8th International Conference on Structures in Fire

Shanghai, China, June 11-13, 2014

A RE-EXAMINATION OF THE MECHANICS OF TENSILE MEMBRANE

ACTION IN COMPOSITE FLOOR SLABS IN FIRE

Ian Burgess*, Shan-Shan Huang

* and Savina Staikova

**

* University of Sheffield, United Kingdom

e-mails: ian.burgess@sheffield.ac.uk, s.huang@sheffield.ac.uk

** TU Darmstadt, Germany

e-mail: sa_vi05@hotmail.com

Keywords: Steel Frames, Fire, Composite Slab, Tensile Membrane Action, BRE Method.

Abstract. This paper presents a re-examination from first principles of the mechanics of tensile

membrane action (TMA) of thin rectangular concrete floor slabs, transversely supported around their

edges. An existing simplified method of assessing the contribution of TMA to the fire resistance of a

composite slab, including unprotected steel downstand beams in its interior area, appears to have some

serious mechanical shortcomings in its fundamental assumptions. This paper describes, and presents

results from, a re-examination of the mechanics of TMA of thin concrete floor slabs in fire conditions,

starting from the same initial state of an optimal small-deflection yield-line hinge pattern. The basic

formulation considers plain flat slabs, but is in no way limited to these, or to isotropic reinforcement. It

is based on a large-deflection plastic analysis. The resulting formulation accounts for the plasticity and

fracture of the reinforcing mesh, which is usually weaker in tension than the slab in which it is embedded,

and the compressive strength of the concrete. It allows the changes in stress patterns around the yield

lines to be monitored, from negligible deflection to complete failure of the slab, and provides a rational

way of predicting when a through-depth tensile crack will occur; in fire conditions this is usually taken as

an integrity failure of the separating function of the floor slab. If necessary the method can then follow

the further development of this cracked mechanism up to full loss of structural load capacity. Like-

against-like comparisons are made against the enhancements predicted by the existing method, and it can

be seen that these are by no means identical; nor are the predictions by one method consistently

conservative relative to the other.

1 INTRODUCTION

The Cardington fire tests [1] conducted in 1995-96 on a purpose-built full-scale composite-framed

building were instrumental in inspiring the upsurge in research interest, which continues to the present, in

the real performance of framed buildings in fires. In the specific context of composite steel-concrete

framing systems the outstanding observation from the six tests in the initial series was that, despite the

fact that steel downstand beams experienced temperatures considerably in excess of their codified critical

temperatures [2], and would have collapsed if tested individually under the same loadings in normal

furnace-test conditions, no composite beam experienced runaway collapse. In the aftermath of the tests it

became apparent that the reason for this apparent enhancement of the strength of the arrays of identical

composite beams which make up floor slabs was the two-way continuity of the concrete slab panels

themselves. The high biaxial curvatures which are engendered by large deflections of heated parts,

effectively vertically supported by cool structure around their edges, cause the appearance of a zone of

‘hydrostatic’ tensile membrane stress in the central area of a panel; this is a two-dimensional analogy to

catenary tension in a cable. However, while a catenary cable requires supports which resist horizontal

pull-in force as well as the vertical load supported, this horizontal reaction in highly-deflected slabs is

Ian Burgess, Shan-Shan Huang and Savina Staikova

provided by a peripheral ring of compressive membrane stress. This combination of membrane stresses is

known as Tensile Membrane Action (TMA), and its existence depends on a panel having good vertical

support around its edges, and on the extent of the deflection of its central region; the panel’s load-carrying

capacity increases with its deflection, subject to the strength of the material of which it is made. In fire

conditions, TMA of the highly deformed concrete slab effectively carries the loading when the strength of

the unprotected downstand steel beam sections has reduced dramatically at high temperatures.

A simplified design method to calculate the strength of a composite slab panel within its allowable

range of deflection, when the strengths of steel downstands have been degraded considerably by high

temperatures, was published by Bailey and Moore [3, 4] of BRE in 2000. This method is based to a very

large extent on a calculation of the enhanced load capacity of concrete slabs at high deflections due to

their membrane strength, which had been published by Hayes [5] in 1968. The method has since then

become widely used in practical fire engineering design in the UK, both in its original form and as a

development published as the software TSLAB [6]. In New Zealand Clifton [7] devised a further variant

of the same rationalization, which is also available as public-domain design software. The European

project FRACOF has recently published reports [8, 9, 10] recommending a design process which is

almost indistinguishable from that given by the generic BRE/Bailey documents. This seems likely to be

adopted in the next round of developments to Eurocode 4 Part 1-2.

The BRE/Bailey method depends directly on the original calculation of enhancement of strength as a

result of increasing deflection, in the form in which it was published by Hayes. It is not possible to

follow the complete sequence of the derivation, since major steps are not reported, but it is clear that

some ad hoc assumptions are made. The method culminates in a deflection-dependent enhancement

factor which multiplies the small-deflection plastic capacity of the slab, as determined by the yield-line

method. Enhancement factors in each of the two principal directions are aggregated from individual

enhancement factors due to the membrane effect and the bending resistance, and then these are combined

in a weighted-mean process, which is hard to explain or to justify, into a single enhancement factor. The

model is based on a slab which has initially formed the optimal yield-line pattern and, on further

deflection, has experienced a through-depth tensile crack across its shorter width at the middle of its

longer length. This is a mechanism which has been observed in tests on loaded thin slabs, and the fact

that the system of forces shown in Fig. 1(a) is used in the calculation indicates that it is the equilibrium of

this cracked system that is being considered. This same assumption was made by Burgess et al. [11]

using different kinematic assumptions. In both cases the enhancement is clearly to the structural load-

carrying capacity after the mid-span tensile crack is in existence, rather than the occurrence of that crack,

which constitutes a compartment integrity failure in the fire case.

Figure 1. Kinematics of (a) Hayes [5] enhancement model and Burgess et al. [7] model.

Since the Hayes calculation was intended only to model structural capacity at ambient temperature it

does not attempt to predict the occurrence of the central through-depth crack, and since the rebar’s

assumed stress-strain curves include no definition of fracture increased deflection will continuously

provide increased enhancement to capacity. The BRE/Bailey method was thus faced with the need to

define a safe limiting deflection at which the through-depth crack still has unbroken rebar crossing it.

(b) Ref. [7]

(a) Hayes [5]

Ian Burgess, Shan-Shan Huang and Savina Staikova

This deflection was rationalised as consisting of two parts; the first due simply to differential thermal

expansion across the depth of the concrete slab, leading to ‘thermal bowing’ under an arbitrary linear

temperature gradient, and the second due to stress-induced mechanical extension of the rebar up to a safe

limiting strain. The safety margin on the latter was originally given by dividing the ambient-temperature

proportional limit strain by 2.4; in the FRACOF recommendations this is reduced to 2.0. However, these

two components are superposed in a totally irrational way; the thermal bowing deflection is based on a

simple beam model which allows horizontal movement of one support, whereas the mechanical strain is

based on a beam which has both ends fixed. These cases simply cannot be superposed.

The work reported in this paper attempts a re-examination of the large-deflection response of a thin

concrete slab, using a yield-line approach to the small-deflection plastic limit as a datum for the behaviour

as deflections increase. This seems quite rational for lightly reinforced slabs, which do not exhibit

tension stiffening and therefore create discrete localised crack patterns which form the yield lines; there is

little incentive for a yield-line pattern to change once it has formed. It is clear that, once a yield-line

pattern has formed at the small-deflection plastic capacity, increased deflection initially simply amplifies

this mode, progressively stretching the rebar (usually mesh) across the widening cracks and changing the

shapes of the concrete compression zones along these yield lines. During this process the bars in either

the x- or y-directions may fracture, the crack-width at which this happens depending on their own

ductility, their positive anchorage points in the concrete, and their bond characteristics. In structural

terms failure may occur when the enhanced load capacity reduces consistently below the applied load; a

temporary reduction which is re-stabilised on further deflection does not constitute structural failure.

Since the method is to be used in the fire context, integrity of the slab as a compartment-separating

element must also be considered. The most severe approach to this limiting condition would be to

assume that integrity is lost when there is no contact between the faces of a crack at any point on the

yield-line pattern. This may be too restrictive, especially for composite slabs cast on profiled steel

decking, and an alternative may be either to specify a minimum acceptable crack width or to identify the

occurrence of the through-depth tensile crack which changes the mechanism, at a specified concrete

tensile strength.

2 EQUILIBRIUM AND KINEMATICS OF YIELD-LINE MECHANISM

A two-way spanning rectangular slab panel of aspect ratio r, which is transversely supported along all

its four edges, is considered. In the present case the slab is considered as isolated (having no continuity

with adjacent panels across its edges). The slab is lightly reinforced with a welded mesh, which for the

purposes of this paper is considered to be isotropic, and the two layers of bars are assumed to lie

effectively at a single mean level within the slab. The transverse loading on the slab is increased until a

plastic yield-line crack pattern forms, in the characteristic arrangement shown in Figure 2.

Figure 2. Small-deflection yield-line mechanism.

rl nl

l

Hogging rotations about edges of panel

Sagging rotations about internal

yield lines

Ian Burgess, Shan-Shan Huang and Savina Staikova

All plastic energy analysis methods are inherently upper-bound, but the optimum yield-line

mechanism, giving the lowest possible failure load intensity, is exact, and is given by:

( √ ) (1)

The general assumptions for the materials involved are that steel rebar across a yield-line only acts

plastically in tension where it is stretched, and that concrete is only active in compression where the

yield-line surfaces overlap, and that in these zones it is at the ultimate stress of the concrete. It is assumed

that the slab facets remain flat but rotate compatibly (creating the same intersection displacement A)

about their respective edge supports. The geometry of the crack opening at certain depths from the top

surface and certain distances from supports, given compatible overlap movements x and y (Figure 3) at

the top surfaces at the slab corners, gives crack opening components at coordinates (x, y) from the slab

corner, and depth z from the top surface, of

(2)

(3)

Figure 3. Crack opening at a certain depth and distance from supports, and corner top surface overlaps.

Since the concrete compression zones must be compatible in the x- and y-projections, and since both

and cause the same lateral deflection at the yield-line intersection, the relationships

and exist.

At any section through the yield lines there can be a combination of concrete in compression, from

the top surface of the slab downwards, and rebar in plastic tension. However the extent of the concrete

compression zone depends on the position and the geometric relationships shown in Figure 3; if the

separation extends from the bottom to the top of the slab at this point there is no compressive block. In

addition, if the reinforcement is located within the compression block at this location, then it is considered

as having no stress; alternatively, if the crack separation at the rebar level exceeds that at which the bars

fracture, then there it clearly has no tensile stress. The key options for concrete compression and steel

tensile yield are shown in Fig. 4, which contains two views of the yield-line crack surface, projected onto

the x-z plane.

Figure 4(a) shows the situation shortly beyond initial yield-line failure, when the concrete stress block

in the central yield line reduces in depth and compressive stress increases at the slab corners. In Figure

4(b) the central compression block has disappeared, which is indicated by a negative value of z2 . Figure

4(c) shows the length xlim,1y beyond which the crack width has become greater than that at which the y-

direction rebar fractures, and xt,1 , marking the point at which the rebar passes into the compression block.

These cases do not cover the entire field of possibilities; under some circumstances the depth z1 of the

compression block may go below the reinforcement depth before any rebar fracture takes place, or before

tt

s

y x

z

y

x

x

y

TOP SURFACE OF SLAB CRACK OPENING AT REBAR LEVEL

Ian Burgess, Shan-Shan Huang and Savina Staikova

compression has ceased in the central yield line. Equally, reinforcement fracture may begin at any stage,

depending on its ductility. A final case of the concrete stress block, not shown in the figure, may occur

when z1 exceeds the depth of the slab, when the stress block becomes trapezoidal.

Figure 4. Key dimensions of concrete compression block and active rebar lengths.

The steel and concrete force components acting on the yield lines are summarised in Figure 5. Since

the reinforcing mesh is aligned with bars in the x and y directions there are two tensile resultants Tx1 and

Ty1 , which are essentially independent of one another, on each diagonal yield line. On the other hand, the

concrete compression block simply presents resolved parts in the x and y directions. On the central yield

line the tensile component Ty2 vanishes instantaneously when the y-rebar fractures; the compression

component Cy2 vanishes when z2 passes from positive to negative.

Figure 5. Force components acting on slab facets.

Equilibrium of the forces and eliminating the concrete shear resultant S gives:

( ) (4)

z1

t

xCA,1

t

z2

Mid-slab yield line

C

Diagonal yield line

S

nl

Ty1

(r/2-n)l

Tx1

Q

z

1

R

-z2

S

Q’

Q”

z1

R’

-z2

R”

xlim,1y

S’

S”

xt,1

Ty2

Mid-span

C2

(a)

(b)

(c)

x1

x

CA,1

xCA,1

x

CA,1

A1y A2

Ian Burgess, Shan-Shan Huang and Savina Staikova

If the plastic strength of the concrete is denoted and the strengths of rebar per unit width of mesh

perpendicular to bars in the x and y directions respectively are and , the forces are:

(5)

(6)

(7)

(8)

(9)

3 SOLUTION FOR DEFLECTIONS AND FORCES

Solution for the amplified yield-line mechanism as the deflection of the central yield line is

incremented from zero has been implemented in a spreadsheet. The force definitions shown in Equations

(5) to (9) are substituted into Equation (4) and expressed as functions of x. The slab width and aspect

ratio, the reinforcement mesh areas in the x and y directions, depth and ductility (fracture strain), and the

strengths of steel and concrete can be changed. The rebar fracture crack widths are based on the defined

fracture strain and a ‘free length’ of bar, which in the simplest case is assumed to be the whole length of

bar in the appropriate direction between adjacent transverse bars. This is logical, since welds at these

points will act as anchors either side of a crack, but is not conservative. If bond is maintained between

the concrete and steel between the anchor points and the crack faces, then the limiting crack-width will

decrease. Some investigation of the effects of using various debonding theories has been done, but will

not be reported in this paper.

In the spreadsheet values of x are calculated as the deflection increases. It is necessary to test these

values for compatibility in a series of 30 scenarios, assuming different concrete stress block shapes and

different reinforcement fracture conditions. This is typical of limit-state analysis; only one scenario

produces exact results, while all others are inherently upper-bound. The cases which are tested are listed

in Table 1.

Table 1. Cases of compression block and rebar assumptions to be tested.

Compression block Reinforcement mesh fractured

None Central y Diag. x Diag. y Central +

diag. x

Diag. x

and y

Full above mesh a1 a1’ a1* a1** a1*’ a1***

below mesh a2 a2’ a2* a2** a2*’ a2***

Triangular above mesh

below mesh

b1 b1’ b1* b1** b1*’ b1***

b2 b2’ b2* b2** b2*’ b2***

Trapezoidal c1 c1’ c1* c1** c1*’ c1***

When x has been evaluated for the correct case it is then possible to use Equations (5) to (9) to

calculate the magnitudes and positions of each of the internal force components at any displacement. The

in-plane bending moments about axes through the points indicated by Q, R and S in Figure 5 can then be

calculated. There is no net force across any of the three lines (Q’Q”, R’R” or S’S”) marked through these

points, and so there are elastic in-plane linearly-varying stress distributions across each of these lines. As

the deflection of the slab increases the maximum value of the concrete tensile stress also increases, and at

some point this will initiate the through-depth tensile crack which constitutes an integrity failure in the

fire limit state.

With respect to the basic enhancement of capacity of the slab with deflection, a simple plastic work

balance method, previously used in [11], is applied. The forces calculated in Equations (5) to (9) all

correspond to plastic action in either rebar or concrete. The corresponding movement distances can be

Ian Burgess, Shan-Shan Huang and Savina Staikova

calculated at the positions of the components; for reinforcement these are the centres of the appropriate

lengths in x or y over which rebar orthogonal to these lengths acts; for concrete they are the centroids of

the appropriate compression stress blocks.

The external work (or loss of potential energy) of the uniformly distributed transverse loading, of

intensity p, is expressed in the same way as for small deflections:

(

) (10)

Since rigid-perfectly plastic behaviour is being assumed for the reinforcing mesh,

(11)

Thus the load capacity of the slab, at any deflection is therefore:

[ (

)]⁄ (12)

4 APPLICATION OF THE PROCEDURE

In order to demonstrate the application of the procedure outlined above, the results for an example

slab will now be considered. The essential data for the slab is given in Table 2 below.

Table 2. Data for the slab example.

Concrete

Minor length l = 6m fc = 30Mpa

Major length rl = 9m

Thickness t = 130mm

Steel

Mesh depth t = 50mm fy = 500Mpa

Mesh area x-dir. = 142mm2/m Ductility = 10%

Mesh area y-dir. = 142mm2/m

Mesh bar spacing = 200mm

The variation of the internal force components on the yield lines is shown in Figure 6(a). The main

phases of behaviour can be seen from the changes in the forces depicted here. At A/t=0.087 the

concrete stress block (C2) on the central yield line vanishes, while the reinforcement across this yield line

(Ty2) remains intact until A/t=1.698, when all the bars across this yield line fracture abruptly. At

A/t=2.497 the y-direction reinforcement across the diagonal yield lines begins to fracture progressively

from the intersections towards the slab corners, followed by the x-direction reinforcement at A/t=2.737.

As deflection increases further it can be seen that all non-zero forces reduce as the remaining

reinforcement is reduced.

Figure 6(b) shows the maximum in-plane tensile bending stresses in the slab at the 3 sections Q’Q”,

R’R” and S’S” as the deflection increases. It can be seen that all three increase until the simultaneous

fracture of the central yield-line reinforcement, with the stress at Q’Q” slightly higher than that at R’R”.

However, there is clearly a value of the maximum tensile stress at which tensile fracture happens and a

through-depth tension crack is initiated. In Eurocode 2 [12] the tensile strength of concrete is related to

its compressive strength largely in terms of tabulated data, which in general terms give values between

6% and 11% of the cylinder strength. For the present example, this range would give a surprisingly low

permissible deflection if used as a limit to the permissible range.

l

rl

Ian Burgess, Shan-Shan Huang and Savina Staikova

Figure 6. (a) Force component values on slab facets; (b) Maximum concrete tensile stresses on sections Q’Q” and

R’R” in trapezoidal facet, S’S” in triangular facet.

In Figure 7(a) the load capacity enhancement factor for the example slab is plotted against the relative

deflection. It can be seen that, for the parameter values defined in Table 2, the enhancement is extremely

close to that given by the generic Hayes or BRE method at the stage before the rebar across the central

yield line has fractured. Beyond this point some strength is gradually regained until the mesh across the

diagonal yield lines begins to fracture, beyond which the load capacity is progressively lost with further

deflection. The limiting deflection prescribed by the generic BRE method, assuming a temperature

difference of 770C between the top and bottom slab surfaces, is marked at A/t=3.0005. Since no

thermal effects have been considered in the enhancement calculation, the BRE limiting deflection bvased

only on the rebar strain limit, without the thermal deflection component, has also been marked.

Figure 7. (a) Load capacity enhancement factor for the example slab (Table 2); (b) Comparison of enhancement

factors for slabs of aspect ratio 1.0, 1.5 and 2.0 with their BRE method enhancement factors.

-100

0

100

200

300

400

500

0 1 2 3 4

Forc

e (k

N))

Deflection/thickness (dA/t)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0

Cap

acit

y e

nh

ance

men

t fa

ctor

Deflection/thickness (A/t)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0

Cap

acit

y e

nh

ance

men

t fa

ctor

Deflection/thickness (A/t)

C

1

Ty1

Tx

1 C

2 T

y2

S

(a)

0

1

2

3

4

5

0 1 2 3 4

Str

ess

(N/m

m2)

Deflection/thickness (A/t)

Q'Q" (b)

R’R”

Q’Q”

(a)

BRE limit

BRE limit

(no thermal)

(b)

r=1.0

r=1.5

r=2.0

r=1.5 r=1.0 r=2.0

Ian Burgess, Shan-Shan Huang and Savina Staikova

In Figure 7(b) the enhancement characteristics for slabs of aspect ratio 1, 1.5 and 2 are compared,

with the Hayes/BRE enhancement curves for the same aspect ratios. It is notable that, although the initial

enhancement curves for r=1.5 are very close, those for r=1.0 and r=2.0 differ considerably. For a square

slab (r=1.0) the BRE method’s enhancement is greater than that of the present approach, whereas for

r=2.0 it is lower. Hence, the BRE method cannot be characterised either as consistently conservative or

unconservative in terms of the enhancement of yield-line capacity that it predicts with deflection.

5 CONCLUSION

This paper has shown the principles of a simplified method of representation of the load-capacity

enhancement in thin slabs due to tensile membrane action at high deflections. Although the major current

use of this phenomenon in structural engineering is to increase the capacity of composite slabs in fire

conditions, the work presented is not directly temperature-related. Heat transfer may affect the results if

the reinforcing mesh heats by conduction to a temperature at which its strength is reduced significantly,

before initial yield-line failure occurs. Once the yield-line cracks have formed, the exposed bars crossing

the cracks will increasingly be affected by convection and radiation; the extent of this heating will depend

on the time of exposure, the width of the crack and the vertical position of the mesh in the concrete.

There is clearly more work to be done before the effect of this heating of reinforcement can be included

logically. Before this is done it may be necessary to represent the bonding of bars either side of a crack in

a more realistic way. Since TMA is of interest when attached steel downstand beams are used across the

interior areas of the slab without fire protection, the effect of rapid heating of these beams, both on the

form of the yield-line mechanism and on the TMA itself, is currently being investigated.

However, the like-against-like comparisons made here have shown that the existing simplified

method, based on Hayes’s work in the 1960s and Bailey’s extensions in the early 2000s, which includes

seemingly illogical assumptions in its treatment of enhancement of load capacity and in its assessment of

the limiting deflection, is inconsistent with the treatment of TMA presented in this paper, which is based

on well-established principles of solid mechanics.

REFERENCES

[1] Newman, G. M., Robinson, J. T. and Bailey, C. G. (2006), Fire Safe Design: A New

Approach to Multi-Storey Steel-Framed Buildings, Second Edition, SCI Publication P288,

The Steel Construction Institute, UK

[2] CEN, EN 1994-1-2:2005: Eurocode 4. Design of composite steel and concrete structures.

Part 1-2: General Rules - Structural fire design. European Committee for Standardization

(CEN), Brussels, 2005.

[3] Bailey, C.G. and Moore, D.B., “The structural behaviour of steel frames with composite

floor slabs subject to fire - Part 1 Theory”, The Structural Engineer , 78 (11), (2000) pp 19-

27.

[4] Bailey, C.G. and Moore, D.B., “The structural behaviour of steel frames with composite

floor slabs subject to fire - Part2: Design”, The Structural Engineer, 78 (11), (2000) pp28-33.

[5] Hayes, B., “Allowing for membrane action in the plastic analysis of rectangular reinforced

concrete slabs”, Magazine of Concrete Research, 20 (65), (1968) pp205-212.

[6] Simms, W.I. and Bake, S., TSLAB V3.0 User Guidance and Engineering Update, SCI

publication P390, 2010.

[7] Clifton, C., Design of composite steel floor systems for severe fires, HERA publication,

Manaukau City, New Zealand (2006).

[8] Vassart, O. and Zhao, B., FRACOF: Fire resistance assessment of partially protected

composite floors: Engineering Background, Arcelor/Mittal & CTICM, 2011.

Ian Burgess, Shan-Shan Huang and Savina Staikova

[9] Vassart, O. and Zhao, B., FRACOF : Fire resistance assessment of partially protected

composite floors: Design guide, Arcelor/Mittal & CTICM, 2011.

[10] Vassart, O. and Zhao, B. (2013), Membrane action of composite structures in case of fire,

ECCS TC3 Report no. 132, 2013.

[11] Burgess, I.W., Dai, X. and Huang, S.-S., 'An Alternative Simplified Model of Tensile

Membrane Action of Slabs in Fire', Proc. Applications of Structural Fire Engineering,

Prague, Czech Republic, (2013) pp361-368.

[12] CEN, EN 1992-1-1:2004: Eurocode 4. Design of composite steel and concrete structures.

Part 1-1: General Rules and rules for buildings. European Committee for Standardization

(CEN), Brussels, 2004.