The question then arises about which time should be used to compute the creep coefficient. To solve this problem the following procedure is proposed:

• Compute fg1 +g2+q(t3, £3) with 9(t,t3) and £3.

• Compute the total deformation by using the following expression:

fg1 +g2+q,t» = fg1,(t1,£1) + fg1+g2(t2,£2) - fq1(t2,£1) + fg1+g2+q(t3,£3) - fg1+g2(t3, £2) (7.65)

This complicated and time-consuming procedure is necessary due to progressive cracking of cross sections. This expression takes into account, for example, that part of the deflection due to gi occurs at time t2 due to the reduction of stiffness produced by the application of load g2. The creep of this extra deflection must therefore be referred to time t2. This is what is achieved by the above expression.

In case a construction live load equivalent to the value of gi+g2+^oiq is assumed to be applied at time ti, the above expression is greatly simplified, since no reduction of stiffness occurs after the application of gi+construction load:

fg1 +g2+q,t» = fg1,(t1,£3) + fg1+g2(t2,£3) - fq1(t2,£3) + fg1+g2+q(t3,£3) - fg1+g2(t3, £3) = fg1,(t1,£3) + fg2+q(t2,£3) + fq(t3,£3) Deflections due to shrinkage

Deflections due to shrinkage are computed for a stiffness corresponding to the quasi-permanent load condition, taking into account an effective modulus with a creep coefficient corresponding to the start of development of shrinkage (i.e. end of curing), ^(t,ts). Also, II and III are calculated using Ec,eff.

7.4.3 Parametric study of slenderness limit Introduction

ENV 1992-1-1:1991 in table 4.14 and MC-90 in table 7.5.2 provide slenderness limits for lightly reinforced and heavily reinforced concrete elements. In both cases, for a simply supported beam the corresponding values are, respectively, L/d=18 and L/d=25. These values are given for a steel yield stress of 400 N/mm2, and are inversely proportional to the steel grade. This means, that the equivalent values for a 500 N/mm2 steel would be L/d=14 and L/d=20.

There have been complaints in the sense that this table is too conservative, or too general. The parametric study described in the following sections, considers a large range of variables affecting the deformation of concrete structures, in order to quantify their influence and study the possibility of including them in the calculation of the slenderness limit. The present proposal for section 7.4 of prEN 1992-1, is based on this study.

The parametric study, which has been carried out according to the procedure described above, considers the influence on the slenderness limit of the following parameters:

• Complex load history. The influence of the values of ti, t2 and t3 on the slenderness limit has been studied.

• Control of total deflections vs. control of deflection which produced cracking of partitions (referred to in this document as active deflection). Slenderness limits are calculated by limiting the total deflection to L/250. It has been investigated whether the limitation of the deflection producing cracking of partitions to L/500 can be more restrictive.

• Influence of relative humidity. The relative humidity affects long term deflections through creep and shrinkage. The influence of this parameter on the slenderness limit has been studied for relative humidity varying from 50 to 80%.

• Real reinforcement vs. required reinforcement. The effect of considering a 5% to 10% increase in the real reinforcement with respect to the required reinforcement determined from U.L.S. analysis has been studied in order to take into account the round-off in detailing.

• Distribution of reinforcement. Reinforcement in real beams is not constant. The influence of the real distribution of reinforcement on the slenderness limit has been studied.

• Concrete grade

• Percentage of self weight (gi), additional deal load (flooring and partitions g2) and quasipermanent live load (^02Q) with respect to the total load (qtot). According to Spanish practice, typical values for these relations could be:

- For one-way slabs,




- For flat slabs,




For this study, as for prEN 1992-1, the slenderness limit is defined as the relationship between the span and the effective depth L/d.

The slenderness limit curves which are presented in the following paragraphs, are given for different reinforcement ratios. The reinforcement ratio, p, is defined as the ratio of tensile reinforcement As, to effective cross section bd. Assumptions for parametric study

The parametric study which follows has been carried out for a simply supported beam with a cross section of 100 x 30 cm2. The cover has been assumed as 1/10 of the total depth. The reference values for which the study is formulated are the following:

- Relative humidity of 70%

- Load history: tM =10/60/365 days

- Permanent load vs. live load: gi = 45%qtot, g2 = 30%qtot and q = 25% of qtot.

- Quasi permanent live load is 30% of characteristic live load

- Tensile and compressive reinforcements are those strictly needed for ULS.

- Concrete Strength: 30 N/mm2

- Steel Yield Stress: 500 N/mm2

- Distribution of reinforcement is considered constant over the beam length.

For each part of the study, one of the above parameters is varied while the others remain constant. Method for determining the slenderness ratio

In order to determine the slenderness ratio, the following steps were taken for each reinforcement ratio:

- A certain span length, L, is assumed.

- Calculation of the ultimate bending moment (Muls). When compression reinforcement is needed in order to yield the tensile reinforcement, compressive reinforcement is provided and taken into account in the calculation of deflections.

- The ultimate load, quLs, is determined from the Muls, assuming a simply supported beam:

qULS L2 '8

- The total service load (qtot) is determined from quLs, and the assumed ratios for g1/qtot, g2/qtot and q/qtot, according to:

quLs - 1.35- (g1 + g2) + 1.5- q - 1.35- (0.45 + 0.30)- qtot + 1.5- 0.25- qtot - 1.38- qtot ^ qtot - 0.72- quLs

- The values of g1, g2 and q are determined from the above ratios.

- The deflection is computed. According to the general method of prEN 1992-1. If the deflection obtained is not L/250 for total deflection or L/500 for active deflection, the procedure is repeated until convergence is achieved. Influence of the dimensions of the cross section used

The cross section assumed for the parametric study is, as stated above, a rectangular cross section of b x h - 100 x 30 cm2. In order to insure that the particular dimensions of the cross section are not important, the slenderness limit for different reinforcement ratios has also been determined for the cross section used by Beeby in [2], all other parameters being those taken as reference (see section 3.2).

This cross section is rectangular of dimensions b x h - 30 x 50 cm2. Figure shows the comparison between both rectangular cross sections. As can be seen no significant difference can be observed.

Figure 7.18. Slenderness ratio for two rectangular cross section of different dimensions Influence of load history

The load history used for the parametric study is described above in 2.2.1. It is assumed that the construction live load is applied at the same time as the self weight so that the section is fully cracked from the beginning. This provides an upper bound estimation for total deflection (not so for active deflection).

The reference load history is: UA¡/t3=10/60/365 days

Two other load histories are considered: UÂ2/t3=7/14/365 days and U/t2/t3=28/90/365 days.

The calculation of the slenderness limit has been carried out in this case for two steel ratios: 0.5% and

The results are shown in Table 7.5. As can be seen, the influence of the load history is very limited. This suggests that simplifications are possible. One such simplification, consisting in considering a single time of loading together with an equivalent creep coefficient is described in detail in section

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