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3rd R.N. Raikar Memorial International Conference and Gettu-Kodur International Symposium Paper Number XX on Advances in Science and Technology of Concrete, 14-15 December 2018, Mumbai, India

The Effects of Continuity on Compressive and Tensile Membrane Action in Lightly Reinforced Concrete Slabs

Ian Burgess

Department of Civil & Structural Engineering, The University of Sheffield, United Kingdom

Billy Chan Arup Ltd, Hong Kong

ABSTRACT

This paper presents analytical approaches to the behaviour at finite deflections of rectangular lightly reinforced concrete slab panels which are subject to different degrees of restraint across their perimeter supporting beams. This edge restraint derives from the panel’s location within the whole floor system of the building storey. The most extreme condition occurs when the panel under consideration is surrounded on all sides by other panels forming a continuous floor; in this case each of the panel’s edges is restrained in terms of both rotation about the edge and in terms of horizontal movement normal to the edge. Interest in these conditions originated after the publication of the new kinematically admissible approach to tensile membrane action by the first author, which produces more consistent and reasonable solutions for large-deflection behaviour than currently accepted methods which are mainly used in dealing with composite slabs in fire conditions. In this method the large-deflection kinematic conditions of the slab facets created by initial yield-line failure are maintained, and the fracture ductility of reinforcing mesh crossing yield lines is used to monitor the progressive fracture of the mesh in these cracks. This progressive fracture causes the enhancement of load capacity to have natural limits, depending largely on the bond characteristics of the bars at a discrete crack-face.

The method, which had previously been used only for isolated panels without edge-restraint, has been extended to model cases in which pairs of parallel boundary edges may have complete restraint, rotational restraint, or may be free to rotate and translate. While rotational restraint alone has only a small effect in strengthening the slab, and tensile membrane action produces similar enhancements of capacity to those in simple slabs, restraint to edge translation causes compressive membrane action to occur in the early stages of the slab deflection. This effect increases the load capacity of the slab almost instantaneously, usually by a large amount. However the effect is unstable, and further finite deflection of the slab causes the load capacity to decline rapidly; at some level of deflection tensile membrane action may begin to enhance the load capacity, provided that there is still sufficient intact mesh across the yield line cracks. The paper illustrates the effects of the different combinations of boundary conditions on the resistance of slab panels.

Comparison with limited experimental evidence suggests that further development of the method might include conservative assumptions to estimate the restraint stiffnesses against translation at the slab edges.

Keywords: compressive membrane action, tensile membrane action, concrete, slabs, yield line theory

1 INTRODUCTION

The Cardington fire tests (Kirby, 1999), 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 (Newman et al., 1999) and would have collapsed if tested individually under the same loadings in normal furnace-test conditions, no composite panel experienced runaway collapse. In the aftermath of the tests it became apparent that the reason for this apparent enhancement was the two-way continuity of the concrete slab panels themselves. The high biaxial curvatures which were generated by large deflections of heated panels, 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 is provided by a peripheral ring of compressive membrane stress in highly deflected slabs. This combination of membrane stresses is known as

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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 constructed.

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 (2000a, 2000b) of BRE. This method is based mainly 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 (1968). The method has become widely used in practical fire engineering design in the UK, both in its original form and in software applications. In New Zealand Clifton (2006) 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 (Vassart & Zhao, 2011a, 2011b) recommending a design process which is almost indistinguishable from that given by the generic BRE/Bailey documents.

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. The model is based on a slab which has initially formed its optimal yield-line pattern, which is retained as deflections increase. The method calculates 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, into a single enhancement factor. The limit of capacity is given by prediction of the occurrence of a through-depth tensile crack across the shorter width at the middle of its longer length. This is a mechanism which has been observed in tests on loaded thin slabs. The assumption of a mechanism based on the optimal yield line pattern was made by Burgess (2017) using different assumptions based on the finite-deflection kinematics of the system of slab facets. This seems quite rational for lightly reinforced slabs, which do not exhibit tension stiffening and therefore create discrete localized crack patterns which form the yield lines; there is no 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-stabilized on further deflection does not constitute structural failure. Since the main motivation for the development is 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 thin rectangular slab panel of aspect ratio r, which is vertically supported along all its four edges, is considered. In the basic 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 Fig. 1.

Figure 1. Small-deflection yield-line mechanism.

rl nl

l

Hogging rotations about edges of panel

Sagging rotations about internal

yield lines

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The optimum yield-line mechanism, giving the lowest possible failure load intensity, is exactly given by:

This can be found either by infinitesimal-deflection equilibrium analysis, or by virtual work. The general assumptions for the materials involved are that steel rebar only acts plastically in tension where it is stretched across a yield-line, and that concrete is only active in compression where the yield-line surfaces overlap. In these zones it acts at the ultimate compressive strength 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

the supported edges, given compatible overlap movements x andy (Fig. 2) 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

2x

xu z

(2)

2

2y

yv z

(3)

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

𝜃 and ∅ cause the same lateral deflection 𝛿𝐴 at the yield-line intersection, the relationships ∆𝑦 = 2𝑛∆𝑥 and ∅ = 2𝑛𝜃 exist.

Figure 2: Geometry of diagonal yield-line crack opening. (a) Crack opening at rebar level; (b) Top surface of slab, including rigid-body movements of triangular and trapezoidal slab facets.

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 plastic tension from rebar. However the extent of the concrete compression zone depends on the position and the geometric relationships shown in Fig. 2; if the separation extends from the bottom to the top of the slab at this point there is no compressive stress 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 it clearly has no tensile stress at this point. The key options for zones of concrete compression and steel tensile yield are shown in Fig. 3, which contains three views of the yield-line crack surface, projected onto the x-z plane.

211 1 3

2n r

r (1)

Top surface (corner)

(b) Plan view (a)

y

x y

x

t t

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Figure 3. Key dimensions of concrete compression block and active rebar lengths. Fig. 3 (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 Fig. 3 (b) the central compression block has disappeared, which is indicated by a negative value of z2. Fig. 3 (c) shows the length xlim,1y beyond which the crack width is 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; the depth z1 of the compression block may be below the reinforcement depth before any rebar fracture takes place, or before 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. Different continuity possibilities for the panel under consideration. As a further extension of the basic approach for lightly reinforced concrete slab panels, this paper now

looks at the effect of continuity with adjacent panels on their large-deflection behaviour under load. Boundary conditions are significant in dictating the slab’s initial yield-line capacity, and they will have a continued influence at finite deflections. The continuity assumptions of a representative panel represent various situations of panels within a uniformly loaded floor, such as those illustrated in Fig. 4. In terms of an analysis on an individual panel (the lighter-shaded panels in the cases of Fig. 4) in a design situation the relevant continuity conditions depend very much on the position of the panel within the whole floor layout. For a corner panel a conservative design approach would be to consider it as completely isolated (Fig. 4 (a)). For panels containing one perimeter edge the corresponding approach considering a symmetric yield-line mechanism would be to assume continuity across the edges perpendicular to the perimeter (Figs 4 (b & c)). An internal

z1

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Mid-slab yield line Diagonal yield line

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(b) Short-edge continuity

(c) Long-edge continuity (d) All-edge continuity

(a) Isolated panel

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panel could be given continuity across all edges (Fig. 4 (d)). Continuity may either be purely rotational, in which case only bending occurs across all yield lines, or it may include the axial restraint provided by a large continuous area of surrounding concrete slab. Only bisymmetric continuity conditions are considered on edges parallel to either the x- or y-axes of the panel. It is worth mentioning that slab continuity is expressly excluded from all the methods developed from Hayes’s treatment.

The effect of in-plane restraint

After Ockleston’s (1955) full scale test, several research studies on laterally restrained reinforced concrete slabs were conducted, the most prominent publications being by Wood (1961) and Park (1964, 1965 and 2000). Wood developed an approach to calculate the enhancement capacity of a reinforced concrete slab due to the two-dimensional arching effect induced in the slab by restraint to outward movement of the lower part of the slab cross-section at its edges. At almost the same time Park (1964) carried out numerous slab tests to investigate CMA; these were mainly of doubly reinforced plain concrete slabs. Park recognized the effect of the stiffness of the panel boundary, and carried out a series of tests on slab panels of various dimensions and reinforcement arrangements, including continuity outside the panel supports. He also developed an analytical approach to estimate the enhancement of capacity of the slab, which was later extended and refined by Park and Gamble (2000). This method has been the basis for approaches applied to different applications; for instance, in utilising unreserved load capacity in bridge decks (Kirkpatrick, 1984) and enhancing punching shear strength in slabs (Kuang and Morley, 1993).

3. METHODOLOGY

This paper presents an analytical approach to the large-deflection behaviour and load capacity of transversely loaded lightly reinforced rectangular concrete slab panels with rotational and/or in-plane continuity across their boundaries. This is based on the fundamental approach developed by Burgess (2017). It is important to restate the basic assumptions of this analysis, since it is utilized throughout the new approach to continuity of both types across boundaries:

After a yield-line pattern of discrete plastic hinges has initially developed at small deflection, this pattern does not change as its deflection increases.

Horizontal equilibrium of all of the flat facets of the slab needs to be maintained by the combination of forces across the yield lines within the slab and across the slab edges.

These forces are created:

as tensile forces from intact reinforcing mesh crossing the yield lines at any section where there is a crack separation at the level of the mesh, and the mesh is assumed to act at its yield stress;

as compressive forces from the concrete stress blocks on any yield line, where the concrete acts at its compressive strength.

The zones of concrete contact along any yield line are defined by the kinematics of the slab facets, which rotate compatibly about the slab edges together with some rigid-body movements.

Reinforcement mesh can fracture across any crack which has a separation greater than the fracture crack-width determined from its ductility, its bond characteristics and the spacing between weld-points to transverse bars.

The application of these principles to slabs with only rotational edge-restraint, and to those with both rotational and in-plane edge-restraint, are both considered. However, the basic horizontal equilibrium equations are identical for all cases.

It is necessary to recognise that the optimum yield-line mechanism of a given slab may have its central yield line aligned in either the x- or y-direction, depending on the slab’s aspect ratio or the orthotropy of its reinforcing mesh; these will be designated as either “x-aligned” or “y-aligned” mechanisms. For a slab with isotropic mesh, this yield line is aligned parallel to the longer sides. Obviously, once the alignment of yield lines has been determined, the subsequent equilibrium states could be analysed in a unified fashion by realigning the positional coordinate system so that “x” is always parallel to the central yield line. However, the two alignments are treated as distinct in the formulation. In either case the yield lines are assumed to be discrete, rather than “smeared” with a finite width, and they divide the slab into four separate facets.

Horizontal equilibrium

The first step is to establish the horizontal equilibrium of all these facets under the forces acting on them. All of these in-plane forces are shown in Fig. 5 for x-aligned conditions. Force equilibrium for the trapezoidal facets gives:

1 2 3 1 2 3sin cosy y y y yT T T C S C C (4)

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Figure 5: The horizontal force system between facets along the yield lines and at edges, for x-aligned mechanisms.

Similarly, force equilibrium in the x-direction for the triangular facets gives:

1 4 1 4cos sinx x xT T C S C (5)

Combining (3) and (4) to eliminate S gives:

1 4 1 2 3 1 2 3 4( )cos ( )sin ( )sin cosx x y y y y y xT T T T T C C C C (6)

The basic movements of the facets have been illustrated in Fig. 2, which shows the rotations; for the

facets which rotate about the y-aligned edges, and for the facets which rotate about the x-aligned edges.

Each facet also has a rigid-body movement (x or y ) perpendicular to its supported edge.

The optimal yield-line pattern is determined by minimizing the load capacity over a range of yield-line patterns defined by the value of nx, at a vanishingly small deflection. In some cases a y-aligned yield-line pattern is found to produce lower load capacities than any x-aligned pattern; this follows the slab aspect ratio when the reinforcing mesh is isotropic. Once the yield-line pattern has been created, its alignment and position are fixed, and do not change as the deflection increases.

Figure 6: Concrete stress blocks and the possible rebar forces on the internal yield lines.

The forces which appear in the horizontal equilibrium condition are the resultant compressions from the concrete stress blocks on the internal yield lines, together with the tensions in the x- and y-directions caused by the intact reinforcing bars which cross the yield lines. These are illustrated in Fig. 6, which shows the crack-surfaces of half of one of the trapezoidal facets for an x-aligned mechanism, and the key neutral axis depths z1 and z2. At every increment of deflection it is necessary to satisfy the horizontal equilibrium Equation (3) for the x-aligned mechanism. However the forces within the equation are calculated in different ways, depending on the concurrent condition of the compressive stress blocks and the bars crossing the yield lines:

At zero deflection the stress blocks are of uniform depth (z1=z2). As the deflection increases z2 decreases progressively, and rapidly becomes negative, indicating that contact has been lost across the

C

S

Ty1

Tx1

Q R

S

Q’

Q”

R’

R” S’ S”

Ty2 Cy2

x

y

1

2

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Cx4

Cy3

Tx4

l/2

rl/2

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nxly

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central yield line and an inner portion of the diagonal yield lines; the stress block is then triangular. At the panel

corners the neutral axis depth z1 initially increases, and may go below the level of the mesh (z1>t). In this analysis any mesh within a compressive zone is ignored. Under some circumstances the corner neutral axis depth can descend below the lower surface of the slab (z1>t), causing the stress block to be trapezoidal.

The mesh crossing the yield lines is initially entirely intact, but when the crack-width at the level of the bars in either direction exceeds the fracture crack-width then the corresponding bars fracture. The initial loss of active rebar is usually a simultaneous fracture of all the y-direction bars crossing the central yield line. At some point either the x- or y-direction bars crossing the diagonal yield lines begin to fracture from the yield-line intersection (a progressive “unzipping” effect), and this is followed by progressive fracture of the orthogonal bars. At large deflection the bars in either direction may be completely fractured at the diagonal yield lines.

The exact form of either Equation (6) depends on the current stress block shape and the rebar fracture state, and so in principle these forms need to be stated for all combinations, although for any particular set of slab details only a few of these cases will apply during the increase of deflection. In total 30 combinations exist for each of the x- and y-aligned mechanisms.

Edge Continuity

Only bisymmetric continuity conditions are considered in this paper, and two distinctly different forms of continuity can exist at panel edges. In the former the adjacent panels are allowed to move laterally so that no resultant force is applied to the panel edges. In the other case the lines of edge support are assumed to remain in their original positions so that there is no rigid-body movement of either the panel’s edges or of any of the adjacent panels. These two cases can respectively be considered as having force-controlled and displacement-controlled boundary conditions. The edge forces created in these two scenarios are illustrated in Fig. 8.

Figure 7: Forces and neutral axis depths at the edges of a continuous rectangular slab.

Rotationally restrained edges

With rotationally restrained boundaries It is assumed that the external panels deflect in the same way as the panel under consideration, but that they can move horizontally outwards so that there is no net horizontal force crossing any edge of the slab. Any mesh tension across an edge is balanced by the compression in the concrete stress block (Fig. 8 (b) at the edge. If this mesh fractures, then external slabs move so that this compressive zone vanishes (Fig. 8 (c)). When the reinforcing bars crossing either edge fracture, then both the tension and compression forces across that edge vanish, and the the neutral axis lies at, or beyond, the lower face of the slab.

ly

nxly

Neutral axis

(depth z3 or z4)

Ty3

Cy3

Cx4

Tx4

zi

8

Figure 8: Panel edge forces with: (i) rotational continuity and (b) unfractured mesh, (c) fractured mesh; (ii) axial and rotational continuity and (e) unfractured mesh, (f) fractured mesh. Original states are shown as (a) and (d).

The boundary forces are irrelevant to horizontal equilibrium, since they are always in balance for this

boundary condition. This does not imply that the edge forces play no part in either the initial yield-line load or the enhancement due to finite deflection. The edge yield lines are clearly involved in the equilibrium at any displacement, although this involvement ends when the edge reinforcement fractures.

4. APPLICATION OF THE MODELS

The method has previously been tested against the current methods, which are all based on the work of Hayes. This has shown that the enhancement of load capacity with deflection differs from predictions given by these methods. There is also the major difference that mesh will fracture abruptly along yield lines orthogonal to the edges of the slab, as well as progressively along diagonal yield lines, both of which can cause the enhancement to be lost. The current methods only deal with isolated panels without continuity across their edges, and so there can be no similar comparisons for cases with edge continuity. It is, however, possible to use the new development to compare the effects of different parameters on the predicted behaviour.

The effects of rotational continuity

The well-documented Garston test conducted by Bailey (2001) at BRE, Garston, was an ambient-temperature slab test intended to simulate the behaviour of a Cardington corner bay at the stage when its central attached unprotected downstand steel beam had lost nearly all of its strength at very high temperatures. The Garston test panel was 9500 mm x 6460 mm in overall dimensions (aspect ratio 1.4706), and 120 mm thick. It had an A142 smooth mesh (isotropic with 6 mm bars at 200 mm spacings) placed at a 69mm effective depth. The concrete had compressive strength of 52 MPa, recorded during the test, and the mesh had a tensile strength of 580 MPa with fracture strain of 0.12. The slab panel was laterally unrestrained, and had no continuity across its edges. This test was used by the first author (Burgess, 2017) to compare the predictions of the current method with those from the method developed by Bailey and Moore. This test is useful here also in doing some basic studies on the effect of edge continuity.

Different rotational conditions at free edges

The effect of rotational continuity at edges which are free to move is illustrated in Figs. 9(a) and 9(b). These plot the load capacity of panels of the configuration given above as the slab deflection increases. In their initial state the four cases have: (i) no continuity (krx = kry = 0); (ii) and (iii) continuity across one set of parallel edges only (krx = 1, kry = 0 or krx = 0, kry = 1); (iv) continuity across all edges (krx = kry = 1). These two figures show the effect of ductility, which in this case is determined by the mesh ductility class and bar surface type. In Fig. 9 (a) the mesh is of the intermediate Ductility Class B and the bars are plain rather than deformed; this gives a fracture crack-width of 7.05 mm. In Fig. 9 (b) the mesh is of the highest Ductility Class

zi

zi

(a)

(b)

(c)

zi

zi

(d)

(e)

(f)

(i) Only rotational continuity

(force-controlled)

(ii) Axial and rotational continuity

(displacement-controlled)

0

0

0

0

0

0

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C, giving a fracture crack-width of 14.91 mm. The most evident effect of the increased ductility is that the peak enhancement is increased, and occurs at a higher displacement. This peak capacity occurs when the mesh begins to “unzip” along the diagonal yield lines. The abrupt reductions in capacity occur when mesh crossing the yield lines along and parallel to the panel’s edges fractures simultaneously. It can be seen that continuity affects mainly the initial yield-line capacity, which increases with the length of the panel boundary that is continuous. The subsequent enhancement of this capacity due to tensile membrane action is very similar for all of the different edge conditions. All-edge continuity gives the highest load capacity and no continuity gives the lowest load capacity, as expected because the boundary forces contribute to the load capacity. It is interesting to see that long-edge continuity at the early stage of deflection gives a higher load capacity than short-edge continuity, until the rebar at the central yield line fractures, beyond which its residual capacity is lower.

(a) (b)

Figure 9: Load capacity of 1.4706 aspect ratio slab with different continuity cases but no axial edge restraint: (a) fracture crack-width 7.05 mm. (b) fracture crack width 14.91 mm.

Continuity of slabs with fixed boundaries

The same slab definitions, based on the Garston slab test, are now used to check the effects of panel boundaries which resist in-plane movement of concrete between adjacent slabs in a continuous floor. Both the x- and y-aligned edges of the panel are restrained in these analyses.

Fig. 10 (a) shows the effect of fixed boundaries in causing compressive membrane action in the slab on the load capacity of the panel. Load capacity is plotted against deflection for panels with restrained edges, for different effective reinforcement ductilities. Again the ductility of reinforcement crossing a yield line is reflected in the crack width at which it fractures, which varies according to its Class (B or C) and whether the bars are deformed or plain. The very low fracture crack width of 1.3 mm is for the well-bonded deformed 6 mm bars, as an example of a reasonably extreme case in which a large amount of bar fracture occurs at low deflections. At the opposite extreme an idealized case is plotted in which no bar fracture occurs across any yield line. It can be seen that for all these cases the initial yield line mechanism occurs at an applied load intensity which is much higher than those for edges which are free to move; in this particular case the initial failure loads are 77.5 kN/m2 with fixed edges and 4.0 kN/m2 with free edges but with mesh continuity. The load capacity-deflection curve for the free-edge case with 14.91 mm fracture crack-width is reproduced on this figure for comparison with the restrained cases.

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acit

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Mid YL

break

Mid YL

break

Short edge

break

Long edge

break

Long edge

break

Short edge

break

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(a)

(b)

Figure 10: Load capacity of 1.4706 aspect ratio slab with in-plane edge restraint: (a) Behaviour with different ductilities (fracture crack-widths), and comparison with unrestrained slab. (b) Idealized restrained and unrestrained cases with infinite ductility.

The transition from compressive to tensile membrane action

It is clear from Fig. 10 (a) that, although slabs with restrained edges have a highly enhanced initial failure load at which a yield-line mechanism forms due to compressive membrane action, this load capacity falls-off extremely rapidly with deflection. It is also apparent that, for any practical mesh ductility across yield lines, the successive fractures of mesh prevent any subsequent increase of load capacity due to tensile membrane action. Although the particular design parameters of the case in question will clearly affect the behaviour, it is likely that this will be the general case. It can be seen that only the limiting case with effectively infinite ductility of reinforcement across yield lines is capable of generating significant enhancement due to TMA. In Fig. 10 (b) a series of infinite-ductility curves is shown, with increasing mesh bar diameters placed at the same mean effective depth in the slab. The initial failure load is almost identical for all of these cases, demonstrating that this load depends almost entirely on the concrete slab’s dimensions, including its thickness. The degree to which the decline of load capacity in CMA changes to enhancement in TMA can be seen to depend on the reinforcement percentage, with the larger mesh areas causing enhancement not only to start at a lower deflection and higher capacity, but also to rise more rapidly. The infinite-ductility enhancements with slab edges free to move are also shown in the figure. These tend almost asymptotically towards the TMA parts of the curves for restrained edges at high deflections. Under normal reinforcement bond conditions fracture crack-widths will clearly limit the amount of enhancement by TMA which is capable of happening after the CMA phase ends, so the curves shown in Fig. 10 (b) are impractical. However, in design specifically aimed at robustness in hazard conditions, ductility could be greatly increased with the use of unbonded, positively anchored reinforcing mesh.

5. CONCLUSIONS

The main motivation for this work has been to develop a mechanically justifiable method to analyse the large-deflection load capacity of concrete and composite slabs. This becomes relevant particularly in the contexts of fire and blast resistance, rather than normal design for slabs at ultimate and serviceability limit states. Under these “hazard” loading conditions avoidance of disproportionate collapse becomes the key criterion, and large deflections are acceptable. The principles, rather than the mathematical detail, of the method have been set out in this paper, including its capability to unify the treatment of both compressive and tensile membrane actions. Previous analytical methods for these two actions have tended to keep them separate, although Wood’s (1961) method considers both phases. The context in which developments of the method can most obviously become relevant to real buildings is in performance-based structural fire engineering design, for composite floors at the stage when steel downstand beam have lost almost all of their strength. Temperatures within the concrete slab at this time are unlikely to be high enough to impair the development of either tensile or compressive membrane action, either of which could enhance the slab’s strength sufficiently to prevent a real collapse.

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Infinite ductility14.91mm (Plain 6mm C)7.05mm (Plain 6mm B)1.3mm (Deformed 6mm B)14.91mm Free edges

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The most significant aspect of this work is that its principles can be applied to extensions of the basic scenario in further developments aimed at improving the realism of the models. These can include edge conditions which better represent the restraint provided by adjacent slabs, and a more realistc concrete stress-block model.

REFERENCES

Bailey, C.G. and Moore, D.B., 2000a, The structural behaviour of steel frames with composite floors slabs subject to fire – Part 1: Theory, The Structural Engineer, 78 (11, pp 19-27.

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