Theory citing drawbacks of the Deformation Theory. The latest researchers like Kemp [10],Kemp et al. [11] and Eyre [12,13] have preferred to use both the theories appropriate to theloading state.Kemp et al. [11] suggested that neither the Deformation Theory nor the incremental Strain

Theory should be exclusively used in materials that crack during loading. Correct flow rule in achanging stress state depends upon the direction of motion along the surface of yield criterion.This means the use of particular flow rule depends upon whether the membrane force is increasing

A BCompressive Membrane

Force

central deflection

Fig. 1. Deformation theory and Strain-Rate Theory apply during increasing and decreasing membrane force

respectively.

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or decreasing during the stress change, see Fig. 1. When the compressive membrane force increases,the neutral axis needs to move into the cracked zone closing up the cracks before stresses candevelop. Thus, stress state requires the recovery of total strain and so is dependent upon the entirestrain history of the closed crack depth. This can be defined only through Deformation Theory,which, therefore, should be used in region A of Fig. 1. When the membrane force starts decreasingafter reaching the maximum, the neutral axis moves back into previously compressed concrete. Asconcrete is assumed to be continuous with perfectly plastic characteristics, the strain incrementswould be independent of the strain history. This may be described by the Strain-rate flow rule undermonotonically straining system (region B in the figure).Eyre [12] proved that the maximum membrane force for a strip case, under either of the flow

rules and regardless of any initial deflection state at the commencement of the membrane action,lies on the same straight line of the rigid-plastic Strain-Rate Theory membrane force–deflectionequation as shown in Fig. 2. Then, for any restraint stiffness, all the elastic–plastic load-deflectioncurves are tangential to the rigid-plastic Strain-Rate Theory load deflection curve as shown inFig. 3 at maximum membrane force. The common point may be used to find an explicit formula

Rigid-plastic strain-rate equationMembrane

Force Elastic-plastic strain-rate theory

Elastic-plastic total-strain theory

central deflection

Fig. 2. Peak under both the theories lie on the rigid plastic strain rate equation line.

Rigid-plastic strain-rate theory load-deflection equation

G1 Explicit load solution by MMFM

G2

Load

S2 S1

Central deflection

Fig. 3. Maximum membrane force method as described by Eyre [13].

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for the load capacity at maximum membrane force. Eyre [13] further showed that the load socalculated underestimates the theoretical maximum load by a convenient safety factor that is selfmodulating, to provide greater margin where stiffness sensitivity and the potential forcatastrophic unloading at collapse is higher. In Fig. 3, margin of safety for curve S1 is higherthan for curve S2 with lower stiffness (G14G2). This method has been termed the ‘‘maximummembrane force method’’ (MMFM). This paper investigates the use of MMFM for a circular slab.

2. Elastic–plastic equation according to Deformation Theory

A conical failure mechanism with a central deflection of wo is considered for a circular clampedslab of depth h and radius R subjected to uniform loading q per unit area. The slab is assumedrigid in bending and its in-plane and surround stiffness are lumped into an equivalent boundaryspring. Deformation of its radial segment is given in Fig. 4. Let r and y describe radial andcircumferential direction along the lateral plane respectively and b be angular rotation of theradial segment in vertical plane. Concrete is assumed to be perfectly plastic in compression andcracks whenever strains become tensile at a yielding section.From Fig. 4,Displacements

w ¼ wo 1�r

R

� �and ur ¼ db � r

w2o

2R2assuming db small

uy ¼ 0,

ba ¼ �wo

Rand bb ¼

wo

R.

Along the radial lines,

ey ¼1

r

duy

dyþ

ur

r¼

db

r�

w2o

2R2, (1)

R

a

a ow

b

Edge

�

�

�

�

b

centre

r , u

Fig. 4. Deformation of the radial segment.

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ky ¼ �1

r

dw

dr¼ �

1

r

d

drwo 1�

r

R

� �h i¼

wo

rR; kr ¼ 0. (2)

The neutral axis distance factor, x, which measures the distance above the mid-surface to theintersection between the rigid slab parts, would be

xb ¼db

hbb

¼ �db

hwo

R

¼Rdb

hwo

and xa ¼ �Rda

hwo

, (3)

xy ¼ey

hky¼

db

r�

w2o

2R2

h1

rRwo

� � ¼1

h

Rdb

wo

�wor

2R

� �¼ xb �

wor

2Rh. (4)

Geometric compatibility:Let e be radial movement of boundary spring due to membrane force effect, as shown in Fig. 5.

Then,

½ðR � da � dbÞ þ e�2 þ w2o ¼ R2.

Neglecting higher orders,

da þ db � e ¼ w2o=2R,

�xa þ xb �e

hðwo=RÞ¼

wo

2h. (5)

Let the dimensionless flexibility parameter of the restraint be defined as

f ¼Rf c

2hS; where

1

S¼

R

Echþ

R

Es

Pas=Ri

. (6)

Here we are adding the flexibilities due to (i) the concrete slab itself considered as a disc ofradius R and (ii) the effect at radius R of circular steel hoops in an edge surround outside R, eachhoop of area as and radius Ri. Other sources of boundary stiffness could be incorporatedappropriately in place of the second term (Fig. 5).

Initial

a

R

�

� Final

e

b

Fig. 5. Geometric compatibility.

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Considering tensile membrane force positive, radial displacement due to lateral flexibility,

e ¼ �Na

S.

Let non-dimensional stress resultants for membrane force and bending moments respectively be

n ¼N

hf c

and m ¼M

h2f c

.

By substitution,

�xa þ xb � 2fhna

wo

¼wo

2h. (7)

2.1. Yield criteria and flow rule for Deformation Theory

The compressive membrane force corresponding to rotation about mid-depth, No, and theultimate moment corresponding to zero normal force, Mo, may be respectively defined as

No ¼ 12

hf c � Ohf c, (8)

Mo ¼ Od

h�

1

2O

� �h2f c, (9)

where

O ¼asf y

hf c

is mechanical reinforcement degree of the slab.

Yield criteria may be written as, Braestrup and Morley [14]:For positive bending at the centre,

b40 : f ðn;mÞ ¼ m � mo þ n no þ12

n�

p0. (10)

For negative bending at support (or edge),

bo0 : f n;mð Þ ¼ �m � m0o þ n n0

o þ12

n�

p0. (11)

The yield condition for flow theory has vectors d

; h b

parallel to the normal to the yield surface, inwhich superposed dots imply differentiation with respect to some monotonically increasingfunction of physical time, Braestrup [8]. However, in total strain Deformation Theory, the totalvectors d; hb would be parallel to the normal qf =qn, qf =qm.So,

dhb

¼qf =qn

qf =qm¼ x. (12)

Hence,

xa ¼ �na � n0o at the edge, (13)

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xb ¼ nb þ no at the centre, (14)

xy ¼ ny þ no for circumferential bending 0prpR.

Substituting Eq. (4) above

ny ¼ xb �wor

2Rh� no.

Substituting Eq. (14) above,

ny ¼ nb �wor

2Rh,

ny ¼ nb; at r ¼ 0,

ny ¼ nb �wo

4h; at r ¼ R. (15)

Substituting Eqs. (13) and (14) into Eq. (7)

no þ n0o þ nb þ na 1þ

2fh

wo

� �¼

wo

2h. (16)

By horizontal equilibrium,

d

drðrnrÞ � ny ¼ 0,

nr ¼ nb �wor

4Rh. (17)

With boundary conditions nr ¼ na at r ¼ R,

nb ¼ na þwo

4h. (18)

Substituting Eq. (18) into Eq. (16),

2na 1þfh

wo

� �¼

wo

4h� ðno þ n0

oÞ,

na ¼

wo

4h� ðno þ n0oÞ

2 1þfh

wo

� � , (19)

nb ¼

wo

4h� ðno � n0oÞ

2 1þfh

wo

� � þwo

4h. (20)

Evidently, when wo ¼ 0,

na ¼ nb ¼ 0.

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At the edge, bo0 : f ðn;mÞ ¼ �ma � m0o þ naðn

0o þ

12

naÞ ¼ 0,

mo ¼ �m0o þ naðn

0o þ no=2Þ.

Substituting from Eq. (13),

ma ¼ �m0o þ

12ðx2a � n0

o2Þ. (21)

Similarly at centre, b40,

mb ¼ mo þ12ðn2o � x2bÞ. (22)

2.2. Equilibrium considerations

Radial moment, mr may be determined by, as given and solved by Braestrup and Morley [14],

d2

dr2ðrmrÞ �

dmy

drþ

1

h

d

drrnr

dw

dr

� �þ

qr

h2f c

¼ 0

with the solution,

mr ¼ �1

6

qr

h2f c

�1

3

r2

R2

wo

h

� �2

þr

R

wo

h

1

2no þ nb

� �þ mb. (23)

At r ¼ R, mr ¼ ma,

1

6

qR

h2f c

¼ ma þ mb �1

3

wo

h

� �2

þwo

h

1

2no þ nb

� �.

Substituting from Eqs. (21) and (22),

1

6

qR2

h2f c

¼ mo þ m0o þ

1

2ðn2o þ n0

o2Þ �

1

2ðx2a þ x2bÞ �

1

3

wo

h

� �2

þwo

h

1

2no þ nb

� �. (24)

Substituting appropriately Eqs. (13), (14) and (20) above, see Annex A,

1

6

qR2

h2f c

¼ mo þ m0o �

11

96

wo

h

� �2

þwono

4h�

wo=4h � ðno þ n0oÞ

2ð1þ fh=woÞ

� �2

þ3w2

o=16h2 � ðwo=hÞðno þ n0oÞ þ ðno þ n0

oÞ2

2ð1þ fh=woÞ. ð25Þ

This is the elastic–plastic equation according to Deformation Theory and should hold true forloading history with increasing membrane force in circular slabs if the basic assumptions arereasonably correct.

2.3. Rigid plastic equation

Braestrup and Morley [14] suggest the following strain-rate solution for restrained circularconcrete slabs:

1

6

qR2

h2f c

¼ mo þ m0o þ

1

4ðno þ n0

oÞ2þ

wo

4hðno þ n0oÞ þ

5

48

wo

h

� �2

� ½a expð�wo=fhÞ � f=4�2,

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where a is a constant depending upon boundary condition. For rigid plastic solution, f ¼ 0,which gives

1

6

qR2

h2f c

¼ mo þ m0o þ

1

4ðno þ n0

oÞ2�

wo

4hðno þ n0oÞ þ

5

48

wo

h

� �2

. (26)

It should be noted that for f ¼ 0 and wo=h ! 0; i.e. just before the onset of the membrane actioneffect, Eqs. (25) and (26) give same values irrespective of the theories considered:

1

6

qR2

h2f c

¼ mo þ m0o þ

1

4ðno þ n0

oÞ2.

Eqs. (25) and (26) are plotted for different degrees of restraint in Fig. 6 with the following valuesassumed to correspond experimental study described later:

R ¼ 470mm; f c ¼ 15MPa,

f y ¼ 230MPa; Ec ¼ 30GPa,

Ey ¼ 200GPa; h ¼ 40mm.

Reinforcement: mesh of 2.45mm diameter at 52mm c/c in tension face with 4mm clear cover.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1

wo /h

qR

2 /fch

2

Strain-rate rigid plastic solution

φ = 0.01

φ = 0.1

φ = 0.5

φ = 1.0

φ = 10.0

Yield line theory prediction

φ = 0

Fig. 6. Plot of Deformation Theory solution and strain-rate rigid plastic solution for different restraint stiffness (for

other parameters, see text).

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These give the following values essential for plotting Fig. 6:

mo ¼ 0:044964; m0o ¼ 0:0,

no ¼ 0:430529; n0o ¼ 0:5.

Fig. 6 shows that, for a circular concrete slab, the elastic–plastic Deformation Theory load-deflection curves do not touch the curve for Strain-Rate Rigid Plastic Theory, in contrast to thestrip case, Eyre [13]. Therefore, the method described therein to define explicit safe load may notbe applicable for circular slabs. It may be noted that both the axes are changed into non-dimensional parameters. Fig. 7 shows the same for slabs restrained by 2-hoops and 6-hoops of f16mm bars around the slab perimeter, again values useful for experiments described later. Theslabs were reinforced with those steel hoops placed at mid-depth, to provide axisymmetric andcalculable boundary restraint stiffness.An interesting feature of the circular case is that, unlike in strip case, the radial membrane force,

at a particular deflection magnitude, is not constant throughout the span. It varies linearly, as maybe noted from Eq. (17). Further, for a particular deflection, membrane force is maximum at theedge (na) and minimum at the centre (nb) and their difference is always a quarter of the currentdeflection parameter, na � nb ¼ �ðwo=4hÞ. (Negative right-hand side means higher compressivemembrane force at the edge). Similar remarks apply to ny, from Eq. (15).It may be suggested that, once the slab’s geometry deviates from the strip type, the membrane

force will vary along the span, with the least magnitude at the geometric centre. Different radialpoints will reach their respective maxima at different deflection values. These individual points

0

0.1

0.20.3

0.4

0.5

0.60.7

0.8

0.9

11.1

1.2

1.3

1.41.5

1.6

1.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

wo /h

qR

2 /fch

2

2 Hoops (φ = 0.301)

(φ = 0)

6 Hoops (φ = 0.123)

Strain-rate rigid plastic solution

Yield line theory prediction

Fig. 7. Plot of Deformation Theory solution and strain-rate rigid plastic solution for 2-hoops and 6-hoops restraint

cases.

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will behave under Deformation Theory first until they reach the maximum membrane force andthen will switch onto Strain-Rate Theory at their corresponding different central deflection values.In strip case, the entire slab reaches its maximum membrane force at a single deflection value.

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6wo /h

N/N

o

nb

na

Region behaving entirely under Deformation

Theory

Max. membrane force deflection value at the edge

Max. membrane force deflection value at the centre

Fig. 8. Membrane force at edge, na, and at centre, nb by Deformation Theory for 2-hoops case (f ¼ 0:301).

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4wo /h

N/N

o

nb

na

Max. membrane force deflection value at the centre

Max. membrane force deflection value at the edge

Region behaving entirely under Deformation

Theory

1.5

Fig. 9. Membrane force at edge, na, and at centre, nb by Deformation Theory for 6-hoops case (f ¼ 0:123).

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Figs. 8 and 9 show the variation of na and nb for 2-hoops and 6-hoops cases, respectively andthe region of application for the two flow theories at local points in the slab. The centre reaches itsmaximum first and the edge, the last, according to the Deformation Theory. Therefore, the entireslab behaves according to the Deformation Theory until the centre reaches the maximum. Beyondthat, the behaviour cannot be predicted with certainty. The figures below relate the fact, andpossibly for this reason, the elastic–plastic Deformation Theory load-deflection curves do nottouch the curve for Strain-Rate Rigid Plastic Theory in Figs. 6 and 7, in contrast to the strip case,Eyre [13].

To find an explicit solution in the region where the slab behaves under both the theories, andthe behaviour cannot be confidently predicted, is complicated and intractable. It may be wise toconsider the region behaving entirely under Deformation Theory for the purpose.The maximum membrane force condition for the centre from Eq. (20) may be derived as

dnb

dðwo=hÞ¼

d

dðwo=hÞ

wo

4h� ðno þ n0oÞ

2 1þfh

wo

� � þwo

4h

0BB@

1CCA ¼ 0,

leading to

wo

h

� �maxF

¼ �fþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif½fþ 4ðno þ n0oÞ�

3

r. (27)

This formula gives an explicit deflection value dependent upon restraint stiffness parameter f andmaterial properties only. This, when substituted into Eq. (25), will give the explicit load capacity

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

wo /h

N/N

o

na

nb

Fig. 10. Membrane force at edge, na, and at centre, nb for f ¼ 2:0.

ARTIC

LEIN

PRES

S

Table 1

Explicit load comparisons with the Deformation Theory peak load and the yield line theory prediction

f wo

h

� maxF

from

Eq. (27)

Corresponding

explicit load qR2

h2f c

wo

hfor peak load

from Eq. (25)

Peak load from

Eq. (25) qR2

h2Fc

Explicit load %

of peak load

Yield line

prediction qR2

h2f c

Explicit load as

multiple of yield

line theory load

0.01 0.102 1.422 0.04 1.471 96.7 0.2698 5.27

0.1 0.257 1.187 0.16 1.238 95.9 0.2698 4.40

0.5 0.339 0.928 0.32 0.928 99.9 0.2698 3.44

0.6 0.329744 0.889372 0.34 NAa NAa 0.2698 3.30

1 0.255 0.734 0.40 NAa NAa 0.2698 2.72

1.5 0.115877 0.498909 0.42 NAa NAa 0.2698 1.85

2 �0.04686 NAb 0.446 NAa NAa 0.2698

10 �3.237 NAb 0.48 NA NA 0.2698 NA

0.1235 (6-hoops) 0.274 1.158 0.18 1.203 96.2 0.2698 4.29

0.3014 (2-hoops) 0.334 1.020 0.28 1.035 98.5 0.2698 3.78

aNA: wo=h for peak load from Eq. (25) is greater than ðwo=hÞmaxF from Eq. (27). Therefore, peak load cannot be calculated since Deformation

Theory ceases to apply exclusively after maximum membrane force at the centre, which occurs at ðwo=hÞmaxF .bNA: Negative value for ðwo=hÞmaxF from Eq. (27) at such high values of f suggest that the central membrane force at centre is never compressive.

See Fig. 10 for f ¼ 2.

S.K

.D

as,

C.T

.M

orley

/In

terna

tion

al

Jo

urn

al

of

Mech

an

ical

Scien

ces4

7(

20

05

)1

62

9–

16

47

1642

ARTICLE IN PRESS


that we intend to calculate. The load capacities so calculated are compared to traditional yield linepredictions in the following table:The figures in Table 1 encourage the use of the formula and it may be noted that the explicit

load so found is normally above 90% of the theoretical peak load.Experimental verification was carried out, Das [15]. The set-up is schematically shown in

Fig. 11. Fig. 12 gives the comparison between the 2-hoops case and corresponding yield lineprediction. Test on corresponding unrestrained slab is also compared.It is evident from Fig. 12 that the explicit solution slightly overestimates the experimental load

capacity. This may be due to the assumption made that radial span is rigid in bending. Estimate ofrestraint stiffness as per Eq. (6) in the experiment can also be attributed to the limitation in theaccurate prediction. The formulation does not account for the initial deflection due to flexure as

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

wo /h

qR

2 /fch

2

Experimental -Unrestrained slab

Yield line prediction

Experimental - 2 hoops slab

Theoretical - 2 hoops

Fig. 12. Experimental comparisons.

Support Support

Reaction plate Rubber ring

Test slab Water bag

Water in

Floor

Base slab

Fig. 11. Test set-up.

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well. The theoretical curve will shift right by an appropriate amount due to this factor. Fig. 12shows, in the experimental case considered, the amount of shift would be 0.5–0.6 wo=h. Despitethe aforementioned limitations, the theoretical curve closely reflects the circular slab behaviourunder compressive membrane action.

3. Conclusion

In absence of an explicit formula that can calculate the load capacity enhancement due tomembrane action, engineers are encouraged to use the effect as an added factor of safety in theirdesign or assessment works at the expense of construction or strengthening cost. The aboveformulations, Eqs. (25) and (27) for circular slabs, give an explicit load value due to compressivemembrane action and avoid the earlier confusion of guessing central deflection value at which thepeak load would possibly occur. Half of the slab depth was usually regarded as a reasonableguess. It is evident from above calculations that this may not be always true in circular slabs.Besides this, the formulae have two major advantages. They use relevant plastic theory that hasconvincing basis. The inputs required are simply material properties and restraint stiffness(Eq. (27)). The formulae avoid the complication of differential parameters and the load socalculated is explicit.Further we see that the deflection value at which any particular radial point in the slab switches

from Deformation Theory to Strain-Rate Theory is sensitive to the radial position of the pointunlike in strip case. The elastic–plastic Deformation Theory load-deflection curves do not touchthe curve for Strain-Rate Rigid Plastic Theory as described by Eyre [13]. However, the maximummembrane force at centre according to Deformation Theory may still be used.The explicit solution may be compared to available peak load capacity only if the central

deflection at which the peak occurs happens to be less that the deflection value found by Eq. (27),ðwo=hÞmaxF . Else, Eq. (25) ceases to apply after ðwo=hÞmaxF . Where comparable, it may be notedthat the formulae are capable of finding an explicit load that is usually more than 90% of theavailable peak capacity. In Table 1, at f ¼ 0:55, the actual peak load capacity is achieved. Atf ¼ 1:861, the formulae give yield line load capacity and at larger value of f than 1.861, thecentral membrane forces are not compressive and hence the formulae may not be applicable. Thismay be noted in Table 1.The formulae do not account the initial deflection due to flexural bending before onset of

membrane action. However, this will not cause any change in the character of the load-deflectioncurve for Eq. (25) but simply a right hand shift equal to the amount of the initial deflection. Themaximum capacity, which is of interest, will not be altered owing to this shift.Excess load capacities available due to compressive membrane effect are usually ignored. The

formulae presented will help the engineers predicting the capacity explicitly as described in thepaper. Some comparisons of tests and theory are given in Fig. 12. One experimental comparisonwas carried out which substantiated their usefulness. However, more experimental comparisonsshould be carried out to confirm their utility.The radial span is assumed to be rigid in bending. This may slightly overestimate the capacity.

The degree of overestimate may be estimated from a data bank of experimental comparisons,which will need to be further carried out.

ARTICLE IN PRESS


Appendix A

A.1. Detailed simplification of Eqs. (24) to (25), Das [15]:

1

6

qR2

h2f c

¼ mo þ m0o þ

1

2ðn2

o þ n0o2Þ �

1

2ðx2a þ x2bÞ �

1

3

wo

h

� �2

þwo

h

1

2no þ nb

� �. (24)

Substituting from Eqs. (13), (14), (19), and (20) appropriately,

1

6

qR2

h2f c

¼ mo þ m0o þ

1

2ðn2

o þ n0o2Þ �

1

3

wo

h

� �2

þwo

h

1

2no þ

wo

4hþ


2ð1þ fh=woÞ

� �

�1

2�


2ð1þ fh=woÞ� n0

o

� �þ


2ð1þ fh=woÞþ

wo

4hþ no

� �� 2

þ1

2� 2 �


2ð1þ fh=woÞ� n0

o

� �wo=4h � ðno þ n0oÞ

2ð1þ fh=woÞþ

wo

4hþ no

� �� ,

1

6

qR2

h2f c

¼ mo þ m0o þ

1

2ðn2

o þ n0o2Þ �

1

3

wo

h

� �2

þ1

4

wo

h

� �2

þwono

2h

þ1

8

wo

h

� �2 1

ð1þ fh=woÞ�

wo

2h

ðno þ n0oÞ

ð1þ fh=woÞ

� ��

1

2

wo

4hþ ðno � n0

oÞ

� �2

�wo=4h � ðno þ n0oÞ

2ð1þ fh=woÞ

� �2

�w2

o

32h2ð1þ fh=woÞ

þwoðno þ n0

oÞ

8hð1þ fh=woÞ�

wono

8hð1þ fh=woÞ

þnoðno þ n0

oÞ

2ð1þ fh=woÞ�

won0o8hð1þ fh=woÞ

þn0

oðno þ n0oÞ

2ð1þ fh=woÞ�

won0o

4h� non0o.

1

6

qR2

h2f c

¼ mo þ m0o þ

1

2n2o þ

1

2n0o

2�

1

12

wo

h

� �2

þwono

2hþ

1

8

wo

h

� �2 1

ð1þ fh=woÞ

�wo

2h

ðno þ n0oÞ

ð1þ fh=woÞ

� ��

1

2

wo

4h

� �2

þ 2wo

4hðno � n0oÞ þ ðn2

o � 2non0o þ n0o

2Þ

� �


2ð1þ fh=woÞ

� �2

�won0

o

4h� non0o

þ

�w2

o

32h2þ

wo

8hðno þ n0oÞ �

wono

8hþ n2

o=2þ non0o=2�won0o8h

þ non0o=2þ n0o

2=2

ð1þ fh=woÞ,

1

6

qR2

h2f c

¼ mo þ m0o þ

1

2n2o þ

1

2n0o

2�

1

12

wo

h

� �2

þwono

2hþ

1

8

wo

h

� �2 1

ð1þ fh=woÞ

ARTICLE IN PRESS


�wo

2h

ðno þ n0oÞ

ð1þ fh=woÞ

� ��

w2o

32h2�

wo

4hðno � n0oÞ �

1

2ðn2o � 2non0o þ n0

o2Þ


2ð1þ fh=woÞ

� �2

þ

�w2

o

32h2þ1

2ðno þ n0

oÞ2�1

22non0o þ non0

o

ð1þ fh=woÞ

�won0o4h

� non0o.

1

6

qR2

h2f c

¼ mo þ m0o �

11

96

wo

h

� �2

þwono

4hþ

won0o

4h�


2ð1þ fh=woÞ

� �2

þ

�w2

o

32h2þ

w2o

8h2�

wo

2hðno þ n0oÞ þ

1

2ðno þ n0oÞ

2

ð1þ fh=woÞ�

won0o4h

.

1

6

qR2

h2f c

¼ mo þ m0o �

11

96

wo

h

� �2

þwono

4h�


2ð1þ fh=woÞ

� �2

þ3w2

o=16h2 � ðwo=hÞðno þ n0oÞ þ ðno þ n0

oÞ2

2ð1þ fh=woÞ. ð25Þ

Appendix B

B.1. Yield line theory prediction

For a circular concrete slab of radius R with positive moment capacity, Mo and negativemoment capacity M 0

o loaded with q per unit area, Johansen [19],

Mo þ M 0o ¼

qR2

6.

Dividing both sides by h2f c,

Mo

h2f c

þM 0

o

h2f c

¼qR2

6h2f c

,

1

6

qR2

h2f c

¼ mo þ m0o.

References

[1] Chana PS, Desai SB. Membrane action and design against punching shear. The Structural Engineer 1992;70(19).

[2] Ontario Ministry of Transportation and Communication. Ontario Highway Design Code, Ontario, Canada, 1983.

ARTICLE IN PRESS


[3] Mufti A, et al. Steel-free concrete bridge deck—the salmon river project. Proceedings CSCE Annual conference in

Edmonton.

[4] Wood RH. Plastic and elastic design of slabs and plates. Thamed and Hudson; 1961.

[5] Christensen KP. The effect of the membrane stresses of the ultimate strength of interior panel in a reinforced

concrete slab. The Structural Engineer 1963;41(8):261–5.

[6] Morley CT. Yield line theory for reinforced concrete slabs at moderately large deflections. Magazine of Concrete

Research 1967;19(61).

[7] Al-Hassani HM. Behaviour of axially restrained concrete slabs. Ph.D thesis, University of London, England, 1978.

[8] Braestrup MW. Dome effect in RC slabs: rigid plastic analysis. Journal of the Structural Division, ASCE 106, no.

ST6. Proceedings Paper 15501; June 1980. p. 1237–53.

[9] Janas M. Large plastic deformations of reinforced concrete slabs. International Journal of Solids and Structures

1968;4:61–74.

[10] Kemp KO. Yield of a square reinforced concrete slab on simple supports allowing membrane forces. The

Structural Engineer, London, England 1967;45(7):235–40.

[11] Kemp KO, Eyre JR, Al-Hassani HM. Plastic flow rules for use in the analysis of compressive membrane action in

concrete slabs. H. Wood memorial conference. Frame and slab structures. London, England: Butterworths; 1989.

[12] Eyre JR. Flow rule in elastically restrained one-way spanning RC slabs. Journal of Structural Engineering

1990;116(12):3251–67.

[13] Eyre JR. Directs assessment of safe strengths of RC slabs under membrane action. Journal of Structural

Engineering 1997;123(10):1331–8.

[14] Braestrup MW, Morley CT. Dome effect in RC slabs: elastic–plastic analysis. Journal of the Structural Division,

ASCE 106, no. ST6. Proceeding Paper 15502; June 1980. p. 1255–62.

[15] Das SK. Compressive membrane action in circular concrete slabs. MPhil thesis, University of Cambridge,

Cambridge, 2001.

[16] Ockelston AJ. Load tests on three storey reinforced concrete building in Johannesburg. The Structural Engineer

1955;33:304–22.

[17] Ockelston AJ. Arching action in reinforced concrete slabs. The Structural Engineer 1958;36:197–201.

[18] Morley CT. Some experiments on circular concrete slabs with lateral restraint. Warsaw: Euromech Conference;

1974.

[19] Johansen KW. Yield line theory. Circular slabs. London: Cement and concrete association; 1962. p. 75–6

[chapter 4].

[20] Janas M. Arching action in elastic–plastic plates. Journal of Structural Mechanics 1973;1(3):277–93.

Further reading

[21] Park P. Ultimate strength of rectangular concrete slabs under short term uniform loading with edges restrained

against lateral movements. Proceedings of Institution of Civil Engineers, vol. 28. London, England; June 1969.

p. 125–50.

compressive membrane action in circular reinforced slabs

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