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Length af member / Height of structure

Figure 5.1. Comparison between imperfection and tolerances. The thick lines represent imperfections, the black line represents basic imperfection do according to expression 5.1 in EN1992-1-1, whereas the grey line represents the lower limit of the mean value 6t for a structure (m = °°J.

The thin lines represent the tolerance according to EN13670; the solid line represents a member in one storey, the dashed line the total inclination of a structure.

Figure 5.1. Comparison between imperfection and tolerances. The thick lines represent imperfections, the black line represents basic imperfection do according to expression 5.1 in EN1992-1-1, whereas the grey line represents the lower limit of the mean value 6t for a structure (m = °°J.

The thin lines represent the tolerance according to EN13670; the solid line represents a member in one storey, the dashed line the total inclination of a structure.

For the total inclination of a column or wall in a structure, the tolerance continues to decrease below the minimum value 1/300 when the number of storeys exceeds 8. The mean value of the imperfection will also decrease in a structure where a number of individual members contribute to the total effect; the lower limit of the mean value (for m = is shown in figure 5.1. The fact that the mean imperfection 0i is sometimes less than the structural tolerance does not mean that the imperfection is on the "unsafe side". The tolerance has to be checked for individual columns and walls, whereas aim represents the average inclination of all vertical members contributing to a certain effect.

5.2.3 Equivalent eccentricity or horizontal force

For isolated members, like in the ENV, the inclination can be transformed to either an equivalent eccentricity (or initial deflection) ei or a horizontal force Hi. This is important for e.g. a pin-ended column, where an inclination has no effect on the column itself. The eccentricity can then represent either an uncertainty in the position of the axial load, or an initial deflection (out-of-straightness). The equivalent eccentricity is linked to the effective length, and the horizontal force should give the same bending moment as the eccentricity (see figure 5.i in EN chapter 5.2):

Equivalent eccentricity: ei = 0i- fc/2 (l0 is the effective length)

Cantilever: e = 0i- l Mi = N- ei = N- 0i-1 = H-1 ^ H =0r N

Pin-ended: ei = 0i-1/2 Mi = N- ei = N- 0i-1/2 = Hi-1/4 ^ H =20- N

5.2.4 Dealing with first order effects

Imperfections can be treated as first order effects, or be added as separate safety elements without any physical meaning, "outside" the second order analysis. With a method like the "curvature method" ("model column" method in ENV), which gives a fixed second order moment independent of the first order moment, there is no difference between the two approaches. In other methods, however, the second order effects depend on the first order effects, and then it does make a difference whether imperfections are treated as first order effects, or added separately.

In 4.3.5.4 P(i) in the ENV, the imperfection is associated with "uncertainties in the prediction of second order effects", which indicates that it is not regarded as a first order effect. The definition of the first order eccentricity in 4.3.5.6.2 further underlines this. On the other hand, formulations in 2.5.i.3 describe the imperfection rather as a first order effect. Thus, the ENV is ambiguous and unclear in this respect.

In the EN it is stated once and for all that imperfections are to be treated as first order effects; see the definition in 5.8.i. This corresponds to a physical interpretation of the imperfection as a deviation in the form of an inclination, an eccentricity or an initial deflection. This is logical, since there is a link between imperfections and tolerances. It is essential to have a clear definition in this respect for the overall analysis of structures, but also for isolated members, when other methods than the curvature method are used; see 5.8.6 and 5.8.7.

5.3 Idealisation of the C5.3 Structural models structure

5.3.1 Structural models for C5.3.1 Classification of structural elements

°veraN analysis For buildings, as a convention, the following provisions apply:

5. a beam is a linear element, for which the span is not less than 3 times the overall section depth. Otherwise it should be considered as a deep beam.

6. a slab is a bidimensional member for which the minimum panel dimension is not less than 5 times the overall slab thickness. Moreover: a slab subjected to dominantly uniformly distributed loads may be considered to be one-way spanning if either (Fig. 4.2):

- it possesses two free (unsupported) and sensibly parallel edges, or

- it is the central part of a sensibly rectangular slab supported on four edges with a ratio of the longer to shorter span greater than 2.

7. ribbed or waffle slabs need not be treated as discrete elements for the purposes of analysis, provided that the flange or structural topping and transverse ribs have sufficient torsional stiffness. This may be assumed provided that (Fig. 4.2):

- the rib spacing does not exceed i500 mm

- the depth of the rib below the flange does not exceed 4 times its width

- the depth of the flange is at least i/i0 of the clear distance between ribs or 50 mm, whichever is the greater

- transverse ribs are provided at a clear spacing not exceeding i0 times the overall depth of the slab.

Trasversal ribs with distance St

Trasversal ribs with distance St

Figure. 5.2.Geometric parameters for slabs s hf hw

Figure. 5.2.Geometric parameters for slabs

The minimum flange thickness of 50 mm may be reduced to 40 mm where permanent blocks are incorporated between the ribs. This exception applies for slabs with clay blocks only. It does not apply for expanded polystyrene blocks.

An exception to this rule is given at [10.9.3(11 )-EC2] in relation to prefabricated slabs without topping, which may be analysed as solid slabs provided that the in situ transverse ribs are provided with continuous reinforcement through the precast longitudinal ribs and at a spacing according to Table [10.1) - EC2].

A column is a member for which the section depth does not exceed 4 times its width and the height is at least 3 times the section depth. Otherwise it should be considered as a wall.

5.3.2 Geometric data 5.3.2.1 Effective width of flanges (all limit states)

C5.3.2 Geometric data

C5.3.2.1 Effective width of flanges of T beams (valid for all limit states)

If a T beam with a relatively wide flange is subjected to bending moment, the width of flange that effectively works with the rib in absorbing the compressive force (effective width) should be assessed.

An exact calculation shows that the actual distribution of compressive stresses has a higher concentration in the part of flange which is close to the rib, and a progressive reduction in the further parts. This implies that the conservation of plane sections is not respected and that the neutral axis is not rectilinear, but is higher on both sides of the rib.

In order to simplify calculations, the actual distribution of stresses is usually replaced by a conventional block, extended to the effective width. This allows the application of the usual design rules, and in particular the assumption that plane sections remain plane.

Effective width is defined at [5.3.2.1-EC2] as a function of the cross section geometry (b, distance between adjacent ribs; bw, width of ribs) and of the distance lo between points of zero moment. Note that the flange depth is not relevant, even if it is expressly cited in (1)P, and that the distance between points of zero moment depends, for continuous beams, on the type of loading Fig. 5.2-EC2 is an example of a continuous beam (subjected to a uniform load distribution) where the lo distance for spans and for parts on supports is identified.

Hence different sections have different effective width. Point (4) makes clear that a constant width may be assumed over the whole span. The value applicable to the span section should be adopted.

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