Civil Engineering Reference
In-Depth Information
is shown dotted where the maximum concrete strain exceeds
10.8
a, the column would have collapsed in material failure before the Òinstability
loadÓ is attained; in Figure 10. 8b, the instability and material failures occur
simultaneously; in Figure 10.8c, material failure occurs.
It is now clear that the major effort required by the method is to obtain
the moment-deflection curves (Wong 1987b, 1988). The stability analysis
described above can be carried out by three methods, listed below in order
of increasing efficiency:
i)
The graphical method (Section 10.4.2)
ii)
The improved graphical method (Section 10.4.3)
iii)
The analytical method (Section 10.4.4)
10.4.2
Stability analysis of columns: graphical method
10.4.2.1
Assumptions and sign convention
The following assumptions and
sign convention are adopted for the graphical method to be described here,
and for the improved graphical method and the analytical method to be
described in Section 10.4.3 and 10.4.4, respectively:
i)
The strains in the concrete and the reinforcing steel are proportional
to the distances from the neutral axis,
ii)
Material failure (i.e. crushing of concrete) occurs when the concrete
strain at the extreme compression fibre reaches a specified value
e
cu
,
which is taken to be 0.0035 as specified in BS 8110 (1985). (Users
of other national Codes of Practice may of course use other values
for
e
cu
at their discretion),
iii)
The tensile strength of the concrete is ignored,
iv)
Compressive stresses and strains are taken to be positive, and
tensile stresses and strains negative.
10.4.2.2
Stress-strain relationships
Figure 10.9
shows the stress-strain
relation for concrete and steel. Expressing the concrete stress
ƒ
as function
e
cu
the area under the concrete stress-strain curve in
Figure 10.9a between
of the strain ratio
e
/
e
=
e
Ó
c
and
e
=
e
c
is
(10.5)
and the corresponding centroidal distance
e
g
(dimensionless; Figure 10.9a) is
(10.6)
The stress-strain relation for steel in Figure 10.9b is that of BS 8110:1985
with the partial safety factor
g
m
set to unity.
