Krq

In chapter 5.8.5 an extended definition of the effective depth d has been introduced, in order to cover cases where there is no unambiguous definition of d; see figure 5.34 where is is the radius of gyration of the total reinforcement area hi2

Figure 5.34.Effective depth in cross sections with reinforcement distributed in direction of bending In order to reduce the curvature in cases where yielding is not reached in the tensile reinforcement, a factor Kr is introduced (same as K2 in ENV 1992-1-1, 4.3.5.6.3):

Figure 5.34.Effective depth in cross sections with reinforcement distributed in direction of bending In order to reduce the curvature in cases where yielding is not reached in the tensile reinforcement, a factor Kr is introduced (same as K2 in ENV 1992-1-1, 4.3.5.6.3):

where n = NEd / (Ac fcd), relative normal force (v in the ENV) NEd is design value of normal force nu = 1 + ro nbal is value of n at maximum moment resistance; the value 0,4 may be used ro = As fyd / (Ac fcd)

As is total area of reinforcement Ac is area of concrete cross section

There is another factor Ki in ENV 1992-1-1, 4.3.5.6.3 (2), which reduces the curvature for values of X between 15 and 35. The purpose of this factor was presumably to avoid discontinuity in cases where second order effects may be ignored. However, second order effects will often be ignored for X between 25 and 35 (see 5.8.3.2), so discontinuities will still occur.

Furthermore, independent of method, there is always a basic discontinuity following from the rule that second order effects may be ignored if they are below a certain limit. For these reasons, the factor K1 has not been included in 5.8.8.

The ENV gives no indication of how to take into account creep in the "model column" method. Comparisons with the general method indicate that in certain cases the method can give unsafe results if allowance for creep is not considered, and the factor Kp has been introduced for this purpose. It has been calibrated against calculations with the general method.

More sophisticated models for estimating the curvature can be found in [2] and [3]. Their background is presented in [1].

Comparison with general method and stiffness method

The result of calculations with curvature according to expressions (8-1) to (8-3) is presented in Appendix 2, in comparison with calculations based on the general method. In the same Appendix calculations with the stiffness method (chapter 7) are also presented and compared.

Using the curvature method for structures

In 5.8.5 there is an indication that the curvature method can be used also for structures, "with proper assumptions concerning the distribution of curvature". This statement is based on [4], where a method is given by which the curvature method can be used also for second order analysis and design of unbraced frames. For details, see [4].

5.8.9 Biaxial bending C5.8.9. Biaxial bending

The general method is suitable for biaxial bending also. The same principles as in uniaxial bending apply, although the complexity of the problem increases.

Simplified methods like the stiffness or curvature method can also be used. They are then used separately for each direction, and if the resulting bending moments fulfil a certain criterion, given in expression (5.38), no further action is necessary.

The criterion in (5.38) is similar to expressions (4.74) and (4.75) in the ENV, 4.3.5.6.4, but there is one important difference: the ENV check concerns only first order eccentricities, whereas in 5.8.9 it concerns total eccentricities including second order effects. The reason for including the second order effects is illustrated in figure 5.35:

Figure 5.35. Example of member with different slenderness in the two directions

Assume for example X = 100 in one direction and X = 20 in the other. Second order effects will then be significant in one direction but negligible in the other. A and B are two examples of the position of the axial load, both fulfilling the criterion for separate checks according to the ENV, based on first order eccentricities. This would be acceptable for case A, since the second order effect will make the total eccentricities even "less biaxial". It is not acceptable for case B, however, since the second order effect will now give total eccentricities outside the "permissible" area. Thus, a first order criterion can be misleading and unsafe.

If criterion (5.38) is not fulfilled, the cross section should be designed for biaxial bending. A simple model for this, "in the absence of an accurate cross section analysis", is given in 6.1:

5.9 Lateral instability of slender beams

5.10 Prestressed members and structures

5.11 Analysis for some particular structural members

f—T 4

(ML 1

1 Mrx J

where Mx/y design moment in the respective direction, including nominal 2nd order moment MRx/y corresponding moment resistance of cross section a exponent

The values of the exponent a are taken from a UK proposal based on [5]. The exponent has been slightly adjusted according to [6]. These values can be used in the absence of more accurate values.

C5.9. Lateral instability of slender beams

Compared to the ENV, the following changes have been made:

1. It is clearly stated that the check of lateral instability of beams is relevant in situations where lateral bracing is lacking. For beams in finished structures, lateral instability is normally prevented by lateral bracing from adjacent members (e.g. floor or roof elements).

2. A lateral deflection l/300 has been introduced as an imperfection to be used in calculations concerning lateral instability and balance at supports.

3. The criterion for neglecting second order effects is different. It is explained and compared to the ENV below.

Expression (5.40) is based on a numerical study [7]. It is technically equivalent to the corresponding criterion in the new DIN 1045 [8], but it has a different mathematical formulation to show the main parameters l/b and h/b more clearly.

Figure 5.36 shows a comparison according to [7] between the numerical results and expression (5.40). The corresponding criterion according to the ENV is also shown. It is quite clear that the ENV criterion does not represent the numerical results very well; it is too conservative in many cases and unsafe in other cases. The DIN model is much better.

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