### Strip Method to Analyze Slab When Openings Arise

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.

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.