Civil Engineering Reference
In-Depth Information
φ
ρ
=
ρ
=
ε
u
=
.
004, f
c
=
.
Figure 3.24
M-N-
relationship for a typical column (
2%,
1%,
0
27
6MPa
(4 000 psi),
f
y
=
413 MPa (60,000 psi), n
=
8, h
=
1.1d)
f
y
=
8. The moment
M
n
has been made nondimensional
by the parameter
f
c
bd
2
, and the axial load
N
n
has been normalized by the capacity of the
concentric load
N
o
.
The solid curve in this nondimensional interaction diagram represents the sectional strength
of the given column under combined bending and axial load. The balanced point is calculated
from Equations (3.139)-(3.141). The tension failure zone is computed by solving Equations
(3.142)-(3.145); and the compression failure zone is determined either by solving Equations
(3.146)-(3.149) or directly by Equation (3.151), depending on the location of the neutral axis.
The dotted curve from the balanced point to the point of pure bending represents the yielding
of the tension steel. The method of calculation for this dotted curve, which is based on Hooke's
laws for both concrete and steel, has not been presented. The reader should be able to derive
the equations required to plot this curve using the equilibrium, Bernoulli compatibility and
Hooke's constitutive laws.
The nondimensional ultimate rotation,
413 MPa (60 000 psi) and
n
=
E
s
/
E
c
=
φ
u
d
, is related to the normalized axial load,
N
/
N
o
,
by the solid curve in Figure 3.24(b). The ultimate rotation
φ
u
d
is calculated from the Bernoulli
compatibility:
=
β
1
ε
u
d
φ
u
d
(3.153)
