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Analysis of Reinforced Concrete Beams in Pure Flexure using ACI 318-02

By T. B. Quimby UAA Civil Engineering Department Spring 2004

The strength analysis of RC sections in flexure is based on the principles of strain compatibility and equilibrium as out lined in ACI 318 10.2. The strength is governed by the forces that will cause the strain in the concrete to equal the crushing strain (.003). The basic design problem is to determine the dimensions and reinforcing for the cross sections such that the limit states (i.e. limiting constraints) of the code and other practical considerations are satisfied. To determine if a beam is feasible, the limiting constraints must be known. This paper summarizes the relevant ACI 318 and practical constraints. ACI 318 9.1.1: Strength, Mu < Mn The determination of Mu is made using the criteria in ACI 318 9.2. The load combination equations are based on ultimate strength concepts and the predictability of the loads, and their probability of simultaneous occurrence. Note that the beam must have the strength to satisfy all the load combination equations. The determination of Mn is determined using principles of strain compatibility and equilibrium according to the assumptions given in ACI 318 10.2. For the purposes of this discussion the strength, Mn, is the bending strength that occurs when Pn = 0. The reduction factor, , is determined using ACI 9.3. New with the 2002 version of ACI 318 is the concept of "tension controlled" and "compression controlled" sections. The definition of "tension controlled" is found in ACI 318 10.3.4. Sections are defined to be tension controlled when the strain in the steel is at or in excess of 0.005 when the concrete reaches its maximum usable strain. This is in excess of the yield strain in the reinforcing steel. When this condition occurs, the reduction factor equals 0.90. The definition of "compression controlled" is found in ACI 318 10.3.3. Sections are compression controlled when the steel does not yield in tension. This means that the steel strain is less than the tensile yield strain. Note that the condition where the steel strain is exactly at its tensile yield strain when the concrete strain is 0.003 is referred to as the "balanced condition". When this condition occurs the reduction factor is either 0.65 or 0.70 depending on the nature of the lateral confinement steel. For properly designed beams, this condition rarely occurs so the discussion on confinement steel will be left for another time.

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There is a range of steel strain for which the section is neither tension nor compression controlled. In this region, the reduction factor is found by a linear interpolation on the actual steel strain between the two limits. ACI 318 10.3.5: Upper Limit on As, s > 0.004 To ensure adequate ductile behavior in beams, ACI 318 requires the steel to have significant strain at the ultimate strength condition (i.e. when the concrete strain is 0.003). Section 10.3.5 prescribes the limit for this state. For practical purposes, it is generally good to keep the steel strain above 0.005 or the reduction factor will need to be reduced. The tensile strain can be derived from strain diagram once the location of the neutral axis, c, is determined. This requirement effectively puts an upper limit on the amount of steel that can be used in the beam. As the amount of steel is increased in a given section the strain in the steel decreases. ACI 318 10.3.5 restricts the amount of steel by restricting the allowable strain in the case of pure flexure. ACI 318 10.5.1: Lower Limit on As As max 3 f c ,200

(

) bf d

w y

(Eq. 1)

A minimum amount of steel is required in all beams. ACI 318 10.5.1 specifies the minimum. Sections 10.5.2 and 10.5.3 give some guidance for implementing 10.5.1 as well as an exception to the minimum steel requirement. Maximum Steel for Deflection Control While not an ACI 318 requirement some authors have indicated that a beam is not likely to have deflection problems when the beam is sized such that the required steel is 0.375 that required for the balanced condition. See the handout on maximum/minimum beams sizes for more discussion on this idea. This condition occurs when the location of the neutral axis, c, equals 0.375 of cb. This can be translated into steel strain limit that can be checked during analysis. From the strain diagram for the balanced condition we can get:

cb = 0.003d f 0.003 + y = Es 87,000d 87,000 + f y

(Eq. 2)

The strain in the steel, given the location of the neutral axis, can be determined from the strain diagram as:

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s =

d -c 0.003 c

(Eq. 3)

Substituting in 0.375cb for c into equation (3) and simplifying the strain in the steel that equals 0.375 of the balanced condition becomes:

s ,at 0.375cb =

f y + 54,375 0.003 32,625

(Eq. 4)

To determine if a beam is LIKELY to have deflection problems or not, the actual steel strain can be compared against this strain. If the actual strain is above this value then the beam is NOT LIKELY to have deflection problems. ACI 318 7.6 Spacing Limits This section should be closely read for a variety of practical spacing limitation. Most of the spacing limits are in place to ensure that the concrete can flow between and around the bars. ACI 318 7.7 Concrete Protection This section should also be closely read for concrete cover requirements. The concrete cover protects the reinforcing steel from corrosion, heat, and other damage.

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