## Yielding Of Steel In Singly Reinforced Be Occurs In

Required ductility (robustness) of concrete structures

Referring again to objective (c) above, the degree to which ductility should be enhanced is debatable. Until the 1990s, research and codes had rightly been preoccupied with overcoming the excessive brittleness and unreliability of ill-reinforced concrete. However, there may have been too much emphasis of creating ductility for ductility's sake. The high cost of design and the complexity of some of the reinforcement of highly ductile concrete has raised the valid question, 'How do we design less ductile structures which are sufficiently reliable in earthquakes?' This question has long been raised regarding structures in regions of lesser seismicity. In any seismic region, the question applies not only to whole structures but also to parts of structures, e.g. beams and columns in buildings where the primary earthquake resisting elements are structural walls.

Methods of adjusting the design loading for different degrees of ductility have been discussed earlier, such that the value of the ductility factor may be chosen in the range from f = 1 (non-ductile) to about f = 6 (highly ductile). Some concrete codes gives recommendations for the design of structures of limited ductility implying a value of f < 3.

Ductility and robustness have been discussed in the general terms of inelastic behaviour in Section 5.4.2, and the problems of analysing inelastic behaviour and hence assessing the required (ductility demand) in a structure have been considered in Section 5.4.7. While most concrete structures are designed by equivalent static analysis and codified reinforcing rules aimed at providing ductility, it is important for designers to understand how the ductility demand arises. This is now discussed using a simplified method of determining hinge rotations in reinforced concrete frames, which involves the assumption of a hinge mechanism (Figure 8.7) and the imposition of an arbitrary lateral deflection ductility factor f on the frame.

As mentioned above, it is preferable that beams should fail before columns (for safety reasons). Considering ten storeys above the column hinges of a column sidesway mechanism, Park (1980) found that for an overall frame deflection ductility factor f = 4, the required section ductility ratio was = 122, which is impossibly high as shown by Figure 10.33. $>u and are the hinge curvatures at ultimate and first yield, respectively. On the other hand, for a beam sidesway mechanism the required section ductility was found to be less than 20.

Having made an estimate of the ductility demands in the structure, the members should be detailed to have the appropriate section ductility, the theory for which is discussed below.

### Available ductility for reinforced concrete members

The available section ductility of a concrete member is most conveniently expressed as the ratio of its curvature at ultimate moment $>u to its curvature at first yield . The expression may be evaluated from first principles, the answers varying with the geometry of the section, the reinforcement arrangement, the loading, and the stress-strain relationships of the steel and the concrete. Various idealizations of the stress-strain relationships give similar values for ductility, and the following methods of determining the available ductility should be satisfactory for most design purposes. It should be noted that the ductility of walls is discussed elsewhere (Section 10.3.2).

(i) Singly reinforced sections

Consider conditions at first yield and ultimate moment as shown in Figure 10.20. Assuming an under-reinforced section, first yield will occur in the steel and the curvature e f <py =-2— =-^--(10.27)

Section At first yield At ultimate

Figure 10.20 Doubly reinforced beam at first yield and ultimate curvatures

Section At first yield At ultimate

Figure 10.20 Doubly reinforced beam at first yield and ultimate curvatures where k = j{(pnf + pn} - pn (10.28)

where p = As/bd = tensile reinforcement ratio, and n = modular ratio = Es/Ec, where Es and Ec are the modulus of elasticity of the steel and the concrete, respectively. Strictly, this formula for k is true for linear elastic concrete behaviour only, i.e. for feu = ^ < 0.7/;

where fy is the steel yield stress and f is the concrete cylinder compressive strength. For higher concrete stresses the true non-linear concrete stress block should be used. Referring again to Figure 10.20, it can be shown that the ultimate curvature is

c a where

Asfy a = —(10.30) 0.85 f'cb and P1, which describes the depth of the equivalent rectangular stress block may be taken as p1 = 0.85 for f'c = 27.6MPa (4 ksi), otherwise

From the above derivation, the available section ductility may be written as

The ultimate concrete strain scu is given various values in different codes for different purposes. For estimating the ductility available from reinforced concrete in a strong

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