B) El criterio de conservación
3.4.2. Cuentas satélites del medio ambiente
3.4.2.3. Aplicación de las cuentas satélites 'O7
Empirical formulas for computing crack widths in reinforced concrete beams may be found in the following national and international codes:
ACI 318-1995
•
Eurocode 2-1992, Eurocode 2-2004
•
114 part 1 ReinfoRced concRete
BS 8110-1985, BS 8110-1997, BS 8007-1987.
•
Based on the laboratory test results from:
10 (full size) simply supported reinforced box beams
•
five (half-scale) simply supported reinforced solid beams
•
three (full size) two-span continuous reinforced box beams
•
12 (full size) simply supported partially prestressed (pretensioned) box beams,
•
Chowdhury and Loo (2001) developed two unified crack-width formulas applicable to both reinforced and prestressed concrete beams. In their approach, the average crack width
wcr=( / )[ . (f Es s 0 6c s− +) 0 1. ( / )]φ ρ Equation 4.7(1) where
fs = the steel stress in MPa, which, for reinforced beams, may be determined using the transformed section approach given in Appendix A
Es = Young’s modulus of the reinforcement (MPa) c = clear concrete cover (mm)
s = average clear spacing between bars (mm) φ = average bar diameter (mm)
ρ = tension reinforcement ratio.
The maximum crack width may simply be computed as
wmax=1 5. wcr Equation 4.7(2)
Equation 4.7(1) is compared with the test results from 30 reinforced and prestressed beams in Figure 4.7(3). With an error range of ±30%, the linear regression process used to derive the average crack-width formula is considered satisfactory.
0.4
Calculated wcr (mm)
Measured wcr (mm)
figure 4.7(3) comparison of equation 4.7(1) with test results from 30 reinforced and prestressed beams
chapter 4 deflection of beams and cRack contRol 115
In a comparative study on the available literature, Chowdhury and Loo (2001, 2002) collated the following simply supported beam test data:
• wcr from 26 reinforced concrete beams by Clark (1956)
• wcr from 16 reinforced concrete beams by Chi and Kirstein (1958)
• wcr from 8 reinforced concrete beams by Hognestad (1962)
• wcr from 9 reinforced concrete beams by Kaar and Mattock (1963)
• wmax from 34 prestressed pre- and post-tensioned beams by Nawy (1984).
Figure 4.7(4) compares these laboratory test results and the predicted values using Equations 4.7(1) and 4.7(2). The two formulas for average and maximum crack widths perform satisfactorily in that almost all the correlation points lie between the ±30% lines.
Note that points located below the diagonal unbroken thick line represent conservative or safe predictions of crack widths. Satisfactory comparison with the test data published by these independent researchers confirms the reliability of the two empirical formulas.
0.9 0.8 0.7
+30% line –30% line
Kaar & Mattock’s beams Hognestad’s beams
Nawy’s beams (14 post-tensioned) Nawy’s beams (20 pre-tensioned)
Chi & Kirstein’s beams Clark’s beams 0.6
0.5 0.4
0.3
Measured crack widths (mm)
0.2
0.1
0.00.0 0.1 0.2 0.3 0.4 0.5
Equation 4.7(1) or 4.7(2) (mm)
0.6 0.7 0.8 0.9
figure 4.7(4) comparison of empirical formulas with all available reinforced and prestressed concrete beam test results
Based on the published test data for reinforced concrete beams (only), Chowdhury and Loo (2001) also compared the performance of Equations 4.7(1) or 4.7(2) with each of the recommendations made in Eurocode 2-1992, ACI 318-1995 and BS 8110-1985 – the results are presented in Figures 4.7(5)a, b and c, respectively. For a better viewing of the comparison graphs, the reader can refer to the original paper.
116 part 1 ReinfoRced concRete
The comparisons found that:
Equation 4.7(1) for average crack widths compares favourably with, or gives more
con-•
servative results than, the prediction procedure recommended in Eurocode 2–1992 Equation 4.7(2) compares reasonably well with the maximum crack-width formula
•
recommended in ACI 318-1995
the maximum crack-width formula given in the British Standards, BS 8110-1985 and
•
BS 8007-1987, tends to underestimate maximum crack widths considerably and consistently.
Equations 4.7(1) and 4.7(2) are considered superior to all of the code procedures, because they are also applicable to partially prestressed beams – the code formulas are restricted to reinforced concrete beams only. In addition, the recommendations in Eurocode 2-2004 and BS 8110-1997 remain the same as in their respective predecessors.
–30% line
Measured wcr (mm)
0.2 0.15 0.1 0.05
00 0.1 0.2
Calculated wcr (mm) Equation 4.7(1)
0.3 0.4 0 0.1 0.2
Calculated wcr (mm) Eurocode 2-1992
Measured wcr (mm)
0.2
Measured wmax (mm)
0.3
Measured wmax (mm)
0.3 0.2 0.1 0 0.1 0.2 0.0
Calculated wmax (mm) 0.4 0.5 0.3
Equation 4.7(2)
0.6 0.7 0 0.1 0.2
Calculated wmax (mm) 0.4 0.5 0.3
ACI 318-1995
0.6 0.7
figure 4.7(5) (a) comparison of equation 4.7(1) with eurocode 2–1992; (b) comparison of equation 4.7(2) with aci 318-1995; (c) comparison of equation 4.7(2) with bs 8110-1985
chapter 4 deflection of beams and cRack contRol 117
figure 4.8(1) neutral axis location for a fully cracked reinforced concrete section figure 4.7(5) (cont.)
4.8 problems
1. For a fully cracked reinforced concrete section as shown in Figure 4.8(1), the neutral axis coefficient is given in Equation A(5) (see Appendix A) as
k= ( )pn2+2pn pn−
Show that this expression can also be obtained by making use of the equilibrium and compatibility conditions (respectively, C = T and εc is proportional to εs).
(Hint: equate f f
c s
from both conditions.)
0.7 (c)
–30% line –30% line
+30% line +30% line
0.6 0.5 0.4
Measured wmax (mm)
0.3
Measured wmax (mm)
0.3 0.2 0.1 0 0.1 0.2 0.0
Calculated wmax (mm) 0.4 0.5 0.3
Equation 4.7(2)
0.6 0.7 0 0.1 0.2
Calculated wmax (mm) 0.4 0.5 0.3
BS 8110-1985
0.6 0.7
118 part 1 ReinfoRced concRete
2. For each of the reinforced concrete sections detailed in Figure 4.8(2), compute the value of kd. Take ′fc=25 MPa and 32 MPa for the sections in Figure 4.8(2)a and b, respectively.
125 125
10048070
75
50 350250
2 N20 4 N36
120
(a) (b)
600
figure 4.8(2) Reinforced concrete sections with (a) fc′=25 mPa and (b) fc′=32 mPa note: all dimensions are in mm.
3. A simply supported rectangular beam, 340 mm wide and 630 mm deep (with d = 568 mm), is part of a floor system that supports a storage area. The steel ratios pt = 0.008 and pc = 0.0025; Es = 200000 MPa and ′fc=32 MPa. For g = 8 kN/m (including self-weight) and q = 8 kN/m, what is the maximum effective span (Lef) beyond which the beam is not considered by AS 3600-2009 as complying with the serviceability requirement for total deflection?
4. A simply supported beam of 12 m effective span having a cross-section as shown in Figure 4.8(2)b, is subjected, in addition to self-weight, to a dead load of 3 kN/m and a live load of 3 kN/m. Take ′ =fc 32 MPa.
(a) What is the short-term maximum deflection?
(b) Does the beam satisfy the Australian Standard’s minimum effective depth requirement for total deflection?
5. Figure 4.8(3) details a cantilever beam which forms part of a domestic floor.
(a) Under the given loading, compute the total maximum deflection.
(b) Does the design satisfy the requirement of span/depth ratio? You may assume that self-weight is the only sustained load, ′ =fc 32 MPa, Ec = 27000 MPa and n = 7.4.
Note: The deflections at the tip of a cantilever are δ =wL EI
4
8 and δ =PL EI
3
3 , where w is the uniformly distributed load, P is the concentrated load and L is the span.
chapter 4 deflection of beams and cRack contRol 119
6. For the flanged beam considered in Problem 15 (Figure 3.11(10)) in Section 3.11, assume that all other details remain unchanged, but pt = 0.015 and pc = 0.005. What is the minimum acceptable d for the section in order to satisfy the alternative ser-viceability requirement of the Australian Standard? You may assume fcm= ′.fc 7. For crack control of the beam in Problem 23 in Section 3.11, what necessary
mea-sures must be taken?
8. The cantilever beam shown in Figure 4.8(4) is to support a dead load (including self-weight) g = 10 kN/m and a live load q = 30 kN/m. Take fc′ = 32 MPa.
(a) Compute the instantaneous deflection.
(b) Does the design satisfy the alternative Australian Standard requirement for total deflection?
Note:
1. Lef = clear span + D/2
2. For a cantilever beam under uniformly distributed load of w over a span L,
Live load = 15 kN 350
pt = 1.2%
pc = 0.6%
5000
6060 530
figure 4.8(3) details of a cantilever beam note: all dimensions are in mm.
a
a 6000
700
360
10 N28
405640 700
3 N28
Section a-a figure 4.8(4) load configuration and sectional details of a cantilever beam
note: all dimensions are in mm.
120 part 1 ReinfoRced concRete
b = 1200
2 N28 45
100
800
8 N28
300 6060
figure 4.8(5) details of a t-section note: all dimensions are in mm.
Δmax=wL EI
4
8 9. For the T-section shown in Figure 4.8(5):
(a) determine the location of the neutral axis of the fully cracked section (i.e. kd) (b) compute the fully cracked moment of inertia, Icr.
10. A reinforced concrete beam has an overall depth of 450 mm, a width of 300 mm, an effective span of 4.5 m and a concrete strength of fc′ = 20 MPa. Tension reinforcement consists of 2480 mm2 of N bars with an effective depth of 400 mm. The dead- and live-load moments are 95 kNm and 103 kNm, respectively. All dead load is in place before the formwork is removed. If 30% of the superimposed live load may be con-sidered sustained, determine whether the deflection of the beam may be concon-sidered satisfactory as per AS 3600 requirements.
11. The reinforced concrete T-beam shown in Figure 4.8(6) is reinforced with N bars at the positions indicated. Design and detail the additional reinforcement required to prevent cracking in the side faces and check that the spacing of the tension reinforce-ment complies with crack control requirereinforce-ments of AS 3600.
chapter 4 deflection of beams and cRack contRol 121
300 150
2 N28
R12 stirrup
2 N32 3 N28 60
60
1200
6090 9060 figure 4.8(6) details of a reinforced concrete t-beam note: all dimensions are in mm.
122