Environmental Engineering Reference
In-Depth Information
The M-
k
curves are shown alternately with and without tension stiffening.
The values in the legend are: 1) normal force N
x
, 2) percentage of reinforcement
r
s
,
3) design value for tensile strength f
ctd
.
The value f
ctd
¼
0 means that tension stiffening was ignored.
In contrast to the rectangular cross-section, the result is continuously curved M-
k
curves - without the distinctive kinks upon reaching the decompression state or the
yield point of the reinforcement. The reasons for this are the annular form of the cross-
section and the reinforcement uniformly distributed around the circumference.
Tension stiffening has a relatively stronger effect with a lower percentage of
reinforcement
r
s
and a lower normal force N
x
(see Figure 3.3). It may certainly be
neglected with high percentages of reinforcement yet still remain on the safe side, see
DIN 1045-1 [33] 8.6.1 (8).
3.4 Deformations and bending moments according
to second-order theory
For a given bending moment diagram M
z
(x), it is possible to determine the associated
course of the curvatures
k
z
(x)
¼k
(M
z
(x)) along the bar axis. This is done with the help
of the M-
k
curves, which depend on the associated normal forces (see Beton-Kalender
2006 [8] for an example of an application).
The yield condition described in Section 3.1, which is based on the model column
method
3)
according to DIN 1045-1 [33], is not reached when calculating the deforma-
tions of a slender tower with the help of second-order theory because the member
already becomes unstable at an earlier stage due to the increasing deformation as the
bending stiffness decreases. Therefore, the portions of the M-
curves beyond the yield
point of the reinforcement are irrelevant for practical applications.
Integrating twice while taking into account the boundary conditions enables us to
calculate the rotations
k
k
z
(x). It
is worth carrying out a numerical integration according to the trapezoidal rule or
Simpson's rule, paying special attention to discontinuities, for example at abrupt
changes in cross-section or for sudden changes of loading.
The final bending moments according to second-order theory are determined itera-
tively. The starting point is the bending moment diagram according to first-order theory
- or possibly the bending moments according to second-order theory based on an
estimated deflection curve.
The convergence criterion is satisfying the equilibrium conditions. This calculation can
generally be carried out easily by hand by comparing the loads acting on the deformed
structure (deflection curve) and the support reactions.
w
z
(x) and the deflection f
y
(x) from the curvatures
3)
The model column method is equivalent to the method based on nominal curvature acc.
to DIN EN 1992-1-1 [23] 5.8.8.
