where (see figure 5.28) M = total moment Mo = first order moment M2 = second order moment N = axial force y = deflection corresponding to 1/r 1/r = curvature corresponding to y l = length c = factor for curvature distribution m

Figure 5.28. Illustration of deformations and moments in a pin-ended column. (In the figure, first order moment is exemplified as the effect of a transverse load. First order moment could also be given by eccentricity of the axial load.)

The difference between the two methods lies in the formulation of the curvature 1/r.

In the stiffness method 1/r is expressed in terms of an estimated nominal flexural stiffness EI:

r EI

The stiffness EI should be defined in such a way that ULS cross section design for the total moment M will give an acceptable end result in comparison with the general method. This includes, among other things, taking account of cracking, creep and non-linear material properties.

In the curvature method, the curvature 1/r is estimated directly, on the basis of assuming yield strain in tensile and compressive reinforcement:

This model overestimates the curvature in those cases where yielding is not reached, giving a too conservative end result. The typical example is where the ultimate load is governed by stability failure, before reaching the cross section resistance. The model may also underestimate the curvature in some cases, since it does not take into account creep. However, various corrections can be introduced to improve the result.

In the following chapters the two simplified methods will be described and compared to the general method.

5.8.7 Method based on nominal stiffness General Nominal stiffness Moment magnification factor

C5.8.7. Method based on stiffness Basic equations

A simple isolated column is considered, e.g. pin-ended with a length l = lo; see figure 5.28. The second order moment can be expressed in the following way, Cf. equation (6-1) and fig. 5.28:

Cn C2

With co and c2 it is possible to consider different distributions of first and second order moments (primarily the corresponding curvatures). Solving for M2 gives l2

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