Most reinforced concrete members are statically indeterminate because they are part of monolithic structure, i.e., in this form of building construction, slabs are cast monolithically with a beam-and –girder floor framing that carries the floor load to columns. A beam is a flexure member. It may be of steel, wood or any other structural materials. Reinforced concrete beams are nonhomogeneous in that they are made of two entirely different materials. A flexural member should suffer a axial compression force less than Ag fc´/10. The member must be at least 10 in wide and its clear span should be least 4 times the effective depth and the width-to-depth ratio should be at least 0.3.
Slabs are used to furnish a flat and useful surface in reinforced concrete construction. It is broad, flat plate, usually horizontal, with top and bottom surfaces parallel or nearly so. It may be supported by reinforced concrete beams, by masonry or reinforced concrete walls, by structural steel members, by directly by columns or continuously by the ground.
Slabs in which the deflected surface is predominantly cylindrical is called one-way slabs spanning in the direction of curvature. This condition arises when slabs are supported on two opposite sides, and those supported on all four sides with the longer span greater than twice the shorter span and also in cantilever slabs. In many cases, rectangular slabs are of such proportions and are supported in such a way that two-way action results. when loaded, such slabs bend into a dished surface i.e., an any point the slab is curved in both principal directions, and since bending moments are proportional to curvature, moments also in both directions.
Concrete slabs may in some cases be carried directly by columns, without the use of beams or girders. Such slabs are called flat plates. In some cases, to reduce the stresses due to shear and negative bending around columns, a thickened slab region in the vicinity of column and flared column tops are incorporated. Such construction is called flat- slab construction.
Occurrence of Opening
In almost all constructions, slab system include openings. These may be of substantial size, as required by stair-ways and elevators shafts, or they may be of smaller dimensions, like those needed to accommodate heating, plumbing, and ventilating risers, floor and roof drains, and access hatches.
These may occur when opening in slabs pass through the effective flange width of T-beams, for installation of building services. Opening are also frequently provided through beam web.
Code’s Recommendation for Placement of Opening
As far as possible, opening in slabs should be located in zones where shear stresses are small and bending moment are below maximum. However, small opening for pipe sleeves etc. can be made anywhere in slab. In case of slabs supported directly by columns, from structural view point, they are best located well away from the columns, preferably in the area common to the slab middle strips. Opening of any size may be provided in slab systems if shown by analysis that the design strength is at least equal to the required strength and that specified limits on deflection meets. For strength the strip method is appropriate.
13.5.1 Opening of any size are permitted in slabs if analysis shows that both strength and deflection are acceptable.
13.5.2 In lieu of special analysis as required by 13.5.1 opening may be provided in slab systems without beams only in accordance with the following.
18.104.22.168 Opening of any size may be located in the area common to intersecting middle strips.
22.214.171.124 In area common to intersecting column strips, not more than 1/8 the width of column strip in either span shall be interrupted by openings.
126.96.36.199 In the area common to one column strip and one middle strip, opening size is limited to 1/4 the width of column strip in either span.
In regions of small shear, as near the middle of a beam span, a horizontal pipe sleeve is not serious. Elsewhere, shear strength must be closely watched and in many places bending strength as well.
Strip Method to analyze Opening
The strip method was first developed by Hillerborg, and later Wood and Armer analyzed the method critically and performed test on slabs designed by this method. They found that a design made by the strip method and reinforced exactly according to moments found, was an exact solution.
The strip method is the simplest one for slabs on simple supports, but continuity can be handled on a basis similar to limit design. The most difficult slabs for this method are slabs supported on columns. For such a case, Hillerborg developed the advanced strip method, using a rectangular element carrying load in two directions to a support at one corner of the element.
The equilibrium equation for slabs is
All elements of this equation, except w, follows the Timoshenko’s notation.
i.e., Mx and My = bending moment about respective directions.
Mxy = twisting moment.
and w = load per unit area on slab.
Hillerborg designs the slab to make Mxy unnecessary, that is he assumes Mxy = 0 and than apportions the load to ∂²Mx/∂x² and ∂²My/∂y², usually at a particular spot wholly to one or to the other. This particular apportionment is more of a convenience than a necessity, however, loads in a particular area are assign to particular slab strips and continuity of the resulting moments and shear must be carefully maintained. Apparent discontinuity in torque or deflection may be disregarded, but a discontinuity in moment or shear is not permitted. Both elastic and plastic analysis concepts are permissible in evaluating moments on strips.
The suitability of the method for slabs with opening is a strong point in its favor. This method have been using for many years by designers, designing by their ‘feel’ for the way the load has most apt to be transferred to the supports. As slabs are normally considerably under-reinforced, it is possible to use certain strips near the opening as small beam simply by increasing the local reinforcement. If the opening is so large that even extra slab steel is inadequate to care for the moment, a real beam is needed around one or more sides of the opening, quite probably spanning to the edge beams. The assumed “beam strips” are drawn as dotted around the opening and the point of inflection as suggested by Wood and Hillerborg are added on the slab. Then different moment for different strip is calculated. Depth can be fixed from loads (moment and shear). The reinforcement should be arranged in bends corresponding to the strips used, this calculation is simple.
Defection at service load must be considered in checking serviceability. In any actual design the service load is available, and it should be on the safe side to use the strip service load moments with EI based on
the cracked section.
Treatment for Opening
Larger opening should be framed by beams to restore, as nearly as possible, the continuity of the slab. The beams should be designed to carry a portion of the floor load, in addition to loads directly applied by partition walls, elevator support beams, stair slabs etc. But, in case of flat plate construction, opening near column render difficulties.
Effects on Strength and Treatment
The effect of openings on the shear strength as well as flexure action (in case of large opening) of slab must be investigate carefully. The effect becomes severe when the openings are within the column strip areas of slabs or within middle strip areas when the opening are closer than 10 times the slab thickness(10h) from a column, measured radial.
To account for reduction of shear strength, it is considered that a portion of the critical section b0 is ineffective which is enclosed by straight lines projecting from the column centroid to the edges of the opening. If two much resisting perimeter is lost, the designer must be sure adequate tow-way bending is really present; otherwise the lower shear permitted in beams become the limiting values for the slab.
For slabs with shear reinforcement, the ineffective portion of the perimeter b0 is one-half of that without shear reinforcement. One-half factor is interpreted to apply equally to shearhead reinforcement and bar or wire reinforcement.
188.8.131.52 Where opening in slabs are located at a distance less than 10 times the slab thickness from a concentrated load or reaction area or when opening in slabs are located within column strip, the critical sections shall be modified as follows:
1) For slabs without shearhead, that part of the perimeter of the critical section that is enclosed by straight lines projecting from the centroid of the column, concentrated load or reaction area and tangent to the boundaries of the opening shall be considered ineffective.
2) for slabs with shearheads, the ineffective portion of the perimeter shall be one-half that defined in (1) above.
Large opening in beams are particularly weakening. They destroy beam action and force the reduced section to act such as a Vierendeel truss ( a truss without diagonals). In such a truss the average bending moment over the length of the opening is resisted by axial compression one chord and tension in the other, with these two forming a couple in the case of pure flexure. Where shear is present the change in the moment over the length of the opening superimpose a reversed bending resistance in each chord, the total of the four end moments on the chords equaling the external shear times the length of the opening. How the shear and these reverse moments are shared by the chords depends on the relative chord stiffness.
Thickening of edge of opening
In case of larger openings, the edge of the slab around the opening may be thickened to enable them to behave like trimmer beams for taking up additional moments and forces. If opening reduce a critical design section for moment, must be maintained by providing extra depth to offset the reduced width.
Function of Steel
The steels used around opening tie the free ends of the bars which are trimmed to form opening. It also serve as reinforcement to make the opening stable against deformation or any other types of failure. This bars take care of the loss of steel in a slab on account of forming opening. Diagonal bars provided in corner control the cracking that will almost inevitably occur there.
With regard to flexural requirements, the total amount of steel required by calculation must be provided, regardless of opening. Any steel interrupted by holes should be matched with an equivalent amount of supplementary reinforcement on either sides, proper lapped to transfer stress by bond.
If minor cracking at the corners of an opening is objectionable, it is always advisable to add one or two diagonal bars at each corner, especially at large openings. Bars are always desirable around Window and door opening in concrete wall slabs, because such reinforcement helps to take care of shrinkage stresses.
184.108.40.206 In case of opening in intersecting middle strips, total amount of reinforcement required for the panel without opening is maintained.
220.127.116.11 In case of opening in intersecting column strips, an amount of reinforcement equivalent to that interrupted by an opening shall be added on the sides of the opening.
18.104.22.168 In case of opening common to one column strip and one middle strip not more than ¼ the reinforcement in either strip shall be interrupted by openings. An amount of reinforcement equivalent to that interrupted by an opening shall be added on the sides of the opening.
22.214.171.124 For members with significant axial tension, requires all shear to be resisted by stirrups (none assign to the concrete).
Reduction in critical design section for moment due to opening, can be overcome providing more closely on each side of the opening to maintain the necessary As. This is possible only when opening is possible to locate where moment is well below the compression capacity of the slab, thereby leaving the arrangement of reinforcement as the only problem. Of course, shear strength must be maintained, but this is rarely a problem except near the columns in flat slab type, as noted previously.
The arrangement of bars around any but minor opening can constitute a real problem. Bars running perpendicular to the face of an opening are not fully effective when simply cut0off at the opening. This is acceptable if there is a beam at the opening to act as reaction for the slab. If there is no beam it is better to fan the bars out or splay them to go around the opening. If fanning or splaying leaves too wide an area without steel, extra bars can be placed parallel to the side of the opening, as indicated.