MECANISMOS DE DEFENSA DEL SUSCRIPTOR O USUARIO EN SEDE DE LA EMPRESA

A model for predicting the maximum final crack width (w∗) in reinforced concrete flexural members based on the tension chord model of Marti et al. (Ref. 32) was recently proposed (Ref. 34). A modified version of that model is presented here and has been shown to provide good agreement with the measured final spacing and width of cracks in reinforced concrete beams and slabs under sustained loads. The notation associated with the model is shown in Fig. 3.8.

Consider a segment of a singly reinforced beam of rectangular section subjected to an in-service bending moment, Ms, greater than the cracking moment, Mcr. The

spacing between the primary cracks is s, as shown in Fig. 3.8a. A typical cross-section between the cracks is shown in Fig. 3.8b and a cross-section at a primary crack is shown in Fig. 3.8c. The cracked beam is idealised as a compression chord of depth kd and width b and a cracked tension chord consisting of the tensile reinforcement of area Ast surrounded by an area of tensile concrete (Act) as shown in Fig. 3.8d. The

centroids of Ast and Act are assumed to coincide at a depth d below the top fibre of

the section.

For the sections containing a primary crack (Fig. 3.8c), Act= 0 and the depth of

the compressive zone, kd, and the second moment of area about the centroidal axis (Icr) may be determined from a cracked section analysis using modular ratio theory

(Eqs 3.14 and 3.17).

Away from the crack, the area of the concrete in the tension chord of Fig. 3.8d (Act)

is assumed to carry a uniform tensile stress (σct) that develops due to the bond stress

(b) Uncracked section (a) Beam elevation

b kd d (1−k)d h Ms s _{Primary crack}

(d) Idealised compression and tension chord model C T Ts + Tc Act Ast Ast Ast s Cracked tension chord

kd b d C (c) Cracked section b

Figure 3.8 Cracked reinforced concrete beam and idealised tension chord model (Ref. 34).

For the tension chord, the area of concrete between the cracks, Act, may be taken

as:

Act= 0.5(D − kd)b∗ (3.45)

where b∗is the width of the section at the level of the centroid of the tensile steel (i.e. at the depth d) but not greater than the number of bars in the tension zone multiplied by 12db. At each crack, the concrete carries no tension and the tensile steel stress is σst1= T/Ast, where:

T=nMs(d− kd) Icr

Ast (3.46)

As the distance z from the crack in the direction of the tension chord increases, the stress in the steel reduces due to the bond shear stressτb between the steel and the

surrounding tensile concrete. For reinforced concrete under service loads, whereσst1

is less than the yield stress fy, Marti et al. (Ref. 33) assumed a rigid-plastic bond shear

stress-slip relationship, withτb= 2.0fctat all values of slip, where fctis the direct tensile

strength of the concrete. To account for the reduction in bond stress with time due to tensile creep and shrinkage, Gilbert (Ref. 34) took the bond stress to beτb= 2.0fctfor

short-term calculations andτb= 1.0fct when the final long-term crack width was to

be determined. Experimental observations (Refs 35 and 36) indicate thatτb reduces

as the stress in the reinforcement increases and, consequently, the tensile stresses in the concrete between the cracks reduces (i.e. tension stiffening reduces with increasing steel stress). In reality, the magnitude ofτbis affected by many factors, including steel

stress, concrete cover, bar spacing, transverse reinforcement (stirrups), lateral pressure, compaction of the concrete, size of bar deformations, tensile creep and shrinkage. It is recommended here that in situations where the concrete cover and the clear spacing between the bars are greater than the bar diameter, the bond stressτbmay be

taken as:

τb= λ1λ2λ3fct (3.47)

whereλ1accounts for the load duration withλ1= 1.0 for short-term calculations and λ1= 0.7 for long-term calculations; λ2is a factor that accounts for the reduction in bond stress as the steel stressσst1(in MPa) increases and is given by (Ref. 36):

λ2= 1.6˙6 − 0.00˙3σst1≥ 0.0 (3.48)

and λ3 is a factor that accounts for the very significant increase in bond stress that has been observed in laboratory tests for small diameter bars (Ref. 35) and may be taken as:

λ3= 7.0 − 0.3db≥ 2.0 (db in mm) (3.49)

An elevation of the tension chord is shown in Fig. 3.9a and the stress variations in the concrete and steel in the tension chord are illustrated in Figs 3.9b and 3.9c, respectively. Following the approach of Marti et al. (Ref. 33), the concrete and steel tensile stresses in Figs 3.9b and 3.9c, where 0< z ≤ s/2, may be expressed as:

σstz= T Ast− 4τbz db (3.50a) T s/2 s/2 Crack Crack

(a) Elevation of tension chord between cracks

z

(b) Tensile concrete stress

(c) Tensile steel stress t_b

sc2

sst1

sst2

sst1

and

σcz=4τbρtcz db

(3.50b)

whereρtc is the reinforcement ratio of the tension chord (= Ast/Act) and db is the

reinforcing bar diameter. Mid-way between the cracks, at z= s/2, the stresses are:

σst2= T As− 2τbs db (3.51a) and σc2= 2τbρtcs db (3.51b)

The maximum crack spacing immediately after loading, s= smax, occurs when σc2= fct, and from Eq. 3.51b:

smax= fctdb

2τbρtc (3.52)

withλ1= 1.0 in Eq. 3.47. If the spacing between two adjacent cracks just exceeds smax, the concrete stress mid-way between the cracks will exceed fctand another crack

will form between the two existing cracks. It follows that the minimum crack spacing is half the maximum value, that is, smin= smax/2.

The instantaneous crack width (wi)tcin the fictitious tension chord is the difference

between the elongation of the tensile steel over the length s and the elongation of the concrete between the cracks and is given by:

(wi)tc= s Es T Ast − τbs db (1+ nρtc)  (3.53)

Depending on the dimensions of the cross-section and the concrete cover, the instantaneous crack width at the bottom concrete surface of the beam or slab, (wi)soffit,

may be different from that given by Eq. 3.53 for the tension chord and may be obtained from: (wi)soffit= kcover(wi)av= kcovers Es T Ast− τbs db (1+ nρtc)  (3.54)

where kcover is a term to account for the dependence of crack width on the clear

concrete cover c and may be taken as:

kcover= D− kd d− kd 5c (D− kd) − 2db 0.3 (3.55)

Under sustained load, additional cracks occur between widely spaced cracks (usually when 0.67smax< s ≤ smax). The additional cracks are due to the combined effect of

tensile creep rupture and shrinkage. As a consequence, the number of cracks increases and the maximum crack spacing reduces with time. The final maximum crack spacing s∗ is only about two-thirds of that given by Eq. 3.52, but the final minimum crack spacing remains about half of the value given by Eq. 3.52.

As previously mentioned, experimental observations indicate thatτbdecreases with

time, probably as a result of shrinkage-induced slip and tensile creep. Hence, the stress in the tensile concrete between the cracks gradually reduces. Furthermore, although creep and shrinkage will cause a small increase in the resultant tensile force T in the real beam and a slight reduction in the internal lever arm, this effect is relatively small and is ignored in the tension chord model presented here. The final crack width is the elongation of the steel over the distance between the cracks minus the extension of the concrete caused byσczplus the shortening of the concrete between the cracks due to

shrinkage. For a final maximum crack spacing of s∗, the final maximum crack width at the member soffit is:

(w∗)soffit= kcovers∗ Es T Ast − τbs∗ db (1+ neρtc)− εshEs  (3.56)

where εsh is the shrinkage strain in the tensile concrete (and is a negative value); ne= Es/Ee= the effective modular ratio; Ee is the effective modulus given by Ee= Ec/(1 + ϕ(t,τ)); Ecand Esare the elastic moduli of the concrete and steel respectively;

andϕ(t,τ) is the creep coefficient of the concrete.

A good estimate of the final maximum crack width is given by Eq. 3.56, where s∗is the maximum crack spacing after all time-dependent cracking has taken place, that is, s∗= 0.67smax, and smax is given by Eq. 3.52. By rearranging Eq. 3.56, the steel stress

on a cracked section corresponding to a particular maximum final crack width (w∗) is given by: fst= w∗Es s∗kcover+ τbs∗ db (1+ neρtc)+ εshEs (3.57)

By substitutingτb(from Eq. 3.47) and s∗= 0.67smaxinto Eq. 3.57 and by selecting a

maximum desired crack width in a particular structure, w∗, the maximum permissible tensile steel stress can be determined.

Example 3.6

The maximum final crack widths determined using Eq. 3.56 are compared with the measured maximum final crack widths for the 12 prismatic, one- way singly reinforced concrete specimens (six beams and six slabs) tested by Gilbert and Nejadi (Ref. 35). The specimens were described in Example 3.5 and their cross-sections are shown in Fig. 3.7 and details provided in Tables 3.9 and 3.10. Typical maximum crack width calculations are provided here for Beam B2-a.

Beam B2-a

Relevant dimensions and properties are: b= 250 mm, D = 333 mm, d = 300 mm, Ast = 400 mm2, Ec= 22,820 MPa, n = Es/Ec = 8.76, ϕ(t,τ) = 1.71, εsh=

−0.000825, Ee= 8420 MPa and ne= 23.8. A cracked section analysis gives kd= 78.8 mm and Icr= 212 × 106mm4.

From Eq. 3.45, the area of concrete in the tension chord is Act= 0.5 × (333 −

78.8) × 250 = 31,780 mm2 and the reinforcement ratio of the tension chord ρtc= Ast/Act= 0.0126. When Ms= 24.8 kNm, the tensile force in the steel on

the cracked section is Eq. 3.46:

T=8.76 × 24.8 × 106(300− 78.8)

212× 106 × 400 = 90.6 kN

and σst1= T

Ast = 226 MPa

For short-term calculations, λ1 = 1.0 and, from Eqs 3.48 and 3.49, λ2 = 1.6˙6 − 0.00˙3 × 226 = 0.91 and λ3 = 7.0 − 0.3 × 16 = 2.2. With fct =

0.6f_c(t)= 2.57 MPa, the instantaneous bond stress is obtained from Eq. 3.47 as τb= 1.0 × 0.91 × 2.2 × 2.57 = 5.15 MPa and the maximum crack spacing

immediately after loading is given by Eq. 3.52:

smax= 2.57 × 16

2× 5.15 × 0.0126= 317 mm

For the calculation of the maximum final crack width, the maximum crack spacing s∗is taken as 2/3 of the instantaneous value and therefore s∗= 2/3 × 317= 211 mm. From Eq. 3.47, for long-term calculations, τb= 0.7 × 0.91 ×

2.2 × 2.57 = 3.60 MPa and, from Eq. 3.55, kcover= 0.967.

The maximum final (long-term) crack width at the soffit of the beam specimen B2-a is obtained from Eq. 3.56:

(w∗)soffit= 0.967 × 211 200,000 90,600 400 − 3.60 × 211 16 (1+ 23.8 × 0.0126) − (−0.000825) × 200,000  = 0.337 mm

The measured maximum final crack width on this specimen after 400 days under load was 0.36 mm.

The measured and calculated maximum final crack widths for all 12 test specimens are compared in Table 3.12. The mean of the ratios of predicted to measured crack widths for the six beam specimens is 1.059, with a coefficient of variation of 22.8 per cent while for the six slab specimens the mean is 0.945, with a coefficient of variation of 17.7 per cent. The agreement between the calculated and measured maximum final crack width for this set of test data is good.

Table 3.12 Comparison of measured and predicted maximum final crack widths (mm)

Specimen B1-a B1-b B2-a B2-b B3-a B3-b S1-a S1-b S2-a S2-b S3-a S3-b

Measured, wmax 0.38 0.18 0.36 0.18 0.28 0.13 0.25 0.20 0.23 0.18 0.20 0.15

Predicted, (w∗)_soffit 0.425 0.262 0.337 0.207 0.212 0.122 0.287 0.196 0.258 0.155 0.162 0.111

(w∗)_soffit/wmax 1.119 1.457 0.935 1.149 0.758 0.936 1.148 0.981 1.123 0.863 0.812 0.742

Example 3.7

A 150 mm thick simply-supported one-way slab located inside a building is to be considered. With appropriate regard for durability, the concrete strength is selected to be f_c= 32 MPa and the cover to the tensile reinforcement is 20 mm. The final shrinkage strain is taken to beεsh= −0.0006. Other relevant material

properties are Ec= 28,600 MPa; n = Es/Ec= 7.0; ϕ(t,τ) = 2.5; fct= 2.04 MPa

and Es= 200 GPa. The effective modulus is therefore Ee= Ec/(1 + ϕ(t,τ)) =

8170 MPa and the effective modular ratio ne= Es/Ee= 24.5. The tensile face

of the slab is to be exposed and the maximum final crack width is to be limited to w∗= 0.3 mm.

After completing the design for strength and deflection control, the required minimum area of tensile steel is 650 mm2_{/m. Under the full service loads, the} maximum in-service sustained moment at mid-span is 20.0 kNm/m. The bar diameter and bar spacing must be determined so that the requirements for crack control are also satisfied.