crack control 4
4.3 short-term (immeDiate) Deflection
4.3.1 Effects of cracking
If reinforced concrete beams are uncracked, deflection analysis is a simple matter and can be done readily using Equations 4.2(1), 4.2(2) or 4.2(3), as the case may be.
However, reinforced concrete beams crack even under service or working load condi-tions and cracking occurs at discrete seccondi-tions along the beam at quite unpredictable positions. Thus, it may be futile to resort to rigorous analytical methods for deflec-tion calculadeflec-tions. Instead, most researchers choose to solve these problems using one semi-empirical method or another. Loo and Wong (1984) studied the relative merits and accuracy of a group of nine such methods. They came to the conclusion that the so-called effective moment of inertia approach is a convenient and accurate one. It is convenient because the traditional deflection formulas (as given in Equations 4.2(1), 4.2(2) and 4.2(3)) are readily applicable with some modifications to the bending rigidity term or EI.
4.2.3 Limits
As part of the serviceability design, it is necessary to control the deflection. Clause 2.3.2(a) of the Standard specifies the limits for beams and slabs. An abridged version of Table 2.3.2 in the Standard is given in Table 4.2(4).
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For a cracked reinforced concrete beam, the E term is replaced by Ec as defined in Equations 2.1(5) or 2.1(6), and for I an effective value Ief should be used, where, in general,
Icr≤Ief ≤ Ig Equation 4.3(1)
in which Ig is the gross moment of inertia of the uncracked beam section and Icr is that of a fully cracked beam (see Appendix A).
4.3.2 Branson’s effective moment of inertia
The formula for calculating the effective moment of inertia (Ief) adopted in the Standard and several other major codes of practice (including the American Concrete Institute) is originally attributed to Branson (see Loo and Wong 1984). The empirical formula, which takes into consideration the stiffening effects of the concrete in tension between cracks (i.e. tension stiffening) is explicit and all-encompassing. That is
I I I I M Branson formula in its original form underestimates the deflection of very lightly rein-forced beams (see Gilbert 2008). The quantities Mcr, Icr and Ms in Equation 4.3(2) are discussed in detail below.
The quantity Ms is the maximum bending moment at the section, based on the short-term serviceability load under consideration. For simply supported beams, Ms may be taken as the midspan moment; for cantilever beams it should be taken as the root moment.
The moment of inertia for a fully cracked section (Icr) can be determined in the usual manner, once the position of the neutral axis of the transformed section is known. A brief discussion on this topic may be found in Appendix A. For the specific case of a singly reinforced rectangular section
I bd
k pn k
cr= 123
[
4 3+12 (1− ) 2]
Equation 4.3(3) in which the elastic neutral axis parameterk= ( )pn2+2pn pn− Equation A(5)
Note that Equation 4.3(3) may be used for a conservative evaluation of Icr for a doubly reinforced rectangular section.
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Finally in Equation 4.3(2), the cracking moment
M I
in which yt is the distance between the neutral axis and the extreme fibre in tension of the uncracked section; fcs is the maximum shrinkage-induced tensile stress on the uncracked section, which may be computed using a rather tedious process as detailed in the Standard in Clause 8.5.3.1 in conjunction with Clause 3.1.7. By virtue of Equation 2.1(2)
M I
This equation is applicable for beam sections of any given shape.
In Clause 3.1.7.2 of the Standard, the formula for shrinkage strains has an accuracy range of ±30%. Relevant provisions in ACI 318–2011 ignore the shrinkage effects.
In view of the cumbersome computational process involved, especially in determin-ing shrinkage-induced stress, Clause 8.5.3.1 of the Standard recommends the followdetermin-ing approximate equations for flanged beams:
For compression. Otherwise, for example, for an inverted T- or L-beam, bef = bw. Similarly, for a rectangular beam bef = bw = b.
It is worth mentioning here that some writers opt for a conservative and easy solution and simply recommend Ief = Icr (see, for example, Warner, Foster and Kilpatrick (2006)).
4.3.3 Load combinations
After replacing EI with EcIef, Equations 4.2(1), 4.2(2) and 4.2(3), which were developed for homogeneous uncracked beams, may be used for cracked reinforced concrete beams.
It may be assumed that, under service load, the principle of superposition holds.
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The load combination formulas recommended in the Standard for serviceability design have been discussed in Section 1.3.2. Using Equation 1.3(8), 1.3(9) or 1.3(10), as the case may be, the dead, live and wind loads can be combined accordingly and the bending moment (Ms) can be readily determined. With this value of Ms, the effective moment of inertia (Ief) is computed using Equation 4.3(2). Needless to say, the deflec-tion under combined loads may be obtained as the sum of the individual effects, each of which is calculated using the same Ief.
4.3.4 Illustrative example
Problem
Given a simply supported beam (with Lef = 10 m, b = 350 mm, d = 580 mm, D = 650 mm and pt = 0.01), compute the midspan deflection under a combination of dead load including self-weight (g = 8 kN/m) and live load (q = 8 kN/m). Take Ec = 26000 MPa, Es = 200000 MPa and ′fc=32 MPa; assume that the beam forms part of a domestic floor system; ignore the shrinkage effects.
Solution
The gross moment of inertia (see Figure 4.3(1))
Ig=350×650 = × mm4 123 8010 106
Equation 4.3(5) : Mcr Ig kNm
=
− = 0 6 32
325 10 6 83 65
. .
With n=200000= and pt=0.01, (from A
26000 7 69. Equation A(5) ppendix A) yields k = 0.3227. In turn, Equation 4.3(3) gives Icr = 3174 × 106 mm4.
350
650 580
pt = 0.01
figure 4.3(1) cross-sectional details of the example simply supported beam note: all dimensions are in mm.
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For domestic floor systems, ψs and ψl are given in Table 1.3(1) as 0.7 and 0.4, respec-tively. For short-term deflection, Equation 1.3(8) governs and we have combined load = g + 0.7q = 8 + 0.7 × 8 = 13.6 kN/m.
The moment at midspan is
Ms=13 6×10 = kNm 82 170 .
Equation 4.3(2) thus gives
Ief
Ig, which is acceptable.
Finally, Equation 4.2(1) in conjunction with Table 4.2(1) gives
Δ = × ×
It may be apparent in Table 4.2(4) that the immediate deflection under dead and live loads is not a criterion for serviceability design. Its calculation, however, is essential in the analysis of long-term deflection as discussed in Section 4.4 and the total deflections under normal or repeated loading (Section 4.6).
4.3.5 Cantilever and continuous beams
The discussion so far in this section has centred mainly on simply supported beams.
For cantilever beams, the procedure remains basically unchanged, except that for Lef Equation 4.2(5) should be used, and Ms in Equation 4.3(2) should be the bending moment at the root of the cantilever, or at the face of the support girder, column or wall, as the case may be.
For continuous beams, the definition of Lef is the same as for simple beams. However, the procedure for computing Ief may require some elaboration. Figure 4.3(2) illustrates typical spans of continuous beams and the corresponding moment diagrams. For the sections i, c and j, the bending moments are Mi, Mc and Mj, respectively, where Mi and Mj are the results from a statically indeterminate analysis of the continuous beam (or out-put from a comout-puter analysis). For an endspan
M M Mj
c= 0−
2 Equation 4.3(8)
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and for an interior span
M M M M
c
i j
= − +
o
( )
2 Equation 4.3(9)
in which
M wL
o
= ef2
8 Equation 4.3(10)
is the maximum moment of a simply supported span under uniformly distributed load (w).
w w
Mj Mj
–Mj
–Mj
+ Mc + Mc
(Mo – Mc) Mo
(a) Endspan (b) Interior span
–Mi Mi
i i
L/2 L/2 L/2 L/2
C j C j
figure 4.3(2) typical spans of continuous beams and the corresponding moment diagrams
With Ms taking the value of Mi, Mc and Mj, respectively, Equation 4.3(2) may be used to compute Ief for the three nominated sections (Ief.i Ief.c and Ief.j). Note that AS 3600-2009, like its predecessor AS 3600-2001, has no provisions for the deflection analysis of con-tinuous beams per se. However, Clause 8.5.3 in AS 3600-1994 recommends that, for an endspan
I I I
ef
ef,c ef,j
=( + )
2 Equation 4.3(11)
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and for an interior span
I I I I
ef
ef,c ef,i ef,j
=(2 + + )
4 Equation 4.3(12)
For computing midspan deflections, these interpolated values of Ief should be used.
Following the assumption that the principle of superposition holds for deflection analy-sis, the midspan deflection of a continuous span, as illustrated in Figure 4.3(2), can be determined by appropriately combining Equations 4.2(1) and/or 4.2(2) with 4.2(3).