Environmental Engineering Reference
In-Depth Information
3
Non-linear material behaviour
3.1 General
DIN 1045-1 [33] 8.6 stipulates that the equilibrium state of loadbearing structures with
bar-type members or walls subjected to axial compression - and in particular the
equilibrium state of these members themselves - has to be verified taking into account
the effects of member deformations when such deformations reduce the load-carrying
capacity by more than 10%. This situation should generally be assumed for slender
towers.
The equilibrium state with respect to the deformed loadbearing structure is verified
by calculating the internal forces according to second-order theory, that is by using a
geometric non-linear analysis. The deformations of the loadbearing structure or the
structural members subjected to axial compression increase disproportionately as
the load increases and so the ultimate limit state is especially critical. The ultimate
load decreases in comparison to a calculation based on first-order theory, or stability
problems occur, depending on the flexibility of the loadbearing structure or the
slenderness of the member being investigated, see [34].
Deformation analyses at the ultimate limit state must take into account the non-linear
material behaviour of reinforced concrete, that is
- the formation of cracks in the tension zones of the member cross-sections,
- the non-linear stress-strain curve for the concrete (Figure 3.1),
- the non-linear stress-strain curve for the reinforcing steel (Figure 3.2) and, if
applicable,
- the non-linear stress-strain curve for the prestressing steel (Figure 3.4).
A geometric and physical non-linear analysis of the internal forces is therefore
necessary. Such an analysis requires an iterative procedure because the changes in
stiffness associated with the load increases have to be recalculated again and again.
That is only possible with the help of computer programs.
Such calculations are very involved and so the model column method may be used when
verifying compression members with square, rectangular or circular cross-sections
that fall within the range of applicability given in DIN 1045-1 [33] 8.6.5. The
model column method converts the calculations according to second-order theory
into a cross-sectional design based on the strains
e
yd
upon yielding of the longitudinal
reinforcement [34].
The deformed loadbearing structure is still just in equilibrium at the yield condition.
However, this limit state, which can be derived from the stress-strain curves of
Section 3.2, is not reached in a slender tower because:
- either a stability failure occurs first in the deformation calculation, or
- the load-carrying capacity of the cross-section determined according to Section 3.5
becomes critical.
