51-44 The Civil Engineering Handbook, Second Edition
where F
my
= the modified yield stress and l
c
= (KL/r
m
p)÷(F
my
/E
m
), in which E
m
is the modified modulus
of elasticity, r
m
is the modified radius of gyration about the axis of buckling, K is the effective length
factor, and L is the laterally unbraced length of a member.
The modified properties F
my
, E
m
, and r
m
account for the effects of concrete and longitudinal reinforcing
bars. The modified radius of gyration r
m
is the radius of gyration of the steel section, and it shall not be
less than 0.3 times the overall thickness of the composite cross section in the plane of buckling. The
modified values F
my
and E
m
are given by the following equations:
(51.71)
and
(51.72)
where F
y
= the yield strength of structural steel, £60 ksi (414 MPa)
F
yr
= the yield strength of longitudinal reinforcement, £60 ksi (414 MPa)
E = the modulus of elasticity of steel
E
c
= the modulus of elasticity of concrete
c
1
, c
2
, and c
3
= the numerical coefficients listed in Table 51.10
Coefficients c
1
, c
2
, and c
3
are higher for filled composite columns than for encased composite columns.
With the steel encasement always available to provide lateral confinement to concrete in filled composite
columns, there is no uncertainty that the contained concrete will reach at least as much strength as that
reached by concrete in unconfined standard concrete cylinders used in determining f
c
¢. In contrast, there
is less uncertainty that an unconfined concrete encasement can attain stress as high as 0.85f
c
¢. If the
unconfined concrete fails to reach 0.85f
c
¢, the longitudinal reinforcement it stabilizes may not reach its
yield stress, F
yr
, either. The values of c
1
and c
2
for encased composite columns are 70% of the values for
filled composite columns, reflecting the higher degree of uncertainty.
To account for the uncertainty regarding the contribution of concrete to the buckling strength of a
composite column, Eq. (51.72) includes the numerical coefficient c
3
, which is equal to 0.4 for filled
composite columns and 0.2 for encased composite columns. These coefficients are consistent with values
recommended in the ACI building code for flexural stiffness, EI, in estimates of inelastic buckling loads.
Concrete loses stiffness at strains near 0.2% and may not be fully effective for stabilizing steel at strains
higher than 0.2%, which translates into steel–stress values of about 60 ksi (414 MPa). The yield stresses
of structural steel (F
y
) and reinforcing bars (F
yr
) used in calculating the strength of composite columns
should not exceed 60 ksi. It is further recommended that the concrete strength f
c
¢ be limited to 10 ksi
(69 MPa) and smaller, since only very few tests are available for composite columns with f
c
¢ in excess of
10 ksi. A lower limit of f
c
¢ = 2.5 ksi (17 MPa) is recommended in order to encourage a degree of quality
control commensurate with this readily available and familiar grade of structural concrete.
Flexural Strength
The nominal flexural strength, M
n
, of a column cross section may be determined from the plastic state
of stress or from an analysis of flexural strength at the ultimate state of strain. For simplicity, the
TA BLE 51.10 Numerical Coefficients for Design
of Composite Columns
Numerical Coefficients
Composite Column Type c
1
c
2
c
3
Concrete-filled pipe and tubing 1.0 0.85 0.4
Concrete-encased shapes 0.7 0.6 0.2
FF
cF A
A
cfA
A
my y
yr r
s
cc
s
=+ +
¢
1
2
EE
cEA
A
m
cc
s
=+
3