## Euro Code Slenderness Moment Magnification

mjmd

Figure 5.22. Effective creep ratio as a function of ratio Ml/Md for a cracked rectangular cross section with tensile reinforcement only, based on d = 0,9h and a = 6. Basic creep coefficient 9 = 3

Figure 5.22. Effective creep ratio as a function of ratio Ml/Md for a cracked rectangular cross section with tensile reinforcement only, based on d = 0,9h and a = 6. Basic creep coefficient 9 = 3

In this case the curves will approach the straight line according to expression (5.19) the higher the reinforcement ratio is. However, curves for low and moderate ratios are also quite close.

### 5.8.4.2.4 Conclusions concerning cross sections

The idea behind the effective creep ratio in 5.8.4 is illustrated in figure 5.20 and demonstrated in three examples. The simple linear relationship according to expression (5.19) is always more or less conservative, but deviations are generally small. Furthermore, in a reinforced section the overall effect of creep on stiffness is reduced with increasing reinforcement, since creep only affects the concrete contribution to the stiffness. Therefore, the effect of deviations on the stiffness will not be as high as it may appear from the above figures.

5.8.4.3 Effect of creep in slender columns 5.8.4.3.1 General

The above conclusion concern only cross sections and are based on linear material behaviour. In this point the relevance of the effective creep ratio for slender columns are examined. A slender column behaves in a non-linear way, due to both material and geometrical non-linearity. A non-linear behaviour similar to the linear one in figure 5.20 is outlined in figure 5.23.

J. Yt. = ^ - y, \ Figure 5.23. Illustration of load history and deformations with nonlinear behaviour

J. Yt. = ^ - y, \ Figure 5.23. Illustration of load history and deformations with nonlinear behaviour

The load history can be divided into three steps:

1. Application of long-term load Ql, immediate deformation yi, calculated for 9ef = 0

2. Long-term load Ql during time t-to, total deformation yi, calculated for 9ef = 9

3. Load increase up to design load Qd, additional deformation y3 - yi, calculated for 9ef = 0

A realistic calculation representing this load history should involve these three steps, including the relevant first order moments or eccentricities for each step. As a simplification, steps 1 and 2 can be combined into one, using a stress-strain diagram with the strains multiplied by (1+9), see 6.4. This corresponds to line AC, and the calculation is then reduced to two steps.

The last step can be calculated in two alternative ways:

a. After calculating point C, the additional load Qd - Ql is added, with deformation starting from yi. See line CD in figure 5.23.

b. After calculating point C, the total load Qd is applied "from scratch", but with yo = yi - yi as an initial deflection added to other first order effects. See line ED in figure 5.23.

Alternative b. will be used in two-step calculations in the following way. The distribution of yo along the column should in principle be the same as the distribution of y2 - yi. For a pin-ended column, however, a sine-shaped or parabolic distribution will be adopted as a simplification.

A further simplification is a one-step calculation, using an effective creep ratio 9ef (line AD in the figure). For the definition of ^ef there are two main options:

a) based on first order moments Mol and Mod, i.e. 9ef = Mol / Mod b) based on total moments Ml and Md, including 2nd order moments, i.e. 9ef = 9- Ml / Md

The relevant deformation parameter in second order analysis is curvature, which depends primarily on bending moment. Therefore, the axial load should not be included in the definition of effective creep ratio.

Alternative b) is the most realistic one, since creep deformations will mainly be governed by total moments. With this alternative, however, iteration is inevitable since second order moments depend on stiffness, which depends on effective creep ratio, which depends on total moments etc. Therefore, alternative a) will be the normal choice in practical design.

Alternative a. is always more or less on the safe side. The reason is that the second order moment is a non-linear function of the axial load. Therefore, the moment increase due to second order effects will be greater under design load than under long-term load, and the ratio Ml/Md will be lower if second order moments are included. This is easy to verify with a magnification factor based on linear material behaviour (see chapter 5.8.7); this tendency will be even stronger in a non-linear analysis.5

5.8.4.3.2 Comparison between one- and two-step calculations

An example will be used to compare the one-step calculation, using the effective creep ratio, with the more realistic two-step calculation. A high slenderness ratio has been chosen, in order to emphasize the effects considered. All calculations below have been done with the general method. (for a general description and discussion of this method, see chapter 5.8.6). Geometric assumption are:

- Concrete C40

- Reinforcement S500

- Rectangular cross section with reinforcement concentrated to opposite sides

- Mechanical reinforcement ratio ro = 0,15 (total reinforcement)

- Edge distance of reinforcement 0,1 h

- Eccentricity eo = 0,08h (same for long-term and design load; no other first order effect)

- Basic creep coefficient 9 = 3

In the following, all axial loads and bending moments are expressed in relative terms, i.e. n = N / Acfcd and m = M / hAfcd. Therefore, no absolute dimensions are used.

1. immediate deformation, calculated with 9= 0: y1/h = 0,0173

2. total deformation, calculated with 9 = 3: y2/h = 0,0819

3. creep deformation: yo/h = 0,0819 - 0,0173 = 0,0646

yo is taken as an initial deflection with parabolic distribution, and is added to the constant first order eccentricity eo given above. The load capacity under this total first order effect and no creep (9 = 0) is calculated. The result is nRd = 0,235

3. yo/h = 0,134 - 0,023 = 0,111 (creep deformation)

The load capacity calculated with eo+ yo and with 9 = 0 is nRd = 0,189

These values are compared to the result of a one-step calculation, using an effective creep ratio based on first order moments.

9ef = 9- Mol / Mod = 9- NL- eo / (ND- eo) = 9- nL / nD = 3 0,100/0,235 = 1,28 6

The load capacity with 9ef = 1,28 is nRd = 0,198

Cf. 0,235 in two-step calculation; thus the result is 16% conservative

5 A "curvature method", giving a fixed 2nd order moment, would lead to the wrong conclusion here.

6 In this particular example the first order moment is proportional to the axial load, therefore the effective creep ratio can be based on axial loads as well as moments. In the general case only moments should be used.

Cf. 0,189 in two-step calculation; thus the result is 12 % conservative

These results are somewhat conservative, as could be expected (the reason is explained above).

Next is a one-step calculation with yet based on total moments.

Total moment under n is mL = a- (eo + 72) = 0,100- (0,08 + 0,0819) = 0,0162

After iteration the following values are found:

Total moment under design load mD = 0,0618

Effective creep ratio yet = y- mL / mD = 3- 0,0162/0,0618 = 0,786

Load capacity with yet = 0,786 is nRd = 0,224 (total moment for this load is mD = 0,0618)

This is within 5 % of the two-step calculation (which gave nRd = 0,235)

Total moment under nL is mL = nL- (e0 + 72) = 0,125- (0,08 + 0,134) = 0,0267 After iteration: mD = 0,0531, yef = y- mL / mD = 3- 0,0267/0,0531 = 1,151, nRd = 0,183 This is within 3 % of the two-step calculation (nRd = 0,189) 5.8.4.3.3 Conclusions

It is conservative to use an effective creep ratio based on first order moments; total moments will give more accurate results. In practical design, however, total moments are much more complicated to use, however, since iteration will be necessary. Therefore, the normal procedure will be to use first order moments. This is further discussed below.

### 5.8.4.4 The effect of creep on slenderness limit

The effect of creep on slenderness limit will be further studied here, comparing the one-step and two-steps methods according to 4.3.1 and 4.3.2. It is thus a complement to clause 3.1, dealing with the slenderness limit in general. It is also a complement to 4.3.2, dealing with creep combined with a high slenderness, since this clause deals with low slenderness ratios.

Table 5.7 shows the results of calculations, based on a slenderness corresponding to the limit for which second order effects may be neglected with yet = 0, see 3.1. The basic parameters are the same as for the example in 4.3.2, except those for which different values are given.

Table 5.7. The effect ot creep tor columns with a low slenderness.

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