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Figure 5.30. Slender member with differing end moments according to figure 7-1 with eo2/h = Mo2/Nh = 0,1 and Nb/N = 2: Comparison between maximum moment according to exact solution and equivalent first order moment (7-5) with magnification factor (7-4). Thick line = exact solution. Upper thin line = equivalent moment with co = 8 Lower thin line = equivalent moment with co = 10

Figure 5.30 shows good agreement with the exact solution for co = 8, whereas for co = 10 slightly unsafe results may arise. Therefore co = 8 is recommended in 5.8.7.3 (2); this is also consistent with the assumption of a constant equivalent first order moment. The example is based on a comparatively high second order effect (N/Nb = o,5), which enhances the differences.

In many cases it is reasonable to assume that first and second order moments have similar distributions, in which case p « 1. Equation (7-4) can then be simplified to

This corresponds to equation (5.3o) in 5.8.7.3. It can be shown that this expression can be used also for structures, provided a global buckling load can be defined. See 5.8.7.4 for global analysis of structures.

5.8.7.3 Estimation of stiffness

The maximum first order moment Mo for different axial forces, slenderness ratios, and reinforcement ratios can be determined using the general method, see chapter 5.8.6.

The values thus obtained can be considered the "correct" ones. With the "stiffness" method, on the other hand, the maximum first order moment can be expressed on the basis of equation (7-4), assuming that the total moment is equal to the ultimate moment resistance Mu for the normal force N:

The following is a simple model for the stiffness, expressed as the sum of separate contributions from concrete and reinforcement:

where Ec, Es = concrete and steel E-moduli respectively lc, Is = moment of inertia of concrete and steel area

The correction factors Kc and Ks can be calibrated using more or less sophisticated models, to give the required agreement between expression (7-7) and the general method. In 5.8.7.2 (2) basically two alternative models are given: a) (expr (5.22)) is a more accurate alternative, valid for reinforcement ratios down to p = 0,002. b) (expr (5.26)) is a simplified alternative, valid only for reinforcement ratios p > 0,01. Thus, for p < 0,01 only a) may be used, for p > 0,01 either method may be used.

where p is the geometrical reinforcement ratio, As/Ac yet is the effective creep ratio, see chapter 4 k1 depends on concrete strength class, see (7-11) k2 depends on axial force and slenderness, see (7-12) k1 =^720 (7-11)

where n is the relative axial force, Neó / (Acfcd)

Ac is the area of concrete cross section

X is the slenderness ratio, k/i

For cases where X is not defined, a simplified alternative to (7-12) is also given (5.25):

More sophisticated models for estimating the stiffness can be found in [2] and [3]. Background, see [1].

The results of calculations with stiffness evaluated according to expressions (7-9) to (7-12) are presented in Appendix 2 of this report, in the form of comparison with calculations done using the general method. The Appendix also compares the curvature method; see chapter 8.

5.8.7.4 Linear analysis of structures

Clause 5.8.7 opens the possibility of using linear second order analysis for structures, using reduced stiffness(es) taking into account the effect of cracking, creep and material nonlinearity in a simplified way. Without this possibility, the only alternative for second order analysis of structures would be non-linear analysis.

When global second order effects are significant, the effects of cracking etc. may be as important as for isolated members. It should also be kept in mind that second order effects may be significant in a structure, even if the geometrical slenderness of individual bracing units is small, in case the braced units carry a comparatively high vertical load.

The paragraphs applicable to structures are 5.8.3.3 (criterion for ignoring global second order effects), 5.8.7.3 (3) and Annex H. Two different approaches can be distinguished, one based on a magnification factor for bending moments, 5.8.7.3 (3), and the other one based on a similar factor for horizontal forces, H.2.

The two approaches are basically the same, but the one based on moments is suitable mainly for structures with bracing units consisting of shear walls without significant global shear deformations, or structures braced by simple cantilever columns, see examples in figure 5.31.

bracing braced members) members

The two approaches are basically the same, but the one based on moments is suitable mainly for structures with bracing units consisting of shear walls without significant global shear deformations, or structures braced by simple cantilever columns, see examples in figure 5.31.

bracing braced members) members

Figure 5.31. Example of structures where a magnification factor can be applied directly to bending moment(s)

in bracing unit(s)

Figure 5.31. Example of structures where a magnification factor can be applied directly to bending moment(s)

in bracing unit(s)

The approach based on magnification of horizontal forces, on the other hand, can be used for all kinds of structures, and it should be used for frames, shear walls with large openings etc. If properly used, it gives the correct second order effects in structural systems like frames, shear walls with or without openings etc; see the schematic example in figure 5.32.

Bracing members Braced members

Bracing members Braced members

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