Theoretical Background > ULS Instability > Nominal curvature method

The analysis of slender isolated columns may be performed with simplified methods such as that based on "nominal curvature method" used in the present program for the typology "rectangular section of columns".

The nominal curvature method (§5.8.8 EC2) is primarily suitable for isolated member with constant normal force NEd and defined length l0 (pre-calculated according to § 5.8.3.2 EC2). The purpose of this method is to give a nominal second order moment M2 (based on effective length l0 and an estimated max curvature) to add to the first order moment M0Ed (inclusive of the effect of imperfections e0=l0/400) so to have a final design moment MEd:

MEd = M0Ed + M2 (5.31)EC2

M2 = NEd ⋅ e2 = NEd (1/r) ⋅ l02/c

where:

1/r = Kr ⋅ Kφ ⋅ 1/r0 = Kr ⋅ Kφ ⋅ εyd/(0.45 ⋅ d) (5.34)EC2

c = 10 for constant cross sections (or less if the first order moment is constant - 8 is a lower limit)

program use 10 always

d = h/2 + is this value for effective depth is to be used if reinforcement is not concentrated on opposite sides

is radius of gyration of the total reinforcement area

Kr correction factor depending on axial load (see (5.36)EC2) : if Kr> 1 then Kr=1

Kφ correction factor for taking account of creep (see (5.37)EC2) : if Kφ< 1 then Kφ=1

The curvature 1/r0 in (5.34) § 5.8.8.3 EC2 can be assessed by the program with the direct formula in § 5.8.8.3 EC2 or by means the moment-curvature diagram as in "model column" method (see ENV). In biaxial bending it is difficult to use the direct formula (5.34) as it is hard to meet the conditions of applicability in (5.38a) and (5.38b). These conditions are always checked by program. The biaxial moment-curvature diagram provided by the program make always possible to check biaxial instability. If curvature 1/r0 is assessed with moment curvature diagram (uniaxial or biaxial), only Kφ is used to correct the curvature as Kr is implicitly taken in account in the diagram.

Slenderness is defined as follows:

λ = l0 /i

where:

l0 is the effective length of the isolated member as defined in §5.8.3.2(2) EC2

i is the radius of gyration of the uncracked concrete section = √(J/A)

Second order effects may be ignored if the slenderness λ is below λlim:

λlim = 20 ⋅ A ⋅ B ⋅ C / √n (5.13)

A = 1/(1+0.2 ⋅ φef)

B = √(1+2ω)

C = 1.7 - rm (if rm is not known C = 0.7 may be used)

φef = φef ⋅M0Eqp/M0Ed effective creep ratio [see (5.19)EC2]

ω = As fyd / (Ac Fcd) where As is the total area of long. bars

n = NEd/(Ac⋅fcd)

Program applies (5.13) setting C = 0.7 always (rm is not assigned) and φef as the input data assigned in the dialog window. The other parameter are directly calculated.

If λ ≤ λlim program stops calculations with a message.