Membrane Action in Concrete Floor Systems
Researchers in the field of reinforced concrete have had a long history of attempting to understand the tensile behavior of structural concrete slab systems and proposing methodologies for quantifying the beneficial effects of catenary action and membrane action. Much of the research conducted in this regard has made its way into ACI 318 provisions for general structural integrity (Hawkins and Mitchell 1979; Mitchell and Cook 1984). Researchers studying the response of structural steel systems to fire have also begun in earnest to understand and capitalize on the inherent robustness present in steel framing systems that is contributed by the concrete deck (Allam et al. 2000; Bailey et al. 2000; Huang et al. 2003a; Huang et al. 2003b).
It has been long recognized that flat plate concrete floor systems have the potential to suffer from disproportionate collapse from a rather simplistic event: punching shear failure at interior and exterior columns (Hawkins and Mitchell 1979; Mitchell and Cook 1984). There was a series of systematic efforts carried out to develop design procedures that could limit the probability of a punching shear failure leading to progressive collapse. The first of such efforts was that conducted by Hawkins and Mitchell (1979). Mitchell and Cook (1984) enhanced this methodology to include procedures that allow quantifying the role of catenary action in the behavior of the floor system and its partnering with membrane action to mitigate progressive collapse in concrete floor system.
M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 239-254. © 2006 Springer. Printed in the Netherlands.
When a concrete floor plate is loaded to the point of inelastic behavior, there is a tendency for the bottom fibers (assuming loading is from the top) to lengthen. This lengthening, however, is restrained by the concrete slab at the perimeter of the panel being loaded. Of course, steel beams in the systems considered in the present study will provide restraint to this outward movement. In the purely theoretical sense, the concrete slab will have a load versus vertical deflection response that exhibits snap through prior to the formation of membrane tension in the system. This hanging net effect cannot take place without significant vertical deformation in the floor system. In the hanging configuration, all sections through the floor plate are subjected to tensile forces and it is imperative that properly developed tension reinforcement exists in the slab and vertical support at the panel edges be maintained.
The beauty of the work of Hawkins and Mitchell (1979) is that the expressions for computing the membrane capacity of concrete floor panels are rather simplistic and include a significant amount of engineering feel. The fundamental assumption of the proposed methodology is that the deformed membrane between supports follows a circular shape. This makes the mathematics tractable and errors are minor when compared to the more correct catenary parabola. The basic slab system and membrane forces considered are schematically shown in Figure 1.
Two slab span directions are assumed: the first is defined as the short direction, l1; and the second is termed the long direction, l2. The reinforcement area on a per unit length basis in the short and long directions are As1 and As 2, respectively. The normal strains in the fibers of the membrane are assumed to be uniform over the membrane thickness and are functions of its curvature. Uniformly distributed loading over the surface of the membrane is assumed and positive loading is taken to be downward. Membrane tension forces (edge tensions) per unit length parallel to the short and long directions are T and T2, respectively. These forces are assumed to be in the direction tangent to the deformed membrane’s midsurface at the edges.
A typical structural mechanics solution procedure (e. g. imposition of vertical equilibrium, ensuring compatibility of deformations, and adherence to constitutive laws for the material) is employed to develop a relationship for the capacity of the tensile membrane that is a function of the edge tension, strain in the membrane (and therefore, vertical deflection) and the panel dimensions. When the panel dimensions differ (i. e. they are rectangular) the membrane capacity of the panel based upon the tensile reinforcement capacity at the edges can be written as (Hawkins and Mitchell 1979),
where: ex is the tensile strain in the membrane fibers parallel to the short direction, which is the dominant membrane direction. If the slab panel is square, there is no dominant direction. As ljl1 increases, the slab panel begins to behave as a single direction membrane (i. e. a catenary).
Concrete slab systems quite often have different reinforcement patterns at the edges than that found in the middle strip areas within the panel span. As a result, if the mid-span reinforcement controls the tensile capacity of the membrane, the vertical load carrying capacity is (Hawkins and Mitchell 1979),
T’ l – + T’-
where: T[ and T2′ are the tensile membrane forces per unit length within the mid-span (positive moment) regions of the panel parallel to the short – and long-directions, respectively.
The strain in the direction parallel to the short and long dimensions of the panel is related to one another as a result of the assumed circular shape of the membrane. If one knows the strain in the direction parallel to the short dimension, the strain in the direction parallel to the long dimension is computed using (Hawkins and Mitchell 1979),
V l1 У
Therefore, once the strains in the two directions are computed (short direction assumed, then long direction computed), the constitutive laws for the reinforcement can be used to determine the state of stress and then the tensile membrane forces on a per foot basis follow.
Once the strain in the direction parallel to the short direction is known, the maximum deflection within the panel can be computed using (Mitchell and Cook 1984),
The vertical deflection is important when assessing the capacity of the membrane. Assuming end anchorage is present, the membrane is capable of carrying more loading in a highly deflected configuration for a fixed tensile force capacity. Therefore, if a large amount of loading is present and there is a fixed tensile capacity for the reinforcement in the membrane (assuming no rupturing of the reinforcement), then there is a tendency for the membrane to continue to deflect vertically to generate greater vertical components in the catenary forces. Therefore, the vertical deflection given by equation (4) can be used to determine if a slab panel will become debris loading for a panel below, or will impede modes of egress from the structure.
Mitchell and Cook (1984) provide an enhanced description of the post-failure response of concrete slab structures that is pertinent to situations considered in the present study. The response of a slab structure after initial failure depends upon the amount and details of the steel reinforcement, the vertical support conditions and the horizontal restraint conditions at the panel edges (Mitchell and Cook 1984). When the slab panel has vertical support surrounding its edges (e. g. steel beams at the perimeter of the panel), the slab is capable of providing its own in-plane compression ring restraint conditions at the perimeter. This compression ring helps to resist the horizontal component of the maximum tensile forces. If the edges of the panel are allowed to deform vertically, then this compression ring cannot form.
When “stiff” beams are present at the perimeter of the slab panel, the membrane action in the slab panel facilitates the slab system hanging off the perimeter beams. When an interior slab panel is considered, the adjacent regions of the floor system will help to restrain the edges of the overloaded panel. Edge or corner panels can develop the necessary compression ring behavior if the edges are supported by beams that have significant flexural stiffness when compared to the slab itself.
Although structural steel floor framing systems are significantly different in many ways than that of a two-way flat plate or flat slab cast-in-place concrete system, there are enough similarities to justify using the theory and expressions developed by Hawkins and Mitchell (1979) and Mitchell and Cook (1984) in assessing the robustness of structural steel framing systems. It is felt that membrane and catenary action are indeed possible within the structural steel framing systems commonly found in buildings. More importantly, it is felt that this catenary and membrane behavior, to a large extent, is inherent in the systems typically constructed. The tension reinforcement present in these systems will need to be quantified and their anchorage discussed prior to detailed examination of ineffective supporting member scenarios.
In composite steel-concrete floor systems, there is typically welded-wire mesh and light gauge steel deck that can be utilized as tension reinforcement within the slab system should membrane and/or catenary action be needed. However, one must understand the usefulness of these components as reinforcing mechanisms in the slab system before one can count on this reinforcement as being inherent sources of membrane and catenary reinforcement for the floor system. The light-gauge steel deck is essentially a unidirectional spanning entity. In the direction parallel to the flutes in the deck, the steel deck is highly likely to be a very useful form of tension reinforcement for facilitating catenary action. However, in the direction orthogonal to the flutes, the steel deck likely has puddle welds or TEK screws that are unlikely to preserve tensile forces within the deck in this orthogonal direction. Furthermore, the fluted nature of the deck results in a tension force that has two distinct elevations at the floor deck soffit. This makes relying on the steel deck providing tensile membrane or catenary reinforcement in two directions very difficult. Therefore, the present analysis assumes that the steel deck provides one-way reinforcement within the floor framing system. It should be noted that if the steel deck panels are not continuous over the supporting beam, a force-transfer mechanism is questionable without edge beams providing vertical support.
The welded-wire fabric present in the floor system is also a source of membrane and catenary tension reinforcement. This steel fabric generally has a slightly elevated yield stress when compared to the usual mild-steel reinforcement. Furthermore, the spacing of the wires in the mesh can change with direction. This reinforcement will be assumed as sufficient to develop catenary and membrane forces if it is considered continuous through the panel perimeter and appropriately lapped.
In the steel building system considered in this study, a panel is defined as having two in-fill beams and two girders bounding a panel of concrete slab. In most cases, the perimeter of the slab panel will have puddle welds or even steel studs connecting the steel deck to the perimeter beams/girders. Furthermore, these perimeter members will have significantly greater flexural stiffness when compared that of the slab. As a result, the slab system can be assumed to develop compression ring anchorage if the perimeter beams remain in tact during a compromising event.