### Diaphragm Flexibility

In addition to supporting gravity loads and to transferring there loads to the columns and walls, roof or floor framing systems act as diaphragms that transfer the lateral forces to the LFR elements. A diaphragm is often assumed to behave like a deep beam: the deck (web) is designed for shear, and the edge framing elements or chords (flanges) are designed for the tensile and compressive forces from bending. A three dimensional analysis that takes into consideration the relative rigidities of the diaphragm and the LFR elements is required in order to obtain the most accurate distribution of the forces in these components. However, a more simple analysis is possible when simplifying assumptions are made concerning the rigidity (or, flexibility) of the diaphragm. For analysis purposes, diaphragms are typically classified as rigid, semi-rigid, and flexible. In details the classification is given in Table-1.

 Flexibility Category Diaphragm Definition Rigid Diaphragm (F<1) A rigid diaphragm is assumed to distribute horizontal forces to the LFR element incorporation to their relative stiffness Semi-rigid Diaphragm( 1≤ F ≤10) Semi-rigid and semi-flexible diaphragm are those that have significant deflection under load but which also have sufficient stiffness to distribute a portion of their load to the LFR elements in proportion to their stiffness. The action is analogous to a continuous concrete beam system of appreciable stiffness on yielding supports. The support reactions are dependent on the relative stiffness of both the diaphragm and the LFR elements Semi-flexible Diaphragm( 10≤ F ≤70) Flexible Diaphragm( 70≤ F ≤150) A flexible diaphragm and a very Flexible Diaphragm are analogous to a shear deflecting continuous beam or series of beams spanning between supports. The supports are considered non-yielding, as the relative stiffness of the LFR elements compared to that of the diaphragm is large. Thus, a flexible diaphragm will distribute the lateral forces to the LFR elements on a tributary load basis. Very Flexible Diaphragm(F >150)

In the case of rigid diaphragms, the forces are distributed to the walls and frames in proportion to their stiffness. A cast-in-place concrete floor is an example of rigid diaphragm. Diaphragm flexibility results in deformations that affect the lateral force transfer to the LFR elements. A very flexible diaphragm, such as that of plywood, will develop a large deformation compared to those of the LFR elements. In such case, the lateral force transferred to the LFR element can be based on tributary areas. The semi-rigid diaphragm, such as steel deck, is partly able to distribute the lateral forces into the LFR elements based on their relative stiffness. This situation is very similar to that of a continuous beam supported on yielding supports.

The rigidity of diaphragm can be ascertained by determining its flexibility factor (F). A slab is considered to be rigid diaphragm if it has a flexibility factor of less than one. A flexibility factor between 1 and 10 is considered to represent a semi-rigid diaphragm, and a factor greater than 10 indicate a flexible diaphragm. The flexibility factor for a concrete diaphragm is defined as:

where,
h  =  thickness of the slab, (in).
wc =  unit weight of the concrete,( pcf)
f′c   = compressive strength of the concrete at 28 days, (psi)

Assuming a minimum allowable slab thickness (h) equal to 3.5 in. and a unit weight of concrete (wc) equal 90 pcf, above equation indicates a cast-in-place concrete slab to be rigid even for a compressive strength (f′c) equal to 1500 psi. since these are extreme value, it is obvious that unless a concrete slab has large openings, it is unlikely that it will be anything but rigid. This simplifies the lateral force transfer analysis, since the forces will be distributed direct proportions to the relative stiffness of the LFR elements. A more refined analysis accounting for diaphragm flexibility would be required for semi-rigid diaphragms.