## Gph

Figure 8.6 illustrates the stress distribution in a prestressed concrete member at and after the decompression stage where the behavior of the prestressed beam starts to resemble that of a reinforced concrete beam.

As the load approaches the limit state at ultimate, the additional strain e3 in the steel reinforcement follows the linear triangular distribution shown in Figure 8.6b, where the maximum compressive strain at the extreme compression fibers is ec = 0.003 in./in. In such a case, the steel strain increment due to overload above the decompression load is es = £c (-7-) (8.37c)

where c is the depth of the neutral axis. Consequently, the total strain in the prestressing steel at this stage becomes es = e1 + e2 + e3 (8.37d)

The corresponding stress fps at nominal strength can be easily obtained from the stress-strain diagram of the steel supplied by the producer. If fpe < 0.50fpu, the ACI Code allows computing fps from the following expression:

where gp can be used as 0.40 for fpy/fpu > 0.85.

8.6.7.1 The Equivalent Rectangular Block and Nominal Moment Strength

1. The strain distribution is assumed to be linear. This assumption is based on Bernoulli's hypothesis that plane sections remain plane before bending and perpendicular to the neutral axis after bending.

2. The strain in the steel and the surrounding concrete is the same prior to cracking of the concrete or yielding of the steel as after such cracking or yielding.

3. Concrete is weak in tension. It cracks at an early stage of loading at about 10% of its compressive strength limit. Consequently, concrete in the tension zone of the section is neglected in the flexural analysis and design computations and the tension reinforcement is assumed to take the total tensile force.

8.6.7.2 Strain Limits Method for Analysis and Design 8.6.7.2.1 General Principles

In this approach, sometimes referred to as the "unified method,'' since it is equally applicable to flexural analysis of prestressed concrete elements, the nominal flexural strength of a concrete member is reached when the net compressive strain in the extreme compression fibers reaches the ACI code-assumed limit 0.003 in./in. It also stipulates that when the net tensile strain in the extreme tension steel, et, is sufficiently large, as discussed in the previous section, at a value equal to or greater than 0.005 in./in., the behavior is fully ductile. The concrete beam section is characterized as tension controlled, with ample warning of failure as denoted by excessive cracking and deflection.

If the net tensile strain in the extreme tension fibers, et, is small, such as in compression members, being equal to or less than a compression-controlled strain limit, a brittle mode of failure is expected, with little warning of such an impending failure. Flexural members are usually tension controlled. Compression members are usually compression controlled. However, some sections, such as those subjected to small axial loads, but large bending moments, the net tensile strain, et, in the extreme tensile fibers, will have an intermediate or transitional value between the two strain limit states, namely, between the compression-controlled strain limit et = fy/Es = 60,000/29 x 106 = 0.002 in./in., and the tension-controlled strain limit et = 0.005 in./in. Figure 8.7 delineates these three zones as well as the variation in the strength reduction factors applicable to the total range of behavior.

For the tension-controlled state, the strain limit et = 0.005 corresponds to reinforcement ratio p/pb = 0.63, where pb is the balanced reinforcement ratio for the balanced strain et = 0.002 in the extreme tensile reinforcement. The net tensile strain et = 0.005 for a tension-controlled state is a single value that applies to all types of reinforcement regardless of whether mild steel or prestressing steel. High reinforcement ratios that produce a net tensile strain less than 0.005 result in a f-factor value lower than 0.90, resulting in less economical sections. Therefore, it is more efficient to add compression

Interpolation on c/dt: Spiral ^ = 0.37 + 0.20/(c/dt) Other ^ = 0.23 + 0.25/(c/dt)

FIGURE 8.7 Strain limit zones and variation of strength reduction factor f with the net tensile strain et [1,5,8].

Interpolation on c/dt: Spiral ^ = 0.37 + 0.20/(c/dt) Other ^ = 0.23 + 0.25/(c/dt)

FIGURE 8.7 Strain limit zones and variation of strength reduction factor f with the net tensile strain et [1,5,8].

reinforcement if necessary or deepen the section in order to make the strain in the extreme tension reinforcement, et > 0.005.

8.6.7.2.1.1 Variation of f as a Function of Strain — Variation of the f value for the range of strain between et = 0.002 and et = 0.005 can be linearly interpolated to give the following expressions:

Tied sections:

Spirally reinforced sections:

8.6.7.2.1.2 Variation of f as a Function of Neutral Axis depth Ratio c/dt — Equations 8.38a and 8.38b can be expressed in terms of the ratio of the neutral axis depth c to the effective depth dt of the layer of reinforcement closest to the tensile face of the section as follows:

Tied sections:

Spirally reinforced sections:

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